Group analysis of an ideal plasticity model

TL;DR

This paper analyzes the symmetry properties of an ideal plasticity model for plane flow, deriving new solutions and classifying symmetries to aid in shaping plastic materials.

Contribution

It introduces six original infinitesimal generators and classifies subalgebras, enabling new analytical solutions for ideal plastic flow.

Findings

Derived algebraic, trigonometric, and elliptic solutions.

Identified solutions depending on arbitrary functions.

Suggested applications in shaping plastic materials.

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 of the system. The infinitesimal generators that span the Lie algebra for this system are obtained, six of which are original to this paper. We completely classify the subalgebras of codimension one and two into conjugacy classes under the action of the symmetry group. We apply the symmetry reduction method in order to obtain invariant and partially invariant solutions. These solutions are of 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 curve and undulate rectangular rods or slabs, or to shape a ring in an ideal plastic material.