4. Modeling of the properties of FGM

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

Vcomposite ceramic <sup>¼</sup> <sup>1</sup>

Ecomposite <sup>¼</sup> Ealloy <sup>V</sup>composite

νalloyVFGM

and <sup>ν</sup>composite <sup>¼</sup> <sup>ν</sup>alloyVcomposite

Figure 2. Variation of modulus of elasticity along the length of the plate.

VFGM

where, <sup>L</sup> is the length of the plate. For <sup>L</sup> <sup>¼</sup> 100 mm, <sup>V</sup>composite

mixtures for the equivalent composite

the FGM may be calculated as [40]

The Poisson's ratio is shown in Figure 4.

νð Þ¼ x

L ð L

0

The variation in volume fraction of ceramic (alumina) in the FGM is shown in Figure 3. The volume fraction for the equivalent composite has also been indicated. Now, using the rule of

we get Ecomposite ¼ 158:04 GPa. The Poisson's ratio for the equivalent composite as well as for

alloy ð Þ<sup>x</sup> Eceramic <sup>þ</sup> <sup>ν</sup>ceramicVFGM

alloy ð Þ<sup>x</sup> Eceramic <sup>þ</sup> <sup>V</sup>FGM

Vcomposite

VFGM

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

alloy <sup>þ</sup> EceramicVcomposite

alloy Eceramic <sup>þ</sup> <sup>ν</sup>ceramicVcomposite

alloy Eceramic <sup>þ</sup> <sup>V</sup>composite

ceramicð Þx dx (11c)

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ceramic (12)

alloy ¼ 61:72%.

175

(13a)

(13b)

ceramic <sup>¼</sup> <sup>38</sup>:28% and <sup>V</sup>composite

ceramicð Þx Ealloy

ceramic Ealloy

ceramic Ealloy

ceramicð Þx Ealloy

Figure 1. Geometry of the FGM plate along with its dimensions.


Table 1. Material properties of aluminum alloy and alumina.

The variation of the elastic modulus for FGM is modeled as

$$E(\mathbf{x}) = E\_{alloy} e^{\alpha x} \text{ where } \alpha \text{ is given as } \alpha = \frac{1}{L} \ln \left( \frac{E\_{cemanic}}{E\_{alloy}} \right) \tag{10}$$

A plot of Eð Þx for L = 100 mm is shown in Figure 2. The fatigue life of FGM has been compared with the same of the aluminum alloy and an equivalent composite of aluminum alloy/alumina. The equivalent composite considered in this example has the same overall volume fractions of aluminum alloy and ceramic as the FGM. The volume fractions of ceramic and aluminum alloy in the FGM are obtained as

$$V\_{cermic}^{\text{FGM}}(\mathbf{x}) = \frac{E(\mathbf{x}) - E\_{\text{alloy}}}{E\_{cermic} - E\_{\text{alloy}}} = \frac{E\_{\text{alloy}}e^{\alpha\_x} - E\_{\text{alloy}}}{E\_{cermic} - E\_{\text{alloy}}} \tag{11a}$$

$$V\_{alloy}^{\text{FGM}}(\mathbf{x}) = \mathbf{1} - V\_{carrier}^{\text{FGM}}(\mathbf{x}) \tag{11b}$$

In this example, the equivalent composite is assumed to have the same amount of metal and ceramic. The volume fraction of alumina in the equivalent composite is calculated as

