Alternative writings of classical elastodynamics equations as a first order symmetric system

TL;DR

This paper investigates various symmetric first order formulations of classical elastodynamics equations in one and two dimensions, aiming to identify forms with symmetric matrices through a systematic symmetrization process.

Contribution

It introduces multiple new symmetric first order systems for elastodynamics by systematically using compatibility, momentum, and constitutive equations, extending previous formulations.

Findings

Multiple symmetric formulations identified in 1D and 2D elastodynamics.

Systematic method for symmetrizing elastodynamics equations.

Conditions for matrices to be symmetric in different formulations.

Abstract

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 ($\sigma$), iii) all three quantities: the velocity, the displacement gradient and the stress, and finally iv) the momentum ($\rho_0 u_t$), 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 ($\boldsymbol \sigma$), iii) all three variables $({\bf u}_t, {\bf e}, \boldsymbol \sigma)$, and finally iv) one more writing utilizing the momentum as well, i.e. $(\rho_0 {\bf u}_t, {\bf u}_t, {\bf e}, \boldsymbol \sigma)$. 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.