**Evaluating the Integrity of Pressure Pipelines by Fracture Mechanics**

Ľubomír Gajdoš and Martin Šperl

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/77358

## **1. Introduction**

Large engineering structures made with the use of sophisticated technology often include material defects and geometrical imperfections. These defects or imperfections do not exert their influence on the initial behaviour of structures designed in accordance with standard rules. Under the action of loading varying in time, however, they can reveal themselves in long-term operation by the initiation and growth of a fatigue crack from a defect root. Similarly, stress corrosion (SC) cracks can develop in a structure when there is an initial stress concentrator and the structure is exposed to both mechanical stress and a corrosion medium. A condition for the growth of a small fatigue crack is that the level of cyclic stress should be above the limit value given by barriers existing in a steel, and a condition for the growth of SC cracks is that the stress is greater than a certain limit value for a specific corrosion medium. It is important to pay due attention to the behaviour of cracks under various gas pipeline loading conditions in different environments, and to the influence of these conditions on the residual strength and life of the gas pipeline. The existence of a crack in the wall of a high-pressure gas pipeline mostly implies a shortened remaining period of reliable operation.

## **2. Theoretical treatment of cracks in pipes**

At the present time, the manufacturing stage of pipes for gas pipelines includes sufficient flaw detection measures, and only products free of detectable material flaws are dispatched for operation. However, there are defects that are not revealed by the required inspection, and which manifest themselves during heavy-duty operation. The most dangerous defect is the occurrence of cracks – these are due to material defects that are difficult to reveal by a standard optical inspection. If the cracks are deep, and spread to a large extent, they can pose a threat to the pipeline operation. Using fracture mechanics it is possible to evaluate

© 2012 Gajdoš and Šperl, 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 Gajdoš and Šperl, 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.

the threat that crack-like defects can pose to the pipeline wall, depending on whether a brittle, quasi-brittle or ductile material is involved. A model description of crack-containing systems, which relies on the stress intensity factor, K*,* can be used for brittle and quasi-brittle fracture, and also for subcritical fatigue growth, corrosion fatigue, and stress corrosion. In these cases, the surface crack is usually located in the field of one of the membrane tensile stress components, or in the field of bending stress, or in a combination of both. The extent of the plastic zone at the crack tip is small in comparison with the dimensions of the crack and the pipeline.

If the gas pipeline is made of a high-toughness material, the plastic strains become extensive before the crack reaches instability. Hence, some elasto-plastic fracture mechanics parameter, such as the J–integral or crack opening displacement, or a two-criterion method, should be employed to assess the threat that the crack poses to the pipeline wall. Although cracks of various directions may occur in the pipe wall, we will consider here only longitudinal cracks, because they are subjected to the biggest stress (hoop stress) in the pipe wall, and they are therefore the most dangerous (when we are considering the parent metal).

## **2.1. Stress intensity factor for a longitudinal through crack in the pipe wall**

The first theoretical solution to the problem of establishing the stress intensity factor for a long cylindrical pipe with a longitudinal through crack under internal pressure was reported by Folias (Folias, 1969) and by Erdogan with Kibler (Erdogan & Kibler, 1969). They managed to show that the problem was analogous to that of a wide plane plate with a through crack. The only adjustment needed for transition to a pipe was to introduce a correction factor to multiply the solution for the plane plate. This factor, frequently referred to as the Folias correction factor and designated by symbol MT, is only a function of the ratio *c Rt* , where *c* is the crack half-length, *R* is the pipe mean radius, and *t* is the pipe wall thickness, and thus it depends only on the geometrical parameters of the crack and the pipe (Fig. 1).

**Figure 1.** Deformation of a pressurized pipe in the vicinity of a longitudinal through crack

Several relations have been reported for determining the Folias correction factor. The following are the most frequently used at the present time:

The Folias relation (Folias, 1970):

284 Applied Fracture Mechanics

and the pipeline.

metal).

(Fig. 1).

the threat that crack-like defects can pose to the pipeline wall, depending on whether a brittle, quasi-brittle or ductile material is involved. A model description of crack-containing systems, which relies on the stress intensity factor, K*,* can be used for brittle and quasi-brittle fracture, and also for subcritical fatigue growth, corrosion fatigue, and stress corrosion. In these cases, the surface crack is usually located in the field of one of the membrane tensile stress components, or in the field of bending stress, or in a combination of both. The extent of the plastic zone at the crack tip is small in comparison with the dimensions of the crack

If the gas pipeline is made of a high-toughness material, the plastic strains become extensive before the crack reaches instability. Hence, some elasto-plastic fracture mechanics parameter, such as the J–integral or crack opening displacement, or a two-criterion method, should be employed to assess the threat that the crack poses to the pipeline wall. Although cracks of various directions may occur in the pipe wall, we will consider here only longitudinal cracks, because they are subjected to the biggest stress (hoop stress) in the pipe wall, and they are therefore the most dangerous (when we are considering the parent

**2.1. Stress intensity factor for a longitudinal through crack in the pipe wall** 

The first theoretical solution to the problem of establishing the stress intensity factor for a long cylindrical pipe with a longitudinal through crack under internal pressure was reported by Folias (Folias, 1969) and by Erdogan with Kibler (Erdogan & Kibler, 1969). They managed to show that the problem was analogous to that of a wide plane plate with a through crack. The only adjustment needed for transition to a pipe was to introduce a correction factor to multiply the solution for the plane plate. This factor, frequently referred to as the Folias correction factor and designated by symbol MT, is only a function of the ratio

 *c Rt* , where *c* is the crack half-length, *R* is the pipe mean radius, and *t* is the pipe wall thickness, and thus it depends only on the geometrical parameters of the crack and the pipe

**Figure 1.** Deformation of a pressurized pipe in the vicinity of a longitudinal through crack

$$M\_T = \sqrt{1 + 1.255\lambda^2 - 0.0135\lambda^4} \tag{1}$$

The Erdogan et al. relation (Erdogan et al., 1977):

$$\begin{aligned} M\_T &= 0.6 + 0.5\lambda + 0.4 \exp(-1.25\lambda) & \quad \text{for} \quad \alpha < 5\\ M\_T &= 1.761 \left(\lambda - 1.9\right)^{0.5} & \quad \text{for} \quad \alpha \ge 5 \end{aligned} \tag{2}$$

where <sup>2</sup> <sup>4</sup> 12 1 

with *ν* denoting Poisson´s ratio

The following relation is the simplest:

$$M\_T = \sqrt{1 + 1.61\lambda^2} \tag{3}$$

However, its validity is limited by the value *λ*<1.

If *c* is the half-length of a longitudinal through crack in the pipe, then the stress intensity factor of such a crack simply reads

$$K\_I = M\_T \sigma\_\wp \sqrt{\pi c} \tag{4}$$

where

*D t* 2 is the hoop stress (*D* and *t* denoting pipe diameter and wall thickness, respectively), and

*MT* is the Folias correction factor

#### **2.2. Stress intensity factor for a longitudinal part-through crack**

Various methods are used for analysing the problem of longitudinal semi-elliptical surface cracks in the wall of cylindrical shells (Fig. 2). As a 3D asymptotic solution to the stress intensity factor is virtually involved, the possibilities offered by accurate analytical procedures are confined to infinite or semi-infinite bodies. Solutions appropriate for finite bodies call for the application of approximate methods, such as the finite element method and the method of boundary integral equations, or various alternative methods (e.g. the weight function method).

The first solutions for semi-elliptical surface cracks in a plate subjected to uniaxial tension or steady bending were derived from solutions for an elliptical plane crack in an infinite 3D body. In order to account for the finite thickness of a body and the plastic zone at the crack tip, correction factors were introduced for the "front" surface and the "rear" surface of the body and for the plastic region at the crack tip (Shah & Kobayashi, 1973). However, solutions by different authors often showed rather considerable disagreement. Scott and Thorpe (Scott & Thorpe, 1981) therefore tested the accuracy of the solutions presented by various authors by measuring changes in the shape of a crack throughout its fatigue growth. They concluded that the best engineering estimation of the stress intensity factor for a partthrough crack in a plate was provided by Newman's solution (Newman, 1973). An adjusted form of this solution for a thin-walled shell is given by:

$$\mathcal{M}\_I = \left[ \mathcal{M}\_F + \left( E\_{\{k\}} \sqrt{c/a} - \mathcal{M}\_F \right) \left( \frac{a}{t} \right)^s \right] \frac{\sigma\_\phi \sqrt{\pi a}}{E\_{\{k\}}} \mathcal{M}\_{TM} \tag{5}$$

where

*MF* is the function depending on the crack geometry (on the ratio *a*/*c*)

$$E\_{\{k\}} = \int\_0^{\pi/2} \sqrt{1 - k^2 \sin^2 \theta} d\theta \text{ is an elliptical integral of the second kind, } k \text{ being } \sqrt{1 - \left(\frac{a}{c}\right)^2}$$

*s* is the function depending on the crack geometry (the ratio *a*/*c*) and the relative crack depth (the ratio *a*/*t*)

$$M\_{TM} = \frac{\left(1 - \frac{a/t}{M\_T}\right)}{\left(1 - a/t\right)}\text{ is the correction factor for the curvature of a cylindrical shell and for an initial field }M\_{TM}$$

increase in stress owing to radial strains in the vicinity of the crack root

In the last relationship, *MT* is the Folias correction factor, determined by any of the relations (1) – (3). The functions *MF* and *s* differ in form for the lowest point of the crack tip (point A in Fig. 2) and for the crack mouth on the surface of the cylindrical shell (point B in Fig. 2).

**Figure 2.** External longitudinal semi-elliptical crack in the wall of a cylindrical shell

#### **2.3. Engineering methods for determining the J integral**

#### *2.3.1. The FC method*

This method was proposed as the Js method in Addendum A16 of the French nuclear code (RCC-MR, 1985). It stems from the second option for describing the transition state between ideally elastic and fully plastic behaviour of a material, i.e. from the function *f2(Lr)* of the R6 method (Milne et al., 1986). This function takes the form:

$$f\_2\left(L\_r\right) = \left(\frac{E\varepsilon\_{ref}}{L\_rR\_e} + \frac{L\_r\,^3R\_e}{2E\varepsilon\_{ref}}\right)^{-1/2} \tag{6}$$

where

286 Applied Fracture Mechanics

where

2

1

1

 

*T*

*a t M*

*a t*

0 1 sin *<sup>k</sup> E kd*

(the ratio *a*/*t*)

*TM*

*M*

2 2

 

body. In order to account for the finite thickness of a body and the plastic zone at the crack tip, correction factors were introduced for the "front" surface and the "rear" surface of the body and for the plastic region at the crack tip (Shah & Kobayashi, 1973). However, solutions by different authors often showed rather considerable disagreement. Scott and Thorpe (Scott & Thorpe, 1981) therefore tested the accuracy of the solutions presented by various authors by measuring changes in the shape of a crack throughout its fatigue growth. They concluded that the best engineering estimation of the stress intensity factor for a partthrough crack in a plate was provided by Newman's solution (Newman, 1973). An adjusted

 

is an elliptical integral of the second kind, *k* being

*MF* is the function depending on the crack geometry (on the ratio *a*/*c*)

increase in stress owing to radial strains in the vicinity of the crack root

**Figure 2.** External longitudinal semi-elliptical crack in the wall of a cylindrical shell

*I F k F TM*

*s* is the function depending on the crack geometry (the ratio *a*/*c*) and the relative crack depth

In the last relationship, *MT* is the Folias correction factor, determined by any of the relations (1) – (3). The functions *MF* and *s* differ in form for the lowest point of the crack tip (point A in Fig. 2) and for the crack mouth on the surface of the cylindrical shell (point B in Fig. 2).

is the correction factor for the curvature of a cylindrical shell and for an

*K M E ca M M*

*k*

(5)

2

<sup>1</sup> *<sup>a</sup> c* 

*s*

*t E* 

*a a*

 

form of this solution for a thin-walled shell is given by:

*Lr* = *σ/σL* (*σ* – applied stress, *σL* – stress at the limit load) *Re* is the yield stress *E* is Young´s modulus *εref* is the reference strain corresponding to the reference (nominal) stress *σref* 

If we identify function *f2(Lr)* with function 1 2 3 *r e <sup>J</sup> f L <sup>J</sup>* and express *Lr* as *σref / Re* and the

elastic J integral *Je* as *K2/ E´*, where E´ *=E* for plane stress state and E´ *=E / (1−ν2)* for plane strain state, we have:

$$J = \frac{K^2}{E'} \left( \frac{E.\mathcal{E}\_{ref}}{\sigma\_{ref}} + \frac{\sigma\_{ref}^{\,^3}}{2E.\mathcal{R}\_e \,^2 \mathcal{E}\_{ref}} \right) \tag{7}$$

The stress *σref* in the above equation is a nominal stress – i.e. a stress acting in the plane where the crack occurs. Taking into consideration the description of the stress-strain dependence by the Ramberg-Osgood relation (8) and adjusting Eq. (7), we obtain the *J*integral in the form (9).

$$\frac{\varepsilon}{\varepsilon\_0} = \frac{\sigma}{\sigma\_0} + \alpha \left(\frac{\sigma}{\sigma\_0}\right)^n \tag{8}$$

$$J\_{\parallel} = \frac{K^2}{E'} \left[ A + \frac{0.5 \left( \sigma / \sigma\_0 \right)^2}{A} \right] \tag{9}$$

where

$$A = 1 + a \left(\frac{\sigma}{\sigma\_0}\right)^{n-1} \tag{10}$$

In the above equations the stress *σ<sup>0</sup>* can be substituted by the yield stress *Re* ; 0 0 *E* ; ,*n* – material constants

As a pipeline is a body of finite dimensions, stress in Eqs. (9) and (10) is a nominal stress – i.e. a stress acting in the plane where the crack occurs. Referring to the R6 method (Milne et al., 1986), this stress for a pipe containing a longitudinal part-through thickness crack may be written as:

$$\sigma = \frac{\sigma\_{\phi}}{1 - \frac{\pi ac}{2t \left(t + 2c\right)}}\tag{11}$$

In eq. (11), 2 *pD t* is the hoop stress, and the meaning of the symbols *a, c,* and *t* is clear from Fig.2.

