A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

On the evolution of isotropic and kinematic hardening with finite plastic deformation Part I: compression/tension loading of OFHC copper cylinders

Compression tests followed by tension tests after re-machining were performed on annealed oxygen-free-high-conductivity copper cylinders. These tests were conducted at nine levels of maximum strain ranging from 5 to 50%. From this data, isotropic and kinematic hardening were calculated using 50, 1000 and 2000 microstrain offset definitions. Both isotropic and kinematic hardening were found to depend on the yield definition. Isotropic hardening, which increased with plastic strain with no signs of saturation, also increased with larger offset definition of yield. Kinematic hardening, which increased to 40% strain and appeared to saturate thereafter, decreased with higher offset definitions of yield.     

Effect of the kinematic hardening on the plastic anisotropy parameters for metallic sheets

The initial plastic anisotropy parameters are conventionally determined from the Lankford strain ratios defined by $r^{\psi }=\frac{{\epsilon }_{22}^{\mathrm{p}\psi }}{{\epsilon }_{33}^{\mathrm{p}\psi }}$ (ψ being the direction of the loading path). They are usually considered as constant parameters that are determined at a given value of the plastic strain far from the early stage of the plastic flow (i.e. equivalent plastic strain of ${\epsilon }_{\mathrm{eq}}^{\mathrm{p}}=0.2\text{\%}$) and typically at an equivalent plastic strain in between 20% and 50% of plastic strain failure (or material ductility). What prompts to question about the relevance of this determination, considering that this ratio does not remain constant, but changes with plastic strain. Accordingly, when the nonlinear evolution of the kinematic hardening is accounted for, the Lankford strain ratios are expected to evolve significantly during the plastic flow.In this work, a parametric study is performed to investigate the effect of the nonlinear kinematic hardening evolution of the Lankford strain ratios for different values of the kinematic hardening parameters. For the sake of clarity, this nonlinear kinematic hardening is formulated together with nonlinear isotropic hardening in the framework of anisotropic Hill-type (1948) yield criterion. Extension to other quadratic or non-quadratic yield criteria can be made without any difficulty. This parametric study is completed by studying the effect of these parameters on simulations of sheet metal forming by large plastic strains.     

Lorentz group SOo(5, 1) for perfect elastoplasticity with large deformation and a consistency numerical scheme

How to effectively deal with non-linearity and accurately fulfill the consistency condition is essential for modeling and computing in plasticity. Utilizing the concepts of two phases and homogeneous coordinates, we obtain a linear representation of a constitutive model of perfect elastoplasticity with large deformation, and, furthermore, a linear irreducible representation, which contains a five-order spin tensor. The underlying vector space is found to be the projective realization of a composite space resulting from a surgery on Minkowski spacetime ⁵⁺¹. The irreducible representation in the vector space admits of the projective proper orthochronous Lorentz group PSO_o(5, 1) in the on (or elastoplastic) phase and the special Euclidean group SE(5) in the off (or elastic) phase. The input path dictates symmetry switching between the two groups. Based on such symmetry a numerical scheme is devised which preserves the consistency condition for every time step. The consistency scheme is shown to be stabler, more accurate, and more efficient than the current numerical schemes developed directly based upon the model itself, because the new scheme preserves the internal symmetry SO_o(5, 1) of the model in the on phase so as to locate the stress point automatically on the yield surface at each time step without iterations at all.