2 2 2 2 2 2 2 2 13 21 13 22 11 22 11 22 2 2 2 2 2 2 0, u u u v v v f f f f g g x y x y x x y y x y y x y y (2d) where u and v are, respectively, the x and y components of the mechanical displacement vector, and are, respectively, the electric and magnetic potentials. The above magneto electro elasticity equations are subjected to the following boundary conditions: 13 13 ,0 , ext x x 33 33 ,0 , ext x x , x a (3a,b) 3 3 ,0 1 , e ext D x k D x 3 3 ,0 1 , m ext B x k B x , x a (3c,d) 13 13 ,0 ,0 , x x 33 33 ,0 ,0 , x x ,x (4a,b) 3 3 ,0 ,0 , D x D x 3 3 ,0 ,0 , B x B x ,x (4c,d) ,0 ,0 , u x u x ,0 ,0 , v x v x , x a (5a,b) ,0 ,0 , x x ,0 ,0 , x x , x a (5c,d) 13 1 , 0, x h 33 1 , 0, x h ,x (6a,b) 3 1 , 0, D x h 3 1 , 0, B x h ,x (6c,d) 13 2 , 0, x h 33 2 , 0, x h ,x (7a,b) 3 2 , 0, D x h 3 2 , 0, B x h ,x (7c,d) Eqs. (3a-d) describe the applied magneto electro mechanical loadings on the crack faces. Eqs. (4a-d) represent the continuity of stresses, electric displacement and magnetic fields along the crack plane. Eqs. (5a-d) describe the continuity of the mechanical displacement and the magnetic and electric potentials outside the crack. Eqs. (6a-d) and (7a-d) represent the free layer’ surfaces boundary conditions. Singular integral equations and their solutions The magneto electro elasticity equations (2a-d) are solved using Fourier transform to yield the mechanical displacement and electric and magnetic potentials in the composite medium. The density functions which represent the discontinuity of the mechanical displacement, electric and magnetic fields across the crack are now introduced 1 , u u x x 2 , v v x x (8a,b) 3 , x x 4 . x x (8c,d) Applying the boundary conditions and after a lengthy analysis, we obtain four coupled singular integral equations in which the unknowns are the density functions 1 2 3 , , and 4. After extracting the Cauchy and logarithmic singularities from the kernels, the four equations take the following form: 0 1 11 12 11 1 13 12 13 1 14 14 13 1 1 , ln 1 1 , ln , 1 ln , , a a a u u v a a a a a a v a a a a a ext a a k k t dt k t x t dt t x t dt t x k k t x t dt t x t dt k t x t dt k t x t dt k t x t dt x , x a (9a)
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