**Fractal Fracture Mechanics Applied to Materials Engineering**

Lucas Máximo Alves and Luiz Alkimin de Lacerda

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52511

## **1. Introduction**

66 Applied Fracture Mechanics

2th Edition.

4319-4344.

– 17755.

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[57] Anderson, T. L. (1995) Fracture Mechanics, Fundamentals and Applications. Crc Press,

[58] Mandelbrot, Benoit B.; Dann E. Passoja & Alvin J. Paullay (1984) Fractal Character of

[59] Lung, C. W. and Z. Q. Mu, (1988)n Fractal Dimension Measured With Perimeter Area Relation and Toughness of Materials, Physical Review B. 1 December , 38( 16): 11781-11784. [60] Bouchaud, Elisabeth (1977) Scaling Properties of Crack. J. Phys: Condens. Matter. 9:

[61] Bouchaud, E.; G. Lapasset and J. Planés (1990) Fractal Dimension of Fractured Surfaces:

[62] Bouchaud, E.; J. P. Bouchaud (1994-I) Fracture Surfaces: Apparent Roughness, Relevant Length Scales, and Fracture Toughness. Physical Review B. 15 December . 50(23): 17752

[63] Bouchaud, Elisabeth (1997) Scaling Properties of Cracks. J. Phy. Condens. Matter. 9:

[64] Mosolov, A. B. (1993) Mechanics of Fractal Cracks In Brittle Solids, Europhysics Letters.

[65] Family & Vicsek (1985), Scaling in steady-state cluster-cluster aggregate. *J. Phys. A* 18: L75. [66] Barabási, Albert – László; H. Eugene Stanley (1995) Fractal Concepts In Surface Growth,

[67] Lopez, Juan M. Miguel A. Rodriguez, and Rodolfo Cuerno (1997) Superroughening

[68] Lopez, Juan M. and Schmittbuhl, Jean (1998) Anomalous scaling of fracture surfaces,

[70] Mishnaevsky Jr., L. L. (1994) A New Approach To the Determination of the Crack Velocity Versus Crack Length Relation. Fatigue Fract. Engng. Mater. Struct. 17(10): 1205-1212. [71] ASTM - E1737. (1996) Standard Test Method For J-Integral Characterization of Fracture

[72] Alves, Lucas Máximo; Rosana Vilarim da Silva and Bernhard Joachim Mokross (2000) In: New Trends In Fractal Aspects of Complex Systems – FACS 2000 – IUPAP International Conference October, 16, 2000m At Universidade Federal de Alagoas – Maceió, Brasil. [73] Alves, Lucas Máximo; Rosana Vilarim da Silva, Bernhard Joachim Mokross (2001) The Influence of the Crack Fractal Geometry on the Elastic Plastic Fracture Mechanics. Physica A: Statistical Mechanics and Its Applications. 12 June 2001, 295,(1/2): 144-148. [74] Alves, Lucas Máximo (2002) Modelamento Fractal da Fratura E Do Crescimento de Trincas Em Materiais. Relatório de Tese de Doutorado Em Ciência E Engenharia de Materiais, Apresentada À Interunidades Em Ciência E Engenharia de Materiais, da Universidade de São Paulo-Campus, São Carlos, Orientador: Bernhard Joachim

[75] Alves, Lucas Máximo (2005) Fractal Geometry Concerned with Stable and Dynamic Fracture Mechanics. Journal of Theorethical and Applied Fracture Mechanics. 44/1:pp: 44-57.

Mokross, Co-Orientador: José de Anchieta Rodrigues, São Carlos – SP.

versus intrinsic anomalous scaling of surfaces, *Phys. Rev. E* 56(4): 3993-3998.

[69] Guy, A. G. (1986) Ciências Dos Materias, Editora Guanabara 435p.

Fracture Surfaces of Metals, Nature (London). 19 April, 308 [5961]: 721-722.

em Ciência e Engenharia de Materiais, São Carlos.

a Universal Value? Europhysics Letters. 13(1): 73-79.

The Classical Fracture Mechanics (CFM) quantifies velocity and energy dissipation of a crack growth in terms of the projected lengths and areas along the growth direction. However, in the fracture phenomenon, as in nature, geometrical forms are normally irregular and not easily characterized with regular forms of Euclidean geometry. As an example of this limitation, there is the problem of stable crack growth, characterized by the *J-R* curve [1, 2]. The rising of this curve has been analyzed by qualitative arguments [1, 2, 3, 4] but no definite explanation in the realm of EPFM has been provided.

Alternatively, fractal geometry is a powerful mathematical tool to describe irregular and complex geometric structures, such as fracture surfaces [5, 6]. It is well known from experimental observations that cracks and fracture surfaces are statistical fractal objects [7, 8, 9]. In this sense, knowing how to calculate their true lengths and areas allows a more realistic mathematical description of the fracture phenomenon [10]. Also, the different geometric details contained in the fracture surface tell the history of the crack growth and the difficulties encountered during the fracture process [11]. For this reason, it is reasonable to consider in an explicit manner the fractal properties of fracture surfaces, and many scientists have worked on the characterization of the topography of the fracture surface using the fractal dimension [12, 13]. At certain point, it became necessary to include the topology of the fracture surface into the equations of the Classical Fracture Mechanics theory [6, 8, 14]. This new "Fractal Fracture Mechanics" (FFM) follows the fundamental basis of the Classical Fracture Mechanics, with subtle modifications of its equations and considering the fractal aspects of the fracture surface with analytical expressions [15, 16].

The objective of this chapter is to include the fractal theory into the elastic and plastic energy released rates *G*0 and 0*J* , in a different way compared to other authors [8, 13, 14, 17, 18, 19]. The non-differentiability of the fractal functions is avoided by developing a differentiable

© 2012 Alves and de Lacerda, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Alves and de Lacerda, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

analytic function for the rugged crack length [20]. The proposed procedure changes the classical *G*<sup>0</sup> , which is linear with the fracture length, into a non-linear equation. Also, the same approach is extended and applied to the Eshelby-Rice non-linear *J-*integral. The new equations reproduce accurately the growth process of cracks in brittle and ductile materials. Through algebraic manipulations, the energetics of the geometric part of the fracture process in the *J*-integral are separated to explain the registered history of strains left on the fracture surfaces. Also, the micro and macroscopic parts of the *J-*integral are distinguished. A generalization for the fracture resistance *J-R* curve for different materials is presented, dependent only on the material properties and the geometry of the fractured surface.

Finally, it is shown how the proposed model can contribute to a better understanding of certain aspects of the standard ASTM test [15].

## **2. Literature review of fractal fracture mechanics**

## **2.1. Background of the fractal theory in fracture mechanics**

Mandelbrot [21] was the first to point out that cracks and fracture surfaces could be described by fractal models. Mecholsky *et al*. [12] and Passoja and Amborski [22] performed one of the first experimental works reported in the literature, using fractal geometry to describe the fracture surfaces. They sought a correlation of the roughness of these surfaces with the basic quantity *D* called fractal dimension.

Since the pioneering work of Mandelbrot *et al.* [23], there have been many investigations concerning the fractality of crack surfaces and the fracture mechanics theory. They analyzed fracture surfaces in steel obtained by Charpy impact tests and used the "slit island analysis" method to estimate their fractal dimensions. They have also shown that *D* was related to the toughness in ductile materials.

Mecholsky *et al.* [12, 24] worked with brittle materials such as ceramics and glass-ceramics, breaking them with a standard three point bending test. They calculated the fractal dimension of the fractured surfaces using Fourier spectral analysis and the "slit island" method, and concluded that the brittle fracture process is a self-similar fractal.

It is known that the roughness of the fracture surface is related to the difficulty in crack growth [25] and several authors attempted to relate the fractal dimension with the surface energy and fracture toughness. Mecholsky *et al.* [24] followed this idea and suggested the dependence between fracture toughness and fractal dimension through

$$K\_{\rm IC} = E \left( D^\* a\_0 \right)^{1/2} \tag{1}$$

where *E* is the elastic modulus of the material, 0*a* is its lattice parameter, \* *D Dd* is the fractional part of the fractal dimension and *d* is the Euclidean projection dimension of the fracture.

Mu and Lung [26] suggested an alternative equation, a power law mathematical relation between the surface energy and the fractal dimension. It will be seen later in this chapter that both suggestions are complementary and are covered by the model proposed in this work.

## **2.2. The elasto-plastic fracture mechanics**

68 Applied Fracture Mechanics

analytic function for the rugged crack length [20]. The proposed procedure changes the classical *G*<sup>0</sup> , which is linear with the fracture length, into a non-linear equation. Also, the same approach is extended and applied to the Eshelby-Rice non-linear *J-*integral. The new equations reproduce accurately the growth process of cracks in brittle and ductile materials. Through algebraic manipulations, the energetics of the geometric part of the fracture process in the *J*-integral are separated to explain the registered history of strains left on the fracture surfaces. Also, the micro and macroscopic parts of the *J-*integral are distinguished. A generalization for the fracture resistance *J-R* curve for different materials is presented,

dependent only on the material properties and the geometry of the fractured surface.

certain aspects of the standard ASTM test [15].

with the basic quantity *D* called fractal dimension.

the toughness in ductile materials.

fracture.

**2. Literature review of fractal fracture mechanics** 

**2.1. Background of the fractal theory in fracture mechanics** 

Finally, it is shown how the proposed model can contribute to a better understanding of

Mandelbrot [21] was the first to point out that cracks and fracture surfaces could be described by fractal models. Mecholsky *et al*. [12] and Passoja and Amborski [22] performed one of the first experimental works reported in the literature, using fractal geometry to describe the fracture surfaces. They sought a correlation of the roughness of these surfaces

Since the pioneering work of Mandelbrot *et al.* [23], there have been many investigations concerning the fractality of crack surfaces and the fracture mechanics theory. They analyzed fracture surfaces in steel obtained by Charpy impact tests and used the "slit island analysis" method to estimate their fractal dimensions. They have also shown that *D* was related to

Mecholsky *et al.* [12, 24] worked with brittle materials such as ceramics and glass-ceramics, breaking them with a standard three point bending test. They calculated the fractal dimension of the fractured surfaces using Fourier spectral analysis and the "slit island"

It is known that the roughness of the fracture surface is related to the difficulty in crack growth [25] and several authors attempted to relate the fractal dimension with the surface energy and fracture toughness. Mecholsky *et al.* [24] followed this idea and suggested the

where *E* is the elastic modulus of the material, 0*a* is its lattice parameter, \* *D Dd* is the fractional part of the fractal dimension and *d* is the Euclidean projection dimension of the

1/2 \* *K E Da IC* <sup>0</sup> (1)

method, and concluded that the brittle fracture process is a self-similar fractal.

dependence between fracture toughness and fractal dimension through

There have been several proposals for including the fractal theory into de fracture mechanics in the last three decades. Williford [17] proposed a relationship between fractal geometric parameters and parameters measured in fatigue tests. Using Williford's proposal Gong and Lai [27] developed one of the first mathematical relationships between the *J-R* curve and the fractal geometric parameters of the fracture surface. Mosolov and Borodich [32] established mathematical relations between the elastic stress field around the crack and the rugged exponent of the fracture surface. Later, Borodich [8, 29] introduced the concept of specific energy for a fractal measurement unit. Carpinteri and Chiaia [30] described the behavior of the fracture resistance as a consequence of its self-similar fractal topology. They used Griffith's theory and found a relationship between the *G-*curve and the advancing crack length and the fractal exponent. Despite the non-differentiability of the fractal functions, they were able to obtain this relationship through a renormalizing method. Bouchaud and Bouchaud [31] also proposed a formulation to correlate fractal parameters of the fracture surface.

Yavari [28] studied the *J*-integral for a fractal crack and showed that it is path-dependent. He conjectured that a *J*-integral fractal should be the rate of release of potential energy per unit of measurement of the fractal crack growth.

Recently, Alves [16] and Alves *et al.* [20] presented a self-affine fractal model, capable of describing fundamental geometric properties of fracture surfaces, including the local and global ruggedness in Griffith´s criterion. In their formulations the fractal theory was introduced in an analytical context in order to establish a mathematical expression for the fracture resistance curve, putting in evidence the influence of the crack ruggedness.

## **3. Postulates of a fracture mechanics with irregularities**

To adapt the CFM, starting from the smooth crack path equations to the rugged surface equations, and using the fractal geometry, it is necessary to establish in the form of postulates the assumptions that underlie the FFM and its correspondence with the CFM.

I. Admissible fracture surfaces

Consider a crack growing along the x-axis direction (Figure 1), deviating from the x-axis path by floating in y-direction. The trajectory of the crack is an admissible fractal if and only if it represents a single-valued function of the independent variable x.

II. Scale limits for a fractal equivalence of a crack

The irregularities of crack surfaces in contrast to mathematical fractals are finite. Therefore, the crack profiles can be assumed as fractals only in a limited scale 0 0 0 max *lLL* [36]. The

lower limit 0 *l* is related to the micro-mechanics of the cracked material and the upper limit 0 max *L* is a function of the geometric size of the body, crack length and other factors.

**Figure 1.** Rugged crack and its projection in the plan of energetic equivalence.

III. Energy equivalence between the rugged crack surface and its projection

Irwin *apud* Cherepanov *et al.* [36] realized the mathematical complexity of describing the fracture phenomena in terms of the complex geometry of the fracture surface roughness in different materials. For this reason, he proposed an energy equivalence between the rough surface path and its projection on the Euclidean plane.

In the energetic equivalence between rugged and projected crack surfaces it is considered that changes in the elastic strain energy introduced by a crack are the same for both rugged and projected paths,

$$
\mathcal{L}\mathcal{L}\_{L0} = \mathcal{L}\mathcal{L}\_L \tag{2}
$$

where the subscript " 0 " denotes quantities in the projected plane. Consequently, the surface energy expended to form rugged fracture surfaces or projected surfaces are also equivalent,

$$
\mathcal{L}I\_{\gamma 0} = \mathcal{L}I\_{\gamma} \tag{3}
$$

IV. Invariance of the equations

Consider a crack of length *L* and the quantities that describe it. Assuming the existence of a geometric operation that transforms the real crack size *L* to an apparent projected size 0 *L* , the length *L* may be described in terms of 0 *L* by a fractal scaling equation, as presented in a previous chapter.

It is claimed that the classical equations of the fracture mechanics can be applied to both rugged and projected crack paths, i.e., they are invariant under a geometric transformation between the rugged and the projected paths. In the crack wrinkling operation (smooth to rough) it is desired to know what will be the form of the fracture mechanics equations for the rough path as a function of the projected length 0 *L* , and their behavior for different roughness degrees and observation scales.

#### V. Continuity of functions

70 Applied Fracture Mechanics

and projected paths,

previous chapter.

IV. Invariance of the equations

roughness degrees and observation scales.

*l* is related to the micro-mechanics of the cracked material and the upper limit

0 max *L* is a function of the geometric size of the body, crack length and other factors.

**Figure 1.** Rugged crack and its projection in the plan of energetic equivalence.

surface path and its projection on the Euclidean plane.

III. Energy equivalence between the rugged crack surface and its projection

Irwin *apud* Cherepanov *et al.* [36] realized the mathematical complexity of describing the fracture phenomena in terms of the complex geometry of the fracture surface roughness in different materials. For this reason, he proposed an energy equivalence between the rough

In the energetic equivalence between rugged and projected crack surfaces it is considered that changes in the elastic strain energy introduced by a crack are the same for both rugged

where the subscript " 0 " denotes quantities in the projected plane. Consequently, the surface energy expended to form rugged fracture surfaces or projected surfaces are also equivalent,

> *U U* 0

Consider a crack of length *L* and the quantities that describe it. Assuming the existence of a geometric operation that transforms the real crack size *L* to an apparent projected size 0 *L* , the length *L* may be described in terms of 0 *L* by a fractal scaling equation, as presented in a

It is claimed that the classical equations of the fracture mechanics can be applied to both rugged and projected crack paths, i.e., they are invariant under a geometric transformation between the rugged and the projected paths. In the crack wrinkling operation (smooth to rough) it is desired to know what will be the form of the fracture mechanics equations for the rough path as a function of the projected length 0 *L* , and their behavior for different

*U U L L* <sup>0</sup> (2)

(3)

lower limit 0

It is considered that the scalar and vector functions that define the irregular surfaces *<sup>A</sup> Axy* , are described by a model (as the fractal model) capable of providing analytical and differentiable functions in the vicinity of the generic coordinate points *P Pxyz* , , , so that it is possible to calculate the surface *roughness*. Thus, it is always possible to define a normal vector in corners.

VI. Transformations from the projected to rugged path equations

As a consequence of the previous two postulates, it can be shown using the chain rule that the relationship between the rates for projected and rugged paths are given by

$$\frac{df\left(L\_0\right)}{dL\_0} = \frac{df\left(L\right)}{dL} \frac{dL}{dL\_0} \tag{4}$$

This result is used to transform the equations from the rugged to the projected path.

### **4. Energies in linear elastic fracture mechanics for irregular media**

The study of smooth, rough, fractal and non-fractal cracks in Fracture Mechanics requires the development of their respective equations of strain and surface energies.

#### **4.1. The elastic strain energy UL for smooth, rugged and fractal cracks**

Consider three identical plates of thickness *t* , with Young's modulus *E*´, subjected to a stress , each of them cracked at its center with a smooth, a rugged and a fractal crack as shown in Figure 2. The area of the unloaded elastic energy due to the introduction of the crack with length *<sup>l</sup> L* is

$$A\_l = m\_l L\_l^2 \tag{5}$$

where *ml* is the shape factor for the smooth crack. The accumulated elastic energy is

$$
\Delta U\_e = \int \frac{\sigma^2}{2E'}dV\tag{6}
$$

Thus, the elastic energy released by the introduction of a smooth crack with length *<sup>l</sup> L* is

$$
\Delta \mathcal{L} I\_l = -m\_l t \frac{\sigma\_l^2 L\_l^2}{2E\_l} \tag{7}
$$

For an elliptical crack the unloaded region can be considered almost elliptical and the shape factor is *ml* , thus

$$
\Delta \mathcal{U}\_l = -t \frac{\pi \sigma\_l^2 L\_l^2}{\mathcal{Z} E\_l'} \tag{8}
$$

**Figure 2.** Griffith model for the crack growth introduced in a plate under stress: a) flat crack and initial length *<sup>l</sup> L* with increase *<sup>l</sup> dL* in size*;* b) rugged crack and initial length *L* with increase *dL* in size; c) fractal crack, showing increase *dL* in size.

Analogously, the area of the unloaded elastic energy due to the introduction of a rugged crack of length *L* is given by

$$A = m \, ^\ast L^2\tag{9}$$

where \* *m* is a shape factor for the rugged crack. Thus, the elastic energy released by the introduction of a rugged crack with length *L* is

$$
\Delta \mathcal{U}\_L = -m^\* t \frac{\sigma\_r^{-2} L^2}{2E'} \tag{10}
$$

Considering that the rugged crack is slightly larger than its projection, then

$$L = \mathfrak{z}L\_0 \tag{11}$$

Consequently, the change of elastic strain energy from the point of view of the projected length 0 *L* can be expressed as:

$$
\Delta \mathcal{U}\_{L0} = -m^\* \, t \frac{\sigma\_0^{-2} L\_0^{-2}}{2E'} \tag{12}
$$

where 0 *<sup>r</sup>* .

#### **4.2. A self-affine fractal model for a crack - LEFM**

To take the roughness into account, it will be inserted in the CFM equations a self-affine fractal model developed in a previous chapter of this book.

*4.2.1. The relationship between strain energies for rugged U*L *and projected U*L0 *cracks in terms of fractal geometry* 

The crack length of the self-affine fractal can be expressed as

$$L = L\_0 \sqrt{1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{L\_0}{l\_0}\right)^{2\left(H - 1\right)}}\tag{13}$$

where *H*0 is the vertically projected crack length and the unloading fractal area of the elastic energy can be expressed as a function of the apparent length,

$$A\_0 = m\_0 L\_0^{-2} \tag{14}$$

And results that

72 Applied Fracture Mechanics

2 2 2 ' *l l*

(8)

<sup>2</sup> *A m L*\* (9)

(10)

(12)

(11)

stress: a) flat crack and

*l*

*E* 

*l*

**Figure 2.** Griffith model for the crack growth introduced in a plate under

c) fractal crack, showing increase *dL* in size.

introduction of a rugged crack with length *L* is

crack of length *L* is given by

length 0 *L* can be expressed as:

where 0 *<sup>r</sup>* . *<sup>L</sup> U t*

initial length *<sup>l</sup> L* with increase *<sup>l</sup> dL* in size*;* b) rugged crack and initial length *L* with increase *dL* in size;

Analogously, the area of the unloaded elastic energy due to the introduction of a rugged

where \* *m* is a shape factor for the rugged crack. Thus, the elastic energy released by the

*<sup>L</sup> U mt*

<sup>0</sup> *L L* 

Consequently, the change of elastic strain energy from the point of view of the projected

<sup>0</sup> \* 2 ' *<sup>L</sup> <sup>L</sup> U mt*

*L*

Considering that the rugged crack is slightly larger than its projection, then

2 2 \* 2 ' *r*

*E* 

> 2 2 0 0

*E* 

$$
\Delta U\_{L0} = 2m^\* t \int \frac{\sigma\_r^2}{2E\_0'} \left( 1 + \left( 2 - H \right) \left( \frac{H\_0}{l\_0} \right)^2 \left( \frac{l\_0}{L\_0} \right)^{2H-2} \right) L\_0 dL\_0 \tag{15}
$$

Therefore, the elastic energy released by the introduction of a crack length 0 *L* is

$$
\Delta \mathcal{U}\_{L0} = -m^\* t \frac{\sigma\_0^{-2} L\_0^{-2}}{2E\_0'} \tag{16}
$$

where 2 22 0 0 0 0 0 1 *H r H l l L* 

Observe that equation (12) is recovered from equation (16) applying the limits *H lL* 00 0 and 1.0 *H* with *<sup>r</sup>* <sup>0</sup> and 0 *E E* ' ' .

To understand the effect of crack roughness on the change of elastic strain energy, one may consider postulates III and IV, thus

$$
\Delta \mathcal{U}\_{Lo} = \Delta \mathcal{U}\_L = -\frac{m^\* \sigma\_r^{-2} L\_0^{-2}}{2E} \left[ 1 + \left( \frac{H\_0}{l\_0} \right)^2 \left( \frac{l\_0}{L\_0} \right)^{2H-2} \right] \tag{17}
$$

It can be noticed that for 1 *H* , which corresponds to a smoother surface, the relationship between the strain energy and the projected length 0 *L* is more linear. While for 0 *H* , which corresponds to a rougher surface, this relationship is increasingly non-linear. This is reasonable since the more ruggedness, more elastic strain per unit of crack length.

#### *4.2.2. Relationship between the applied stress on the rough and projected crack lengths*

Comparing (8), (10) and (12), one has

$$
\Delta U\_{L0} = \Delta U\_{LI} \left(\frac{m^\*}{\pi}\right) = \Delta U\_L \tag{18}
$$

Then, from postulate III, i.e., the following relationship is valid only for the situation of free loading without crack growth.

$$\frac{\sigma\_0^2}{E\_0'} = \frac{\sigma\_r^2}{E'} \left(\frac{L}{L\_0}\right)^2\tag{19}$$

Using equation (13) in (19), one has the resilience as a function of the projected length 0 *L*

$$\frac{\sigma\_0^2}{E\_0} = \frac{1}{\sqrt{2}} \frac{\sigma^2}{E} \left[ 1 + \left( \frac{H\_0}{l\_0} \right)^2 \left( \frac{l\_0}{L\_0} \right)^{2H-2} \right] \tag{20}$$

Or, the rugged length *L* can be written in terms of the projected length 0 *L* , thus

$$L = \sqrt{\frac{E'}{E\_0'}} \left(\frac{\sigma\_0}{\sigma\_r}\right) L\_0 \tag{21}$$

Since the elasticity modulus is independent of the crack path, one has

$$
\sigma\_0 L\_0 = \sigma\_r L \tag{22}
$$

Substituting equation (13) in equation (22), one has the relationship between stresses on the rough and projected surfaces,

$$
\sigma\_0 = \frac{\sigma\_r}{\sqrt{2}} \left[ 1 + \left( \frac{H\_0}{l\_0} \right)^2 \left( \frac{l\_0}{L\_0} \right)^{2H-2} \right]^{1/2} \tag{23}
$$

This last result is still incomplete since it is not valid for crack propagation. For its correction it will be considered that the elastic energy released rate *G* can be expressed as a function of *G*0 according to equation (4).

## *4.2.3. The surface energy U*<sup>0</sup> *for smooth, rugged and projected cracks in accordance with fractal geometry*

The surface energy of a smooth and a rugged crack are, respectively, given by

Fractal Fracture Mechanics Applied to Materials Engineering 75

$$
\Delta \mathbf{L} I\_{\gamma l} = \mathbf{2} \{ L\_l t \} \boldsymbol{\gamma}\_l \tag{24}
$$

$$
\Delta \mathbf{L} I\_{\gamma l} = \mathbf{2} \boldsymbol{\gamma}\_l \mathbf{t} \mathbf{L}\_l \tag{25}
$$

and

74 Applied Fracture Mechanics

Comparing (8), (10) and (12), one has

loading without crack growth.

rough and projected surfaces,

*G*0 according to equation (4).

*fractal geometry* 

which corresponds to a rougher surface, this relationship is increasingly non-linear. This is

\* *L Ll L <sup>m</sup> UU U* 

(18)

(19)

(21)

(22)

(23)

(20)

reasonable since the more ruggedness, more elastic strain per unit of crack length.

0

0

 

<sup>1</sup> <sup>1</sup> 2

Or, the rugged length *L* can be written in terms of the projected length 0 *L* , thus

1 2

The surface energy of a smooth and a rugged crack are, respectively, given by

Since the elasticity modulus is independent of the crack path, one has

0

*E E lL*

*4.2.2. Relationship between the applied stress on the rough and projected crack lengths* 

Then, from postulate III, i.e., the following relationship is valid only for the situation of free

2 2 2

*<sup>r</sup> L*

 2 22 2 2 0 0 0 0 0 0

> 0 0

0

0 0 *r L L* 

Substituting equation (13) in equation (22), one has the relationship between stresses on the

*<sup>r</sup> H l*

0 0

 

0 0

*l L*

This last result is still incomplete since it is not valid for crack propagation. For its correction it will be considered that the elastic energy released rate *G* can be expressed as a function of

*4.2.3. The surface energy U*<sup>0</sup> *for smooth, rugged and projected cracks in accordance with* 

1/2 2 22

*H*

' *r <sup>E</sup> L L E*

'

 

*H l*

*H*

0 0 ' '

*E EL*

Using equation (13) in (19), one has the resilience as a function of the projected length 0 *L*

 

$$
\begin{aligned}
\Delta \mathcal{U}\_{\gamma L} &= \mathcal{Z} \{ Lt \} \mathcal{V}\_r \\
\Delta \mathcal{U}\_{\gamma L} &= \mathcal{Z} \mathcal{Y}\_r t L
\end{aligned}
\tag{25}
$$

Using equation (11), the surface energy of the projected length 0 *L* is given by

$$\begin{aligned} \Delta \mathcal{U}\_{\neq 0} &= \mathcal{Z} \{ t \mathcal{L}\_0 \} \mathcal{V}\_0 \\ \Delta \mathcal{U}\_{\neq 0} &= \mathcal{Z} \chi\_0 t \mathcal{L}\_0 \end{aligned} \tag{26}$$

where 0 *<sup>r</sup>* . The surface energy equation (25) can be rewritten in terms of the projected length 0 *L* of a self-affine fractal crack

$$
\Delta \mathcal{U}\_{\gamma 0} = 2 \gamma\_r t \mathcal{L}\_0 \sqrt{1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H-2}} \tag{27}
$$

To see the influence of crack roughness on the surface energy, one may consider postulates III and IV, thus

$$\mathcal{U}\_{\gamma 0} = \mathcal{U}\_{\gamma} = \frac{2\gamma\_r L\_0}{\sqrt{2}} \sqrt{\mathbf{1} + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H-2}} \tag{28}$$

### **5. Stable or quasi-static fracture mechanics to the rough path**

In this section, a review of the conceptual changes introduced by Irwin (1957) in Griffith's theory (1920) is presented considering an irregular fracture surface, taking into account the postulates previously proposed. The purpose of this section is to use the mathematical formalism of Linear Elastic Fracture Mechanics for stable growth of smooth cracks, generalizing it to the case of an irregular rough crack.

#### **5.1. The Griffith energy balance in terms of fractal geometry**

According to Griffith´s energy balance, one has

$$d\mathcal{L}I\_T = d\left(\mathcal{U}\_i + \mathcal{U}\_L - F + \mathcal{U}\_\gamma\right) \le 0 \tag{29}$$

whilst

$$F - \mathcal{U}\_L \ge \mathcal{U}\_\gamma \tag{30}$$

Where *UT* is the total energy, *Ui* is the initial potential elastic energy, *F* is the work done by external forces, *UL* is the change of elastic energy stored in the body caused by the introduction of the crack length 0 *L* and *U*is the energy released to form the fracture surfaces.

One can now add the contributions of *UL*<sup>0</sup> and *U* <sup>0</sup> to reproduce Griffith´s energy balance in a fractal vision. In other words,

$$
\Delta \mathcal{U}\_T = \mathcal{U}\_i + \Delta \mathcal{U}\_L + \Delta \mathcal{U}\_\gamma - F \tag{31}
$$

and

$$\frac{d}{dL\_l}(\mathcal{U}\_l + -\frac{\pi \sigma^2 L\_0^2}{2E} \left[ 1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H-2} \right] + \frac{2\gamma L\_o}{\sqrt{2}} \sqrt{1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H-2}} - F) \le 0 \tag{32}$$

This new result is shown in Figure 3, which is analogous to the traditional Griffith energy balance graphs, but distorted due to the roughness of the fracture surface. Observe that for a reference total energy value the roughness of the crack surface tends to increase the critical size of the fracture 0*<sup>C</sup> L* compared to a material with a smooth fracture *lC C L L* . This is due to the roughness being a result of the interaction of the crack with the microstructure of the material.

**Figure 3.** Griffith´s energy balance in the view of the fractal geometry of fracture surface roughness.

## **5.2. The modification of Irwin in Griffith´s energy balance theory for smooth, rugged and projected cracks**

Irwin found from Griffith´s instability equation, given by (29), that this instability should take place by varying the crack length, so

Fractal Fracture Mechanics Applied to Materials Engineering 77

$$\frac{d}{dL} \left( \mathcal{U}\_i + \mathcal{U}\_L + \mathcal{U}\_\gamma - F \right) \le 0 \tag{33}$$

which can be rewritten as

76 Applied Fracture Mechanics

and

the material.

of the crack length 0 *L* and *U*

*i l*

**rugged and projected cracks** 

take place by varying the crack length, so

One can now add the contributions of *UL*<sup>0</sup> and *U*

balance in a fractal vision. In other words,

Where *UT* is the total energy, *Ui* is the initial potential elastic energy, *F* is the work done by external forces, *UL* is the change of elastic energy stored in the body caused by the introduction

*UU U UF Ti L*

2 22 2 22 2 2 0 00 0 0

*<sup>d</sup> L Hl L Hl U F*

This new result is shown in Figure 3, which is analogous to the traditional Griffith energy balance graphs, but distorted due to the roughness of the fracture surface. Observe that for a reference total energy value the roughness of the crack surface tends to increase the critical size of the fracture 0*<sup>C</sup> L* compared to a material with a smooth fracture *lC C L L* . This is due to the roughness being a result of the interaction of the crack with the microstructure of

**Figure 3.** Griffith´s energy balance in the view of the fractal geometry of fracture surface roughness.

**5.2. The modification of Irwin in Griffith´s energy balance theory for smooth,** 

Irwin found from Griffith´s instability equation, given by (29), that this instability should

*dL E lL l L*

0 0 0 0 <sup>2</sup> ( <sup>1</sup> <sup>1</sup> ) 0 <sup>2</sup> <sup>2</sup>

 

is the energy released to form the fracture surfaces.

*o*

*H H*

(31)

<sup>0</sup> to reproduce Griffith´s energy

(32)

$$\frac{d}{dL}(F - \mathcal{U}\_L) \ge \frac{d\mathcal{U}\_\gamma}{dL} \tag{34}$$

since *Ui* is constant. On the left hand side of equation (34), *<sup>L</sup> dF dL dU dL* is the amount of energy that remains available to increase crack extension by an amount *dL* . On the right hand side of equation (34), *dU dL* is the surface energy that must be released to form the rugged crack surfaces. This energy is the crack growth resistance.

Deriving equation (30) with respect to the projected crack length 0 *L* , one has

$$\frac{d}{dL\_0}(F - \mathcal{U}\_L) \ge \frac{d\mathcal{U}\_\mathcal{V}}{dL\_0} \tag{35}$$

Considering postulate II, one can apply the derivation chain rule and obtain

$$\frac{d}{dL}(F - \mathcal{U}\_L)\frac{dL}{dL\_0} \ge \frac{d\mathcal{U}\_\gamma}{dL}\frac{dL}{dL\_0} \tag{36}$$

Considering the following cases:

i. Fixed grips condition with *F constant* : since 2 2 0 0 \* 2' *UU m LE L L* decreases with the crack length, and using equations (10) and (25) in (36), one can derive

$$\frac{m^\*\sigma\_r^{-2}L}{E'} \ge 2\gamma\_r. \tag{37}$$

Or, by using equations (17) and (26) in (35), one finds

$$\frac{m^\* \sigma\_r^{-2} L\_0}{2E} \left( 1 + \left( 2 - H \right) \left( \frac{H\_0}{l\_0} \right)^2 \left( \frac{l\_0}{L\_0} \right)^{2H - 2} \right) \ge 2\gamma\_0. \tag{38}$$

ii. Condition of constant loading or stress, where necessarily 2 *<sup>L</sup> F U* , since 2 2 0 0 \* 2' *UU m LE L L* increases with the work of external forces, and using equations (10) and (25) in (36), one can find

$$\frac{m^\*\sigma\_r^{-2}L}{E'} \ge 2\gamma\_r. \tag{39}$$

Or, by using equations (17) and (26) in (35), one has

$$\frac{m^\* \sigma\_r^{-2} L\_0}{2E} \left( 1 + \left( 2 - H \right) \left( \frac{H\_0}{l\_0} \right)^2 \left( \frac{l\_0}{L\_0} \right)^{2H - 2} \right) \ge 2\gamma\_0. \tag{40}$$

Irwin defined the elastic energy released rate *G* and the fracture resistance *R* in equation (34), like

$$G \equiv \frac{d(F - \mathcal{U}\_L)}{dL} \tag{41}$$

and

$$R \equiv \frac{d\mathcal{U}\_{\mathcal{Y}}}{dL}.\tag{42}$$

These definitions can be extended to the terms in equation (35), so

$$G\_0 \ge R\_0 \tag{4.5} \\ \tag{4.6} \\ \mathbf{G}\_0 \equiv \frac{d(\mathbf{F} - \mathbf{U}\_{L0})}{d\mathbf{L}\_0} \tag{43}$$

and

$$R\_0 \equiv \frac{d\mathcal{U}\_{\mathcal{I}^0}}{d\mathcal{L}\_0}.\tag{44}$$

Notice that the proposal made by Irwin extended the concept of specific energy *eff* to the concept of R-curve given by equation (42), allowing to consider situations where the microstructure of the material interacts with the crack tip. In this way, it is assumed that the surface energy is dependent on the direction of crack growth.

Finally, using equations (41) and (42) in (36), the Griffith-Irwin criterion is obtained,

$$\mathcal{G}\frac{dL}{dL\_0} \ge \mathcal{R}\frac{dL}{dL\_0}.\tag{45}$$

## **5.3. Comparative analysis between smooth, projected and rugged fracture quantities**

Based on the results of the previous section, further analyses of the magnitudes of the Fracture Mechanics are performed in order to obtain a mathematical reformulation for an irregular or rugged Fracture Mechanics.

*5.3.1. Relationship between the elastic energy released rate rates for smooth, projected and rugged cracks* 

Using the chain rule, it is possible to write *G*0 in terms of *G* ,

$$G\_0 = G \frac{dL}{dL\_0} \tag{46}$$

The energetics equivalence between the rugged surface and its projection establishes that the energy per unit length along the rugged path is equal to the energy per unit length along the projected path. Notice that

$$\frac{d\mathcal{U}\_{L0}}{dL\_0} \ge \frac{d\mathcal{U}\_L}{dL} \tag{47}$$

since 0 *dL dL* 1 , therefore,

$$\mathbb{G}\_0 \ge \mathbb{G}.\tag{48}$$

The elastic energy released rates for the projected and rugged paths are, respectively

$$G\_0 = \frac{d\mathcal{U}\_{L0}}{d\mathcal{L}\_0} = \frac{m^\*\sigma\_0^2 L\_0}{E\_0'} \tag{49}$$

and

78 Applied Fracture Mechanics

(34), like

and

and

**quantities** 

irregular or rugged Fracture Mechanics.

Or, by using equations (17) and (26) in (35), one has

2 '

*mL H l <sup>r</sup> <sup>H</sup> E l L*

2 22 2 0 0 0

\* 1 2 2 .

Irwin defined the elastic energy released rate *G* and the fracture resistance *R* in equation

( ) *<sup>L</sup> dF U <sup>G</sup> dL*

. *dU*

*dL* 

( ) *<sup>L</sup> dF U <sup>G</sup> dL*

0

0

0 . *dU*

*dL* 

*R*

0

0

Notice that the proposal made by Irwin extended the concept of specific energy *eff*

the concept of R-curve given by equation (42), allowing to consider situations where the microstructure of the material interacts with the crack tip. In this way, it is assumed that

> 0 0 . *dL dL G R dL dL*

Based on the results of the previous section, further analyses of the magnitudes of the Fracture Mechanics are performed in order to obtain a mathematical reformulation for an

*R*

Finally, using equations (41) and (42) in (36), the Griffith-Irwin criterion is obtained,

**5.3. Comparative analysis between smooth, projected and rugged fracture** 

These definitions can be extended to the terms in equation (35), so

the surface energy is dependent on the direction of crack growth.

*G R* 0 0 <sup>0</sup>

 

0 0

*H*

0

(41)

(42)

(43)

(44)

(45)

to

(40)

$$\mathbf{G} = \frac{d\mathbf{U}\_L}{dL} = \frac{m^\* \sigma\_r^{-2} \mathbf{L}}{E^\prime}. \tag{50}$$

Combining these expressions and including, for comparison, the elastic energy released rate for a smooth path, one has for infinitesimal crack lengths,

$$\mathbf{G}\_0 = \mathbf{G}\_l \frac{d\mathbf{L}\_l}{d\mathbf{L}\_0} \left(\frac{m\*}{\pi}\right) = \mathbf{G} \frac{d\mathbf{L}}{d\mathbf{L}\_0} \tag{51}$$

Considering that the smooth crack length is equal to the projected crack length, one has

$$G\_0 = G\_l \left(\frac{m\*}{\pi}\right) = G\frac{dL}{dL\_0} \tag{52}$$

Observe that the difference between the elastic energy released rate for the smooth, rugged and projected cracks is the ruggedness added on crack during its growth. Using a thermodynamic model for the crack propagation, it can be concluded that a rugged crack dissipates more energy than a smooth crack propagating at the same speed.

The elastic energy released rate *G*0 can be written in terms of a fractal geometry,

$$G\_0 = \frac{m^\* \sigma\_r^{-2}}{E^\prime} L\_0 \left[ 1 + (2 - H) \left( \frac{H\_0}{l\_0} \right)^2 \left( \frac{l\_0}{L\_0} \right)^{2H - 2} \right] \tag{53}$$

#### **5.4. The crack growth resistance R for smooth, projected and rough paths**

Considering a plane strain condition, crack growth resistance for a smooth crack is given by

$$R\_l = \frac{d\mathcal{U}\_{pl}}{dL\_l} \tag{54}$$

Substituting equation (24) in equation (54), one finds

$$R\_l = \mathcal{D}\boldsymbol{\gamma}\_l \tag{55}$$

Observe that if the fracture path is smooth, the specific surface energy *<sup>l</sup>* is a cleavage surface energy and does not necessarily depend on the crack length. This model is only valid for brittle crystalline materials where the plastic strain at the crack tip does not absorb sufficient energy to cause dependence between fracture toughness and crack length.

Similarly, for a rugged crack, the fracture resistance to propagation is given by

$$R = \mathcal{D}\boldsymbol{\gamma}\_r \tag{56}$$

The concept of fracture growth resistance for the projected surface is given by

$$R\_0 = \frac{dLI\_\gamma}{dL\_0} \tag{57}$$

and substituting equation (26) in equation (57), one has

$$R\_0 = \mathcal{D}\boldsymbol{\gamma}\_0 \tag{58}$$

Again, this model is valid for ideally brittle materials where there is almost no plastic strain at the crack tip. It basically corresponds to the model presented by Griffith, with a modified interpretation introduced by Irwin with the *G R* curve concept.

## **5.5. Relationship between rugged** *R* **and projected** *R*0 **fracture resistances**

Using the chain rule, and admitting Irwin´s energetic equivalence represented by equation (3), the projected fracture resistance can be written on the basis of the resistance of the real surface,

Fractal Fracture Mechanics Applied to Materials Engineering 81

$$R\_0 = R \frac{dL}{dL\_0} \tag{59}$$

where 0 *dL dL* / is derived from equation (13),

$$\frac{dL}{dL\_0} = \frac{1 + (2 - H) \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H - 2}}{\sqrt{2\left(1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H - 2}\right)}} \ge 1\tag{60}$$

Therefore, the crack growth resistance ( *R* -curve), which is defined for a flat projected surface, is given substituting equation (56) and equation (60) in equation (59),

$$R\_0 = 2\gamma\_r \frac{1 + (2 - H) \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H - 2}}{\sqrt{2\left(1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H - 2}\right)}}\tag{61}$$

## **5.6. Final remarks about equivalent quantities of smooth, rugged and projected fracture surfaces**

It is important to emphasize that the energetic equivalence between the rugged surface crack path and its projection was considered such that the developed equations of the Fracture Mechanics for the flat plane path are still valid in the absence of any roughness.

However, if a flat and smooth fracture *<sup>l</sup> L* is considered with the same length of a projected fracture 0 *L* , the energetic quantities and their derivatives have the following relationship,

$$dL\_{Ll} \le dL\_{L0} \to \frac{d\mathcal{U}\_L}{dL\_l} \le \frac{d\mathcal{U}\_{L0}}{dL\_0} \to G\_l \le G\_0 \tag{62}$$

and

80 Applied Fracture Mechanics

The elastic energy released rate *G*0 can be written in terms of a fractal geometry,

\* 1 (2 ) '

*<sup>m</sup> <sup>r</sup> H l G LH*

**5.4. The crack growth resistance R for smooth, projected and rough paths** 

*l*

*R*

Observe that if the fracture path is smooth, the specific surface energy *<sup>l</sup>*

Similarly, for a rugged crack, the fracture resistance to propagation is given by

The concept of fracture growth resistance for the projected surface is given by

sufficient energy to cause dependence between fracture toughness and crack length.

*E l L*

Considering a plane strain condition, crack growth resistance for a smooth crack is given by

*dU*

*dL* 

2 *Rl l* 

surface energy and does not necessarily depend on the crack length. This model is only valid for brittle crystalline materials where the plastic strain at the crack tip does not absorb

> 2 *R <sup>r</sup>*

0

*R*

**5.5. Relationship between rugged** *R* **and projected** *R*0 **fracture resistances** 

0

*dU*

*dL* 

0 0 *R* 2

Again, this model is valid for ideally brittle materials where there is almost no plastic strain at the crack tip. It basically corresponds to the model presented by Griffith, with a modified

Using the chain rule, and admitting Irwin´s energetic equivalence represented by equation (3), the projected fracture resistance can be written on the basis of the resistance of the real

0 0

Substituting equation (24) in equation (54), one finds

and substituting equation (26) in equation (57), one has

surface,

interpretation introduced by Irwin with the *G R* curve concept.

2 22 2

 

*l*

*l*

0 0

*H*

(54)

(55)

(56)

(58)

(57)

is a cleavage

(53)

0 0

$$d\mathcal{U}\_{\gamma l} \le \mathcal{U}\_{\gamma 0} \to \frac{d\mathcal{U}\_{\gamma l}}{d\mathcal{L}\_l} \le \frac{d\mathcal{U}\_{\gamma 0}}{d\mathcal{L}\_0} \to R\_l \le R\_{\gamma 0'} \tag{63}$$

which have produced conflicting conclusions in the literature [37, 38, 46]. Since the energy for the smooth length 0 *<sup>l</sup> <sup>L</sup>* is smaller than the energy for the projected 0 *<sup>L</sup>* or rough *<sup>L</sup>* lengths, one has

$$\text{CLI}\_{Ll} \le \text{LI}\_L \to \text{G}\_l \le \text{G} \frac{dL}{dL\_0} \tag{64}$$

and

$$\mathcal{U}\_{\gamma l} \le \mathcal{U}\_{\gamma} \to \mathcal{R}\_{\gamma l} \le \mathcal{R} \frac{dL}{dL\_0} \tag{65}$$

In postulate III it was assumed that the rugged crack path satisfies the same energetic conditions of the plan path, but in the LEFM this roughness is not taken into account, causing discrepancies between theory and experiments. For example, it has not been possible to explain by an analytical function in a definitive way the growth of the *G R* curve. The proposed introduction of the term 0 *dL dL* / allows correcting this problem.

## **6. The elastic-plastic fractal fracture mechanics**

The non-linear elastic plastic energy released rate 0*J* for a crack of plane projected path can be extended from the Irwin-Orowan approach. They introduced the specific energy of plastic strain *<sup>p</sup>* on the elastic energy released rate *G*0 to describe the fracture phenomenon with considerable plastic strain at the crack tip. Thus, it is possible to define the elastic plastic energy released rate in an analogous way to the definition of the elastic energy released rate,

$$J\_o \equiv \frac{d(F - \mathcal{U}\_{Vo})}{d\mathcal{L}\_o} \tag{66}$$

where *UVo* is the volumetric strain energy given by the sum of the elastic and plastic (*Upl* ) contributions to the strain energy in the material.

