ABSTRACT
ABSTRACT: We present in this paper a new assumed strain finite element formulation (or B-bar method) for the locking-free simulation of nearly incompressible elastic and inelastic solids in the finite deformation dynamic range that also preserves the conservation/dissipation properties of the so-called energy-dissipative momentum-conserving (EDMC) time-stepping algorithms. The general setting of finite strain plasticity is considered, including hyperelastic models as a particular case. The main motivation of this work is to avoid the nonlinear numerical instabilities observed in classical numerical schemes with unbounded growth of the energy (even in the plastic case) by introducing the exact dissipation/conservation of the energy in the discrete system by design. The incorporation of the conservation laws of linear and angular momenta, and the preservation of the associated relative equilibria, is also obtained. The paper identifies the conditions that the linearized strain operator (or, simply, the B-bar operator as it is usually known) has to satisfy for the preservation of these properties in time. These conditions require the definition of the assumed strain operator, originally developed by with spatial considerations only, accounting for the temporal discretization in the definition of the associated strain variations. As a result, we arrive to a fully discrete system in space and time that shows exactly all these conservation/dissipation laws of the underlying physical system, including the exact plastic dissipation of the energy, with exact energy conservation for elastic steps. Numerical simulations are presented to illustrate the performance of the new formulation.
