A first-order energy-dissipative momentum-conserving scheme for elasto-plasticity using the variational updates formulation

In a previous paper [L. Noels, L. Stainier, J.-P. Ponthot, An energy momentum conserving algorithm using the variational formulation of visco-plastic updates, Int. J. Numer. Methods Engrg. 65 (2006) 904–942] the authors demonstrated the efficiency of the variational formulation of elasto-plastic updates to develop energy–momentum conserving time integration algorithms. Indeed, within such a framework, the stress tensor always derives from an incremental potential, even when plastic behavior is considered. Therefore the verification of the conservation of energy in the non-linear range can easily be demonstrated: the sum of the reversible stored energy and irreversible dissipated energy exactly corresponds to the work of the external forces applied to the structure. Although this formulation was shown to be accurate and robust, the introduction of numerical dissipation for high-frequency numerical modes can be necessary to simulate complex phenomena. In this work, we propose a modification of the variational updates framework to introduce this numerical property, leading to a new energy-dissipative momentum-conserving time-integration algorithm for elasto-plasticity.