Some basic issues in traditional Eulerian formulations of finite elastoplasticity

In traditional Eulerian formulations of finite elastoplasticity, there are some basic issues that still need clarification and further investigation. Among them are the characterization of initial anisotropy for scalar- and tensor-valued constitutive functions, plastic consistency conditions involving time differentiation of anisotropic yield functions, a unified loading criterion for hardening, softening, and perfectly plastic behaviour, etc. Sometimes, it is thought that a satisfactory and complete treatment for these issues could not be achieved within a framework of traditional Eulerian formulation. In this work, efforts are made towards explaining and clarifying these basic issues. Introducing the notion of an Eulerian type rotation-conjugate group of the initial material symmetry group, we show that consistent Eulerian formulations of scalar- and tensorvalued constitutive functions may be achieved for any given type of initial anisotropy. With such consistent Eulerian formulations, we derive plastic consistency conditions in a corotating frame associated with the foregoing rotation-conjugate group. As to the loading conditions for hardening, softening, and perfectly plastic behaviour, we recall and study the unified criterion proposed by Hill [J Mech Phys Solids 6 (1958) 236]. It is pointed out that the tangential elastic stiffness tensor in this unified criterion has to fulfill rather complicated integrability conditions for the elastic rate equation with an objective stress rate. It appears to be far from being a simple or even trivial matter to give an explicit form of the tangential elastic stiffness tensor meeting these conditions. It is indicated that this issue may be resolved by using the logarithmic stress rate. With this rate, an explicit form of Hill's unified loading criterion, together with an explicit form of tangential elastic stiffness tensor, is presented in terms of a general complementary hyperelastic potential.