**4. Fracture Criteria**

Incompatibility formulation is applied to extracted *J*-integral [26, 27].

$$
\sigma\_{\vec{\eta}\vec{\}} = d\_{\vec{\eta}kl}(\infty)\varepsilon\_{kl} \quad , \qquad \varepsilon\_{\vec{\eta}} \neq \frac{1}{2} \left( u\_{i\vec{\eta}} + u\_{\vec{\rho},i} \right) \; , \; \sigma\_{\vec{\eta}\vec{\eta}} = 0 \tag{41}
$$

where

$$
\sigma\_{i\dot{j}} = C\_{i\dot{j}kl}(\infty)\sigma\_{kl} \quad (i,j,k,l = \mathbf{1}, \mathbf{2}, \mathbf{3})\tag{42}
$$

with

$$\mathbf{C} = \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{16} \\\\ \mathbf{C}\_{12} & \mathbf{C}\_{22} & \mathbf{C}\_{26} \\\\ \mathbf{C}\_{16} & \mathbf{C}\_{26} & \mathbf{C}\_{66} \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{1111} & \mathbf{C}\_{1122} & 2\mathbf{C}\_{1112} \\\\ \mathbf{C}\_{2211} & \mathbf{C}\_{2222} & 2\mathbf{C}\_{2212} \\\\ 2\mathbf{C}\_{1211} & 2\mathbf{C}\_{1222} & 4\mathbf{C}\_{1212} \end{bmatrix} \tag{43}$$

From **Figure 3**:

$$J = \int\_{A} \left( \sigma\_{\vec{\eta}} u\_{i,1} - w \delta\_{\vec{\eta}} \right) q\_{\cdot \cdot j} dA + \int\_{A} \left( \sigma\_{\vec{\eta}} u\_{i,1} - w \delta\_{\vec{\eta}} \right)\_{,j} q dA \tag{44}$$

*w* is the strain energy density:

$$w = \frac{1}{2} \sigma\_{\vec{\eta}} \varepsilon\_{\vec{\eta}} \tag{45}$$

**Figure 3.** *An integral contour at the tip of the crack.*

The interaction integral and J integral can be defined [28–30]:

$$\begin{split} M &= \int\_{A} \left\{ \sigma\_{\vec{\eta}} u\_{i,1}^{a\text{aux}} + \sigma\_{\vec{\eta}}^{a\text{aux}} u\_{i,1} - \frac{1}{2} \left( \sigma\_{ik} u\_{ik}^{a\text{aux}} + \sigma\_{ik}^{a\text{aux}} u\_{i} \right) \delta\_{\vec{\eta}} \right\} q\_{\cdot \cdot \vec{f}} dA \\ &+ \int\_{A} \left\{ \sigma\_{\vec{\eta}} \left( c\_{ijkl}^{\text{tip}} - c\_{ijkl}(\varkappa) \right) \sigma\_{kl,1}^{a\text{aux}} \right\} q dA \end{split} \tag{46}$$

$$J\_{local} = \left(K\_I^2 + K\_{II}^2\right) / E\_{tip} \tag{47}$$

$$J\_{local}^{\epsilon} = \frac{\left(K\_I + K\_I^{\text{aux}}\right)^2 + \left(K\_{II} + K\_{II}^{\text{aux}}\right)^2}{E\_{tip}} = J\_{local} + J\_{local}^{\text{aux}} + \mathcal{M}\_{local} \tag{48}$$

$$J\_{local}^{\text{aux}} = \left[ \left( K\_I^{\text{aux}} \right)^2 + \left( K\_{II}^{\text{aux}} \right)^2 \right] / E\_{tip} \tag{49}$$

and *M*local is given by

$$\mathcal{M}\_{local} = \mathcal{Z} \left( \mathcal{K}\_{I} \mathcal{K}\_{I}^{aux} + \mathcal{K}\_{II} \mathcal{K}\_{II}^{aux} \right) / E\_{tip} \tag{50}$$

$$\begin{array}{ll} \mathbf{K}\_{I} = \mathbf{M}\_{\text{local}}^{(1)} \mathbf{E}\_{\text{tip}} / 2, & \quad \left( \mathbf{K}\_{I}^{\text{aux}} = \mathbf{1.0}, \mathbf{K}\_{I}^{\text{aux}} = \mathbf{0.0} \right), \mathbf{K}\_{II} \\\ = \mathbf{M}\_{\text{local}}^{(2)} \mathbf{E}\_{\text{tip}} / 2, & \quad \left( \mathbf{K}\_{I}^{\text{aux}} = \mathbf{0.0}, \mathbf{K}\_{I}^{\text{aux}} = \mathbf{1.0} \right). \end{array} \tag{51}$$

The Eq. (51) is used to calculate the stress intensity factors during the fracture analysis in functionally graded materials. The crack propagation criterion (Maximum hoop stress) was applied by the depend on procedure that was adopted by Erdogan and Sih [31].
