System analysis of dynamic problems of anisotropic plasticity theory

Dynamic problems are the least studied area of plasticity theory. These problems arise in various fields of engineering and science, but the complexity of the original differential equations do not allow to develop accurate solutions and correctly solve numerical boundary value problems. This is even more typical of dynamic equations of anisotropic plasticity. Anisotropy reduces the group of symmetries allowed by the equations, and therefore narrows the number of invariant solutions. One-dimensional dynamic plasticity problems are well studied, but two-dimensional problems cause insurmountable mathematical difficulties due to the nonlinearity of the basic equations, even in the isotropic case. The study of the symmetries of the plasticity equations allowed us to find some exact solutions. The most known solution was found by B. D. Annin, who described the unsteady compression of a plastic layer made of isotropic material by rigid plates. Annin's solution is linear in two spatial variables, however, it includes arbitrary functions of time. Symmetries are also used in the proposed work. Point symmetries are first calculated for dynamic plasticity equations in the anisotropic case and are presented in the paper. The Lie algebra generated by the found symmetries appeared to be infinite-dimensional. This circumstance made it possible to apply the method of constructing new classes of nonstationary solutions. Symmetry can transform the exact solution of stationary dynamic equations in non-stationary solutions. The framed solutions include arbitrary functions and arbitrary constants. The outline of the article is as follows: according to the method of Lie group of point symmetries allowed by the equations of anisotropic plasticity is calculated. Two classes of new stationary invariant solutions are framed. These stationary solutions, by means of transformations generated by point symmetries, are transformed into new non-stationary solutions. In conclusion, a new self-similar solution of unsteady equations of anisotropic plasticity is framed; Annin's solution is generalized for the anisotropic case. The framed solutions can be used to describe the compression of plastic material between rigid plates, as well as to test programs, designed to solve anisotropic plastic problems.