Analytical Solution of Cracked Functionally Graded Magneto-Electro-Elastic Half-Plane Under Impact Loading

Abstract

The distributed dislocation technique is developed for the transient analysis of functionally graded magneto-electro-elastic half-plane where cracks are parallel/perpendicular with respect to the half-plane boundary. Laplace and Fourier transforms are employed to solve the governing equations leading to a system of Cauchy singular integral equations on the Laplace transform domain. The dynamic stress intensity factor history can be calculated by numerical Laplace transform inversion of the solution of the integral equations. Numerical results are provided to show the effect of the length and position of the cracks, geometry of interacting between the cracks, the magnitude and direction of magnetoelectrical loads and functionally graded constant on the resulting DSIFs. Also, the obtained solutions can be used as a Green’s function to solve dynamic problems involving multiple parallel finite cracks.

Appendix

The unknown coefficients in (12) may be determined as:

$$A_{1} (\omega ,s) = - \frac{{b_{z} (s)[\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{1} }}(\lambda - \gamma_{1} ){\text{e}}^{{ - 2\gamma_{1} h}}$$

$$A_{2} (\omega ,s) = \frac{{b_{z} (s)[\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{1} }}(\lambda + \gamma_{1} )$$

$$A_{3} (\omega ,s) = - \frac{{b_{z} (s)[\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{1} }}(\lambda - \gamma_{1} )({\text{e}}^{{ - 2\gamma_{1} h}} - 1)$$

$$B_{1} (\omega ,s) = - \frac{{[b_{\phi } (s) - \alpha_{2} b_{z} (s)][\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{2} }}(\lambda - \gamma_{2} ){\text{e}}^{{ - 2\gamma_{2} h}}$$

$$B_{2} (\omega ,s) = \frac{{[b_{\phi } (s) - \alpha_{2} b_{z} (s)][\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{2} }}(\lambda + \gamma_{2} )$$

$$B_{3} (\omega ,s) = - \frac{{[b_{\phi } (s) - \alpha_{2} b_{z} (s)][\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{2} }}(\lambda - \gamma_{2} )({\text{e}}^{{ - 2\gamma_{2} h}} - 1)$$

$$C_{1} (\omega ,s) = - \frac{{[b_{\psi } (s) - \alpha_{3} b_{z} (s)][\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{2} }}(\lambda - \gamma_{2} ){\text{e}}^{{ - 2\gamma_{2} h}}$$

$$C_{2} (\omega ,s) = \frac{{[b_{\psi } (s) - \alpha_{3} b_{z} (s)][\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{2} }}(\lambda + \gamma_{2} )$$

$$C_{3} (\omega ,s) = - \frac{{[b_{\psi } (s) - \alpha_{3} b_{z} (s)][\pi \delta (\omega ) - {i \mathord{\left/ {\vphantom {i \omega }} \right. \kern-0pt} \omega }]}}{{2\gamma_{2} }}(\lambda - \gamma_{2} )({\text{e}}^{{ - 2\gamma_{2} h}} - 1)$$