#### **6.1. Influence of ruggedness in elastic plastic solids with low ductility**

Considering elastic plastic materials with low ductility where the effect of the plastic term is small compared to the elastic term, one can define a crack growth resistance as

<sup>2</sup> ( ) , *Ro Ro K fv <sup>J</sup> <sup>E</sup>* (67)

where *f*( ) *v* is a function that defines the testing condition. For plane stress *f v* 1 , and for plane strain <sup>2</sup> *f v v* 1 and *KRo* is the fracture toughness resistance curve.

Due to the ruggedness, the crack grows an amount 0 *dL dL* and correcting equation (59), one has

$$R\_o = \frac{d\mathbf{U}\_\gamma}{dL}\frac{dL}{dL\_o} = \left(2\boldsymbol{\gamma}\_e + \boldsymbol{\gamma}\_p\right)\frac{dL}{dL\_o}.\tag{68}$$

Similarly,

Fractal Fracture Mechanics Applied to Materials Engineering 83

$$J\_o = \frac{d(F - U\_V)}{dL} \frac{dL}{dL\_o}.\tag{69}$$

The energy balance proposed by Griffith-Irwin-Orowan, for stable fracture, is

$$J\_o = \mathbb{R}\_o.\tag{70}$$

Therefore, for plane stress or plane strain conditions, one can write from equation (61) that,

$$J\_{Ro} = \left(2\gamma\_e + \gamma\_p\right) \frac{dL}{dL\_v} = \frac{K\_{Ro}^{-2}f(v)}{E} \tag{71}$$

Thus,

82 Applied Fracture Mechanics

plastic strain *<sup>p</sup>*

released rate,

one has

Similarly,

0

(65)

(66)

*K fv <sup>J</sup> <sup>E</sup>* (67)

(68)

*dL*

on the elastic energy released rate *G*0 to describe the fracture phenomenon

*l l dL UU RR*

 

In postulate III it was assumed that the rugged crack path satisfies the same energetic conditions of the plan path, but in the LEFM this roughness is not taken into account, causing discrepancies between theory and experiments. For example, it has not been possible to explain by an analytical function in a definitive way the growth of the *G R* curve. The proposed introduction of the term 0 *dL dL* / allows correcting this problem.

The non-linear elastic plastic energy released rate 0*J* for a crack of plane projected path can be extended from the Irwin-Orowan approach. They introduced the specific energy of

with considerable plastic strain at the crack tip. Thus, it is possible to define the elastic plastic energy released rate in an analogous way to the definition of the elastic energy

( ) *Vo*

*dF U <sup>J</sup> dL*

where *UVo* is the volumetric strain energy given by the sum of the elastic and plastic (*Upl* )

Considering elastic plastic materials with low ductility where the effect of the plastic term is

where *f*( ) *v* is a function that defines the testing condition. For plane stress *f v* 1 , and

Due to the ruggedness, the crack grows an amount 0 *dL dL* and correcting equation (59),

2 . *<sup>o</sup> e p*

 

*dU dL dL <sup>R</sup> dL dL dL*

*o o*

<sup>2</sup> ( ) , *Ro*

*o*

*o*

**6.1. Influence of ruggedness in elastic plastic solids with low ductility** 

small compared to the elastic term, one can define a crack growth resistance as

*Ro*

for plane strain <sup>2</sup> *f v v* 1 and *KRo* is the fracture toughness resistance curve.

**6. The elastic-plastic fractal fracture mechanics** 

contributions to the strain energy in the material.

and

$$K\_{Ro} = \sqrt{\frac{\left(2\gamma\_e + \gamma\_p\right)E}{f(\upsilon)}\frac{dL}{dL\_o}}.\tag{72}$$

Knowing that fracture toughness is given by

$$K\_{Co} = \sqrt{\frac{\left(2\mathcal{V}\_e + \mathcal{V}\_p\right)E}{f(v)}},\tag{73}$$

one has,

$$K\_{Ro} = K\_{Co} \sqrt{\frac{dL}{dL\_o}}.\tag{74}$$

From the Classical Fracture Mechanics, the fracture resistance for the loading mode I, is given by

$$K\_{I\text{Ro}} = Y\_o \left(\frac{L\_o}{w}\right) \sigma\_f \sqrt{L\_{o\text{ \textquotedblleft}}} \tag{75}$$

where *<sup>o</sup> o <sup>L</sup> <sup>Y</sup> w* is a function that defines the shape of the specimen (CT, SEBN, etc) and the type of test (traction, flexion, etc), and *<sup>f</sup>* is the fracture stress. Considering the case when 0 0*<sup>C</sup> L L* , then *K K IR IC* 0 0 and the fracture toughness for the loading mode I is given by

$$K\_{ICo} = Y\_o \left(\frac{L\_{oc}}{w}\right) \sigma\_f \sqrt{L\_{oc}}.\tag{76}$$

Therefore, from equation (72) the fracture toughness curve for the loading mode I is given by

$$K\_{I\text{Ro}} = K\_{I\text{Co}} \sqrt{\frac{dL}{dL\_o}}.\tag{77}$$

Substituting equation (75) and equation (76) in equation (77), one has

$$\frac{d\mathcal{L}}{dL\_o} = Y\_o^2 \left(\frac{L\_o}{w}\right) \frac{\sigma\_f^{-2} L\_o f(v)}{\left(2\gamma\_e + \gamma\_p\right)E} \,\tag{78}$$

Observe that according to the right hand side of equation (78), the ruggedness 0 *dL dL* is determined by the condition of the test (plane strain or stress), the shape of the sample (CT, SEBN, etc), the type of test (traction, flexion, etc) and kind of material.

Considering the fracture surface as a fractal topology, one observes that the characteristics of the fracture surface listed above in equation (78) are all included in the ruggedness fractal exponent *H*. Substituting equation (60) in equation (71), one obtains

$$J\_{Ro} = \left(2\gamma\_e + \gamma\_p\right) \frac{1 + \left(2 - H\right)\left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{\Delta L\_0}\right)^{2H - 2}}{\sqrt{1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{\Delta L\_0}\right)^{2H - 2}}}.\tag{79}$$

which is non-linear in the crack extension 0 *L* . It corresponds to the classical equation (70) corrected for a rugged surface with Hurst's exponent *H*. Experimental results [1, 2] show that *J0* and the crack resistance *R*0 rise non-linearly and it is well known that this rising of the *J-R* curve is correlated to the ruggedness of the cracked surface [3, 4].

## **6.2. The** 0*J* **Eshelby-Rice integral for rugged and plane projected crack paths**

The *J-*integral concept of Eshelby-Rice is a non-linear extension of the definition given by Irwin-Orowan, for the linear elastic plastic energy released rate. In this context the potential energy 0 is defined as

$$
\Pi\_0 = \int\_{V\_0} \mathcal{W}dV\_0 - \int\_{\mathcal{C}} \vec{T}\vec{\omega}ds\_\prime \tag{80}
$$

where *W* the energy density integral in the in the volume *V*0 encapsulated by the boundary *C* with tractions *T* and displacements *u* , and *s* is the distance along the boundary *<sup>C</sup>* , as shown in Figure 4.

Accordingly,

$$J\_0 = -\frac{d\Pi\_0}{dL\_0} = -\frac{d}{dL\_0}\left(\int\_V \mathcal{W}dV\_0 - \int\_\mathcal{C} \vec{T} \,\vec{u}ds\right) \tag{81}$$

(83)

where 0 *dL* is the incremental growth of the crack length. In the two-dimensional case, where the fracture surface is characterized by a crack with length 0 *L* and a unit thickness body, one has *dV dxdy* and

84 Applied Fracture Mechanics

. *IRo ICo*

*dL K K*

*o e p dL <sup>L</sup> L fv <sup>Y</sup> dL w E*

Observe that according to the right hand side of equation (78), the ruggedness 0 *dL dL* is determined by the condition of the test (plane strain or stress), the shape of the sample (CT,

Considering the fracture surface as a fractal topology, one observes that the characteristics of the fracture surface listed above in equation (78) are all included in the ruggedness fractal

which is non-linear in the crack extension 0 *L* . It corresponds to the classical equation (70) corrected for a rugged surface with Hurst's exponent *H*. Experimental results [1, 2] show that *J0* and the crack resistance *R*0 rise non-linearly and it is well known that this rising of

1

**6.2. The** 0*J* **Eshelby-Rice integral for rugged and plane projected crack paths** 

0

0 0 0 0

and displacements *u*

The *J-*integral concept of Eshelby-Rice is a non-linear extension of the definition given by Irwin-Orowan, for the linear elastic plastic energy released rate. In this context the potential

> 0 0 . , *V C WdV T uds*

where *W* the energy density integral in the in the volume *V*0 encapsulated by the boundary

*d d <sup>J</sup> WdV T uds*

<sup>0</sup>

*V C*

.

*dL dL* (81)

*Ro e p <sup>H</sup>*

2 .

1 2

Substituting equation (75) and equation (76) in equation (77), one has

SEBN, etc), the type of test (traction, flexion, etc) and kind of material.

exponent *H*. Substituting equation (60) in equation (71), one obtains

the *J-R* curve is correlated to the ruggedness of the cracked surface [3, 4].

 

*J*

energy 0 is defined as

*C* with tractions *T*

shown in Figure 4.

Accordingly,

*o*

*o*

2 22

*H*

0 0 0 0 2 22

0 0 0 0

 

*H l l L*

*H l <sup>H</sup> l L*

2 <sup>2</sup> ( ) , <sup>2</sup> *f o o*

 

*dL* (77)

(78)

(79)

(80)

, and *s* is the distance along the boundary *<sup>C</sup>* , as

$$J\_0 \equiv -\frac{d\Pi\_0}{dL\_0} = -\left(\int\_V \mathcal{W} \frac{d\mathbf{x}}{dL\_0} dy - \int\_\mathcal{C} \vec{T} \cdot \frac{\partial \vec{u}}{\partial L\_0} ds\right). \tag{82}$$

For a fixed boundary *C* , 0 *d dL d dx* , and the 0*J* -integral for the plane projected crack path can be written only in terms of the boundary,

<sup>0</sup> . . *V C <sup>u</sup> J Wdy T ds*

*x*

**Figure 4.** Boundary around to the rugged crack tip where is defined the *J-*Integral [43].

Now, the *J-R* Eshelby-Rice integral theory is modified to include the fracture surface ruggedness. Initially, equation (82) is rewritten,

$$J\_0 = -\left(\int\_V \mathcal{W} \frac{d\mathbf{x}}{dL} \frac{dL}{dL\_0} dy - \int\_\mathcal{C} \vec{T}. \frac{\partial \vec{u}}{\partial L} \frac{dL}{dL\_0} ds\right). \tag{84}$$

From postulate IV, the new *J-*integral on the rugged crack path is given by

$$J \equiv -\frac{d\Pi}{dL} = -\left(\int\_{V} W \frac{d\boldsymbol{x}^\*}{dL} d\boldsymbol{y}^\* - \int\_{\mathcal{C}} \vec{T}. \frac{\partial \vec{u}}{\partial L} ds\right) \tag{85}$$

where the \* symbol represents coordinates with respect to the rugged path. So, in an analogous way to the *J*-integral for the projected crack path given by equation (85), since *d dL d dx* \* , one has

$$J = \int\_{V} \mathcal{W} dy^\* - \int\_{\mathbb{C}} \vec{T} \cdot \frac{\partial \vec{u}}{\partial \mathbf{x}^\*} ds. \tag{86}$$

Returning to equation (82) and considering postulate III along with the derivative chain rule and substituting equation (85), one has

$$J\_0 \equiv -\frac{d\Pi}{dL}\frac{dL}{dL\_0} = -\left(\int\_V \mathcal{W}\frac{d\mathbf{x}^\*}{dL} dy^\* - \int\_\mathcal{C} \vec{T}\_\cdot \frac{\partial \vec{u}}{\partial L} ds\right) \frac{dL}{dL\_0}.\tag{87}$$

Comparing (84) with equation (87) and considering that the rugged crack is a result of a transformation in the volume of the crack, analogous to the "*bakers´ transformation*" of the projected crack over the Euclidian plane, it can be concluded that

$$\begin{aligned} dx^\* \, dy^\* &= \frac{\partial(x^\*, y^\*)}{\partial(x, y)} dx dy\\ \frac{dx^\*}{dL} dy &\overset{dL}{dL\_o} = \frac{dx}{dL} \frac{dL}{dL\_o} dy \end{aligned} \tag{88}$$

which show the equivalence between the volume elements,

$$dV = d\mathbf{x}^\* \, dy^\* = d\mathbf{x} dy.\tag{89}$$

Therefore, the ruggedness 0 *dL dL* / of the rugged crack path does not depend on the volume *V*, nor on the boundary *C* and nor on the infinitesimal element length *ds* or *dy* . Thus, it must depend only on the characteristics of the rugged path described by the crack on the material. Finally, the integral in equation (84) can be written as

$$J\_0 = -\left(\int\_V W \frac{d\mathbf{x}}{dL} dy - \int\_\mathcal{C} \vec{T}. \frac{\partial \vec{u}}{\partial L} ds\right) \frac{dL}{dL\_0} \tag{90}$$

where the infinitesimal increment / cos *<sup>i</sup> dx dL* accompanies the direction of the rugged path *L* , as show in Figure 4. Thus,

$$J = \int\_{V} \mathcal{W} dy \cos \theta\_i - \int\_{\mathcal{C}} \vec{T}. \frac{\partial \vec{u}}{\partial x} \cos \theta\_i ds. \tag{91}$$

Observe that the *J*-integral for the rugged crack path given by equation (91) differs from the *J*-integral for the plane projected crack path given by equation (83) by a fluctuating term, cos *<sup>i</sup>* inside the integral. It can be observed that the energetic and geometric parts of the fracture process are separated and put in evidence the influence of the ruggedness of the material in the elastic plastic energy released rate,

$$J\_0 \equiv J \frac{dL}{dL\_0}.\tag{92}$$

It must be pointed out that this relationship is general and the introduction of the fractal approach to describe the ruggedness is just a particular way of modeling.

#### **6.3. Fractal theory applied to J-R curve model for ductile materials**

86 Applied Fracture Mechanics

and substituting equation (85), one has

0

projected crack over the Euclidian plane, it can be concluded that

which show the equivalence between the volume elements,

material. Finally, the integral in equation (84) can be written as

0

where the infinitesimal increment / cos *<sup>i</sup> dx dL*

material in the elastic plastic energy released rate,

path *L* , as show in Figure 4. Thus,

cos *<sup>i</sup>* 

Returning to equation (82) and considering postulate III along with the derivative chain rule

0 0

*o o*

.

cos . cos . *i i*

 inside the integral. It can be observed that the energetic and geometric parts of the fracture process are separated and put in evidence the influence of the ruggedness of the

0

It must be pointed out that this relationship is general and the introduction of the fractal

*x*

 

*<sup>u</sup> J Wdy T ds*

Observe that the *J*-integral for the rugged crack path given by equation (91) differs from the *J*-integral for the plane projected crack path given by equation (83) by a fluctuating term,

*V C d dL dx u dL <sup>J</sup> W dy T ds dL dL dL L dL* 

Comparing (84) with equation (87) and considering that the rugged crack is a result of a transformation in the volume of the crack, analogous to the "*bakers´ transformation*" of the

( \*, \*) \* \* (,)

*x y dx dy dxdy x y*

*dx dL dx dL dy dy dL dL dL dL*

Therefore, the ruggedness 0 *dL dL* / of the rugged crack path does not depend on the volume *V*, nor on the boundary *C* and nor on the infinitesimal element length *ds* or *dy* . Thus, it must depend only on the characteristics of the rugged path described by the crack on the

> *V C dx u dL J W dy T ds dL L dL*

> > *V C*

0

approach to describe the ruggedness is just a particular way of modeling.

\* \*

\* \*. .

(87)

(88)

*dV dx dy dxdy* \*\* . (89)

0

(90)

accompanies the direction of the rugged

(91)

. *dL J J dL* (92)

This section includes the formalism of fractal geometry in the EPFM to describe the roughness effects on the fracture mechanical properties of materials. For this purpose the classical expression of the elastic-plastic energy released rate was modified by introducing the fractality (roughness) of the cracked surface. With this procedure the classical expression (49) of LEFM, linear with the crack length, is changed into a non-linear equation (53), which reproduces with precision the quasi-static crack propagation process in ductile materials.

Observe that the quasi-static crack growth condition is obtained with Griffith fracture criterion, doing 0 0 *J R* and 00 00 *dJ dL dR dL* / / . In this case, it is concluded that the *J-R* curve is given by Griffith criterion 2 *eff J* in equations (92) and (59). Therefore, for a selfaffine crack with *H l* 0 0 , one has

$$J\_0 = 2\gamma\_{eff} \frac{1 + (2 - H) \left(\frac{l\_0}{L\_0}\right)^{2H - 2}}{\sqrt{2\left(1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_0}{L\_0}\right)^{2H - 2}\right)}}\tag{93}$$

This model shows in unambiguous way how different morphologies (roughness) are correlated with the *J-R* curve growth. Given the energy equivalence between rough and projected surfaces for the crack path, the *J-R* curve increases due to the influence of the roughness, which has not been computed previously with the classical equations of EPFM.

The *J-*integral on the rugged crack path is a specific characteristic of the material and can be considered as being proportional to *CJ* [15], on the onset of crack extension, since in this case it has the rugged crack length greater than the projected crack length *L L* <sup>0</sup> . Thus,

$$J\_o \sim J\_\mathbb{C} \frac{d\mathcal{L}}{dL\_o}.\tag{94}$$

Substituting the fractal crack model proposed in equation (60), one has

$$J\_o \sim I\_C \frac{1 + \left(2 - H\right) \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_o}{\Delta L\_o}\right)^{2H - 2}}{\sqrt{1 + \left(\frac{H\_0}{l\_0}\right)^2 \left(\frac{l\_o}{\Delta L\_o}\right)^{2H - 2}}},\tag{95}$$

corroborating that the surface specific energy is related to the critical fracture resistance.

$$J\_{\mathbb{C}} \sim \left(2\gamma\_{e} + \gamma\_{p}\right). \tag{96}$$

#### *6.3.1. Case – 1. Ductile self-similar limit*

The local self-similar limit can be calculated applying the condition *H Ll* 0 00 in equation (79), obtaining

$$J\_{Ro} = \left(2\gamma\_e + \gamma\_p\right) \left(2 - H\right) \left(\frac{l\_o}{\Delta L\_o}\right)^{H-1} \tag{97}$$

or, with *D H* 2 , one has

$$J\_0 = 2\gamma\_{\rm eff} D \left(\frac{l\_0}{L\_0}\right)^{1-D}.\tag{98}$$

This result corresponds to the one found by Mu and Lung [26, 37] for ductile materials. Equation (98) is shown in Figure 5, where *J-R* curves are calculated for different values of the fractal dimension *D* . 2 *eff* <sup>=</sup> <sup>2</sup> 10.0 / *KJ m* is adopted and 0 0 *L l* is the crack length in 0 *l* units. This figure shows very clearly how the surface morphology (characterized by *D* ) determines the shape of the *J-R* curve at the beginning of the crack growth.

**Figure 5.** *J-R* curves calculated according to the projected crack length 0 *L* , for a fracture of unit thickness, and fractal dimensions *D* 1.0,1.1,1.3,1.5,1.7 and 2.0 with <sup>2</sup> 2 10 / *<sup>e</sup> KJ m* .

In Figure 6, *J-R* curves with fractal dimension 1.3 *D* are calculated according to the projected length 0 *L* for different measuring rulers 0 *l* , showing how the morphology of rugged surface cracks is best described for small values of 0 *l* , causing the pronounced rising of *J-R* curve. Figure 6 and equation (98) show that the initial crack resistance is correlated to the surface morphology characterized by dimension *D* , in accordance with the literature.

The self-similar limit of *J-R* curve, given by equation (98), is valid only for regions near the onset of the crack growth in brittle materials ( *H L* 0 0 ). This is due to the hardening of the material, which gives rise to ruggedness of the fracture surface.

In the case of ductile materials, the length of the work hardening zone *H*0 affects an increasingly greater area of the material as the crack propagates, but the self-similar limit *HL l* <sup>000</sup> is still valid.

**Figure 6.** *J-R* curves calculated in function of the projected crack length 0 *L* with different ruler lengths 0 *l* 0.0001,0.001,0.01,0.1 and 1.0*mm* , for a fracture of unit thickness, fractal dimension *D* 1.3 and <sup>2</sup> 2 10 / *<sup>e</sup> KJ m* .

However, in the case of brittle materials (ceramics), after the initial stage of hardening, the crack maintains this state in a region of length *H*<sup>0</sup> , very short if compared to the crack length 0 *L* , generating a self-similar fractal structure only when the crack length 0 *L* is small, in the order of 0 *l* , i.e., *HLl* 0 00 . When the crack length 0 *L* becomes much larger than the initial size of the hardening region *H*0 present at the onset of crack growth, the self-similar limit is not valid, and the self-affine (or global) limit of fracture becomes valid.

#### *6.3.2. Case – 2. Brittle self-affine limit*

It is easy verify that in stable crack growth, where 0 0 *J R* , using equations (59) and (79), one has 0 *dL dL* / 1 when *L* . The global self-affine limit of 0*J* can be calculated applying the condition when the observation scale corresponds to a rather small amplitude of the crack, similar in size to the crack increment, i.e., when *H lL* 00 0 in equation (79), resulting the linear elastic expression

$$J\_0 = \mathcal{D} \mathcal{Y}\_{\rm eff} \tag{99}$$

where 0 0 *J G* and

88 Applied Fracture Mechanics

equation (79), obtaining

or, with *D H* 2 , one has

the fractal dimension *D* . 2 *eff*

*6.3.1. Case – 1. Ductile self-similar limit* 

The local self-similar limit can be calculated applying the condition *H Ll* 0 00 in

1

<sup>=</sup> <sup>2</sup> 10.0 / *KJ m* is adopted and 0 0 *L l* is the crack length in 0

*<sup>e</sup> KJ m* .

*l* , showing how the morphology of

*l* , causing the pronounced rising

(97)

*l*

. (98)

*H o*

*o*

1 0

*D*

0

 

This result corresponds to the one found by Mu and Lung [26, 37] for ductile materials. Equation (98) is shown in Figure 5, where *J-R* curves are calculated for different values of

units. This figure shows very clearly how the surface morphology (characterized by *D* )

**Figure 5.** *J-R* curves calculated according to the projected crack length 0 *L* , for a fracture of unit thickness, and fractal dimensions *D* 1.0,1.1,1.3,1.5,1.7 and 2.0 with <sup>2</sup> 2 10 /

In Figure 6, *J-R* curves with fractal dimension 1.3 *D* are calculated according to the

of *J-R* curve. Figure 6 and equation (98) show that the initial crack resistance is correlated to the surface morphology characterized by dimension *D* , in accordance with the literature.

The self-similar limit of *J-R* curve, given by equation (98), is valid only for regions near the onset of the crack growth in brittle materials ( *H L* 0 0 ). This is due to the hardening of the

*<sup>l</sup> J H <sup>L</sup>*

2 2

2

*eff <sup>l</sup> J D <sup>L</sup>* 

*Ro e p*

0

determines the shape of the *J-R* curve at the beginning of the crack growth.

projected length 0 *L* for different measuring rulers 0

rugged surface cracks is best described for small values of 0

material, which gives rise to ruggedness of the fracture surface.

 

$$G\_{Ro} = \mathcal{D}\boldsymbol{\gamma}\_e + \boldsymbol{\gamma}\_p. \tag{100}$$

This result corresponds to a classic one in Fracture Mechanics, which is the general case valid for brittle materials as glass and ceramics.

## **7. Experimental analyses**

## **7.1. Ceramic, metallic and polyurethane samples**

The analyzed ceramic samples were produced by Santos [19] and Mazzei [41]. The raw material used for its production was an alumina powder A-1000SG by ALCOA with 99% purity. Specimens of dimensions 52 8 4 *mm mm mm* were sintered at 1650 °*C* for 2 hours, showing average 7 mm grain sizes. Their average mechanical properties are shown in Table 1 with elastic modulus E = 300 *GPa* and rupture stress 340 *<sup>f</sup> MPa* .

The analyzed metallic samples were multipass High Strength Low Alloy (HSLA) steel weld metals and standard DCT specimens. HSLA are divided in two groups based on the welding process utilized and the microstructural composition. The first group (*A1* and *A2* welds) is composed of *C-Mn Ti-Killed* weld metals and were joined by a manual metal arc process. The second group (*B1* and *B2* welds), joined by a submerged arc welding process, is also a *C-Mn Ti-Killed* weld metal, but with different alloying elements added to increase the hardenability. Mechanical properties of both welds and DCT metals are listed in Table 1.


**Table 1.** Data extracted from experimental testing of *J-R* curves obtained by compliance method.

The analyzed polymeric samples are a two-component Polyurethane, consisting of 1:1 mixture of polyol and prepolymer. The polyol was synthesized from oil and the prepolymer from diphenyl methane diisocyanate (MDI). Their mechanical properties are shown in Table 1.

## **7.2. Fracture tests**

90 Applied Fracture Mechanics

*Material Sample* 

Metals

Polymers

*f (MPa)*

PU0,5 40,70 0.8

PU1,0 40,70 0.8

*E* 

**7. Experimental analyses** 

valid for brittle materials as glass and ceramics.

**7.1. Ceramic, metallic and polyurethane samples** 

1 with elastic modulus E = 300 *GPa* and rupture stress 340

This result corresponds to a classic one in Fracture Mechanics, which is the general case

The analyzed ceramic samples were produced by Santos [19] and Mazzei [41]. The raw material used for its production was an alumina powder A-1000SG by ALCOA with 99% purity. Specimens of dimensions 52 8 4 *mm mm mm* were sintered at 1650 °*C* for 2 hours, showing average 7 mm grain sizes. Their average mechanical properties are shown in Table

The analyzed metallic samples were multipass High Strength Low Alloy (HSLA) steel weld metals and standard DCT specimens. HSLA are divided in two groups based on the welding process utilized and the microstructural composition. The first group (*A1* and *A2* welds) is composed of *C-Mn Ti-Killed* weld metals and were joined by a manual metal arc process. The second group (*B1* and *B2* welds), joined by a submerged arc welding process, is also a *C-Mn Ti-Killed* weld metal, but with different alloying elements added to increase the hardenability. Mechanical properties of both welds and DCT metals are listed in Table 1.

Ceramic Alumina 340 300 0,030 0.4956 424,2477056 0,7975 0,0096

**Table 1.** Data extracted from experimental testing of *J-R* curves obtained by compliance method.

The analyzed polymeric samples are a two-component Polyurethane, consisting of 1:1 mixture of polyol and prepolymer. The polyol was synthesized from oil and the prepolymer from diphenyl methane diisocyanate (MDI). Their mechanical properties are shown in Table 1.

A1CT2 516,00 1,34 291,60 0,48256 635,3313677 0,71 0,01 A2SEB2 537,00 3,63 174,67 0,36264 573,1747828 0,77 0,01 B1CT6 771,00 16,64 40,61 0,22634 650,1446157 0,77 0,02 B2CT2 757,00 1,96 99,22 0,26553 691,3971955 0,58 0,05 DCT1 554,001,7197 227,00 0,40487 624,8021278 - DCT2 530,001,6671 211,47 0,3995 593,7576222 - DCT3 198,750,3902 318,00 1,00000 352,2752029 -

*(GPa) JIC(exp)(KJ/m2)L0C(exp)(mm) KIC (MPa.m1/2) H (exp)* 

0.0 8,10 0,29951 39,47980593 0,47 ± 0,07

0.0 3,00 0,23685 35,10799599 0,50 ± 0,05

*<sup>f</sup> MPa* .

A standard three-point bending test was performed on alumina specimens, SE(B), notched plane. Low speed and constant prescribed displacement 1 *mm/min* was employed to obtain stable propagation. The *R*-curve was obtained using LEFM equations and fracture results are shown in Table 1.

The fracture toughness evaluation of metallic samples was executed using the *J*-integral concept and the elastic compliance technique with partial unloadings of 15% of the maximum load. For weld metals the *J-R* curve tests were performed by the compliance and multi-test techniques. Tests were executed in a MTS810 (Material Test System) system at ambient temperature, according to standard ASTM E1737-96 [15]. A single edge notch bending SENB and compact tension CT were used. One *J-R* curve for each tested specimen was retrieved and fracture results are shown in Table 1.

To obtain the fractured surfaces of polymeric materials, fracture toughness tests were performed by multiple specimen technique using the concept of *J-R* curve according to ASTM D6068-2002 [42]. However, these tests were different from the ones used for weld metals, due to the viscoelasticity of the polymers. The used nomenclatures PU0,5 and PU1,0 mean the loading rate used during the test, 0,5 *mm/min* and 1,0 *mm/min*, respectively. Fracture results are shown in Table 1.

## **7.3. Fractal analyses of fractured specimens**

The fractured surfaces of ceramic samples were obtained with a Rank Taylor Hobson profilometer (Talysurf model 120) and an HP 6300 scanner. The fractal analyses to obtain the Hurst dimensions were made by methods, such as Counting Box, Sand Box and Fourier transform. The fracture surface analysis of metallic and polymeric samples were executed using scanning electronic microscopy SEM and the analyses to obtain the Hurst exponents were made with the Contrast Islands Fractal Analysis. Fractal dimension results are shown in the last column of Table 1.

## **7.4. G-R and J-R curve tests and fitting with self-similar and self-affine fractal models**

A characteristic load-displacement result in the Alumina ceramic sample is shown in Figure 7. Observe that the stiffness of the material at the first deflection region is constant, corresponding to the elastic modulus of the material. However, as the crack propagates, the stiffness varies significantly.

The corresponding *G-R* curve test is shown in Figure 8. It can be seen that at the onset of crack growth ( 0 0*<sup>C</sup> L L* ), the behavior of this material is self-similar, as previously discussed. However, the results in the wider range of crack lengths ( 0 0 0 max *<sup>C</sup> L LL* ) show that this material behave according to the self-affine model. Finally, at the end of *G-R* curve ( <sup>0</sup> *L* ) the behavior is explained by the influence of the shape function <sup>0</sup> *YL w*/ used in the testing methodology [41].

**Figure 7.** Load (*X*) versus displacement (*u*) for a *G-R* curve test in a ceramic sample [41].

*J-R* curves obtained from standard metallic specimens provided by ASTM standard testing are shown in Figure 9 along with the fitting with the proposed fractal models. Fitting results with these samples, named DCT1, DCT2 and DCT3, are a consistent validation of the applied fractal models. The fitting results of the self-similar and self-affine models coincide and are not distinguishable in Figure 9.

**Figure 8.** *G-R* curve fitted with the self-similar model (equation (97)) and the self-affine model (equation (100)) for the Alumina sample [41].

the testing methodology [41].

and are not distinguishable in Figure 9.

(equation (100)) for the Alumina sample [41].

( <sup>0</sup> *L* ) the behavior is explained by the influence of the shape function <sup>0</sup> *YL w*/ used in

**Figure 7.** Load (*X*) versus displacement (*u*) for a *G-R* curve test in a ceramic sample [41].

**Figure 8.** *G-R* curve fitted with the self-similar model (equation (97)) and the self-affine model

*J-R* curves obtained from standard metallic specimens provided by ASTM standard testing are shown in Figure 9 along with the fitting with the proposed fractal models. Fitting results with these samples, named DCT1, DCT2 and DCT3, are a consistent validation of the applied fractal models. The fitting results of the self-similar and self-affine models coincide

**Figure 9.** *J-R* curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for steel samples DCT1, DCT2 and DCT3 [43].

Typical testing results performed to obtain *J-R* curves of metallic weld materials are shown in Figure 10 and Figure 11. In all results, *J-R* curves measured experimentally were fitted using models given by equations (93) and (97), where the factor 2 *e p* was obtained by adjusting the 0 *l* and *H* values for each different sample, by the self-similar and the selfaffine models.

The *J-R* curves for the tested polymeric specimens are shown in Figure 12 and Figure 13. Reasonably good results were obtained despite the greater dispersion of data.

**Figure 10.** *J-R* curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for HSLA-Mn/Ti steel (sample A1CT2).

**Figure 11.** *J-R* curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for HSLA-Mn/Ti steel (sample B2CT2) killed with titanium and other alloy elements to increase hardenability [43].

**Figure 12.** *J-R* curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for the poliurethane polymer PU0,5.

elements to increase hardenability [43].

**Figure 11.** *J-R* curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for HSLA-Mn/Ti steel (sample B2CT2) killed with titanium and other alloy

**Figure 12.** *J-R* curve fitted with the self-similar model shown in equation (97) and the self-affine model

shown in equation (93) for the poliurethane polymer PU0,5.

**Figure 13.** *J-R* curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for the poliurethane polymer PU1,0.

After the experimental *J-R* curves were fitted using equation (79) and equation (97), values of 2 *eff , H* and <sup>0</sup> *l* were determined and are shown in Table 2 and Table 3. With 0 2 *R eff J* , the value of the crack size 0 *eff L* was calculated and it corresponded to the specific surface energy. Using the experimental values of 0 , *IC C J L* and *H* given in Table 1, the values of the constants in the last column of Table 2 and Table 3 were calculated.


**Table 2.** Fitting data of *J-R* curves with the self- similar model [43].

A good level of agreement is seen between measured Hurst's exponents *H* at Table 1 and theoretical ones shown in Table 2 and Table 3. Larger differences in metals can be attributed to the quality of the fractographic images, which did not present well defined "Contrast Islands".


**Table 3.** Fitting data of *J-R* curves with the self- affine model [43].

## **7.5. Complementary discussion**

The proposed fractal scaling law (self-affine or self-similar) model is well suited for the elastic-plastic experimental results. However, the self-similar model in brittle materials appears to underestimate the values of specific surface energy *eff* and the minimum size of the microscopic fracture 0 *l* , although not affecting the value of the Hurst exponent *H* .

For a self-affine natural fractal such as a crack, the self-similar limit approach is only valid at the beginning of the crack growth process [39], and the self-affine limit is valid for the rest of the process. It can be observed from the results that the ductile fracture is closer to selfsimilarity while the brittle fracture is closer to self-affinity.

Equation (79) represents a self-affine fractal model and demonstrates that apart from the coefficient *H* , there is a certain "universality" or, more accurately, a certain "generality" in the *J-R* curves. This equation can be rewritten using a factor of universal scale, 0 0 *l L*/ *,* as

$$\underbrace{f(2\gamma\_{\varepsilon}+\gamma\_{p},I\_{0})}\_{\text{energy}} = \frac{I\_{o}}{2(2\gamma\_{\varepsilon}+\gamma\_{p})} = \underbrace{\frac{1+(2-H)\varepsilon^{2H-2}}{\sqrt{2\left(1+\varepsilon^{2H-2}\right)}}}\_{\text{geometric}} = \text{g}\{\varepsilon,H\}}\tag{101}$$

which is a valid function for all experimental results shown in Figure 14. It shows the existent relation between the energetic and geometric components of the fracture resistance of the material. The greater the material energy consumption in the fracture, straining it plastically, the longer will be its geometric path and more rugged will be the crack.

96 Applied Fracture Mechanics

Metals

Polymers

Material Sample <sup>2</sup> 2 / 

**7.5. Complementary discussion** 

the microscopic fracture 0

A good level of agreement is seen between measured Hurst's exponents *H* at Table 1 and theoretical ones shown in Table 2 and Table 3. Larger differences in metals can be attributed to the quality of the fractographic images, which did not present well defined "Contrast Islands".

*l mm*

Ceramic Alumina 0,0301871 1,000 0,2493645 0,2493645 1,00000 0,03018707

0

*L*

A1CT2 160,640 0,609 0,24422 0,105004 2,413408 387,700806 A2SEB2 102,750 0,442 0,31002 0,140040 2,993092 307,535922 B1CT6 22,980 0,700 0,08123 0,033873 2,757772 63,385976 B2CT2 57,978 0,705 0,10304 0,042893 2,529433 146,651006 DCT1 129,850 0,599 0,23309 0,100540 2,511844 326,184445 DCT2 118,850 0,624 0,20167 0,086294 2,512302 298,592197 DCT3 178,810 0,612 0,5282 0,226901 1,778386 318,000000

PU0,5 7,500 0,664 0,56541 0,238775 1,618852 12,150370 PU1,0 1,690 0,649 0,10898 0,046244 2,938220 4,971102

The proposed fractal scaling law (self-affine or self-similar) model is well suited for the elastic-plastic experimental results. However, the self-similar model in brittle materials

For a self-affine natural fractal such as a crack, the self-similar limit approach is only valid at the beginning of the crack growth process [39], and the self-affine limit is valid for the rest of the process. It can be observed from the results that the ductile fracture is closer to self-

Equation (79) represents a self-affine fractal model and demonstrates that apart from the coefficient *H* , there is a certain "universality" or, more accurately, a certain "generality" in the *J-R* curves. This equation can be rewritten using a factor of universal scale, 0 0

> 

<sup>0</sup> 2 2 1 (2 ) (2 , ) (, ) 2(2 ) 2 1

*energetic geometric*

which is a valid function for all experimental results shown in Figure 14. It shows the existent relation between the energetic and geometric components of the fracture resistance

 

*<sup>o</sup> e p <sup>H</sup> e p*

<sup>0</sup> 2 *eff*

*l H*

1/ 1

*l* , although not affecting the value of the Hurst exponent *H* .

 

 

2 2

*H*

*J H f J g H* (101)

*H*

1

*C*

2 2

*eff H*

*H l*

1 0

and the minimum size of

*l L*/ *,* as

*H* 1 *C C J L = constant*

*eff KJ m H theo* <sup>0</sup>

**Table 3.** Fitting data of *J-R* curves with the self- affine model [43].

appears to underestimate the values of specific surface energy *eff*

similarity while the brittle fracture is closer to self-affinity.

 

**Figure 14.** Generalized *J-R* curves for different materials, modelled using the self-affine fractal geometry, in function of the scale factor 0 0 *l L* of the crack length [43].

In the self-similar limit *l LH* <sup>000</sup> , equation (97) is applicable and the energetic and geometric components are put in evidence in the equation below,

$$J\_0 = \underbrace{(2\gamma\_{eo} + \gamma\_p)}\_{energy\,tic} \underbrace{(2 - H) \left(\frac{l\_0}{\Delta L\_0}\right)^{H-1}}\_{geom\,metric} \tag{102}$$

From equation (102), an expression can be derived which results in a constant value associated to each material,

$$\underbrace{J\_0 \Delta L\_0}\_{macrossopic}^{H-1} = \underbrace{(2\gamma\_{e0} + \gamma\_p)(2 - H)l\_0^{H-1}}\_{microssopic} = \text{(const)}\_{material} \tag{103}$$

It is possible to conclude that the macroscopic and microscopic terms on the left and righthand sides of equation (103) are both equal to a constant, suggesting the existence of a fracture fractal property valid for the beginning of crack growth, and justified experimentally and theoretically. These constant values were calculated for each point in each *J-R* curve for the tested materials. The average value for each material is listed in the last column of Table 2 and Table 3. Observe that this new property is uniquely determined by the process of crack growth, depending on the exponent *H* , the specific surface energy 2 *e p* and the minimum crack length 0 *l* .

This new constant can be understood as a "fractal energy density" and it is a physical quantity that takes into account the ruggedness of the fracture surface and other physical properties. Its existence can explain the reason for different problems encountered when defining the value of fracture toughness *KIC* . This constant can be used to complement the information yielded by the fracture toughness, which depends on several factors, such as the thickness *B* of the specimen, the shape or size of the notch, etc. To solve this problem, ASTM E1737-96 [15] establishes a value for the crack length *a* (approximately 0.5 / 0.7 *a W* and, 0.5 *B W* , where *W* is the width of the specimen) for obtaining the fracture toughness *KIC* , in order to maintain the small-scale yielding zone.

As shown in equation (103), a relationship exists between the specific surface energy 2 *eff* and the minimum crack size 0 *l* in the considered observation scale 0 0 *l L*/ . In Figure 15, it can be observed that the consideration of a minimum size for the fracture 01 *l* on a grain should mean the effective specific energy of the fracture 1 2 *eff* in this scale. In a similar way, the consideration of a minimum size of fracture in a different scale, like one that involves several polycrystalline grains 02 03 *l l*, etc.., should take into account the value of an effective specific energy in this other scale, 2 3 2 ,2 *eff eff ,* etc., in such a way that

**Figure 15.** Microstructural aspects of the observation scale with different 0 *l* ruler sizes, for the fractal scaling of fracture [43].

$$2\gamma\_{\rm ef}(2 - H\_1)\mathbb{I}\_{o1}^{\ \ H\_1 - 1} = 2\gamma\_{\rm eff}(2 - H\_2)\mathbb{I}\_{o2}^{\ \ H\_2 - 1} = const.\tag{104}$$

although 01 02 03 *lll* and 123 222 *eff eff eff* . So, the constant does not depend on the single rule of measurement 0 *l* used in the fractal model, but it depends on the kind of material used in the testing.

Another interpretation of equation (102) can be made by splitting the elastic and plastic terms,

Fractal Fracture Mechanics Applied to Materials Engineering 99

$$J\_0 = \underbrace{2\,\gamma\_e(\mathfrak{L} - H) \left(\frac{l\_0}{\Delta L\_0}\right)^{H-1}}\_{\text{elastic}} + \underbrace{\gamma\_p(\mathfrak{L} - H) \left(\frac{l\_0}{\Delta L\_0}\right)^{H-1}}\_{\text{plastic}}.\tag{105}$$

For the particular situation where 0 *IC J J* and 0 0*<sup>C</sup> L L* , it can be derived from equation (97),

$$J\_{IC} = \left(2\gamma\_e + \gamma\_p\right) \left(2 - H\right) \left(\frac{l\_0}{L\_{0C}}\right)^{H-1} \tag{106}$$

and from equation (72),

*l* on a grain

*l L*/ . In Figure 15,

in this scale. In a similar way,

*l* ruler sizes, for the fractal

98 Applied Fracture Mechanics

and the minimum crack size 0

several polycrystalline grains 02 03

scaling of fracture [43].

although 01 02 03

terms,

single rule of measurement 0

material used in the testing.

specific energy in this other scale, 2 3 2 ,2 *eff eff*

This new constant can be understood as a "fractal energy density" and it is a physical quantity that takes into account the ruggedness of the fracture surface and other physical properties. Its existence can explain the reason for different problems encountered when defining the value of fracture toughness *KIC* . This constant can be used to complement the information yielded by the fracture toughness, which depends on several factors, such as the thickness *B* of the specimen, the shape or size of the notch, etc. To solve this problem, ASTM E1737-96 [15] establishes a value for the crack length *a* (approximately 0.5 / 0.7 *a W* and, 0.5 *B W* , where *W* is the width of the specimen) for obtaining the

As shown in equation (103), a relationship exists between the specific surface energy 2 *eff*

the consideration of a minimum size of fracture in a different scale, like one that involves

1 2 1 1 1 11 2 22 2 (2 ) 2 (2 ) , *H H ef H lo eff H l const <sup>o</sup>*

(104)

*l* used in the fractal model, but it depends on the kind of

. So, the constant does not depend on the

 

> 

Another interpretation of equation (102) can be made by splitting the elastic and plastic

*l* in the considered observation scale 0 0

*,* etc., in such a way that

*l l*, etc.., should take into account the value of an effective

fracture toughness *KIC* , in order to maintain the small-scale yielding zone.

**Figure 15.** Microstructural aspects of the observation scale with different 0

 

*lll* and 123 222 *eff eff eff* 

should mean the effective specific energy of the fracture 1 2 *eff*

it can be observed that the consideration of a minimum size for the fracture 01

$$K\_{IC} = \sqrt{(2\gamma\_c + \gamma\_p)E(2 - H)\left(\frac{l\_o}{L\_{oc}}\right)^{H - 1}}\tag{107}$$

Therefore, using the fact that once the experimental value of *IC J* is determined and the fitting of *J-R* curve has already yielded the values 0 2 , *e p l* and *H* for the material, the value 0*<sup>C</sup> L* can be calculated.

Fracture Mechanics science was originally developed for the study of isotropic situations and homogeneous bodies.

At the microscopic level, the elastic material is modeled considering Einstein's solid harmonic approximation where Hooke's law is employed for the force between the chemical bonds of the atoms or molecules [48]. Therefore, the elastic theory is used to make linear approximations and it does not involve micro structural effects of the material.

At the mesoscopic level the equation of energy used for the fracture does not take into account effects at the atomic scale involving non-homogeneous situations [47]. Based on the arguments of the last paragraphs, it becomes clear why Herrmman *et al*. [49] needed to include statistical weights, as a crack growth criterion, for the break of chemical bonds in fracture simulations, as a form of portraying micro structural aspects of the fracture (defects) when using finite difference and finite element methods in computational models.

At the macroscopic level, on the other hand, Griffith's theory uses a thermodynamic energy balance. It is important to remember that the linear elastic theory of fracture developed by Irwin and Westergaard and the Griffith's theory are differential theories for the macroscopic scale, which means they are punctual in their local limit. These two approaches involve the micro structural aspects of the fracture, since they take a larger infinitesimal local limit than the linear elastic theory at the atomic and mesoscopic scales. This infinitesimal macroscopic scale is big enough to include *1015* particles as the lower thermodynamic limit, where the physical quantity Fracture Resistance (*J-R* Curve) portrays aspects of the interaction of the crack with the microstructure of the material.

In this chapter, Classical Fracture Mechanics was modified directly using fractal theory, without taking into account more basic formulations, such as the interaction force among particles, or Lamé's energy equation in the mesoscopic scale as a form to include the ruggedness in the fracture processes.

