Abstract

For an elastic-plastic isotropic homogeneous body, two complete closed-form solutions are presented: simple shear and diamond-shaped pure shear. In the literature of the last century, simple shear was sometimes also referred to as rectilinear shear. And diamond-shaped pure shear has a double-symmetrical shear kinematics in which only the materially co-rotated pure Cauchy shear stress is active. Here, these two solutions are derived from a multiplicative logarithmic strain space formulation, which in turn is based on commutative-symmetrical stretch-tensor products. An elasto-plasticity formulation with commutative-symmetrical stretch-tensor products does not require a (questionable) intermediate (zero-stress) configuration, since it is formulated exclusively with proper-Eulerian or proper-Lagrangean tensors (in particular of logarithmic strain) for the total as well as for the elastic and for the plastic deformation. When the Lagrangean principal axes rotate under finite shear deformations, as is the case with simple shear, the proper-Lagrangean tensors of logarithmic Hoger stress and back-rotated Kirchhoff stress must be distinguished. This is accomplished here by transforming two symmetric second-order tensors through a double contraction with a fourth-order tensor. Because pure shear exhibits statically double-symmetrical shear stresses, but simple shear is kinematically not double-symmetrical, a solution of diamond-shaped pure shear is also discussed, which is kinematically double-symmetrical as well. The presented considerations are generally valid for isotropic logarithmic-hyperelastic stress-strain relations and for isochoric plasticity with isotropic hardening. Due to the presented shear kinematics, the plastic deformation can be specified directly and must not be integrated by means of a plastic flow rule. However, it is shown how the time derivatives of the plastic deformation and the corresponding work-conjugate stresses can be calculated in a multiplicative logarithmic strain space formulation. In the case of diamond-shaped pure shear, the plastic flow rule corresponds to that of Prandtl–Reuß.

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