| , A.N. Bulygin, Yu.V. Pavlov |
|
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.
Keywords: complex lattice; nonlinear model; plane deformation; complex representation of solution; nonautonomous sine-Gordon equation. |
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