The use of the fractality in the fracture surface to quantify the physical process of energy dissipation was approached with two different proposals. The first was given by Mu and Lung [26, 37], who proposed a phenomenological exponential relation between crack length and the elastic energy released rate in the following form

$$G\_{IC} = G\_{I0} \varepsilon^{1-D} \,\prime \,\, \tag{108}$$

where is the length of the measurement rule. The second proposal was given by Mecholsky *et al*. [24] and Mandelbrot *et al*. [23], who suggested an empirical relation between the fractional part of the fractal dimension *D* \* and fracture toughness *KIC* ,

$$K\_{\rm IC} \sim A \left( D^\* \right)^{1/2} \tag{109}$$

where *A* 0 0 *E l* is a constant and 0 *E* is the stiffness modulus and 0 *l* is a parameter that has a unit length (an atomic characteristic length). The elastic energy released rate is then given by,

$$G\_0 = El\_0 D^\* \tag{110}$$

where <sup>2</sup> <sup>0</sup> / *G KE C IC* is the critical energy released rate.

The authors cited above used the Slit Island Method in their measurements of the fractal dimension *D* and it is important to emphasize that both proposals have plausible arguments, in spite of their mathematical differences. Observe that in the proposal of Mu and Lung [26, 37] the fractal dimension appears in the exponent of the scale factor, while in the proposal of Mecholsky *et al*. [24] and Mandelbrot *et al*. [23] the fractal dimension appears as a multiplying term of the scale factor.

The mathematical expression proposed in this work, equation (93) and equation (97), for the case 0 0 *J G* , is compatible with the two proposals above and can be seen as a unification of these two different approaches in a single mathematical expression. In other words, the two previous proposals are complementary views of the problem according to the expression deduced in this chapter.

A careful experimental interpretation must be done from results obtained in a *J-R* curve test. The authors mentioned above worked with the concept of *G* , valid for brittle materials, and not with the concept of *J* valid for ductile materials. The experimental results show that for the case of metallic materials the fitting with their expressions are only valid in the initial development of the crack because of the self-similar limit, while self-affinity is a general characteristic of the whole fracture process [39].

The plane strain is a mathematical condition that allows defining a physical quantity called *KIC* , which doesn't depend on the thickness of the material. The measure of an average crack size along the thickness of the material, according to ASTM E1737-96 [15], is taken as an average of the crack size at a certain number of profiles along the thickness. In this way, any self-affine profile, among all the possible profiles that can be obtained in a fracture surface, are statistically equivalent to each other, and give a representative average for the Hurst exponent.

100 Applied Fracture Mechanics

where

where <sup>2</sup>

ruggedness in the fracture processes.

and the elastic energy released rate in the following form

In this chapter, Classical Fracture Mechanics was modified directly using fractal theory, without taking into account more basic formulations, such as the interaction force among particles, or Lamé's energy equation in the mesoscopic scale as a form to include the

The use of the fractality in the fracture surface to quantify the physical process of energy dissipation was approached with two different proposals. The first was given by Mu and Lung [26, 37], who proposed a phenomenological exponential relation between crack length

Mecholsky *et al*. [24] and Mandelbrot *et al*. [23], who suggested an empirical relation

a unit length (an atomic characteristic length). The elastic energy released rate is then given by,

The authors cited above used the Slit Island Method in their measurements of the fractal dimension *D* and it is important to emphasize that both proposals have plausible arguments, in spite of their mathematical differences. Observe that in the proposal of Mu and Lung [26, 37] the fractal dimension appears in the exponent of the scale factor, while in the proposal of Mecholsky *et al*. [24] and Mandelbrot *et al*. [23] the fractal dimension appears

The mathematical expression proposed in this work, equation (93) and equation (97), for the case 0 0 *J G* , is compatible with the two proposals above and can be seen as a unification of these two different approaches in a single mathematical expression. In other words, the two previous proposals are complementary views of the problem according to the expression

A careful experimental interpretation must be done from results obtained in a *J-R* curve test. The authors mentioned above worked with the concept of *G* , valid for brittle materials, and not with the concept of *J* valid for ductile materials. The experimental results show that for the case of metallic materials the fitting with their expressions are only valid in the initial development of the crack because of the self-similar limit, while self-affinity is a general

between the fractional part of the fractal dimension *D* \* and fracture toughness *KIC* ,

where *A* 0 0 *E l* is a constant and 0 *E* is the stiffness modulus and 0

<sup>0</sup> / *G KE C IC* is the critical energy released rate.

as a multiplying term of the scale factor.

characteristic of the whole fracture process [39].

deduced in this chapter.

1 <sup>0</sup> , *<sup>D</sup> G G IC I* 

is the length of the measurement rule. The second proposal was given by

(108)

1/2 ~ \* *K AD IC* (109)

0 0 *G El D* \* (110)

*l* is a parameter that has

The crack height (corresponding to the opening crack test CTOD) follows a power law with the scales, *h v* 0 0 *l L* and can be written as,

$$\frac{\Delta H\_0}{h\_0} = \left(\frac{\Delta L\_0}{l\_0}\right)^{1-H} \tag{111}$$

This relation shows that, while the measurement of the number of units of the crack length *N Ll <sup>h</sup>* 0 0 in the growth direction grows linearly, the number of units of the crack height units *N Hl <sup>v</sup>* 0 0 grows with a power of 1 *H* . If it is considered that the inverse of the number of crack increments in the growth direction <sup>1</sup> *N lL <sup>h</sup>* 0 0 is also a measure of strain of the material, as the crack grows, and considering that the number of crack height increments can be a measure of the amount of the piling up dislocation, in agreement with equation (111), then the normal stress is of the type [44, 45]

$$
\sigma \sim \mathfrak{s}^{-H} \tag{112}
$$

Observe that this relation shows a homogeneity in the scale of deformations, similar to the power law hardening equation [34]. This shows that the fractal scaling of a rugged fracture surface is related to the power law of the hardening. It is possible that the fractality of the rugged fracture surface is a result of the accumulation of the pilled up dislocations in the hardening of the material before the crack growth.

In all three situations (metallic, polymer and ceramic) the presence of microvoids, or other microstructural defects, cooperate with the formation of ruggedness on the fracture surface. This ruggedness on the way it was modeled records the "history" of crack growth being responsible for the difficulty encountered by the crack to propagate, thus defining the crack growth resistance. In EPFM literature, the rising of *J-R* curve for a long time has been associated with the interposition of plane stress and plane strain conditions generating the unique morphology of the fracture surface ruggedness [1, 2]. In metals this rising has been associated with the growth and coalescence of microvoids [2]. However, the Fractal EPFM has proposed that the morphology of the fracture surface, characterized by parameters of fractal geometry, explains in a simple and direct way the rising of the *J-R* curves.

The success of fracture fractal modeling between the *J-R* curve and the exponent *H* can be attributed to the following fact: a fracture occurs only after a process of hardening in the material, even minimal. Such a process follows a power law [35], self-similar [33], of the stress applied, with the strain , as shown in equation (166). It is therefore possible to associate the *elasto-plastic energy released rate J* which is an energetic quantity with the applied stress , which is an *energy density*, and the *fracture length* <sup>0</sup> *L* with strain, and *l l* / and the ruggedness exponent *H* with the strain hardening exponent " *n* " [15]. As the strain hardening occurs before the onset of crack growth, it is evident that its physical result appears registered in the fracture surface in terms of ruggedness, created in the process of crack growth. This process of crack growth admits a fractal scaling in terms of the projected surface 0 *L* , so it is possible that the effect of its prior work hardening is responsible for the further self-affinity of fracture valid at the beginning of crack growth. This is because in the limit of the beginning of crack growth, the fractal scaling relationship is a self-similar power law, analogous to the power law hardening relationship [8, 33].

The technical standards ASTM E813 [40] and ASTM E1737-96 [15] suggest an exponential fitting of the type

$$J\_0 = \mathbb{C}\_1 \Delta L\_0^{\mathbb{C}\_2} \tag{113}$$

for the *J-R* curves. They do not supply any explanation for the nature of the coefficients for this fitting. However, by comparing equation (113) with equation (97), it can be concluded that <sup>1</sup> 1 0 2 2 *<sup>H</sup> C Hl eff* and 2 *C H* <sup>1</sup> , which explains the physical nature of this parameters;

## **8. Conclusions**

The theory presented in this chapter introduces fractal geometry (to describe ruggedness) in the formalism of classical EPFM. The resulting model is consistent with the experimental results, showing that fractal geometry has much to contribute to the advance of this particular science.

It was shown that the rising of the *J-R* curve is due to the non-linearity in Griffith-Irwin-Orowan's energy balance when ruggedness is taken into account. The idea of connecting the morphology of a fracture with physical properties of the materials has been done by several authors and this connection is shown in this chapter with mathematical rigor.

It is important to emphasize that the model proposed in this chapter illuminates the nature of the coefficients for the fitting proposed by the fractal model, which is the true influence of ruggedness in the rising of the *J-R* curve. The application of this model in the practice of fracture testing can be used in future, since the techniques for obtaining the experimental parameters, 0 *l H*, , and *eff* can be accomplished with the necessary accuracy.

The method for obtaining the *J-R* curves proposed in this chapter does not intend to substitute the current experimental method used in Fracture Mechanics, as presented by the ASTM standards. However, it can give a greater margin of confidence in experimental results, and also when working with the microstructure of the materials. For instance, in search of new materials with higher fracture toughness, once the model explains micro and macroscopically the behavior of *J-R* curves.

It is well known that the fracture surfaces in general are multifractal objects [9] and the treatment presented here applies only to monofractals surfaces. However, for purposes of demonstrating the ruggedness influence on the phenomenology of Fracture Mechanics, through the models presented in this chapter, the obtained results were satisfactory. The generalization by multifractality is a matter to be discussed in future work.

## **Author details**

102 Applied Fracture Mechanics

stress applied,

applied stress

fitting of the type

parameters;

**8. Conclusions** 

particular science.

parameters, 0

*l H*, , and *eff* 

that <sup>1</sup> 1 0 2 2 *<sup>H</sup> C Hl eff* 

with the strain

material, even minimal. Such a process follows a power law [35], self-similar [33], of the

associate the *elasto-plastic energy released rate J* which is an energetic quantity with the

 *l l* / and the ruggedness exponent *H* with the strain hardening exponent " *n* " [15]. As the strain hardening occurs before the onset of crack growth, it is evident that its physical result appears registered in the fracture surface in terms of ruggedness, created in the process of crack growth. This process of crack growth admits a fractal scaling in terms of the projected surface 0 *L* , so it is possible that the effect of its prior work hardening is responsible for the further self-affinity of fracture valid at the beginning of crack growth. This is because in the limit of the beginning of crack growth, the fractal scaling relationship is a self-similar power law, analogous to the power law hardening relationship [8, 33].

The technical standards ASTM E813 [40] and ASTM E1737-96 [15] suggest an exponential

0 10

for the *J-R* curves. They do not supply any explanation for the nature of the coefficients for this fitting. However, by comparing equation (113) with equation (97), it can be concluded

The theory presented in this chapter introduces fractal geometry (to describe ruggedness) in the formalism of classical EPFM. The resulting model is consistent with the experimental results, showing that fractal geometry has much to contribute to the advance of this

It was shown that the rising of the *J-R* curve is due to the non-linearity in Griffith-Irwin-Orowan's energy balance when ruggedness is taken into account. The idea of connecting the morphology of a fracture with physical properties of the materials has been done by several

It is important to emphasize that the model proposed in this chapter illuminates the nature of the coefficients for the fitting proposed by the fractal model, which is the true influence of ruggedness in the rising of the *J-R* curve. The application of this model in the practice of fracture testing can be used in future, since the techniques for obtaining the experimental

The method for obtaining the *J-R* curves proposed in this chapter does not intend to substitute the current experimental method used in Fracture Mechanics, as presented by the ASTM standards. However, it can give a greater margin of confidence in experimental

can be accomplished with the necessary accuracy.

authors and this connection is shown in this chapter with mathematical rigor.

2

and 2 *C H* <sup>1</sup> , which explains the physical nature of this

, which is an *energy density*, and the *fracture length* <sup>0</sup> *L* with strain, and

, as shown in equation (166). It is therefore possible to

*<sup>C</sup> J CL* (113)

Lucas Máximo Alves

*GTEME – Grupo de Termodinâmica, Mecânica e Eletrônica dos Materiais, Departamento de Engenharia de Materiais, Universidade Estadual de Ponta Grossa, Uvaranas, Ponta Grossa – PR, Brazil* 

Luiz Alkimin de Lacerda

*LACTEC – Instituto de Tecnologia para o Desenvolvimento, Departamento de Estruturas Civis, Centro Politécnico da Universidade Federal do Paraná, Curitiba – PR, Brazil* 

## **9. References**


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Toughness. pp.1-24.

Carlos, São Carlos.

Cerâmica 42(275).

and Its Applications. 295(1/2): 144-148.

America Ceramic Society 22: pp. 127-134.

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W. H. Freeman and Company.

[9] Xie, H.; Wang, J-A.; Stein, E. (1998) Direct Fractal Measurement and Multifractal

[10] Herrmann, H.J.; Stéphane, R. (1990) Statistical Models For the Fracture of Disordered Media, Random Materials and Processes. In: Series Editors: H. Eugene Stanley and

[11] Rodrigues, J.A.; Pandolfelli, V.C (1998) Insights on the Fractal-Fracture Behaviour

[12] Mecholsky, J. J.; Passoja, D.E.; Feinberg-Ringel, K.S. (1989) Quantitative Analysis of Brittle Fracture Surfaces Using Fractal Geometry, J. Am. Ceram. Soc. 72(1): 60-65. [13] Tanaka, M. (1996) Fracture Toughness and Crack Morphology in Indentation Fracture of

[14] Xie, H. (1989) The Fractal Effect of Irregularity of Crack Branching on the Fracture

[15] ASTM E1737 (1996) Standard Test Method For J-Integral Characterization of Fracture

[16] Alves, L.M. (2005) Fractal Geometry Concerned with Stable and Dynamic Fracture Mechanics. Journal of Theoretical and Applied Fracture Mechanics. 44(1): 44-57. [17] Williford, R. E. (1990) Fractal Fatigue. Scripta Metallurgica et Materialia. 24: 455-460. [18] Chelidze, T.; Gueguen, Y. (1990) Evidence of Fractal Fracture, (Technical Note) Int. J.

[19] Dos Santos, S.F. (1999) Aplicação do Conceito de Fractais para Análise do Processo de Fratura de Materiais Cerâmicos, Dissertação de Mestrado, Universidade Federal de São

[20] Alves, L.M.; Silva, R.V.; Mokross, B.J. (2001) The Influence of the Crack Fractal Geometry on the Elastic Plastic Fracture Mechanics. Physica A: Statistical Mechanics

[21] Mandelbrot, B.B. (1977) Fractals: Form Chance and Dimension, San Francisco, Cal-USA:

[23] Mandelbrot, B.B.; Passoja, D.E.; Paullay, A.J. (1984) Fractal Character of Fracture

[24] Mecholsky, J.J.; Mackin, T.J.; Passoja, D.E. (1988) Self-Similar Crack Propagation In Brittle Materials. In: Advances In Ceramics, Fractography of Glasses and Ceramics, the American Ceramic Society, Inc. J. Varner and V. D. Frechette editors. Westerville, Oh:

[25] Rodrigues, J.A.; Pandolfelli, V.C. (1996) Dimensão Fractal e Energia Total de Fratura.

[26] Mu, Z.Q.; Lung, C.W. (1988) Studies on the Fractal Dimension and Fracture Toughness

[27] Gong, B.; Lai, Z.H. (1993) Fractal Characteristics of *J-R* Resistance Curves of Ti-6Al-4V

[28] Yavari, A. (2002) The Mechanics of Self-Similar and Self-Afine Fractal Cracks, Int.

Toughness of Brittle Materials. International Journal of Fracture. 41: 267-274.

Properties of Fracture Surfaces, Physics Letters A. 242: 41-50.

Etienne Guyon editors. Amsterdam: North-Holland.

Brittle Materials. Journal of Materials Science. 31: 749-755.

Rock. Mech Min. Sci & Geomech Abstr. 27(3): 223-225.

[22] Passoja, D.E.; Amborski, D.J. (1978) In Microsstruct. Sci. 6: 143-148.

Surfaces of Metals, Nature (London), 308 [5961]: 721-722.

Relationship. Materials Research. 1(1): 47-52.


[49] Herrmann, H.J., Kertész, J.; De Arcangelis, L. (1989) Fractal Shapes of Deterministic Cracks, Europhys. Lett. 10(2): 147-152.

**Section 2** 

**Fracture of Biological Tissues** 

106 Applied Fracture Mechanics

Cracks, Europhys. Lett. 10(2): 147-152.

[49] Herrmann, H.J., Kertész, J.; De Arcangelis, L. (1989) Fractal Shapes of Deterministic

## **Chapter 4**

## **Fracture of Dental Materials**

Karl-Johan Söderholm

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48354

## **1. Introduction**

Finding a material capable of fulfilling all the requirements needed for replacing lost tooth structure is a true challenge for man. Many such restorative materials have been explored through the years, but the ideal substitute has not yet been identified. What we use today for different restorations are different metals, polymers and ceramics as well as combinations of these materials. Many of these materials work well even though they are not perfect. For example, by coating and glazing a metal crown shell with a ceramic, it is possible to make a strong and aesthetic appealing crown restoration. This type of crown restoration is called a porcelain-fused-to-metal restoration, and if such crowns are properly designed, they can also be soldered together into so called dental bridges. The potential problem with these crowns is that the ceramic coating may chip with time, which could require a complete remake of the entire restoration. Another popular restorative material consists of a mixture of ceramic particles and curable monomers forming a so called dental composite resin. These composites resins can be bonded to cavity walls and produce aesthetic appealing restorations. A potential problem with these restorations is that they shrink during curing and sometimes debond and fracture. In addition to porcelain-fused-to-metal crowns and composites, allceramic and metallic restorations as well as polymer based dentures are also commonly used. These constructions have their inherent limitations too.

The reason it is difficult to make an ideal dental material is because such a material has to be biocompatible, strong, aesthetic, corrosion resistant and reasonable easy to process, properties that are difficult to find in one single material. Besides, material as well as processing costs of such a material should be relatively low in order to make the use of the material wide among all social-economical groups. That demand makes the ideal material identification process even more challenging.

Today, dentistry to a great deal is driven by aesthetic demands, restricting the selectable materials mainly to tooth colored materials. Because of that demand, dentists are moving

© 2012 Söderholm, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Söderholm, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

away from traditional metallic restorations with high fracture toughness values, toward resin based composites and all-ceramic restorations with rather low fracture toughness values. Modern all-ceramic restorations consist of core structures made by fracture tough ceramics such as alumina and partly stabilized zirconia. However, the rather opaque appearance of these two ceramics often requires that they are veneered with less fracture tough but more aesthetic appealing ceramics. The use of more aesthetic appealing materials has not increased the longevity of dental restorations, but in some cases when composite resins are being used, the move toward bonded composites might have increased the way tooth structures can be preserved. The benefit of such usage is that it decreases the amount of tooth structure needed to be removed during preparation and can therefore increase the longevity of the tooth.

The intention with this chapter is to give an overview of some fundamental fracture mechanics aspect of aesthetic restorative materials such as dental ceramics and dental composite resins, as well as some fracture mechanics considerations related to the way ceramics and composites are bonded to the tooth via a cement/adhesive. However, before addressing these man-made materials, the two most important dental materials, the biologically developed materials, enamel and dentin, will be discussed. An insight into the fracture mechanics of these two substrates clearly shows how sophisticated Nature was when these two biologic materials evolved. An understanding of enamel and dentin shows quite clearly where the limitations and short-comings are with the man-made dental materials, and may help us in developing better restorative dental materials.

## **2. The tooth**

Nature provided animals and humans with teeth to be used for digesting food, but also as tools for hunting and self-defense. To fulfill these functions, Nature developed enamel to become the hardest biological tissue. Tooth enamel ranks 5 on Mohs hardness scale, where steel is ranked 4.5 and thus slightly softer than enamel. Its Young's modulus is 83 GPa, which falls between aluminum (69 GPa) and bronze (96-120 GPa)[1]. The enamel can be described as the whitish looking shell covering the visible part of a tooth positioned in the alveolar socket (Figure 1).

Regarding enamel and dentin, the first hard tissue to form is dentin, produced by newly differentiated odontoblasts. The first formed dentin layer is called mantle dentin and is approximately 150 μm thick and contains loosely packed coarse collagen fibrils surrounded by precipitated hydroxyapatite crystals [2]. Tiny side-branching channels oriented parallel to the dentin-enamel-junction (DEJ) and connected to the protoplasmatic extension of the odontoblasts are parts of the mantel dentin. The mantle dentin matrix is slightly less (4 vol- %) mineralized then the rest of the finally formed dentin.

As the odontoblasts move away from the DEJ, each of them leaves a cell extension protruding from the odontoblasts to the DEJ with the side-branching channels of the mantle dentin. These cell extensions may remain in contact with the DEJ during the formation of dentin as well as during the lifetime of the tooth, and they form channels through the dentin as the odontoblasts move inwards toward the pulp. The secreted collagen fibers, which are mainly oriented perpendicularly to the dentinal tubules, act as nucleisation centers for hydroxyapatite crystallites precipitating as the odontoblasts migrate inwards (Figure 2). The dentin formed by these collagen fibers represents the so called intertubular dentin (Figure 3). However, surrounding the odontoblastice processes are thin layers of collagen oriented parallel to the odontoblastic processes. These collagen layers are also mineralized and form the so called peritubular dentin, which is denser than the intertubular dentin located between the peritubular dentin tubules. An important difference between enamel and dentin is that dentin, in contrast to enamel, is a living tissue as long as the pulp is alive, while the enamel becomes a completely dead tissue as soon as the outer layer of the enamel has formed and the ameloblasts degraded.

110 Applied Fracture Mechanics

**2. The tooth** 

alveolar socket (Figure 1).

%) mineralized then the rest of the finally formed dentin.

away from traditional metallic restorations with high fracture toughness values, toward resin based composites and all-ceramic restorations with rather low fracture toughness values. Modern all-ceramic restorations consist of core structures made by fracture tough ceramics such as alumina and partly stabilized zirconia. However, the rather opaque appearance of these two ceramics often requires that they are veneered with less fracture tough but more aesthetic appealing ceramics. The use of more aesthetic appealing materials has not increased the longevity of dental restorations, but in some cases when composite resins are being used, the move toward bonded composites might have increased the way tooth structures can be preserved. The benefit of such usage is that it decreases the amount of tooth structure needed

to be removed during preparation and can therefore increase the longevity of the tooth.

materials, and may help us in developing better restorative dental materials.

The intention with this chapter is to give an overview of some fundamental fracture mechanics aspect of aesthetic restorative materials such as dental ceramics and dental composite resins, as well as some fracture mechanics considerations related to the way ceramics and composites are bonded to the tooth via a cement/adhesive. However, before addressing these man-made materials, the two most important dental materials, the biologically developed materials, enamel and dentin, will be discussed. An insight into the fracture mechanics of these two substrates clearly shows how sophisticated Nature was when these two biologic materials evolved. An understanding of enamel and dentin shows quite clearly where the limitations and short-comings are with the man-made dental

Nature provided animals and humans with teeth to be used for digesting food, but also as tools for hunting and self-defense. To fulfill these functions, Nature developed enamel to become the hardest biological tissue. Tooth enamel ranks 5 on Mohs hardness scale, where steel is ranked 4.5 and thus slightly softer than enamel. Its Young's modulus is 83 GPa, which falls between aluminum (69 GPa) and bronze (96-120 GPa)[1]. The enamel can be described as the whitish looking shell covering the visible part of a tooth positioned in the

Regarding enamel and dentin, the first hard tissue to form is dentin, produced by newly differentiated odontoblasts. The first formed dentin layer is called mantle dentin and is approximately 150 μm thick and contains loosely packed coarse collagen fibrils surrounded by precipitated hydroxyapatite crystals [2]. Tiny side-branching channels oriented parallel to the dentin-enamel-junction (DEJ) and connected to the protoplasmatic extension of the odontoblasts are parts of the mantel dentin. The mantle dentin matrix is slightly less (4 vol-

As the odontoblasts move away from the DEJ, each of them leaves a cell extension protruding from the odontoblasts to the DEJ with the side-branching channels of the mantle dentin. These cell extensions may remain in contact with the DEJ during the formation of dentin as well as during the lifetime of the tooth, and they form channels through the dentin as the odontoblasts move inwards toward the pulp. The secreted collagen fibers, which are mainly

**Figure 1.** Drawing showing a tooth attached in its alveolar socket. The root of the tooth is attached to the alveolar socket via collagen fibers, the so called periodontal ligament. Blood vessels and nerves enter the pulp chamber via the apical opening.

The formation of mantle dentin triggers the ameloblasts to start secreting enamel proteins on the newly formed mantle dentin. The first hydroxyapatite crystals that form on the mantle dentin are randomly packed in this first formed enamel and interdigitated with the crystallites of dentin. Eventually the dentin crystallites present in the mantel dentin act as nucleation sites for the first enamel crystallites.

After the first layer of structureless enamel has formed, the ameloblasts move away from the DEJ, which permits the formation of the so called Tomes' processes, which form at the ends the ameloblasts closest to the DEJ. When the Tomes' processes are established, the enamel rods start developing (Figure 4).

**Figure 2.** The hard tissues are formed by the odontoblasts (dentin) and ameloblasts (enamel). During the development of the tooth, epithelial cells have formed a bell shaped enamel organ. Inside that bell is connective tissue that shows active budding of capillaries. At a certain stage, the fibroblasts in contact with the epithelium bell become highly differentiated and develop into odontoblasts and form the first layer of dentin. That layer stimulates the epithelium cells in contact with the dentin at the DEJ to differentiate into ameloblasts and form enamel. As a consequence, the two cell types move in opposite direction as they form dentin (blue arrows) and enamel (red arrows). When the ameloblasts reach the outer cells of the enamel organ they start degrading and lose vitality. At the same time, the dentin has increased in thickness and the epithelial diaphragm with odontoblasts have grown downwards and developed the root and the pulp chamber (see Figure 1).

**Figure 3.** The top drawing represents a cross-section of dentin, perpendicular to the peritubular dentin (A). The lower drawing represents a plane parallel to the odontoblastic processes (C) and cut along a-a. In the peritubular dentin, collagen fibers represented by pink lines are present parallel to the odontoblastic processes. Hydroxyapatite precipitate along these fibers, and together they form the so called peritubular dentin (A). Collagen precipitates perpendicular to the odontoblastic processes too, and when hydroxyapatite precipitate in that matrix, the intertubular dentin (B) is formed.

**Figure 2.** The hard tissues are formed by the odontoblasts (dentin) and ameloblasts (enamel). During the development of the tooth, epithelial cells have formed a bell shaped enamel organ. Inside that bell is connective tissue that shows active budding of capillaries. At a certain stage, the fibroblasts in contact with the epithelium bell become highly differentiated and develop into odontoblasts and form the first layer of dentin. That layer stimulates the epithelium cells in contact with the dentin at the DEJ to differentiate into ameloblasts and form enamel. As a consequence, the two cell types move in opposite direction as they form dentin (blue arrows) and enamel (red arrows). When the ameloblasts reach the outer cells of the enamel organ they start degrading and lose vitality. At the same time, the dentin has increased in thickness and the epithelial diaphragm with odontoblasts have grown downwards and

**Figure 3.** The top drawing represents a cross-section of dentin, perpendicular to the peritubular dentin (A). The lower drawing represents a plane parallel to the odontoblastic processes (C) and cut along a-a. In the peritubular dentin, collagen fibers represented by pink lines are present parallel to the odontoblastic processes. Hydroxyapatite precipitate along these fibers, and together they form the so called peritubular dentin (A). Collagen precipitates perpendicular to the odontoblastic processes too, and

when hydroxyapatite precipitate in that matrix, the intertubular dentin (B) is formed.

developed the root and the pulp chamber (see Figure 1).

**Figure 4.** After the first layer of structureless enamel has formed on the mantel dentin, the ameloblast differentiate its end closest to the precipitated enamel into the so called Tomes' process. This unit can be described as a concave formation from which hydroxyapatite crystallites precipitate. The c-axis of these crystallites are perpendicular to the surface of Tomes's process, explaining the the well organized precipitation of the hydroxyapatite crystallites in each enamel rod.

The secretion from the peripheral site of the Tomes' process results in the formation of what is referred to as the enamel matrix wall. These walls enclose pits into which the Tomes' processes fit. These sites are then filled with matrix proteins acting as nucleating agents for the hydroxylapatite crystallites. The crystallites that precipitate in these two matrices (the matrix wall and the central pit) have different orientation. It is important to emphasize that the final wall and pit enamel have the same composition. The only difference is the orientation of the crystallites in these two enamel types.

A cross-section of the enamel rods reveals that the individual rods have a key-hole shaped structure (Figure 5).

As the ameloblasts move toward their final destiny, they produce enamel rods that are somewhat wavy and interwoven (Figure 6). Independent on these waves, the enamel rods form angles that are roughly perpendicular to the outer as well as inner surfaces of the enamel shell (Figure 7). The hard enamel can be described as a hard shield protecting the underlying dental tissue of the visible part of the tooth.

Enamel consists mainly of hydroxyapatite crystallites, which are oriented in very well organized larger bundles of crystallites. These larger bundles are referred to as enamel rods. Each enamel rod is made by enamel forming cells, the so called ameloblasts. The diameters of the rods range from 4-8 μm. During enamel formation, the ameloblasts secret different proteins (amelogenins and enamelins), which act as nucleating agents for the hydroxyl

apatite crystallites. During enamel formation, the ameloblasts move from the dentin-enamel junction (DEJ) to the surface of the final enamel crown. When the enamel shell has reached its final shape, the ameloblasts degenerate and die, explaining why mature enamel is a nonvital tissue made up by ~85 vol-% hydroxyapatite and 15 vol-% proteins and water [2].

**Figure 5.** Cross section of enamel rods shows the key-hole structutre (blue). The longitudinal orientation of the hydroxyapatite crystallites can be explained by considering how Tomes' process controls the crystallite orientation (Figure 4). Figure redrawn after [3].

**Figure 6.** The keyhole shaped rods become more and more interwoven as the rods approaches the DEJ. Redrawn from [4]. The intervowen structure shown in the drawing is also characteristic for the cusp tips, where that type of enamel is called "gnarled enamel".

apatite crystallites. During enamel formation, the ameloblasts move from the dentin-enamel junction (DEJ) to the surface of the final enamel crown. When the enamel shell has reached its final shape, the ameloblasts degenerate and die, explaining why mature enamel is a nonvital tissue made up by ~85 vol-% hydroxyapatite and 15 vol-% proteins and water [2].

**Figure 5.** Cross section of enamel rods shows the key-hole structutre (blue). The longitudinal orientation of the hydroxyapatite crystallites can be explained by considering how Tomes' process controls the

**Figure 6.** The keyhole shaped rods become more and more interwoven as the rods approaches the DEJ. Redrawn from [4]. The intervowen structure shown in the drawing is also characteristic for the cusp

crystallite orientation (Figure 4). Figure redrawn after [3].

tips, where that type of enamel is called "gnarled enamel".

**Figure 7.** The bottom left drawing shows the orientation of the rods along the long axis of the tooth. As seen from both drawings, there is a continous shift in orientation resulting in the S-shaped orientation. If the line joining points 1, 11 and 21 along the S-shaped curve represents a plane forming 90 degrees to the enamel surface, it is seen from the drawing that there is a difference in rod orientation that can be described as 90 ± 10 degrees. Redrawn from [2].

The pulp chamber is a cavity inside the dentin formed by the surrounding dentin. The pulp chamber contains soft tissue, blood vessels and nerves and is lined by the odontoblasts. As the tooth grew older, the odontoblasts continue to produce dentin, causing the size of the pulp chamber to decrease with age.

As seen from the properties presented in Table 1, enamel has lower fracture toughness than dentin, but significantly higher hardness and modulus of elasticity. These properties suggest that enamel is a highly brittle material that should easily chip away from the dentin. Fortunately that is not the case. The reason can be related to a firm enamel-dentin attachment as well the sophisticated anisotropic composite structure of both enamel and dentin. If cracks propagate through the enamel, they often stop before they reach the enamel-dental interface, and if they continue propagating they usually stop when they reach the enamel-dentin interface. That explains why fractured teeth are not as common as one otherwise would expect by considering force and fatigue levels teeth have to withstand.


**Table 1.** Highest and lowest reported values in Xu et al.'s study[5], except for the fracture toughness value of dentin which is from El Mowafy and Watts study[6]. Identified variations relate to the anisotropic nature of enamel and dentin as well as variations among teeth.

## **2.1. Fracture mechanical aspects of enamel**

As discussed earlier, the tooth can be described as a rather complicated composite structure developed to serve the user. Nature adapted the principle that teeth must be hard and rigid in order to generate sufficiently high local stress levels. These stresses are capable of penetrating tissues during hunting and fighting, but also capable of crushing hard food. At the same time, enamel has also been designed to limit the inherent brittle nature of hydroxyapatite by dispersing propagating cracks and thereby resist some brittle failures.

By orienting the rods on the cusp tips along the axis of the tooth, a parallel model composite is formed in that region (Figure 8). At the same time, by orienting the rods more or less perpendicular to the long axis of the tooth in the remaining parts of the crown, a series model composite is formed in that part of the crown. These models are valid under the assumption the load is in an axial direction. Since the parallel model results in a stiffer combination than a series model material, the tooth has been designed so that rigidity is optimized in the chewing/biting direction and flexing in a direction perpendicular to that direction.

As the stiffness of the parallel model exceeds the stiffness of the series model, we can understand how such a design assists an animal attacking another animal. During such an attack, the canines of the attacking animal may penetrate the tissue of the attacked animal, but that bite may not necessarily result in an instant kill. During the biting action, the attacking animal benefits from the stiffness of the canines (parallel model behavior of the tip of the canines make the tooth stiff like a steel arrow). However, if the attacked animal was not killed instantaneously, most likely it will try to get loose from the attacking animal's jaws. During that attempt the risk of fracturing the canines of the attacking animal increases. However, thanks to the rod and tubule orientations in the cervical and mid crown regions, the material characteristics of the enamel in these bendable regions are represented by the series model, thereby allowing the tooth to flex somewhat and absorb mechanical energy rather than fracture.

116 Applied Fracture Mechanics

Hard tissue Modulus of elasticity (GPa)

As seen from the properties presented in Table 1, enamel has lower fracture toughness than dentin, but significantly higher hardness and modulus of elasticity. These properties suggest that enamel is a highly brittle material that should easily chip away from the dentin. Fortunately that is not the case. The reason can be related to a firm enamel-dentin attachment as well the sophisticated anisotropic composite structure of both enamel and dentin. If cracks propagate through the enamel, they often stop before they reach the enamel-dental interface, and if they continue propagating they usually stop when they reach the enamel-dentin interface. That explains why fractured teeth are not as common as one otherwise would expect by considering force and fatigue levels teeth have to withstand.

Fracture toughness

Hardness (GPa)

(MPa m1/2)

Enamel 78 ± 1 to 98 ± 4 0.44 ± 0.04 to 1.55 ± 0.29 2.83 ± 0.10 to 3.74 ± 0.48 Dentin 18 ± 1 to 22 ± 1 3.08 ± 0.33 0.53 ± 0.01 to 0.63 ± 0.03

**Table 1.** Highest and lowest reported values in Xu et al.'s study[5], except for the fracture toughness value of dentin which is from El Mowafy and Watts study[6]. Identified variations relate to the aniso-

As discussed earlier, the tooth can be described as a rather complicated composite structure developed to serve the user. Nature adapted the principle that teeth must be hard and rigid in order to generate sufficiently high local stress levels. These stresses are capable of penetrating tissues during hunting and fighting, but also capable of crushing hard food. At the same time, enamel has also been designed to limit the inherent brittle nature of hydroxyap-

By orienting the rods on the cusp tips along the axis of the tooth, a parallel model composite is formed in that region (Figure 8). At the same time, by orienting the rods more or less perpendicular to the long axis of the tooth in the remaining parts of the crown, a series model composite is formed in that part of the crown. These models are valid under the assumption the load is in an axial direction. Since the parallel model results in a stiffer combination than a series model material, the tooth has been designed so that rigidity is optimized in the chewing/biting direction and flexing in a direction perpendicular to that direction.

As the stiffness of the parallel model exceeds the stiffness of the series model, we can understand how such a design assists an animal attacking another animal. During such an attack, the canines of the attacking animal may penetrate the tissue of the attacked animal, but that bite may not necessarily result in an instant kill. During the biting action, the attacking animal benefits from the stiffness of the canines (parallel model behavior of the tip of the canines make the tooth stiff like a steel arrow). However, if the attacked animal was not killed instantaneously, most likely it will try to get loose from the attacking animal's jaws. During that attempt the risk of fracturing the canines of the attacking animal increases.

atite by dispersing propagating cracks and thereby resist some brittle failures.

tropic nature of enamel and dentin as well as variations among teeth.

**2.1. Fracture mechanical aspects of enamel** 

**Figure 8.** A tooth loaded in axial direction (blue arrow) responds in two ways. On the cusp tips, the rods and the tubules are oriented parallel with the load and resulting in a material which modulus can be predicted from a parallel model prediction. In the cervical region of the tooth, the modulus can be predicted from a series model.

Some basic science studies have been conducted to study the fracture behavior of enamel and dentin through the years. In one such study [7] the investigators studied a mandibular molar tooth restored with different Class II amalgam preparations. By use of finite element analysis, the stress distribution induced along the internal edges as a result of occlusal loading was calculated, and by use of Paris law the cyclic crack growth rate of sub-surface flaws located along the dentinal internal edges was determined. Based on the assumptions used in their calculations, they claimed that flaws located within the dentin along the buccal and lingual internal edges can reduce the fatigue life of restored teeth significantly. Sub-surface cracks as short as 25 μm were capable of promoting tooth fracture well within 25 years from the time of restoration placement. Furthermore, cracks longer than 100 μm reduced the fatigue life of the tooth to less than 5 years. Consequently, sub-surface cracks introduced

during cavity preparation with conventional dental burs may serve as a principal source for premature restoration failure.

As the hardest and one of the most durable load bearing tissues of the body, enamel has attracted considerable interest from both material scientists and clinical practitioners due to its excellent mechanical properties. In a recent article [8] possible mechanisms responsible for the excellent mechanical properties of enamel were explored and summarized. What these authors emphasized was the hierarchical structure and the nanomechanical properties of the minor protein macromolecular components. The experimental and numerical results supported the made assumptions. For example, enamel showed to have lower elastic modulus, higher energy absorption ability and greater indentation creep behavior than sintered hydroxyapatite material. These findings suggest that the structural and compositional characteristics of the minor protein component significantly regulate the mechanical properties of enamel in order to better match its functional needs.

The fascinating aspect of enamel is that its structure seems to have evolved and adapted to the need of the user of the teeth. For example, in some recent publications [9, 10], these issues have been discussed. Lucas et al. [9] proposed a model based on how fracture and deformation concepts of teeth may be adapted to the mechanical demands of diet, while Constantino et al [10] used that model by examining existing data on the food mechanical properties and enamel morphology of great apes (Pan, Pongo, and Gorilla). They paid particular attention to whether the consumption of fallback foods plays a key role in influencing great ape enamel morphology. Their results suggest that so is the case, and that their findings may explain the evolution of the dentition of extinct hominins.

Along these lines, Lee et al.[11] did a comparative study of human and great ape molar tooth enamel. They used nano-indentation techniques to map profiles of elastic modulus and hardness across sections from the enamel–dentin junction to the outer tooth surface. The measured data profiles overlapped between species, suggesting a degree of commonality in material properties. Using established deformation and fracture relations, critical loads to produce function-threatening damage in the enamel of each species were calculated for characteristic tooth sizes and enamel thicknesses. The results suggest that differences in load-bearing capacity of molar teeth in primates are less a function of underlying material properties than of morphology.

From the above studies, it is quite clear that Nature has adapted the structure of enamel to resist fractures. In a study by Bajaj [4] the crack growth resistance behavior and fracture toughness of human tooth enamel was determined. The results were quantified using incremental crack growth measures and conventional fracture mechanics. The results revealed that enamel undergoes an increase in crack growth resistance (i.e. rising R-curve) with crack extension from the outer to the inner enamel, and that the rise in toughness is a function of distance from the dentin enamel junction (DEJ). The outer enamel exhibited the lowest apparent toughness (0.67± 0.12 MPa m0.5), and the inner enamel exhibited a rise in the growth toughness from 1.13 MPa m0.5/mm to 3.93 MPa m0.5/mm. The maximum crack growth resistance at fracture (i.e. fracture toughness (KC)) ranged from 1.79 to 2.37 MPa m0.5. Crack growth in the inner enamel was accompanied by a host of mechanisms operating from the micro- to the nano-scale. Decussation in the inner enamel promoted crack deflection and twist, resulting in a reduction of the local stress intensity at the crack tip (Figures 6 and 7). In addition, extrinsic mechanisms such as bridging by unbroken ligaments of the tissue and the organic matrix promoted crack closure. Micro-cracking due to loosening of prisms was also identified as an active source of energy dissipation. The unique microstructure of enamel in the decussated region promotes crack growth toughness that is approximately three times that of dentin and over ten times that of bone.

In addition to the micro- and nano-structure of enamel, the tooth anatomy by itself is such that it has adapted to force conditions present in the oral cavity. Anderson et al. [12] modeled what they believed drove the initial evolution of the cingulum. Recent work on physical modeling of fracture mechanics has shown that structures which approximate mammalian dentition (hard enamel shell surrounding a softer/tougher dentine interior) undergo specific fracture patterns dependent on the material properties of the food items [9, 13]. Soft materials result in fractures occurring at the base of the stiff shell away from the contact point due to heightened tensile strains. These tensile strains occur around the margin in the region where cingula develop. In Anderson et al.'s [12] study, they tested whether the presence of a cingulum structure would reduce the tensile strains seen in enamel using basic finite element models of bilayered cones. Finite element models of generic cone shaped ''teeth'' were created both with and without cingula of various shapes and sizes. Various forces were applied to the models to examine the relative magnitudes and directions of average maximum principal strain in the enamel. The addition of a cingulum greatly reduces tensile strains in the enamel caused by ''soft-food'' forces. The relative shape and size of the cingulum has a strong effect on strain magnitudes as well. Scaling issues between shapes are explored and show that the effectiveness of a given cingulum to reducing tensile strains is dependent on how the cingulum is created. Partial cingula, which only surround a portion of the tooth, are shown to be especially effective at reducing strain caused by asymmetrical loads, and shed new light on the potential early function and evolution of mammalian dentitions.

#### **2.2. Fracture mechanical aspects of dentin**

118 Applied Fracture Mechanics

premature restoration failure.

properties than of morphology.

during cavity preparation with conventional dental burs may serve as a principal source for

As the hardest and one of the most durable load bearing tissues of the body, enamel has attracted considerable interest from both material scientists and clinical practitioners due to its excellent mechanical properties. In a recent article [8] possible mechanisms responsible for the excellent mechanical properties of enamel were explored and summarized. What these authors emphasized was the hierarchical structure and the nanomechanical properties of the minor protein macromolecular components. The experimental and numerical results supported the made assumptions. For example, enamel showed to have lower elastic modulus, higher energy absorption ability and greater indentation creep behavior than sintered hydroxyapatite material. These findings suggest that the structural and compositional characteristics of the minor protein component significantly regulate the

The fascinating aspect of enamel is that its structure seems to have evolved and adapted to the need of the user of the teeth. For example, in some recent publications [9, 10], these issues have been discussed. Lucas et al. [9] proposed a model based on how fracture and deformation concepts of teeth may be adapted to the mechanical demands of diet, while Constantino et al [10] used that model by examining existing data on the food mechanical properties and enamel morphology of great apes (Pan, Pongo, and Gorilla). They paid particular attention to whether the consumption of fallback foods plays a key role in influencing great ape enamel morphology. Their results suggest that so is the case, and that

Along these lines, Lee et al.[11] did a comparative study of human and great ape molar tooth enamel. They used nano-indentation techniques to map profiles of elastic modulus and hardness across sections from the enamel–dentin junction to the outer tooth surface. The measured data profiles overlapped between species, suggesting a degree of commonality in material properties. Using established deformation and fracture relations, critical loads to produce function-threatening damage in the enamel of each species were calculated for characteristic tooth sizes and enamel thicknesses. The results suggest that differences in load-bearing capacity of molar teeth in primates are less a function of underlying material

From the above studies, it is quite clear that Nature has adapted the structure of enamel to resist fractures. In a study by Bajaj [4] the crack growth resistance behavior and fracture toughness of human tooth enamel was determined. The results were quantified using incremental crack growth measures and conventional fracture mechanics. The results revealed that enamel undergoes an increase in crack growth resistance (i.e. rising R-curve) with crack extension from the outer to the inner enamel, and that the rise in toughness is a function of distance from the dentin enamel junction (DEJ). The outer enamel exhibited the lowest apparent toughness (0.67± 0.12 MPa m0.5), and the inner enamel exhibited a rise in the growth toughness from 1.13 MPa m0.5/mm to 3.93 MPa m0.5/mm. The maximum crack growth resistance at fracture (i.e. fracture toughness (KC)) ranged from 1.79 to 2.37 MPa m0.5.

mechanical properties of enamel in order to better match its functional needs.

their findings may explain the evolution of the dentition of extinct hominins.

Dentin is not as brittle as enamel. However, considering that enamel rests on dentin, and that cracks may propagate through the enamel, it is important to understand the fracture mechanics of dentin.

Human dentin is known to be susceptible to failure under repetitive cyclic fatigue loading. Nalla et al. [14] addressed the paucity of fatigue data through a systematic investigation of the effects of prolonged cyclical loading on human dentin. They performed the evaluations in an environment of ambient temperature and where the dentin was kept in a Hank's balanced salt solution. The results they got were discussed in the context of possible mechanisms of fatigue damage and failure. The stiffness loss data collected were used to deduce crack velocities and the thresholds for such cracking. They concluded that the

presence of small (on the order of 250 μm) incipient flaws in human dentin will not radically affect their useful life as Arola et al.[7] claimed.