#### *2.3.2. GS method*

The GS method was derived by Gajdoš and Srnec (Gajdoš & Srnec, 1994) on the basis of the limit transition of the *J*-integral, formally expressed for a semi-circular notch, to a crack, with the variation of the strain energy density along the notch circumference being approximated by the third power of the cosine function of the polar angle. If the stress-strain dependence is further expressed by the Ramberg-Osgood relation (8), with *0 = 0 / E*, ( *, n* – material constants), we can arrive at Eq. (12)

$$J = \frac{K^2}{E'} \left[ 1 + \frac{2\alpha n}{\left(n+1\right)} \left(\frac{\sigma}{\sigma\_0}\right)^{n-1} \right] \tag{12}$$

where is the nominal stress in the reduced cross-section of a body. For a pipe containing a longitudinal part-through thickness crack it may be determined by relation (11).

### **3. Consideration of the constraint**

As mentioned above, the situation existing at the crack tip in conditions of small-scale yielding can be characterized by a single fracture parameter (e.g. K, J or ). This parameter can be used as a fracture criterion, independent of geometry. However, single-parameter fracture mechanics fails in cases of developed plasticity, where fracture toughness is a function not only of the material, but also of the dimensions and the geometry of the specimen. It is well known from the theory of fracture mechanics that for small-scale yielding the maximum stress existing at the crack tip in a non-hardening material is about 30, where 0 is the yield stress. Single-parameter fracture mechanics apparently does not apply to non-hardening materials under fully plastic conditions, because the stress and strain fields in the vicinity of the crack tip are affected by configurations of both the body and the crack. The situation is more favourable in hardening materials, where singleparameter fracture mechanics may approximately apply also for the developed plasticity, provided that the body maintains a high level of stress triaxiality.

The reported experimental studies suggest that the configuration of the specimen and the crack (the crack depth and the specimen dimensions, in particular) affect the fracture toughness in a brittle state. However, the fact that this configuration can also influence the R-curve of ductile materials is not so well known.

Generally, the bigger the dimension of the crack, the smaller the resistance of the material to fracture will be. The R-curve obtained on specimens with rather long cracks is, as a rule, below the R-curve obtained on specimens with rather short cracks. For this reason, standards require that the relative crack lengths be within a comparatively narrow range of values for valid values of fracture toughness *Jin*.

#### **3.1. The J – Q theory**

288 Applied Fracture Mechanics

be written as:

In eq. (11), 2

*2.3.2. GS method* 

from Fig.2.

where 

 

*pD t*

constants), we can arrive at Eq. (12)

**3. Consideration of the constraint** 

,*n* – material constants

As a pipeline is a body of finite dimensions, stress

In the above equations the stress *σ<sup>0</sup>* can be substituted by the yield stress *Re* ; 0 0

i.e. a stress acting in the plane where the crack occurs. Referring to the R6 method (Milne et al., 1986), this stress for a pipe containing a longitudinal part-through thickness crack may

> <sup>1</sup> 2 2 *ac tt c*

The GS method was derived by Gajdoš and Srnec (Gajdoš & Srnec, 1994) on the basis of the limit transition of the *J*-integral, formally expressed for a semi-circular notch, to a crack, with the variation of the strain energy density along the notch circumference being approximated by the third power of the cosine function of the polar angle. If the stress-strain dependence

As mentioned above, the situation existing at the crack tip in conditions of small-scale yielding can be characterized by a single fracture parameter (e.g. K, J or ). This parameter can be used as a fracture criterion, independent of geometry. However, single-parameter fracture mechanics fails in cases of developed plasticity, where fracture toughness is a function not only of the material, but also of the dimensions and the geometry of the specimen. It is well known from the theory of fracture mechanics that for small-scale yielding the maximum stress existing at the crack tip in a non-hardening material is about 30, where 0 is the yield stress. Single-parameter fracture mechanics apparently does not apply to non-hardening materials under fully plastic conditions, because the stress and strain fields in the vicinity of the crack tip are affected by configurations of both the body

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

*<sup>K</sup> <sup>n</sup> <sup>J</sup> <sup>E</sup> <sup>n</sup>*

longitudinal part-through thickness crack it may be determined by relation (11).

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

1

is the nominal stress in the reduced cross-section of a body. For a pipe containing a

0

*n*

 

is further expressed by the Ramberg-Osgood relation (8), with

 

is the hoop stress, and the meaning of the symbols *a, c,* and *t* is clear

 *E* ;

*, n* – material

(12)

in Eqs. (9) and (10) is a nominal stress –

(11)

*0 = 0 / E*, (  Some researchers dealing with fracture mechanics tried to extend the theory of fracture mechanics beyond the boundaries of the assumptions of single-parameter fracture mechanics, introducing other parameters to provide a more accurate characterization of conditions at the crack tip. One of the parameters is the so-called T-stress, which is a uniform stress acting axially (in the direction of the *x-*axis) in front of the crack tip in an isotropic elastic material loaded by the first mode, i.e. the opening mode, of the load. In this case, the stress field in front of the crack tip may be written as:

$$
\sigma\_{ij} = \frac{K\_I}{\sqrt{2\pi r}} f\_{ij} \left(\Theta\right) + T \,\delta\_{1i}\delta\_{1j} \tag{13}
$$

The elastic T-stress heavily affects the shape of the plastic zone and the stress deep in this zone. T-stress values are linked with the stress biaxiality ratio, , defined as

$$
\beta = \frac{T\sqrt{\pi a}}{K\_I} \tag{14}
$$

It can be mentioned by way of illustration that the stress biaxiality ratio equals -1 for a through crack in an infinite plate loaded by a normal stress applied far away from the crack plane. By implication, this remote stress, , induces a T-stress in the direction of the *x*-axis, whose magnitude is -. In an elastic case, positive values of the T-stress generally lead to a high constraint under fully elastic conditions, whereas a geometry with a negative T-stress leads to a rapid drop in the constraint as the load rises. For different geometries, the stress biaxiality ratio can be used as a qualitative index for a relative constraint at the crack tip.

The so-called J – Q theory provides another approach to the extension of single-parameter fracture mechanics beyond the conditions of its validity. This theory aims to describe the stress field at the crack tip deep in the plastic zone. It is a well-known fact that if the smallstrain theory is used, the stress field at the crack tip in the plastic zone can be described by a power series, in which the so-called HRR solution is the leading term (Hutchinson, 1968), (Rice & Rosengren, 1968). The other terms of higher magnitudes, when summed up, provide a difference stress field, which approximately corresponds to a uniform hydrostatic shift of the stress field in front of the crack tip. It has become customary to designate the amplitude of this approximate difference stress field with letter Q, according to its authors O´Dowd and Shih (O´Dowd & Shih, 1991). O´Dowd and Shih defined the Q parameter as:

$$Q \equiv \frac{\sigma\_{yy} - \left(\sigma\_{yy}\right)\_{HRR \text{ or } T=0}}{\sigma\_0} \tag{15}$$

for = 0 and <sup>0</sup> <sup>2</sup> *<sup>r</sup> J* 

The parameter is equal to zero (Q = 0) under small-scale yielding conditions, but it acquires negative values as the load (and in consequence the strain) grows. Classical singleparameter fracture mechanics assumes that fracture toughness is a material constant. However, the J-Q theory suggests that the critical value of the *J*-integral for a given material depends on the Q parameter – i.e. *Jc* = *Jc(Q)* – and that fracture toughness is thus not some single-value quantity, but rather a function that defines the critical values of the *J*-integral and the Q parameter (Shih et al., 1993). Although the relation between critical J-integral values and the Q parameter shows a considerable scatter, the critical value of the *J*-integral tends in general to drop as the Q parameter increases in value.

The theory of single-parameter fracture mechanics assumes that the fracture toughness values obtained on laboratory specimens can be applied to a body. However, two-parameter approaches, such as the J-Q theory, reveal that the specimen must be tested at the same constraint as that of the body with a crack. In other words, the two geometries must have the same Q value at the moment of fracture, so that the corresponding critical values of the *J*-integral, *Jcr*, will be equal to each other. Since *Jcr* values are often scattered to a large extent, we cannot make a clear-cut prediction of this quantity. It is only possible to predict a certain range of plausible *Jcr* values for a given body or structure.

It should also be noted that the J-Q approach is only descriptive, and not predictive. This implies that the Q parameter quantifies the constraint at the crack tip, without providing any indication of the particular influence of the constraint on the fracture toughness. Twoparameter theories cannot be strictly correct as far as their universality is concerned, because they assume two degrees of freedom. Recent research into the influence of the constraint at the crack tip on fracture toughness indicates that geometries with a low constraint can in many cases be judged by a two-parameter theory, and geometries with a high constraint can be judged by a single-parameter theory (Ainsworth & O´Dowd, 1995).

### **3.2. Plastic constraint factor on yielding**

A simple procedure based on the use of the so-called plastic constraint factor on yielding, C, can be applied to determine the fracture conditions in a thin-walled pressure pipeline. The factor is given by the ratio of the stress needed to obtain plastic macrostrains under constraint conditions to the yield stress at a homogeneous uniaxial state of stress (Gajdoš et al., 2004). The C factor can be expressed by the relation (16)

$$C\_{\text{in}} = \frac{\sigma\_1}{\sigma\_{\text{HHH}}} \tag{16}$$

where *HMH* , the Huber-Mises-Hencky stress, is put equal to the yield stress.

290 Applied Fracture Mechanics

 = 0 and <sup>0</sup> <sup>2</sup> *<sup>r</sup> J* 

for 

power series, in which the so-called HRR solution is the leading term (Hutchinson, 1968), (Rice & Rosengren, 1968). The other terms of higher magnitudes, when summed up, provide a difference stress field, which approximately corresponds to a uniform hydrostatic shift of the stress field in front of the crack tip. It has become customary to designate the amplitude of this approximate difference stress field with letter Q, according to its authors O´Dowd

<sup>0</sup>

(15)

0

The parameter is equal to zero (Q = 0) under small-scale yielding conditions, but it acquires negative values as the load (and in consequence the strain) grows. Classical singleparameter fracture mechanics assumes that fracture toughness is a material constant. However, the J-Q theory suggests that the critical value of the *J*-integral for a given material depends on the Q parameter – i.e. *Jc* = *Jc(Q)* – and that fracture toughness is thus not some single-value quantity, but rather a function that defines the critical values of the *J*-integral and the Q parameter (Shih et al., 1993). Although the relation between critical J-integral values and the Q parameter shows a considerable scatter, the critical value of the *J*-integral

The theory of single-parameter fracture mechanics assumes that the fracture toughness values obtained on laboratory specimens can be applied to a body. However, two-parameter approaches, such as the J-Q theory, reveal that the specimen must be tested at the same constraint as that of the body with a crack. In other words, the two geometries must have the same Q value at the moment of fracture, so that the corresponding critical values of the *J*-integral, *Jcr*, will be equal to each other. Since *Jcr* values are often scattered to a large extent, we cannot make a clear-cut prediction of this quantity. It is only possible to predict a certain

It should also be noted that the J-Q approach is only descriptive, and not predictive. This implies that the Q parameter quantifies the constraint at the crack tip, without providing any indication of the particular influence of the constraint on the fracture toughness. Twoparameter theories cannot be strictly correct as far as their universality is concerned, because they assume two degrees of freedom. Recent research into the influence of the constraint at the crack tip on fracture toughness indicates that geometries with a low constraint can in many cases be judged by a two-parameter theory, and geometries with a high constraint can

A simple procedure based on the use of the so-called plastic constraint factor on yielding, C, can be applied to determine the fracture conditions in a thin-walled pressure pipeline. The

*yy yy HRR or T <sup>Q</sup>*

 

and Shih (O´Dowd & Shih, 1991). O´Dowd and Shih defined the Q parameter as:

tends in general to drop as the Q parameter increases in value.

range of plausible *Jcr* values for a given body or structure.

**3.2. Plastic constraint factor on yielding** 

be judged by a single-parameter theory (Ainsworth & O´Dowd, 1995).

Let us now consider the state of stress at the crack tip in a thick-walled body, where the stress perpendicular to the crack plane, 1, and the stress in the direction of the crack, 2, are equal, and the stress in the direction of the thickness of the body, 3, is governed by the expression 3=(1 + 2). Then, based on the HMH criterion and assumed elastic conditions ( 0.33), the plastic constraint factor *C* 3. If the stress in the thickness direction, 3, falls between 21 and zero (thin-walled body), the value of the plastic constraint factor will range between *C* = 1 and *C* = 3. This data can be used to assess the fracture conditions in gas pipelines with surface part-through cracks, employing a C-factor which has to be experimentally determined. After the C factor has been determined, the value of *C*0 would be used instead of the yield stress 0 in relations for calculating the *J*-integral. The C factor was experimentally investigated at the Institute of Theoretical and Applied Mechanics of the Academy of Sciences of the Czech Republic in the framework of a broader research project on the reliability and operational safety of high pressure gas pipelines. Fracture conditions were investigated on five pipe bodies, made of steels X52, X65 and X70, with cyclinginduced cracks. Data on the pipe bodies that were used, the cracks in the walls, and the mechanical and fracture-mechanical material properties of the bodies are given in Table 1.


