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) 50 37). This paper substantially uses the logarithmic rate (Acta Mechanica 124 (1997a) 89), which is equally based on strong mathematical and physical pr inciples and therefore may contrast to classical approaches of cited kinds. Stress responses to large simple shear and torsional deformations in elasto plastic bodies are studied by applying the seff-consistent kinematic harden ing J(2)-flow model based on the logarithmic tensor rate, recently establis hed by these authors (Int. J. Plasticity 15 (1999) 479). The application of the logarithmic stress rate in the elastic rate equation of hypoelastic ty pe results in an exact finite hyperelastic solution in terms of Hencky's lo garithmic strain. The plastic solution is composed of two parts: the back s tress and the effective stress (the Kirchhoff stress reduced by the back st ress). It is shown that the evolution equation of the back stress with the logarithmic rate is integrable to deliver a closed-form relation between th e back stress and Hencky's logarithmic strain and the current stress. Moreo ver, the effective stress is shown to be governed by a first-order nonlinea r ordinary differential equation with a small dimensionless material parame ter multiplying the highest derivative, for which the initial condition is related to the elastic-plastic transition and prescribed in terms of the ju st-mentioned small parameter. A singular perturbation solution for the just -mentioned equation is derived by utilizing the method of matched expansion s. 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 l arge deformations at issue, it is demonstrated that, merely with three comm only known classical material constants, i.e., the elastic shear modulus, t he initial tensile yield stress and the hardening modulus, the simple kinem atic hardening J(2)-flow model with the logarithmic rate may supply satisfa ctory explanations for salient features of complex behaviour in experimenta l observation. (C) 2001 Elsevier Science Ltd. All rights reserved.