Multiple cracks analysis in a FG orthotropic layer with FGPM coating under anti-plane loading

Abstract

An analytical solution to the fracture analysis of a functionally graded (FG) orthotropic substrate with FG piezoelectric coating weakened by a screw dislocation is carried out. The material properties are assumed to vary exponentially through the thickness of the layers. The problem is solved under various types of anti-plane shear and in-plane electric loadings. At first, by considering a single-screw dislocation at crack location, an analytical solution is developed. Next, by using the Fourier transform, the problem is reduced to a system of singular integral equations with Cauchy-type singularities. Then, by computing the dislocation densities, both the stress intensity factors and the stress fields at the crack tips under different electromechanical loadings are determined. In this investigation, various examples are solved to show the applicability of the proposed solution by studying the effects of the cracks configurations, material properties, and non-homogeneity parameter on the stress intensity factors.

Appendices

Appendix A

The functions $T_{ij}$ given in Eq. (17a) and (17b)

$$\begin{aligned} T_{11} & = {\text{e}}^{(y + \eta )\lambda } \sinh [(h_{2} + \eta )\chi_{1} ] \\ T_{12} & = G_{y} \chi_{2} \cosh (h_{1} \chi_{2} )[\chi_{1} \cosh (y\chi_{1} ) + \lambda \sinh (y\chi_{1} )] \\ T_{13} & = \left[ {G_{y} \kappa \chi_{1} \cosh (y\chi_{1} ) + \left( {G_{y} \kappa \lambda + s^{2} \alpha_{1} } \right)\sinh (y\chi_{1} )} \right]\sinh (h_{1} \chi_{2} ) \\ \end{aligned}$$

(32)

$$\begin{aligned} T_{21} & = {\text{e}}^{(y + \eta )\lambda } s\sinh [(h_{2} + \eta )\chi_{1} ] \\ T_{22} & = G_{y} \chi_{2} g^{2} \cosh (h_{1} \chi_{2} )\sinh (y\chi_{1} ) \\ T_{23} & = \left[ {\alpha_{1} \chi_{1} \cosh (y\chi_{1} ) + \left( {G_{y} \kappa g^{2} - \alpha_{1} \lambda } \right)\sinh (y\chi_{1} )} \right]\sinh (h_{1} \chi_{2} ) \\ \end{aligned}$$

(33)

$$\begin{aligned} T_{31} & = {\text{e}}^{(y + \eta )\lambda } (\chi_{1} \cosh [(h_{2} + y)\chi_{1} ] + \lambda \sinh \,[(h_{2} + y)\chi_{1} ]) \\ T_{32} & = \chi_{2} G_{y} g^{2} \cosh (h_{1} \chi_{2} )\sinh (\eta \chi_{1} ) \\ T_{33} & = \left[ {\alpha_{1} \chi_{1} \cosh (\eta \chi_{1} ) + \left( {G_{y} \kappa g^{2} - \alpha_{1} \lambda } \right)\sinh (\eta \chi_{1} )} \right]\sinh (h_{1} \chi_{2} ) \\ \end{aligned}$$

(34)

$$\begin{aligned} T_{41} & = {\text{e}}^{(y + \eta )\lambda } s\sinh [(h_{2} + \eta )\chi_{1} ] \\ T_{42} & = g^{2} G_{y} \,\chi_{2} \cosh (h_{1} \chi_{2} )\sinh (\eta \chi_{1} ) \\ T_{43} & = \left[ {\alpha_{1} \chi_{1} \cosh (\eta \chi_{1} ) + \left( {G_{y} \kappa g^{2} - \alpha_{1} \lambda } \right)\sinh (\eta \chi_{1} )} \right]\sinh (h_{1} \chi_{2} ) \\ \end{aligned}$$

(35)

Appendix B

The functions $A_{i} ,C_{i}$ and $F$ given in Eqs. (20) and (21)

$$\begin{aligned} A_{1} & = 2\chi_{2} y(\chi_{1} - \lambda )({\text{e}}^{{ - (h_{2} + y)\chi_{1} + y\lambda - h_{1} \kappa }} + {\text{e}}^{{(\lambda + \chi_{1} )y + h_{2} \chi_{1} - h_{1} \kappa }} ) \\ A_{2} & = {\text{e}}^{{(h_{2} - y)\lambda - y\chi_{1} - h_{1} \chi_{2} }} (\chi_{2} + \kappa )\left[ {(\chi_{2} - \kappa )\alpha_{1} + y(\lambda - \chi_{1} )} \right] \\ A_{3} & = {\text{e}}^{{(h_{2} + y)\lambda - y\chi_{1} - h_{1} \chi_{2} }} (\chi_{2} - \kappa )\left[ {(\chi_{2} + \kappa )\alpha_{1} - y(\lambda - \chi_{1} )} \right] \\ A_{4} & = {\text{e}}^{{(h_{2} + 3y)\lambda - y\chi_{1} + h_{1} \chi_{2} }} (\chi_{2} - \kappa )\left[ {(\chi_{2} + \kappa )\alpha_{1} - y(\lambda + \chi_{1} )} \right] \\ A_{5} & = {\text{e}}^{{(h_{2} + y)\lambda + y\chi_{1} - h_{1} \chi_{2} }} (\chi_{2} + \kappa )\left[ {(\chi_{2} - \kappa )\alpha_{1} + y(\lambda + \chi_{1} )} \right] \\ A_{6} & = {\text{e}}^{{ - (h_{2} + y)\chi_{1} + y\lambda - h_{1} \kappa }} 2D_{0} e_{15} y\left[ {\left( { - 1 + {\text{e}}^{{2\left( {h2 + y} \right)\chi_{1} }} } \right)\lambda + \left( {1 + {\text{e}}^{{2\left( {h_{2} + y} \right)\chi_{1} }} } \right)\chi_{1} } \right] \\ A_{7} & = ( - \,\chi_{2} + {\text{e}}^{{h_{1} \kappa }} [\chi_{2} \cosh (h_{1} \chi_{2} ) - \kappa \sinh (h_{1} \chi_{2} )]) \\ \end{aligned}$$

