Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity

TL;DR

This paper explores how conservation laws and hodograph transformations can be used to solve boundary value problems in plane plasticity, providing methods to construct solutions and characteristic lines for hyperbolic PDE systems.

Contribution

It establishes a link between conservation laws and hodograph transformations for solving quasilinear PDEs in plasticity, including methods to find solutions and characteristic lines.

Findings

Conservation laws facilitate solving the Cauchy problem for hyperbolic PDEs.

Hodograph transformation linearizes the system, aiding in solution construction.

Examples from gas dynamics and plasticity illustrate the methods.

Abstract

For the hyperbolic system of quasilinear first-order partial differential equations, linearizable by hodograph transformation, the conservation laws are used to solve the Cauchy problem. The equivalence of the initial problem for quasilinear system and the problem for conservation laws system permits to construct the characteristic lines in domains, where Jacobian of hodograph transformations is equal to zero. Moreover, the conservation laws give all solutions of the linearized system. Some examples from the gas dynamics and theory of plasticity are considered.