**2. Field Equations**

In elastic materials, Hooke's law can be employed in the following equation that clears the relationship between the stresses and strains in specific material.

$$
\boldsymbol{\sigma} = \mathbf{D} \mathbf{e} \tag{1}
$$

**σ** and **ε** are the stress and strain respectively; while **D** represents the material matrix that varied with the displacement *x* in the functionally graded materials can be find as:

$$\mathbf{D} = \frac{E(\mathbf{x})}{\mathbf{1} - \nu^2} \begin{bmatrix} \mathbf{1} & \nu(\mathbf{x}) & \mathbf{0} \\ \nu & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & (\mathbf{1} - \nu(\mathbf{x}))/2 \end{bmatrix} \tag{\text{Plane stress}} \tag{2}$$

and

$$\mathbf{D} = \frac{E(\mathbf{x})(1-\nu(\mathbf{x}))}{(1+\nu(\mathbf{x}))(1-2\nu(\mathbf{x}))} \begin{bmatrix} 1 & \frac{\nu(\mathbf{x})}{1-\nu(\mathbf{x})} & \mathbf{0} \\\\ \frac{\nu(\mathbf{x})}{1-\nu(\mathbf{x})} & \mathbf{1} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \frac{1-2\nu(\mathbf{x})}{2(1-\nu(\mathbf{x}))} \end{bmatrix} \tag{\text{Plane strain}} \tag{3}$$

Where E and *ν* represent Young's Modulus and Poisson's ratio that varied with the displacement *x* respectively, Eqs. (2) and (3) are useful because they give a change in the properties of the material at each point of the material in relation to the displacement.

Eq. (1) can be put as:

$$\mathbf{e} = \mathbf{C}\boldsymbol{\sigma} = \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{16} \\ \mathbf{C}\_{12} & \mathbf{C}\_{22} & \mathbf{C}\_{26} \\ \mathbf{C}\_{16} & \mathbf{C}\_{26} & \mathbf{C}\_{66} \end{bmatrix} \begin{Bmatrix} \sigma\_{\mathbf{x}\mathbf{x}} \\ \sigma\_{\mathbf{yy}} \\ \sigma\_{\mathbf{xy}} \end{Bmatrix} \tag{4}$$

*XEFGM Fracture Analysis of Functionally Graded Materials under Mixed Mode… DOI: http://dx.doi.org/10.5772/intechopen.98765*

where **C** represents the compliance matrix. In FGM

$$\mathbf{E} = \mathbf{E}(\mathbf{x}\_1, \mathbf{x}\_2) = \mathbf{E}(\mathbf{x}) \tag{5}$$

$$\mathfrak{v} = \mathfrak{v}(\mathbf{x\_1}, \mathbf{x\_2}) = \mathfrak{v}(\mathbf{x}) \tag{6}$$

In the above two equations, it is clear that the Young's Modulus and Poisson's ratio change with the displacement and this gives the impression for the behave of properties of the functionally graded materials as shown in **Figure 1**.

The stresses and displacements in functionally graded materials near the crack tip can be obtained by depending on angular stress and displacement functions as [2]:

$$\sigma\_{\rm 11} = \frac{1}{\sqrt{2\pi \mathbf{r}}} \left[ \mathbf{K}\_{\rm I} \mathbf{f}\_{\mathbf{11}}^{\rm I}(\boldsymbol{\Theta}) + \mathbf{K}\_{\rm I} \mathbf{f}\_{\mathbf{11}}^{\rm II}(\boldsymbol{\Theta}) \right] \tag{7}$$

$$\sigma\_{22} = \frac{1}{\sqrt{2\pi\mathbf{r}}} \left[ \mathbf{K}\_{\mathrm{I}} \mathbf{f}\_{22}^{\mathrm{I}}(\boldsymbol{\Theta}) + \mathbf{K}\_{\mathrm{II}} \mathbf{f}\_{22}^{\mathrm{II}}(\boldsymbol{\Theta}) \right] \tag{8}$$

$$\sigma\_{12} = \frac{1}{\sqrt{2\pi r}} \left[ \mathbf{K}\_{\mathrm{I}} \mathbf{f}\_{12}^{\mathrm{I}}(\boldsymbol{\Theta}) + \mathbf{K}\_{\mathrm{II}} \mathbf{f}\_{12}^{\mathrm{II}}(\boldsymbol{\Theta}) \right] \tag{9}$$

$$\mathbf{u}\_1 = \frac{\mathbf{1}}{\mathbf{G}\_{\mathrm{tip}}} \sqrt{\frac{\mathbf{r}}{2\pi}} [\mathbf{K}\_{\mathrm{l}} \mathbf{g}\_1^{\mathrm{l}}(\boldsymbol{\theta}) + \mathbf{K}\_{\mathrm{l}l} \mathbf{g}\_1^{\mathrm{ll}}(\boldsymbol{\theta})] \tag{10}$$

$$\mathbf{u}\_2 = \frac{\mathbf{1}}{\mathbf{G}\_{\mathrm{tip}}} \sqrt{\frac{\mathbf{r}}{2\pi}} [\mathbf{K}\_{\mathrm{l}} \mathbf{g}\_2^{\mathrm{l}}(\theta) + \mathbf{K}\_{\mathrm{ll}} \mathbf{g}\_2^{\mathrm{ll}}(\theta)] \tag{11}$$

$$\mathbf{G\_{tip}} = \frac{\mathbf{E\_{tip}}}{\left[\mathbf{2}\left(\mathbf{1} + \nu\_{\rm tip}\right)\right]} \tag{12}$$

**Figure 1.** *FGM body with crack,*

The mechanical properties in Eq. (11) to (13) will be extracted at the tip of crack. f I ijð Þ<sup>θ</sup> , fII ijð Þ<sup>θ</sup> , g<sup>I</sup> ijð Þ<sup>θ</sup> and gII ijð Þθ ð Þ i, j ¼ 1, 2 represent the standard angular functions [17, 18]. Eqs. (7) to (13) can calculate the values of stresses and displacements at each point of the material supported by the enrichment functions.