Fatigue Fracture of Functionally Graded Materials Under Elastic-Plastic Loading Conditions Using Extended… http://dx.doi.org/10.5772/intechopen.72778 175

$$V\_{crumic}^{compposite} = \frac{1}{L} \int\_{0}^{L} V\_{crumic}^{FGM}(\mathbf{x}) \, d\mathbf{x} \tag{11c}$$

where, <sup>L</sup> is the length of the plate. For <sup>L</sup> <sup>¼</sup> 100 mm, <sup>V</sup>composite ceramic <sup>¼</sup> <sup>38</sup>:28% and <sup>V</sup>composite alloy ¼ 61:72%. The variation in volume fraction of ceramic (alumina) in the FGM is shown in Figure 3. The volume fraction for the equivalent composite has also been indicated. Now, using the rule of mixtures for the equivalent composite

$$E\_{compposite} = E\_{alloy} V\_{alloy}^{compposite} + E\_{carrier} V\_{carrier}^{compposite} \tag{12}$$

we get Ecomposite ¼ 158:04 GPa. The Poisson's ratio for the equivalent composite as well as for the FGM may be calculated as [40]

$$\nu(\mathbf{x}) = \frac{\nu\_{\text{alloy}} V\_{\text{alloy}}^{\text{FGM}}(\mathbf{x}) \, E\_{\text{carrier}} + \nu\_{\text{carrier}} V\_{\text{carrier}}^{\text{FGM}}(\mathbf{x}) \, E\_{\text{alloy}}}{V\_{\text{alloy}}^{\text{FGM}}(\mathbf{x}) \, E\_{\text{carrier}} + V\_{\text{carrier}}^{\text{FGM}}(\mathbf{x}) \, E\_{\text{alloy}}} \tag{13a}$$

$$\mathbf{and} \quad \nu\_{\text{composite}} = \frac{\nu\_{\text{alloy}} V\_{\text{alloy}}^{\text{compposite}} E\_{\text{crammic}} + \nu\_{\text{cramic}} V\_{\text{carrier}}^{\text{compposite}} E\_{\text{alloy}}}{V\_{\text{alloy}}^{\text{compposite}} E\_{\text{cramic}} + V\_{\text{carrier}}^{\text{compposite}} E\_{\text{alloy}}} \tag{13b}$$

The Poisson's ratio is shown in Figure 4.

The variation of the elastic modulus for FGM is modeled as

�

<sup>α</sup> <sup>x</sup> where <sup>α</sup> is given as <sup>α</sup> <sup>¼</sup> <sup>1</sup>

Material properties Aluminum alloy Alumina

Elastic modulus Eð Þ GPa 70 300 Poisson's ratio, ν 0.33 0.21

Paris law parameter, mð Þx 3 10

A plot of Eð Þx for L = 100 mm is shown in Figure 2. The fatigue life of FGM has been compared with the same of the aluminum alloy and an equivalent composite of aluminum alloy/alumina. The equivalent composite considered in this example has the same overall volume fractions of aluminum alloy and ceramic as the FGM. The volume fractions of ceramic and aluminum alloy

> E xð Þ� Ealloy Eceramic � Ealloy

ceramic. The volume fraction of alumina in the equivalent composite is calculated as

alloy ð Þ¼ <sup>x</sup> <sup>1</sup> � <sup>V</sup>FGM

In this example, the equivalent composite is assumed to have the same amount of metal and

VFGM

<sup>L</sup> ln Eceramic Ealloy � �

C) <sup>25</sup> � <sup>10</sup>�<sup>6</sup> 8.2 � <sup>10</sup>�<sup>6</sup>

<sup>m</sup><sup>p</sup> ð Þ�<sup>m</sup> <sup>10</sup>�<sup>12</sup> 2.8 � <sup>10</sup>�<sup>10</sup>

<sup>m</sup><sup>p</sup> ð Þ <sup>29</sup> 3.5

ceramicð Þx (11b)

<sup>¼</sup> Ealloye<sup>α</sup> <sup>x</sup> � Ealloy Eceramic � Ealloy

(10)

(11a)

E xð Þ¼ Ealloye

Table 1. Material properties of aluminum alloy and alumina.

Figure 1. Geometry of the FGM plate along with its dimensions.

VFGM ceramicð Þ¼ x

in the FGM are obtained as

Coefficient of thermal expansion γ (/

Paris law parameter C in m=cycle MPa ffiffiffiffi

Fracture toughness KIC MPa ffiffiffiffi

174 Contact and Fracture Mechanics

Figure 2. Variation of modulus of elasticity along the length of the plate.

