Lie Group Symmetry Applied to the Computation of Convex Plasticity Constitutive Equation

Abstract

This paper delivers several new types of representations of the convex plasticity equation and realizes them by numerical discretizations. In terms of the Gaussian unit vector and the Weingarten map techniques in differential geometry, we prove that the plastic equation exhibits a Lie group symmetry. We convert the nonlinear constitutive equations to a quasilinear equations system X = AX, X ∈ Mⁿ⁺¹, A ∈ so(n,1) in local. In this way the inherent symmetry of the constitutive model of convex plasticity is brought out. The underlying structure is found to be a cone in the Minkowski space Mⁿ⁺¹ on which the proper orthochronous Lorentz group SO_o(n,1) left acts. Based on the group properties some numerical methods are developed, which together with a post-projecting method can update the stress points on the yield surface at every time increment.