Abstract

Motivated from standard procedures in linear wave equations, we write the equations of classical elastodynamics as a linear symmetric hyperbolic system in terms of the displacement gradient ( \({\mathbf{u}}_{\mathbf{x}}\) ) and the velocity ( \({\mathbf{u}}_{t}\) ); this is in contrast with common practice, where the stress tensor and the velocity are used as the basic variables. We accomplish our goal by a judicious use of the compatibility equations. The approach using the stress tensor and the velocity requires use of the time differentiated constitutive law as a field equation; the present approach is devoid of this need. The symmetric form presented here is based on a Cartesian decomposition of the variables and the differential operators that does not alter the Hamiltonian structure of classical elastodynamics. We comment on the differences of our approach with that using the stress tensor in terms of the differentiability of the coefficients and the differentiability of the solution. Our analysis is confined to classical elastodynamics, namely geometrically and materially linear anisotropic elasticity which we treat as a linear theory per se and not as the linearization of the nonlinear theory. We, nevertheless, comment on the symmetrization processes of the nonlinear theories and the potential relation of them with the present approach.

中文翻译：

---

经典弹性动力学作为线性对称双曲系统 $({\mathbf{u}}_{\mathbf{x}}, {\mathbf{u}}_{t})$

摘要

受线性波动方程标准程序的启发，我们根据位移梯度 ( \({\mathbf{u}}_{\mathbf{x}}\) )将经典弹性动力学方程写为线性对称双曲系统速度 ( \({\mathbf{u}}_{t}\) );这与通常的实践相反，通常的实践使用应力张量和速度作为基本变量。我们通过明智地使用兼容性方程来实现我们的目标。使用应力张量和速度的方法需要使用时间微分本构律作为场方程；目前的做法没有这种需要。这里提出的对称形式基于变量和微分算子的笛卡尔分解，不会改变经典弹性动力学的哈密顿结构。我们评论了我们的方法与使用应力张量的方法在系数的可微性和解的可微性方面的差异。我们的分析仅限于经典弹性动力学，即几何和材料线性各向异性弹性，我们将其视为线性理论本身，而不是非线性理论的线性化。尽管如此，我们还是对非线性理论的对称化过程以及它们与当前方法的潜在关系进行了评论。