In the present article, a covariant spacetime formalism is used to model the behavior of viscoelastic and hyperelastic solids, within a thermodynamical framework. The latter aims to ensure the validity of thermodynamics second principle and to derive reversible or irreversible models for thermomechanics. The use of the Lie derivative is of particular interest to achieve such goals. Covariance enables to address finite deformation. Coupled to a covariant finite element analysis, it allows numerical simulations that simultaneously ensure the physical balance of energy and momentum for thermomechanical applications. Different mechanical loadings are considered, bending or uniaxial extension ones, with quasi-static or time exponential or time cyclic evolution. We also provide quantification of the different performances of the numerical simulations, and show the advantages and drawbacks of the spacetime approach.

中文翻译：

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关于使用多维微分几何来模拟粘弹性或超弹性结构的协变行为，并通过时空有限元分析的数值模拟进行说明

在本文中，协变时空形式主义用于在热力学框架内模拟粘弹性和超弹性固体的行为。后者旨在确保热力学第二原理的有效性并导出热力学的可逆或不可逆模型。使用李导数对于实现这些目标特别有意义。协方差能够解决有限变形问题。与协变有限元分析相结合，它允许进行数值模拟，同时确保热机械应用中能量和动量的物理平衡。考虑不同的机械载荷，弯曲或单轴延伸载荷，具有准静态或时间指数或时间循环演化。我们还提供了数值模拟不同性能的量化，并展示了时空方法的优点和缺点。