Analytical perturbation solution for large simple shear problems in elastic-perfect plasticity with the logarithmic stress rate

Abstract

The large simple shear deformations in elastic-perfectly plastic bodies are studied using the self-consistent elastic-perfectly plasticJ2-flow model based on the logarithmic stress rate, recently established by these authors [2]. The application of the logarithmic stress rate in the elastic rate equation of hypoelastic type leads to an exact finite hyperelastic solution. The plastic solution is shown to be governed by a first-order nonlinear ordinary differential equation with a small dimensionless material parameter multiplying the highest derivative, for which the initial condition is related to the elastic-plastic transition and prescribed in terms of the just-mentioned small parameter. A singular perturbation solution is derived for large plastic strain by utilizing the method of matched expansions. The solution obtained is shown to be in a satisfactory manner close to the numerical solution by a Runge-Kutta integration procedure with high accuracy. Remarks are given to explain a phenomenon of instability concerning the shear stress.