The coefficient of thermal expansion for the FGM by the rule of mixtures is calculated as

The value of the coefficient of thermal expansion for the equivalent composite is calculated

The fracture toughness of the FGM as well as the equivalent composite may be expressed as a

q

IC <sup>þ</sup> <sup>K</sup>ceramic IC 2

The Paris law parameters are assumed to have exponential variation in a manner similar to the

, where, <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>

, where, <sup>ς</sup> <sup>¼</sup> <sup>1</sup>

For the equivalent composite, we find the location at which the volume fraction of ceramic in the FGM is same as that of the equivalent composite. This location x may be found by either

present example x ¼ 56 mm. The Paris law parameters of the equivalent composite is assumed

Ccomposite ¼ Calloye

mcomposite ¼ malloye

<sup>α</sup> ln Ecomposite

elastic modulus. Thus, the variation in the parameters of Paris equation is taken as

ϑ x

ς x

function of the volume fraction of the ceramic by the following formula given by [41]

alloy ð Þþ <sup>x</sup> <sup>γ</sup>ceramicVFGM

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

alloy <sup>þ</sup> <sup>γ</sup>ceramicVcomposite

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>V</sup>FGM

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>V</sup>composite ceramic <sup>q</sup>

ceramicð Þx

�

�

<sup>L</sup> ln Cceramic

<sup>L</sup> ln mceramic

ceramic � � <sup>q</sup>

� � q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VFGM ceramicð Þx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vcomposite

Ealloy � �, where <sup>α</sup> is defined in Eq. (10). For the

Calloy � � (16)

malloy � � (17)

<sup>ϑ</sup> <sup>x</sup> (18a)

<sup>ς</sup> <sup>x</sup> (18b)

/ �

ceramicð Þx (14a)

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ceramic (14b)

C. A variation of coefficient of

(15a)

177

(15b)

<sup>γ</sup>ðÞ ¼ <sup>x</sup> <sup>γ</sup>alloyVFGM

The coefficient of thermal expansion for the equivalent composite is given by

<sup>γ</sup>composite <sup>¼</sup> <sup>γ</sup>alloyVcomposite

using Eq. (14b) and is found to be <sup>γ</sup>composite = 18.57 � <sup>10</sup>�<sup>6</sup>

Kalloy

Kcomposite IC <sup>¼</sup> <sup>K</sup>alloy

þ Kalloy

> þ Kalloy

The variation of fracture toughness is shown in Figure 6.

C xð Þ¼ Calloye

m xð Þ¼ malloye

IC <sup>þ</sup> <sup>K</sup>ceramic IC 2

IC � <sup>K</sup>ceramic IC 2

IC � <sup>K</sup>ceramic IC 2

thermal expansion for the FGM is shown in Figure 5.

KICð Þ¼ x

Figure 5 or by using the formula <sup>x</sup> <sup>¼</sup> <sup>1</sup>

to be same as that of the FGM at x ¼ x. Thus,

Figure 3. Variation of volume fraction of ceramic along the length of the plate.

Figure 4. Variation of Poisson's ratio along the length of the plate.

The coefficient of thermal expansion for the FGM by the rule of mixtures is calculated as

$$\gamma(\mathbf{x}) \, \, = \gamma\_{\text{alloy}} V\_{\text{alloy}}^{\text{FGM}}(\mathbf{x}) + \gamma\_{\text{carrier}} V\_{\text{carrier}}^{\text{FGM}}(\mathbf{x}) \tag{14a}$$

The coefficient of thermal expansion for the equivalent composite is given by

$$\mathcal{Y}\_{\text{composite}} = \mathcal{Y}\_{\text{alloy}} V\_{\text{alloy}}^{\text{compposite}} + \mathcal{Y}\_{\text{carrier}} V\_{\text{carrier}}^{\text{compposite}} \tag{14b}$$

The value of the coefficient of thermal expansion for the equivalent composite is calculated using Eq. (14b) and is found to be <sup>γ</sup>composite = 18.57 � <sup>10</sup>�<sup>6</sup> / � C. A variation of coefficient of thermal expansion for the FGM is shown in Figure 5.

