2. Calculation of SIF for FGM

Cracks/flaws are inevitable in all engineering materials. Loading under severe environmental conditions may either initiate new cracks or may cause the propagation of pre-existing cracks in the structures. Theoretically, fracture can be defined as the breaking or rupturing of a

Composite materials manifested in the middle of the twentieth century. Composites are naturally occurring or engineered materials made from two or more constituents with different chemical or physical properties distinct boundary among constituents. Lightweight composite materials with high strength to weight and stiffness to weight ratios have been used successfully in aircraft industry and other engineering applications. Under high temperature conditions the strength of

FGMs can be referred as multiphase composite materials in which the composition or microstructure or both are spatially varied which lead to a certain gradation in the local material properties. FGMs can be defined as multi-phase composites. FGMs are synthesized such that they own continuous variations in volume fractions of their components in space to return a pre-established composition. FGMs possess continuously varying properties in one or more than one direction and the form non-homogeneous macrostructure due to these variations. By gradually varying the volume fraction of the constituents, FGMs exhibit a smooth and continuous change from one surface to another, thus reducing interface problems, and minimizing thermal stress concentrations. The ceramic phase of FGMs provides a good resistance to heat, while the metal phase provides a

strong mechanical performance and hence reduces the possibility of catastrophic failure.

The major advantages of FGM over conventional materials are firstly, FGM satisfies the working conditions for which it is specifically developed. Secondly, it is economical as it reduces material costs for particular engineering applications. Thirdly, it can reduce the magnitude of residual and thermal stresses generated under working conditions. Finally, FGMs exhibit better fracture toughness and bond strength. This is normally achieved by using a ceramic layer connected with a metallic layer. FGMs have wide area of engineering applications like in the computer circuit and aerospace industries. FGMs have typical applications is in aircraft

In general, all structural components are subjected to thermo-mechanical cyclic load. The fatigue life of these components is generally predicted without considering the effect of defects/discontinuities present in component. However, FGMs are commonly made by sintering process, which are porous in nature. These discontinuities at the vicinity of a major crack tip lead to increase the effective SIF at the major crack tip due to which the life of the components get depreciated. Hence, the analysis of FGMs in the vicinity of discontinuities becomes very important from the design point of view. To widen the spectrum of applications

Over the years, greater understanding of fracture mechanics has undoubtedly prevented a significant number of structural failures. Fracture mechanics approach for the design of structures includes flaw size as one of the key variables. Fracture toughness replaces strength of material as a relevant material attribute, and its evaluation is mainly done in composites using the J-integral approach [1]. Failure of FGM has always been a trending domain of research for

scientists and engineers due to the wide spectra of their engineering applications.

material resulting into its separation into two or more pieces.

170 Contact and Fracture Mechanics

and automotive industries as thermal barrier coatings (TBCs).

of FGMs, the fatigue/fracture behavior should be properly evaluated.

the metal is deteriorated whereas, ceramics have excellent resistance to heat.

A domain based interaction integral approach can be used for calculating the stress intensity factors for homogeneous, bi-layer and functionally graded materials under thermal as well as mechanical loading. In this chapter, interaction integral approach will be extended to calculate the SIFs for FGM and bi-layered FGM under mechanical loads. The interaction integral is calculated based on J-integral. The J-integral for an elastic body subjected to thermomechanical load is given as,

$$J = \oint\_{\Gamma\_{\sigma}} \left( \tilde{W} \,\delta\_{1\dot{\jmath}} - \sigma\_{i\dot{\jmath}} \,\frac{\partial u\_i}{\partial \mathbf{x}\_1} \right) n\_{\dot{\jmath}} d\Gamma \tag{1}$$

Where, a is the crack length and N is the number of loading cycles. C and m are material properties to find the rate of crack growth. At each crack tip, the local direction of crack growth θ<sup>c</sup> can be calculated by the maximum principal stress theory [37]. Crack is assumed to grow in a direction perpendicular to the maximum principal stress. Thus, by enforcing the condition

Fatigue Fracture of Functionally Graded Materials Under Elastic-Plastic Loading Conditions Using Extended…

KI �

In the numerical example, the crack growth value Δa is assumed and the corresponding number of cycles ΔN is computed from Eq. (9). When multiple crack tips are present, the crack growth value Δa is assumed for the most dominant crack tip, corresponding ΔN is computed and then at the other crack tips the crack growth is computed corresponding to ΔN. Eventually, when the maximum value of KIeq for any crack tip becomes more than the fracture toughness KIC at corresponding location then the simulation is terminated. At this point, the

In this chapter, the results have been presented for a FGM plate as shown in Figure 1. The FGM plate is manufactured by reinforcing an alloy with ceramic. The volume fraction of ceramic is varied in the x-direction to get a material property variation in the x-direction. It is assumed that at x ¼ 0 the FGM have the properties of the alloy and at x ¼ L properties of ceramic. The major crack is always taken at the center of the FGM plate in the x-direction. The interface, when present is also in the same direction. The material properties of the aluminum

0 @

2 � �

For stable crack propagation, the generalized Paris' law for FGM is given as da

<sup>θ</sup><sup>c</sup> <sup>¼</sup> 2tan�<sup>1</sup>

According to this criterion, the equivalent mode-I SIF is obtained as

KIeq <sup>¼</sup> KI cos<sup>3</sup> <sup>θ</sup><sup>c</sup>

where, Cð Þx and mð Þx are the functions of the location.

total number of cycles elapsed is the fatigue life of the FGM.

alloy and alumina used in FGM are tabulated in Table 1 [38, 39].

