Nonlinear Model of Deformation of Crystalline Media Allowing for Martensitic Transformations: Plane Deformation

Abstract

This article is devoted to development of mathematical solutions of statics equations of plane nonlinear deformation of crystalline media with a complex lattice allowing for martensitic transformations. Statics equations comprised of a set of four coupled nonlinear equations are reduced to a set of separate equations. The macrodisplacement vector is sought in the Papkovich-Neuber form. The microdisplacement vector is determined by the sine-Gordon equation with a variable coefficient (amplitude) before the sine and Poisson’s equation. For the case of constant amplitude the class of doubly periodic solutions has been determined which are expressed via elliptical Jacobian functions. It has been demonstrated that nonlinear theory leads to a combination of solutions describing fragmentation of the crystalline medium, occurrence of structural imperfections of various types, phase transformations, and other peculiarities of deformation which occur under the action of intensive loads and are not described by classical continuum mechanics.