The fracture toughness of the FGM as well as the equivalent composite may be expressed as a function of the volume fraction of the ceramic by the following formula given by [41]

$$\begin{split} K\_{\text{IC}}(\mathbf{x}) &= \frac{K\_{\text{IC}}^{\text{alloy}} + K\_{\text{IC}}^{\text{carrier}}}{2} \\ &+ \frac{K\_{\text{IC}}^{\text{alloy}} - K\_{\text{IC}}^{\text{carrier}}}{2} \left( \sqrt{1 - V\_{\text{cramic}}^{\text{FQM}}(\mathbf{x})} - \sqrt{V\_{\text{cramic}}^{\text{FQM}}(\mathbf{x})} \right) \\ &K\_{\text{IC}}^{\text{compposite}} = \frac{K\_{\text{IC}}^{\text{alloy}} + K\_{\text{IC}}^{\text{carrier}}}{2} \\ &+ \frac{K\_{\text{IC}}^{\text{alloy}} - K\_{\text{IC}}^{\text{carrier}}}{2} \left( \sqrt{1 - V\_{\text{cramic}}^{\text{compposite}}} - \sqrt{V\_{\text{cramic}}^{\text{compposite}}} \right) \end{split} \tag{15b}$$

The variation of fracture toughness is shown in Figure 6.

Figure 3. Variation of volume fraction of ceramic along the length of the plate.

176 Contact and Fracture Mechanics

Figure 4. Variation of Poisson's ratio along the length of the plate.

The Paris law parameters are assumed to have exponential variation in a manner similar to the elastic modulus. Thus, the variation in the parameters of Paris equation is taken as

$$\mathsf{C}(\mathbf{x}) = \mathsf{C}\_{\text{alloy}} \mathcal{e}^{\otimes \mathbf{x}}, \text{ where, } \mathfrak{G} = \frac{1}{L} \ln \left( \frac{\mathsf{C}\_{\text{cemanic}}}{\mathsf{C}\_{\text{alloy}}} \right) \tag{16}$$

$$m(\mathbf{x}) = m\_{\text{alloy}} e^{\boldsymbol{\varsigma} \cdot \mathbf{x}}, \text{ where, } \boldsymbol{\varsigma} = \frac{1}{L} \ln \left( \frac{m\_{\text{carrier}}}{m\_{\text{alloy}}} \right) \tag{17}$$

For the equivalent composite, we find the location at which the volume fraction of ceramic in the FGM is same as that of the equivalent composite. This location x may be found by either Figure 5 or by using the formula <sup>x</sup> <sup>¼</sup> <sup>1</sup> <sup>α</sup> ln Ecomposite Ealloy � �, where <sup>α</sup> is defined in Eq. (10). For the present example x ¼ 56 mm. The Paris law parameters of the equivalent composite is assumed to be same as that of the FGM at x ¼ x. Thus,

$$\mathsf{C}\_{\text{composite}} = \mathsf{C}\_{\text{alloy}} e^{\theta \cdot \mathsf{T}} \tag{18a}$$

$$m\_{\text{composite}} = m\_{\text{alloy}} e^{\varepsilon \overline{\mathbf{z}}} \tag{18b}$$

Figure 5. Variation of coefficient of thermal expansion along the length of the plate.

The values of <sup>C</sup> and <sup>m</sup> for the equivalent composite comes out to be Ccomposite <sup>¼</sup> <sup>2</sup>:<sup>34</sup> � <sup>10</sup>�<sup>11</sup> <sup>m</sup><sup>=</sup> cycle MPa ffiffiffiffi <sup>m</sup><sup>p</sup> ð Þ�<sup>m</sup> and mcomposite <sup>¼</sup> <sup>5</sup>:88. The variation of <sup>C</sup> and <sup>m</sup> are shown in Figures 7 and <sup>8</sup> respectively.