**Table 1.** Summary of data on the assessment of the fracture behaviour of model pipe bodies

The rows in the table show the following data (top to bottom): body diameter *D*, body wall thickness *t*, half-length of a longitudinal part-through crack *c*, crack depth *a*, relative crack depth *a/t*, aspect ratio *a/c* of a semi-elliptical crack, fracture pressure *p*, ratio of fracture pressure *p* and pressure *p0.2* corresponding to the hoop stress at the yield stress, yield stress in the circumferential direction of the body 0, Ramberg-Osgood constant , Ramberg-Osgood exponent *n*, plastic constraint factor *C*, *J*-integral critical value *Jcr*, determined as *Jm* (corresponding to attaining the maximum force at the "force – force point displacement" curve), T-stress to yield stress ratio *T/*0, and the *Q* parameter. Values of 0, and *n* were derived from tensile tests, and the values of *Jcr* were derived from fracture tests run on CT specimens. Fracture pressure values *p* were read at the moment the ligament under the crack in the pipe body ruptured. Values for the plastic constraint factor on yielding, *C*, were determined on the basis of the *J*-integral in such a way that agreement was reached between the predicted and experimentally established fracture parameters for the given crack and fracture toughness of the material. The *J*-integral value was calculated using the GS method (Gajdoš & Srnec, 1994), on the one hand, and on the basis of the French nuclear code (RCC-MR, 1985), on the other.

It should be noted that in determining the C factor, the critical J-integral value established on CT specimens was considered – namely *Jcr* = 439 N/mm for steel X70, *Jcr* = 432 N/mm for steel X65 and *Jcr* = 487 N/mm for steel X52. It was found by a computational analysis of the CT specimens, employed to construct the R curve, that the Q parameter for these specimens was Q = 0.267. A comparison of this with the Q parameter for pipe bodies (Q -0.55 -0.65) reveals that the constraint in the CT specimens was much higher. This implies that the real fracture toughness – i.e. the critical value of the J-integral, *Jcr* – was higher in the pipe bodies. The real C factor for a cracked pipe body is lower, so that the *J-a* curve for a pipe body is steeper than the curve for CT specimens with a greater C factor (Gajdoš & Šperl, 2011). Due to this, the J-integral for the axial part-through crack reaches the corresponding higher fracture toughness (for a lower constraint) for the same crack depth as the J-integral with a higher C factor reaches lower fracture toughness (determined on CT specimens). The situation is illustrated in Fig.3.

The normalized T–stress values in Table 1 were obtained using the plane solution – i.e. a solution for a crack of infinite length oriented longitudinally along the pipe. The problem was solved at the Institute of Physics of Materials, Brno, by the finite element method. The solution consisted of two steps: (i) a corresponding FEM network was established and corresponding boundary conditions were formulated for each crack depth, (ii) the magnitudes of the stress intensity factor and the T-stress were calculated for each FEM network by means of the CRACK2D FEM system with hybrid crack elements. The Q parameter values were derived from the *Q – T/*0 curves obtained by O´Dowd and Shih (O´Dowd & Shih, 1991), by modified boundary layer analysis for different values of the strain coefficient (Ramberg-Osgood exponent, *n*). Strictly speaking, the *Q* parameter values from Table 1 do not correspond accurately to the values for the examined cracks, because the T-stresses were not computed for real semi-elliptical cracks, but for cracks spreading along the entire length of the pipe body (*a/c ≈ 0*). Nevertheless, due to the fact that the ratio of the depth to the surface half-length of the examined cracks (*a/c*) was close to zero (*a/c=0.053÷0.14*), we can assume that the differences between the real values of the *Q* parameter and the values listed in Table 1 will be small.

292 Applied Fracture Mechanics

MR, 1985), on the other.

situation is illustrated in Fig.3.

The rows in the table show the following data (top to bottom): body diameter *D*, body wall thickness *t*, half-length of a longitudinal part-through crack *c*, crack depth *a*, relative crack depth *a/t*, aspect ratio *a/c* of a semi-elliptical crack, fracture pressure *p*, ratio of fracture pressure *p* and pressure *p0.2* corresponding to the hoop stress at the yield stress, yield stress in the circumferential direction of the body 0, Ramberg-Osgood constant , Ramberg-Osgood exponent *n*, plastic constraint factor *C*, *J*-integral critical value *Jcr*, determined as *Jm* (corresponding to attaining the maximum force at the "force – force point displacement" curve), T-stress to yield stress ratio *T/*0, and the *Q* parameter. Values of 0, and *n* were derived from tensile tests, and the values of *Jcr* were derived from fracture tests run on CT specimens. Fracture pressure values *p* were read at the moment the ligament under the crack in the pipe body ruptured. Values for the plastic constraint factor on yielding, *C*, were determined on the basis of the *J*-integral in such a way that agreement was reached between the predicted and experimentally established fracture parameters for the given crack and fracture toughness of the material. The *J*-integral value was calculated using the GS method (Gajdoš & Srnec, 1994), on the one hand, and on the basis of the French nuclear code (RCC-

It should be noted that in determining the C factor, the critical J-integral value established on CT specimens was considered – namely *Jcr* = 439 N/mm for steel X70, *Jcr* = 432 N/mm for steel X65 and *Jcr* = 487 N/mm for steel X52. It was found by a computational analysis of the CT specimens, employed to construct the R curve, that the Q parameter for these specimens was Q = 0.267. A comparison of this with the Q parameter for pipe bodies (Q -0.55 -0.65) reveals that the constraint in the CT specimens was much higher. This implies that the real fracture toughness – i.e. the critical value of the J-integral, *Jcr* – was higher in the pipe bodies. The real C factor for a cracked pipe body is lower, so that the *J-a* curve for a pipe body is steeper than the curve for CT specimens with a greater C factor (Gajdoš & Šperl, 2011). Due to this, the J-integral for the axial part-through crack reaches the corresponding higher fracture toughness (for a lower constraint) for the same crack depth as the J-integral with a higher C factor reaches lower fracture toughness (determined on CT specimens). The

The normalized T–stress values in Table 1 were obtained using the plane solution – i.e. a solution for a crack of infinite length oriented longitudinally along the pipe. The problem was solved at the Institute of Physics of Materials, Brno, by the finite element method. The solution consisted of two steps: (i) a corresponding FEM network was established and corresponding boundary conditions were formulated for each crack depth, (ii) the magnitudes of the stress intensity factor and the T-stress were calculated for each FEM network by means of the CRACK2D FEM system with hybrid crack elements. The Q parameter values were derived from the *Q – T/*0 curves obtained by O´Dowd and Shih (O´Dowd & Shih, 1991), by modified boundary layer analysis for different values of the strain coefficient (Ramberg-Osgood exponent, *n*). Strictly speaking, the *Q* parameter values from Table 1 do not correspond accurately to the values for the examined cracks, because the T-stresses were not computed for real semi-elliptical cracks, but for cracks spreading along the entire length of the pipe body (*a/c ≈ 0*). Nevertheless, due to the fact that the ratio

**Figure 3.** Schematic J-a dependence, (i) for a CT specimen, and (ii) for a pipe with an axial part-through crack

The figures shown in Table 1 provide an idea of the nature of the changes both in the plastic constraint factor, *C,* and in the *Q* parameter brought about by changes in the relative crack length, *a/t*. The diagrams shown in Figs. 4 and 5 can be obtained on the basis of the graphic representation of the pairs *C – a/t* and *Q – a/t*.

These diagrams clearly show the trends of the changes in the two parameters with a change in the relative crack depth, *a/t*. It follows that, in the range of relative depths examined here (*a/t* = 0.57 to 0.72), the plastic constraint factor, *C,* and the *Q* parameter are a growing function of the relative crack depth, *a/t,* the *Q* – *a/t* dependence being rather weak. Expressed simply (i.e. linearly), the following relations are involved:

$$C = 2.56a/t + 0.53\tag{17}$$

$$Q = 0.32 \, a \not{/} t - 0.83 \,\tag{18}$$

The high scatter of the *C* and *Q* values is (i) due to differences in the cross section dimensions of the DN800 and DN1000 pipes and (ii) due to different values of the strain exponent *n* in the Ramberg-Osgood relation, because the pipes were made of three different materials.

**Figure 4.** Plastic constraint factor, *C*, as affected by the relative crack depth, *a/t*

**Figure 5.** Parameter Q, as affected by the relative crack depth, *a/t*

Table 1 lists explicit values of *Q* and *C* for all examined cracks in the pipes that were used, and thus a graphic representation of the *C* – *Q* relation can be plotted (Fig. 6). In the region where the established values of parameter *Q* for the examined pipe bodies are found, the *C – Q* relation can be most simply described by the linear relation:

$$\mathbf{C} = \mathbf{3}.4 \,\, \mathbf{Q} + \mathbf{4}.\mathbf{3} \tag{19}$$

**Figure 6.** Dependence of the plastic constraint factor, *C*, on parameter *Q*

The relation implies that the plastic constraint factor, *C,* decreases with a decreasing value (increasing negative value) of parameter *Q*. The observed scatter of the experimental points is mainly due to inaccuracies of the T-stress estimate, which result from the substitution of the real conditions of cracks of certain lengths by the plane solution used in the task (crack along the entire length of the body).

## **4. Fracture toughness**

294 Applied Fracture Mechanics

1





Q parametr



1,2

1,4

1,6

1,8

plastic constraint factor C

2

2,2

2,4

Pipes: X52 - 820/10.2 X65 - 820/10.7

2,6

**Figure 4.** Plastic constraint factor, *C*, as affected by the relative crack depth, *a/t*

Pipes: X52 - 820/10.2 X65 - 820/10.7

X70 - 1018/11.7 X70

0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,75 0,8 relative crack depth a/t

0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,75 relative crack depth, *a/t*

X70 - 1018/11.7 X70 X70

X70

X65

C = 2.56(a/t)+0.53

X52

Q = 0.32(a/t) - 0.83

X65 X65

X52

X65

**Figure 5.** Parameter Q, as affected by the relative crack depth, *a/t*

If we are to evaluate the strength reliability and the remaining life of gas pipelines, we need to get an accurate picture of the properties of the material that the gas pipelines are made of. In the case of gas pipelines operated for different periods of time, we should be aware that the properties of the material of a used pipeline will be different from the initial properties.

In order to pass a qualified judgement on the reliability of a gas pipeline, we should know the true properties that the material displays at the time when the gas pipeline is being examined. The fracture properties can be characterized with sufficient generality by the fracture toughness, determined by quantities *Jin, J0.2,* or *Jm*, where *Jin* is the so-called initiation magnitude of the J integral for a stable subcritical crack extension; *J0.2* is the J magnitude corresponding to the real crack extension *∆a* = 0.2 mm, and *Jm* is the magnitude of the J integral corresponding to attaining the maximum force at the "force – force point displacement" curve. We should point here to two aspects of fracture toughness that can be encountered when dealing with pressure pipelines. One of them is the effect of pipe band straightening, and the other is the effect of stress corrosion cracks on fracture toughness.

## **4.1. The effect of straightening**

Fracture toughness tests are carried out with fracture mechanical specimens, e.g. single edge notched bend (SENB) specimens or compact tension (CT) specimens. Both types are plane specimens. When investigating the integrity of thin-walled pressure pipelines, we face the problem of ensuring the planeness of the semiproducts for manufacturing the fracture mechanical specimens. The only way is press straightening of pipe bands taken from the pipe that is under investigation. As a consequence of the plastic deformation that the semiproduct undergoes during straightening, internal stresses are induced not only in the semiproduct but also in the final specimens. Therefore there are still some doubts about the reliability of the fracture toughness characteristics obtained with straightened specimens. In order to verify this matter, Gajdoš and Šperl (Gajdoš & Šperl, 2012) carried out an experimental investigation of fracture toughness, as determined using press straightened CT specimens and curved CT specimens, manufactured directly from a pipe band, i.e. ensuring that their natural curvature and wall thickness were preserved.

The so-called curved CT specimens (see Fig. 7) to some extent simulate the stress conditions in the pipe wall upon loading by internal pressure. In order to apply a circumferential force on these specimens, we used a special testing rig, similar to that developed by Evans (Evans et al., 1995). The rig is shown in Fig. 8.

**Figure 7.** The shape and dimensions of the curved CT specimens

**Figure 8.** The testing rig for circumferential loading of a curved CT specimen

**4.1. The effect of straightening** 

et al., 1995). The rig is shown in Fig. 8.

magnitude of the J integral for a stable subcritical crack extension; *J0.2* is the J magnitude corresponding to the real crack extension *∆a* = 0.2 mm, and *Jm* is the magnitude of the J integral corresponding to attaining the maximum force at the "force – force point displacement" curve. We should point here to two aspects of fracture toughness that can be encountered when dealing with pressure pipelines. One of them is the effect of pipe band straightening, and the other is the effect of stress corrosion cracks on fracture toughness.

