Analysis of a line defect in a multilayered smart structure by the image method

In this paper, the continuous image dislocation method is used to develop approximate, explicit closed-form solutions for a line defect (dislocation) in a multilayered smart structure consisting of dissimilar layers described by different constitutive equations. Three techniques, instrumental to the success of the method, are used: (1) proper decomposition of the original problem into subproblems with strategic allocation of the distributed images, (2) exact evaluations of certain integral transforms via the residue theory, and (3) elimination of all singular integrals from the governing integral equations. The solutions can be used as Green's functions for problems of dislocation pileups, cracks, dislocation walls, and finite boundaries. To demonstrate the method, the image force on a generalized dislocation, as characterized by the three components of the Burgers vector and a component representing discontinuity in the electric potential, in a gold-PZT-platinum-silicon multilayer is calculated. The results show that the image force depends significantly on the components of the generalized Burgers vector, the position of the dislocation and the thicknesses of the layers.