Plastic behaviour for FGM can be modeled using Ramberg Osgood equation [42]

$$
\varepsilon = \frac{\sigma}{E} + \left(\frac{\sigma}{H}\right)^{1/n} \tag{19}
$$

KIeq max > KIC. Simulation continues until this condition is met. Here, KIeq max is the equivalent SIF for mode-I at principal crack tip and KIC is the material property called fracture toughness

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179

or critical SIF. KIC for FGM is given by [35]

Figure 6. Variation of fracture toughness along the length of the plate.

Figure 7. Variation of Cð Þx along the length of the plate.

Here, H is the strength coefficient and n is the strain hardening exponent. The value of n ¼ 0:0946 is used for the present example. The values of the parameters of Paris equation are taken as <sup>C</sup> <sup>¼</sup> <sup>3</sup> � <sup>10</sup>�<sup>11</sup> and <sup>m</sup> <sup>¼</sup> 3. In actual case the path of crack growth is curved but in this study the linear crack growth path is taken. Linear crack extension length Δa for an edge crack is kept constant. For a center crack maximum crack extension length Δamax is kept on principal crack tip. The principle crack tip is the crack tip where ΔKIeq maximum. Crack increment at the other crack tip is given by:

$$
\Delta a = \Delta a\_{\text{max}} \left( \frac{\Delta K\_{\text{le}}}{\Delta K\_{\text{le}} \\_ {\text{max}}} \right)^{m} \tag{20}
$$

The crack tip extension at the principal crack tip is Δamax and at the other crack tip extension is smaller. The crack extension takes place KIeq max < KIC . Crack becomes unstable when KIeq max > KIC. Simulation continues until this condition is met. Here, KIeq max is the equivalent SIF for mode-I at principal crack tip and KIC is the material property called fracture toughness or critical SIF. KIC for FGM is given by [35]

Figure 6. Variation of fracture toughness along the length of the plate.

The values of <sup>C</sup> and <sup>m</sup> for the equivalent composite comes out to be Ccomposite <sup>¼</sup> <sup>2</sup>:<sup>34</sup> � <sup>10</sup>�<sup>11</sup> <sup>m</sup><sup>=</sup>

Plastic behaviour for FGM can be modeled using Ramberg Osgood equation [42]

Figure 5. Variation of coefficient of thermal expansion along the length of the plate.

Δa ¼ Δamax

<sup>ε</sup> <sup>¼</sup> <sup>σ</sup> E þ

<sup>m</sup><sup>p</sup> ð Þ�<sup>m</sup> and mcomposite <sup>¼</sup> <sup>5</sup>:88. The variation of <sup>C</sup> and <sup>m</sup> are shown in Figures 7 and <sup>8</sup>

σ H � �<sup>1</sup>=<sup>n</sup>

ΔKIeq ΔKIeq max � �<sup>m</sup>

The crack tip extension at the principal crack tip is Δamax and at the other crack tip extension is smaller. The crack extension takes place KIeq max < KIC . Crack becomes unstable when

Here, H is the strength coefficient and n is the strain hardening exponent. The value of n ¼ 0:0946 is used for the present example. The values of the parameters of Paris equation are taken as <sup>C</sup> <sup>¼</sup> <sup>3</sup> � <sup>10</sup>�<sup>11</sup> and <sup>m</sup> <sup>¼</sup> 3. In actual case the path of crack growth is curved but in this study the linear crack growth path is taken. Linear crack extension length Δa for an edge crack is kept constant. For a center crack maximum crack extension length Δamax is kept on principal crack tip. The principle crack tip is the crack tip where ΔKIeq maximum. Crack increment at the

(19)

(20)

cycle MPa ffiffiffiffi

178 Contact and Fracture Mechanics

other crack tip is given by:

respectively.