Kruzic et al. [15] investigated the fracture toughness properties of dentin in terms of resistance-curve (R-curve) behavior, i.e., fracture resistance increase with crack extension. Of particular interest was the identification of relevant toughening mechanisms involved in the crack growth. Their study was conducted on elephant dentin, and they compared hydrated and dehydrated dentin. Crack bridging by uncracked ligaments, observed directly by microscopy and X-ray tomography, was identified as a major toughening mechanism. Further experimental evidence were provided by compliance-based experiments. In addition, with hydration, dentin was observed to display significant crack blunting leading to a higher overall fracture resistance than in the dehydrated material. In this paper they show how uncracked bridges remain behind the propagating crack, giving the dentin some fracture toughness.

Bajaj et al. [16] used striations resulting from fatigue crack growth in the dentin of human teeth to identify difference between young and old dentin. They used compact tension (CT) specimens obtained from the coronal dentin of molars from young (17 ≤ age ≤ 37 years) and senior (age ≥ 50 years) individuals, and exposed the dentin to cyclic Mode I loads. Striations evident on the fracture surfaces were examined using a scanning electron microscope and contact profilometer. Fatigue crack growth striations that developed in vivo were also examined on fracture surfaces of restored molars. The average spacing in the dentin of seniors (130 ± 23 μm) was significantly larger (p < 0.001) than that in young dentin (88 ± 13 μm). Fatigue striations in the restored teeth exhibited features that were consistent with those that developed in vitro and a spacing ranging from 59 to 95 μm. Unlike metals, the striations in dentin developed after a period of cyclic loading that ranged from 1 x 103 to 1 x 105 cycles. The study showed that the cracks tend to propagate perpendicular towards the orientation of the tubules, and climb along a plane tangential to the peritubular cuffs and then continue perpendicularly to the tubules.

Yan et al. [17] showed that rather than using a linear-elastic fracture mechanics (LEFM)(KC) that ignores plastic deformation and tend to underestimate the fracture toughness, a plastic fracture mechanics (EPFM)(KJC) approach was used. The presence of collagen (approximately 30% by volume) was assumed to enhance the toughening mechanisms in dentin. By comparing the values of the fracture toughness values estimated using either LEFM or EPFM, they found that the KC and KJC values of plane parallel as well as antiplane parallel specimens were different. The fracture toughness estimated based on KJC was significantly greater than that estimated based on KC (32.5% on average; p<0.001). In addition, KJC of antiplane parallel specimens was significantly greater than that of in-plane parallel specimens. Consequently, in order to critically evaluate the fracture toughness of human dentin, EPFM should be employed rather than LEFM.

## **3. Man made dental materials**

By considering the sophistication of the biological materials enamel and dentin, it is easy to understand why it is such a challenge to identify a man made material that can compete with the biological hard tissues. In addition to their mechanical properties, such a material should be biocompatible, aesthetic, corrosion resistant, easy to process and reasonable inexpensive, making such an identification extremely challenging. Of these properties, strength values within a group of materials are often used by manufacturers in their marketing and by the dentist when it comes to selecting a product. Unfortunately, strength by itself may not be the best parameter to choose. The reason is that strength is a conditional rather than an inherent material property [18]. Strength data alone should therefore not be used to extrapolate and predict the performance of a structure. Instead, they should be used together with the microstructure of the material, processing history, testing methodology, testing environment and failure mechanism. Structural failures are determined by additional failure probability variables in concert with strength that describe stress distributions, flaw size distributions, which can contribute to either single or multiple failure modes. Lifetime predictions require additional information about the time dependence of slow crack growth. Basic fracture mechanics principles and Weibull failure modeling are important to consider.

To make dental treatment even more challenging, just consider how dentists cut teeth and use different materials. During the cutting process, flaws of different sizes are most likely induced in the remaining tooth structure. Flaws and different defects are also most likely induced during handling and insertion of different materials. The impact of such flaws can be devastating for any material, particularly for brittle ceramic materials. To show how different surface treatments can affect the strength properties, Table 2 has been included to show how different surface treatments of glass can affect its strength [19]. A severely sandblasted glass lose as much as 67% of its original strength, while a drawn silica fiber tested in vacuum is 400 times stronger than the glass, a difference that can be related to the presence of water molecules in air.


**Table 2.** Effect of surface treatments on the strength of glass

120 Applied Fracture Mechanics

toughness.

perpendicularly to the tubules.

be employed rather than LEFM.

**3. Man made dental materials** 

affect their useful life as Arola et al.[7] claimed.

presence of small (on the order of 250 μm) incipient flaws in human dentin will not radically

Kruzic et al. [15] investigated the fracture toughness properties of dentin in terms of resistance-curve (R-curve) behavior, i.e., fracture resistance increase with crack extension. Of particular interest was the identification of relevant toughening mechanisms involved in the crack growth. Their study was conducted on elephant dentin, and they compared hydrated and dehydrated dentin. Crack bridging by uncracked ligaments, observed directly by microscopy and X-ray tomography, was identified as a major toughening mechanism. Further experimental evidence were provided by compliance-based experiments. In addition, with hydration, dentin was observed to display significant crack blunting leading to a higher overall fracture resistance than in the dehydrated material. In this paper they show how uncracked bridges remain behind the propagating crack, giving the dentin some fracture

Bajaj et al. [16] used striations resulting from fatigue crack growth in the dentin of human teeth to identify difference between young and old dentin. They used compact tension (CT) specimens obtained from the coronal dentin of molars from young (17 ≤ age ≤ 37 years) and senior (age ≥ 50 years) individuals, and exposed the dentin to cyclic Mode I loads. Striations evident on the fracture surfaces were examined using a scanning electron microscope and contact profilometer. Fatigue crack growth striations that developed in vivo were also examined on fracture surfaces of restored molars. The average spacing in the dentin of seniors (130 ± 23 μm) was significantly larger (p < 0.001) than that in young dentin (88 ± 13 μm). Fatigue striations in the restored teeth exhibited features that were consistent with those that developed in vitro and a spacing ranging from 59 to 95 μm. Unlike metals, the striations in dentin developed after a period of cyclic loading that ranged from 1 x 103 to 1 x 105 cycles. The study showed that the cracks tend to propagate perpendicular towards the orientation of the tubules, and climb along a plane tangential to the peritubular cuffs and then continue

Yan et al. [17] showed that rather than using a linear-elastic fracture mechanics (LEFM)(KC) that ignores plastic deformation and tend to underestimate the fracture toughness, a plastic fracture mechanics (EPFM)(KJC) approach was used. The presence of collagen (approximately 30% by volume) was assumed to enhance the toughening mechanisms in dentin. By comparing the values of the fracture toughness values estimated using either LEFM or EPFM, they found that the KC and KJC values of plane parallel as well as antiplane parallel specimens were different. The fracture toughness estimated based on KJC was significantly greater than that estimated based on KC (32.5% on average; p<0.001). In addition, KJC of antiplane parallel specimens was significantly greater than that of in-plane parallel specimens. Consequently, in order to critically evaluate the fracture toughness of human dentin, EPFM should

By considering the sophistication of the biological materials enamel and dentin, it is easy to understand why it is such a challenge to identify a man made material that can compete By use of Griffith's equation[20], one can show how the stress level is affected by flaw size and surface energy and explain the results presented in Table 2. That equation further shows that any processing step affecting the size, orientation or distribution of flaws will affect the measured strength of materials, particularly brittle materials. It also shows how environmental conditions may affect surface energy and thereby also the strength.

### **3.1. Fracture mechanics aspects of ceramics**

Clinical experience suggests that all-ceramic crowns may not be as durable as their porcelain-fused-to-metal counterparts, particularly when placed on molar teeth. The reason

relates to the brittleness of ceramics, making them prone for chipping and fracturing [21-27]. In the 1980s and 1990s, crowns were fabricated as enamel-like monoliths from micaceous glass-ceramics (Dicor, Dentsply/Caulk, Milford, DE) and high leucite porcelains (IPS Empress, Ivoclar, Schaan, Lichtenstein), but these ceramics showed unacceptably high failure rates and were soon replaced by improved ceramics [28, 29]. Subsequent crown design has focused on retention of porcelain as an aesthetic veneer fired to much stronger alumina-based ceramics, either glassinfiltrated (InCeram, Vita Zahnfabrik, Bad S.ackingen, Germany) or pure and dense (Procera, Nobel Biocare, Goteborg, Sweden) alumina, as supporting cores. Although alumina-based crowns continued to replace metal-based crowns, failure rates remained an issue [30]. During the past 15 years, ultra-strong core ceramics, e.g. yttria-stabilized zirconia (Y-TZP) and alumina-matrix composites (AMC)[31] have gained in popularity but have yet to be documented regarding their clinical long-term success.

Clinically, bulk fractures are the reported cause of all ceramic crown failure whether the crown is a monolith or a layered structure. According to a fractographic evaluation by Thompson et al.[32], in which they evaluated fractured and recovered Dicor and Cerestore crowns, they found that failures generally did not ensue from damage at the occlusal surface. Instead, for Dicor the cracks emerged from the internal surface, while in the case of Cerestore, the initiation occurred at the porcelain/core interface inside the core materials. In other studies it has been shown that radial cracks are initially contained within the inner core layer, but subsequently propagate to the core boundaries, ultimately causing irretrievable failure. This failure mechanism raises an interesting question: If the ceramic core materials are so strong, why do the cracks not originate in the weak outer porcelain? In the case of porcelain-fused-to-metal, porcelain failures do seem to occur preferentially in the porcelain, although there is some indication that such failures may be preceded by plasticity in the ductile metal [33]. That in turn raises the question: What are the important material parameters that govern these failure modes in crown structures, and how may they be optimized? Maybe McLean's [33] suggestion from 1983 that layered all-ceramic crowns should perform well if the core fracture strength exceeded the yield strength of base metal alloys (about 400– 500MPa for gold).

Before diving deeper into the fracture strength of the core, let us accept that there are several factors which can be associated with crack initiation and propagation in dental ceramic restorations. These factors include: (a) shape of the restoration; (b) micro-structural inhomogeneities; (c) size and distribution of surface flaws; (d) residual stresses and stress gradients, induced by polishing and/or thermal processing; (e) the environment in contact with the restoration; (f) ceramic/cement interfacial features; (g) thickness and thickness variation of the restoration; (h) elastic module of restoration components; and (i) magnitude and orientation of applied loads. The possible interactions among these variables complicate the interpretation of failure analysis observations, explaining why fracture behavior of allceramic crowns is rather tricky problem to understand.

Even though McLean's[34] suggestion that the core fracture strength exceeded the yield strength of base metal alloys (about 400–500MPa for gold) might be tempting to adopt to, it is very important to realize that ceramics, in contrast to metals, are brittle materials, and that strength is more of a "conditional" than an inherent material property, and strength data alone cannot be directly extrapolated to predict structural performance [18]. Strength data, particularly of brittle materials, are meaningful when placed into context via knowledge of material microstructure, processing history, testing methodology, testing environment and failure mechanism(s). Lifetime predictions require additional information about the time dependence of slow crack-growth. Basic fracture mechanics principles and Weibull failure modeling are key factors to consider as well as the role of interfacial stresses. Thus, in order to understand the actual clinical failure mode it is absolutely necessary to consider all the variables listed in the previous paragraph until results from in vitro strength testing can be considered to have any clinical value.

122 Applied Fracture Mechanics

success.

500MPa for gold).

relates to the brittleness of ceramics, making them prone for chipping and fracturing [21-27]. In the 1980s and 1990s, crowns were fabricated as enamel-like monoliths from micaceous glass-ceramics (Dicor, Dentsply/Caulk, Milford, DE) and high leucite porcelains (IPS Empress, Ivoclar, Schaan, Lichtenstein), but these ceramics showed unacceptably high failure rates and were soon replaced by improved ceramics [28, 29]. Subsequent crown design has focused on retention of porcelain as an aesthetic veneer fired to much stronger alumina-based ceramics, either glassinfiltrated (InCeram, Vita Zahnfabrik, Bad S.ackingen, Germany) or pure and dense (Procera, Nobel Biocare, Goteborg, Sweden) alumina, as supporting cores. Although alumina-based crowns continued to replace metal-based crowns, failure rates remained an issue [30]. During the past 15 years, ultra-strong core ceramics, e.g. yttria-stabilized zirconia (Y-TZP) and alumina-matrix composites (AMC)[31] have gained in popularity but have yet to be documented regarding their clinical long-term

Clinically, bulk fractures are the reported cause of all ceramic crown failure whether the crown is a monolith or a layered structure. According to a fractographic evaluation by Thompson et al.[32], in which they evaluated fractured and recovered Dicor and Cerestore crowns, they found that failures generally did not ensue from damage at the occlusal surface. Instead, for Dicor the cracks emerged from the internal surface, while in the case of Cerestore, the initiation occurred at the porcelain/core interface inside the core materials. In other studies it has been shown that radial cracks are initially contained within the inner core layer, but subsequently propagate to the core boundaries, ultimately causing irretrievable failure. This failure mechanism raises an interesting question: If the ceramic core materials are so strong, why do the cracks not originate in the weak outer porcelain? In the case of porcelain-fused-to-metal, porcelain failures do seem to occur preferentially in the porcelain, although there is some indication that such failures may be preceded by plasticity in the ductile metal [33]. That in turn raises the question: What are the important material parameters that govern these failure modes in crown structures, and how may they be optimized? Maybe McLean's [33] suggestion from 1983 that layered all-ceramic crowns should perform well if the core fracture strength exceeded the yield strength of base metal alloys (about 400–

Before diving deeper into the fracture strength of the core, let us accept that there are several factors which can be associated with crack initiation and propagation in dental ceramic restorations. These factors include: (a) shape of the restoration; (b) micro-structural inhomogeneities; (c) size and distribution of surface flaws; (d) residual stresses and stress gradients, induced by polishing and/or thermal processing; (e) the environment in contact with the restoration; (f) ceramic/cement interfacial features; (g) thickness and thickness variation of the restoration; (h) elastic module of restoration components; and (i) magnitude and orientation of applied loads. The possible interactions among these variables complicate the interpretation of failure analysis observations, explaining why fracture behavior of all-

Even though McLean's[34] suggestion that the core fracture strength exceeded the yield strength of base metal alloys (about 400–500MPa for gold) might be tempting to adopt to, it

ceramic crowns is rather tricky problem to understand.

Natural teeth as well as most modern ceramic restorations can be described as layered structures. In the case of teeth the layers are enamel and dentin, while in the case of all ceramics a core ceramic and a porcelain coating. There are also unlayered ceramics in use, but since they are resting on a cement layer and dentin, even they can be described as layered structures. In a study by Jung et al. [35], they determined whether coating thickness and coating/substrate mismatch are key factors in the determination of contact induced damage in clinically relevant bilayer composites. They studied crack patterns in two bilayer systems conceived to simulate crown and tooth structures, at opposite extremes of elastic/plastic mismatch. In one case they looked at porcelain on glass-infiltrated alumina ("soft/hard"), and glass-ceramic on resin composite ("hard/soft"). Hertzian contacts were used to investigate the evolution of fracture damage in the coating layers, as functions of contact load and coating thickness. The crack patterns differed radically in the two bilayer systems: In the porcelain coatings, cone cracks initiate at the coating top surface; in the glass-ceramic coatings, cone cracks again initiate at the top surface, but additionally, upward-extending transverse cracks initiate at the internal coating/substrate interface, where the latter were dominant. This study revealed that the substrate has a profound influence on the damage evolution to ultimate failure in bilayer systems. It was also found that the cracks were highly stabilized in both systems, with wide ranges between the loads to initiate first cracking and to cause final failure, implying damage-tolerant structures. Finite element modeling was used to evaluate the tensile stresses responsible for the different crack types.

In a follow up study, Jung et al.[36] assumed that the lifetimes of dental restorations are limited by the accumulation of contact damage introduced during chewing, and that the strengths of dental ceramics are significantly lower after multi-cycle loading than after single-cycle loading. To test that hypothesis, they looked at indentation damage and associated strength degradation from multi-cycle contacts using spherical indenters in water. They evaluated four dental ceramics: "aesthetic" ceramics porcelain and micaceous glass-ceramic (MGC), and "structural" ceramics--glass-infiltrated alumina and yttriastabilized tetragonal zirconia polycrystal (Y-TZP) They found that at large numbers of contact cycles, all materials showed an abrupt transition in damage mode, consisting of strongly enhanced damage inside the contact area and attendant initiation of radial cracks outside. This transition in damage mode is not observed in comparative static loading tests, attesting to a strong mechanical component in the fatigue mechanism. Radial cracks, once formed, lead to rapid degradation in strength properties, signaling the end of the useful lifetime of the material. Strength degradation from multi-cycle contacts were examined in the test materials, after indentation at loads from 200 to 3000 N up to 106 cycles. Degradation occurs in the porcelain and MGC after ~ 104 cycles at loads as low as 200 N; comparable degradation in the alumina and Y-TZP requires loads higher than 500 N, well above the clinically significant range.

In another study from the same year, Drummond et al. [37] evaluated the flexure strength under static and cyclic loading and determined the fracture toughness under static loading of six restorative ceramic materials. Their intent was primary to compare four leucite (K2O•Al2O3•4SiO2) strengthened feldspathic (pressable) porcelains to a low fusing feldspathic porcelain and an experimental lithium disilicate containing ceramic. All materials were tested as a control in air and distilled water (without aging) and after three months aging in air or distilled water to determine flexure strength and fracture toughness. A staircase approach was used to determine the cyclic flexure strength. The mean flexure strength for the controls in air and water (without aging or cyclic loading) ranged from 67 to 99 MPa, except the experimental ceramic that was twice as strong with mean flexure strength of 191– 205 MPa. For the mean fracture toughness, the range was 1.1–1.9 MPa m0.5 with the experimental ceramic being 2.7 MPa m 0.5. The effect of testing in water and aging for three months caused a moderate reduction in the mean flexure strength (6–17%), and a moderate to severe reduction in the mean fracture toughness (5–39%). The largest decrease (15–60%) in mean flexure strength was observed when the samples were subjected to cyclic loading. The conclusion they draw from the study was that the lithium disilicate containing ceramic had significantly higher flexure strength and fracture toughness when compared to the four pressable leucite strengthened ceramics and the low fusing conventional porcelain. All of the leucite containing pressable ceramics did provide an increase in mean flexure strength (17–19%) and mean fracture toughness (3–64%) over the conventional feldspathic porcelain. Further, the influence of testing environment and loading conditions implies that these ceramic materials in the oral cavity might be susceptible to cyclic fatigue, resulting in a significant decrease in the survival time of all-ceramic restorations.

The studies conducted by Jung et al. [35, 36] were followed up by Rhee et al. [38] who approached the onset of competing fracture modes in ceramic coatings on compliant substrates from Hertzian-like contacts. They paid special attention to a deleterious mode of radial cracking that initiates at the lower coating surface beneath the contact, in addition to traditional cone cracking and quasiplasticity in the near contact area. The critical load relations were expressed in terms of well-documented material parameters (elastic modulus, toughness, hardness, and strength) and geometrical parameters (coating thickness and sphere radius). Data from selected glass, Al2O3 and ZrO2 coating materials on polycarbonate substrates were used to demonstrate the validity of the relations. The formulation provides a basis for designing ceramic coatings with optimum damage resistance.

Deng et al. [39] used spherical indenters on flat ceramic coating layers bonded to compliant substrates. They identified critical loads needed to produce various damage modes, cone cracking, and quasi-plasticity at the top surfaces and radial cracking at the lower (inner) surfaces are measured as a function of ceramic-layer thickness. The characteristic features of these were;

i. Cone cracks initiate from the top surface outside the contact circle, where the Hertzian tensile stress level reaches its maximum [40, 41]. The crack first grows downward as a shallow, stable surface ring, resisted by the material toughness T (KIC), before popping into full cone geometry at load

PC = A(T2/E)r

124 Applied Fracture Mechanics

clinically significant range.

attesting to a strong mechanical component in the fatigue mechanism. Radial cracks, once formed, lead to rapid degradation in strength properties, signaling the end of the useful lifetime of the material. Strength degradation from multi-cycle contacts were examined in the test materials, after indentation at loads from 200 to 3000 N up to 106 cycles. Degradation occurs in the porcelain and MGC after ~ 104 cycles at loads as low as 200 N; comparable degradation in the alumina and Y-TZP requires loads higher than 500 N, well above the

In another study from the same year, Drummond et al. [37] evaluated the flexure strength under static and cyclic loading and determined the fracture toughness under static loading of six restorative ceramic materials. Their intent was primary to compare four leucite (K2O•Al2O3•4SiO2) strengthened feldspathic (pressable) porcelains to a low fusing feldspathic porcelain and an experimental lithium disilicate containing ceramic. All materials were tested as a control in air and distilled water (without aging) and after three months aging in air or distilled water to determine flexure strength and fracture toughness. A staircase approach was used to determine the cyclic flexure strength. The mean flexure strength for the controls in air and water (without aging or cyclic loading) ranged from 67 to 99 MPa, except the experimental ceramic that was twice as strong with mean flexure strength of 191– 205 MPa. For the mean fracture toughness, the range was 1.1–1.9 MPa m0.5 with the experimental ceramic being 2.7 MPa m 0.5. The effect of testing in water and aging for three months caused a moderate reduction in the mean flexure strength (6–17%), and a moderate to severe reduction in the mean fracture toughness (5–39%). The largest decrease (15–60%) in mean flexure strength was observed when the samples were subjected to cyclic loading. The conclusion they draw from the study was that the lithium disilicate containing ceramic had significantly higher flexure strength and fracture toughness when compared to the four pressable leucite strengthened ceramics and the low fusing conventional porcelain. All of the leucite containing pressable ceramics did provide an increase in mean flexure strength (17–19%) and mean fracture toughness (3–64%) over the conventional feldspathic porcelain. Further, the influence of testing environment and loading conditions implies that these ceramic materials in the oral cavity might be susceptible to cyclic fatigue, resulting in a sig-

The studies conducted by Jung et al. [35, 36] were followed up by Rhee et al. [38] who approached the onset of competing fracture modes in ceramic coatings on compliant substrates from Hertzian-like contacts. They paid special attention to a deleterious mode of radial cracking that initiates at the lower coating surface beneath the contact, in addition to traditional cone cracking and quasiplasticity in the near contact area. The critical load relations were expressed in terms of well-documented material parameters (elastic modulus, toughness, hardness, and strength) and geometrical parameters (coating thickness and sphere radius). Data from selected glass, Al2O3 and ZrO2 coating materials on polycarbonate substrates were used to demonstrate the validity of the relations. The formulation provides

Deng et al. [39] used spherical indenters on flat ceramic coating layers bonded to compliant substrates. They identified critical loads needed to produce various damage modes, cone

nificant decrease in the survival time of all-ceramic restorations.

a basis for designing ceramic coatings with optimum damage resistance.

with A = 8.6x103 from fits to data from monolithic ceramics with known toughness [42]

ii. Quasiplasticity initiates when the maximum shear stress in the Hertzian near field exceeds Y/2, with yield stress Y ~ H/3 determined by the material hardness H (load/projected area, Vickers indentation)[43]. The critical load is PY = DH(H/E)2r2

with D = 0.85 from fits to data for monolithic ceramics with known hardness [42].

iii. Radial cracks initiate spontaneously from a starting flaw at the lower ceramic surface when the maximum tensile stress in this surface equals the bulk flexure strength σF, at load

PR = BσFd2/log(EC/ES)

with d being the ceramic layer thickness and B = 2.0 from data fits to well-characterized ceramic-based bilayer systems [38].

Thus, given basic material parameters, one can in principal make priori predictions of the critical loads for any given bilayer system. Note that PC and PY are independent of layer thickness d, whereas PR is independent of sphere radius r. These relations, within the limits of certain underlying assumptions, have been verified for model ceramic/substrate bilayer systems [38, 44]. They claimed that these damage modes, especially radial cracking, were directly relevant to the failure of all-ceramic dental crowns. The critical load data were analyzed with the use of explicit fracture-mechanics relations, expressible in terms of routinely measurable material parameters (elastic modulus, strength, toughness, hardness) and essential geometrical variables (layer thickness, contact radius).

Lawn et al. [45] conducted tests on model flat-layer specimens fabricated from various dental ceramic combinations bonded to dentin-like polymer substrates in bilayer (ceramic/polymer) and trilayer (ceramic/ceramic/polymer) configurations. The specimens were loaded at their top surfaces with spherical indenters, simulating occlusal function. The onset of fracture was observed in situ using a video camera system mounted beneath the transparent polymer substrate. Critical loads to induce fracture and deformation at the ceramic top and bottom surfaces were measured as functions of layer thickness and contact duration. Radial cracking at the ceramic undersurface occurred at relatively low loads, especially in thinner layers. Fracture mechanics relations were used to confirm the experimental data trends, and to provide explicit dependencies of critical loads in terms of key variables (material—elastic modulus, hardness, strength and toughness; geometric—layer thicknesses and contact radius). Tougher, harder and (especially) stronger materials show superior damage resistance. Critical loads depend strongly (quadratically) on crown net thickness. The analytic relations provided a seemingly sound basis for the materials design of next-generation dental crowns.

## **3.2. Fracture mechanics aspects of dental composites**

Dental composite resins consist of ceramic filler particles, usually within a size range of 1-5 μm and mixed with nano-sized (20-40 nm) particles. These inorganic filler particles are silane coated and mixed with a curable monomer to form a viscous paste that can be inserted into a prepared cavity, whereupon it can be shaped and cured. During curing, the silane coated particles bond chemically with the polymer matrix. The filler fraction in dental composites rarely exceeds 60-65 vol-% because of problems with having higher volumes of randomized packed filler particles. Depending on filler size and filler size distribution, it is possible to make different types of dental composites. Since the total filler surface area per gram filler increases as the filler size decreases, finer particles tie up more resin, causing the viscosity of the material to increase fastest with filler fraction of smallest particles. Because of that phenomenon, composites with the finest filler particles tend to contain the lowest filler volume. The modulus of elasticity of a dental composite can roughly be estimated by determine the theoretical modulus of both the series as well as parallel models, and assume that the modulus of the composite for a certain filler fraction falls somewhere between these boundaries.

When the first modern dental composites were introduced during the 60s, it soon became clear that their wear resistance when used on load bearing surfaces was not high enough to be able to resist wear on occlusal surfaces. As a consequence, research performed during the 60s to the 80s focused on finding a solution to the wear problem as well as developing an understanding of the wear mechanism of these materials. During that era, it became clear that some of the key factors associated with composite wear were the quality of the filler matrix bond as well as the filler particle size and distribution. At a symposium supported by 3M in 1984 [46], research findings revealed that the best posterior composites at that time had reached a wear resistance of the commonly used amalgams.

During the research involving wear of composites, researchers had identified that cracks sometimes developed in regions in contact with an opposing cusp. The wear in those regions were often described as two-body wear, while the more general and less dramatic wear occurring on other surfaces were described as a three-body wear caused by abrasive particles sliding over the composite surface during chewing. When it came to the so-called two-body wear, it seemed reasonable to assume that during cusp sliding, micro-cracks could be induced. Another possible wear mechanism induced in the contact region could also be fatigue wear, triggered by a Hertzian failure [47]. In both these cases, microscopic flaws would develop, and these flaws would then contribute to an accelerated wear in these regions. In 1988, Roulet [48] claimed that fractures within the body of restorations and at the margins were a major problem regarding the failure of posterior composites.

However, during the 70s and 80s, the focus on dental composites were related to what clinicians perceived as being the major reasons for failures, which included wear, recurrent caries and discolorations. The notion that flaws were involved in the wear process led Truong and Tyas [49] to determine stable crack growth in dental composites. They did so by use of a double-torsion technique to establish the relationship between the stress intensity factor (SIF) KI and the crack velocity (v) for commercial and experimental composites. They tested dry, water-saturated and ethanol/water (3:1 v/v) saturated specimens. At a given crack velocity, the difference between the KI of a dry specimen and that of a water-saturated specimen was attributed solely to the change of Young's modulus caused by the plasticizing effect of water. However, microcracking occurring during immersion in an ethanol/water mixture resulted in an excessive drop of KI values from the dry state to ethanol/water mixture saturated state for Estilux Posterior and Occlusin samples, while little effect of fluids on KI could be observed on P10 and P30. The investigators tried to theoretically predict the wear of the composites, based on the assumption that microcracking occurs in the subsurface layer due to cyclic and impact stresses. Based on that assumption, three criteria for good wear resistance would be: (a) high fracture toughness (high critical SIF, KlC) and larger threshold crack length (at); (b) small inherent flaw size (ao) and (c) high crazing stress (σc). Based in these assumptions and the results of this study, the wear resistance of tested commercial composites should be: Occlusin > P10 > Estilux Posterior > P30 = Ful-Fil > Profile > Silux --~ Isomolar > Concept.

126 Applied Fracture Mechanics

dental crowns.

boundaries.

resistance. Critical loads depend strongly (quadratically) on crown net thickness. The analytic relations provided a seemingly sound basis for the materials design of next-generation

Dental composite resins consist of ceramic filler particles, usually within a size range of 1-5 μm and mixed with nano-sized (20-40 nm) particles. These inorganic filler particles are silane coated and mixed with a curable monomer to form a viscous paste that can be inserted into a prepared cavity, whereupon it can be shaped and cured. During curing, the silane coated particles bond chemically with the polymer matrix. The filler fraction in dental composites rarely exceeds 60-65 vol-% because of problems with having higher volumes of randomized packed filler particles. Depending on filler size and filler size distribution, it is possible to make different types of dental composites. Since the total filler surface area per gram filler increases as the filler size decreases, finer particles tie up more resin, causing the viscosity of the material to increase fastest with filler fraction of smallest particles. Because of that phenomenon, composites with the finest filler particles tend to contain the lowest filler volume. The modulus of elasticity of a dental composite can roughly be estimated by determine the theoretical modulus of both the series as well as parallel models, and assume that the modulus of the composite for a certain filler fraction falls somewhere between these

When the first modern dental composites were introduced during the 60s, it soon became clear that their wear resistance when used on load bearing surfaces was not high enough to be able to resist wear on occlusal surfaces. As a consequence, research performed during the 60s to the 80s focused on finding a solution to the wear problem as well as developing an understanding of the wear mechanism of these materials. During that era, it became clear that some of the key factors associated with composite wear were the quality of the filler matrix bond as well as the filler particle size and distribution. At a symposium supported by 3M in 1984 [46], research findings revealed that the best posterior composites at that time

During the research involving wear of composites, researchers had identified that cracks sometimes developed in regions in contact with an opposing cusp. The wear in those regions were often described as two-body wear, while the more general and less dramatic wear occurring on other surfaces were described as a three-body wear caused by abrasive particles sliding over the composite surface during chewing. When it came to the so-called two-body wear, it seemed reasonable to assume that during cusp sliding, micro-cracks could be induced. Another possible wear mechanism induced in the contact region could also be fatigue wear, triggered by a Hertzian failure [47]. In both these cases, microscopic flaws would develop, and these flaws would then contribute to an accelerated wear in these regions. In 1988, Roulet [48] claimed that fractures within the body of restorations and at the

**3.2. Fracture mechanics aspects of dental composites** 

had reached a wear resistance of the commonly used amalgams.

margins were a major problem regarding the failure of posterior composites.

In a study from 1991, Higo et al. [50] used a fracture mechanics approach to investigate the fracture toughness behavior of three commercial composite resins for dental use named Clearfil photo posterior, P-50 and Occlusin. The outcome of that study was that Occlusin exhibited higher fracture toughness values than any other resin when employing a ring specimen test procedure. However, when an indentation method was used, comparable fracture toughness values for all three resins were produced.

As a fracture mechanics approaches became more popular in attempts to estimating lifetimes of dental restorative materials, it became important to have available data on the fatigue behavior of these materials. At the end of the 90s, efforts at estimation included several untested assumptions related to the equivalence of flaw distributions sampled by shear, tensile, and compressive stresses. However, environmental influences on material properties were so far not accounted for to any greater extent, and it was unclear if fatigue limits existed. In a study by Baran et al. [51], they characterized the shear and flexural strengths of three resins used as matrices in dental restorative composite materials by use of Weibull parameters. They found that shear strengths were lower than flexural strengths, liquid sorption had a profound effect on characteristic strengths, and the Weibull shape parameter obtained from shear data differed for some materials from that obtained in flexure. In shear and flexural fatigue, a power law relationship applied for up to 250 000 cycles; no fatigue limits were found, and the data thus implied only one flaw population is responsible for failure. Again, liquid sorption adversely affected strength levels in most materials (decreasing shear strengths and flexural strengths by factors of 2–3) and to a greater extent than did the degree of cure or material chemistry.

In a study by Manhart et al. [52], they determined some mechanical properties of three packable composites (Solitaire, Surefil, ALERT), a packable ormocer (Definite), an advanced hybrid composite (Tetric Ceram) and an ionreleasing composite (Ariston pHc) in vitro (Table 3). As seen from that table, the properties of these composites differed significantly, which could be related to differences in filler particle size and shape distributions among the different materials. Their study suggested that fracture and wear behavior of the composite resins would be highly influenced by the filler system. They found that ALERT had the highest fracture toughness value, but also the highest wear rate, which they related to the fiber like particles used in that material. Overall, Surefil demonstrated good fracture mechanics parameters and low wear rate, which they suggested could be related to their more particle shaped filler particles. This study suggested that fracture and wear behavior of the composite resins are highly influenced by the filler system.



Considering the importance of being able to perform life-time predictions of dental composites, McCool et al. [53] , continued their research from 1998 [51], by comparing the lifetime predictions resulting from two methods of fatigue testing: dynamic and cyclic fatigue. To do so they made model composites, in which one variable was the presence of a silanizing agent. They tested their specimens in 4-point flexure, using a cyclic fatigue frequency of 5 Hz, while their dynamic fatigue testing spanned seven decades of stress rate application. Data were reduced and the crack propagation parameters for each material were calculated from both sets of fatigue data. These parameters were then used to calculate an equivalent static tensile stress for a 5-year survival time. The 5-year survival stresses predicted by dynamic fatigue data were approximately twice those predicted by cyclic fatigue data. In the absence of filler particle silanization, the survival stress was reduced by half. Aging in a water-ethanol solution reduced the survival stresses by a factor of four to five. One of the conclusions drawn from this study is that cyclic fatigue is a more conservative means of predicting lifetimes of resin-based composites.

The notion that there is a correlation between wear resistance and fracture toughness was to some degree rejected by Ruddel et al.[54]. In their study they produced pre-polymerized fused-fiber filler modified composite particles and determined their effectiveness by incorporating these fibers into composites. The results revealed that these particles decreased both flexural strength and fracture toughness, but improved wear performance. The SEM evaluations did not suggest that porosities had been incorporated during particle incorporation. Instead, fractures were transgranular through the reinforcing particles. Microscopic flaws observed in the new particles most likely explain the lower strength and toughness values. This study is important, because it shows that a composite with improved wear resistance could also suffer from an increase in fracture risk.

128 Applied Fracture Mechanics

Composite material

In a study by Manhart et al. [52], they determined some mechanical properties of three packable composites (Solitaire, Surefil, ALERT), a packable ormocer (Definite), an advanced hybrid composite (Tetric Ceram) and an ionreleasing composite (Ariston pHc) in vitro (Table 3). As seen from that table, the properties of these composites differed significantly, which could be related to differences in filler particle size and shape distributions among the different materials. Their study suggested that fracture and wear behavior of the composite resins would be highly influenced by the filler system. They found that ALERT had the highest fracture toughness value, but also the highest wear rate, which they related to the fiber like particles used in that material. Overall, Surefil demonstrated good fracture mechanics parameters and low wear rate, which they suggested could be related to their more particle shaped filler particles. This study suggested that fracture and wear behavior of

Flexural modulus

Considering the importance of being able to perform life-time predictions of dental composites, McCool et al. [53] , continued their research from 1998 [51], by comparing the lifetime predictions resulting from two methods of fatigue testing: dynamic and cyclic fatigue. To do so they made model composites, in which one variable was the presence of a silanizing agent. They tested their specimens in 4-point flexure, using a cyclic fatigue frequency of 5 Hz, while their dynamic fatigue testing spanned seven decades of stress rate application. Data were reduced and the crack propagation parameters for each material were calculated from both sets of fatigue data. These parameters were then used to calculate an equivalent static tensile stress for a 5-year survival time. The 5-year survival stresses predicted by dynamic fatigue data were approximately twice those predicted by cyclic fatigue data. In the absence of filler particle silanization, the survival stress was reduced by half. Aging in a water-ethanol solution reduced the survival stresses by a factor of four to five. One of the conclusions drawn from this study is that cyclic fatigue is a more conservative means of

The notion that there is a correlation between wear resistance and fracture toughness was to some degree rejected by Ruddel et al.[54]. In their study they produced pre-polymerized fused-fiber filler modified composite particles and determined their effectiveness by incor-

Fracture toughness KIC (MN m-1/2)

Mean wear rate (μm3 cycle-1)

(GPa)

Solitare 81.6 (10.0) 4.4 (0.3) 1.4 (0.2) 1591 Definite 103.0 (19.9) 6.3 (0.9) 1.6 (0.3) 2763 Surefil 132.0 (14.3) 9.3 (0.9) 2.0 (0.2) 3028 ALERT 124.7 (22.1) 12.5 (2.1) 2.0 (0.2) 8275 Tetric Ceram 107.6 (11.4) 6.8 (0.5) 2.0 (0.1) 5417 Ariston pHc 118.1 (10.5) 7.3 (0.8) 1.9 (0.2) 7194

the composite resins are highly influenced by the filler system.

Flexural strength

**Table 3.** Some properties of six dental composite materials [52].

predicting lifetimes of resin-based composites.

(MPa)

During the past 10 years, it has become clear that fracture is a major reason for clinical failure of dental composites. Many clinical fractures are likely to be preceded by slow subcritical crack propagation. To study the slow sub-critical crack propagation, Loughran [55] used notched composite (Z100, 3M ESPE) specimens and fatigued them in a four-point bending test using a load cycle at 5 Hz between 25 and 230 N until failure. Displacement and load were recorded during the fatigue tests and used to derive crack propagation based on beam-compliance. What they found was that the number of cycles until failure ranged between 34 and 82,481. In the last 1500 cycles prior to final fracture, the beam compliance increased consistently, indicating sub-critical crack propagation. From the compliance change they calculated that the crack length increased 8% (77 ± 14 μm) before final failure. The crack growth rate during sub-critical crack propagation was determined as a function of the stress intensity for the last 1500 cycles before fracture. The importance of this study was that they found that the fatigue lifetime varied widely, and that stable crack growth existed prior to fracture consistently. This consistency allowed formulation of stress-based crack propagation relationships that can be used in concert with numerical simulations to predict composite restoration performance. The large variation found for specimen lifetime was attributed to the initiation process that precedes sub-critical crack propagation.

As mentioned earlier, during the early 80s, dentists regarded poor wear resistance tendency to be associated with recurrent caries and restoration discolorations as the key shortcomings with dental composites. Today, that perception has changed quite considerable. By improved filler technology and silanization methods, the poor wear resistance is no longer a major clinical problem. Improved adhesives, now making it possible to bond composites to both enamel and dentin, have decreased the risk for recurrent caries. The use of more stable chemicals and smoother composite surfaces caused by the use of finer filler particles has decreased the magnitude of restoration discolorations. In other words, what were regarded as major shortcomings with posterior composites are no longer regarded as major weaknesses. Of course, these shortcomings have not yet been completely eliminated, so there is still room for improvements. However, as the composites have been improved, another shortcoming has been identified as now being the biggest problem, namely fractures[48]. In a recently published clinical study [56], in which two composites were evaluated over a 22 year period, the authors claimed that the most common reason for failures of posterior composites were fractures. That study suggests that further understanding of the fracture mechanical behavior of dental composites is needed.

## **3.3. Fracture mechanics aspects of cements and adhesives**

In order to attach restorations such as composite fillings, inlays/onlays, crowns and bridges, different cements/adhesives have been used in dentistry through the years. The oldest but still used cement is the zincphosphate cement, which was introduced about 150 year ago and consists mainly of a zincoxide powder mixed with phosphoric acid. During setting, that cement goes through an acid-base reaction during which a salt and water is formed. The way this cement works is simply by etching the surfaces of the tooth and the surface of the restoration the cement is in contact with, a process that occurs as the cement sets, whereupon zincphosphate crystallites precipitate into the etched surface regularities as the cement sets. With that mechanism a mechanical interlocking is established, explaining the retention of the cemented restoration.

In addition to the zinc phosphate cement, other cements such as silicate, zincsilico phosphate, polycarboxylate and glass ionomer cements have been used. In the case of the silicate and zincsilico phosphate cements, phosphoric acid is used in both cases, while the powders of these two cements are either a silicate glass powder or a mixture of that powder with a zinc phosphate powder. When it comes to the polycarboxylate and the glass ionomer cements, the powders are either the zinc oxide powder or the glass powder used in the silicate cement, while the acid has been replaced with a polyalceonic acid. The polyalkeoinic acid, often polyacrylic acid, is capable of reacting with the powder through an acid-base reaction, but also with the dentin or enamel surface. During that reaction the –COO of the polyalkeonic acid can interact with ions such as the Ca2+ present in the tooth surface and form some ionic interaction. Compared to the zinc phosphate and silicate cements, the polycarboxylate and glass ionomer cements were introduced to dentistry during the 60s and the 70s. Regarding the ability to bond to hard tooth tissues, it is generally assumed that zinc phosphate, zincsilico phosphate, and silicate cements only bond via micromechanical retention, while polycarboxylate and glass ionomer cements bond both via a micromechanical retention as well ionic surface interaction.

The idea to develop some kind of chemical bond to dental hard tissues was however introduced before the zincpolycarboxylate and glass ionomer cements had been invented. The first idea to use some kind of chemical interaction to form a bond to the hard dental tissues was introduced during the late 40s when Hagger [57] suggested that a molecule that had a phosphate group capable of interaction with Ca2+ at the tooth surface and a methylmethacrylate group capable of forming a covalent bond to a curing methacrylate based filling materials could form such a bond. Unfortunately, the molecule Hagger used to achieve such a bond did not show to be very efficient. However, when Buonocore in 1955 [58] explored the possibility to first etch the enamel surface with a phosphoric acid, then rinse and dry and coat the acid roughened surface with a curable resin, it became possible to achieve a predictable bond to enamel.

Buonocore's idea was not widely accepted initially, because dentists feared that the phosphoric acid, particularly if it came in contact with exposed dentin surface, would cause pulp irritation and eventually pulp death. Such pulp reactions were known to occur, particularly when the more slow setting silicate cement was used. As a consequence it would take several years until Buonocore's acid-etch approach took off. A major contributor for teaching dentists how to use enamel etching and composite resins was 3M, who during the 60s had expanded their products to dentistry.

130 Applied Fracture Mechanics

retention of the cemented restoration.

dictable bond to enamel.

**3.3. Fracture mechanics aspects of cements and adhesives** 

micromechanical retention as well ionic surface interaction.

In order to attach restorations such as composite fillings, inlays/onlays, crowns and bridges, different cements/adhesives have been used in dentistry through the years. The oldest but still used cement is the zincphosphate cement, which was introduced about 150 year ago and consists mainly of a zincoxide powder mixed with phosphoric acid. During setting, that cement goes through an acid-base reaction during which a salt and water is formed. The way this cement works is simply by etching the surfaces of the tooth and the surface of the restoration the cement is in contact with, a process that occurs as the cement sets, whereupon zincphosphate crystallites precipitate into the etched surface regularities as the cement sets. With that mechanism a mechanical interlocking is established, explaining the

In addition to the zinc phosphate cement, other cements such as silicate, zincsilico phosphate, polycarboxylate and glass ionomer cements have been used. In the case of the silicate and zincsilico phosphate cements, phosphoric acid is used in both cases, while the powders of these two cements are either a silicate glass powder or a mixture of that powder with a zinc phosphate powder. When it comes to the polycarboxylate and the glass ionomer cements, the powders are either the zinc oxide powder or the glass powder used in the silicate cement, while the acid has been replaced with a polyalceonic acid. The polyalkeoinic acid, often polyacrylic acid, is capable of reacting with the powder through an acid-base reaction, but also with the dentin or enamel surface. During that reaction the –COO-

polyalkeonic acid can interact with ions such as the Ca2+ present in the tooth surface and form some ionic interaction. Compared to the zinc phosphate and silicate cements, the polycarboxylate and glass ionomer cements were introduced to dentistry during the 60s and the 70s. Regarding the ability to bond to hard tooth tissues, it is generally assumed that zinc phosphate, zincsilico phosphate, and silicate cements only bond via micromechanical retention, while polycarboxylate and glass ionomer cements bond both via a

The idea to develop some kind of chemical bond to dental hard tissues was however introduced before the zincpolycarboxylate and glass ionomer cements had been invented. The first idea to use some kind of chemical interaction to form a bond to the hard dental tissues was introduced during the late 40s when Hagger [57] suggested that a molecule that had a phosphate group capable of interaction with Ca2+ at the tooth surface and a methylmethacrylate group capable of forming a covalent bond to a curing methacrylate based filling materials could form such a bond. Unfortunately, the molecule Hagger used to achieve such a bond did not show to be very efficient. However, when Buonocore in 1955 [58] explored the possibility to first etch the enamel surface with a phosphoric acid, then rinse and dry and coat the acid roughened surface with a curable resin, it became possible to achieve a pre-

Buonocore's idea was not widely accepted initially, because dentists feared that the phosphoric acid, particularly if it came in contact with exposed dentin surface, would cause pulp irritation and eventually pulp death. Such pulp reactions were known to occur,

of the

To spread the usage and the acceptance of enamel etching and resin bonding as well as their composite resin, 3M sponsored a symposium entitled "The Acid Etch Technique" in 1975. The presentations presented at that symposium were published in a book [59] that was then widely distributed by representatives for the company. By having prominent researchers presenting papers related to the acid etch technique, a lot of misperceptions could be eliminated and the enamel etch technique became generally accepted [60]. When it came to testing enamel bonding, most in vitro studies relied on morphology achieved by use of SEM and different strength tests of which shear bond testing soon became the most popular.

