Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation

This paper is concerned with the formulation of a phenomenological model of finite elasto-plasticity valid for small elastic strains for initially isotropic polycrystalline material. As a basic we assume the multiplicative split $F=F_e F_p$ of the deformation gradient into elastic and plastic part. A key feature of the model is the introduction of an independent field of 'elastic' rotations $R_e$ which eliminate the remaining geometrical nonlinearities coming from finite elasticity in the presence of small elastic strains. In contrast to micro-polar theories an evolution equation for $R_e$ is presented which relates $R_e$ to $F_e$ making use of a new device found by the author to perform the polar decomposition asymptotically. The model is shown to be invariant under both change of frame and rotation of the so called intermediate configuration. The corresponding equilibrium equations at frozen plastic and viscoelastic configuration constitute then a linear, elliptic system with nonconstant coefficients which makes this model amenable to a rigorous mathematical analysis. The introduced hysteresis effects within the elastic region are related to viscous elastic rotations of the grains of the polycrystal due to internal friction at the grain boundaries and constitute as such a rate dependent transient texture effect. The inclusion of work hardening will be addressed in future work.