A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
A first order hyperbolic framework for large strain computational solid dynamics. Part III: Thermo-elasticity
复制标题
DOI：
10.1016/j.cma.2020.113505
复制
发表时间：
2021-01-01
影响因子：
7.2
通讯作者：
Ghavamian, Ataollah
中科院分区：
文献类型：
--
作者：
Bonet, Javier;Lee, Chun Hean;Ghavamian, Ataollah
在第I部分（Bonet等人，2015）和II（Gil等人，2016），提出了一种新的计算框架，用于可压缩和几乎/真正不可压缩等温超弹性中大应变快固体动力学的数值分析。该方法利用了一个系统的一阶总拉格朗日守恒定律制定的线性动量和变形措施的三重组成的变形梯度张量，其系数和雅可比矩阵。此外，考虑多凸本构关系，以保证系统的双曲性，并显示对称化所需的凸熵函数（每单位未变形体积的动能和应变能之和）的存在。在这篇新的论文中，该框架被扩展到更一般的情况下，热弹性纳入热力学第一定律作为一个额外的守恒定律，写在熵（适用于光滑的解决方案）或总能量密度（适用于不连续的解决方案）的系统。该文件进一步加强了以下关键的创新。首先，给出了在参考温度下测量的内能密度和熵的充分条件，以保证内能密度关于由形变测度和熵组成的三元组扩展集的从头算多凸性.其次，研究了系统的特征值结构，作为双曲性的证明，目的是获得正确的时间步长的显式时间积分器的界限。两个完善的热弹性模型的应用：Mie-Gruneisen和修改的熵弹性。第三，使用polyconvex内部能量本构关系，使广义凸熵函数的定义，即弹道能量，和相关的熵通量，允许对称的系统的守恒定律的熵共轭领域。第四，并与以前的系列文件，一个明确的稳定彼得罗夫-伽辽金框架的数值解的热弹性系统的守恒律时，考虑熵作为一个未知的系统。最后，一系列的数值例子，以评估所提出的配方的适用性和鲁棒性。(C)2020 Elsevier B. V.保留所有权利。
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. (C) 2020 Elsevier B.V. All rights reserved.