Figure 7. Variation of Cð Þx along the length of the plate.

Figure 8. Variation of mð Þx along the length of the plate.

$$K\_{\rm IC}(\mathbf{x}) = K\_{\rm IC}^{\rm certain} \left[ \frac{E(\mathbf{x})}{1 - \nu\_{\rm FQM}^2} \left\{ V\_m(\mathbf{x}) \frac{1 - \nu\_{\rm alloy}^2}{E\_{\rm alloy}} \left( \frac{K\_{\rm IC}^{\rm alloy}}{K\_{\rm IC}^{\rm certain}} \right)^2 + (1 - V\_m(\mathbf{x})) \frac{1 - \nu\_{\rm certain}^2}{E\_{\rm fermionic}} \right\} \right]^{1/2} \tag{21}$$

where KICð Þ<sup>x</sup> is the fracture toughness of the FGM at point x. <sup>K</sup>alloy IC and <sup>K</sup>ceramic IC are the fracture toughness of the alloy and ceramic, while νalloy and νceramic are Poisson's ratios for the alloy and ceramic respectively. Vmð Þx denotes the volume fraction for the alloy at point x.

The constitutive relation for the elastic-plastic material is given as

$$
\sigma(\mathfrak{u}) = D\_{\mathfrak{e}\mathfrak{p}}(\mathfrak{x}) \varepsilon(\mathfrak{u}) \tag{22}
$$

Deð Þ¼ x

as discussed under:

following relation is obtained,

Plastic modulus Η is given as

tensor. So, we must have <sup>F</sup>ð Þ¼ <sup>σ</sup> <sup>σ</sup><sup>2</sup>

Eð Þx f g 1 � 2νð Þx f g 1 þ νð Þx

Total strain increment is the sum of elastic and plastic strains

Elastic incremental strain and stress is determined

1 � νð Þx νð Þx 0 νð Þx 1 � νð Þx 0

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

The incremental theory of plasticity [43] has been used to model the elastic-plastic constitutive relation for a material. An incremental stress vector dσ and incremental strain vector dε are such that dσ ¼ Dep:dε. Where Dep is the elastic-plastic constitutive matrix, which is determined

where σ is the stress tensor and σ is the equivalent stress, F and f are two different failure functions. By the flow rule the incremental strain is related to the gradient of a function known plastic potential. If the plastic potential function and the failure function is same, then the

> <sup>Η</sup> <sup>¼</sup> <sup>d</sup><sup>σ</sup> dε<sup>p</sup>

According to the Von Mises criteria, F ¼ J2, where J<sup>2</sup> is the second invariant of deviatoric stress

dσ ¼ De dε � dε<sup>p</sup>

<sup>¼</sup> <sup>∂</sup><sup>f</sup> ∂σ : ∂σ ∂ε<sup>p</sup> : ∂ε<sup>p</sup> <sup>∂</sup><sup>w</sup> : ∂w ∂ε<sup>p</sup> :∂ε<sup>p</sup>

<sup>3</sup> , Thus Eqs. (26) and (27) result in

For a given strain energy δw, and according to the definitionof dε<sup>p</sup> we must have,

After taking the derivatives from both sides of failure criteria equation

∂F ∂σ dσ � � 0 0 <sup>1</sup> � <sup>2</sup>νð Þ<sup>x</sup>

2

dε ¼ dε<sup>e</sup> þ dε<sup>p</sup> (24)

dσ ¼ Dedε<sup>e</sup> (25)

Fð Þ¼ σ fð Þ σ (26)

dε<sup>p</sup> ¼ ∇F:d λ (27)

δw ¼ σ:dε<sup>p</sup> (29)

� � (30)

� � (31)

3 7 7

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<sup>5</sup> <sup>¼</sup> <sup>D</sup> (23b)

