A discussion of stress rates in finite deformation problems.

*(English)*Zbl 0546.73031Summary: It has recently been shown that the finite elastic-plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed, while the solution for isotropic hardening gives a monotonically increasing stress. This paper analyzes this response on the basis of continuum mechanical descriptions of the problem. Three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress. In light of the equations for the evolution of the yield surface, the hypoelastic solution to the simple shear problem for each of the three stress rates is presented. It is shown that the use of the Jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution. An alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress.

##### MSC:

74B20 | Nonlinear elasticity |

74C99 | Plastic materials, materials of stress-rate and internal-variable type |