Consistent Eulerian and Lagrangian variational formulations of non-linear kinematic hardening for solid media undergoing large strains and shocks

In this paper, two Eulerian and Lagrangian variational formulations of non-linear kinematic hardening are derived in the context of finite thermoplasticity. These are based on the thermo-mechanical variational framework introduced by Heuzé and Stainier (2022), and follow the concept of pseudo-stresses introduced by Mosler and Bruhns (2009). These formulations are derived from a thermodynamical framework and are based on the multiplicative split of the deformation gradient in the context of hyperelasticity. Both Lagrangian and Eulerian formulations are derived in a consistent manner via some transport associated with the mapping, and use quantities consistent with those updated by the set of conservation or balance laws written in these two cases. These Eulerian and Lagrangian formulations aims at investigating the importance of non-linear kinematic hardening for bodies submitted to cyclic impacts in dynamics, where Bauschinger and/or ratchetting effects are expected to occur. Continuous variational formulations of the local constitutive problems as well as discrete variational constitutive updates are derived in the Eulerian and Lagrangian settings. The discrete updates are coupled with the second order accurate flux difference splitting finite volume method, which permits to solve the sets of conservation laws. A set of test cases allow to show on the one hand the good behavior of variational constitutive updates, and on the other hand the good consistency of Lagrangian and Eulerian numerical simulations.