181

(28)

where x is the vector of x and y-coordinates, Depð Þx is elastic-plastic constitutive matrix varying in x-direction. The elastic constitutive matrix can be written for plane stress condition as

$$D\_{\varepsilon}(\mathbf{x}) = \frac{E(\mathbf{x})}{\left\{1 - \nu(\mathbf{x})^2\right\}} \begin{bmatrix} 1 & \nu(\mathbf{x}) & 0\\ \nu(\mathbf{x}) & 1 & 0\\ 0 & 0 & \frac{1 - \nu(\mathbf{x})}{2} \end{bmatrix} = D \tag{23a}$$

and for plane strain condition as

Fatigue Fracture of Functionally Graded Materials Under Elastic-Plastic Loading Conditions Using Extended… http://dx.doi.org/10.5772/intechopen.72778 181

$$D\_{\varepsilon}(\mathbf{x}) = \frac{E(\mathbf{x})}{\{1 - 2\nu(\mathbf{x})\}\{1 + \nu(\mathbf{x})\}} \begin{bmatrix} 1 - \nu(\mathbf{x}) & \nu(\mathbf{x}) & 0\\ \nu(\mathbf{x}) & 1 - \nu(\mathbf{x}) & 0\\ 0 & 0 & \frac{1 - 2\nu(\mathbf{x})}{2} \end{bmatrix} = D \tag{23b}$$

The incremental theory of plasticity [43] has been used to model the elastic-plastic constitutive relation for a material. An incremental stress vector dσ and incremental strain vector dε are such that dσ ¼ Dep:dε. Where Dep is the elastic-plastic constitutive matrix, which is determined as discussed under:

Total strain increment is the sum of elastic and plastic strains

$$d\varepsilon = d\varepsilon\_{\varepsilon} + d\varepsilon\_{p} \tag{24}$$

Elastic incremental strain and stress is determined

$$d\sigma = D\_\epsilon d\varepsilon\_\epsilon \tag{25}$$

$$F(\sigma) = f(\overline{\sigma}) \tag{26}$$

where σ is the stress tensor and σ is the equivalent stress, F and f are two different failure functions. By the flow rule the incremental strain is related to the gradient of a function known plastic potential. If the plastic potential function and the failure function is same, then the following relation is obtained,

$$d\varepsilon\_p = \nabla F.d\,\lambda\,\tag{27}$$

Plastic modulus Η is given as

KICð Þ¼ <sup>x</sup> <sup>K</sup>ceramic IC

180 Contact and Fracture Mechanics

E xð Þ <sup>1</sup> � <sup>ν</sup><sup>2</sup> FGM

Figure 8. Variation of mð Þx along the length of the plate.

Deð Þ¼ x

and for plane strain condition as

2 4

Vmð Þx

where KICð Þ<sup>x</sup> is the fracture toughness of the FGM at point x. <sup>K</sup>alloy

The constitutive relation for the elastic-plastic material is given as

Eð Þx 1 � νð Þx <sup>2</sup> n o

8 < :

<sup>1</sup> � <sup>ν</sup><sup>2</sup> alloy Ealloy

and ceramic respectively. Vmð Þx denotes the volume fraction for the alloy at point x.

Kalloy IC Kceramic IC

toughness of the alloy and ceramic, while νalloy and νceramic are Poisson's ratios for the alloy

where x is the vector of x and y-coordinates, Depð Þx is elastic-plastic constitutive matrix varying in x-direction. The elastic constitutive matrix can be written for plane stress condition as

1 νð Þx 0 νð Þx 1 0 0 0 <sup>1</sup> � <sup>ν</sup>ð Þ<sup>x</sup>

2

!<sup>2</sup>

þ ð Þ 1 � Vmð Þx

σð Þ¼ u Depð Þx εð Þ u (22)

3 7 7 <sup>1</sup> � <sup>ν</sup><sup>2</sup> ceramic Eceramic

IC and <sup>K</sup>ceramic

9 = ;

IC are the fracture

<sup>5</sup> <sup>¼</sup> <sup>D</sup> (23a)