Even though enamel bonding was a major advance in dentistry, the ability to bond to dentin was not resolved when enamel bonding took off. Because most surfaces exposed during tooth preparations of cavities and crowns consist of dentin, a reliable dentin bonding was still needed in order to truly bond different restorative materials to dentin. However, dentin bonding was much more complicated to achieve than enamel bonding. In contrast to enamel, dentin is a living tissue and therefore much more demanding than enamel when it came to biocompatibility of chosen materials. Besides, dentin contains much more water, making it difficult to adapt more or less hydrophobic materials to the dentin surface.

Parallel to these events, Bowen had already during the 60s initiated research to develop resin systems capable of bonding to cut dentin surfaces [61]. The basic principle behind his ideas was that the adhesives should contain a reactive group capable of reacting with Caions present on the tooth surface, and then react with the resin when the resin cured. When these adhesives, often referred to as the first generation of dentin adhesives were explored, it soon became clear that a cut dentin surface was coated with a so called smear layer. That smear layer consisted of a few microns thick layer of smeared collagen in which fractured hydroxyapatite crystallites were embedded. It was soon clear that the first generation of adhesives developed a weak bond to the tooth surface, and that the bond was weak and worked for a short time period only, mainly because the bonds formed to the smear layer, or the bonds between the smear layer and the dentin were too weak to resist loading.

During the 70s, the dental community discussed the effect the smear layer had on bonding and whether or not it should remain on the dentin surface. Some researchers viewed it as beneficial, since the vital dentin channels were sealed off, decreasing the risk of pulp irritations caused by the restorative material. As a consequence, somewhat more acidic adhesive systems were developed, capable of removing some of the smear layer but retaining some smear serving as protective layer. The adhesives that fell into this class are often referred to as the 2nd generation adhesives.

At the end of the 70th, a major break-through occurred. That break-through consisted of a clinical study performed by Fusayama at al. [62], in which they claimed that by etching both

enamel and dentin, they were able to bond composites to dentin without having any problem with pulps responding to the etching procedure. There is no doubt that Fusayama et al.'s finding was looked upon with enormous skepticism. Their explanation that resin infiltrated the tubules and thereby formed resin tags that contributed to the retention was also questioned. It was first when Nakabaiashi [63] came out with his hybridization explanation, suggesting that the resin infiltrated the etched dentin surface and formed a hybrid layer consisting of partly dissolved dentin, as dentin etching started to become accepted .

These two studies[62, 63] opened the door for more aggressive dentin etching resulting in the 3rd generation adhesives. Etching dentin with phosphoric acid was still not the general trend. Instead, weaker conditioners such as EDTA and citric acids were used[64]. However, at the end of the 80th, some bonding systems had occurred on the market that used the same etchant for both enamel and dentin. The success of these adhesives, the so called 4th generation adhesives, took of during the early 90s, when both Kanca [65] as well as Gwinnett [66] in two independent studies claimed that by leaving the dentin moist before priming, they could better infiltrate the collagen layer with the primer and thereby achieve better bonding to dentin.

Simultaneous with these trends related to bonded composite, it had also been noticed that by etching the surface of ceramic restorations located at the dentin surface with hydrofluoric acid and then silane coat the etched surface, it was possible to bond ceramic restorations to tooth surfaces. Such an approach resulted in a significantly lower risk of ceramic fracture than compared to the use of more traditional cements, including polycarboxylate and glass ionomer cements. By use of the information presented under the ceramic section in this chapter, it is quite easy to explain why resin bonded ceramics performed so well by considering fracture mechanics. In the case of the more traditional cements, they can be described as having brittle properties with limited ability to form strong bonds to the ceramic surface. In the case of the phosphoric acid based cements they did not form any strong bonds to the tooth surface neither. By realizing that even a ceramic restoration can flex during chewing, one can visualize the development of shear stresses at the ceramic-cement interface, and that these stresses can trigger a crack growth along the ceramic-cement interface. In the case of the resin bonded ceramics, the shrinkage of the resin cement initially induced some compressive stress in the ceramic surface adjacent to the resin cement. If a crack propagates to the resin interface in such a case, the more ductile nature of the resin cement will not as easily allow the crack to propagate along the ceramic-cement interface. Besides, after the load has been removed, the resin will because of its polymerization shrinkage, try to force the fractured ceramic in contact with its fractured surfaces. Thus, in this case, a ceramic fracture may occur, but in contrast to a fracture in a ceramic cemented with more traditional cement, one may not end up with a detectable catastrophic failure.

From a fracture mechanics point of view, there is no doubt that the adhesive joint is the most challenging region. The reason relates simply to practical problems such as minimizing the incorporation of defects in this region. In addition, the fact that the adhesive shrinks and induces shrinkage stresses between the tooth and the adhesive, as well as between the adhesive and the restorative material, does not make the situation manageable, which is further complicated by differences in mechanical/physical properties of the different materials forming a joint. In the following section we will approach the adhesive joint in an attempt to identify different challenges associated with this region.

132 Applied Fracture Mechanics

accepted .

better bonding to dentin.

enamel and dentin, they were able to bond composites to dentin without having any problem with pulps responding to the etching procedure. There is no doubt that Fusayama et al.'s finding was looked upon with enormous skepticism. Their explanation that resin infiltrated the tubules and thereby formed resin tags that contributed to the retention was also questioned. It was first when Nakabaiashi [63] came out with his hybridization explanation, suggesting that the resin infiltrated the etched dentin surface and formed a hybrid layer consisting of partly dissolved dentin, as dentin etching started to become

These two studies[62, 63] opened the door for more aggressive dentin etching resulting in the 3rd generation adhesives. Etching dentin with phosphoric acid was still not the general trend. Instead, weaker conditioners such as EDTA and citric acids were used[64]. However, at the end of the 80th, some bonding systems had occurred on the market that used the same etchant for both enamel and dentin. The success of these adhesives, the so called 4th generation adhesives, took of during the early 90s, when both Kanca [65] as well as Gwinnett [66] in two independent studies claimed that by leaving the dentin moist before priming, they could better infiltrate the collagen layer with the primer and thereby achieve

Simultaneous with these trends related to bonded composite, it had also been noticed that by etching the surface of ceramic restorations located at the dentin surface with hydrofluoric acid and then silane coat the etched surface, it was possible to bond ceramic restorations to tooth surfaces. Such an approach resulted in a significantly lower risk of ceramic fracture than compared to the use of more traditional cements, including polycarboxylate and glass ionomer cements. By use of the information presented under the ceramic section in this chapter, it is quite easy to explain why resin bonded ceramics performed so well by considering fracture mechanics. In the case of the more traditional cements, they can be described as having brittle properties with limited ability to form strong bonds to the ceramic surface. In the case of the phosphoric acid based cements they did not form any strong bonds to the tooth surface neither. By realizing that even a ceramic restoration can flex during chewing, one can visualize the development of shear stresses at the ceramic-cement interface, and that these stresses can trigger a crack growth along the ceramic-cement interface. In the case of the resin bonded ceramics, the shrinkage of the resin cement initially induced some compressive stress in the ceramic surface adjacent to the resin cement. If a crack propagates to the resin interface in such a case, the more ductile nature of the resin cement will not as easily allow the crack to propagate along the ceramic-cement interface. Besides, after the load has been removed, the resin will because of its polymerization shrinkage, try to force the fractured ceramic in contact with its fractured surfaces. Thus, in this case, a ceramic fracture may occur, but in contrast to a fracture in a ceramic cemented with more traditional

From a fracture mechanics point of view, there is no doubt that the adhesive joint is the most challenging region. The reason relates simply to practical problems such as minimizing the incorporation of defects in this region. In addition, the fact that the adhesive shrinks and

cement, one may not end up with a detectable catastrophic failure.

When it comes to the failure mechanism at the dentin resin interface, there are certain questions that need to be addressed. These questions include: (1) does failure at the human dentin-resin interface occur by a cohesive or an adhesive mechanism? (2) is the failure mechanism accompanied by a plastic deformation, and if so how important is it? To address these questions, Lin and Douglas [67] performed a computational analysis and fractography of two different bonding systems: Scotchbond- (SB2) and Scotchbond-Multipurpose (SBM). The difference between these two systems is that SB2 consists of a mixture of primer and a so called bonding resin, while SBM uses the same primer and bonding resin, but in contrast to SB2 they are placed as separate systems on the dentin surface. According to their estimates, the dentin-resin interracial fracture toughness (GIC), for the SB2 and for the SBM were 30.22 ± 5.61 and 49.56 ± 7.65 J m-2, respectively, which were significantly different (p < 0.01). Both SB2 and SBM interfaces with dentin displayed significant degrees of plasticity (0.15 and 0.19) which were beneficial to crack resistance. Thus, correcting for the plasticity, the GIC for SB2 and for SBM increased to 42.83 ± 7.75 and 74.97 ± 10.47 J m-2, respectively. The fractography of the two systems reflected these numeric differences. SB2 showed largely interfacial adhesive failure, while SBM showed adhesive-cohesive failure with occasional dentin adhesions attached to the composite interface and vice versa.

In another study, Toparli [68] determined the reliability and validity of the adhesive bond toughness of dentin/composite resin interfaces from the standpoint of fracture mechanics. The fracture toughness (KIC) and fracture energy (JIC) values of two different composite resins (Brilliant Dentin and P50) were determined. The fracture toughness and energy values obtained experimentally for Brilliant Dentin were found to be higher than those for P50. It was seen that calculated J values (Jadh and Jres) changed with the crack length; but the effective fracture energy (Jeff) was independent of the crack length, as expected. The applied fracture energy (Jappl) and effective fracture energy (Jeff) are considerably smaller than the experimentally determined JIC values of composite resins. The important finding was that the bonded interface tends to produce microscopic flaws which could act as critical stress risers promoting interfacial failures. The initiation and propagation of such flaws under the mastication forces can be followed by fracture toughness (KIC) or fracture energy (JIC) in linear elastic fracture mechanics (LEFM).

The effect of crack growth at a resin bonded metal interface after storage in water was studied by Moulin [69], who found that the adherence energy dramatically decreased with time in water. The slope of the regression straight line appeared to be a good criterion for evaluating the durability of the alloy/adhesive interface. The study revealed the importance of silica coating the metal surface and, especially, the effectiveness of the Rocatec system upon the degree of hydrolytic degradation. The study showed how the development of cracks depends upon surface treatment.

Adhesion at the titanium–porcelain interface using a fracture mechanics approach has also been used to investigate the bonding mechanism of such systems [70]. In that study they used specimens of five different titanium–porcelain and one base metal–porcelain bonding systems on which they performed a four-point bending interfacial delaminating test. The pre-cracked specimen was subjected to load and the strain energy release rate (G) was calculated from the critical load to induce stable crack extension in each system. The strain energy release rate of titanium–porcelain with a Gold Bonder interface layer was highest among the five different systems. No attempt was made to explain the experimental findings.

In two studies by Ichim et al. [71, 72] they looked at a typical non-carious cervical lesion, a so called abfraction, treated with a glass ionome or a combination of glassionomer and composite. The approach they used was that they used a nonlinear fracture mechanical approach simulated by use of FEA. They used a novel Rankine and rotating crack model to trace the fracture failure process of the cervical restorations. The approach involves an automatic insertion of an initial crack, mesh updating for crack propagation and self contact at the cracked interface. The results were in good agreement with published clinical data, in terms of the location of the fracture failure of the simulated restoration and the inadequacy of the dental restoratives for abfraction lesions.

In their second study [72] they investigated the influence of the elastic modulus (E) on the failure of cervical restorative materials and tried to identify an E value that would minimize mechanical failure under clinically realistic loading conditions. What they found was that the restorative materials currently used in non-carious cervical lesions are largely unsuitable in terms of resistance to fracture of the restoration. They suggested that the elastic modulus of such a material should be in the range of 1 GPa rather than several GPa that is usually the case.

Despite an obvious advantage to approach adhesives and their performance from a fracture mechanics point of view, traditional bond studies usually focus on bond strength values. By comparing such strength values, it is noticed that large variations exist among different reports. These variations are due to differences among operators, but also on the day a certain tester performed a test. The standard deviation is 25-50 % of the mean value, which suggests that defects present in the adhesive region may be of a bigger concern than the true adhesive strength.

In an attempt to resolve the questions related to the large variability in strength values and their clinical meaning, The Academy of Dental Materials at their annual meeting in 2010, focused that meeting on the value of bond strength measurements. In one presentation, Scherrer et al. [73] presented a literature search based on all dentin bond strength data obtained for six adhesives evaluated with four tests (shear, microshear, tensile and microtensile) and critically analyzed the results with respect to average bond strength, coefficient of variation, mode of failure and product ranking. The PubMed search was carried out for the years between 1998 and 2009. The six adhesive resins that were selected included three step systems (OptiBond FL, Scotch Bond Multi-Purpose Plus), two-step (Prime & Bond NT, Single Bond, Clearfil SE Bond) and one step (Adper Prompt L Pop). By pooling the results from the 147 references, it was revealed an ongoing high scatter in the bond strength data regardless which adhesive and which bond test was used. Coefficients of variation remained high (20–50%) even with the microbond test. The reported modes of failure for all tests still included a high number of cohesive failures. The ranking of the adhesives seemed to be dependent on the test method being used. The scatter in dentin bond strength data, independent of used test, confirmed Finite Element Analysis predicting non-uniform stress distributions due to a number of geometrical, loading, material properties and specimens preparation variables. The study reopened the question whether an interfacial fracture mechanics approach to analyze the dentin–adhesive bond would not be more appropriate for obtaining better agreement among dentin bond related papers.

134 Applied Fracture Mechanics

adhesive strength.

Adhesion at the titanium–porcelain interface using a fracture mechanics approach has also been used to investigate the bonding mechanism of such systems [70]. In that study they used specimens of five different titanium–porcelain and one base metal–porcelain bonding systems on which they performed a four-point bending interfacial delaminating test. The pre-cracked specimen was subjected to load and the strain energy release rate (G) was calculated from the critical load to induce stable crack extension in each system. The strain energy release rate of titanium–porcelain with a Gold Bonder interface layer was highest among the

In two studies by Ichim et al. [71, 72] they looked at a typical non-carious cervical lesion, a so called abfraction, treated with a glass ionome or a combination of glassionomer and composite. The approach they used was that they used a nonlinear fracture mechanical approach simulated by use of FEA. They used a novel Rankine and rotating crack model to trace the fracture failure process of the cervical restorations. The approach involves an automatic insertion of an initial crack, mesh updating for crack propagation and self contact at the cracked interface. The results were in good agreement with published clinical data, in terms of the location of the fracture failure of the simulated restoration and the inadequacy

In their second study [72] they investigated the influence of the elastic modulus (E) on the failure of cervical restorative materials and tried to identify an E value that would minimize mechanical failure under clinically realistic loading conditions. What they found was that the restorative materials currently used in non-carious cervical lesions are largely unsuitable in terms of resistance to fracture of the restoration. They suggested that the elastic modulus of such a material should be in the range of 1 GPa rather than several GPa that is usually the case.

Despite an obvious advantage to approach adhesives and their performance from a fracture mechanics point of view, traditional bond studies usually focus on bond strength values. By comparing such strength values, it is noticed that large variations exist among different reports. These variations are due to differences among operators, but also on the day a certain tester performed a test. The standard deviation is 25-50 % of the mean value, which suggests that defects present in the adhesive region may be of a bigger concern than the true

In an attempt to resolve the questions related to the large variability in strength values and their clinical meaning, The Academy of Dental Materials at their annual meeting in 2010, focused that meeting on the value of bond strength measurements. In one presentation, Scherrer et al. [73] presented a literature search based on all dentin bond strength data obtained for six adhesives evaluated with four tests (shear, microshear, tensile and microtensile) and critically analyzed the results with respect to average bond strength, coefficient of variation, mode of failure and product ranking. The PubMed search was carried out for the years between 1998 and 2009. The six adhesive resins that were selected included three step systems (OptiBond FL, Scotch Bond Multi-Purpose Plus), two-step (Prime & Bond NT, Single Bond, Clearfil SE Bond) and one step (Adper Prompt L Pop). By pooling the results from

five different systems. No attempt was made to explain the experimental findings.

of the dental restoratives for abfraction lesions.

In another paper presented at that meeting, Soderholm [74] emphasized the benefits of using fracture mechanics approaches when it comes to studying dental adhesives. In his review, different general aspects of fracture mechanics and adhesive joints were reviewed, serving as a foundation for a review of fracture toughness studies performed on dental adhesives. The dental adhesive studies were identified through a MEDLINE search using "dental adhesion testing AND enamel OR dentin AND fracture toughness" as search strategy. The outcome of the review revealed that fracture toughness studies performed on dental adhesives are complex, both regarding technical performance as well as achieving good discriminating ability between different adhesives. The review also suggested that most fracture toughness tests of adhesives performed in dentistry are not totally reliable because they usually did not consider the complex stress pattern at the adhesive interface. However, despite these limitations, the review strongly supports the notion that the proper way of studying dental adhesion is by use a fracture mechanics aproach.

In a study by Howard and Soderholm [75] they used a fracture mechanics approach previously described by Pilliar and Tam [76-80] to test the hypothesis that a self-etching adhesive is more likely to fail at the dentin adhesive interface than an etch-and-rinse adhesive. What they found was that the fracture toughness values (KIC) of the two adhesives were not significantly different. The rather high frequency of mixed failures did not support the hypothesis that the dentin-adhesive interface is clearly less resistant to fracture than the adhesive–composite interface. The finding that cracks occurred in 6–8% in the composite suggests that defects within the composite or at the adhesive–composite interface are important variables to consider in adhesion testing.

In a recently published study by Ausello et al. [81], they used FEA and fatigue mechanic laws to estimate the fatigue damage of a restored molar. The simulated restoration consisted of an indirect class II MOD cavity preparation restored with a composite. Fatigue simulation was performed by combining a preliminary static FEA simulation with classical fatigue mechanical laws. It was found that regions with the shortest fatigue-life were located around the fillets of the class II MOD cavity, where the static stress was highest.

From the above papers, it becomes clear that adhesion tests utilized in dentistry are unable to separate the effects of adhesive composition, substrate properties, joint geometry and type of loading on the measured bond strength. This makes it difficult for the clinician to identify the most suitable adhesive for a given procedure and for the adhesive manufacturer to optimize its composition. To come to grip with these challenges, Jancar [82] proposed an adhesion test protocol based on the fracture mechanics to generate data for which separation of the effect of composition from that of the joint geometry on the shear (τa) and tensile (σa) bond strengths was possible for five commercial dental adhesives. The adhesive thicknesses (h) used varied from 15 to 500 μm, and the commercial adhesives had fracture toughness values (KIC) ranging from 0.3 to 1.6 MPa m1/2. They used double lap joint (DLJ) and modified compact tension (MCT) specimens which were conditioned for 24 h in 37°C distilled water, then dried in a vacuum oven at 37°C for 24 h prior to testing. Both τa and σ<sup>a</sup> increased with increasing adhesive thickness, exhibiting a maximum bond strength at the optimum thickness (hopt). For h < hopt, both τa and σa were proportional to h, and, above hopt, both τa and σa decreased with h−4/10 in agreement with the fracture mechanics predictions. Hence, two geometry-independent material parameters, Ψ and (Hc/Q), were found to characterize τa and σa over the entire thickness interval. The results seem important, because it suggests that the adhesion tests currently used in dentistry provide the geometry dependent bond strength, and such data cannot be used either for prediction of clinical reliability of commercial dental adhesives or for development of new ones. Instead, the proposed test protocol allowed one to determine two composition-only dependent parameters determining τa and σa. A simple proposed procedure can then be used to estimate the weakest point in clinically relevant joints always exhibiting varying adhesive thickness and, thus, to predict the locus of failure initiation. Moreover, this approach can also be used to analyze the clinical relevance of the fatigue tests of adhesive joints.

In a recent paper by Kotousove [83] a conceptual framework utilizing interfacial fracture mechanics and Toya's solution for a partially delaminated circular inclusion in an elastic matrix was used , which can be applied (with caution) to approximate polymer curing induced cracking about composite resins for Class I cavity restorations. The findings indicated that: (I) most traditional shear tests are not appropriate for the analysis of the interfacial failure initiation; (II) material properties of the restorative and tooth material have a strong influence on the energy release rate; (III) there is a strong size effect; and (IV) interfacial failure once initiated is characterized by unstable propagation along the interface almost completely encircling the composite. The importance of this study is that it analyses the reliability of composite Class I restorations and provides an adequate interpretation of recent adhesion debonding experimental results utilizing tubular geometry of specimens. The approach clearly identifies the critical parameters including; curing strain, material module, size and interfacial strain energy release rate for reliable development of advanced restorative materials.

In a similar approach, Yamamoto [84] calculated stresses produced by polymerization contraction in regions surrounding a dental resin composite restoration. Initial cracks were made with a Vickers indenter at various distances from the edge of a cylindrical hole in a soda-lime glass disk. Indentation crack lengths were measured parallel to tangents to the hole edge. Resin composites (three brands) were placed in the hole and polymerized (two light irradiation protocols) at equal radiation exposures. The crack lengths were remeasured at 2 and 10 min after irradiation. Radial tensile stresses due to polymerization contraction at the location of the cracks (σ-crack) were calculated from the incremental crack lengths and the fracture toughness KC of the glass. Contraction stresses at the composite–glass bonded interface (σ-interface) were calculated from σ-crack on the basis of the simple mechanics of an internally pressurized thick-walled cylinder. The greater the distance or the shorter the time following polymerization, the smaller was σ-crack. Distance, material, irradiation protocol and time significantly affected σ-crack. Two-step irradiation resulted in a significant reduction in the magnitude of σ-interface for all resin composites. The contraction stress in soda-lime glass propagated indentation cracks at various distances from the cavity, enabling calculation of the contraction stresses.

## **4. Conclusion**

136 Applied Fracture Mechanics

type of loading on the measured bond strength. This makes it difficult for the clinician to identify the most suitable adhesive for a given procedure and for the adhesive manufacturer to optimize its composition. To come to grip with these challenges, Jancar [82] proposed an adhesion test protocol based on the fracture mechanics to generate data for which separation of the effect of composition from that of the joint geometry on the shear (τa) and tensile (σa) bond strengths was possible for five commercial dental adhesives. The adhesive thicknesses (h) used varied from 15 to 500 μm, and the commercial adhesives had fracture toughness values (KIC) ranging from 0.3 to 1.6 MPa m1/2. They used double lap joint (DLJ) and modified compact tension (MCT) specimens which were conditioned for 24 h in 37°C distilled water, then dried in a vacuum oven at 37°C for 24 h prior to testing. Both τa and σ<sup>a</sup> increased with increasing adhesive thickness, exhibiting a maximum bond strength at the optimum thickness (hopt). For h < hopt, both τa and σa were proportional to h, and, above hopt, both τa and σa decreased with h−4/10 in agreement with the fracture mechanics predictions. Hence, two geometry-independent material parameters, Ψ and (Hc/Q), were found to characterize τa and σa over the entire thickness interval. The results seem important, because it suggests that the adhesion tests currently used in dentistry provide the geometry dependent bond strength, and such data cannot be used either for prediction of clinical reliability of commercial dental adhesives or for development of new ones. Instead, the proposed test protocol allowed one to determine two composition-only dependent parameters determining τa and σa. A simple proposed procedure can then be used to estimate the weakest point in clinically relevant joints always exhibiting varying adhesive thickness and, thus, to predict the locus of failure initiation. Moreover, this approach can

also be used to analyze the clinical relevance of the fatigue tests of adhesive joints.

rate for reliable development of advanced restorative materials.

In a recent paper by Kotousove [83] a conceptual framework utilizing interfacial fracture mechanics and Toya's solution for a partially delaminated circular inclusion in an elastic matrix was used , which can be applied (with caution) to approximate polymer curing induced cracking about composite resins for Class I cavity restorations. The findings indicated that: (I) most traditional shear tests are not appropriate for the analysis of the interfacial failure initiation; (II) material properties of the restorative and tooth material have a strong influence on the energy release rate; (III) there is a strong size effect; and (IV) interfacial failure once initiated is characterized by unstable propagation along the interface almost completely encircling the composite. The importance of this study is that it analyses the reliability of composite Class I restorations and provides an adequate interpretation of recent adhesion debonding experimental results utilizing tubular geometry of specimens. The approach clearly identifies the critical parameters including; curing strain, material module, size and interfacial strain energy release

In a similar approach, Yamamoto [84] calculated stresses produced by polymerization contraction in regions surrounding a dental resin composite restoration. Initial cracks were made with a Vickers indenter at various distances from the edge of a cylindrical hole in a soda-lime glass disk. Indentation crack lengths were measured parallel to tangents to the hole edge. Resin composites (three brands) were placed in the hole and polymerized (two light irradiation protocols) at equal radiation exposures. The crack lengths were remeasured By reviewing enamel, dentin and their interfacial bond, it is obvious that the tooth evolved in such a way it would be able to function in an optimal way without fracturing. With the sophisticated structure of both enamel and dentin, it becomes quite clear that existing man made restorative materials are far from optimal in comparison to the biological hard tissues. The crack growth risk in ceramics needs to be reduced, something that can be achieved by use of fracture tough ceramics such as alumina and zirconia. Unfortunately, as shown in Lawn et al.'s[45] paper, rather extensive removal of existing tooth structure needs to be performed in order to minimize future failures. Such an approach, though, does not make sense if one considers that a more sophisticated material is removed in order to replace it with an inferior material.

When it comes to dental composites, we have now reached a point when fractures of composites are being judged as the most common reason for composite failures [48, 56]. To come to grip with that problem, our understanding of the fracture mechanics of dental composites needs to be improved. The same is true regarding cements/adhesives. The particulate filled resins we are now using are rather primitive when compared to both enamel and dentin. However, it seems as this group of materials has the highest chance to evolve and approach the properties of enamel and dentin.

By looking at dentistry from a fracture mechanics point of view, it becomes quite clear that traditional dentistry suffer from some major processing problems. The first problem is that restorations are individual units that differ in shape and size. These different sizes and shapes result in different levels and locations of localized stresses. The second problem is that restorations are placed by individual dentists working under different conditions and introducing different amounts and types of flaws during the different dental procedures. Considering that the theoretical strength is several magnitudes stronger than the real strength values due to the presence of defects in materials suggest that processing defects, located in a material or at an interface is a significant dental problem. The third problem is partly self-inflicted. During dental education, students learn to copy the anatomy of natural teeth. The pits and fissures present in natural teeth act naturally as stress concentrators, but because of the sophisticated structure of a substrate such as enamel, such pits and fissures may in fact act as crack stoppers. Take for example a crack that might propagate along a cusp toward the central fissure. When that crack reaches the bottom of that fissure, the thickness of the enamel decreases and one can assume that the crack will not continue to propagate up along the other cusp with increasing enamel thickness. In the case of a manmade crown or filling, the fissure will not serve the same protective purpose. However, because dental students are trained to reproduce the sharp anatomic details, sharp anatomic fissures are often regarded as a sign of good competence, while in reality such details will rather facilitate crack growth.

Based on the information provided in this chapter it seems reasonable to suggest that future dental students should receive more training in fracture mechanics in order to better understand how handling and design may affect the final outcome of a restorative procedure Besides, with such a knowledge they would be able to communicate better with other scientists and thereby facilitate the development of better restorative materials.

## **Author details**

Karl-Johan Söderholm *College of Dentistry, University of Florida, Gainesville, Florida, USA* 

## **5. References**


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**Author details** 

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Based on the information provided in this chapter it seems reasonable to suggest that future dental students should receive more training in fracture mechanics in order to better understand how handling and design may affect the final outcome of a restorative procedure Besides, with such a knowledge they would be able to communicate better with

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## **Fracture Mechanics Based Models of Structural and Contact Fatigue**

Ilya I. Kudish

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48511

## **1. Introduction**

The subsurface initiated contact fatigue failure is one of the dominating mechanisms of failure of moving machine parts involved in cyclic motion. Structural fatigue failure may be of surface or subsurface origin. The analysis of a significant amount of accumulated experimental data obtained from field exploitation and laboratory testing provides undisputable evidence of the most important factors affecting contact fatigue [1]. It is clear that the factors affecting contact fatigue the most are as follows (a) acting cyclic normal stress and frictional stress (detailed lubrication conditions, surface roughness, etc.) which in part are determined by the part geometry, (b) distribution of residual stress versus depth, (c) initial statistical defect/crack distribution versus defect size, and location, (d) material elastic and fatigue parameters as functions of materials hardness, etc., (e) material fracture toughness, (f) material hardness versus depth, (g) machining and finishing operations, (h) abrasive contamination of lubricant and residual surface contamination, (i) non-steady cyclic loading regimes, etc. In case of structural fatigue the list of the most important parameters affecting fatigue performance is similar. None of the existing contact or structural fatigue models developed for prediction of contact fatigue life of bearings and gears as well as other structures takes into account all of the above operational and material conditions. Moreover, at best, most of the existing contact fatigue models are only partially based on the fundamental physical and mechanical mechanisms governing the fatigue phenomenon. Most of these models are of empirical nature and are based on assumptions some of which are not supported by experimental data or are controversial as it is in the case of fatigue models for bearings and gears [1]. Some models involve a number of approximations that usually do not reflect the actual processes occurring in material.

Therefore, a comprehensive mathematical models of contact and structural fatigue failure should be based on clearly stated mechanical principles following from the theory of elasticity, lubrication theory of elastohydrodynamic contact interactions, and fracture mechanics. Such

©2012 Kudish, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Kudish, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

models should take into consideration all the parameters described in items (a)-(e) and beyond. The advantage of such comprehensive models would be that the effect of variables such as steel cleanliness, externally applied stresses, residual stresses, etc. on contact and structural fatigue life could be examined as single or composite entities.

The goal of this chapter is to provided fracture mechanics based models of contact and structural fatigue. Historically, one of the most significant problems in realization of such an approach is the availability of simple but sufficiently precise solutions for the crack stress intensity factors. To overcome this difficulty some problems of fracture mechanics will be analyzed and their solutions will be represented in an analytical form acceptable for the further usage in modeling of contact and structural fatigue. In particular, a problem for an elastic half-plain weakened with a number of subsurface cracks and loaded with contact normal and frictional stresses as well with residual stress will be formulated and its asymptotic analytical solution will be presented. The latter solutions for the crack stress intensity factors are expressed in terms of certain integrals of known functions. These solutions for the stress intensity factors will be used in formulation of a two-dimensional contact fatigue model. In addition to that, a three-dimensional model applicable to both structural and contact fatigue will be formulated and some examples will be given. The above models take into account the parameters indicated in items (a)-(e) and are open for inclusion of the other parameters significant for fatigue.

In particular, these fatigue models take into account the statistical distribution of inclusions/cracks over the volume of the material versus their size and the resultant stress acting at the location of every inclusion/crack. Some of the main assumptions of the models are that (1) fatigue process in any machine part or structural unit runs in a similar manner and it is a direct reflection of the acting cyclic stresses and material properties and (2) the main part of fatigue life corresponds to the fatigue crack growth period, i.e. fatigue crack initiation period can be neglected. Furthermore, the variation in the distribution of cracks over time due to their fatigue growth is accounted for which is absent in all other existing fatigue models. The result of the above fatigue modeling is a simple relationship between fatigue life and cyclic loading, material mechanical parameters and its cleanliness as well as part geometry.

## **2. Three-dimensional model of contact and structural fatigue**

The approach presented in this section provides a unified model of contact and structural fatigue of materials [1, 2]. The model development is based on a block approach, i.e. each block of the model describes a certain process related to fatigue and can be easily replaced by another block describing the same process differently. For example, accumulation of new more advanced knowledge of the process of fatigue crack growth may provide an opportunity to replaced the proposed here block dealing with fatigue crack growth with an improved one. Two examples of the application of this model to structural fatigue are provided.

### **2.1. Initial statistical defect distribution**

It is assumed that material defects are far from each other and practically do not interact. However, in some cases clusters of nonmetallic inclusions located very close to each other are observed. In such cases these defect clusters can be represented by single defects of approximately the same size. Suppose there is a characteristic size *L<sup>σ</sup>* in material that is determined by the typical variations of the material stresses, grain and surface geometry. It is also assumed that there is a size *L <sup>f</sup>* in material such that *Ld* � *L <sup>f</sup>* � *Lσ*, where *Ld* is the typical distance between the material defects. In other words, it is assumed that the defect population in any such volume *L*<sup>3</sup> *<sup>f</sup>* is large enough to ensure an adequate statistical representation. It means that any parameter variations on the scale of *L <sup>f</sup>* are indistinguishable for the fatigue analysis purposes and that in the further analysis any volume *L*<sup>3</sup> *<sup>f</sup>* can be represented by its center point (*x*, *y*, *z*).

Therefore, there is an initial statistical defect distribution in the material such that each defect can be replaced by a subsurface penny-shaped or a surface semi-circular crack with a radius approximately equal to the half of the defect diameter. The usage of penny-shaped subsurface and semi-circular surface cracks is advantageous to the analysis because such fatigue cracks maintain their shape and their size is characterized by just one parameter. The orientation of these crack propagation will be considered later. The initial statistical distribution is described by a probabilistic density function *f*(0, *x*, *y*, *z*, *l*0), such that *f*(0, *x*, *y*, *z*, *l*0)*dl*0*dxdydz* is the number of defects with the radii between *l*<sup>0</sup> and *l*<sup>0</sup> + *dl*<sup>0</sup> in the material volume *dxdydz* centered about point (*x*, *y*, *z*). The material defect distribution is a local characteristic of material defectiveness. The model can be developed for any specific initial distribution *f*(0, *x*, *y*, *z*, *l*0). Some experimental data [3] suggest a log-normal initial defect distribution *f*(0, *x*, *y*, *z*, *l*0) versus the defect initial radius *l*<sup>0</sup>

$$f(0, x, y, z, l\_0) = 0 \text{ if } l\_0 \le 0,$$

$$f(0, x, y, z, l\_0) = \frac{\rho(0, x, y, z)}{\sqrt{2\pi\sigma\_{ln}l\_0}} \exp\left[-\frac{1}{2}(\frac{\ln\left(l\_0\right) - \mu\_{ln}}{\sigma\_{ln}})^2\right] if \, l\_0 > 0,$$

where *μln* and *σln* are the mean value and standard deviation of the crack radii, respectively.

#### **2.2. Direction of fatigue crack propagation**

2 Will-be-set-by-IN-TECH

models should take into consideration all the parameters described in items (a)-(e) and beyond. The advantage of such comprehensive models would be that the effect of variables such as steel cleanliness, externally applied stresses, residual stresses, etc. on contact and

The goal of this chapter is to provided fracture mechanics based models of contact and structural fatigue. Historically, one of the most significant problems in realization of such an approach is the availability of simple but sufficiently precise solutions for the crack stress intensity factors. To overcome this difficulty some problems of fracture mechanics will be analyzed and their solutions will be represented in an analytical form acceptable for the further usage in modeling of contact and structural fatigue. In particular, a problem for an elastic half-plain weakened with a number of subsurface cracks and loaded with contact normal and frictional stresses as well with residual stress will be formulated and its asymptotic analytical solution will be presented. The latter solutions for the crack stress intensity factors are expressed in terms of certain integrals of known functions. These solutions for the stress intensity factors will be used in formulation of a two-dimensional contact fatigue model. In addition to that, a three-dimensional model applicable to both structural and contact fatigue will be formulated and some examples will be given. The above models take into account the parameters indicated in items (a)-(e) and are open for inclusion

In particular, these fatigue models take into account the statistical distribution of inclusions/cracks over the volume of the material versus their size and the resultant stress acting at the location of every inclusion/crack. Some of the main assumptions of the models are that (1) fatigue process in any machine part or structural unit runs in a similar manner and it is a direct reflection of the acting cyclic stresses and material properties and (2) the main part of fatigue life corresponds to the fatigue crack growth period, i.e. fatigue crack initiation period can be neglected. Furthermore, the variation in the distribution of cracks over time due to their fatigue growth is accounted for which is absent in all other existing fatigue models. The result of the above fatigue modeling is a simple relationship between fatigue life and cyclic loading, material mechanical parameters and its cleanliness as well as part geometry.

The approach presented in this section provides a unified model of contact and structural fatigue of materials [1, 2]. The model development is based on a block approach, i.e. each block of the model describes a certain process related to fatigue and can be easily replaced by another block describing the same process differently. For example, accumulation of new more advanced knowledge of the process of fatigue crack growth may provide an opportunity to replaced the proposed here block dealing with fatigue crack growth with an improved one.

It is assumed that material defects are far from each other and practically do not interact. However, in some cases clusters of nonmetallic inclusions located very close to each other

**2. Three-dimensional model of contact and structural fatigue**

Two examples of the application of this model to structural fatigue are provided.

structural fatigue life could be examined as single or composite entities.

of the other parameters significant for fatigue.

**2.1. Initial statistical defect distribution**

It is assumed that the duration of the crack initiation period is negligibly small in comparison with the duration of the crack propagation period. It is also assumed that linear elastic fracture mechanics is applicable to small fatigue cracks. The details of the substantiation of these assumptions can be found in [1]. Based on these assumptions in the vicinity of a crack the stress intensity factors completely characterize the material stress state. The normal *k*<sup>1</sup> and shear *k*<sup>2</sup> and *k*<sup>3</sup> stress intensity factors at the edge of a single crack of radius *l* can be represented in the form [4]

$$\begin{aligned} k\_1 &= F\_1(\mathbf{x}, \mathbf{y}, z, \mathbf{a}, \boldsymbol{\beta}) \sigma\_1 \sqrt{\pi l}, \ k\_2 = F\_2(\mathbf{x}, \mathbf{y}, z, \mathbf{a}, \boldsymbol{\beta}) \sigma\_1 \sqrt{\pi l}, \\ k\_3 &= F\_3(\mathbf{x}, \mathbf{y}, z, \mathbf{a}, \boldsymbol{\beta}) \sigma\_1 \sqrt{\pi l}, \end{aligned} \tag{2}$$

where *σ*<sup>1</sup> is the maximum of the local tensile principal stress, *F*1, *F*2, and *F*<sup>3</sup> are certain functions of the point coordinates (*x*, *y*, *z*) and the crack orientation angles *α* and *β* with respect to the coordinate planes. The coordinate system is introduced in such a way that the *x*- and

*y*-axes are directed along the material surface while the *z*-axis is directed perpendicular to the material surface.

The resultant stress field in an elastic material is formed by stresses *σx*(*x*, *y*, *z*), *σy*(*x*, *y*, *z*), *σz*(*x*, *y*, *z*), *τxz*(*x*, *y*, *z*), *τxy*(*x*, *y*, *z*), and *τzy*(*x*, *y*, *z*). Some regions of material are subjected to tensile stress while other regions are subjected to compressive stress. Conceptually, there is no difference between the phenomena of structural and contact fatigue as the local material response to the same stress in both cases is the same and these cases differ in their stress fields only. As long as the stress levels do not exceed the limits of applicability of the quasi-brittle linear fracture mechanics when plastic zones at crack edges are small the rest of the material behaves like an elastic solid. The actual stress distributions in cases of structural and contact fatigue are taken in the proper account. In contact interactions where compressive stress is usually dominant there are still zones in material subjected to tensile stress caused by contact frictional and/or tensile residual stress [1, 5].

Experimental and theoretical studies show [1] that after initiation fatigue cracks propagate in the direction determined by the local stress field, namely, perpendicular to the local maximum tensile principle stress. Therefore, it is assumed that fatigue is caused by propagation of penny-shaped subsurface or semi-circular surface cracks under the action of principal maximum tensile stresses. Only high cycle fatigue phenomenon is considered here. On a plane perpendicular to a principal stress the shear stresses are equal to zero, i.e. the shear stress intensity factors *k*<sup>2</sup> = *k*<sup>3</sup> = 0. To find the plane of fatigue crack propagation (i.e. the orientation angles *α* and *β*), which is perpendicular to the maximum principal tensile stress, it is necessary to find the directions of these principal stresses. The latter is equivalent to solving the equations

$$k\_2(N, \mathfrak{a}, \mathfrak{z}, l, \mathfrak{x}, y, z) = 0, \; k\_3(N, \mathfrak{a}, \mathfrak{z}, l, \mathfrak{x}, y, z) = 0. \tag{3}$$

Usually, there are more than one solution sets to these equations at any point (*x*, *y*, *z*). To get the right angles *α* and *β* one has to chose the solution set that corresponds to the maximum tensile principal stress, i.e. maximum of the normal stress intensity factor *k*1(*N*, *l*, *x*, *y*, *z*). That guarantees that fatigue cracks propagate in the direction perpendicular to the maximum tensile principal stress if *α* and *β* are chosen that way.

For steady cyclic loading for small cracks *k*<sup>20</sup> = *k*2/ <sup>√</sup>*<sup>l</sup>* and *<sup>k</sup>*<sup>30</sup> <sup>=</sup> *<sup>k</sup>*3/ <sup>√</sup>*<sup>l</sup>* are independent from the number of cycles *N* and crack radius *l* together with equation (3) lead to the conclusion that for cyclic loading with constant amplitude the angles *α* and *β* characterizing the plane of fatigue crack growth are independent from *N* and *l*. Thus, angles *α* and *β* are functions of only crack location, i.e. *α* = *α*(*x*, *y*, *z*) and *β* = *β*(*x*, *y*, *z*). For the most part of their lives fatigue cracks created and/or existed near material defects remain small. Therefore, penny-shaped subsurface cracks conserve their shape but increase in size.

Even for the case of an elastic half-space it is a very difficult task to come up with sufficiently precise analytical solutions for the stress intensity factors at the edges of penny-shaped subsurface or semi-circular surface crack of arbitrary orientation at an arbitrary location (*x*, *y*, *z*). However, due to the fact that practically all the time fatigue cracks remain small and exsert little influence on the material general stress state the angles *α* and *β* can be determined in the process of calculation of the maximum tensile principle stress *σ*. The latter is equivalent to solution of equations (3). As soon as *σ* is determined for a subsurface crack its normal stress intensity factor *k*<sup>1</sup> can be approximated by the normal stress intensity factor for the case of a single crack of radius *l* in an infinite space subjected to the uniform tensile stress *σ*, i.e. by *k*<sup>1</sup> = 2*σ* <sup>√</sup>*l*/*π*.

#### **2.3. Fatigue crack propagation**

4 Will-be-set-by-IN-TECH

*y*-axes are directed along the material surface while the *z*-axis is directed perpendicular to the

The resultant stress field in an elastic material is formed by stresses *σx*(*x*, *y*, *z*), *σy*(*x*, *y*, *z*), *σz*(*x*, *y*, *z*), *τxz*(*x*, *y*, *z*), *τxy*(*x*, *y*, *z*), and *τzy*(*x*, *y*, *z*). Some regions of material are subjected to tensile stress while other regions are subjected to compressive stress. Conceptually, there is no difference between the phenomena of structural and contact fatigue as the local material response to the same stress in both cases is the same and these cases differ in their stress fields only. As long as the stress levels do not exceed the limits of applicability of the quasi-brittle linear fracture mechanics when plastic zones at crack edges are small the rest of the material behaves like an elastic solid. The actual stress distributions in cases of structural and contact fatigue are taken in the proper account. In contact interactions where compressive stress is usually dominant there are still zones in material subjected to tensile stress caused by contact

Experimental and theoretical studies show [1] that after initiation fatigue cracks propagate in the direction determined by the local stress field, namely, perpendicular to the local maximum tensile principle stress. Therefore, it is assumed that fatigue is caused by propagation of penny-shaped subsurface or semi-circular surface cracks under the action of principal maximum tensile stresses. Only high cycle fatigue phenomenon is considered here. On a plane perpendicular to a principal stress the shear stresses are equal to zero, i.e. the shear stress intensity factors *k*<sup>2</sup> = *k*<sup>3</sup> = 0. To find the plane of fatigue crack propagation (i.e. the orientation angles *α* and *β*), which is perpendicular to the maximum principal tensile stress, it is necessary to find the directions of these principal stresses. The latter is equivalent to solving

Usually, there are more than one solution sets to these equations at any point (*x*, *y*, *z*). To get the right angles *α* and *β* one has to chose the solution set that corresponds to the maximum tensile principal stress, i.e. maximum of the normal stress intensity factor *k*1(*N*, *l*, *x*, *y*, *z*). That guarantees that fatigue cracks propagate in the direction perpendicular to the maximum

the number of cycles *N* and crack radius *l* together with equation (3) lead to the conclusion that for cyclic loading with constant amplitude the angles *α* and *β* characterizing the plane of fatigue crack growth are independent from *N* and *l*. Thus, angles *α* and *β* are functions of only crack location, i.e. *α* = *α*(*x*, *y*, *z*) and *β* = *β*(*x*, *y*, *z*). For the most part of their lives fatigue cracks created and/or existed near material defects remain small. Therefore, penny-shaped

Even for the case of an elastic half-space it is a very difficult task to come up with sufficiently precise analytical solutions for the stress intensity factors at the edges of penny-shaped subsurface or semi-circular surface crack of arbitrary orientation at an arbitrary location (*x*, *y*, *z*). However, due to the fact that practically all the time fatigue cracks remain small and exsert little influence on the material general stress state the angles *α* and *β* can be determined in the process of calculation of the maximum tensile principle stress *σ*. The latter is equivalent to solution of equations (3). As soon as *σ* is determined for a subsurface crack its normal stress

*k*2(*N*, *α*, *β*, *l*, *x*, *y*, *z*) = 0, *k*3(*N*, *α*, *β*, *l*, *x*, *y*, *z*) = 0. (3)

<sup>√</sup>*<sup>l</sup>* and *<sup>k</sup>*<sup>30</sup> <sup>=</sup> *<sup>k</sup>*3/

<sup>√</sup>*<sup>l</sup>* are independent from

material surface.

the equations

frictional and/or tensile residual stress [1, 5].

tensile principal stress if *α* and *β* are chosen that way. For steady cyclic loading for small cracks *k*<sup>20</sup> = *k*2/

subsurface cracks conserve their shape but increase in size.

