Abstract In Parts I (Bonet et al., 2015) and II (Gil et al., 2016) of this series, a novel computational framework was presented for the numerical analysis of large strain fast solid dynamics in compressible and nearly/truly incompressible isothermal hyperelasticity. The methodology exploited the use of a system of first order Total Lagrangian conservation laws formulated in terms of the linear momentum and a triplet of deformation measures comprised of the deformation gradient tensor, its co-factor and its Jacobian. Moreover, the consideration of polyconvex constitutive laws was exploited in order to guarantee the hyperbolicity of the system and show the existence of a convex entropy function (sum of kinetic and strain energy per unit undeformed volume) necessary for symmetrisation. In this new paper, the framework is extended to the more general case of thermo-elasticity by incorporating the first law of thermodynamics as an additional conservation law, written in terms of either the entropy (suitable for smooth solutions) or the total energy density (suitable for discontinuous solutions) of the system. The paper is further enhanced with the following key novelties. First, sufficient conditions are put forward in terms of the internal energy density and the entropy measured at reference temperature in order to ensure ab-initio the polyconvexity of the internal energy density in terms of the extended set comprised of the triplet of deformation measures and the entropy. Second, the study of the eigenvalue structure of the system is performed as proof of hyperbolicity and with the purpose of obtaining correct time step bounds for explicit time integrators. Application to two well-established thermo-elastic models is presented: Mie–Gruneisen and modified entropic elasticity. Third, the use of polyconvex internal energy constitutive laws enables the definition of a generalised convex entropy function, namely the ballistic energy, and associated entropy fluxes, allowing the symmetrisation of the system of conservation laws in terms of entropy-conjugate fields. Fourth, and in line with the previous papers of the series, an explicit stabilised Petrov–Galerkin framework is presented for the numerical solution of the thermo-elastic system of conservation laws when considering the entropy as an unknown of the system. Finally, a series of numerical examples is presented in order to assess the applicability and robustness of the proposed formulation.

中文翻译：

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大应变计算固体动力学的一阶双曲线框架。第三部分：热弹性

摘要 在本系列的第一部分 (Bonet et al., 2015) 和第二部分 (Gil et al., 2016) 中，提出了一种新的计算框架，用于可压缩和几乎/真正不可压缩等温的大应变快速固体动力学的数值分析。超弹性。该方法利用一阶总拉格朗日守恒定律系统的使用，该系统是根据线性动量和由变形梯度张量、其辅因子和雅可比矩阵组成的变形量度三元组制定的。此外，为了保证系统的双曲性并显示对称化所必需的凸熵函数（每单位未变形体积的动能和应变能之和）的存在，利用了对多凸本构律的考虑。在这篇新论文中，通过将热力学第一定律作为附加守恒定律，该框架扩展到更一般的热弹性情况，根据熵（适用于平滑解）或总能量密度（适用于不连续解）系统的。该论文通过以下主要新颖性得到进一步增强。首先，在参考温度下测量的内部能量密度和熵方面提出了充分条件，以确保内部能量密度在由变形措施三重态组成的扩展集和熵。第二，系统的特征值结构的研究是作为双曲线的证明进行的，目的是为显式时间积分器获得正确的时间步长边界。介绍了对两个完善的热弹性模型的应用：Mie-Gruneisen 和修正的熵弹性。第三，使用多凸内能本构法则可以定义广义凸熵函数，即弹道能量和相关的熵通量，允许根据熵共轭场对守恒定律系统进行对称化。第四，与该系列之前的论文一致，当将熵视为系统的未知数时，为热弹性守恒定律系统的数值解提出了一个显式稳定的 Petrov-Galerkin 框架。最后，