Symmetry group analysis of an ideal plastic flow

TL;DR

This paper analyzes the symmetry properties of an ideal plastic flow system to derive analytical solutions, classify subalgebras, and explore invariant solutions that can inform practical shaping processes.

Contribution

It provides a comprehensive symmetry analysis and classification of solutions for ideal plastic flow equations, including new invariant and partially invariant solutions.

Findings

Derived Lie symmetry generators for the system

Classified subalgebras up to codimension two

Obtained explicit solutions of algebraic, trigonometric, and elliptic types

Abstract

In this paper, we study the Lie point symmetry group of a system describing an ideal plastic plane flow in two dimensions in order to find analytical solutions. The infinitesimal generators that span the Lie algebra for this system are obtained. We completely classify the subalgebras of up to codimension two in conjugacy classes under the action of the symmetry group. Based on invariant forms, we use Ansatzes to compute symmetry reductions in such a way that the obtained solutions cover simultaneously many invariant and partially invariant solutions. We calculate solutions of the algebraic, trigonometric, inverse trigonometric and elliptic type. Some solutions depending on one or two arbitrary functions of one variable have also been found. In some cases, the shape of a potentially feasible extrusion die corresponding to the solution is deduced. These tools could be used to thin, curve, undulate or shape a ring in an ideal plastic material.