Alternative writings of classical elastodynamic equations as a first-order symmetric system

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 (1) the velocity ( $u_t$ ) and the displacement gradient ( $u_x$ ), (2) the velocity and stress (σ), (3) all three quantities: the velocity, the displacement gradient and the stress, and finally, (4) 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 (1) the velocity ( ${\mathbf{u}}_t$ ) and the strain tensor (e), (2) the velocity and the stress tensor ( σ ), (3) all three variables $({\mathbf{u}}_t,\mathbf{e},\mathit{\sigma })$ , and finally, (4) one more writing utilizing the momentum as well, i.e., $({\rho }_0{\mathbf{u}}_t,{\mathbf{u}}_t,\mathbf{e},\mathit{\sigma })$ . 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 (q being the vector representing the variables in each writing)

$$A\frac{\partial \mathbf{q}}{\mathrm{\partial t}}+\sum _{i=1}^n{B}_i\frac{\partial \mathbf{q}}{\partial x_i}=0,$$

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.