4. Modeling of the properties of FGM

KIsinθ<sup>c</sup> þ KIIð Þ¼ 3cosθ<sup>c</sup> � 1 0 (6)

1

2

dN <sup>¼</sup> <sup>C</sup>ð Þ<sup>x</sup> <sup>Δ</sup>KIeq � �<sup>m</sup>ð Þ<sup>x</sup> (9)

A (7)

http://dx.doi.org/10.5772/intechopen.72778

173

� � (8)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 <sup>I</sup> <sup>þ</sup> <sup>8</sup>K<sup>2</sup> II <sup>q</sup>

> 2 � � sin <sup>θ</sup><sup>c</sup>

4KII

� <sup>3</sup>KIIcos<sup>2</sup> <sup>θ</sup><sup>c</sup>

that the local shear stress is zero for θ ¼ θc,

The solution of Eq. (6) gives

For the interaction integral calculation of an elastic body, consider two equilibrium states i.e. state 1, the actual state with given boundary conditions and state 2, an auxiliary state of the cracked body. The parameters for auxiliary state are represented with superscript a. The final expression for the interaction integral takes the form [35]

$$\begin{aligned} M\_{12} &= \int\_{A\_o} \left( \sigma\_{\vec{\eta}} \frac{\partial u\_i^a}{\partial \mathbf{x}\_1} + \sigma\_{i\vec{\eta}}^a \frac{\partial u\_i}{\partial \mathbf{x}\_1} - \sigma\_{ik}^a \varepsilon\_{ik}^m \delta\_{1j} \right) \frac{\partial q}{\partial \mathbf{x}\_j} \, dA + \\ &\int\_{A\_o} \left( \sigma\_{i\vec{\eta}} \left( S\_{i\vec{\eta}kl}^{tip} - S\_{i\vec{\eta}kl}(\mathbf{x}) \right) \frac{\partial \sigma\_{kl}^a}{\partial \mathbf{x}\_1} \right) q \, dA \end{aligned} \tag{2}$$

where, the auxiliary field for the FGM may be taken from [36] as.

$$\sigma\_{i\bar{\jmath}}^{a} = \mathfrak{C}\_{i\bar{\jmath}k}^{t\bar{\jmath}p} \frac{1}{2} \left( \frac{\partial u\_k^a}{\partial \mathbf{x}\_l} + \frac{\partial u\_l^a}{\partial \mathbf{x}\_k} \right) , \varepsilon\_{i\bar{\jmath}}^{a} = \mathfrak{S}\_{i\bar{\jmath}k}(\mathbf{x}) \sigma\_{kl}^{a} \text{ and } \ \varepsilon\_{i\bar{\jmath}}^{a} \neq \frac{1}{2} \left( \frac{\partial u\_i^a}{\partial \mathbf{x}\_j} + \frac{\partial u\_j^a}{\partial \mathbf{x}\_i} \right) \tag{3}$$

The SIFs are calculated from the interaction integral as [36]:

Mode-I SIF is given as,

$$K\_{l} = \frac{M\_{12}E^\* \cosh^2(\pi \varepsilon^{t\dot{\varphi}})}{2} \text{ with } K\_{l}^{a} = 1 \text{ and } K\_{ll}^{a} = 0 \tag{4a}$$

Mode-II SIF is given as,

$$K\_{ll} = \frac{M\_{12}E^\* \cosh^2\left(\pi\varepsilon^{t\dot{p}}\right)}{2} \text{with } K\_l^a = 0 \quad \text{and} \ K\_{ll}^a = 1 \tag{4b}$$

$$\text{where, } E^\* = \frac{2\overline{E}\_i \overline{E}\_2}{E\_1 + E\_2} \text{ with } \overline{E}\_i = \begin{cases} E\_i^{\text{tip}} & \text{for plane stress} \\ E\_i^{\text{tip}} / \left(1 - \left(\nu\_i^{\text{tip}}\right)^2\right) & \text{for plane strain} \end{cases} \text{with } i = 1, 2.$$

#### 3. Fatigue crack growth

Here we use Paris law for stable crack propagation, the generalized Paris's law is given as:

$$\frac{da}{dN} = \mathcal{C} \left(\mathcal{A} K\_{\mathbb{I}eq} \right)^m \tag{5}$$

