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

The large simple shear deformations in elastic-perfectly plastic bodies are studied using the self-consistent elastic-perfectly plastic J(2)-flow mode l based on the logarithmic stress rate, recently established by these autho rs [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 o rdinary differential equation with a small dimensionless material parameter multiplying the highest derivative, for which the initial condition is rel ated 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 sol ution obtained is shown to be in a satisfactory manner close to the numeric al solution by a Runge-Kutta integration procedure with high accuracy. Rema rks are given to explain a phenomenon of instability concerning the shear s tress.