Use of conservation laws to solve the problem of load wave in an elastoplastic rod

The process of propagation of plastic deformations in a semi-infinite elastic plastic rod caused by dynamic loading applied to the end of the rod, which is not decreasing in time, is considered. The equations are written in the Lagrangian coordinate system. It is assumed that during deformation there is no lateral bulging of the rod and that the influence of transverse deformations of the rod on the process of propagation of longitudinal waves is negligible. At the initial moment, the rod is in a deformed and dormant state. Small deformations of the rod are considered. The density of the rod during deformation does not change. The only non-zero component of the stress tensor will be the component along the ox axis, non-zero components of the strain tensor will be the components along the Ox, Oy axes. As a result, a system of two quasilinear first-order homogeneous equations is constructed. The equations are hyperbolic. They are built for performance and ratio on them. Next, the equations are written in terms of Riemann invariants. For the equations constructed, the conservation laws are found in the case when the current remaining depends only on the functions sought. As a result, a system of linear equations with coefficients depending only on the required functions is obtained. Тhe construction of conservation laws is reduced to the solution of the boundary value problem for the known Euler-Poisson-Darboux equations. This problem is solved with the help of Riemann functions. The conservation laws allowed us to find the coordinates of the points of intersection of characteristics, and thus to solve the problem. In conclusion, the article considers the case when one of the characteristics crosses the line on which the initial conditions are given. In this case, as is known, the Cauchy problem cannot be solved. This leads to a procedure which, with the help of conservation laws, makes it possible to find out the solvability of the Cauchy problem. It is reduced to the solution of a simple integral equation by the method of successive approximations.