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.

中文翻译：

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对于承受大应变和冲击的固体介质的非线性运动硬化的一致欧拉和拉格朗日变分公式


在本文中，在有限热塑性的背景下推导出了非线性运动硬化的两种欧拉和拉格朗日变分公式。这些基于 Heuzé 和 Stainier （2022） 引入的热机械变分框架，并遵循 Mosler 和 Bruhns （2009） 引入的伪应力概念。这些公式源自热力学框架，并基于超弹性背景下变形梯度的乘法分裂。拉格朗日公式和欧拉公式都是通过与映射相关的某种传输以一致的方式得出的，并且使用的数量与在这两种情况下编写的守恒定律或平衡定律集更新的数量一致。这些欧拉和拉格朗日公式旨在研究非线性运动硬化对动力学中受到循环冲击的物体的重要性，预计其中会出现 Bauschinger 和/或棘轮效应。局部本构问题的连续变分公式以及离散变分本构更新是在 Eulerian 和 Lagrangian 设置中推导出的。离散更新与二阶精确磁通量差分分裂有限体积方法相结合，该方法允许求解守恒定律集。一组测试用例一方面可以显示变分本构更新的良好行为，另一方面可以显示拉格朗日和欧拉数值模拟的良好一致性。