Fatigue cracks propagate at every point of the material stressed volume *V* at which max *<sup>T</sup>* (*k*1) <sup>&</sup>gt; *kth*, where the maximum is taken over the duration of the loading cycle *T* and *kth* is the material stress intensity threshold. There are three distinct stages of crack development: (a) growth of small cracks, (b) propagation of well–developed cracks, and (c) explosive and, usually, unstable growth of large cracks. The stage of small crack growth is the slowest one and it represents the main part of the entire crack propagation period. This situation usually causes confusion about the duration of the stages of crack initiation and propagation of small cracks. The next stage, propagation of well–developed cracks, usually takes significantly less time than the stage of small crack growth. And, finally, the explosive crack growth takes almost no time.

A relatively large number of fatigue crack propagation equations are collected and analyzed in [6]. Any one of these equations can be used in the model to describe propagation of fatigue cracks. However, the simplest of them which allows to take into account the residual stress and, at the same time, to avoid the usage of such an unstable characteristic as the stress intensity threshold *kth* is Paris's equation

$$\frac{d\mathbf{l}}{d\mathbf{N}} = \mathbf{g}\_0 (\max\_{-\infty < \mathbf{x} < \infty} \triangle k\_1)^{\mathbf{n}} \text{ } l \mid\_{\mathbf{N} = 0} = l\_{\mathbf{0} \nu} \tag{4}$$

where *g*<sup>0</sup> and *n* are the parameters of material fatigue resistance and *l*<sup>0</sup> is the crack initial radius. Notice, that in cases of loading and relaxation such as in contact fatigue �*k*<sup>1</sup> = *k*1.

Fatigue cracks propagate until they reach their critical size with radius *lc* for which *k*<sup>1</sup> = *Kf* (*Kf* is the material fracture toughness, i.e. to the radius of *lc* = (*Kf* /*k*10)2). After that their growth becomes unstable and very fast. Usually, the stage of explosive crack growth takes just few loading cycles.

It can be shown that the number of loading cycles needed for a crack to reach its critical radius is almost independent from the material fracture toughness *Kf* . This conclusion is supported by direct numerical simulations. For the further analysis, it is necessary to determine for a crack its initial radius *l*0*c*, which after *N* loading cycles reaches the critical size of *lc*. Solving the initial-value problem (4) one obtains the formula

$$l\_{0\emptyset} = \left\{ l\_c^{\frac{2-n}{2}} + N(\frac{n}{2} - 1)g\_0 \left[ \max\_{-\infty < x < \infty} \triangle k\_{10} \right]^n \right\}^{\frac{2}{2-n}} \tag{5}$$

where *l*0*<sup>c</sup>* depends on *N*, *x*, *y*, and *z*. Obviously, for *n* > 2 and fixed *x*, *y*, and *z* the value of *l*0*<sup>c</sup>* is a decreasing function of *N*. It is important to keep in mind that *l*0*c*(*N*, *x*, *y*, *z*) is minimal where *k*10(*x*, *y*, *z*) is maximal, which, in turn, happens where the material tensile stress reaches its maximum.

#### **2.4. Crack propagation statistics**

To describe crack statistics after the crack initiation stage is over it is necessary to make certain assumptions. The simplest assumptions of this kind are: the existing cracks do not heal and new cracks are not created. In other words, the number of cracks in any material volume remains constant in time. Based on a practically correct assumption that the defect distribution is initially scarce, the coalescence of cracks and changes in the general stress field are possible only when cracks have already reached relatively large sizes. However, this may happen only during the last stage of crack growth the duration of which is insignificant for calculation of fatigue life. Therefore, it can be assumed that over almost all life span of fatigue cracks their orientations do not change. This leads to the equation for the density of crack distribution *f*(*N*, *x*, *y*, *z*, *l*) as a function of crack radius *l* after *N* loading cycles in a small parallelepiped *dxdydz* with the center at the point with coordinates (*x*, *y*, *z*)

$$f(N, \mathbf{x}, y, z, l)dl = f(0, \mathbf{x}, y, z\_{\prime}l\_{0})dl\_{0\prime} \tag{6}$$

which being solved for *f*(*N*, *x*, *y*, *z*, *l*) gives

$$f(N, \mathbf{x}, y, z, l)dl = f(0, \mathbf{x}, y, z, l\_0) \frac{dl\_0}{dl} \,\prime \tag{7}$$

where *l*<sup>0</sup> and *dl*0/*dl* as functions of *N* and *l* can be obtain from the solution of (4) in the form

$$\begin{array}{c} l\_0 = \{l^{\frac{2-n}{2}} + N(\frac{n}{2}-1)\mathbf{g}\_0[\max\_{-\infty < x < \infty} \triangle k\_{10}]^n\}^{\frac{2}{2-n}}, \\\\ \frac{dl\_0}{dl} = \{1 + N(\frac{n}{2}-1)\mathbf{g}\_0[\max\_{-\infty < x < \infty} \triangle k\_{10}]^n I^{\frac{n-2}{2}}\}^{\frac{n}{n-2}}. \end{array} \tag{8}$$

Equations (7) and (8) lead to the expression for the crack distribution function *f* after *N* loading cycles

$$\begin{aligned} f(N, x, y, z, l) &= f(0, x, y, z, l\_0(N, l, y, z)) \{1 \\ &+ N(\frac{n}{2} - 1) g\_0 [\max\_{-\infty < x < \infty} \triangle k\_{10}]^n I^{\frac{n-2}{2}}\}^{\frac{n}{n-2}}, \end{aligned} \tag{9}$$

where *l*0(*N*, *l*, *y*, *z*) is determined by the first of the equations in (8).

Formula (9) leads to a number of important conclusions. The fatigue crack distribution function *f*(*N*, *x*, *y*, *z*, *l*) depends on the initial crack distribution *f*(0, *x*, *y*, *z*, *l*0) and it changes with the number of applied loading cycles *N* in such a way that the crack volume density *ρ*(*N*, *x*, *y*, *z*) remains constant. Because of crack growth the crack distribution *f*(*N*, *x*, *y*, *z*, *l*) widens with respect to *l* with number of loading cycles *N*.

#### **2.5. Local fatigue damage accumulation**

Let us design the measure of material fatigue damage. If at a certain point (*x*, *y*, *z*) after *N* loading cycles radii of all cracks *l* < *lc* then there is no damage at this point and the material local survival probability *p*(*N*, *x*, *y*, *z*) = 1. On the other hand, if at this point after *N* loading cycles radii of all cracks *l* ≥ *lc*, then all cracks reached the critical size and the material at this point is completely damaged and the local survival probability *p*(*N*, *x*, *y*, *z*) = 0. Obviously, the more fatigue cracks with larger radii *l* exist at the point the lower is the local survival probability *p*(*N*, *x*, *y*, *z*). It is reasonable to assume that the material local survival probability *p*(*N*, *x*, *y*, *z*) is a certain monotonic measure of the portion of cracks with radius *l* below the critical radius *lc*. Therefore, *p*(*N*, *x*, *y*, *z*) can be represented by the expressions

$$p(N, \mathbf{x}, y, z) = \frac{1}{\rho} \int\_0^{l\_c} f(N, \mathbf{x}, y, z, l) dl \text{ if } f(0, \mathbf{x}, y, z, l\_0) \neq 0,$$

$$p(N, \mathbf{x}, y, z) = 1 \text{ otherwise},\tag{10}$$

$$\rho = \rho(N, \mathbf{x}, y, z) = \underset{0}{\overset{\text{\textquotedbl{}}}{\text{\${}}}} f(N, \mathbf{x}, y, z, l) dl = \rho(0, \mathbf{x}, y, z).$$

Obviously, the local survival probability *p*(*N*, *x*, *y*, *z*) is a monotonically decreasing function of the number of loading cycles *N* because fatigue crack radii *l* tend to grow with the number of loading cycles *N*.

To calculate *p*(*N*, *x*, *y*, *z*) from (10) one can use the specific expression for *f* determined by (9). However, it is more convenient to modify it as follows

$$p(N, \mathbf{x}, y, z) = \frac{1}{\rho} \int\_0^{l\_0} f(0, \mathbf{x}, y, z, l\_0) dl\_0 \text{ if } f(0, \mathbf{x}, y, z, l\_0) \neq 0,\tag{11}$$

$$p(N, \mathbf{x}, y, z) = 1 \text{ otherwise},$$

where *l*0*<sup>c</sup>* is determined by (5) and *ρ* is the initial volume density of cracks. Thus, to every material point (*x*, *y*, *z*) is assigned a certain local survival probability *p*(*N*, *x*, *y*, *z*), 0 ≤ *p*(*N*, *x*, *y*, *z*) ≤ 1.

Equations (11) demonstrate that the material local survival probability *p*(*N*, *x*, *y*, *z*) is mainly controlled by the initial crack distribution *f*(0, *x*, *y*, *z*, *l*0), material fatigue resistance parameters *g*<sup>0</sup> and *n*, and external contact and residual stresses. Moreover, the material local survival probability *p*(*N*, *x*, *y*, *z*) is a decreasing function of *N* because *l*0*<sup>c</sup>* from (5) is a decreasing function of *N* for *n* > 2.

#### **2.6. Global fatigue damage accumulation**

6 Will-be-set-by-IN-TECH

To describe crack statistics after the crack initiation stage is over it is necessary to make certain assumptions. The simplest assumptions of this kind are: the existing cracks do not heal and new cracks are not created. In other words, the number of cracks in any material volume remains constant in time. Based on a practically correct assumption that the defect distribution is initially scarce, the coalescence of cracks and changes in the general stress field are possible only when cracks have already reached relatively large sizes. However, this may happen only during the last stage of crack growth the duration of which is insignificant for calculation of fatigue life. Therefore, it can be assumed that over almost all life span of fatigue cracks their orientations do not change. This leads to the equation for the density of crack distribution *f*(*N*, *x*, *y*, *z*, *l*) as a function of crack radius *l* after *N* loading cycles in a small parallelepiped

*f*(*N*, *x*, *y*, *z*, *l*)*dl* = *f*(0, *x*, *y*, *z*, *l*0) *dl*<sup>0</sup>

where *l*<sup>0</sup> and *dl*0/*dl* as functions of *N* and *l* can be obtain from the solution of (4) in the form

<sup>2</sup> <sup>−</sup> <sup>1</sup>)*g*0[ max <sup>−</sup>∞<*x*<∞�*k*10]

<sup>2</sup> <sup>−</sup> <sup>1</sup>)*g*0[ max <sup>−</sup>∞<*x*<∞�*k*10]

Equations (7) and (8) lead to the expression for the crack distribution function *f* after *N*

*f*(*N*, *x*, *y*, *z*, *l*) = *f*(0, *x*, *y*, *z*, *l*0(*N*, *l*, *y*, *z*)){1

Formula (9) leads to a number of important conclusions. The fatigue crack distribution function *f*(*N*, *x*, *y*, *z*, *l*) depends on the initial crack distribution *f*(0, *x*, *y*, *z*, *l*0) and it changes with the number of applied loading cycles *N* in such a way that the crack volume density *ρ*(*N*, *x*, *y*, *z*) remains constant. Because of crack growth the crack distribution *f*(*N*, *x*, *y*, *z*, *l*)

Let us design the measure of material fatigue damage. If at a certain point (*x*, *y*, *z*) after *N* loading cycles radii of all cracks *l* < *lc* then there is no damage at this point and the material local survival probability *p*(*N*, *x*, *y*, *z*) = 1. On the other hand, if at this point after *N* loading cycles radii of all cracks *l* ≥ *lc*, then all cracks reached the critical size and the material at this point is completely damaged and the local survival probability *p*(*N*, *x*, *y*, *z*) = 0. Obviously,

<sup>2</sup> <sup>−</sup> <sup>1</sup>)*g*0[ max <sup>−</sup>∞<*x*<∞�*k*10]

*f*(*N*, *x*, *y*, *z*, *l*)*dl* = *f*(0, *x*, *y*, *z*, *l*0)*dl*0, (6)

*<sup>n</sup>*} <sup>2</sup> <sup>2</sup>−*<sup>n</sup>* ,

*nl n*−2 <sup>2</sup> } *<sup>n</sup> <sup>n</sup>*−<sup>2</sup> .

*nl n*−2 <sup>2</sup> } *<sup>n</sup> <sup>n</sup>*−<sup>2</sup> ,

*dl* , (7)

(8)

(9)

**2.4. Crack propagation statistics**

*dxdydz* with the center at the point with coordinates (*x*, *y*, *z*)

*l*<sup>0</sup> = {*l*

*dl*<sup>0</sup>

loading cycles

2−*n* <sup>2</sup> + *N*( *<sup>n</sup>*

*dl* <sup>=</sup> {<sup>1</sup> <sup>+</sup> *<sup>N</sup>*( *<sup>n</sup>*

+*N*( *<sup>n</sup>*

widens with respect to *l* with number of loading cycles *N*.

**2.5. Local fatigue damage accumulation**

where *l*0(*N*, *l*, *y*, *z*) is determined by the first of the equations in (8).

which being solved for *f*(*N*, *x*, *y*, *z*, *l*) gives

The survival probability *P*(*N*) of the material as a whole is determined by the local probabilities of all points of the material at which fatigue cracks are present. It is assumed that the material fails as soon as it fails at just one point. It is assumed that the initial crack distribution in the material is discrete. Let *pi*(*N*) = *p*(*N*, *xi*, *yi*, *zi*), *i* = 1, . . . , *Nc*, where *Nc* is the total number of points in the material stressed volume *V* at which fatigue cracks are present. Then based on the above assumption the material survival probability *P*(*N*) is equal to

$$P(N) = \prod\_{i=1}^{N\_c} p\_i(N). \tag{12}$$

Obviously, probability *P*(*N*) from (12) satisfies the inequalities

$$[p\_{\mathfrak{m}}(\mathcal{N})]^{\mathcal{N}\_{\ell}} \le P(\mathcal{N}) \le p\_{\mathfrak{m}}(\mathcal{N}), \ p\_{\mathfrak{m}}(\mathcal{N}) = \min\_{\mathcal{V}} p(\mathcal{N}, \mathfrak{x}, \mathcal{y}, \mathcal{z}).\tag{13}$$

In (13) the right inequality shows that the survival probability *P*(*N*) is never greater than the minimum value *pm*(*N*) of the local survival probability *p*(*N*, *x*, *y*, *z*) over the material stressed volume *V*.

An analytical substantiation for the assumption that the first pit is created by the cracks from a small material volume with the smallest survival probability *pm*(*N*) is provided in [1]. Moreover, the indicated analysis also validates one of the main assumptions of the model that new cracks are not being created. Namely, if new cracks do get created in the process of loading, they are very small and have no chance to catch up with already existing and propagating larger cracks. The graphical representation of this fact is given in Fig. 1 [1]. In this figure fatigue cracks are initially randomly distributed over the material volume with respect to their normal stress intensity factor *k*<sup>1</sup> and are allowed to grow according to Paris' law (see (4)) with sufficiently high value of *n* = 6.67 − 9. In Fig. 1 the values of the normal stress intensity factor *k*<sup>1</sup> are shown at different time moments (*k*<sup>0</sup> and *L*<sup>0</sup> are the characteristic normal stress intensity factor and geometric size of the solid). These graphs clearly show that a crack with the initially larger value of the normal stress intensity factor *k*<sup>1</sup> propagates much faster than all other cracks, i.e. the value of its *k*<sup>1</sup> increases much faster than the values of *k*<sup>1</sup> for all other cracks, which are almost dormant. As a result of that, the crack with the initially larger value of *k*<sup>1</sup> reaches its critical size way ahead of other cracks. This event determines the time and the place where fatigue failure occurs initially. Therefore, in spite of formula (12) which indicates that all fatigue cracks have influence on the survival probability *P*(*N*), for high values of *n* the material survival probability *P*(*N*) is a local fatigue characteristic, and it is determined by the material defect with the initially highest value of the stress intensity factor *k*1. The higher the power *n* is the more accurate this approximation is.

Therefore, assuming that it is a very rare occurrence when more than one fatigue initiation/spall happen simultaneously, it can be shown that at the early stages of the fatigue process the material global survival probability *P*(*N*) is determined by the minimum of the local survival probability *pm*(*N*)

$$P(\text{N}) = p\_{\text{\textquotedblleft}}(\text{N}), \ p\_{\text{\textquotedblleft}}(\text{N}) = \min\_{\text{V}} p(\text{N}, x, y, z), \tag{14}$$

where the maximum is taken over the (stressed) volume *V* of the solid.

If the initial crack distribution is taken in the log-normal form (1) then

$$P(N) = p\_m(N) = \frac{1}{2} \left\{ 1 + \varepsilon r f\left[ \min\_V \frac{\ln l\_{0c}(N, y, z) - \mu\_{\text{li}}}{\sqrt{2} \sigma\_{\text{li}}} \right] \right\},\tag{15}$$

where *er f*(*x*) is the error integral [8]. Obviously, the local survival probability *pm*(*N*) is a complex combined measure of applied stresses, initial crack distribution, material fatigue parameters, and the number of loading cycles.

In cases when the mean *μln* and the standard deviation *σln* are constants throughout the material formula (15) can be significantly simplified

$$P(N) = p\_m(N) = \frac{1}{2} \left\{ 1 + \text{erf} \left[ \frac{\ln \min\_V l\_{\mathbb{K}}(N, y, z) - \mu\_{\text{ln}}}{\sqrt{2} \sigma\_{\text{ln}}} \right] \right\}. \tag{16}$$

**Figure 1.** Illustration of the growth of the initially randomly distributed normal stress intensity factor *k*<sup>1</sup> with time *N* as initially unit length fatigue cracks grow (after Tallian, Hoeprich, and Kudish [7]). Reprinted with permission from the STLE.

To determine fatigue life *N* of a contact for the given survival probability *P*(*N*) = *P*∗, it is necessary to solve the equation

$$p\_{\mathfrak{m}}(N) = P\_\*.\tag{17}$$

#### **2.7. Fatigue life calculation**

8 Will-be-set-by-IN-TECH

In (13) the right inequality shows that the survival probability *P*(*N*) is never greater than the minimum value *pm*(*N*) of the local survival probability *p*(*N*, *x*, *y*, *z*) over the material stressed

An analytical substantiation for the assumption that the first pit is created by the cracks from a small material volume with the smallest survival probability *pm*(*N*) is provided in [1]. Moreover, the indicated analysis also validates one of the main assumptions of the model that new cracks are not being created. Namely, if new cracks do get created in the process of loading, they are very small and have no chance to catch up with already existing and propagating larger cracks. The graphical representation of this fact is given in Fig. 1 [1]. In this figure fatigue cracks are initially randomly distributed over the material volume with respect to their normal stress intensity factor *k*<sup>1</sup> and are allowed to grow according to Paris' law (see (4)) with sufficiently high value of *n* = 6.67 − 9. In Fig. 1 the values of the normal stress intensity factor *k*<sup>1</sup> are shown at different time moments (*k*<sup>0</sup> and *L*<sup>0</sup> are the characteristic normal stress intensity factor and geometric size of the solid). These graphs clearly show that a crack with the initially larger value of the normal stress intensity factor *k*<sup>1</sup> propagates much faster than all other cracks, i.e. the value of its *k*<sup>1</sup> increases much faster than the values of *k*<sup>1</sup> for all other cracks, which are almost dormant. As a result of that, the crack with the initially larger value of *k*<sup>1</sup> reaches its critical size way ahead of other cracks. This event determines the time and the place where fatigue failure occurs initially. Therefore, in spite of formula (12) which indicates that all fatigue cracks have influence on the survival probability *P*(*N*), for high values of *n* the material survival probability *P*(*N*) is a local fatigue characteristic, and it is determined by the material defect with the initially highest value of the stress intensity

*<sup>V</sup> <sup>p</sup>*(*N*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*). (13)

*<sup>V</sup> <sup>p</sup>*(*N*, *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*), (14)

, (15)

. (16)

[*pm*(*N*)]*Nc* <sup>≤</sup> *<sup>P</sup>*(*N*) <sup>≤</sup> *pm*(*N*), *pm*(*N*) = min

factor *k*1. The higher the power *n* is the more accurate this approximation is.

*P*(*N*) = *pm*(*N*), *pm*(*N*) = min

2 1 + *er f* min *V*

2 1 + *er f*

where the maximum is taken over the (stressed) volume *V* of the solid. If the initial crack distribution is taken in the log-normal form (1) then

*P*(*N*) = *pm*(*N*) = <sup>1</sup>

*P*(*N*) = *pm*(*N*) = <sup>1</sup>

parameters, and the number of loading cycles.

material formula (15) can be significantly simplified

local survival probability *pm*(*N*)

Therefore, assuming that it is a very rare occurrence when more than one fatigue initiation/spall happen simultaneously, it can be shown that at the early stages of the fatigue process the material global survival probability *P*(*N*) is determined by the minimum of the

where *er f*(*x*) is the error integral [8]. Obviously, the local survival probability *pm*(*N*) is a complex combined measure of applied stresses, initial crack distribution, material fatigue

In cases when the mean *μln* and the standard deviation *σln* are constants throughout the

ln *l*0*<sup>c</sup>*( √ *N*,*y*,*z*)−*μln* 2*σln*

 ln min*<sup>V</sup> <sup>l</sup>*0*<sup>c</sup>*(*N*,*y*,*z*)−*μln* <sup>√</sup>2*σln*

Obviously, probability *P*(*N*) from (12) satisfies the inequalities

volume *V*.

Suppose the material failure occurs at point (*x*, *y*, *z*) with the probability 1 − *P*(*N*). That actually determines the point where in (16) the minimum over the material volume *V* is reached. Therefore, at this point in (16), the operation of minimum over the material volume *V* can be dropped. By solving (16) and (17), one gets

$$\begin{split} N &= \{ (\frac{\boldsymbol{n}}{2} - 1)\boldsymbol{\varrho}\_{0} [\underset{-\infty < \boldsymbol{x} < \infty}{\max} \triangle k\_{10}]^{\boldsymbol{n}} \}^{-1} \{ \exp[(1 - \frac{\boldsymbol{n}}{2})(\mu\_{\boldsymbol{ln}} \\\\ &+ \sqrt{2} \sigma\_{\boldsymbol{ln}} \epsilon \boldsymbol{r} f^{-1} (2P\_{\ast} - 1))] - l\_{\boldsymbol{c}}^{\frac{2-\mu}{2}} \}\_{\prime} \end{split} \tag{18}$$

where *er f* <sup>−</sup>1(*x*) is the inverse function to the error integral *er f*(*x*). Assuming that the material initially is free of damage, i.e., when *P*(0) = 1, one can simplify the latter equation. Discounting the very tail of the initial crack distribution, one gets max *<sup>V</sup> l*<sup>0</sup> ≤ *lc*. Thus, for well–developed cracks and, in many cases, even for small cracks, the second term in (5) for *l*0*<sup>c</sup>* dominates the first one. It means that the dependence of *l*0*<sup>c</sup>* on *lc* and, therefore, on the material fracture toughness *Kf* can be neglected. Then equation (18) can be approximated by

$$N = \{ (\frac{n}{2} - 1) \emptyset\_0 [\max\_{-\infty < x < \infty} \triangle k\_{10}]^n \}^{-1} \{ \exp[(1 - \frac{n}{2})(\mu\_{ln} $$
 
$$+ \sqrt{2} \sigma\_{ln} \text{erf}^{-1}(2P\_\* - 1)) \} \}. \tag{19}$$

#### 10 Will-be-set-by-IN-TECH 154 Applied Fracture Mechanics

Taking into account that in the case of contact fatigue *k*<sup>10</sup> is proportional to the maximum contact pressure *qmax* and that it also depends on the friction coefficient *λ* and the ratio of residual stress *q*<sup>0</sup> and *qmax* = *pH* as well as taking into account the relationships *μln* = ln *<sup>μ</sup>*<sup>2</sup> <sup>√</sup>*μ*<sup>2</sup>+*σ*<sup>2</sup> , *<sup>σ</sup>ln* <sup>=</sup> ln [1 + ( *<sup>σ</sup> <sup>μ</sup>* )2] [1] (where *<sup>μ</sup>* and *<sup>σ</sup>* are the regular initial mean and standard deviation) one arrives at a simple analytical formula

$$\begin{split} N &= \frac{\mathbb{C}\_0}{(n-2)g\_0 p\_H^n} (\frac{\sqrt{\mu^2 + \sigma^2}}{\mu^2})^{\frac{\mu}{2}-1} \\\\ &\times \exp[(1-\frac{\eta}{2})\sqrt{2\ln[1+(\frac{\sigma}{\mu})^2]} \text{erf}^{-1}(2P\_\*-1)], \end{split} \tag{20}$$

where *C*<sup>0</sup> depends only on the friction coefficient *λ* and the ratio of the residual stress *q*<sup>0</sup> and the maximum Hertzian pressure *pH*. Finally, assuming that *σ* � *μ* from (20) one can obtain the formula

$$N = \frac{\mathbb{C}\_0}{(n-2)\mathbb{g}\_0 p\_H^u \mu^{\frac{\mu}{2}-1}} \exp[(1-\frac{\mu}{2})\frac{\sqrt{2}\sigma}{\mu}erf^{-1}(2P\_\*-1)) ]. \tag{21}$$

Also, formulas (20) and (21) can be represented in the form of the Lundberg-Palmgren formula (see [1] and the discussion there).

Formula (21) demonstrates the intuitively obvious fact that the fatigue life *N* is inversely proportional to the value of the parameter *g*<sup>0</sup> that characterizes the material crack propagation resistance. Equation (21) exhibits a usual for roller and ball bearings as well as for gears dependence of the fatigue life *N* on the maximum Hertzian pressure *pH*. Thus, from the well–known experimental data for bearings the range of *n* values is 20/3 ≤ *n* ≤ 9. Keeping in mind that usually *σ* � *μ*, for these values of *n* contact fatigue life *N* is practically inverse proportional to a positive power of the mean crack size, i.e. to *μ <sup>n</sup>* <sup>2</sup> <sup>−</sup>1. Therefore, fatigue life *N* is a decreasing function of the initial mean crack (inclusion) size *μ*. This conclusion is valid for any material survival probability *P*<sup>∗</sup> and is supported by the experimental data discussed in [1]. In particular, ln *<sup>N</sup>* is practically a linear function of ln *<sup>μ</sup>* with a negative slope 1 <sup>−</sup> *<sup>n</sup>* 2 which is in excellent agreement with the Timken Company test data [9]. Keeping in mind that *<sup>n</sup>* <sup>&</sup>gt; 2, at early stages of fatigue failure, i.e. when *er f* <sup>−</sup>1(2*P*<sup>∗</sup> <sup>−</sup> <sup>1</sup>) <sup>&</sup>gt; 0 for *<sup>P</sup>*<sup>∗</sup> <sup>&</sup>gt; 0.5, one easily determines that fatigue life *N* is a decreasing function of the initial standard deviation of crack sizes *σ*. Similarly, at late stages of fatigue failure, i.e. when *P*<sup>∗</sup> < 0.5, the fatigue life *N* is an increasing function of the initial standard deviation of crack sizes *σ*. According to (21), for *P*<sup>∗</sup> = 0.5 fatigue life *N* is independent from *σ*, however, according to (20), for *P*<sup>∗</sup> = 0.5 fatigue life *N* is a slowly increasing function of *σ*. By differentiating *pm*(*N*) obtained from (16) with respect to *σ*, one can conclude that the dispersion of *P*(*N*) increases with *σ*.

The stress intensity factor *k*<sup>1</sup> decreases as the magnitude of the compressive residual stress *q*<sup>0</sup> increases and/or the magnitude of the friction coefficient *λ* decreases. Therefore, in (20) and 21) the value of *C*<sup>0</sup> is a monotonically decreasing function of residual stress *q*<sup>0</sup> and friction coefficient *λ*.

Being applied to bearings and/or gears the described statistical contact fatigue model can be used as a research and/or engineering tool in pitting modeling. In the latter case, some of the model parameters may be assigned certain fixed values based on the scrupulous analysis of steel quality and quality and stability of gear and bearing manufacturing processes.

In case of structural fatigue the Hertzian stress in formulas (20) and (21) should be replaced the dominant stress acting on the part while constant *C*<sup>0</sup> would be dependent on the ratios of other external stresses acting on the part at hand to the dominant stress in a certain way (see examples of torsional and bending fatigue below).

#### **2.8. Examples of torsional and bending fatigue**

10 Will-be-set-by-IN-TECH

Taking into account that in the case of contact fatigue *k*<sup>10</sup> is proportional to the maximum contact pressure *qmax* and that it also depends on the friction coefficient *λ* and the ratio of residual stress *q*<sup>0</sup> and *qmax* = *pH* as well as taking into account the relationships *μln* =

2 ln[1 + ( *<sup>σ</sup>*

<sup>2</sup> <sup>−</sup><sup>1</sup> exp[(<sup>1</sup> <sup>−</sup> *<sup>n</sup>*

where *C*<sup>0</sup> depends only on the friction coefficient *λ* and the ratio of the residual stress *q*<sup>0</sup> and the maximum Hertzian pressure *pH*. Finally, assuming that *σ* � *μ* from (20) one can obtain

Also, formulas (20) and (21) can be represented in the form of the Lundberg-Palmgren formula

Formula (21) demonstrates the intuitively obvious fact that the fatigue life *N* is inversely proportional to the value of the parameter *g*<sup>0</sup> that characterizes the material crack propagation resistance. Equation (21) exhibits a usual for roller and ball bearings as well as for gears dependence of the fatigue life *N* on the maximum Hertzian pressure *pH*. Thus, from the well–known experimental data for bearings the range of *n* values is 20/3 ≤ *n* ≤ 9. Keeping in mind that usually *σ* � *μ*, for these values of *n* contact fatigue life *N* is practically inverse

*N* is a decreasing function of the initial mean crack (inclusion) size *μ*. This conclusion is valid for any material survival probability *P*<sup>∗</sup> and is supported by the experimental data discussed in [1]. In particular, ln *<sup>N</sup>* is practically a linear function of ln *<sup>μ</sup>* with a negative slope 1 <sup>−</sup> *<sup>n</sup>*

which is in excellent agreement with the Timken Company test data [9]. Keeping in mind that *<sup>n</sup>* <sup>&</sup>gt; 2, at early stages of fatigue failure, i.e. when *er f* <sup>−</sup>1(2*P*<sup>∗</sup> <sup>−</sup> <sup>1</sup>) <sup>&</sup>gt; 0 for *<sup>P</sup>*<sup>∗</sup> <sup>&</sup>gt; 0.5, one easily determines that fatigue life *N* is a decreasing function of the initial standard deviation of crack sizes *σ*. Similarly, at late stages of fatigue failure, i.e. when *P*<sup>∗</sup> < 0.5, the fatigue life *N* is an increasing function of the initial standard deviation of crack sizes *σ*. According to (21), for *P*<sup>∗</sup> = 0.5 fatigue life *N* is independent from *σ*, however, according to (20), for *P*<sup>∗</sup> = 0.5 fatigue life *N* is a slowly increasing function of *σ*. By differentiating *pm*(*N*) obtained from (16) with

The stress intensity factor *k*<sup>1</sup> decreases as the magnitude of the compressive residual stress *q*<sup>0</sup> increases and/or the magnitude of the friction coefficient *λ* decreases. Therefore, in (20) and 21) the value of *C*<sup>0</sup> is a monotonically decreasing function of residual stress *q*<sup>0</sup> and friction

Being applied to bearings and/or gears the described statistical contact fatigue model can be used as a research and/or engineering tool in pitting modeling. In the latter case, some of the model parameters may be assigned certain fixed values based on the scrupulous analysis of

steel quality and quality and stability of gear and bearing manufacturing processes.

2 ) √ 2*σ*

*N* = *<sup>C</sup>*<sup>0</sup> (*n*−2)*g*<sup>0</sup> *<sup>p</sup><sup>n</sup> H* ( √*μ*<sup>2</sup>+*σ*<sup>2</sup> *<sup>μ</sup>*<sup>2</sup> ) *n* <sup>2</sup> −1

2 ) 

*<sup>H</sup><sup>μ</sup> <sup>n</sup>*

proportional to a positive power of the mean crack size, i.e. to *μ <sup>n</sup>*

respect to *σ*, one can conclude that the dispersion of *P*(*N*) increases with *σ*.

*<sup>μ</sup>* )2] [1] (where *<sup>μ</sup>* and *<sup>σ</sup>* are the regular initial mean and standard

*<sup>μ</sup>* )2]*er f* <sup>−</sup>1(2*P*<sup>∗</sup> <sup>−</sup> <sup>1</sup>)],

*<sup>μ</sup> er f* <sup>−</sup>1(2*P*<sup>∗</sup> <sup>−</sup> <sup>1</sup>))]. (21)

<sup>2</sup> <sup>−</sup>1. Therefore, fatigue life

(20)

2

ln *<sup>μ</sup>*<sup>2</sup>

the formula

coefficient *λ*.

<sup>√</sup>*μ*<sup>2</sup>+*σ*<sup>2</sup> , *<sup>σ</sup>ln* <sup>=</sup>

ln [1 + ( *<sup>σ</sup>*

deviation) one arrives at a simple analytical formula

<sup>×</sup> exp[(<sup>1</sup> <sup>−</sup> *<sup>n</sup>*

*N* = *<sup>C</sup>*<sup>0</sup> (*n*−2)*g*<sup>0</sup> *<sup>p</sup><sup>n</sup>*

(see [1] and the discussion there).

Suppose that in a beam material the defect distribution is space-wise uniform and follows equation (1). Also, let us assume that the residual stress is zero.

First, let us consider torsional fatigue. Suppose a beam is made of an elastic material with elliptical cross section (*a* and *b* are the ellipse semi-axes, *b* < *a*) and directed along the *y*-axis. The beam is under action of torque *My* about the *y*-axis applied to its ends. The side surfaces of the beam are free of stresses. Then it can be shown (see Lurye [10], p. 398) that

$$\mathfrak{tr}\_{\mathbf{x}\mathbf{y}} = -\frac{2G\gamma a^2}{a^2 + b^2} \mathbf{z} \; \mathfrak{r}\_{\mathbf{z}\mathbf{y}} = \frac{2G\gamma b^2}{a^2 + b^2} \mathbf{x} \; \sigma\_{\mathbf{x}} = \sigma\_{\mathbf{y}} = \sigma\_{\mathbf{z}} = \mathfrak{r}\_{\mathbf{x}\mathbf{z}} = \mathfrak{0} \tag{22}$$

where *G* is the material shear elastic modulus, *G* = *E*/[2(1+ *ν*)] (*E* and *ν* are Young's modulus and Poisson's ratio of the beam material), and *γ* is a dimensionless constant. By introducing the principal stresses *<sup>σ</sup>*1, *<sup>σ</sup>*2, and *<sup>σ</sup>*<sup>3</sup> that satisfy the equation *<sup>σ</sup>*<sup>3</sup> <sup>−</sup> (*τ*<sup>2</sup> *xy* + *τ*<sup>2</sup> *zy*)*σ* = 0, one obtains that

$$
\sigma\_1 = -\sqrt{\tau\_{xy}^2 + \tau\_{zy}^2}, \ \sigma\_2 = 0, \ \sigma\_3 = \sqrt{\tau\_{xy}^2 + \tau\_{zy}^2}.\tag{23}
$$

For the case of *<sup>a</sup>* <sup>&</sup>gt; *<sup>b</sup>* the maximum principal tensile stress *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>2*Gγa*2*<sup>b</sup> <sup>a</sup>*<sup>2</sup>+*b*<sup>2</sup> is reached at the surface of the beam at points (0, *y*, ±*b*) and depending on the sign of *My* it acts in one of the directions described by the directional cosines

$$\cos(\mathfrak{a}, \mathfrak{x}) = \mp \frac{\sqrt{2}}{2}, \cos(\mathfrak{a}, \mathfrak{y}) = \pm \frac{\sqrt{2}}{2}, \cos(\mathfrak{a}, \mathfrak{z}) = 0,\tag{24}$$

where *α* is the direction along one of the principal stress axes. For the considered case of elliptic beam, the moments of inertia of the beam elliptic cross section about the *x*- and *y*-axes, *Ix* and *Iz* as well as the moment of torsion *My* applied to the beam are as follows (see Lurye [10], pp. 395, 399) *Ix* = *πab*3/4, *Iz* = *πa*3*b*/4, *My* = *GγC*, *C* = 4*Ix Iz*/(*Ix* + *Iz*). Keeping in mind that according to Hasebe and Inohara [11] and Isida [12], the stress intensity factor *k*<sup>1</sup> for an edge crack of radius *l* and inclined to the surface of a half-plane at the angle of *π*/4 (see (24)) is *k*<sup>1</sup> = 0.705 | *σ*<sup>1</sup> | √ *πl*, one obtains *k*<sup>10</sup> = 1.41 <sup>√</sup>*<sup>π</sup>* |*My*| *ab*<sup>2</sup> . Then, fatigue life of a beam under torsion follows from substituting the expression for *k*<sup>10</sup> into equation (19)

$$N = \frac{2}{(n-2)g\_0} \{ \frac{1.257ab^2}{|M\_y|} \}^n \gspace{0} (\mu, \sigma),\tag{25}$$

$$g(\mu, \sigma) = (\frac{\sqrt{\mu^2 + \sigma^2}}{\mu^2})^{\frac{n-2}{2}} \exp[(1 - \frac{n}{2})\sqrt{2\ln[1 + (\frac{\sigma}{\mu})^2]} \sigma f^{-1}(2P\_\* - 1)]. \tag{26}$$

Now, let us consider bending fatigue of a beam/console made of an elastic material with elliptical cross section (*a* and *b* are the ellipse semi-axes) and length *L*. The beam is directed along the *y*-axis and it is under the action of a bending force *Px* directed along the *x*-axis which is applied to its free end. The side surfaces of the beam are free of stresses. The other end *y* = 0 of the beam is fixed. Then it can be shown (see Lurye [10]) that

$$
\sigma\_{\mathbf{x}} = \sigma\_{\mathbf{z}} = 0,\\
\sigma\_{\mathbf{y}} = -\frac{p\_{\mathbf{v}}}{I\_{\mathbf{z}}} \mathbf{x}(L - y),
$$

$$
\tau\_{\mathbf{x}2} = 0,\\
\tau\_{\mathbf{xy}} = \frac{p\_{\mathbf{v}}}{2(1 + \nu)I\_{\mathbf{z}}} \frac{2(1 + \nu)a^2 + b^2}{3a^2 + b^2} \{a^2 - \mathbf{x}^2 - \frac{(1 - 2\nu)a^2 \mathbf{z}^2}{2(1 + \nu)a^2 + b^2}\},
\tag{27}
$$

$$
\tau\_{\mathbf{zy}} = -\frac{p\_{\mathbf{v}}}{(1 + \nu)I\_{\mathbf{z}}} \frac{(1 + \nu)a^2 + \nu b^2}{3a^2 + b^2} \mathbf{x} \mathbf{z},
$$

where *Iz* is the moment of inertia of the beam cross section about the *z*-axis. By introducing the principal stresses that satisfy the equation *<sup>σ</sup>*<sup>3</sup> <sup>−</sup> *<sup>σ</sup>yσ*<sup>2</sup> <sup>−</sup> (*τ*<sup>2</sup> *xy* + *τ*<sup>2</sup> *zy*)*σ* = 0, one can find that

$$\begin{aligned} \sigma\_1 &= \frac{1}{2} [\sigma\_y - \sqrt{\sigma\_y^2 + 4(\tau\_{xy}^2 + \tau\_{zy}^2)}], \ \sigma\_2 = 0, \\\\ \sigma\_3 &= \frac{1}{2} [\sigma\_y + \sqrt{\sigma\_y^2 + 4(\tau\_{xy}^2 + \tau\_{zy}^2)}]. \end{aligned} \tag{28}$$

The tensile principal stress *<sup>σ</sup>*<sup>1</sup> reaches its maximum <sup>4</sup>|*Px*|*<sup>L</sup> <sup>π</sup>a*<sup>2</sup> *<sup>b</sup>* at the surface of the beam at one of the points (±*a*, 0, 0) (depending on the sign of load *Px*) and is acting along the *y*-axis - the axis of the beam. Based on equations (28) and the solution for the surface crack inclined to the surface of the half-space at angle of *π*/2 (see Hasebe and Inohara [11] and Isida [12]), one obtains *k*<sup>10</sup> = 4.484 <sup>√</sup>*<sup>π</sup>* |*Px*|*L <sup>a</sup>*2*<sup>b</sup>* . Therefore, bending fatigue life of a beam follows from substituting the expression for *k*<sup>10</sup> into equation (19)

$$N = \frac{2}{(n-2)g\_0} \{ \frac{0.395a^2b}{|P\_\mathrm{x}|L} \}^n g(\mu, \sigma),\tag{29}$$

where function *g*(*μ*, *σ*) is determined by equation (26).

In both cases of torsion and bending, fatigue life is independent of the elastic characteristic of the beam material (see formulas (25), (29), and (26)), and it is dependent on fatigue parameters of the beam material (*n* and *g*0), the initial defect distribution (i.e. on *μ* and *σ*), the geometry of the beam cross section (*a* and *b*), and its length *L* and the applied loading (*Px* or *My*).

In a similar fashion the model can be applied to contact fatigue if the stress field is known. A more detailed analysis of contact fatigue is presented below for a two-dimensional case.

#### **3. Contact problem for an elastic half-plane weakened by straight cracks**

A general theory of a stress state in an elastic plane with multiple cracks was proposed in [13]. In this section this theory is extended to the case of an elastic half-plane loaded by contact and residual stresses [1]. A study of lubricant-surface crack interaction, a discussion of the difference between contact fatigue lives of drivers and followers, the surface and subsurface initiated fatigue as well as fatigue of rough surfaces can be found in [1].

The main purpose of the section is to present formulations for the contact and fracture mechanics problems for an elastic half-plane weakened by subsurface cracks. The problems for surface cracks in an elastic lubricated half-plane are formulated and analyzed in [1]. The problems are reduced to systems of integro-differential equations with nonlinear boundary conditions in the form of alternating equations and inequalities. An asymptotic (perturbation) method for the case of small cracks is applied to solution of the problem and some numerical examples for small cracks are presented.

12 Will-be-set-by-IN-TECH

2(1+*ν*)*a*<sup>2</sup>+*b*<sup>2</sup>

where *Iz* is the moment of inertia of the beam cross section about the *z*-axis. By introducing

(1+*ν*)*Iz*

 *σ*2 *<sup>y</sup>* + 4(*τ*<sup>2</sup>

of the points (±*a*, 0, 0) (depending on the sign of load *Px*) and is acting along the *y*-axis - the axis of the beam. Based on equations (28) and the solution for the surface crack inclined to the surface of the half-space at angle of *π*/2 (see Hasebe and Inohara [11] and Isida [12]), one

{ 0.395*a*2*<sup>b</sup>*

In both cases of torsion and bending, fatigue life is independent of the elastic characteristic of the beam material (see formulas (25), (29), and (26)), and it is dependent on fatigue parameters of the beam material (*n* and *g*0), the initial defect distribution (i.e. on *μ* and *σ*), the geometry of the beam cross section (*a* and *b*), and its length *L* and the applied loading (*Px* or *My*).

In a similar fashion the model can be applied to contact fatigue if the stress field is known. A more detailed analysis of contact fatigue is presented below for a two-dimensional case.

**3. Contact problem for an elastic half-plane weakened by straight cracks**

A general theory of a stress state in an elastic plane with multiple cracks was proposed in [13]. In this section this theory is extended to the case of an elastic half-plane loaded by contact and residual stresses [1]. A study of lubricant-surface crack interaction, a discussion of the difference between contact fatigue lives of drivers and followers, the surface and subsurface

The main purpose of the section is to present formulations for the contact and fracture mechanics problems for an elastic half-plane weakened by subsurface cracks. The problems for surface cracks in an elastic lubricated half-plane are formulated and analyzed in [1]. The

*Iz x*(*L* − *y*),

<sup>3</sup>*a*<sup>2</sup>+*b*<sup>2</sup> {*a*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> <sup>−</sup> (1−2*ν*)*a*2*z*<sup>2</sup>

(1+*ν*)*a*<sup>2</sup>+*νb*<sup>2</sup> <sup>3</sup>*a*<sup>2</sup>+*b*<sup>2</sup> *xz*,

*xy* + *τ*<sup>2</sup>

*xy* + *τ*<sup>2</sup> *zy*)].

*<sup>a</sup>*2*<sup>b</sup>* . Therefore, bending fatigue life of a beam follows from substituting

<sup>2</sup>(1+*ν*)*a*<sup>2</sup>+*b*<sup>2</sup> },

*zy*)*σ* = 0, one can find that

*<sup>π</sup>a*<sup>2</sup> *<sup>b</sup>* at the surface of the beam at one

<sup>|</sup>*Px*|*<sup>L</sup>* }*ng*(*μ*, *<sup>σ</sup>*), (29)

*xy* + *τ*<sup>2</sup>

*zy*)], *σ*<sup>2</sup> = 0,

(27)

(28)

*<sup>σ</sup><sup>x</sup>* <sup>=</sup> *<sup>σ</sup><sup>z</sup>* <sup>=</sup> 0, *<sup>σ</sup><sup>y</sup>* <sup>=</sup> <sup>−</sup> *Px*

2(1+*ν*)*Iz*

*<sup>τ</sup>zy* <sup>=</sup> <sup>−</sup> *Px*

 *σ*2 *<sup>y</sup>* + 4(*τ*<sup>2</sup>

<sup>2</sup> [*σ<sup>y</sup>* +

of the beam is fixed. Then it can be shown (see Lurye [10]) that

the principal stresses that satisfy the equation *<sup>σ</sup>*<sup>3</sup> <sup>−</sup> *<sup>σ</sup>yσ*<sup>2</sup> <sup>−</sup> (*τ*<sup>2</sup>

<sup>2</sup> [*σ<sup>y</sup>* −

*N* = <sup>2</sup> (*n*−2)*g*<sup>0</sup>

initiated fatigue as well as fatigue of rough surfaces can be found in [1].

