The solutions of nonlinear equations of plane deformation of the crystal media allowing martensitic transformations: complex representation for macrofield equations


Mathematical methods of the solution of the equations of statics of plane nonlinear deformation of the crystal media with a complex lattice allowing martensitic transformations are developed. The equations of a statics represent system of four connected nonlinear equations. The vector of macroshifts is looked in the Papkovish-Neuber form. The system of the connected nonlinear equations is reduced to system of the separate equations. The vector of microshifts can be found from the sine-Gordon equation with variable coefficient (amplitude) before the sine and Poisson equation. The class of doubly periodic solutions expressing in the Jacobi elliptic functions is found for a case of constant amplitude. It is shown that the nonlinear theory possesses a set of solutions which describe fragmentation of the crystal medium, emergence of defects of structure of different types, phase transformations and other topological features of the deformation which are implemented under the influence of intensive power loadings and which can't be described by classical mechanics of the continuous medium. Features of the found solutions are discussed.