Fracture toughness tests are carried out with fracture mechanical specimens, e.g. single edge notched bend (SENB) specimens or compact tension (CT) specimens. Both types are plane specimens. When investigating the integrity of thin-walled pressure pipelines, we face the problem of ensuring the planeness of the semiproducts for manufacturing the fracture mechanical specimens. The only way is press straightening of pipe bands taken from the pipe that is under investigation. As a consequence of the plastic deformation that the semiproduct undergoes during straightening, internal stresses are induced not only in the semiproduct but also in the final specimens. Therefore there are still some doubts about the reliability of the fracture toughness characteristics obtained with straightened specimens. In order to verify this matter, Gajdoš and Šperl (Gajdoš & Šperl, 2012) carried out an experimental investigation of fracture toughness, as determined using press straightened CT specimens and curved CT specimens, manufactured directly from a pipe band, i.e. ensuring

The so-called curved CT specimens (see Fig. 7) to some extent simulate the stress conditions in the pipe wall upon loading by internal pressure. In order to apply a circumferential force on these specimens, we used a special testing rig, similar to that developed by Evans (Evans

that their natural curvature and wall thickness were preserved.

**Figure 7.** The shape and dimensions of the curved CT specimens

It is clear that the testing rig is tied with only certain cross – sectional dimensions of a pipe. In the case considered here, the dimensions corresponded to a pipe 266 mm in outside diameter and 8 mm in wall thickness. The material of the pipe was low-C steel CSN 411353. Static tests of the steel provided the following results: Rp0.2 = 286 MPa; Rm = 426 MPa; A5 = 31%; Z = 54%. The Ramberg-Osgood constants had the following values: *α* = 6.23; *n* = 5.87; *σ<sup>0</sup>* = 286 MPa.

First, fracture toughness tests were carried out by an ordinary procedure, as specified in the ASTM standard (E 1820-01, 2001), on CT specimens manufactured from a press-straightened band taken from the pipe.

The result in the form of an R-curve is presented in Fig. 9. One point (designated by a triangle) has not been included in the regression analysis because it was outside the valid area of the diagram. The positions of *Jin* and *J0.2* at the R-curve are clearly defined from the construction of the R-curve, the blunting line and the 0.2 offset line; the position of *Jm* is also indicated in the diagram, and it represents the mean of six values obtained on specimens where the maximum force was attained in loading the specimens. The R-curve determined by the least-square method is described by a power function (20):

$$J = 327.05 \left(\Delta a\right)^{0.6406} \tag{20}$$

**Figure 9.** R curve for CT specimens manufactured from a press-straightened semi-product

Eight specimens were used for fracture-mechanical tests of curved CT specimens. Cracks were cycled up at a frequency of 4 Hz, using the testing rig, Fig. 8. The stress state in the inner side of a specimen was bigger than in the outer side, because of the bending moment induced by the out-of-axis action of the vertical component of the tangential force with regard to the intersection of the middle cylindrical area of the specimen with the symmetry plane of the specimen. For this reason, the growth rate of the fatigue crack was higher in the inner side than in the outer side.

This resulted in uneven length of the fatigue crack on the two sides of a specimen after finishing the cycling up. On one half of the specimens, a slant front of the starting notch was therefore made in such a way that the notch was 1 mm deeper on the outer side. By this operation, a much more even front of the fatigue crack was obtained. This is clearly demonstrated in Figs. 10 and 11, which show the fracture surfaces of specimens with a straight front and a slant front of the starting notch. In the two photographs, we can observe areas corresponding to the notch, fatigue, static crack extension and final break after the specimens were cooled down in liquid nitrogen.

On the basis of the finite element analysis and the compliance measurements made by Evans (Evans et al., 1995), it was concluded that the use of standard expressions for determining K factor will not cause error greater than 4% for curved CT specimens. By proceeding in the same way as in standard *J – ∆a* testing, an R-curve was obtained for curved CT specimens. It is described by a power function (21), and is presented in Fig. 12.

$$J = 278.21 \text{(}\Delta a\text{)}^{0.525} \tag{21}$$

**Figure 10.** Fracture surface – straight front of the notch

0

100

200

300

J (N/mm)

400

500

600

inner side than in the outer side.

Jin = 16,5 N/mm

specimens were cooled down in liquid nitrogen.

**Figure 9.** R curve for CT specimens manufactured from a press-straightened semi-product

J = 4Rfs.∆a Rfs = 353 MPa

J0,2 = 154 N/mm

Eight specimens were used for fracture-mechanical tests of curved CT specimens. Cracks were cycled up at a frequency of 4 Hz, using the testing rig, Fig. 8. The stress state in the inner side of a specimen was bigger than in the outer side, because of the bending moment induced by the out-of-axis action of the vertical component of the tangential force with regard to the intersection of the middle cylindrical area of the specimen with the symmetry plane of the specimen. For this reason, the growth rate of the fatigue crack was higher in the

Jm = 261 N/mm

**J = 327,05(∆a)0,6406**

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 2,2 2,4 delta a (mm)

This resulted in uneven length of the fatigue crack on the two sides of a specimen after finishing the cycling up. On one half of the specimens, a slant front of the starting notch was therefore made in such a way that the notch was 1 mm deeper on the outer side. By this operation, a much more even front of the fatigue crack was obtained. This is clearly demonstrated in Figs. 10 and 11, which show the fracture surfaces of specimens with a straight front and a slant front of the starting notch. In the two photographs, we can observe areas corresponding to the notch, fatigue, static crack extension and final break after the

On the basis of the finite element analysis and the compliance measurements made by Evans (Evans et al., 1995), it was concluded that the use of standard expressions for determining K factor will not cause error greater than 4% for curved CT specimens. By proceeding in the same way as in standard *J – ∆a* testing, an R-curve was obtained for curved CT specimens. It is described by a power function (21), and is presented in Fig. 12.

0.525 *J a* 278.21 (21)

**Figure 11.** Fracture surface – slant front of the notch

A comparison of the two R-curves shows that the decline of the R-curve obtained with the curved CT specimens is less than the decline of the R-curve obtained with plane, i.e. straightened, CT specimens. The higher decline of the R-curve with the straightened CT specimens is most probably connected with work hardening of a semiproduct during straightening. In the mathematical description of the R-curve of the curved CT specimens, not only the exponent but also the constant is less than for the standard R-curve. This means that the standard R-curve is situated above the R-curve of the curved CT specimens. However, the lower position of the R- curve for the curved CT specimens does not mean significantly lower magnitudes of the fracture toughness characteristics. For example, the *Jm* value is lower by 1.1%, the *J0.2* is lower by less than 3%, and the magnitude *Jin* is even higher than the respective characteristics for plane (straightened) CT specimens. In absolute units, the difference is 2.9 N/mm for *Jm* and 4.6 N/mm for *J0.2*. There is a significant difference in *Jin* , namely 29.7 N/mm in favour of the curved CT specimens.

**Figure 12.** R-curve for curved CT specimens

By accounting the scatter of the results in the form of the *J - ∆a* points, caused both by a natural process of subcritical crack growth and by inaccuracies in determining the J-integral and, in particular, the crack extension during monotonic loading of a specimen, it can be stated with a high level of reliability that the fracture toughness of a pipe material determined on straightened CT specimens is practically the same as the fracture toughness determined on curved CT specimens.

### **4.2. The effect of stress corrosion**

(Gajdoš et al., 2011) investigated the stress corrosion fracture toughness of gas pipeline material, and compared it with fatigue fracture toughness. The material used for the investigation was a low-C steel according to CSN 411353 (equivalent to ASTM A519), containing 0.17% C, 0.035% P, 0.035% S. The test CT specimens were manufactured from a real pipe section cut out from a DN 150 gas pipeline 4.5 mm in wall thickness while it was being repaired after 20 years of operation. Before the CT specimens were manufactured, the pipe section was press straightened. Owing to the small thickness of the specimens (a low constraint), the fracture toughness values cannot be qualified to represent the real fracture toughness values. However, they can be used as a comparative measure of fracture toughness, thus enabling quantification of the effect of stress corrosion cracks on the apparent fracture toughness.

The CT specimens were first cyclically loaded by a routine procedure used in determining fracture toughness; the only difference was that the cycling was stopped when the growth of the fatigue crack reached approximately the magnitude ∆aFA ≈ 1.5 mm. After that, the CT specimens were put into the stress-corrosion (SC) crack generator with an acidic solution according to the NACE Standard (NACE Standard TM0177, 2005). This solution consisted of 50 g NaCl (sodium chloride) + 5 g CH3COOH (acetic acid) + 945 g H2O, and during the generating process it was bubbled by H2S (hydrogen sulphide). A constant force F of 3 kN was applied to the specimens. The corresponding level of the nominal stress (tension and bending) at the fatigue crack tip exceeded the yield stress Rp0.2 by about 25%. The crack length increment due to stress-corrosion ∆aSC was determined with the help of the relations for elastic crack-edge displacements at CT specimens. In total, three groups of CT specimens were prepared. The first group (A) was the reference group; the specimens from this group contained only the fatigue crack. The second group of CT specimens (B) contained specimens that were left freely in air at the indoor temperature for two weeks after being removed from the SC crack generator, and were then subjected to fracture toughness tests. The specimens from the third group (C) were tested immediately after they had been removed from the SC crack generator (the time difference between testing the first specimen and the last specimen being approximately 20 minutes).

300 Applied Fracture Mechanics

0

100

200

300

J (N/mm)

400

500

600

**Figure 12.** R-curve for curved CT specimens

Jin = 46,2 N/mm

J = 4 Rfs (∆ a) Rfs = 353 MPa

J0,2 = 149,4 N/mm

determined on curved CT specimens.

**4.2. The effect of stress corrosion** 

apparent fracture toughness.

By accounting the scatter of the results in the form of the *J - ∆a* points, caused both by a natural process of subcritical crack growth and by inaccuracies in determining the J-integral and, in particular, the crack extension during monotonic loading of a specimen, it can be stated with a high level of reliability that the fracture toughness of a pipe material determined on straightened CT specimens is practically the same as the fracture toughness

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 2,2 2,4 delta a (mm)

Jm = 258,1 N/mm

**J = 278.21 (∆ a)0.525**

(Gajdoš et al., 2011) investigated the stress corrosion fracture toughness of gas pipeline material, and compared it with fatigue fracture toughness. The material used for the investigation was a low-C steel according to CSN 411353 (equivalent to ASTM A519), containing 0.17% C, 0.035% P, 0.035% S. The test CT specimens were manufactured from a real pipe section cut out from a DN 150 gas pipeline 4.5 mm in wall thickness while it was being repaired after 20 years of operation. Before the CT specimens were manufactured, the pipe section was press straightened. Owing to the small thickness of the specimens (a low constraint), the fracture toughness values cannot be qualified to represent the real fracture toughness values. However, they can be used as a comparative measure of fracture toughness, thus enabling quantification of the effect of stress corrosion cracks on the

The CT specimens were first cyclically loaded by a routine procedure used in determining fracture toughness; the only difference was that the cycling was stopped when the growth of the fatigue crack reached approximately the magnitude ∆aFA ≈ 1.5 mm. After that, the CT The results confirmed that the fracture resistance of a component (given by the apparent fracture toughness) depends not only on the material of the component and on the crack tip constraint (the thickness of the wall of the component) but also on the origin of the crack (fatigue, stress corrosion), and thus on the corresponding crack growth mechanism. In contradiction with the opinion that low-C steels are not susceptible to stress corrosion cracking our results showed that under conditions specified in (NACE Standard TM0177, 2005) stress corrosion cracks can also be generated from fatigue cracks in low-C steels such as CSN 411353. Unlike a fatigue crack, the occurrence of a stress-corrosion crack in a component means a significant decrease in the fracture toughness characteristics while the crack is exposed to stress corrosion conditions, and a partial "recovery" of the fracture toughness when the stress corrosion conditions are removed. The results for all three groups of specimens are summarized in Fig. 13.

**Figure 13.** A bar chart of the J integral values for specimens of groups A, B and C

As this figure shows, the stress corrosion fracture toughness characteristics for the low-C steel CSN 411353 were lower than the fatigue fracture toughness characteristics by a factor ranging between 4.5 ( *Jm* value) and 5.7 ( *Jin* value). However, a two-week recovery period made it possible to recover their fracture properties to some extent, namely the J-integral *Jm* to almost 80%, the J-integral *J0.2* to about 60%, and the J-integral *Jin* to about 22% of the fatigue crack J-integral values. It follows from here that in evaluating the reliability of gas pipelines it is always necessary to examine the character of the cracks in the pipe wall, and in the case of stress corrosion cracks to take into account that the fracture toughness can be drastically lower than the values determined on specimens with cracks of fatigue origin.

## **5. Burst tests**

An experimental verification of the fracture conditions of gas pipelines can be made most accurately on a test pipe body cut out of the gas pipeline to be examined. When deciding on the length of the test pipe body, we should bear in mind that the working length of the body (characterized by the absence of stress effects from welded-on bottoms) will be shorter by *2* x *2.5(Rt) 3.5(Dt).* It is usually sufficient for the distance between the welds of dished bottoms to be at least *3.5D*. This length permits a number of starting cuts to be placed axially along the length of the body. The cuts are made to initiate crack growth when the body is subsequently pressurized by a fluctuating pressure. The cuts can be made in several ways, one of which uses a thin grinding wheel. The smallest real functional thickness of such a wheel is about 1.2 mm, and the corresponding width of the cuts made with it is approximately 1.5 mm. Depending on the type of pipes of which gas pipelines are built (seamless, spirally welded, longitudinally welded), the starting cuts can be provided in the base material, in the transition region or in the weld metal, their orientation being axial, circumferential or along the spiral weld.