3 5

1=2

(21)

$$H = \frac{d\overline{\sigma}}{d\overline{\varepsilon}\_p} \tag{28}$$

For a given strain energy δw, and according to the definitionof dε<sup>p</sup> we must have,

$$
\delta w = \overline{\sigma}.d\overline{\varepsilon}\_p\tag{29}
$$

According to the Von Mises criteria, F ¼ J2, where J<sup>2</sup> is the second invariant of deviatoric stress tensor. So, we must have <sup>F</sup>ð Þ¼ <sup>σ</sup> <sup>σ</sup><sup>2</sup> <sup>3</sup> , Thus Eqs. (26) and (27) result in

$$d\sigma = D\_{\varepsilon} \{ d\varepsilon - d\varepsilon\_p \} \tag{30}$$

After taking the derivatives from both sides of failure criteria equation

$$\left(\frac{\partial F}{\partial \sigma}d\sigma\right) = \left(\frac{\partial f}{\partial \overline{\sigma}}.\frac{\partial \overline{\sigma}}{\partial \overline{\varepsilon}\_p}.\frac{\partial \overline{\varepsilon}\_p}{\partial w}.\frac{\partial w}{\partial \overline{\varepsilon}\_p}.\partial \epsilon\_p\right) \tag{31}$$

For simplicity we take <sup>∂</sup><sup>F</sup> <sup>∂</sup><sup>σ</sup> <sup>¼</sup> <sup>a</sup>, <sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>σ</sup> ¼ a

$$a.d\sigma = \overline{a}.H.\left(\frac{1}{\sigma}\right)\sigma.d\varepsilon\_p\tag{32}$$

6.1. Example 1

<sup>Δ</sup><sup>a</sup> <sup>¼</sup> <sup>a</sup>

A rectangular FGM plate of length (L) 100 mm. and height (D) 200 mm. with 100% aluminum alloy on left side and 100% ceramic (alumina) on right side is considered. Property variation is taken in x-direction, where x = 0 to x = 100 mm. The plate with a major edge crack of length a = 20 mm is analyzed under plane strain condition in the presence of multiple discontinuities. In all simulations, the plate dimensions, initial crack length and material properties are taken to be same. The properties of FGM, composites and aluminum alloy are already described in Table 1. The material properties of the inclusions are taken as Ε ¼ 20 GPa and ν ¼ 0:2. The plate domain is discretized using uniformly distributed 117 nodes in x-direction and 235 nodes in y-direction. The fatigue crack growth analysis is performed by taking a crack increment of

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

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

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10= 2 mm. A cyclic tensile load varying from σmax ¼ 70 MPa to σmin ¼ 0 MPa is applied in all the simulations. The geometric discontinuities like holes, inclusions and minor cracks are added in the plate in addition to the major edge or center crack to analyze their effect on the fatigue life of the material. The fatigue life of the FGM, equivalent composite and aluminum

Figures 9 and 10 show a plate with a major edge crack of length a = 20 mm at the left and right edge respectively. These plates have been analyzed under plane strain condition using a uniform mesh of 117 by 235 nodes. The plots of the fatigue life for different materials are shown in Figure 11. From these figures, it is seen that the equivalent composite withstands 7885 cycles before it fails while the FGM with crack on alloy side undergoes 15,561 cycles and

alloy are obtained under mode-I loading, and are compared with each other.

6.2. Plate with a major edge crack under linear elastic condition

Figure 9. Plate with an edge crack on the alloy rich side under mode-I loading.

dλ is calculated by omitting dσ between Eqs. (30) and (31) and substituting dε<sup>p</sup> from Eq. (27). By substituting dλ in Eq. (27), the final form of material matrix is obtained as [43]

$$D\_{ep} = D\_c - D\_p \tag{33}$$

$$\text{where } D\_{\mathcal{V}} = \frac{Da \ a^T D}{\frac{\overline{a}}{\overline{\sigma}} H \sigma^T + a^T D a} \tag{34}$$