(36)

$$\begin{aligned} C_{1} & = 2g^{2} s^{2} \chi_{2} y({\text{e}}^{{(\lambda + \chi_{1} )y + h_{2} \chi_{1} - h_{1} \kappa }} - {\text{e}}^{{(\lambda - \chi_{1} )y - h_{2} \chi_{1} - h_{1} \kappa }} ) \\ C_{2} & = {\text{e}}^{{(h_{2} + y)\lambda + y\chi_{1} + h_{1} \chi_{2} }} [\alpha_{1} (\chi_{2} + \kappa ) - y(\lambda + \chi_{1} )](\chi_{2} - \kappa )(\chi_{1} - \lambda ) \\ C_{3} & = {\text{e}}^{{(h_{2} + y)\lambda + y\chi_{1} - h_{1} \chi_{2} }} s^{2} [g^{2} (\chi_{2} + \kappa )y + \alpha_{1} (\chi_{1} - \lambda )] \\ C_{4} & = {\text{e}}^{{(h_{2} + y)\lambda - y\chi_{1} + h_{1} \chi_{2} }} s^{2} [g^{2} (\chi_{2} - \kappa )y + \alpha_{1} (\chi_{1} + \lambda )] \\ C_{5} & = {\text{e}}^{{(h_{2} + y)\lambda - y\chi_{1} - h_{1} \chi_{2} }} s^{2} [g^{2} (\chi_{2} + \kappa )y - \alpha_{1} (\chi_{1} + \lambda )] \\ C_{6} & = {\text{e}}^{{ - (h_{2} + y)\chi_{1} + y\lambda - h_{1} \kappa }} 2D_{0} \left( { - \,1 + {\text{e}}^{{2(h_{2} + y)\chi_{1} }} } \right){\text{e}}_{15} g^{2} s^{2} y \\ C_{7} & = ( - \,\chi_{2} + {\text{e}}^{{h_{1} \kappa }} [\chi_{2} \cosh (h_{1} \chi_{2} ) - \kappa \sinh (h_{1} \chi_{2} )]) \\ \end{aligned}$$

(37)

$$F = 4d_{11} \pi sG_{y} \left( {g^{2} G_{y} \chi_{2} \cosh (h_{1} \chi_{2} )\sinh (h_{2} \chi_{1} ) + \left[ {\alpha_{1} \chi_{1} \cosh (h_{2} \chi_{1} ) + \left( {\alpha_{1} \lambda - g^{2} \kappa G_{y} } \right)\sinh (h_{2} \chi_{1} )} \right]\sinh (h_{1} \chi_{2} )} \right)$$

(38)

Appendix C

The kernels mentioned in Eq. (24) are given as

$$\begin{aligned} k_{ij} & = - \frac{1}{\pi }\left( {g^{2} G_{y} \left[ {\int\limits_{0}^{ + \infty } {\left\{ {\frac{{T_{21} }}{E}[T_{22} - T_{23} ] - \frac{{{\text{e}}^{{ - sg(y_{i} - y_{j} )}} }}{2}} \right\}} } \right.\sin [s(x_{i} - x_{j} )]{\text{d}}s + \left. {\frac{{(x_{i} - x_{j} )}}{{2[(x_{i} - x_{j} )^{2} + g^{2} (y_{i} - y_{j} )^{2} ]}}} \right]\cos \theta } \right. \\ \quad - \left. {G_{x} \left[ {\int\limits_{0}^{ + \infty } {\left\{ {\frac{{T_{11} }}{E}[T_{12} - T_{13} ] + \frac{{{\text{e}}^{{ - sg(y_{i} - y_{j} )}} }}{2}} \right\}} } \right.\cos [s(x_{i} - x_{j} )]{\text{d}}s - \left. {\frac{{g(y_{i} - y_{j} )}}{{2[(x_{i} - x_{j} )^{2} + g^{2} (y_{i} - y_{j} )^{2} ]}}} \right]\sin \theta } \right) \\ \end{aligned}$$

$$\begin{aligned} k_{ij} & = \frac{1}{\pi }\left( {g^{2} G_{y} \left[ {\int\limits_{0}^{ + \infty } {\left\{ {\frac{{T_{41} }}{E}[T_{42} - T_{43} ] - \frac{{{\text{e}}^{{sg(y_{i} - y_{j} )}} }}{2}} \right\}} } \right.\sin [s(x_{i} - x_{j} )]{\text{d}}s + \left. {\frac{{(x_{i} - x_{j} )}}{{2[(x_{i} - x_{j} )^{2} + g^{2} (y_{i} - y_{j} )^{2} ]}}} \right]\cos \theta } \right. \\ \quad - \left. {G_{x} \left[ {\int\limits_{0}^{ + \infty } {\left\{ {\frac{{T_{31} }}{E}[T_{32} - T_{33} ] - \frac{{{\text{e}}^{{sg(y_{i} - y_{j} )}} }}{2}} \right\}} } \right.\cos [s(x_{i} - x_{j} )]{\text{d}}s + \left. {\frac{{g( - y_{i} + y_{j} )}}{{2[(x_{i} - x_{j} )^{2} + g^{2} (y_{i} - y_{j} )^{2} ]}}} \right]\sin \theta } \right) \\ \end{aligned}$$

(39)