A phase-field model of dislocations on parallel slip planes

Abstract

We expand the Peierls-Nabarro phase-field model of dislocations on one active slip plane introduced by Koslowski, Cuitiño, and Ortiz to the case of multiple parallel slip planes embedded into a homogeneous anisotropic crystal. We deduce the leading-order behavior of the energy as lattice size and slip plane spacing tend to zero. Under a logarithmic rescaling, the limit energy in the sense of Γ-convergence takes the form of a line-tension functional supported on dislocation lines featuring interactions between parallel dislocation lines in different slip planes. An optimal dislocation configuration is shown to contain a two-scale microstructure. <br /> This is an extension of a result by Conti, Garroni, and Müller. We are able to treat anisotropic materials with possibly nonpositive interaction kernels, and obtain the leading order energy for non-dilute dislocations in a special geometry. We use the theory of functions of bounded variation, the fractional Sobolev space H^1/2, and linear elasticity theory. We show some new results using iterated mollification and multiscale analysis, and use a modified ball construction for an extension result in BV.