The Analytical Solutions For Doubly Periodic Line Inclutions In A Magnetoelectroelastic Solid Under Antiplane Deformation Mode

From the viewpoint of inhomogeneity in solids, line inclusion is two extremes in the flat two-dimensional inclusions. line inclusion tends to crack and rigid line respectively when the youngs modulus E→0and E→∞.Line inclusions, which are inevitable defects in magnetoelectroelastic materials, have a significant impact on local magnetoelectroelastic field and macroscopic effective properities, even lead to the smart structures invalidation. Therefore, the study of the line inclusions has a very important significance. This paper dedicate to the research on the magnetoelectroelastic materials with doubly periodic unequal line inclusions under far-field antiplane mechanical load coupled with inplane electro-magnetic load, and providing valuable reference to the engineering applications and further developed of magnetoelectroelastic materials.By using permeable boundary condition (for rigid lines) and impermeable boundary condition (for cracks) respectively and the analysis of symmetry and anti symmetry, the stress, electric potential and magnetic potential function on the boundary of 1/2 cell of crack and rigid line are obtained. Then, by using elliptical function theory and conformal mapping technique of Jacobi sn function, transfers the 1/2 cell (a rectangle region on physical plane z) into the upper-half plane (on mapping planeζ). Finally, by using Keldych-Sedov formula, far-field equilibrium condition and homogeneous field theorem, the closed form solutions of the stress field, stress intensity factor and the magnetoelectroelastic effective modulus are calculated. Programming by symbolic computation software Mathematica, the numerical solution of the stress intensity factor and the magnetoelectroelastic effective modulus are obtained. The examples indicate that the length ratio of the unequal line inclusion and the micro-structural parameters of permutation have a impact on the stress intensity factor and the magnetoelectroelastic effective modulus. Numerical analysis show that the length ratio of the unequal line inclusion and the micro-structural parameters of permutation have a impact on properties of magnetoelectroelastic material.