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Large simple shear and torsion problems in kinematic hardening elasto-plasticity with logarithmic rate. (English) Zbl 1046.74011

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, established by these authors. The application of the logarithmic stress rate equation of hypoelasic 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. 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 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 elasto-plastic transition and prescribed in terms of the above-mentioned small parameter. A singular perturbation solution for the corresponding equation is derived by utilizing the method of matched expansions. For 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 logarithmic rate may supply satisfactory explanations for salient features of complex behavior in experimental observation.

MSC:

74C20 Large-strain, rate-dependent theories of plasticity
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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