### **5.1. Preparation of test pipe bodies**

It is appropriate to relate the surface length of the cuts to the wall thickness of the pipe body. Testing the body for the danger posed by so-called long cracks should be carried out with crack lengths not exceeding twenty times the wall thickness of the pipe body. The situation with the depth of the starting cuts is different. The depth of an initiated fatigue crack must be at least 0.5 mm along the whole perimeter of the cut tip, so that the cut with the initiated crack at its tip can be considered as a crack after the pipe body has been subjected to cycling. This value follows from the work done by Smith and Miller (Smith & Miller, 1977). If such a crack *at* in size finds itself in a notch root defined by depth *av* and radius of the roundness (see Fig. 14), this configuration can be regarded as a surface crack *ae* in depth, where

$$\begin{aligned} a\_e &= \left( 1 + 7.69 \sqrt{\frac{a\_v}{\rho}} \right) a\_t \quad \text{for} \quad a\_t < 0.13 \sqrt{a\_v \rho} \\\ a\_e &= a\_v + a\_t \quad \text{for} \quad a\_t \ge 0.13 \sqrt{a\_v \rho} \end{aligned} \tag{22}$$

It is evident that for 0.13 *t v a a* , a cut with a crack along the perimeter of the cut tip can be taken for a crack with a depth of *av + at*. For the cut width *2* = 1.5–2.0 mm and the notch depth *av* = 6–10 mm (in relation to the wall thickness), we find that the fatigue increment of the size of the initiated crack, *at*, should be greater than about 0.5 mm.

**Figure 14.** Substitution of a notch with a crack by the equivalent crack

302 Applied Fracture Mechanics

**5. Burst tests** 

circumferential or along the spiral weld.

**5.1. Preparation of test pipe bodies** 

x *2.5(Rt) 3.5*

As this figure shows, the stress corrosion fracture toughness characteristics for the low-C steel CSN 411353 were lower than the fatigue fracture toughness characteristics by a factor ranging between 4.5 ( *Jm* value) and 5.7 ( *Jin* value). However, a two-week recovery period made it possible to recover their fracture properties to some extent, namely the J-integral *Jm* to almost 80%, the J-integral *J0.2* to about 60%, and the J-integral *Jin* to about 22% of the fatigue crack J-integral values. It follows from here that in evaluating the reliability of gas pipelines it is always necessary to examine the character of the cracks in the pipe wall, and in the case of stress corrosion cracks to take into account that the fracture toughness can be drastically lower than the values determined on specimens with cracks of fatigue origin.

An experimental verification of the fracture conditions of gas pipelines can be made most accurately on a test pipe body cut out of the gas pipeline to be examined. When deciding on the length of the test pipe body, we should bear in mind that the working length of the body (characterized by the absence of stress effects from welded-on bottoms) will be shorter by *2*

bottoms to be at least *3.5D*. This length permits a number of starting cuts to be placed axially along the length of the body. The cuts are made to initiate crack growth when the body is subsequently pressurized by a fluctuating pressure. The cuts can be made in several ways, one of which uses a thin grinding wheel. The smallest real functional thickness of such a wheel is about 1.2 mm, and the corresponding width of the cuts made with it is approximately 1.5 mm. Depending on the type of pipes of which gas pipelines are built (seamless, spirally welded, longitudinally welded), the starting cuts can be provided in the base material, in the transition region or in the weld metal, their orientation being axial,

It is appropriate to relate the surface length of the cuts to the wall thickness of the pipe body. Testing the body for the danger posed by so-called long cracks should be carried out with crack lengths not exceeding twenty times the wall thickness of the pipe body. The situation with the depth of the starting cuts is different. The depth of an initiated fatigue crack must be at least 0.5 mm along the whole perimeter of the cut tip, so that the cut with the initiated crack at its tip can be considered as a crack after the pipe body has been subjected to cycling. This value follows from the work done by Smith and Miller (Smith & Miller, 1977). If such a crack *at* in size finds itself in a notch root defined by depth *av* and radius of the roundness

(see Fig. 14), this configuration can be regarded as a surface crack *ae* in depth, where

*v*

*a*

 

*e vt t v*

*a a a for a a*

1 7.69 0.13

*e ttv*

*a a for a a*

0.13

*(Dt).* It is usually sufficient for the distance between the welds of dished

(22)

As described in paragraph 3.2, three test pipe bodies, made of X52, X65 and X70 steels, were provided with working slits and so-called check slits, which were of the same surface length as the working slits but their depth was greater. These check slits functioned as a safety measure to prevent cracks that developed at the working slits from penetrating through the pipe wall. For illustration, a DN1000 test pipe body with a working length of 3.5 m is shown in Fig. 15. The check slits are denoted in Fig. 15 by a supplementary letter K. The material of the test pipe body is a thermo-mechanically treated steel X70 according to API specification. The pipe is spirally welded, the weld being inclined at an angle of = 62° to the pipe axis. It is provided with starting cuts oriented either axially or in the direction of the strip axis (i.e. in the direction of the spiral) and then along or inside the spiral weld. The cuts differ in length (*2c* = 115 mm or 230mm) and in depth (*a* = 5, 6.5, 7, and 7.5 mm). We are particularly interested in axial (longitudinal) slits situated aside welds, because these are sites where axial cracks will be formed in the basic material of the pipe.

Efforts were made in the fracture tests to keep the circumferential fracture stress below the yield stress, because the operating stress in gas pipelines is virtually around one half of the yield stress (and does not exceed two-thirds of the yield stress even in intrastate highpressure gas transmission pipelines). Calculations reveal that in order to comply with this, the depth of the axial semi-elliptical cracks should be greater than one half of the wall thickness. Oblique cracks should be even deeper, as the normal stress component opening these cracks is smaller. If the crack depth is to have a certain magnitude before the fracture test is begun, the depth of the starting slit should be smaller than this magnitude by the fatigue extension of the crack along the perimeter of the slit tip. At the same time, we should bear in mind that the greater the fatigue extension of the crack, the better the agreement with a real crack.

**Figure 15.** Test pipe body with the starting cuts marked

## **5.2. Prediction of fracture parameters**

After the starting slits were made, the test pipes were subjected to water pressure cycling to produce fatigue cracks in the tips of the starting slits. The cycling was carried out in a pressurizing system, which included a high-pressure water pump, a collecting tank, a regulator designed to control the amount of water that was supplied and, consequently, the rate at which the pressure is increased in the pipe section. This was effected by opening bypass valves.

In cycling the cracks, the water pressure fluctuated between pmin = 1.5 MPa and pmax = 5.3 MPa, and the number of pressure cycles was between 3 000 and 4 000. The period of a cycle was approximately 150 seconds. The cycling continued until a crack initiated in one of the check slits became a through crack. This moment was easy to detect, because it was accompanied by a water leak. By choosing an appropriate difference between the depths of the working slits and the check slits it was possible to obtain a working crack depth (= starting slit depth + fatigue crack extension) of approximately the required size. To run a test for a fracture, however, it was necessary to remove the check slit which had penetrated through the wall of the test pipe from the body shell and to repair the shell, e.g. by welding a patch in it.

After removing the check slit with a crack which penetrated through the wall, and repairing the shell of the test pipe, the pipe was loaded by increasing the water pressure to burst. The test procedure, which was common for all test pipes, will now be briefly described for the DN1000 pipe shown in Fig. 15. As the figure suggests, slits A, A´, B and B´ were oriented along the axis of the pipe. The nominal length of notches B, B´ was twice as long as notches A, A´, but notches B, B´ were shallower. As was mentioned above, the cracks at the slit tips were extended by fluctuating water pressure, and this proceeded until the cracks from the check slits (BK, BK´) grew through the wall and a water leak developed. Then the damaged parts of the shell were cut out, patches were welded in their place, and the test pipe was monotonically loaded to fracture at the location of crack B or B´. The burst of the test pipe at crack B is shown in Figs. 16 and 17 (as a detail). A part of the fracture surface is shown in Fig. 18.

**Figure 16.** Burst initiated on slit B with a fatigue crack

304 Applied Fracture Mechanics

pass valves.

a patch in it.

**Figure 15.** Test pipe body with the starting cuts marked

After the starting slits were made, the test pipes were subjected to water pressure cycling to produce fatigue cracks in the tips of the starting slits. The cycling was carried out in a pressurizing system, which included a high-pressure water pump, a collecting tank, a regulator designed to control the amount of water that was supplied and, consequently, the rate at which the pressure is increased in the pipe section. This was effected by opening by-

In cycling the cracks, the water pressure fluctuated between pmin = 1.5 MPa and pmax = 5.3 MPa, and the number of pressure cycles was between 3 000 and 4 000. The period of a cycle was approximately 150 seconds. The cycling continued until a crack initiated in one of the check slits became a through crack. This moment was easy to detect, because it was accompanied by a water leak. By choosing an appropriate difference between the depths of the working slits and the check slits it was possible to obtain a working crack depth (= starting slit depth + fatigue crack extension) of approximately the required size. To run a test for a fracture, however, it was necessary to remove the check slit which had penetrated through the wall of the test pipe from the body shell and to repair the shell, e.g. by welding

After removing the check slit with a crack which penetrated through the wall, and repairing the shell of the test pipe, the pipe was loaded by increasing the water pressure to burst. The test procedure, which was common for all test pipes, will now be briefly described for the DN1000 pipe shown in Fig. 15. As the figure suggests, slits A, A´, B and B´ were oriented along the axis of the pipe. The nominal length of notches B, B´ was twice as long as notches A, A´, but notches B, B´ were shallower. As was mentioned above, the cracks at the slit tips were extended by fluctuating water pressure, and this proceeded until the cracks from the check slits (BK, BK´) grew through the wall and a water leak developed. Then the damaged parts of the shell were cut out, patches were welded in their place, and the test pipe was monotonically

**5.2. Prediction of fracture parameters** 

**Figure 17.** Burst initiated on slit B – a detail

**Figure 18.** A part of the fracture surface of crack B (fatigue region ~ 2.4 mm)

Evidently, at the instant of fracture the crack spread not only through the remaining ligament, but also lengthwise. After removing the part of the pipe shell with crack B, a patch was welded in and the second burst test followed. Table 2 extracts from Table 1 the numerical values of the geometrical parameters, the J-integral fracture values, the Ramberg-Osgood constants, the fracture pressure and the fracture depth for cracks B and B´, respectively.


**Table 2.** Some characteristics referring to crack B and crack B´

It should be noted that Table 2 includes the Ramberg-Osgood constants for the circumferential direction of the test pipe, with the crack oriented axially in the pipe. This is because the stress-strain properties perpendicular to the crack plane are crucial in determining the J-integral for an axial crack. The stress-strain dependence in the circumferential direction should therefore be taken into account where an axial orientation of the crack is concerned. The most important fracture test results from the viewpoint of the fracture conditions are the magnitudes of the fracture pressure, *pf*, and the fracture depth, *af*, for a given crack length *2c*. It follows from Table 2 that *pf* = 9.55 MPa and *af* = 7.1 mm for crack B, and *pf* = 9.86 MPa and *af* = 6.7 mm for crack B´. These values are also shown in the last two columns of Table 1.

Now let us predict the fracture conditions according to engineering approaches, and compare the prediction results with the real fracture parameter values (pressure, crack depth). The procedure for verifying the engineering methods for the predictions involves determining either the fracture stress for a given (fracture) crack depth, or the fracture crack depth for a given (fracture) pressure. To illustrate this, we select the latter case – i.e. determining the fracture depth of a crack for a given (fracture) pressure. Fig. 19 shows the J-integral vs. crack B depth dependences, as determined by the FC and GS predictions for the fracture hoop stress given by the measured fracture pressure. When using equations (9), (10), and (12) to determine J-integrals, the following parameters were used for the calculation: *D* = 1018 mm; *t* = 11.7 mm ; *p* = *pf* = 9.55 MPa; *c* = 115 mm; α = 5.92; *n* = 9.62; σ0 = 2.07×536 = 1110 MPa (i.e. C = 2.07). Fig. 20 shows similar dependences for crack B´.

**Figure 19.** Prediction of the fracture depth for crack B ( *p* = *pf* = 9.55 MPa and C = 2.07)

respectively.

CRACK DIMENSIONS

FRACTURE TOUGHNESS

FRACTURE PRESSURE

last two columns of Table 1.

dependences for crack B´.

