Orientational anisotropy and strength-differential effect in orthotropic elasto-plastic materials

This paper is devoted to elasto-plastic orthotropic model within the multiplicative elasto-plasticity, which describes the change of the orthotropic axes, i.e., orientational anisotropy and the strength-differential effect, when the yield condition is pressure insensitive and dependent on the third invariant of the stress. The orthotropy directions are characterized by Euler angles within the constitutive framework with small elastic strains, large elastic rotations and large plastic distortions. The presence of the plastic spin makes possible the description of the orientational anisotropy. We make herein an attempt to develop a hardening model, which includes the kinematic hardening given by law adapted to orthotropic material and isotropic hardening, and complies with the experimentally observed plastic yield and flow behaviour reported by in tension–compression–tension and compression–tension–compression. By pushing away to the actual configuration the material response, the rate form of the model with the objective derivatives expressed via the elastic rotations is characterized by a differential system for the following unknowns: the Cauchy stress, plastic part of deformation, tensorial and scalar hardening variables and Euler angles. We present the rate elasto-plastic model with a plastic spin in the case of an in-plane rotation of the orthotropy direction, and a plane stress, respectively. In the plane stress, the equation for the rate of the strain in the normal direction is first derived and subsequently the modified expression for the plastic multiplier associated with an in-plane rate of the deformation becomes available. Numerical simulations for the homogeneous deformation process on the sheets and comparisons with experimental data make possible a selection among the plastic spins introduced in this paper, aiming at obtaining a good agreement with the experiments performed for an in-plane stress state by . When the shear deformation of the plate is numerically simulated, the stabilization of the orientational anisotropy occurs in the presence of the plastic spin, in contrast with the unreasonable behaviour produced in the absence of the plastic spin.