Theory of dislocation loops in multilayered anisotropic solids with magneto-electro-elastic couplings

Abstract

This article presents a general theory on dislocation loops in multilayered anisotropic solids with magneto-electro-elastic couplings. First, a novel extended point-force Green's function is developed that not only guarantees the numerical stability but also admits a direct interpretation of the image method in multilayered solids. Then, based on the new Green's function, the classical theory of dislocation loops in pure-elastic infinite solids is systematically generalized by accounting for both the surface/interface conditions and the magneto-electro-elastic couplings. Two major contributions are made in the generalization: (i) the extended displacement and stress fields induced by an arbitrary dislocation loop are derived and expressed concisely as a simple line integral along the closed dislocation curve; (ii) the interaction energy between two arbitrary dislocation loops is also derived and expressed compactly as double line integrals along the two closed dislocation curves. Based on the new line-integral expressions for multilayered solids, special configurations such as straight dislocation segments and elliptic dislocation loops are analytically studied in detail; and more strikingly, elegant line-integral expressions for dislocation loops in bi-materials or in a semi-infinite solid are obtained by simple reduction. All the expressions derived in this article are compared with available solutions in existing literature and the correctness of them is fully verified. As novel applications, the present theory is utilized to calculate the self-energy of interfacial elliptic dislocation loops efficiently, which clearly shows that both the surface/interface conditions and the multi-field couplings have a considerable influence on the self-energy of interfacial dislocation loops. The line-integral based formulae developed here for dislocation-loop problems will not only enrich the theory of magneto-electro-elasticity, but also have potential applications in three-dimensional discrete dislocation dynamics simulations associated with multilayered thin-film structures.