RAMBERG-OSGOOD PARAMETERS

**Table 2.** Some characteristics referring to crack B and crack B´

Evidently, at the instant of fracture the crack spread not only through the remaining ligament, but also lengthwise. After removing the part of the pipe shell with crack B, a patch was welded in and the second burst test followed. Table 2 extracts from Table 1 the numerical values of the geometrical parameters, the J-integral fracture values, the Ramberg-Osgood constants, the fracture pressure and the fracture depth for cracks B and B´,

Characteristics Crack B Crack B´

half-length, *c* (mm) 115 127 depth in fracture, *af* (mm) 7.1 6.7

*Jcr* = *Jm* (N/mm) 439 439

*pf* (MPa) 9.55 9.86

It should be noted that Table 2 includes the Ramberg-Osgood constants for the circumferential direction of the test pipe, with the crack oriented axially in the pipe. This is because the stress-strain properties perpendicular to the crack plane are crucial in determining the J-integral for an axial crack. The stress-strain dependence in the circumferential direction should therefore be taken into account where an axial orientation of the crack is concerned. The most important fracture test results from the viewpoint of the fracture conditions are the magnitudes of the fracture pressure, *pf*, and the fracture depth, *af*, for a given crack length *2c*. It follows from Table 2 that *pf* = 9.55 MPa and *af* = 7.1 mm for crack B, and *pf* = 9.86 MPa and *af* = 6.7 mm for crack B´. These values are also shown in the

Now let us predict the fracture conditions according to engineering approaches, and compare the prediction results with the real fracture parameter values (pressure, crack depth). The procedure for verifying the engineering methods for the predictions involves determining either the fracture stress for a given (fracture) crack depth, or the fracture crack depth for a given (fracture) pressure. To illustrate this, we select the latter case – i.e. determining the fracture depth of a crack for a given (fracture) pressure. Fig. 19 shows the J-integral vs. crack B depth dependences, as determined by the FC and GS predictions for the fracture hoop stress given by the measured fracture pressure. When using equations (9), (10), and (12) to determine J-integrals, the following parameters were used for the calculation: *D* = 1018 mm; *t* = 11.7 mm ; *p* = *pf* = 9.55 MPa; *c* = 115 mm; α = 5.92; *n* = 9.62; σ0 = 2.07×536 = 1110 MPa (i.e. C = 2.07). Fig. 20 shows similar

α / *n* /*σ0* (MPa) 5.92 / 9.62 /536 5.92 / 9.62 /536

**Figure 20.** Prediction of the fracture depth for crack B´ ( *p* = *pf* = 9.86 MPa and C = 2.0)

The same computational parameters as those employed in the case of crack B were used in the equations to determine the J-integral according to the FC and GS methods, with the exception of the fracture pressure (*pf* = 9.86 MPa ), the crack half-length (*c* = 127 mm) and factor C ( C = 2.0). As is evident from Fig. 19, the intersection of the straight line *J* = *Jcr* = 439 N/mm with the two *J* − *a* curves gives the value *acr* ≈ 7.05 mm, which is well consistent with crack depth B *acr* = 7.1 mm, established experimentally. Similarly, the intersection of the straight line *J* = *Jcr* = 439 N/mm with the *J*−*a* curves according to the FC and GS procedures in Fig. 20 shows the fracture crack depth *acr* to be virtually identical to the experimentally found fracture depth *af* = 6.7 mm. For other test pipes, namely DIA 820/10.7, made of X65 steel, and DIA 820/10.2, made of X52 steel, various magnitudes of the plastic constraint factor C were obtained to achieve good agreement of the geometric parameters at fracture with the experimental parameters. They are illustrated in Fig. 4. The conclusion can thus be drawn that very good agreement of the fracture parameter values predicted by the FC and GS engineering approaches with the values found experimentally can be achieved when using the plastic constraint factor on yielding, *C*, at the level *C* = 2. If a higher value of the C factor provides more precise results, the use of the value C = 2 will yield a conservative result.

## **6. Conclusion**

A specific fracture-mechanics-based procedure for assessing the integrity of pressurized thin-walled cylindrical shells made from steels includes a theoretical treatment for cracks in pipes. On the basis of both experimental work and a fracture-mechanical evaluation of experimental results, an engineering method has been worked out for assessing the geometrical parameters of critical axial crack-like defects in a high-pressure gas pipeline wall for a given internal pressure of a gas. The method makes use of simple approximate expressions for determining fracture parameters *K, J*, and it accommodates the crack tip constraint effects by means of the so-called plastic constraint factor on yielding. Involving this in the fracture analysis leads to multiplication of the uniaxial yield stress by this factor in the expression for determining the J-integral. Two independent approximate equations for determining the J-integral provided very close assessments of the critical geometrical dimensions of part-through axial cracks. With the use of the crack assessment method, the critical gas pressure in a pipeline can also be determined for a given crack geometry.

The fracture toughness with which the J-integral is compared in fracture analysis is determined using fracture mechanics specimens (e.g. CT, SENB and others). Experiments made on press-straightened CT specimens and on curved CT specimens with a natural curvature, made from pipe 266/8 mm of low-C steel CSN 411353, showed that straightening a pipe band prior to the machining of CT specimens had a practically negligible effect on the fracture toughness characteristics (*J0.2*, *Jm*). However, experiments with fracture toughness testing of specimens with stress corrosion cracks, formed by the hydrogen mechanism, showed a dramatic reduction of all fracture toughness characteristics in comparison with fracture toughness determined on specimens with fatigue cracks, e.g. the quantities *Jin, J0.2* and *Jm* dropped to 17.5%, 18.5%, and 22.3%, respectively. A partial "recovery" of fracture toughness characteristics was observed when the stress corrosion conditions were removed.

## **Author details**

Ľubomír Gajdoš and Martin Šperl *Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, Czech Republic* 

## **Acknowledgement**

Financial support from Research Plan AV0Z 20710524 and from grant-funded projects GACR P105/10/2052 and P105/10/P555 are highly appreciated.

#### **7. References**

308 Applied Fracture Mechanics

**6. Conclusion** 

**Author details** 

*Czech Republic* 

**Acknowledgement** 

Ľubomír Gajdoš and Martin Šperl

result.

drawn that very good agreement of the fracture parameter values predicted by the FC and GS engineering approaches with the values found experimentally can be achieved when using the plastic constraint factor on yielding, *C*, at the level *C* = 2. If a higher value of the C factor provides more precise results, the use of the value C = 2 will yield a conservative

A specific fracture-mechanics-based procedure for assessing the integrity of pressurized thin-walled cylindrical shells made from steels includes a theoretical treatment for cracks in pipes. On the basis of both experimental work and a fracture-mechanical evaluation of experimental results, an engineering method has been worked out for assessing the geometrical parameters of critical axial crack-like defects in a high-pressure gas pipeline wall for a given internal pressure of a gas. The method makes use of simple approximate expressions for determining fracture parameters *K, J*, and it accommodates the crack tip constraint effects by means of the so-called plastic constraint factor on yielding. Involving this in the fracture analysis leads to multiplication of the uniaxial yield stress by this factor in the expression for determining the J-integral. Two independent approximate equations for determining the J-integral provided very close assessments of the critical geometrical dimensions of part-through axial cracks. With the use of the crack assessment method, the

critical gas pressure in a pipeline can also be determined for a given crack geometry.

*Institute of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic,* 

GACR P105/10/2052 and P105/10/P555 are highly appreciated.

Financial support from Research Plan AV0Z 20710524 and from grant-funded projects

The fracture toughness with which the J-integral is compared in fracture analysis is determined using fracture mechanics specimens (e.g. CT, SENB and others). Experiments made on press-straightened CT specimens and on curved CT specimens with a natural curvature, made from pipe 266/8 mm of low-C steel CSN 411353, showed that straightening a pipe band prior to the machining of CT specimens had a practically negligible effect on the fracture toughness characteristics (*J0.2*, *Jm*). However, experiments with fracture toughness testing of specimens with stress corrosion cracks, formed by the hydrogen mechanism, showed a dramatic reduction of all fracture toughness characteristics in comparison with fracture toughness determined on specimens with fatigue cracks, e.g. the quantities *Jin, J0.2* and *Jm* dropped to 17.5%, 18.5%, and 22.3%, respectively. A partial "recovery" of fracture toughness characteristics was observed when the stress corrosion conditions were removed.


## **Fracture Analysis of Generator Fan Blades**

Mahmood Sameezadeh and Hassan Farhangi

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54122

## **1. Introduction**

310 Applied Fracture Mechanics

(1981), pp. 291 – 309

9, (1973), pp. 133-146

Testing and Materials, Philadelphia, pp. 2-20

*Mechanical Sciences*, Vol.19, pp. 11-22

RCC – MR (1985). *Design and Construction Rules for Mechanical Components of FBR Nuclear* 

Rice, J. R. & Rosengren G. F. (1968). Plane Strain Deformation near a Crack Tip in a Power-Law Hardening Material. *Journal of the Mechanics and Physics of Solids*, Vol.16, pp. 1-12 Scott, P. M. & Thorpe, T. W. (1981). A Critical Review of Crack Tip Stress Intensity Factors for Semi – Elliptical Cracks. *Fatigue of Engineering Materials and Structures*, Vol.4, No.4,

Shah, R. C. & Kobayashi, A. S. (1973). Stress Intensity Factors for an Elliptical Crack Approaching the Surface of a Semi – Infinite Solid. *International Journal of Fracture*, Vol.

Shih, C. F.; O´Dowd, N. P. & Kirk, M. T. (1993). A Framework for Quantifying Crack Tip Constraint. In: *Constraint Effects in Fracture*. ASTM STP 1171, American Society for

Smith, R. A. & Miller, K. J. (1977). Fatigue Cracks at Notches. *International Journal of* 

*Island*. First Edition (AFCEN–3-5 Av. De Friedeland Paris 8), (1985)

Critical gas turbine rotating component, such as turbine blades, compressor disks, spacers and cooling fan blades are subjected to cyclic stresses during engine start-up, operation and shut-down. The lifetime of these components are usually established on the basis of probabilistic crack initiation criterion for a known fracture-critical location (Koul & Dainty, 1992). Therefore, periodic inspections are carried out to detect the probable cracks and prevent suddenly fractures.

Shaft driven rotating fans are commonly utilized to provide the required cooling for generators. These fans circulate cooling gas, air or hydrogen, throughout the machine to maintain the electrical windings at safe operating temperatures. Cooling air is circulated in a closed cycle, in a way that after passage of air through rotor, it is heated and exhausted from top of the generator, which then passes through a cooler, which would cool it down using water flow. Cool air again flows towards rotor and by use of fans, which are installed on retaining ring at the generator sides, is blown around the rotor. Each fan is comprised of several blades, which have been separated by using spacers. In Fig. 1, overall plan of generator and air cycle is shown (Moussavi et al., 2009).

Failure of a rotating fan inside a generator will cause extensive damage. The stored rotational energy in a fan that lets loose will typically destroy the stator winding, sometimes damage the stator core and cause damage to other rotor components such as retaining rings, the rotor winding and possibly even the rotor forging (Moore, 2002). Fan blades are regularly inspected during overhauls by visual and dye penetrant inspections and are required to be replaced due to defects caused by crack, corrosion and impact.

This chapter reports the failure investigation of a rotating axial flow fan of the Iran Montazer-Ghaem-VI 123 MW capacity generator unit. The unit was equipped with two rotating fans, one at each end namely at the turbine side and the exciter side of the generator. The failed fan consisting of 11 blades was mounted on the generator-rotor at the

turbine end, and had a total service life of about 41000 hours prior to the failure. The fan rotational speed was 3000 revolutions per minute (rpm) and the maximum operating temperature of the blades was 90°C.

**Figure 1.** Generator diagram (Moussavi et al., 2009)

Initial investigation pointed out that three blades were fractured and several others were cracked just about 11 hours after resuming operation following the last major overhaul, causing extensive damage to the generator unit specially the stator windings. The failure of the blades was investigated using fractographic and microstructural characterization techniques as well as mechanical evaluations to identify the root cause of the failure. Two similar failures at this kind of fan that caused extensive damage to generator units have been reported from Iran. After some investigations, the corresponding company changed the mounting angle of blades from 19º to 14º to solve the problem of the fans (Iran Power Plant Repair Comany [IPPRC], 2003, 2004).

## **2. Experimental procedure**

Visual inspections were taken on the generator parts especially on the fan blades and the effect of accident on them was studied. Three kinds of blades were found in the turbine casing after the accident: fractured blades, cracked blades and un-cracked blades. The failure was at the turbine side of the generator and according to the visual inspections, the fan blades at the excitor side were not damaged. Dye penetrant non-destructive test was used for detection of surface cracks on the blades. Chemical analysis of the fan blade material was conducted using optical emission spectroscopy. Brinell hardness measurements were taken on the sections prepared from the airfoils as well as on the base of all blades. All measurements were carried out using a 5 mm ball at a load of 1.23 kN. Longitudinal round tensile specimens were machined from the roots and tested according to ASTM E8M. Fatigue specimens were prepared from root of the fractured blades and rotary bending test was done in accordance with DIN 50113.

Longitudinal and transverse specimens were cut from the airfoils for scanning electron microscopy (SEM) and metallography. The metallography samples were prepared by using standard metallographic techniques and etched with modified Keller's reagent. The microstructure of the blade material was analyzed using an optical microscope and an Oxford MV2300 SEM equipped with an energy dispersive spectroscopy (EDS) facility. Fractographic studies were performed using visual examination. Following visual examination of the failed blades, portions of the fracture surfaces were cut for fractographic studies using SEM. All the specimens used for material characterization tests were prepared from the airfoils and bases of the fractured fan blades.

The stresses acting on the blades in steady state condition and at the time of final fracture were estimated using linear elastic fracture mechanics, finite element method (FEM) and fractographic results. At the end, a three-dimensional crack growth software was utilized to assess the crack growth rate and fatigue life in a simplified model of airfoils.

## **3. Results and discussion**

## **3.1. Visual inspections**

312 Applied Fracture Mechanics

temperature of the blades was 90°C.

**Figure 1.** Generator diagram (Moussavi et al., 2009)

Plant Repair Comany [IPPRC], 2003, 2004).

