A brief introduction to one-dimensional elasticity theory and elastic wave propagation was given in Section 2.12. In this chapter we will explore the full three-dimensional elasticity equations in the context of elastic wave propagation, or elastodynamics. There are many references available on the basic theory of linear and nonlinear elastodynamics (e.g., though often not in the first-order hyperbolic form we need. In this chapter the equations, eigenstructure, and Riemann solutions are written out in detail for several different variants of the linear problem.

The notation and terminology for these equations differs widely between different fields of application. Much of the emphasis in the literature is on steady-state problems, or elastostatics, in which the goal is to determine the deformation of an object and the internal stresses that result from some applied force. These boundary-value problems are often posed as second-order or fourth-order elliptic equations. We will concentrate instead on the hyperbolic nature of the first-order time-dependent problem, and the eigenstructure of this system. This is important in many wave-propagation applications such as seismic modeling in the earth or the study of ultrasound waves propagating through biological tissue. For small deformations, linear elasticity can generally be used. But even this case can be challenging numerically, since most practical problems involve heterogeneous materials and complicated geometry. High-resolution finite volume methods are well suited to these problems, since interfaces between different materials are handled naturally in the process of solving Riemann problems.