Symmetry group analysis of an ideal plastic flow

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

Some symmetry group aspects of a perfect plane plasticity system

“…and is similar to (9). Eliminating r from the above system, we obtain one equation for the function u = tan θ p :…”

“…The interest to the system of plane plasticity has been recently renewed. In [9] the known symmetries, admitted by (1), (5) were used to determine some solutions in the form of a propagation wave. In [10] the complete Lie algebra of symmetries for (1), (5) is calculated.…”

Symmetry groups of non-stationary planar ideal plasticity

“…The importance of such a study resides in the immediate applicability of the results, as was done in [9,10] for the stationary case. New solutions of plasticity problems such as that given in system (1) are essential for the development and efficiency of certain industrial procedures such as sheet rolling and extrusion.…”

“…If we introduce the force of the form (10) into equation (9) and into the differential consequences with respect to u and v of equations (4), and imposing the condition that the coefficients c 0 , c 1 , τ 2 (t) and τ 3 (t) all vanish, we find that the components of the force do not depend on u and v. In this case, equation (9) implies that τ 1 (t) is a constant, which we denote c 2 . The force (10) then reduces to a monogenic type, i.e.…”

“…Solutions were found involving simple and double Riemann waves by using the method of characteristics. However, as is often the case with this method, solutions rely on numerical integration in order to obtain velocity components u and v. A study of the system (1) from the group-theoretical point of view has been done for the stationary case [9,10]. Such a study has never been carried out for the non-stationary case and for different types of force involved in the system.…”