**2. Experimental procedure** 

turbine end, and had a total service life of about 41000 hours prior to the failure. The fan rotational speed was 3000 revolutions per minute (rpm) and the maximum operating

Initial investigation pointed out that three blades were fractured and several others were cracked just about 11 hours after resuming operation following the last major overhaul, causing extensive damage to the generator unit specially the stator windings. The failure of the blades was investigated using fractographic and microstructural characterization techniques as well as mechanical evaluations to identify the root cause of the failure. Two similar failures at this kind of fan that caused extensive damage to generator units have been reported from Iran. After some investigations, the corresponding company changed the mounting angle of blades from 19º to 14º to solve the problem of the fans (Iran Power

Visual inspections were taken on the generator parts especially on the fan blades and the effect of accident on them was studied. Three kinds of blades were found in the turbine casing after the accident: fractured blades, cracked blades and un-cracked blades. The failure was at the turbine side of the generator and according to the visual inspections, the fan blades at the excitor side were not damaged. Dye penetrant non-destructive test was used for detection of surface cracks on the blades. Chemical analysis of the fan blade material was conducted using optical emission spectroscopy. Brinell hardness measurements were taken on the sections prepared from the airfoils as well as on the base of all blades. All Visual inspections indicated that the accident has led to three different categories for the fan blades of the turbine side (Table 1). Dye penetrant testing revealed the cracked blades which did not completely fracture during the accident. A photograph of the fractured blades, labelled in according with their location on the fan is shown in Fig. 2.

All the examinations of mounting clearances, tightening and locking of the blades and the air guide mounting showed no defect also, there were no sign of foreign bodies in the turbine casing.


**Table 1.** Visual examination of turbine side blades

### **3.2. Materials characterization**

#### *3.2.1. Chemical composition*

The chemical composition of the fan blades is given in Table 2. The closest standard aluminum alloy found in the literature is AA 2124 which is a wrought and heat treatable alloy (American Society for Metals [ASM], 1990). This alloy derives its strength mainly from second phase particles which are distributed in the matrix through a precipitation hardening process.

**Figure 2.** Fractured fan blades


**Table 2.** Chemical composition of the fan blade material

## *3.2.2. Hardness*

Brinell macrohardness measurements carried out on different sections of airfoils showed that the hardness was about 133 ± 5 HB, which was essentially uniform along various sections.

## *3.2.3. Tensile properties*

The average values of yield stress, tensile strength, and elongation are 380 MPa, 510 MPa, and 22%, respectively.

The tensile properties and the hardness number of the blades material are all within the standard range reported for Aluminum alloy 2124 (ASM, 1990). The results indicate no degradation in the mechanical properties of the fan blades during service operation.

### *3.2.4. Fatigue test*

The measured lifetime versus applied stress from the rotaty bending fatigue test is presented in Fig. 3. A SEM micrograph taken from the fracture surface of a tested specimen is shown in Fig. 4. Presence of several second phase particles on the surface is obvious. This kind of large particles can accelerate the fatigue crack initiation and affect the fatigue behaviour of the blade material.

**Figure 3.** Fatigue test S-N curve

314 Applied Fracture Mechanics

hardening process.

**Figure 2.** Fractured fan blades

**Table 2.** Chemical composition of the fan blade material

Element

*3.2.2. Hardness* 

*3.2.3. Tensile properties* 

and 22%, respectively.

*3.2.4. Fatigue test* 

sections.

second phase particles which are distributed in the matrix through a precipitation

AlloyAl Cu Mg Mn Si Fe Zn

2124 --Base-- 3.8 – 4.9 1.2 – 1.8 0.30 – 0.9 0.20 max 0.30 max 0.25 max

Brinell macrohardness measurements carried out on different sections of airfoils showed that the hardness was about 133 ± 5 HB, which was essentially uniform along various

The average values of yield stress, tensile strength, and elongation are 380 MPa, 510 MPa,

The tensile properties and the hardness number of the blades material are all within the standard range reported for Aluminum alloy 2124 (ASM, 1990). The results indicate no

The measured lifetime versus applied stress from the rotaty bending fatigue test is presented in Fig. 3. A SEM micrograph taken from the fracture surface of a tested specimen

degradation in the mechanical properties of the fan blades during service operation.

Blade Base 4.19 1.72 0.62 0.167 0.11 0.03

**Figure 4.** Fracture surface of a fatigue test specimen

## *3.2.5. Microstructure*

Typical microstructures of the blade material in the longitudinal and transverse sections of a cracked airfoil are shown in Fig. 5. The microstructure consists of elongated grains and second phase particles in the longitudinal direction. Various types of second phase particles can be identified in the SEM micrograph, shown in Fig. 6. The large and elongated particle on the micrograph was subjected to EDS analysis. The composition of the particle contained iron, copper and aluminum which is consistent with the β-phase (Al7Cu2Fe) particles common to aluminum alloys (Merati, 2005).

**Figure 5.** Microstructures of longitudinal (a) and transverse (b) sections of a cracked blade

**Figure 6.** SEM micrograph showing large second phase particles

## **3.3. Fractography**

The fracture location of the broken blades can be identified from Fig. 2. It can be seen that the fracture had occurred close to the transition radius between the blade airfoil and the blade root. All the fracture surfaces exhibited very similar macroscopic features. A representative fractograph of the fracture surface of blade No. 8 is shown in Fig. 7a. It is observed to consist of two distinct regions at low magnification, a semi-elliptical and smooth region which is oriented normal to the blade axis and exhibits a macroscopically brittle appearance, and an outer region with a rougher and more ductile appearance. The transition from semi-elliptical region to the outer region can be clearly identified in Fig. 7b. In this region the remaining cross section of the blade failed by tensile overload.

316 Applied Fracture Mechanics

*3.2.5. Microstructure* 

common to aluminum alloys (Merati, 2005).

Typical microstructures of the blade material in the longitudinal and transverse sections of a cracked airfoil are shown in Fig. 5. The microstructure consists of elongated grains and second phase particles in the longitudinal direction. Various types of second phase particles can be identified in the SEM micrograph, shown in Fig. 6. The large and elongated particle on the micrograph was subjected to EDS analysis. The composition of the particle contained iron, copper and aluminum which is consistent with the β-phase (Al7Cu2Fe) particles

**Figure 5.** Microstructures of longitudinal (a) and transverse (b) sections of a cracked blade

The fracture location of the broken blades can be identified from Fig. 2. It can be seen that the fracture had occurred close to the transition radius between the blade airfoil and the blade root. All the fracture surfaces exhibited very similar macroscopic features. A

**Figure 6.** SEM micrograph showing large second phase particles

**3.3. Fractography** 

**Figure 7.** (a) Fracture surface of blade No. 8, (b) transition from fatigue to tensile overload fracture

Faint beach marks indicative of the progressive nature of crack growth in the semi-elliptical region can be seen in Fig. 8. Parallel microscopic fracture surface markings can also be observed in this region at higher magnifications in SEM micrograph as shown in Fig. 9. A schematic drawing of the fan with the location of damaged blades is presented in Fig. 10. Visual inspections and fractographic assessments revealed that the sequence of fracture for the broken blades has been blade No. 1, blade No. 8 and blade No. 11 respectively (Sameezadeh, 2005).

**Figure 8.** Beach marks on the fracture surface near the leading edge of the blade

**Figure 9.** SEM fractograph showing parallel fracture surface markings

**Figure 10.** Schematic drawing of the fan with location of damaged blades

Table 3 shows the crack lengths which are measured on the surface of cracked blades. Schematic drawings of the fracture surfaces of three broken fan blades which show multiple crack initiation sites as well as the macroscopic crack growth paths are presented in Fig. 11. Also, For the first and the second fractured blades (blades No. 1 & No. 8), primary crack initiation sites are located on the concave side of the airfoils near the centre where the crosssectional area is high. Primary cracks have coalesced during fatigue crack growth to form shallow semi-elliptical crack geometry and have propagated to reach the final cracks. In


addition, several small semi-elliptical cracks are also shown to have initiated from the opposite convex side of the airfoils.

**Table 3.** Measured lengths of surface cracks

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**Figure 9.** SEM fractograph showing parallel fracture surface markings

**Figure 10.** Schematic drawing of the fan with location of damaged blades

Table 3 shows the crack lengths which are measured on the surface of cracked blades. Schematic drawings of the fracture surfaces of three broken fan blades which show multiple crack initiation sites as well as the macroscopic crack growth paths are presented in Fig. 11. Also, For the first and the second fractured blades (blades No. 1 & No. 8), primary crack initiation sites are located on the concave side of the airfoils near the centre where the crosssectional area is high. Primary cracks have coalesced during fatigue crack growth to form shallow semi-elliptical crack geometry and have propagated to reach the final cracks. In Based on the fractographic observations, fatigue cracking is singled out as the primary fracture mechanism involved in the failure of the fan blades. Final fracture regions constitute only about 25–30 % of the fracture surfaces of the blades. Therefore, fatigue cracking of the blades can be considered to have occurred under high cycle fatigue conditions.

According to Fig. 11 the presence of shallow semi-elliptical cracks on both concave and convex sides of the airfoils is indicative of the influences of considerable bending stresses during crack propagation.

An SEM micrograph showing one of the typical primary crack initiation sites is presented in Fig. 12a. A secondary electron mode micrograph of this region revealed a large second phase particle with a length of about 100 µm at the crack origin as shown in Fig. 12b. The EDS spectrum of this particle identified that its composition is very similar to the intermetallic particle. Presence of the large particles at the origin of the crack have been shown to act as preferred fatigue crack nucleation sites in such alloys, despite the fact that they represent a small fraction of the particle population (Kung, 1979; Merati, 2005).

Crack propagation mode near the initiation region and throughout the fatigue fracture surface was predominantly transgranular and formation of secondary cracks was very limited. Examination of the airfoils near the crack initiation sites showed no apparent defects due to corrosion or foreign object impact.

Dye penetrant non-destructive testing that was carried out on the blades after the accident identified that some of the non-fractured blades have been cracked. The crack surface of one of these blades (blade No. 7) was disclosed as shown in Fig. 13 by cutting the remaining cross-section of the airfoil and opening carefully. In this figure the semi-elliptical fatigue region is obvious. The crack surface is covered by the penetrant material and a higher magnification view identifies the transition of the penetrant material out of the fatigue region that has different microscopic features, so it can be recognized as the final fracture zone. Therefore, the final fracture stage could not be completed because of the generator stoppage after the fracture of the three blades.

**Figure 11.** Schematic drawings of the fracture surfaces: (a) blade No. 1, (b) blade No. 8 and (c) blade No. 11

**Figure 12.** (a) Typical micrograph showing a primary crack initiation site and (b) a crack nucleating particle at the origin of the crack

**Figure 13.** Opened crack surface of blade No. 7

Crack initiation and growth from the convex side and also decrease in the fatigue region in comparison with the fractured blades, prove that the fluctuation of the stresses and the maximum stress had been highly increased during the accident probably due to excessive vibrations.

## **3.4. Stress analysis of the blades**

### *3.4.1. Steady state stresses*

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No. 11

**Figure 11.** Schematic drawings of the fracture surfaces: (a) blade No. 1, (b) blade No. 8 and (c) blade

The most important stresses acting at the transition radius in the critical cross-section of the airfoils, under normal operating conditions, consist of a tensile stress component due to centrifugal forces and a bending stress component introduced by the action of the air flow pressure. The tensile stress depends on the rotational speed (*N*, 50 revolutions per second), mass of the airfoils (*m*, 0.51 kg), distance from the center of rotation (*r*, 0.615 m) and the cross-sectional area (*A*, 0.00155 m2) as given by the following expression (Bleier, 1997):

$$
\sigma\_{\rm t} = \frac{mr(2\pi N)^2}{A} \tag{1}
$$

The tensile stress component due to the centrifugal forces is calculated from the above relationship to be about 20 MPa, which is essentially constant during operation.

A 3D finite element model was used to simulate the normal operation of the fan blades. The results revealed that the maximum total stress acting on the blade under normal condition was about 27 MPa which occurred close to the transition radius between the airfoil and the root where the cracks initiated in fractured blades (Fig. 14) (Ataei, 2006). The bending stress introduced by the action of the air flow pressure are less important (Cohen, 1987) and was estimated as about 7 MPa. According to the results, the steady state stresses are very low in comparison with the strength of the blade material therefore, it can be concluded that the fracture accident was happened during an abnormal condition.

**Figure 14.** Finite element simulation of the fan blade, which shows the stress distribution under normal operating condition (Ataei, 2006)

### *3.4.2. Failure stresses*

The low aspect ratio of primary fatigue cracks suggests that crack propagation was influenced by significantly higher bending stress levels than that caused by air flow pressure alone. Moreover, since mode I loading condition was dominant and the fatigue crack growth plane was normal to the tensile stress axis, individual components of stress intensity factors due to tension and bending can be added to obtain the total crack tip stress intensity. Accordingly, the failure stress at the time of final fracture and the magnitude of additional bending stresses which influenced the fatigue cracking process can be estimated by applying the superposition principle and fracture toughness data using the following equation (Anderson, 1995; Broek, 1995):

322 Applied Fracture Mechanics

operating condition (Ataei, 2006)

*3.4.2. Failure stresses* 

1997):

and the cross-sectional area (*A*, 0.00155 m2) as given by the following expression (Bleier,

*mr N* (2 ) *A* 

The tensile stress component due to the centrifugal forces is calculated from the above

A 3D finite element model was used to simulate the normal operation of the fan blades. The results revealed that the maximum total stress acting on the blade under normal condition was about 27 MPa which occurred close to the transition radius between the airfoil and the root where the cracks initiated in fractured blades (Fig. 14) (Ataei, 2006). The bending stress introduced by the action of the air flow pressure are less important (Cohen, 1987) and was estimated as about 7 MPa. According to the results, the steady state stresses are very low in comparison with the strength of the blade material therefore, it can be concluded that the

**Figure 14.** Finite element simulation of the fan blade, which shows the stress distribution under normal

The low aspect ratio of primary fatigue cracks suggests that crack propagation was influenced by significantly higher bending stress levels than that caused by air flow pressure alone. Moreover, since mode I loading condition was dominant and the fatigue crack

t

relationship to be about 20 MPa, which is essentially constant during operation.

fracture accident was happened during an abnormal condition.

