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

Abstract

Summary.The large simple shear deformations in elastic-perfectly plastic bodies are studied using the self-consistent elastic-perfectly plastic J2-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 hypoetastic 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.

On instability of constitutive models for isotropic elastic-perfectly plastic material behaviour at finite deformations

“…The observed theoretically unexpected instability in elastic-plastic transition in the case of elastic-perfectly plastic material behaviour prediction was first detected by Moss 30 and later approved by Szabo´and Balla 31 and Bruhns et al 32,33 The numerical results obtained by the present research are in accordance with the analytical solutions for the Log-rate 32 and are qualitatively in a very good agreement with analytical and numerical results for the Jaumann and Green-Naghdi rates, respectively, presented in Moss. 30 According to Moss, 30 the reason why the instability phenomenon occurs does not lie in the choice of the objective rate which is used in the constitutive relation (for details see Moss 30 ).…”

“…In Bruhns et al, 33 the authors concluded that this unexpected result is due to the absence of hardening. It can be observed from the consistency condition that in this case, the rate of the shear stress and the rate of the normal stress have to be of an opposite sign.…”

“…The behaviour of two different materials is examined in this work. Concerning the first material, in order to compare the results for the elastic-perfectly plastic case with the corresponding presented in Moss 30 and Bruhns et al, 32,33 the material parameters are taken from the just mentioned references, i.e. the Young's modulus and Poisson's ratio are 210 GPa and 0.3, respectively.…”

“…The scalar functions g(k) and c(k) that are introduced in the constitutive model, for the elastic-perfectly plastic case, regarding the relation (33), are…”

“…These results are marked as ABAQUS on the following graphs. The results of the analytical solution for the finite simple shear problem using the logarithmic rate, following the instructions from Bruhns et al, 33 are also presented on the graphs, labelled as Analytical.…”