经典弹性动力学方程作为一阶对称系统的另一种写法
Alternative writings of classical elastodynamics equations as a first order symmetric system
AI总结：
本文研究经典线性各向异性弹性动力学方程的一阶对称化，通过逆向结合相容方程、动量方程和时间微分本构定律，给出一维与二维下多组变量形式。
摘要：
我们探索将经典弹性动力学方程改写为一阶对称系统的另一种形式。在一维情形下，我们给出了关于以下变量的对称写法：i）速度（$u_t$）和位移梯度（$u_x$）；ii）速度和应力（$σ$）；iii）所有三个量：速度、位移梯度和应力；最后 iv）动量（$ρ_0 u_t$）、速度、位移梯度和应力。在二维情形下，我们给出了关于以下变量的类似写法：i）速度（${\bf u}_t$）和应变张量（$\bf e$）；ii）速度和应力张量（$\boldsymbol σ$）；iii）所有三个变量 $({\bf u}_t,{\bf e},\boldsymbol σ)$；最后 iv）另一种同样利用动量的写法，即 $(ρ_0{\bf u}_t,{\bf u}_t,{\bf e},\boldsymbol σ)$。我们通过恰当地利用相容方程、动量方程以及时间微分本构定律来实现目标。这是以一种逆向方式完成的：我们首先将初始方程写成如下形式的一阶系统（${\bf q}$ 是表示每种写法中变量的向量）$$A \frac{\partial{\bf q}}{\partial t}+\sum_{i=1}^n B_i\frac{\partial{\bf q}}{\partial x_i}=0,$$$n=1,2$ 取决于我们处于一维还是二维。然后，我们考察矩阵 $B_i$ 的对称形式是什么，以及应使用相容方程、动量方程和时间微分本构定律的哪些组合，以便矩阵 $B_i$ 的对称形式出现在系统中。这一“对称化”过程会改变矩阵 $A$，如果所得矩阵 $A$ 是对称的，我们的目标就实现了。我们的分析限于经典弹性动力学，即几何和材料均为线性的各向异性弹性。
英文摘要：
We explore alternative writings of the equations of classical elastodynamics as a first order symmetric system. In the one dimensional case we present symmetric writings with respect to: i) the velocity ($u_t$) and the displacement gradient ($u_x$), ii) the velocity and stress ($σ$), iii) all three quantities: the velocity, the displacement gradient and the stress, and finally iv) the momentum ($ρ_0 u_t$), the velocity, the displacement gradient and the stress. In the two dimensional case we present similar writings with respect to: i) the velocity (${\bf u}_t$) and the strain tensor ($\bf e$), ii) the velocity and the stress tensor ($\boldsymbol σ$), iii) all three variables $({\bf u}_t, {\bf e}, \boldsymbol σ)$, and finally iv) one more writing utilizing the momentum as well, i.e. $(ρ_0 {\bf u}_t, {\bf u}_t, {\bf e}, \boldsymbol σ)$. We accomplish our goal by judiciously using the compatibility equations as well as the momentum equation and the time differentiated constitutive law. This is done in an inverse way: we start by writing our initial equations as a first order system of the form ($\bf q$ being the vector representing the variables in each writing) $$ A \frac{\partial {\bf q}}{\partial t}+\sum_{i=1}^n B_i \frac{\partial {\bf q}}{\partial x_i}=0, $$ with $n=1, 2$ depending on whether we are in 1 or 2 dimensions. We then check what are the symmetric forms of matrices $B_i$ and which combinations of the compatibility equations, the momentum equations and the time differentiated constitutive law should be used in order the symmetric form of matrices $B_i$ to appear into the system. This "symmetrization" process alters matrix $A$ and if the resulting matrix $A$ is symmetric our goal is accomplished. Our analysis is confined to classical elastodynamics, namely geometrically and materially linear anisotropic elasticity.
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