*σ*<sup>3</sup> = <sup>1</sup>

*σ*<sup>1</sup> = <sup>1</sup>

The tensile principal stress *<sup>σ</sup>*<sup>1</sup> reaches its maximum <sup>4</sup>|*Px*|*<sup>L</sup>*

where function *g*(*μ*, *σ*) is determined by equation (26).

obtains *k*<sup>10</sup> = 4.484

<sup>√</sup>*<sup>π</sup>*


the expression for *k*<sup>10</sup> into equation (19)

*τxz* = 0, *τxy* = *Px*

Let us introduce a global coordinate system with the *x*0-axis directed along the half-plane boundary and the *y*0-axis perpendicular to the half-plane boundary and pointed in the direction outside the material. The half-plane occupies the area of *<sup>y</sup>*<sup>0</sup> <sup>≤</sup> 0. Let us consider a contact problem for a rigid indenter with the bottom of shape *y*<sup>0</sup> = *f*(*x*0) pressed into the elastic half-plane (see Fig. 2). The elastic half-plane with effective elastic modulus *<sup>E</sup>*� (*E*� <sup>=</sup> *<sup>E</sup>*/(<sup>1</sup> <sup>−</sup> *<sup>ν</sup>*2), *<sup>E</sup>* and *<sup>ν</sup>* are the half-plane Young's modulus and Poisson's ratio) is weakened by *N* straight cracks. The crack faces are frictionless. Besides the global coordinate system we will introduce local orthogonal coordinate systems for each straight crack of half-length *lk* in such a way that their origins are located at the crack centers with complex coordinates *z*<sup>0</sup> *<sup>k</sup>* <sup>=</sup> *<sup>x</sup>*<sup>0</sup> *<sup>k</sup>* <sup>+</sup> *iy*<sup>0</sup> *<sup>k</sup>*, *k* = 1, . . . , *N*, the *xk*-axes are directed along the crack faces and the *yk*-axes are directed perpendicular to them. The cracks are inclined to the positive direction of the *x*0-axis at the angles *αk*, *k* = 1, . . . , *N*. All cracks are considered to be subsurface. The faces of every crack may be in partial or full contact with each other. The indenter is loaded by a normal force *P* and may be in direct contact with the half-plane or separated from it by a layer of lubricant. The indenter creates a pressure *p*(*x*0) and frictional stress *τ*(*x*0) distributions. The frictional stress *τ*(*x*0) between the indenter and the boundary of the half-plane is determined by the contact pressure *p*(*x*0) through a certain relationship. The cases of dry and fluid frictional stress *τ*(*x*0) are considered in [1]. At infinity the half-plane is loaded by a tensile or compressive (residual) stress *σ*<sup>∞</sup> *<sup>x</sup>*<sup>0</sup> <sup>=</sup> *<sup>q</sup>*<sup>0</sup> which is directed along the *x*0-axis. In this formulation the problem is considered in [1].

Then the problem is reduced to determining of the cracks behavior. Therefore, in dimensionless variables

$$\begin{array}{c} (\mathbf{x}\_{n}^{0'}, y\_{n}^{0'}) = (\mathbf{x}\_{n}^{0}, y\_{n}^{0}) / \widetilde{b}, \ (p\_{n}^{0'}, \mathbf{r}\_{n}^{0'}, p\_{n}') = (p\_{n}^{0}, \mathbf{r}\_{n}^{0}, p\_{n}) / \widetilde{q}, \\\ (\mathbf{x}\_{n}^{\prime}, t^{\prime}) = (\mathbf{x}\_{n}, t) / l\_{n\nu} \ (v\_{n}^{\prime}, u\_{n}^{\prime}) = (v\_{n}, u\_{n}) / \widetilde{v\_{n}}, \ \widetilde{v\_{n}} = \frac{4\widetilde{q}l\_{n}}{E}, \\\ (k\_{1\nu}^{\pm \prime}, k\_{2\nu}^{\pm \prime}) = (k\_{1\nu}^{\pm}, k\_{2\nu}^{\pm}) / (\widetilde{q}\sqrt{l\_{n}}) \end{array} \tag{30}$$

the equations of the latter problem for an elastic half-plane weakened by cracks and loaded by contact and residual stresses have the following form [1]

$$\begin{split} \int\_{-1}^{1} \frac{\nu\_{k}'(t)dt}{t - \mathbf{x}\_{k}} &+ \sum\_{m=1}^{N} \delta\_{m} \int [\nu\_{m}'(t)A\_{km}'(t, \mathbf{x}\_{k}) - u\_{m}'(t)B\_{km}'(t, \mathbf{x}\_{k})]dt \\ &= \pi \rho\_{mk}(\mathbf{x}\_{k}) + \pi \rho\_{k}^{0}(\mathbf{x}\_{k}), \; \nu\_{k}(\pm 1) = 0, \\ \int \frac{u\_{k}'(t)dt}{t - \mathbf{x}\_{k}} &+ \sum\_{m=1}^{N} \delta\_{m} \int [\nu\_{m}'(t)A\_{km}'(t, \mathbf{x}\_{k}) - u\_{m}'(t)B\_{km}^{0}(t, \mathbf{x}\_{k})]dt \\ &= \pi \pi\_{k}^{0}(\mathbf{x}\_{k}), \; u\_{k}(\pm 1) = 0, \\ \end{split} \tag{31}$$

$$p\_{k}^{0} - i\pi\_{k}^{0} = -\frac{1}{\pi} \int [p(t)\overline{D}\_{k}(t, \mathbf{x}\_{k}) + \tau(t)\overline{\mathbf{G}}\_{k}(t, \mathbf{x}\_{k})]dt - \frac{1}{2}q^{0}(1 - e^{-2i\boldsymbol{\alpha}\_{k}}), \tag{32}$$

$$p\_{nk}(\mathbf{x}\_k) = 0, \ v\_k(\mathbf{x}\_k) > 0; \ p\_{nk}(\mathbf{x}\_k) \le 0, \ v\_k(\mathbf{x}\_k) = 0, \ k = 1, \dots, N\_\prime$$

**Figure 2.** The general view of a rigid indenter in contact with a cracked elastic half-plane. where the kernels in these equations are described by formulas

*Akm* = *Rkm* + *Skm*, *Bkm* = −*i*(*Rkm* − *Skm*), (*A<sup>r</sup> km*, *<sup>B</sup><sup>r</sup> km*, *<sup>D</sup><sup>r</sup> <sup>k</sup>*, *<sup>G</sup><sup>r</sup> <sup>k</sup>*) = *Re*(*Akm*, *Bkm*, *Dk*, *Gk*), (*A<sup>i</sup> km*, *<sup>B</sup><sup>i</sup> km*, *<sup>D</sup><sup>i</sup> <sup>k</sup>*, *<sup>G</sup><sup>i</sup> <sup>k</sup>*) = *Im*(*Akm*, *Bkm*, *Dk*, *Gk*), *Dk*(*t*, *xk*) = *<sup>i</sup>* 2 <sup>−</sup> <sup>1</sup> *<sup>t</sup>*−*Xk* <sup>+</sup> <sup>1</sup> *t*−*Xk* <sup>−</sup> *<sup>e</sup>*−2*iα<sup>k</sup>* (*Xk*−*Xk*) (*t*−*Xk* )<sup>2</sup> , *Gk*(*t*, *xk*) = <sup>1</sup> 2 <sup>1</sup> *<sup>t</sup>*−*Xk* <sup>+</sup> <sup>1</sup>−*e*−2*iα<sup>k</sup> t*−*Xk* <sup>−</sup> *<sup>e</sup>*−2*iα<sup>k</sup>* (*t*−*Xk*) (*t*−*Xk* )<sup>2</sup> , *Rnk*(*t*, *xn*)=(<sup>1</sup> <sup>−</sup> *<sup>δ</sup>nk*)*Knk*(*t*, *xn*) + *<sup>e</sup>iα<sup>k</sup>* 2 <sup>1</sup> *Xn*−*Tk* + *<sup>e</sup>*−2*iα<sup>n</sup> Xn*−*Tk* +(*Tk* − *Tk*) <sup>1</sup>+*e*−2*iα<sup>n</sup>* (*Xn*−*Tk* )<sup>2</sup> <sup>+</sup> <sup>2</sup>*e*−2*iα<sup>n</sup>* (*Tk*−*Xn*) (*Xn*−*Tk* )<sup>3</sup> , *Snk*(*t*, *xn*)=(<sup>1</sup> <sup>−</sup> *<sup>δ</sup>nk*)*Lnk*(*t*, *xn*) + *<sup>e</sup>*−*iα<sup>k</sup>* 2 *Tk*−*Tk* (*Xn*−*Tk* )<sup>2</sup> <sup>+</sup> <sup>1</sup> *Xn*−*Tk* <sup>+</sup> *<sup>e</sup>*−2*iα<sup>n</sup>* (*Tk*−*Xn*) (*Xn*−*Tk* )<sup>2</sup> , *Knk*(*tk*, *xn*) = *<sup>e</sup>iα<sup>k</sup>* 2 <sup>1</sup> *Tk*−*Xn* <sup>+</sup> *<sup>e</sup>*−2*iα<sup>n</sup> Tk*−*Xn* , (33)

$$L\_{nk}(t\_{k\prime}\mathbf{x}\_{\eta}) = \frac{e^{-i\alpha\_{k}}}{2} \left[ \frac{1}{\overline{T}\_{k} - \overline{X}\_{\eta}} - \frac{T\_{k} - X\_{\eta}}{(\overline{T}\_{k} - \overline{X}\_{\eta})^{2}} e^{-2i\alpha\_{n}} \right],$$

$$T\_{k} = t e^{i\alpha\_{k}} + z\_{k\prime}^{0} \ \mathbf{X}\_{\eta} = \mathbf{x}\_{\eta} e^{i\alpha\_{\eta}} + z\_{\eta\prime}^{0} \ \mathbf{k}, \eta = \mathbf{1}, \ldots, N,$$

where *vk*(*xk*), *uk*(*xk*), and *pnk*(*xk*), *k* = 1, . . . , *N*, are the jumps of the normal and tangential crack face displacements and the normal stress applied to crack faces, respectively, *a* and *b* are the dimensionless contact boundaries, *<sup>δ</sup><sup>k</sup>* is the dimensionless crack half-length, *<sup>δ</sup><sup>k</sup>* = *lk*/˜ *b*, *δnk* is the Kronecker tensor (*δnk* <sup>=</sup> 0 for *<sup>n</sup>* �<sup>=</sup> *<sup>k</sup>*, *<sup>δ</sup>nk* <sup>=</sup> 1 for *<sup>n</sup>* <sup>=</sup> *<sup>k</sup>*), *<sup>i</sup>* is the imaginary unit, *<sup>i</sup>* <sup>=</sup> √−1.

For simplicity primes at the dimensionless variables are omitted. The characteristic values *q*˜ and ˜ *b* that are used for scaling are the maximum Hertzian pressure *pH* and the Hertzian contact half-width *aH*

$$p\_H = \sqrt{\frac{E'P}{\pi R}},\ a\_H = 2\sqrt{\frac{RP}{\pi E'}}\tag{34}$$

where *R* can be taken as the indenter curvature radius at the center of its bottom.

To simplify the problem formulation it is assumed that for small subsurface cracks (i.e. for *δ*<sup>0</sup> � 1, *δ*<sup>0</sup> = max 1≤*k*≤*N δk*) the pressure *p*(*x*0) and frictional stress *τ*(*x*0) are known and are close to the ones in a contact of this indenter with an elastic half-plane without cracks. It is worth mentioning that cracks affect the contact boundaries *a* and *b* and the pressure distribution *<sup>p</sup>*(*x*0) as well as each other starting with the terms of the order of *<sup>δ</sup>*<sup>0</sup> � 1.

Therefore, for the given shape of the indenter *f*(*x*0), pressure *p*(*x*0), frictional stress functions *τ*(*x*0), residual stress *q*0, crack orientation angles *α<sup>k</sup>* and sizes *δk*, and the crack positions *z*<sup>0</sup> *k* , *k* = 1, . . . , *N*, the solution of the problem is represented by crack faces displacement jumps *uk*(*xk*), *vk*(*xk*), and the normal contact stress *pnk*(*xk*) applied to the crack faces (*k* = 1, . . . , *N*). After the solution of the problem has been obtained, the dimensionless stress intensity factors *k*± <sup>1</sup>*<sup>k</sup>* and *k*<sup>±</sup> <sup>2</sup>*<sup>k</sup>* are determined according to formulas

$$ik\_{1n}^{\pm} + ik\_{2n}^{\pm} = \mp \lim\_{\mathbf{x}\_n \to \pm 1} \sqrt{1 - \mathbf{x}\_n^2} [v\_n'(\mathbf{x}\_n) + iu\_n'(\mathbf{x}\_n)], \ 0 \le n \le N. \tag{35}$$

#### **3.1. Problem solution**

14 Will-be-set-by-IN-TECH

*y*

*P*

*yn*

**Figure 2.** The general view of a rigid indenter in contact with a cracked elastic half-plane.

*Akm* = *Rkm* + *Skm*, *Bkm* = −*i*(*Rkm* − *Skm*),

*<sup>t</sup>*−*Xk* <sup>+</sup> <sup>1</sup> *t*−*Xk*

*<sup>t</sup>*−*Xk* <sup>+</sup> <sup>1</sup>−*e*−2*iα<sup>k</sup> t*−*Xk*

where the kernels in these equations are described by formulas

*km*, *<sup>D</sup><sup>r</sup> <sup>k</sup>*, *<sup>G</sup><sup>r</sup>*

*km*, *<sup>D</sup><sup>i</sup> <sup>k</sup>*, *<sup>G</sup><sup>i</sup>*

> 2 <sup>−</sup> <sup>1</sup>

2 <sup>1</sup>

*Rnk*(*t*, *xn*)=(<sup>1</sup> <sup>−</sup> *<sup>δ</sup>nk*)*Knk*(*t*, *xn*) + *<sup>e</sup>iα<sup>k</sup>*

*Snk*(*t*, *xn*)=(<sup>1</sup> <sup>−</sup> *<sup>δ</sup>nk*)*Lnk*(*t*, *xn*) + *<sup>e</sup>*−*iα<sup>k</sup>*

<sup>1</sup>+*e*−2*iα<sup>n</sup>*

, *Knk*(*tk*, *xn*) = *<sup>e</sup>iα<sup>k</sup>*

(*A<sup>r</sup> km*, *<sup>B</sup><sup>r</sup>*

(*A<sup>i</sup> km*, *<sup>B</sup><sup>i</sup>*

*Dk*(*t*, *xk*) = *<sup>i</sup>*

*Gk*(*t*, *xk*) = <sup>1</sup>

+(*Tk* − *Tk*)

<sup>+</sup> *<sup>e</sup>*−2*iα<sup>n</sup>* (*Tk*−*Xn*) (*Xn*−*Tk* )<sup>2</sup>

*xi xe*

*qo*

*y=f(x)*

*xn n*

*<sup>k</sup>*) = *Re*(*Akm*, *Bkm*, *Dk*, *Gk*),

*<sup>k</sup>*) = *Im*(*Akm*, *Bkm*, *Dk*, *Gk*),

2 <sup>1</sup> *Xn*−*Tk*

(*Xn*−*Tk* )<sup>3</sup>

2

2 <sup>1</sup>

(*Xn*−*Tk* )<sup>2</sup> <sup>+</sup> <sup>2</sup>*e*−2*iα<sup>n</sup>* (*Tk*−*Xn*)

<sup>−</sup> *<sup>e</sup>*−2*iα<sup>k</sup>* (*Xk*−*Xk*) (*t*−*Xk* )<sup>2</sup>

<sup>−</sup> *<sup>e</sup>*−2*iα<sup>k</sup>* (*t*−*Xk*) (*t*−*Xk* )<sup>2</sup>

*Tk*−*Tk*

 ,

 ,

+ *<sup>e</sup>*−2*iα<sup>n</sup> Xn*−*Tk* (33)

*Xn*−*Tk*

 ,

(*Xn*−*Tk* )<sup>2</sup> <sup>+</sup> <sup>1</sup>

*Tk*−*Xn* ,

*Tk*−*Xn* <sup>+</sup> *<sup>e</sup>*−2*iα<sup>n</sup>*

*\*n +*

*F*

*qo*

*x*

Solution of this problem is associated with formidable difficulties represented by the nonlinearities caused by the presence of the free boundaries of the crack contact intervals and the interaction between different cracks. Under the general conditions solution of this problem can be done only numerically. However, the problem can be effectively solved with the use of just analytical methods in the case when all cracks are small in comparison with the characteristic size ˜ *b* of the contact region, i.e., when *δ*<sup>0</sup> = max 1≤*k*≤*N δ<sup>k</sup>* � 1. In this case, it can be shown that the influence of the presence of cracks on the contact pressure is of the order of *O*(*δ*0) and with the precision of *O*(*δ*0) the crack system in the half-plane is subjected to the action of the contact pressure *p*0(*x*0) and frictional stress *τ*0(*x*0) that are obtained in the absence of cracks. The further simplification of the problem is achieved under the assumption that cracks are small in comparison to the distances between them, i.e.

$$z\_n^0 - z\_k^0 \gg \delta\_0, \ n \ne k, \ n, k = 1, \dots, N. \tag{36}$$

The latter assumption with the precision of *O*(*δ*<sup>2</sup> <sup>0</sup> ), *δ*<sup>0</sup> � 1, provides the conditions for considering each crack as a single crack in an elastic half-plane while the crack faces are loaded by certain stresses related to the contact pressure *p*0(*x*0), contact frictional stress *τ*0(*x*0), and the residual stress *q*0. The crucial assumption for simple and effective analytical solution of the considered problem is the assumption that all cracks are subsurface and much smaller in size than their distances to the half-plane surface

$$z\_k^0 - \overline{z}\_k^0 \gg \delta\_{0\prime} \text{ } k = 1, \dots, N. \tag{37}$$

Essentially, that assumption permits to consider each crack as a single crack in a plane (not a half-plane) with faces loaded by certain stresses related to *p*0(*x*0), *τ*0(*x*0), and *q*0.

Let us assume that the frictional stress *τ*0(*x*0) is determined by the Coulomb law of dry friction which in dimensionless variables can be represented by

$$
\pi\_0(\mathbf{x}^0) = -\lambda \, p\_0(\mathbf{x}^0),
\tag{38}
$$

where *λ* is the coefficient of friction. In (38) we assume that *λ* ≥ 0, and, therefore, the frictional stress is directed to the left. It is well known that for small friction coefficients *λ* the distribution of pressure is very close to the one in a Hertzian frictionless contact. Therefore, the expression for the pressure *p*0(*x*0) in the absence of cracks with high accuracy can be taken in the form

$$p\_0(\mathbf{x}^0) = \sqrt{1 - (\mathbf{x}^0)^2}.\tag{39}$$

Let us consider the process of solution of the pure fracture mechanics problem described by equations (31)-(33), (35), (38), (39). For small cracks, i.e. for *δ*<sup>0</sup> � 1, the kernels from (32) and (33 ) are regular functions of *t*, *xn*, and *xk* and they can be represented by power series in *δ<sup>k</sup>* � 1 and *δ<sup>n</sup>* � 1 as follows

$$\{A\_{km}(t, \mathbf{x}\_k), B\_{km}(t, \mathbf{x}\_k)\}$$

$$= \sum\_{j+n=0, j, n \ge 0}^{\infty} (\delta\_k \mathbf{x}\_k)^j (\delta\_m t)^n \{A\_{kmjn}, B\_{kmjn}\},$$

$$\{D\_k(t, \mathbf{x}\_k), \mathbf{G}\_k(t, \mathbf{x}\_k)\} = \sum\_{j=0}^{\infty} (\delta\_k \mathbf{x}\_k)^j \{D\_{kj}, \mathbf{G}\_{kj}\}.\tag{41}$$

In (40) and (41) the values of *Akmjn* and *Bkmjn* are independent of *δk*, *δm*, *xk*, and *t* while the values of *Dkj*(*t*) and *Gkj*(*t*) are independent of *δ<sup>k</sup>* and *xk*. The values of *Akmjn* and *Bkmjn* are certain functions of constants *αk*, *αm*, *x*<sup>0</sup> *<sup>k</sup>* , *<sup>y</sup>*<sup>0</sup> *<sup>k</sup>* , *<sup>x</sup>*<sup>0</sup> *<sup>m</sup>*, and *y*<sup>0</sup> *<sup>m</sup>* while the values of *Dkj*(*t*) and *Gkj*(*t*) are certain functions of *αk*, *x*<sup>0</sup> *<sup>k</sup>* , and *<sup>y</sup>*<sup>0</sup> *<sup>k</sup>*. Therefore, for *δ*<sup>0</sup> � 1 the problem solution can be sought in the form

$$\{\upsilon\_{k\prime}\mu\_k\} = \sum\_{j=0}^{\infty} \delta\_k^j \{\upsilon\_{kj}(\mathbf{x}\_k), \mu\_{kj}(\mathbf{x}\_k)\}, \ p\_{nk} = \sum\_{j=0}^{\infty} \delta\_0^j p\_{nkj}(\mathbf{x}\_k) \tag{42}$$

where functions *vkj*, *ukj*, *pnkj* have to be determined in the process of solution. Expanding the terms of the equations (31)-(33), and (35) and equating the terms with the same powers of *δ*<sup>0</sup> we get a system of boundary-value problems for integro-differential equations of the first kind which can be easily solved by classical methods [1, 14]. We will limit ourselves to determining only the first two terms of the expansions in (42) in the case of Coulomb's friction law given by (38) and (39). Without getting into the details of the solution process (which can be found in [1]) for the stress intensity factors *k*± <sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup> <sup>2</sup>*<sup>n</sup>* we obtain the following analytical formulas [1]

$$k\_1^{\pm} = c\_0^r \pm \frac{1}{2} \delta\_0 c\_1^r + \dots \text{ if } c\_0^r > 0, \ k\_1^{\pm} = 0 \text{ if } c\_0^r < 0,$$

$$k\_1^{\pm} = \frac{\sqrt{3} \delta\_0}{9} c\_1^r [\pm \Gamma - 3\theta(c\_1^r)] \sqrt{\frac{1 \pm \theta(c\_1^r)}{1 \pm 3\theta(c\_1^r)}} + \dots \text{ if } c\_0^r = 0 \text{ and } c\_1^r \neq 0,$$

$$k\_2^{\pm} = c\_0^i \pm \frac{1}{2} \delta\_0 c\_1^r + \dots,$$

$$c\_j = \frac{1}{\pi} \int\_{-1}^1 [p(\mathbf{x}) \overline{D}\_j(\mathbf{x}) + \tau(\mathbf{x}) \overline{G}\_j(\mathbf{x})] d\mathbf{x} + \frac{\delta\_0}{2} q^0 (1 - e^{-2i\mathbf{a}}), \ j = 1, 2,$$

$$\mathcal{c}\_{\dot{j}}^r = \text{Re}(\mathcal{c}\_{\dot{j}}) , \; \mathcal{c}\_{\dot{j}}^i = \text{Im}(\mathcal{c}\_{\dot{j}}) , \; \mathcal{c}\_{\dot{j}}^r$$

where the kernels are determined according to the formulas

16 Will-be-set-by-IN-TECH

considering each crack as a single crack in an elastic half-plane while the crack faces are loaded by certain stresses related to the contact pressure *p*0(*x*0), contact frictional stress *τ*0(*x*0), and the residual stress *q*0. The crucial assumption for simple and effective analytical solution of the considered problem is the assumption that all cracks are subsurface and much smaller in

Essentially, that assumption permits to consider each crack as a single crack in a plane (not a

Let us assume that the frictional stress *τ*0(*x*0) is determined by the Coulomb law of dry friction

where *λ* is the coefficient of friction. In (38) we assume that *λ* ≥ 0, and, therefore, the frictional stress is directed to the left. It is well known that for small friction coefficients *λ* the distribution of pressure is very close to the one in a Hertzian frictionless contact. Therefore, the expression for the pressure *p*0(*x*0) in the absence of cracks with high accuracy can be taken

Let us consider the process of solution of the pure fracture mechanics problem described by equations (31)-(33), (35), (38), (39). For small cracks, i.e. for *δ*<sup>0</sup> � 1, the kernels from (32) and (33 ) are regular functions of *t*, *xn*, and *xk* and they can be represented by power series in

{*Akm*(*t*, *xk*), *Bkm*(*t*, *xk*)}

In (40) and (41) the values of *Akmjn* and *Bkmjn* are independent of *δk*, *δm*, *xk*, and *t* while the values of *Dkj*(*t*) and *Gkj*(*t*) are independent of *δ<sup>k</sup>* and *xk*. The values of *Akmjn* and *Bkmjn* are

∑ *j*=0

*<sup>m</sup>*, and *y*<sup>0</sup>

*<sup>k</sup>*{*vkj*(*xk*), *ukj*(*xk*)}, *pnk* <sup>=</sup> <sup>∞</sup>

where functions *vkj*, *ukj*, *pnkj* have to be determined in the process of solution. Expanding the terms of the equations (31)-(33), and (35) and equating the terms with the same powers of *δ*<sup>0</sup> we get a system of boundary-value problems for integro-differential equations of the first kind

(*δmt*)*n*{*Akmjn*, *Bkmjn*},

(*δkxk*)*<sup>j</sup>*

(*δkxk* )*<sup>j</sup>*

*<sup>k</sup>* , *<sup>y</sup>*<sup>0</sup> *<sup>k</sup>* , *<sup>x</sup>*<sup>0</sup>

{*Dk*(*t*, *xk*), *Gk*(*t*, *xk*)} <sup>=</sup> <sup>∞</sup>

*<sup>k</sup>* , and *<sup>y</sup>*<sup>0</sup>

∑ *j*=0 *δ j*

half-plane) with faces loaded by certain stresses related to *p*0(*x*0), *τ*0(*x*0), and *q*0.

<sup>0</sup> ), *δ*<sup>0</sup> � 1, provides the conditions for

*<sup>k</sup>* � *δ*0, *k* = 1, . . . , *N*. (37)

*<sup>τ</sup>*0(*x*0) = <sup>−</sup>*λp*0(*x*0), (38)

*<sup>p</sup>*0(*x*0) = <sup>1</sup> <sup>−</sup> (*x*0)2. (39)

{*Dkj*, *Gkj*}. (41)

*<sup>m</sup>* while the values of *Dkj*(*t*) and *Gkj*(*t*)

<sup>0</sup> *pnkj*(*xk* ) (42)

*<sup>k</sup>*. Therefore, for *δ*<sup>0</sup> � 1 the problem solution can be

∑ *j*=0 *δ j* (40)

The latter assumption with the precision of *O*(*δ*<sup>2</sup>

size than their distances to the half-plane surface

which in dimensionless variables can be represented by

<sup>=</sup> <sup>∞</sup> ∑ *j*+*n*=0;*j*,*n*≥0

in the form

*δ<sup>k</sup>* � 1 and *δ<sup>n</sup>* � 1 as follows

certain functions of constants *αk*, *αm*, *x*<sup>0</sup>

{*vk*, *uk*} <sup>=</sup> <sup>∞</sup>

are certain functions of *αk*, *x*<sup>0</sup>

sought in the form

*z*0 *<sup>k</sup>* <sup>−</sup> *<sup>z</sup>*<sup>0</sup>

*D*0(*x*) = *<sup>i</sup>* 2 <sup>−</sup> <sup>1</sup> *<sup>x</sup>*−*z*<sup>0</sup> <sup>+</sup> <sup>1</sup> *<sup>x</sup>*−*z*<sup>0</sup> <sup>−</sup> *<sup>e</sup>*−2*i<sup>α</sup>*(*z*<sup>0</sup>−*z*<sup>0</sup>) (*x*−*z* <sup>0</sup> )<sup>2</sup> , *G*0(*x*) = <sup>1</sup> 2 <sup>1</sup> *x*−*z*<sup>0</sup> +1−*e*−2*i<sup>α</sup> <sup>x</sup>*−*z*<sup>0</sup> <sup>−</sup> *<sup>e</sup>*−2*i<sup>α</sup>*(*x*−*z*<sup>0</sup>) (*x*−*z* <sup>0</sup> )<sup>2</sup> , *<sup>D</sup>*1(*x*) = *ie*−*i<sup>α</sup>* 2(*x*−*z* <sup>0</sup> )<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−2*i<sup>α</sup>* −2*e*−2*i<sup>α</sup>*(*z*<sup>0</sup>−*z*<sup>0</sup>) *<sup>x</sup>*−*z*<sup>0</sup> + *iei<sup>α</sup>* 2 <sup>−</sup> <sup>1</sup> (*x*−*z*<sup>0</sup>)<sup>2</sup> <sup>+</sup> *<sup>e</sup>*−2*i<sup>α</sup>* (*x*−*z*<sup>0</sup> )<sup>2</sup> , *<sup>G</sup>*1(*x*) = *<sup>e</sup>*−*i<sup>α</sup>* <sup>2</sup>(*x*−*z*<sup>0</sup> )<sup>2</sup> 1 <sup>−</sup>*e*−2*i<sup>α</sup>* <sup>−</sup> <sup>2</sup>*e*−2*i<sup>α</sup>*(*x*−*z*<sup>0</sup>) *<sup>x</sup>*−*z*<sup>0</sup> + *<sup>e</sup>i<sup>α</sup>* 2 <sup>1</sup> (*x*−*z*<sup>0</sup>)<sup>2</sup> <sup>+</sup> *<sup>e</sup>*−2*i<sup>α</sup>* (*x*−*z* <sup>0</sup> )<sup>2</sup> , (44)

and *θ*(*x*) is the step function (*θ*(*x*) = −1 for *x* < 0 and *θ*(*x*) = 1 for *x* ≥ 0).

#### **3.2. Comparison of analytical asymptotic and numerical solutions for small subsurface cracks**

Let us compare the asymptotically (*k*± <sup>1</sup>*<sup>a</sup>* and *k*<sup>±</sup> 2*a*) and numerically (*k*<sup>±</sup> <sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup> <sup>2</sup>*n*) obtained solutions of the problem for the case when *<sup>y</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.4, *<sup>δ</sup>*<sup>0</sup> <sup>=</sup> 0.1, *<sup>α</sup>* <sup>=</sup> *<sup>π</sup>*/2, *<sup>λ</sup>* <sup>=</sup> 0.1, and *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.005. The numerical method used for calculating *<sup>k</sup>*<sup>±</sup> <sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup> <sup>2</sup>*<sup>n</sup>* is described in detail in [1]. Both the numerical and asymptotic solutions are represented in Fig. 3. It follows from Fig. 3 that the asymptotic and numerical solutions are almost identical except for the region where the numerically obtained *k*<sup>+</sup> <sup>1</sup> (*x*0) is close to zero. The difference is mostly caused by the fact that the used asymptotic solution involve only two terms, i.e., the accuracy of these asymptotic solutions is *O*(*δ*<sup>2</sup> <sup>0</sup> ) for small *δ*0. However, according to the two-term asymptotic solutions the maximum values of *k*± <sup>1</sup> differ from the numerical ones by no more than 1.4%. One can expect to get much higher precision if *<sup>δ</sup>*<sup>0</sup> <sup>&</sup>lt; 0.1 and <sup>|</sup> *<sup>y</sup>*<sup>0</sup> |� *<sup>δ</sup>*0.

Therefore, formulas (43) and (44) provide sufficient precision for most possible applications and can be used to substitute for numerically obtained values of *k*± <sup>1</sup> and *k*<sup>±</sup> 2 .

**Figure 3.** Comparison of the two-term asymptotic expansions *k*± <sup>1</sup>*<sup>a</sup>* and *k*<sup>±</sup> <sup>2</sup>*<sup>a</sup>* with the numerically calculated stress intensity factors *k*± <sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup> <sup>2</sup>*<sup>n</sup>* obtained for *<sup>y</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.4, *<sup>δ</sup>*<sup>0</sup> <sup>=</sup> 0.1, *<sup>α</sup>* <sup>=</sup> *<sup>π</sup>*/2, *<sup>λ</sup>* <sup>=</sup> 0.1, and *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.005. Solid curves are numerical results while dashed curves are asymptotical results. (*k*<sup>−</sup> <sup>1</sup>*<sup>n</sup>* group 1, *k*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* group 2, *k*<sup>−</sup> <sup>2</sup>*<sup>n</sup>* group 3, *<sup>k</sup>*<sup>+</sup> <sup>2</sup>*<sup>n</sup>* group 4) (after Kudish [15]). Reprinted with permission of the STLE.

#### **3.3. Stress intensity factors** *k*± <sup>1</sup>*<sup>n</sup>* **and** *k*<sup>±</sup> <sup>2</sup>*<sup>n</sup>* **behavior for subsurface cracks**

Some examples of the behavior of the stress intensity factors *k*± <sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup> <sup>2</sup>*<sup>n</sup>* for subsurface cracks are presented below.

It is important to keep in mind that for the cases of no friction (*λ* = 0) and compressive or zero residual stress (*q*<sup>0</sup> <sup>≤</sup> 0) all subsurface cracks are closed and, therefore, at their tips *<sup>k</sup>*<sup>±</sup> <sup>1</sup>*<sup>n</sup>* = 0.

Let us consider the case when the residual stress *q*<sup>0</sup> is different from zero. The residual stress influence on *k*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* results in increase of *<sup>k</sup>*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* for a tensile residual stress *<sup>q</sup>*<sup>0</sup> <sup>&</sup>gt; 0 or its decrease for a compressive residual stress *q*<sup>0</sup> < 0 of the material region with tensile stresses. From formulas (43), (44), and Fig. 4 (obtained for *y*<sup>0</sup> *<sup>n</sup>* = −0.2, *α<sup>n</sup>* = *π*/2, and *δ<sup>n</sup>* = 0.1) follows that for all *x*<sup>0</sup> *<sup>n</sup>* and for increasing residual stress *q*<sup>0</sup> (see the curves marked with 3 and 5 that correspond to *λ* = 0.1, *q*<sup>0</sup> = 0.04, and *λ* = 0.2, *q*<sup>0</sup> = 0.02, respectively) the stress intensity factor *k*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* is a non-decreasing function of *<sup>q</sup>*0. Moreover, if at some material point *<sup>k</sup>*<sup>+</sup> 1*n*(*q*<sup>0</sup> <sup>1</sup>) > 0 for some residual stress *q*<sup>0</sup> <sup>1</sup>, then *<sup>k</sup>*<sup>+</sup> 1*n*(*q*<sup>0</sup> <sup>2</sup>) <sup>&</sup>gt; *<sup>k</sup>*<sup>+</sup> 1*n*(*q*<sup>0</sup> <sup>1</sup>) for *<sup>q</sup>*<sup>0</sup> <sup>2</sup> <sup>&</sup>gt; *<sup>q</sup>*<sup>0</sup> <sup>1</sup> (compare curves marked with 1 and 2 with curves marked with 3 and 4 as well as with curves marked with 5 and 6, respectively). Similarly, for all *x*<sup>0</sup> *<sup>n</sup>* when the magnitude of the compressive residual stress (*q*<sup>0</sup> <sup>&</sup>lt; 0) increases (see curves marked with 4 and 6 that correspond to *<sup>λ</sup>* <sup>=</sup> 0.1, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.01 and

18 Will-be-set-by-IN-TECH

2

2 4 6 8 10

*<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.005. Solid curves are numerical results while dashed curves are asymptotical results. (*k*<sup>−</sup>

It is important to keep in mind that for the cases of no friction (*λ* = 0) and compressive or zero

Let us consider the case when the residual stress *q*<sup>0</sup> is different from zero. The residual stress

for a compressive residual stress *q*<sup>0</sup> < 0 of the material region with tensile stresses. From

correspond to *λ* = 0.1, *q*<sup>0</sup> = 0.04, and *λ* = 0.2, *q*<sup>0</sup> = 0.02, respectively) the stress intensity

with 1 and 2 with curves marked with 3 and 4 as well as with curves marked with 5 and

(*q*<sup>0</sup> <sup>&</sup>lt; 0) increases (see curves marked with 4 and 6 that correspond to *<sup>λ</sup>* <sup>=</sup> 0.1, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.01 and

1*n*(*q*<sup>0</sup>

<sup>1</sup>*<sup>n</sup>* is a non-decreasing function of *<sup>q</sup>*0. Moreover, if at some material point *<sup>k</sup>*<sup>+</sup>

<sup>2</sup>) <sup>&</sup>gt; *<sup>k</sup>*<sup>+</sup>

*<sup>n</sup>* and for increasing residual stress *q*<sup>0</sup> (see the curves marked with 3 and 5 that

<sup>1</sup>) for *<sup>q</sup>*<sup>0</sup>

<sup>2</sup> <sup>&</sup>gt; *<sup>q</sup>*<sup>0</sup>

*<sup>n</sup>* when the magnitude of the compressive residual stress

residual stress (*q*<sup>0</sup> <sup>≤</sup> 0) all subsurface cracks are closed and, therefore, at their tips *<sup>k</sup>*<sup>±</sup>

−4

<sup>2</sup>*<sup>a</sup>* with the numerically

<sup>1</sup>*<sup>n</sup>* group

<sup>1</sup>*<sup>n</sup>* = 0.

1*n*(*q*<sup>0</sup>

<sup>1</sup> (compare curves marked

<sup>1</sup>) > 0

<sup>2</sup>*<sup>n</sup>* for subsurface cracks

x o

<sup>2</sup>*<sup>n</sup>* obtained for *<sup>y</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.4, *<sup>δ</sup>*<sup>0</sup> <sup>=</sup> 0.1, *<sup>α</sup>* <sup>=</sup> *<sup>π</sup>*/2, *<sup>λ</sup>* <sup>=</sup> 0.1, and

<sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup>

<sup>1</sup>*<sup>n</sup>* for a tensile residual stress *<sup>q</sup>*<sup>0</sup> <sup>&</sup>gt; 0 or its decrease

*<sup>n</sup>* = −0.2, *α<sup>n</sup>* = *π*/2, and *δ<sup>n</sup>* = 0.1) follows

<sup>1</sup>*<sup>a</sup>* and *k*<sup>±</sup>

<sup>2</sup>*<sup>n</sup>* **behavior for subsurface cracks**

<sup>2</sup>*<sup>n</sup>* group 4) (after Kudish [15]). Reprinted with permission of the STLE.

1

−3

−2

−1

0

1 x 10−3

k2 ±

0

<sup>2</sup>*<sup>n</sup>* group 3, *<sup>k</sup>*<sup>+</sup>

calculated stress intensity factors *k*±

**3.3. Stress intensity factors** *k*±

1, *k*<sup>+</sup>

<sup>1</sup>*<sup>n</sup>* group 2, *k*<sup>−</sup>

are presented below.

influence on *k*<sup>+</sup>

that for all *x*<sup>0</sup>

for some residual stress *q*<sup>0</sup>

6, respectively). Similarly, for all *x*<sup>0</sup>

factor *k*<sup>+</sup>

**Figure 3.** Comparison of the two-term asymptotic expansions *k*±

Some examples of the behavior of the stress intensity factors *k*±

<sup>1</sup>, then *<sup>k</sup>*<sup>+</sup>

1*n*(*q*<sup>0</sup>

<sup>1</sup>*<sup>n</sup>* results in increase of *<sup>k</sup>*<sup>+</sup>

formulas (43), (44), and Fig. 4 (obtained for *y*<sup>0</sup>

<sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup>

<sup>1</sup>*<sup>n</sup>* **and** *k*<sup>±</sup>

0.5

1

1.5

2

2.5 x 10−3

k1 ±

4

3

**Figure 4.** The dependence of the normal stress intensity factor *k*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* on *<sup>x</sup>*<sup>0</sup> *<sup>n</sup>* for *δ<sup>n</sup>* = 0.1, *α* = *π*/2, *y*0 *<sup>n</sup>* <sup>=</sup> <sup>−</sup>0.2 and different levels of the residual stress *<sup>q</sup>*0. Curves 1 and 2 are obtained for *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0 and *<sup>λ</sup>* <sup>=</sup> 0.1 and *<sup>λ</sup>* <sup>=</sup> 0.2, respectively. Curves 3 and 4 are obtained for *<sup>λ</sup>* <sup>=</sup> 0.1, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0.04 and *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.01, respectively, while curves 5 and 6 are obtained for *<sup>λ</sup>* <sup>=</sup> 0.2, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0.01 and *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.03, respectively (after Kudish [5]). Reprinted with permission of Springer.

*<sup>λ</sup>* <sup>=</sup> 0.2, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.03, respectively) and at some material point *<sup>k</sup>*<sup>+</sup> 1*n*(*q*<sup>0</sup> <sup>1</sup>) > 0 for some residual stress *q*<sup>0</sup> <sup>1</sup> then *<sup>k</sup>*<sup>+</sup> 1*n*(*q*<sup>0</sup> <sup>2</sup>) <sup>&</sup>lt; *<sup>k</sup>*<sup>+</sup> 1*n*(*q*<sup>0</sup> <sup>1</sup>) for *<sup>q</sup>*<sup>0</sup> <sup>2</sup> <sup>&</sup>lt; *<sup>q</sup>*<sup>0</sup> <sup>1</sup>. Based on the fact that the normal stress intensity factor *k*0<sup>+</sup> <sup>1</sup>*<sup>n</sup>* is positive only in the near surface material layer, we can make a conclusion that this layer increases in size and, starting with a certain value of tensile residual stress *q*<sup>0</sup> > 0, crack propagation becomes possible at any depth beneath the half-plane surface. For increasing compressive residual stresses, the thickness of the material layer where *k*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* > 0 decreases.

The analysis of the results for subsurface cracks following from formulas (43) and (44) shows [1] that the values of the stress intensity factors *k*± <sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup> <sup>2</sup>*<sup>n</sup>* are insensitive to even relatively large variations in the behavior of the distributions of the pressure *p*(*x*0) and frictional stress *τ*(*x*0). In particular, the stress intensity factors *k*<sup>±</sup> <sup>1</sup>*<sup>n</sup>* and *k*<sup>±</sup> <sup>2</sup>*<sup>n</sup>* for the cases of dry and fluid (lubricant) friction as well as for the cases of constant pressure and frictional stress are very close to each other as long as the normal force (integral of *p*(*x*0) over the contact region) and the friction force (integral of *τ* over the contact region) applied to the surface of the half-plane are the same [1].

Qualitatively, the behavior of the normal stress intensity factors *k*± <sup>1</sup>*<sup>n</sup>* for different angles of orientation *αn* is very similar while quantitatively it is very different. An example of that is presented for a horizontal (*α<sup>n</sup>* = 0) subsurface crack in Fig. 5 and for a subsurface crack perpendicular to the half-plane boundary (*α<sup>n</sup>* = *π*/2) in Fig 6 for *p*(*x*0) = *π*/4, *τ*(*x*0) = <sup>−</sup>*λp*(*x*0), *<sup>y</sup>*<sup>0</sup> *<sup>n</sup>* <sup>=</sup> <sup>−</sup>0.2, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0 for *<sup>λ</sup>* <sup>=</sup> 0.1 and *<sup>λ</sup>* <sup>=</sup> 0.2.

**Figure 5.** The dependence of the normal stress intensity factor *k*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* on the coordinate *<sup>x</sup>*<sup>0</sup> *<sup>n</sup>* for the case of the boundary of half-plane loaded with normal *<sup>p</sup>*(*x*0) = *<sup>π</sup>*/4 and frictional *<sup>τ</sup>*(*x*0) = <sup>−</sup>*λp*(*x*0) stresses, *y*0 *<sup>n</sup>* <sup>=</sup> <sup>−</sup>0.2, *<sup>α</sup><sup>n</sup>* <sup>=</sup> 0, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0: *<sup>λ</sup>* <sup>=</sup> 0.1 - curve marked with 1, *<sup>λ</sup>* <sup>=</sup> 0.2 - curve marked with 2 (after Kudish and Covitch [1]). Reprinted with permission from CRC Press.

For the same loading and crack parameters the behavior of the shear stress intensity factor *k*± 2*n* is represented in Fig. 7 and 8. It is important to observe that the shear stress intensity factors *k*± <sup>2</sup>*<sup>n</sup>* are insensitive to changes of the friction coefficient *λ*.

Fig. 5 and 6 clearly show that for subsurface cracks with angle *α<sup>n</sup>* = *π*/2 for zero or tensile residual stress *q*<sup>0</sup> the normal stress intensity factors *k*<sup>±</sup> <sup>1</sup>*<sup>n</sup>* are significantly higher (by two orders of magnitude) than the ones for *α<sup>n</sup>* = 0. For *α<sup>n</sup>* = 0 and *α<sup>n</sup>* = *π*/2 the orders of magnitude of the shear stress intensity factors *k*± <sup>2</sup>*<sup>n</sup>* are the same (see Fig. 7 and 8). Moreover, from these graphs it is clear that the normal stress intensity factors *k*± <sup>1</sup>*<sup>n</sup>* are significantly influenced by the friction coefficient *λ* while the shear stress intensity factors *k*± <sup>2</sup>*<sup>n</sup>* are insensitive to the value of the friction coefficient *λ*.

Obviously, in the single-term approximation the behavior of *k*0<sup>−</sup> <sup>1</sup>*<sup>n</sup>* and *<sup>k</sup>*0<sup>−</sup> <sup>2</sup>*<sup>n</sup>* is identical to the one of *k*0<sup>+</sup> <sup>1</sup>*<sup>n</sup>* and *<sup>k</sup>*0<sup>+</sup> <sup>2</sup>*<sup>n</sup>* , respectively. Generally, the difference between *<sup>k</sup>*0<sup>−</sup> <sup>1</sup>*<sup>n</sup>* and *<sup>k</sup>*0<sup>+</sup> <sup>1</sup>*<sup>n</sup>* as well as between *k*0<sup>−</sup> <sup>2</sup>*<sup>n</sup>* and *<sup>k</sup>*0<sup>+</sup> <sup>2</sup>*<sup>n</sup>* is of the order of magnitude of *δ*<sup>0</sup> � 1.