Where, a is the crack length and N is the number of loading cycles. C and m are material properties to find the rate of crack growth. At each crack tip, the local direction of crack growth θ<sup>c</sup> can be calculated by the maximum principal stress theory [37]. Crack is assumed to grow in a direction perpendicular to the maximum principal stress. Thus, by enforcing the condition that the local shear stress is zero for θ ¼ θc,

$$K\_l \sin \theta\_\varepsilon + K\_{ll} (\Im \cos \theta\_\varepsilon - 1) = 0 \tag{6}$$

The solution of Eq. (6) gives

J ¼ ∮ Γo

expression for the interaction integral takes the form [35]

ð

σij ∂ua i ∂x<sup>1</sup> <sup>þ</sup> <sup>σ</sup><sup>a</sup> ij ∂ui ∂x<sup>1</sup> � <sup>σ</sup><sup>a</sup> ikε<sup>m</sup> ik δ1<sup>j</sup>

σij Stip

, ε<sup>a</sup>

Ao

ð

Ao

∂ua k ∂xl þ ∂ua l ∂xk � �

The SIFs are calculated from the interaction integral as [36]:

KI <sup>¼</sup> <sup>M</sup>12E<sup>∗</sup> cosh<sup>2</sup> πεtip � �

KII <sup>¼</sup> <sup>M</sup>12E<sup>∗</sup> cosh<sup>2</sup> πεtip � �

Etip

8 ><

>:

Etip

<sup>i</sup> = 1 � ν

where, the auxiliary field for the FGM may be taken from [36] as.

M<sup>12</sup> ¼

σa ij <sup>¼</sup> <sup>C</sup>tip ijkl 1 2

Mode-I SIF is given as,

172 Contact and Fracture Mechanics

Mode-II SIF is given as,

where, <sup>E</sup><sup>∗</sup> <sup>¼</sup> <sup>2</sup>E<sup>1</sup> <sup>E</sup><sup>2</sup>

E1þE<sup>2</sup>

3. Fatigue crack growth

with Ei ¼

<sup>W</sup><sup>~</sup> <sup>δ</sup>1<sup>j</sup> � <sup>σ</sup>ij

� �

For the interaction integral calculation of an elastic body, consider two equilibrium states i.e. state 1, the actual state with given boundary conditions and state 2, an auxiliary state of the cracked body. The parameters for auxiliary state are represented with superscript a. The final

� � ∂q

kl ∂x<sup>1</sup>

kl and ε<sup>a</sup>

q dA

ijkl � Sijklð Þx � � ∂σ<sup>a</sup>

ij <sup>¼</sup> Sijklð Þ<sup>x</sup> <sup>σ</sup><sup>a</sup>

<sup>2</sup> with <sup>K</sup><sup>a</sup>

<sup>2</sup> with <sup>K</sup><sup>a</sup>

tip i � �<sup>2</sup> � �

Here we use Paris law for stable crack propagation, the generalized Paris's law is given as:

dN <sup>¼</sup> <sup>C</sup> <sup>Δ</sup>KIeq

da

<sup>i</sup> for plane stress

� �

∂xj dAþ

ij 6¼ <sup>1</sup> 2

<sup>I</sup> <sup>¼</sup> 1 and <sup>K</sup><sup>a</sup>

<sup>I</sup> <sup>¼</sup> 0 and <sup>K</sup><sup>a</sup>

for plane strain

∂ua i ∂xj þ ∂u<sup>a</sup> j ∂xi

!

with i ¼ 1, 2

� �<sup>m</sup> (5)

II ¼ 0 (4a)

II ¼ 1 (4b)

∂ui ∂x<sup>1</sup>

njdΓ (1)

(2)

(3)

$$\theta\_{\varepsilon} = \text{ } 2 \text{tan}^{-1} \left( \frac{\text{K}\_{I} - \sqrt{\text{K}\_{I}^{2} + 8 \text{K}\_{II}^{2}}}{4 \text{K}\_{II}} \right) \tag{7}$$

According to this criterion, the equivalent mode-I SIF is obtained as

$$K\_{leq} = K\_l \cos^3\left(\frac{\theta\_\varepsilon}{2}\right) - 3K\_{ll}\cos^2\left(\frac{\theta\_\varepsilon}{2}\right)\sin\left(\frac{\theta\_\varepsilon}{2}\right) \tag{8}$$

For stable crack propagation, the generalized Paris' law for FGM is given as

$$\frac{da}{dN} = \left. \mathbb{C}(\mathbf{x}) \left( \Delta K\_{Ieq} \right)^{m(\mathbf{x})} \right. \tag{9}$$

where, Cð Þx and mð Þx are the functions of the location.

In the numerical example, the crack growth value Δa is assumed and the corresponding number of cycles ΔN is computed from Eq. (9). When multiple crack tips are present, the crack growth value Δa is assumed for the most dominant crack tip, corresponding ΔN is computed and then at the other crack tips the crack growth is computed corresponding to ΔN. Eventually, when the maximum value of KIeq for any crack tip becomes more than the fracture toughness KIC at corresponding location then the simulation is terminated. At this point, the total number of cycles elapsed is the fatigue life of the FGM.
