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

R. Bagheri

doi:10.1007/s40997-019-00331-x

Iranian Journal of Science and Technology, Transactions of Mechanical Engineering

pp 1–15 | Cite as

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

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R. Bagheri

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First Online: 27 November 2019

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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.

Keywords

Magneto-electro-elastic half-plane Multiple defects Screw dislocation Dynamic stress intensity factors Transient anti-plane loading

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Appendix

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

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

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

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

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

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

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

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

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

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

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1.Department of Mechanical Engineering, Mechatronics Faculty, Karaj BranchIslamic Azad UniversityKarajIran

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

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