Volume-preserving energy–momentum schemes for isochoric multiplicative plasticity

Abstract

This paper presents a new energy-dissipative momentum-conserving algorithm for multiplicative finite strain plasticity ( F = F e F p) that also preserves exactly the plastic volume for isochoric plastic models. The new algorithm exhibits exactly the conservation laws of linear and angular momentum of the underlying physical problem as well as its energy evolution. A strictly positive energy dissipation, in fact the exact energy dissipation, is obtained by design during plastic steps while enforcing the plastic consistency (i.e. the yield condition) on the final stress appearing in the equations of motion. Exact energy conservation is attained, in particular, during elastic steps. The aforementioned preservation of the plastic volume is obtained by a new treatment of the geometric structure behind the considered multiplicative models of finite strain plasticity. Namely, we present a new approximation of the reference and elastic metrics whose contractions with the incremental total and elastic strains lead exactly to the increment of the total and elastic natural volumetric strains, respectively. The elastic metric is defined in the intermediate configuration, defined itself by the proper discrete approximation in time of the plastic deformation gradient F p. The new algorithm extends to the plastic range existing energy–momentum-conserving schemes for nonlinear elastic problems, but incorporating a new modified elastic stress formula consistent with this new geometric setting. The inherited conservation laws of momenta and, especially, the non-negative character of the energy dissipation leads to an improved performance over existing, more classical schemes showing numerical instabilities in the considered highly nonlinear geometric setting of large deformations and strains. Several numerical simulations are presented illustrating these properties.