Large simple shear and torsion problems in kinematic hardening elasto-plasticity with logarithmic rate

Large simple shear and torsion problems in plasticity have been the object of a large number of papers. Sophisticated schemes have been developed (e.g. J. Appl. Mech. 50 (1983) 561) that overcome problems encountered (cf. e.g. J. Mech. Phys. Solids 48 (2000) 2231; Int. J. Solids Struct. 37 (2000) 5037). This paper substantially uses the logarithmic rate (Acta Mechanica 124 (1997a) 89), which is equally based on strong mathematical and physical principles and therefore may contrast to classical approaches of cited kinds. Stress responses to large simple shear and torsional deformations in elastoplastic bodies are studied by applying the self-consistent kinematic hardening J 2-flow model based on the logarithmic tensor rate, recently established by these authors (Int. J. Plasticity 15 (1999) 479). The application of the logarithmic stress rate in the elastic rate equation of hypoelastic type results in an exact finite hyperelastic solution in terms of Hencky's logarithmic strain. The plastic solution is composed of two parts: the back stress and the effective stress (the Kirchhoff stress reduced by the back stress). It is shown that the evolution equation of the back stress with the logarithmic rate is integrable to deliver a closed-form relation between the back stress and Hencky's logarithmic strain and the current stress. Moreover, the effective stress 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 for the just-mentioned equation is derived by utilizing the method of matched expansions. With the analytical solution derived, it is possible to make a detailed study of the coupling effect of material properties, including the elastic, yielding and hardening properties, on elastic–plastic responses. For the large deformations at issue, it is demonstrated that, merely with three commonly known classical material constants, i.e., the elastic shear modulus, the initial tensile yield stress and the hardening modulus, the simple kinematic hardening J 2-flow model with the logarithmic rate may supply satisfactory explanations for salient features of complex behaviour in experimental observation.