2

(1)

$$K\_{\rm IC} = K\_t + K\_b \tag{2}$$

$$K\_{\rm IC} = F \sigma\_{\rm t} \sqrt{\frac{\pi a}{Q}} + FH \sigma\_{\rm b} \sqrt{\frac{\pi a}{Q}} \tag{3}$$

where the fracture toughness of the material was taken as 31.9 MPa m from the literature (ASM, 1990) and the values for F, H and Q geometrical parameters can be calculated for each specific crack geometry under combined tension and bending stresses using Newman– Raju equations (Newman & Raju, 1984).

According to the fractographic results the final fatigue crack of the first broken blade (blade No.1) at the time of final fracture was assumed as a quarter-elliptical corner crack with <sup>6</sup> *a* 12 10 m and <sup>6</sup> *c* 83 10 m dimensions, thus the values for F, H and Q parameters in Eq. (3) were calculated as 5.02, 0.20 and 1.07 respectively. Using the above data and a 20 MPa tensile stress, the bending stress can be estimated by the following calculations:

$$K\_{\rm t} = F \sigma\_{\rm t} \sqrt{\frac{\pi a}{Q}} = 18.8 \quad \text{MPa} \sqrt{\text{m}} \tag{4}$$

$$K\_{\rm b} = HF \sigma\_{\rm b} \sqrt{\frac{\pi a}{Q}} = K\_{\rm IC} - K\_{\rm t} = 13.1 \text{ MPa} \sqrt{\text{m}} \tag{5}$$

$$
\sigma\_{\rm b} = 69.5 \text{ MPa} \tag{6}
$$

The bending stress at the time of fracture of the first broken blade was estimated as about 70 MPa and the total stress acting on the airfoil is thus about 90 MPa (adding a 20 MPa tensile stress). Subtracting the bending stress due to air flow pressure (7 MPa) from this total magnitude of bending stresses, an additional bending stress of 63 MPa is estimated to have influenced the failure of the fan blades. This significant magnitude of additional bending stress, probably caused by excessive vibrations, can explain the early initiation and rapid growth of fatigue cracks to final fracture in the fan blades, which occurred after only a short period of operation following the last overhaul.

The final fatigue crack of the second fractured blade (blade No. 8) is similar to blade No.1 and can be assumed as a quarter-elliptical corner crack too with <sup>6</sup> *a* 11 10 m and <sup>6</sup> *c* 71 10 m dimensions. According that, the values for F, H and Q parameters were calculated as 3.74, 0.24 and 1.07 respectively for this crack. With similar calculations as before the bending stress at the time of final fracture for blade No. 8 can be estimated as 115 MPa and the total stress acting on this airfoil is about 135 MPa.

Finally, for the third fractured blade (blade No. 11), the final fatigue crack shape is different and can be assumed as a semi-elliptical surface crack with <sup>6</sup> *a* 11 10 m and <sup>6</sup> *c* 55 10 m dimensions. After calculation of the crack geometry parameters (F=1.84, H=0.66, Q=1.10) and using the Eq. (3), the bending stress at the time of final fracture for this blade was estimated as 118 MPa. By adding a 20 MPa tensile stress, the total stress is thus about 138 MPa. The details of above calculations can be found elsewhere (Sameezadeh, 2005; Ataei, 2006). Table 4 shows the summary of stress analysis results for the fan blades. According to the estimated failure stresses for the fractured blades it should be noted that the significant magnitude of additional stresses acting on the blades and leading to the premature and catastrophic failure of the fan, possibly have been due to aerodynamical disturbances that have resulted in a state of resonant condition of vibration. Additionally, changes in blade installation conditions, such as the level of torque tightening applied to the fixing bolts, which can influence the blade natural frequency, may be regarded as a contributing factor to fan blade failure shortly after overhaul.


**Table 4.** Summary of stress analysis results of the fan blades

### **3.5. Simulation of fatigue crack growth**

Fatigue crack growth rates in a model of the airfoils, under the action of estimated loads, were computed using FRANC3D/BES crack propagation software. The FRANC3D (FRacture ANalysis Code for 3D problems)/BES (boundary element solver) software developed at cornell university, is capable of evaluating stress intensity factors (SIF) along 3D crack fronts. This software utilizes boundary elements and linear elastic fracture mechanics. The displacements and stress intensity factors are calculated on the crack leading edge to obtain crack propagation trajectories and growth rates. It was assumed throughout the calculations that linear elastic fracture mechanics conditions hold (Carter et al., 2000; Cornell Fracture Group [CFG], 2002).

A simplified 3D model of airfoils was created using a geometry pre-processor program called OSM (Object Solid Modeler). A boundary element model of the geometry, consisting of triangular and square elements was then meshed within the FRANC3D program, and the stresses were applied by the model boundary conditions.

324 Applied Fracture Mechanics

calculated as 3.74, 0.24 and 1.07 respectively for this crack. With similar calculations as before the bending stress at the time of final fracture for blade No. 8 can be estimated as 115

Finally, for the third fractured blade (blade No. 11), the final fatigue crack shape is different and can be assumed as a semi-elliptical surface crack with <sup>6</sup> *a* 11 10 m and <sup>6</sup> *c* 55 10 m dimensions. After calculation of the crack geometry parameters (F=1.84, H=0.66, Q=1.10) and using the Eq. (3), the bending stress at the time of final fracture for this blade was estimated as 118 MPa. By adding a 20 MPa tensile stress, the total stress is thus about 138 MPa. The details of above calculations can be found elsewhere (Sameezadeh, 2005; Ataei, 2006). Table 4 shows the summary of stress analysis results for the fan blades. According to the estimated failure stresses for the fractured blades it should be noted that the significant magnitude of additional stresses acting on the blades and leading to the premature and catastrophic failure of the fan, possibly have been due to aerodynamical disturbances that have resulted in a state of resonant condition of vibration. Additionally, changes in blade installation conditions, such as the level of torque tightening applied to the fixing bolts, which can influence the blade natural frequency, may be regarded as a

> **Total bending stress (MPa)**

**Normal condition** 20 7 0 27 **Blade No. 1** 20 70 63 90 **Blade No. 8** 20 115 108 135 **Blade No. 11** 20 118 111 138

Fatigue crack growth rates in a model of the airfoils, under the action of estimated loads, were computed using FRANC3D/BES crack propagation software. The FRANC3D (FRacture ANalysis Code for 3D problems)/BES (boundary element solver) software developed at cornell university, is capable of evaluating stress intensity factors (SIF) along 3D crack fronts. This software utilizes boundary elements and linear elastic fracture mechanics. The displacements and stress intensity factors are calculated on the crack leading edge to obtain crack propagation trajectories and growth rates. It was assumed throughout the calculations that linear elastic fracture mechanics conditions hold (Carter et al., 2000; Cornell Fracture

**Additional bending stress (MPa)** 

**Maximum stress (MPa)** 

MPa and the total stress acting on this airfoil is about 135 MPa.

contributing factor to fan blade failure shortly after overhaul.

**Centrifugal tensile stress (MPa)** 

**Table 4.** Summary of stress analysis results of the fan blades

**3.5. Simulation of fatigue crack growth** 

Group [CFG], 2002).

To start the crack growth simulation in FRANC3D an initial crack was introduced into the model. The initial crack was assumed to be a semi-elliptical surface crack with 0.1 *<sup>a</sup> c* , based on previous fractographic findings. Initial crack length was computed from the threshold stress intensity factor range based on the equation (Hertzberg, 1989):

$$
\Delta K\_{\rm th} = Y \Delta \sigma \sqrt{\pi a\_0} \tag{7}
$$

Taking *K*th from the literature and values for the first and the second fractured blades based on the computed failure stresses and the positive portion of the stress cycle, and using the calculated values of Y from Newman–Raju equations (Newman & Raju, 1984), the initial crack lengths below which crack growth is arrested, 0*a* , were computed to be approximately <sup>5</sup> 48 10 m and <sup>5</sup> 22 10 m for the first and the second fractured blades respectively.

Based on the stresses applied by the model boundary conditions, the initial crack was grown in a series of crack propagation steps. Fig. 15a is a simplified 3D model of the airfoil that was meshed 2D (Boundary elements) in FRANC3D for simulating fatigue crack growth in fractured blades. Fig. 15b shows simulated fatigue crack in blade No. 8 after three steps of propagation.

Stress intensity factors at each step were calculated within FRANC3D and the fatigue crack growth curves were calculated using the Paris power law relationship given by (Broek, 1995):

$$\frac{\text{d}a}{\text{d}N} = \text{C}\Lambda K^n \tag{8}$$

The empirical constants C and n were computed by fitting the near threshold crack growth data for AA 2124 aluminum alloy (Department of Defense, 1998) as <sup>13</sup> <sup>n</sup> *C* 9.5 10 m.(MPa m ) and *n* 4.9 for *da dN* expressed in m.cycle-1 at a load ratio of *R* 0.1 assumed in this case. The results of crack growth simulations in the first and the second fractured blades, where initial cracks are grown in small steps to their final dimensions are plotted in Fig. 16. These curves show the crack size as a function of the number of applied stress cycles. It can be seen that the number of cycles required to propagate initial cracks to their final dimensions for the first and the second fractured blades are just about <sup>6</sup> 1.3 10 and <sup>5</sup> 8 10 cycles respectively.

Stress calculations showed that the total stress acting on the third fractured blade (blade No. 11) is higher than for the two other fractured blades. Therefore, it can be assumed that the fatigue crack growth life of this blade is the shortest one and has the least portion on the total fatigue crack growth life of the failed fan. Thus, the simulation of blade No. 11 fatigue crack growth life was not performed and was ignored.

**Figure 15.** (a) A 3D model of the airfoil with 2D mesh for simulating fatigue crack growth in FRANC3D software and (b) simulated fatigue crack in blade No. 8 after three steps of propagation

**Figure 16.** Crack length as a function of number of cycles in: (a) the first fractured blade (blade No. 1) and (b) the second fractured blade (blade No. 8)

As mentioned before, it is assumed that the excessive vibrations of the fan blades resulted from a resonant condition. Therefore, modal analysis was performed and natural frequencies and corresponding mode shapes were calculated. The results show that the first bending vibrational mode is a possible cause of the blades' failure. This first natural frequency calculated as 649 Hz (Ataei, 2006) So, the fatigue crack growth time for the first fractured blade with respect to the first natural frequency value (649 Hz) can be calculated as below:

$$t\_{\rm g1} = \frac{1.3 \times 10^6}{649 \times 3600} = 0.56 \quad hour \quad \text{\textdegree} \tag{9}$$

and similarly for the second fractured blade:

$$t\_{\rm g2} = \frac{8 \times 10^5}{649 \times 3600} = 0.34 \quad hour \tag{10}$$

Thus the total fatigue crack growth time is about one hour. This lifetime is within the 11 hours of actual operating period following the last overhaul, which ended with the failure of fan blades.

### **4. Conclusion**

326 Applied Fracture Mechanics

Stress calculations showed that the total stress acting on the third fractured blade (blade No. 11) is higher than for the two other fractured blades. Therefore, it can be assumed that the fatigue crack growth life of this blade is the shortest one and has the least portion on the total fatigue crack growth life of the failed fan. Thus, the simulation of blade No. 11 fatigue

**Figure 15.** (a) A 3D model of the airfoil with 2D mesh for simulating fatigue crack growth in FRANC3D

software and (b) simulated fatigue crack in blade No. 8 after three steps of propagation

crack growth life was not performed and was ignored.


5. Based on crack growth simulations in the first and the second fractured blades, the total fatigue crack growth time was calculated as only about one hour, which is within the 11 hours of actual operating period following the last overhaul, ending with the failure of fan blades.

## **Nomenclature**


## **Author details**

Mahmood Sameezadeh and Hassan Farhangi *University of Tehran, College of Engineering, Iran* 

## **Acknowledgement**

328 Applied Fracture Mechanics

fan blades.

**Nomenclature** 

*a* crack depth, m

*A* cross-sectional area, m2

*a0* initial crack depth, m

*KIC* critical stress intensity factor, MPa m

*N* rotational speed, revolutions per second

*r* distance from center of rotation, m *tg* fatigue crack growth time, hours

*Kt* stress intensity factor due to tension, MPa m *Kb* stress intensity factor due to bending, MPa m

*K*th threshold stress intensity factor range, MPa m

*K* stress intensity factor range, MPa m

Mahmood Sameezadeh and Hassan Farhangi *University of Tehran, College of Engineering, Iran* 

*N* crack growth rate , m.cycle-1

*C* empirical material constant *n* empirical material constant

**Author details** 

*a/c* crack aspect ratio

*m* mass of blade, kg

*Q* crack shape factor

<sup>t</sup> *σ* tensile stress, MPa <sup>b</sup> *σ* bending stress, MPa Δ*σ* stress range, MPa

d d *a*

c length for corner cracks and half-length for surface cracks

*F* boundary correction factor on stress intensity factor for remote tension

*H* bending multiplier on stress intensity factor for remote bending

5. Based on crack growth simulations in the first and the second fractured blades, the total fatigue crack growth time was calculated as only about one hour, which is within the 11 hours of actual operating period following the last overhaul, ending with the failure of

The authors are grateful to Eng. H. Vahedi and Dr. E. Poursaeidi of the Iran Power Plant Repairs Company (IPPRC) for their support and stimulating discussions. The authors also acknowledge the helpful assistances of Eng. M. Vatanara and Eng. P. Ataei during the study.

#### **5. References**