#### **3.4. Lubricant-surface crack interaction. Stress intensity factors** *k*± <sup>1</sup>*<sup>n</sup>* **and** *k*<sup>±</sup> 2*n* **Behavior**

The process of lubricant-surface crack interaction is very complex and the details of the problem formulation, the numerical solution approach, and a comprehensive analysis of the results can be found in [1]. Therefore, here we will discuss only the most important features of this phenomenon. It is well known that the presence of lubricant between surfaces in contact

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0 2.5 5 7.5

boundary of half-plane loaded with normal *<sup>p</sup>*(*x*0) = *<sup>π</sup>*/4 and frictional *<sup>τ</sup>*(*x*0) = <sup>−</sup>*λp*(*x*0) stresses,

*<sup>n</sup>* <sup>=</sup> <sup>−</sup>0.2, *<sup>α</sup><sup>n</sup>* <sup>=</sup> 0, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0: *<sup>λ</sup>* <sup>=</sup> 0.1 - curve marked with 1, *<sup>λ</sup>* <sup>=</sup> 0.2 - curve marked with 2 (after Kudish

For the same loading and crack parameters the behavior of the shear stress intensity factor *k*±

is represented in Fig. 7 and 8. It is important to observe that the shear stress intensity factors

Fig. 5 and 6 clearly show that for subsurface cracks with angle *α<sup>n</sup>* = *π*/2 for zero or tensile

of magnitude) than the ones for *α<sup>n</sup>* = 0. For *α<sup>n</sup>* = 0 and *α<sup>n</sup>* = *π*/2 the orders of magnitude

<sup>2</sup>*<sup>n</sup>* , respectively. Generally, the difference between *<sup>k</sup>*0<sup>−</sup>

The process of lubricant-surface crack interaction is very complex and the details of the problem formulation, the numerical solution approach, and a comprehensive analysis of the results can be found in [1]. Therefore, here we will discuss only the most important features of this phenomenon. It is well known that the presence of lubricant between surfaces in contact

<sup>2</sup>*<sup>n</sup>* is of the order of magnitude of *δ*<sup>0</sup> � 1.

**3.4. Lubricant-surface crack interaction. Stress intensity factors** *k*±

2

1

x n o

<sup>1</sup>*<sup>n</sup>* on the coordinate *<sup>x</sup>*<sup>0</sup>

<sup>2</sup>*<sup>n</sup>* are the same (see Fig. 7 and 8). Moreover, from these

<sup>1</sup>*<sup>n</sup>* and *<sup>k</sup>*0<sup>−</sup>

<sup>1</sup>*<sup>n</sup>* are significantly higher (by two orders

<sup>1</sup>*<sup>n</sup>* are significantly influenced by the

<sup>2</sup>*<sup>n</sup>* are insensitive to the value of

<sup>1</sup>*<sup>n</sup>* and *<sup>k</sup>*0<sup>+</sup>

<sup>1</sup>*<sup>n</sup>* **and** *k*<sup>±</sup>

<sup>2</sup>*<sup>n</sup>* is identical to the

2*n*

<sup>1</sup>*<sup>n</sup>* as well as

*<sup>n</sup>* for the case of the

2*n*

0

*y*0

*k*±

**Figure 5.** The dependence of the normal stress intensity factor *k*<sup>+</sup>

and Covitch [1]). Reprinted with permission from CRC Press.

<sup>2</sup>*<sup>n</sup>* are insensitive to changes of the friction coefficient *λ*.

graphs it is clear that the normal stress intensity factors *k*±

friction coefficient *λ* while the shear stress intensity factors *k*±

Obviously, in the single-term approximation the behavior of *k*0<sup>−</sup>

residual stress *q*<sup>0</sup> the normal stress intensity factors *k*<sup>±</sup>

of the shear stress intensity factors *k*±

the friction coefficient *λ*.

<sup>1</sup>*<sup>n</sup>* and *<sup>k</sup>*0<sup>+</sup>

<sup>2</sup>*<sup>n</sup>* and *<sup>k</sup>*0<sup>+</sup>

one of *k*0<sup>+</sup>

**Behavior**

between *k*0<sup>−</sup>

2.5

5

k1n <sup>±</sup> <sup>⋅</sup>104

**Figure 6.** The dependence of the normal stress intensity factor *k*<sup>+</sup> <sup>1</sup>*<sup>n</sup>* on the coordinate *<sup>x</sup>*<sup>0</sup> *<sup>n</sup>* for the case of the boundary of half-plane loaded with normal *<sup>p</sup>*(*x*0) = *<sup>π</sup>*/4 and frictional *<sup>τ</sup>*(*x*0) = <sup>−</sup>*λp*(*x*0) stresses, *y*0 *<sup>n</sup>* <sup>=</sup> <sup>−</sup>0.2, *<sup>α</sup><sup>n</sup>* <sup>=</sup> *<sup>π</sup>*/2, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0: *<sup>λ</sup>* <sup>=</sup> 0.1 - curve marked with 1, *<sup>λ</sup>* <sup>=</sup> 0.2 - curve marked with 2 (after Kudish and Covitch [1]). Reprinted with permission from CRC Press.

is very beneficial as it reduces the contact friction and wear and facilitates better heat transfer from the contact. However, in some cases the lubricant presence may play a detrimental role. In particular, in cases when the elastic solid (half-plane) has a surface crack inclined toward the incoming high contact pressure transmitted through the lubricant. Such a crack may open up and experience high lubricant pressure applied to its faces. This pressure creates the normal stress intensity factor *k*− <sup>1</sup>*<sup>n</sup>* far exceeding the value of the stress intensity factors *k*± <sup>1</sup>*<sup>n</sup>* for comparable subsurface cracks while the shear stress intensity factors for surface *k*<sup>−</sup> 2*n* and and subsurface *k*± <sup>2</sup>*<sup>n</sup>* cracks remain comparable in value. That becomes obvious from the comparison of the graphs from Fig. 4-8 with the graphs from Fig. 10 and 9.

In cases when a surface crack is inclined away from the incoming high lubricant pressure the crack does not open up toward the incoming lubricant with high pressure and it behaves similar to a corresponding subsurface crack, i.e. its normal stress intensity factor *k*− <sup>1</sup>*<sup>n</sup>* in its value is similar to the one for a corresponding subsurface crack. It is customary to see the normal stress intensity factor for surface cracks which open up toward the incoming high lubricant pressure to exceed the one for comparable subsurface cracks by two orders of magnitude. In such cases the compressive residual stress has very little influence on the crack behavior due to domination of the lubricant pressure.

The possibility of high normal stress intensity factors for surface cracks leads to serious consequences. In particular, it explains why fatigue life of drivers is usually significantly higher than the one for followers [1]. Also, it explains why under normal circumstances fatigue failure is of subsurface origin [1].

**Figure 7.** The dependence of the shear stress intensity factor *k*<sup>+</sup> <sup>2</sup>*<sup>n</sup>* on the coordinate *<sup>x</sup>*<sup>0</sup> *<sup>n</sup>* for the case of the boundary of half-plane loaded with normal *<sup>p</sup>*(*x*0) = *<sup>π</sup>*/4 and frictional *<sup>τ</sup>*(*x*0) = <sup>−</sup>*λp*(*x*0) stresses, *y*0 *<sup>n</sup>* <sup>=</sup> <sup>−</sup>0.2, *<sup>α</sup><sup>n</sup>* <sup>=</sup> 0, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0: *<sup>λ</sup>* <sup>=</sup> 0.1 - curve marked with 1, *<sup>λ</sup>* <sup>=</sup> 0.2 - curve marked with 2 (after Kudish and Covitch [1]). Reprinted with permission from CRC Press.

#### **4. Stress intensity factors and directions of fatigue crack propagation**

The process of fatigue failure is usually subdivided into three major stages: the nucleation period, the period of slow pre-critical fatigue crack growth, and the short period of fast unstable crack growth ending in material loosing its integrity. The durations of the first two stages of fatigue failure depend on a number of parameters such as material properties, specific environment, stress state, temperature, etc. Usually, the nucleation period is short [1]. We are interested in the main part of the process of fatigue failure which is due to slow pre-critical crack growth. In these cases max (*k*<sup>+</sup> <sup>1</sup> , *k*<sup>−</sup> <sup>1</sup> ) < *Kf* , where *Kf* is the material fracture toughness.

During the pre-critical fatigue crack growth cracks remain small. Therefore, they can be modeled by small straight cuts in the material. For such small subsurface cracks it is sufficient to use the one-term asymptotic approximations *k*± <sup>1</sup> = *k*<sup>1</sup> and *k*<sup>±</sup> <sup>2</sup> = *k*<sup>2</sup> for the stress intensity factors from (43) and (44) which in dimensional variables take the form

$$k\_1 = \sqrt{7}[Y^r + q^0 \sin^2 a] \theta[Y^r + q^0 \sin^2 a], \ k\_2 = \sqrt{7}[Y^i - \frac{q^0}{2} \sin 2a],$$

$$Y = \frac{1}{\pi} \int\_{-a\_H}^{a\_H} [p(t)\overline{D}\_0(t) + \tau(t)\overline{G}\_0(t)]dt,\ \tau = -\lambda p,$$

$$\{Y^r, Y^i\} = \{\operatorname{Re}(Y), \operatorname{Im}(Y)\},\tag{45}$$

$$D\_0(t) = \frac{i}{2} \left[ -\frac{1}{t - X} + \frac{1}{t - X} - \frac{e^{-2i\epsilon}(\overline{X} - X)}{(t - \overline{X})^2} \right],$$

$$G\_0(t) = \frac{1}{2} \left[ \frac{1}{t - X} + \frac{1 - e^{-2i\epsilon}}{t - \overline{X}} - \frac{e^{-2i\epsilon}(t - X)}{(t - \overline{X})^2} \right],\ X = x + iy,$$

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boundary of half-plane loaded with normal *<sup>p</sup>*(*x*0) = *<sup>π</sup>*/4 and frictional *<sup>τ</sup>*(*x*0) = <sup>−</sup>*λp*(*x*0) stresses,

**4. Stress intensity factors and directions of fatigue crack propagation**

*<sup>n</sup>* <sup>=</sup> <sup>−</sup>0.2, *<sup>α</sup><sup>n</sup>* <sup>=</sup> 0, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0: *<sup>λ</sup>* <sup>=</sup> 0.1 - curve marked with 1, *<sup>λ</sup>* <sup>=</sup> 0.2 - curve marked with 2 (after Kudish

The process of fatigue failure is usually subdivided into three major stages: the nucleation period, the period of slow pre-critical fatigue crack growth, and the short period of fast unstable crack growth ending in material loosing its integrity. The durations of the first two stages of fatigue failure depend on a number of parameters such as material properties, specific environment, stress state, temperature, etc. Usually, the nucleation period is short [1]. We are interested in the main part of the process of fatigue failure which is due to slow

<sup>1</sup> , *k*<sup>−</sup>

<sup>1</sup> = *k*<sup>1</sup> and *k*<sup>±</sup>

*<sup>t</sup>*−*<sup>X</sup>* <sup>−</sup> *<sup>e</sup>*−2*i<sup>α</sup>*(*X*−*X*) (*t*−*X*)<sup>2</sup>

 ,

, *X* = *x* + *iy*,

During the pre-critical fatigue crack growth cracks remain small. Therefore, they can be modeled by small straight cuts in the material. For such small subsurface cracks it is sufficient

*<sup>l</sup>*[*Y<sup>r</sup>* <sup>+</sup> *<sup>q</sup>*<sup>0</sup> sin2 *<sup>α</sup>*]*θ*[*Y<sup>r</sup>* <sup>+</sup> *<sup>q</sup>*<sup>0</sup> sin2 *<sup>α</sup>*], *<sup>k</sup>*<sup>2</sup> <sup>=</sup> <sup>√</sup>*l*[*Y<sup>i</sup>* <sup>−</sup> *<sup>q</sup>*<sup>0</sup>

[*p*(*t*)*D*0(*t*) + *τ*(*t*)*G*0(*t*)]*dt*, *τ* = −*λp*,

} = {*Re*(*Y*), *Im*(*Y*)},

*<sup>t</sup>*−*<sup>X</sup>* <sup>−</sup> *<sup>e</sup>*−2*i<sup>α</sup>*(*t*−*X*) (*t*−*X*)<sup>2</sup>

x n o

<sup>2</sup>*<sup>n</sup>* on the coordinate *<sup>x</sup>*<sup>0</sup>

<sup>1</sup> ) < *Kf* , where *Kf* is the material fracture

<sup>2</sup> = *k*<sup>2</sup> for the stress intensity

(45)

<sup>2</sup> sin 2*α*],

*<sup>n</sup>* for the case of the

2

1

−4

**Figure 7.** The dependence of the shear stress intensity factor *k*<sup>+</sup>

and Covitch [1]). Reprinted with permission from CRC Press.

pre-critical crack growth. In these cases max (*k*<sup>+</sup>

to use the one-term asymptotic approximations *k*±

*Y* = <sup>1</sup> *π aH* −*aH*

*G*0(*t*) = <sup>1</sup> 2 <sup>1</sup>

*<sup>k</sup>*<sup>1</sup> <sup>=</sup> <sup>√</sup>

factors from (43) and (44) which in dimensional variables take the form

{*Yr* , *Y<sup>i</sup>*

*<sup>t</sup>*−*<sup>X</sup>* <sup>+</sup> <sup>1</sup>−*e*−2*i<sup>α</sup>*

*D*0(*t*) = *<sup>i</sup>* 2 <sup>−</sup> <sup>1</sup> *<sup>t</sup>*−*<sup>X</sup>* <sup>+</sup> <sup>1</sup>

*y*0

toughness.

−2

0

2

k 2n <sup>+</sup> <sup>⋅</sup><sup>10</sup>

**Figure 8.** The dependence of the shear stress intensity factor *k*<sup>+</sup> <sup>2</sup>*<sup>n</sup>* on the coordinate *<sup>x</sup>*<sup>0</sup> *<sup>n</sup>* for the case of the boundary of half-plane loaded with normal *<sup>p</sup>*(*x*0) = *<sup>π</sup>*/4 and frictional *<sup>τ</sup>*(*x*0) = <sup>−</sup>*λp*(*x*0) stresses, *y*0 *<sup>n</sup>* <sup>=</sup> <sup>−</sup>0.2, *<sup>α</sup><sup>n</sup>* <sup>=</sup> *<sup>π</sup>*/2, *<sup>q</sup>*<sup>0</sup> <sup>=</sup> 0: *<sup>λ</sup>* <sup>=</sup> 0.1 - curve marked with 1, *<sup>λ</sup>* <sup>=</sup> 0.2 - curve marked with 2 (after Kudish and Covitch [1]). Reprinted with permission from CRC Press.

where *i* is the imaginary unit (*i* <sup>2</sup> <sup>=</sup> <sup>−</sup>1), *<sup>θ</sup>*(*x*) is a step function: *<sup>θ</sup>*(*x*) = 0, *<sup>x</sup>* <sup>≤</sup> 0 and *<sup>θ</sup>*(*x*) = 1, *x* > 0. It is important to mention that according to (45) for subsurface cracks the quantities of *k*<sup>10</sup> = *k*1*l* <sup>−</sup>1/2 and *k*<sup>20</sup> = *k*2*l* <sup>−</sup>1/2 are functions of *x* and *y* and are independent from *l*.

Numerous experimental studies have established the fact that at relatively low cyclic loads materials undergo the process of pre-critical failure while the rate of crack growth *dl*/*dN* (*N* is the number of loading cycles) in the predetermined direction is dependent on *k*± <sup>1</sup> and *Kf* . A number of such equations of pre-critical crack growth and their analysis are presented in [6]. However, what remains to be determined is the direction of fatigue crack growth.

Assuming that fatigue cracks growth is driven by the maximum principal tensile stress (see the section on Three-Dimensional Model of Contact and Structural Fatigue) we immediately obtain the equation

$$k\_2^{\pm}(N, \mathfrak{x}, y, l, \mathfrak{a}^{\pm}) = 0,\tag{46}$$

which determines the orientation angles *α*± of a fatigue crack growth at the crack tips. Due to the fact that a fatigue crack remains small during its pre-critical growth (i.e. practically during its entire life span) and being originally modeled by a straight cut with half-length *l* at the point with coordinates (*x*, *y*) the crack remains straight, i.e. the crack direction is characterized by one angle *α* = *α*<sup>+</sup> = *α*−. This angle is practically independent from crack half-length *l* because *k*± <sup>2</sup> = *k*<sup>±</sup> 20 √ *l*, where *k*± <sup>20</sup> is almost independent from *l* for small *l* (see (45)). The dependence of *k*± <sup>2</sup> on the number of loading cycles *N* comes only through the dependence of the crack half-length *l* on *N*. Therefore, the crack angle *α* is just a function of the crack location (*x*, *y*).

**Figure 9.** Distributions of the stress intensity factors *k*− <sup>1</sup> (curve 1) and *k*<sup>−</sup> <sup>2</sup> (curve 2) in case of a surface crack: *α* = 0.339837, *δ*<sup>0</sup> = 0.3, and other parameters as in the previous case of a surface crack (after Kudish [15]). Reprinted with permission of the STLE.

In particular, according to (45) and (46) at any point (*x*, *y*) there are two angles *α*<sup>1</sup> and *α*<sup>2</sup> along which a crack may propagate which are determined by the equation in dimensional variables

$$\tan 2\alpha = -\frac{2y \int\_{-4H}^{4H} (t - x)T(t, x, y)dt}{\frac{4}{2}q^{0} + \int\_{-4H}^{4H} [(t - x)^{2} - y^{2}]T(t, x, y)dt}, \; T(t, x, y) = \frac{yp(t) + (t - x)\tau(t)}{[(t - x)^{2} + y^{2}]^{2}}.\tag{47}$$

Along these directions *k*<sup>1</sup> reaches its extremum values. The actual direction of crack propagation *α* is determined by one of these two angles *α*<sup>1</sup> and *α*<sup>2</sup> for which the value of the normal stress intensity factor *k*1(*N*, *x*, *y*, *l*, *α*) is greater.

A more detailed analysis of the directions of fatigue crack propagation can be found in [1].

### **5. Two-dimensional contact fatigue model**

In a two-dimensional case compared to a three-dimensional case a more accurate description of the contact fatigue process can be obtained due to the fact that in two dimensions it is relatively easy to get very accurate formulas for the stress intensity factors at crack tips [1, 5]. The rest of the fatigue modeling can be done the same way as in the three-dimensional case with few simple changes. In particular, in a two-dimensional case of contact fatigue only subsurface originated fatigue is considered and cracks are modeled by straight cuts with half-length *l*. That gives the opportunity to use equations (45) and (47) for stress intensity

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(*t*−*x*)*T*(*t*,*x*,*y*)*dt*

[(*t*−*x*)<sup>2</sup>−*y*<sup>2</sup>]*T*(*t*,*x*,*y*)*dt*

crack: *α* = 0.339837, *δ*<sup>0</sup> = 0.3, and other parameters as in the previous case of a surface crack (after

In particular, according to (45) and (46) at any point (*x*, *y*) there are two angles *α*<sup>1</sup> and *α*<sup>2</sup> along which a crack may propagate which are determined by the equation in dimensional variables

Along these directions *k*<sup>1</sup> reaches its extremum values. The actual direction of crack propagation *α* is determined by one of these two angles *α*<sup>1</sup> and *α*<sup>2</sup> for which the value of

A more detailed analysis of the directions of fatigue crack propagation can be found in [1].

In a two-dimensional case compared to a three-dimensional case a more accurate description of the contact fatigue process can be obtained due to the fact that in two dimensions it is relatively easy to get very accurate formulas for the stress intensity factors at crack tips [1, 5]. The rest of the fatigue modeling can be done the same way as in the three-dimensional case with few simple changes. In particular, in a two-dimensional case of contact fatigue only subsurface originated fatigue is considered and cracks are modeled by straight cuts with half-length *l*. That gives the opportunity to use equations (45) and (47) for stress intensity

2

<sup>1</sup> (curve 1) and *k*<sup>−</sup>

1

xo

, *T*(*t*, *x*, *y*) = *yp*(*t*)+(*t*−*x*)*τ*(*t*)

<sup>2</sup> (curve 2) in case of a surface

[(*t*−*x*)<sup>2</sup>+*y*<sup>2</sup>]<sup>2</sup> . (47)

0

0.5

1

2

**Figure 9.** Distributions of the stress intensity factors *k*−

Kudish [15]). Reprinted with permission of the STLE.

2*y a H* −*aH*

*π* <sup>2</sup> *q*<sup>0</sup>+ *a H* −*aH*

the normal stress intensity factor *k*1(*N*, *x*, *y*, *l*, *α*) is greater.

**5. Two-dimensional contact fatigue model**

tan 2*α* = −

1

1.5

2

k1 − , k2 −

2.5

**Figure 10.** Distributions of the stress intensity factors *k*− <sup>1</sup> (curve 1) and *k*<sup>−</sup> <sup>2</sup> (curve 2) in case of a surface crack: *<sup>α</sup>* <sup>=</sup> *<sup>π</sup>*/6, *<sup>λ</sup>* <sup>=</sup> 0.1, and *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>0.5 (after Kudish and Burris [16]). Reprinted with permission from Kluwer Academic Publishers.

factors *k*1, *k*2, and crack angle orientations *α*. The rest of contact fatigue modeling follows exactly the derivation presented in the section on Three-Dimensional Model of Contact and Structural Fatigue. Therefore, the fatigue life *N* with survival probability *P*<sup>∗</sup> of a contact subjected to cyclic loading can be expressed in the form [1, 17]

$$\begin{split} N &= \{ (\frac{n}{2} - 1)g\_0 [\max\_{-\infty < x < \infty} k\_{10}]^n \}^{-1} \{ \exp[(1 - \frac{n}{2})(\mu\_{ln} \\\\ &+ \sqrt{2} \sigma\_{ln} \epsilon r f^{-1} (2P\_\* - 1)) \} - l\_c^{\frac{2-\mu}{2}} \} , \end{split} \tag{48}$$

where *er f* <sup>−</sup>1(*x*) is the inverse function to the error integral *er f*(*x*) [8].

First, let us consider the model behavior in some simple cases. If *f*(0, *x*, *y*, *z*, *l*0) is a uniform crack distribution over the material volume *V* (except for a thin surface layer where *f* = 0). Then based on (11) it can be shown that *p*(*N*, *x*, *y*, *z*) reaches its minimum at the points where *k*<sup>10</sup> and the principal tensile stress reach their maximum values. This leads to the conclusion that the material local failure probability (1 − *p*) reaches its maximum at the points with maximal tensile stress. Therefore, for a uniform initial crack distribution *f*(0, *x*, *y*, *z*, *l*0) the survival probability *P*(*N*) from (16) is determined by the material local survival probability at the points at which the maximal tensile stress is attained.

However, the latter conclusion is not necessarily correct if the initial crack distribution *f*(0, *x*, *y*, *z*, *l*0) is not uniform over the material volume. Suppose, *k*10(*x*, *y*, *z*) is maximal at the point (*xm*, *ym*, *zm*) and at the initial time moment *N* = 0 at some point (*x*∗, *y*∗, *z*∗) there exist cracks larger than the ones at the point (*xm*, *ym*, *zm*), namely,

$$\int\_0^{l\_\varepsilon} fdl\_0 \mid\_{\left(\mathfrak{x}\_\*,\mathcal{Y}\_\*,\mathcal{Z}\_\*\right)} < \int\_0^{l\_\varepsilon} fdl\_0 \mid\_{\left(\mathfrak{x}\_m,\mathcal{Y}\_m,\mathcal{Z}\_m\right)} \cdot \mathcal{Z}$$

Then after a certain number of loading cycles *N* > 0 the material damage at point (*x*∗, *y*∗, *z*∗) may be greater than at point (*xm*, *ym*, *zm*), where *l*0*<sup>c</sup>* reaches its maximum value. Therefore, fatigue failure may occur at the point (*x*∗, *y*∗, *z*∗) instead of the point (*xm*, *ym*, *zm*), and the material weakest point is not necessarily is the material most stressed point.

If *μln* and *σln* depend on the coordinates of the material point (*x*, *y*, *z*), then there may be a series of points where in formula (16) for the given number of loading cycles *N* the minimum over the material volume *V* is reached. The coordinates of such points may change with *N*. This situation represents different potentially competing fatigue mechanisms such as pitting, flaking, etc. The occurrence of fatigue damage at different points in the material depends on the initial defect distribution, applied stresses, residual stress, etc.

In the above model of contact fatigue the stressed volume *V* plays no explicit role. However, implicitly it does. In fact, the initial crack distribution *f*(0, *x*, *y*, *z*, *l*0) depends on the material volume. In general, in a larger volume of material, there is a greater chance to find inclusions/cracks of greater size than in a smaller one. These larger inclusions represent a potential source of pitting and may cause a decrease in the material fatigue life of a larger material volume.

Assuming that *μln* and *σln* are constants, and assuming that the material failure occurs at the point (*x*, *y*, *z*) with the failure probability 1 − *P*(*N*) following the considerations of the section on Three-Dimensional Model of Contact and Structural Fatigue from (48) we obtain formulas (19)-(21). Actually, fatigue life formulas (20) and (21) can be represented in the form of the Lundberg-Palmgren formula, i.e.

$$N = \frac{\mathbb{C}\_\*}{p\_H^{n\_H}} \tag{49}$$

where parameter *n* can be compared with constant *c*/*e* in the Lundberg-Palmgren formula [1]. The major difference between the Lundberg-Palmgren formula and formula (49) derived from this model of contact fatigue is the fact that in (49) constant *C*<sup>∗</sup> depends on material defect parameters *<sup>μ</sup>*, *<sup>σ</sup>*, coefficient of friction *<sup>λ</sup>*, residual stresses *<sup>q</sup>*0, and probability of survival *<sup>P</sup>*<sup>∗</sup> in a certain way while in the Lundberg-Palmgren formula the constant *C*<sup>∗</sup> depends only on the depth *z*<sup>0</sup> of the maximum orthogonal stress, stressed volume *V*, and probability of survival *P*∗.

Let us analyze formula (21). It demonstrates the intuitively obvious fact that the fatigue life *N* is inverse proportional to the value of the parameter *g*<sup>0</sup> that characterizes the material crack propagation resistance. So, for materials with lower crack propagation rate, the fatigue life is higher and vice versa. Equation (21) exhibits a usual for gears and roller and ball bearings dependence of fatigue life *N* on the maximum Hertzian pressure *pH*. Thus, from the well-known experimental data for bearings, the range of *n* values is 20/3 ≤ *n* ≤ 9. Keeping in mind that usually *σ* � *μ*, for these values of *n* contact fatigue life *N* is practically inverse proportional to a positive power of the crack initial mean size, i.e., to *μn*/2−1. Therefore, fatigue life *N* is a decreasing function of the initial mean crack (inclusion) size *μ* and ln *N* = −(*n*/2 − 1)ln *μ* + *constant*. This conclusion is valid for any value of the material survival

26 Will-be-set-by-IN-TECH

the point (*xm*, *ym*, *zm*) and at the initial time moment *N* = 0 at some point (*x*∗, *y*∗, *z*∗) there

*l c* 0

Then after a certain number of loading cycles *N* > 0 the material damage at point (*x*∗, *y*∗, *z*∗) may be greater than at point (*xm*, *ym*, *zm*), where *l*0*<sup>c</sup>* reaches its maximum value. Therefore, fatigue failure may occur at the point (*x*∗, *y*∗, *z*∗) instead of the point (*xm*, *ym*, *zm*), and the

If *μln* and *σln* depend on the coordinates of the material point (*x*, *y*, *z*), then there may be a series of points where in formula (16) for the given number of loading cycles *N* the minimum over the material volume *V* is reached. The coordinates of such points may change with *N*. This situation represents different potentially competing fatigue mechanisms such as pitting, flaking, etc. The occurrence of fatigue damage at different points in the material depends on

In the above model of contact fatigue the stressed volume *V* plays no explicit role. However, implicitly it does. In fact, the initial crack distribution *f*(0, *x*, *y*, *z*, *l*0) depends on the material volume. In general, in a larger volume of material, there is a greater chance to find inclusions/cracks of greater size than in a smaller one. These larger inclusions represent a potential source of pitting and may cause a decrease in the material fatigue life of a larger

Assuming that *μln* and *σln* are constants, and assuming that the material failure occurs at the point (*x*, *y*, *z*) with the failure probability 1 − *P*(*N*) following the considerations of the section on Three-Dimensional Model of Contact and Structural Fatigue from (48) we obtain formulas (19)-(21). Actually, fatigue life formulas (20) and (21) can be represented in the form of the

> *N* = *<sup>C</sup>*<sup>∗</sup> *pn H*

where parameter *n* can be compared with constant *c*/*e* in the Lundberg-Palmgren formula [1]. The major difference between the Lundberg-Palmgren formula and formula (49) derived from this model of contact fatigue is the fact that in (49) constant *C*<sup>∗</sup> depends on material defect parameters *<sup>μ</sup>*, *<sup>σ</sup>*, coefficient of friction *<sup>λ</sup>*, residual stresses *<sup>q</sup>*0, and probability of survival *<sup>P</sup>*<sup>∗</sup> in a certain way while in the Lundberg-Palmgren formula the constant *C*<sup>∗</sup> depends only on the depth *z*<sup>0</sup> of the maximum orthogonal stress, stressed volume *V*, and probability of survival

Let us analyze formula (21). It demonstrates the intuitively obvious fact that the fatigue life *N* is inverse proportional to the value of the parameter *g*<sup>0</sup> that characterizes the material crack propagation resistance. So, for materials with lower crack propagation rate, the fatigue life is higher and vice versa. Equation (21) exhibits a usual for gears and roller and ball bearings dependence of fatigue life *N* on the maximum Hertzian pressure *pH*. Thus, from the well-known experimental data for bearings, the range of *n* values is 20/3 ≤ *n* ≤ 9. Keeping in mind that usually *σ* � *μ*, for these values of *n* contact fatigue life *N* is practically inverse proportional to a positive power of the crack initial mean size, i.e., to *μn*/2−1. Therefore, fatigue life *N* is a decreasing function of the initial mean crack (inclusion) size *μ* and ln *N* = −(*n*/2 − 1)ln *μ* + *constant*. This conclusion is valid for any value of the material survival

, (49)

*f dl*<sup>0</sup> |(*xm*,*ym*,*zm*) .

exist cracks larger than the ones at the point (*xm*, *ym*, *zm*), namely,

the initial defect distribution, applied stresses, residual stress, etc.

material volume.

*P*∗.

Lundberg-Palmgren formula, i.e.

*f dl*<sup>0</sup> <sup>|</sup>(*x*∗,*y*∗,*z*∗)<sup>&</sup>lt;

material weakest point is not necessarily is the material most stressed point.

*l c* 0

**Figure 11.** Bearing life-inclusion length correlation (after Stover and Kolarik II[18], COPYRIGHT The Timken Company 2012).

probability *P*<sup>∗</sup> and is supported by the experimental data obtained at the Timken Company by Stover and Kolarik II [18] and represented in Fig. 11 in a log-log scale. If *P*<sup>∗</sup> > 0.5 then *er f* <sup>−</sup>1(2*P*<sup>∗</sup> <sup>−</sup> <sup>1</sup>) <sup>&</sup>gt; 0 and (keeping in mind that *<sup>n</sup>* <sup>&</sup>gt; 2) fatigue life *<sup>N</sup>* is a decreasing function of the initial standard deviation of crack sizes *σ*. Similarly, if *P*<sup>∗</sup> < 0.5, then fatigue life *N* is an increasing function of the initial standard deviation of crack sizes *σ*. According to (20), for *P*<sup>∗</sup> = 0.5 fatigue life *N* is independent from *σ*, and, according to (21), for *P*<sup>∗</sup> = 0.5 fatigue life *N* is a slowly increasing function of *σ*. By differentiating *pm*(*N*) obtained from (16) with respect to *σ*, we can conclude that the dispersion of *P*(*N*) increases with *σ*.

From (45) (also see Kudish [1]) follows that the stress intensity factor *k*<sup>1</sup> decreases as the magnitude of the compressive residual stress *q*<sup>0</sup> increases and/or the magnitude of the friction coefficient *λ* decreases. Therefore, it follows from formulas (45) that *C*<sup>0</sup> is a monotonically decreasing function of the residual stress *q*<sup>0</sup> and friction coefficient *λ*. Numerical simulations of fatigue life show that the value of *C*<sup>0</sup> is very sensitive to the details of the residual stress distribution *q*<sup>0</sup> versus depth.

Let us choose a basic set of model parameters typical for bearing testing: maximum Hertzian pressure *pH* = 2 *GPa*, contact region half-width in the direction of motion *aH* = 0.249 *mm*, friction coefficient *<sup>λ</sup>* <sup>=</sup> 0.002, residual stress varying from *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>237.9 *MPa* on the surface to *q*<sup>0</sup> = 0.035 *MPa* at the depth of 400 *μm* below it, fracture toughness *Kf* varying between 15 and 95 *MPa* · *<sup>m</sup>*1/2, *<sup>g</sup>*<sup>0</sup> <sup>=</sup> 8.863 *MPa*−*<sup>n</sup>* · *<sup>m</sup>*1−*n*/2 · *cycle*−1, *<sup>n</sup>* <sup>=</sup> 6.67, mean of crack initial half-lengths *μ* = 49.41 *μm* (*μln* = 3.888+ *ln*(*μm*)), crack initial standard deviation *σ* = 7.61 *μm* (*σln* = 0.1531). Numerical results show that the fatigue life is practically independent from the material fracture toughness *Kf* , which supports the assumption used for the derivation of formulas (19)-(21). To illustrate the dependence of contact fatigue life on some of the model

**Figure 12.** Pitting probability 1 − *P*(*N*) calculated for the basic set of parameters (solid curve) with *μ* = 49.41 *μm*, *σ* = 7.61 *μm* (*μln* = 3.888 + ln(*μm*), *σln* = 0.1531), for the same set of parameters and the increased initial value of crack mean half-lengths (dash-dotted curve) *μ* = 74.12 *μm* (*μln* = 4.300 + ln(*μm*), *σln* = 0.1024), and for the same set of parameters and the increased initial value of crack standard deviation (dotted curve) *σ* = 11.423 *μm* (*μln* = 3.874 + ln(*μm*), *σln* = 0.2282) (after Kudish [17]). Reprinted with permission from the STLE.

**Figure 13.** Pitting probability 1 − *P*(*N*) calculated for the basic set of parameters including *λ* = 0.002 (solid curve) and for the same set of parameters and the increased friction coefficient (dashed curve) *λ* = 0.004 (after Kudish [17]). Reprinted with permission from the STLE.

28 Will-be-set-by-IN-TECH

7E7 8E8 9E9

**Figure 12.** Pitting probability 1 − *P*(*N*) calculated for the basic set of parameters (solid curve) with *μ* = 49.41 *μm*, *σ* = 7.61 *μm* (*μln* = 3.888 + ln(*μm*), *σln* = 0.1531), for the same set of parameters and the

(*μln* = 4.300 + ln(*μm*), *σln* = 0.1024), and for the same set of parameters and the increased initial value of crack standard deviation (dotted curve) *σ* = 11.423 *μm* (*μln* = 3.874 + ln(*μm*), *σln* = 0.2282) (after

1E7 1E8 1E9 1E10

**Figure 13.** Pitting probability 1 − *P*(*N*) calculated for the basic set of parameters including *λ* = 0.002 (solid curve) and for the same set of parameters and the increased friction coefficient (dashed curve)

increased initial value of crack mean half-lengths (dash-dotted curve) *μ* = 74.12 *μm*

1−P(N)

N

N

0

Kudish [17]). Reprinted with permission from the STLE.

0

*λ* = 0.004 (after Kudish [17]). Reprinted with permission from the STLE.

0.25

0.5

0.75

1

0.25

0.5

0.75

1

1−P(N)

**Figure 14.** Pitting probability 1 − *P*(*N*) calculated for the basic set of parameters (solid curve) and for the same set of parameters and changed profile of residual stress *q*<sup>0</sup> (dashed curve) in such a way that at points where *q*<sup>0</sup> is compressive its magnitude is unchanged and at points where *q*<sup>0</sup> is tensile its magnitude is doubled (after Kudish [17]). Reprinted with permission from the STLE.

parameters, just one parameter from the basic set of parameters will be varied at a time and graphs of the pitting probability 1 − *P*(*N*) for these sets of parameters (basic and modified) will be compared. Figure 12 shows that as the initial values of the mean *μ* of crack half-lengths and crack standard deviation *σ* increase contact fatigue life *N* decreases. Similarly, contact fatigue life decreases as the magnitude of the tensile residual stress and/or friction coefficient increase (see Fig. 4 and 13). The results show that the fatigue life does not change when the magnitude of the compressive residual stress is increased/decreased by 20% of its base value while the tensile portion of the residual stress distribution remains the same. Obviously, that is in agreement with the fact that tensile stresses control fatigue. Moreover, the fatigue damage occurs in the region with the resultant tensile stresses close to the boundary between tensile and compressive residual stresses. However, when the compressive residual stress becomes small enough the acting frictional stress may supersede it and create new regions with tensile stresses that potentially may cause acceleration of fatigue failure.


**Table 1.** Relationship between the tapered bearing fatigue life *N*15.9 and the initial inclusion size mean and standard deviation (after Kudish [17]). Reprinted with permission from the STLE.

Let us consider an example of the further validation of the new contact fatigue model for tapered roller bearings based on a series of approximate calculations of fatigue life. The

main simplifying assumption made is that bearing fatigue life can be closely approximated by taking into account only the most loaded contact. The following parameters have been used for calculations: *pH* <sup>=</sup> 2.12 *GPa*, *aH* <sup>=</sup> 0.265 *mm*, *<sup>λ</sup>* <sup>=</sup> 0.002, *<sup>g</sup>*<sup>0</sup> <sup>=</sup> 6.009 *MPa*−*<sup>n</sup>* · *<sup>m</sup>*1−*n*/2 · *cycle*−1, *<sup>n</sup>* <sup>=</sup> 6.67, the residual stress varied from *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>237.9 *MPa* on the surface to *q*<sup>0</sup> = 0.035 *MPa* at the depth of 400 *μm* below the surface, fracture toughness *Kf* varied between 15 and 95*MPa* · *<sup>m</sup>*1/2. The crack/inclusion initial mean half-length *<sup>μ</sup>* varied between 49.41 and 244.25 *μm* (*μln* = 3.888 − 5.498 + *ln*(*μm*)), the crack initial standard deviation varied between *σ* = 7.61 and 37.61 *μm* (*σln* = 0.1531). The results for fatigue life *N*15.9 (for *P*(*N*15.9) = *P*<sup>∗</sup> = 0.159) calculations are given in the Table 1 and practically coincide with the experimental data obtained by The Timken Company and presented in Fig. 19 by Stover, Kolarik II, and Keener [**?** ] (in the present text this graph is given as Fig. 11). One must keep in mind that there are certain differences in the numerically obtained data and the data presented in the above mentioned Fig. 11 due to the fact that in Fig. 11 fatigue life is given as a function of the cumulative inclusion length (sum of all inclusion lengths over a cubic inch of steel) while in the model fatigue life is calculated as a function of the mean inclusion length.

It is also interesting to point out that based on the results following from the new model, bearing fatigue life can be significantly improved for steels with the same cumulative inclusion length but smaller mean half-length *μ* (see Fig. 12). In other words, fatigue life of a bearing made from steel with large number of small inclusions is higher than of the one made of steel with small number of larger inclusions given that the cumulative inclusion length is the same in both cases. Moreover, bearing and gear fatigue lives with small percentage of failures (survival probability *P* > 0.5) for steels with the same cumulative inclusion length can also be improved several times if the width of the initial inclusion distribution is reduced, i.e., when the standard deviation *σ* of the initial inclusion distribution is made smaller (see Figure 12). Figures 13 and 14 show that the elevated values of the tensile residual stress are much more detrimental to fatigue life than greater values of the friction coefficient.

Finally, the described model is flexible enough to allow for replacement of the density of the initial crack distribution (see (1)) by a different function and of Paris's equation for fatigue crack propagation (see (4)) by another equation. Such modifications would lead to results on fatigue life varying from the presented above. However, the methodology, i.e., the way the formulas for fatigue life are obtain and the most important conclusions will remain the same.

This methodology has been extended on the cases of non-steady cyclic loading as well as on the case of contact fatigue of rough surfaces [1]. Also, this kind of modeling approach has been applied to the analysis of wear and contact fatigue in cases of lubricant contaminated by rigid abrasive particles and contact surfaces charged with abrasive particles [19] as well as to calculation of bearing wear and contact fatigue life [20].

In conclusion we can state that the presented statistical contact and structural fatigue models take into account the most important parameters of the contact fatigue phenomenon (such as normal and frictional contact and residual stresses, initial statistical defect distribution, orientation of fatigue crack propagation, material fatigue resistance, etc.). The models allows for examination of the effect of variables such as steel cleanliness, applied stresses, residual stress, etc. on contact fatigue life as single or composite entities. Some analytical results illustrating these models and their validation by the experimentally obtained fatigue life data for tapered bearings are presented.

## **6. Closure**

30 Will-be-set-by-IN-TECH

main simplifying assumption made is that bearing fatigue life can be closely approximated by taking into account only the most loaded contact. The following parameters have been used for calculations: *pH* <sup>=</sup> 2.12 *GPa*, *aH* <sup>=</sup> 0.265 *mm*, *<sup>λ</sup>* <sup>=</sup> 0.002, *<sup>g</sup>*<sup>0</sup> <sup>=</sup> 6.009 *MPa*−*<sup>n</sup>* · *<sup>m</sup>*1−*n*/2 · *cycle*−1, *<sup>n</sup>* <sup>=</sup> 6.67, the residual stress varied from *<sup>q</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>237.9 *MPa* on the surface to *q*<sup>0</sup> = 0.035 *MPa* at the depth of 400 *μm* below the surface, fracture toughness *Kf* varied between 15 and 95*MPa* · *<sup>m</sup>*1/2. The crack/inclusion initial mean half-length *<sup>μ</sup>* varied between 49.41 and 244.25 *μm* (*μln* = 3.888 − 5.498 + *ln*(*μm*)), the crack initial standard deviation varied between *σ* = 7.61 and 37.61 *μm* (*σln* = 0.1531). The results for fatigue life *N*15.9 (for *P*(*N*15.9) = *P*<sup>∗</sup> = 0.159) calculations are given in the Table 1 and practically coincide with the experimental data obtained by The Timken Company and presented in Fig. 19 by Stover, Kolarik II, and Keener [**?** ] (in the present text this graph is given as Fig. 11). One must keep in mind that there are certain differences in the numerically obtained data and the data presented in the above mentioned Fig. 11 due to the fact that in Fig. 11 fatigue life is given as a function of the cumulative inclusion length (sum of all inclusion lengths over a cubic inch of steel) while in the model fatigue life is calculated as a function of the mean inclusion length. It is also interesting to point out that based on the results following from the new model, bearing fatigue life can be significantly improved for steels with the same cumulative inclusion length but smaller mean half-length *μ* (see Fig. 12). In other words, fatigue life of a bearing made from steel with large number of small inclusions is higher than of the one made of steel with small number of larger inclusions given that the cumulative inclusion length is the same in both cases. Moreover, bearing and gear fatigue lives with small percentage of failures (survival probability *P* > 0.5) for steels with the same cumulative inclusion length can also be improved several times if the width of the initial inclusion distribution is reduced, i.e., when the standard deviation *σ* of the initial inclusion distribution is made smaller (see Figure 12). Figures 13 and 14 show that the elevated values of the tensile residual stress are

much more detrimental to fatigue life than greater values of the friction coefficient.

calculation of bearing wear and contact fatigue life [20].

for tapered bearings are presented.

Finally, the described model is flexible enough to allow for replacement of the density of the initial crack distribution (see (1)) by a different function and of Paris's equation for fatigue crack propagation (see (4)) by another equation. Such modifications would lead to results on fatigue life varying from the presented above. However, the methodology, i.e., the way the formulas for fatigue life are obtain and the most important conclusions will remain the same. This methodology has been extended on the cases of non-steady cyclic loading as well as on the case of contact fatigue of rough surfaces [1]. Also, this kind of modeling approach has been applied to the analysis of wear and contact fatigue in cases of lubricant contaminated by rigid abrasive particles and contact surfaces charged with abrasive particles [19] as well as to

In conclusion we can state that the presented statistical contact and structural fatigue models take into account the most important parameters of the contact fatigue phenomenon (such as normal and frictional contact and residual stresses, initial statistical defect distribution, orientation of fatigue crack propagation, material fatigue resistance, etc.). The models allows for examination of the effect of variables such as steel cleanliness, applied stresses, residual stress, etc. on contact fatigue life as single or composite entities. Some analytical results illustrating these models and their validation by the experimentally obtained fatigue life data

The chapter presents a detailed analysis of a number of plane crack mechanics problems for loaded elastic half-plane weakened by a system of cracks. Surface and subsurface cracks are considered. All cracks are considered to be straight cuts. The problems are analyzed by the regular asymptotic method and numerical methods. Solutions of the problems include the stress intensity factors. The regular asymptotic method is applied under the assumption that cracks are far from each other and from the boundary of the elastic solid. It is shown that the results obtained for subsurface cracks based on asymptotic expansions and numerical solutions are in very good agreement. The influence of the normal and tangential contact stresses applied to the boundary of a half-plane as well as the residual stress on the stress intensity factors for subsurface cracks is analyzed. It is determined that the frictional and residual stresses provide a significant if not the predominant contribution to the problem solution. Based on the numerical solution of the problem for surface cracks in the presence of lubricant the physical nature of the "wedge effect" (when lubricant under a sufficiently large pressure penetrates a surface crack and ruptures it) is considered. Solution of this problem also provides the basis for the understanding of fatigue crack origination site (surface versus subsurface) and the difference of fatigue lives of drivers and followers. New twoand three-dimensional models of contact and structural fatigue are developed. These models take into account the initial crack distribution, fatigue properties of the solids, and growth of fatigue cracks under the properly determined combination of normal and tangential contact and residual stresses. The formula for fatigue life based on these models can be reduced to a simple formula which takes into account most of the significant parameters affecting contact and structural fatigue. The properties of these contact fatigue models are analyzed and the results based on them are compared to the experimentally obtained results on contact fatigue for tapered bearings.

### **7. References**


*Processing XXVII: Proc. 31st Mechanical Working and Steel Processing Conf.*, Chicago, Illinois, October 22-25, 1989, Iron and Steel Soc., Inc., 431-440.

