Incollection,
Fundamental aspects of numerical methods for the propagation of multi-dimensional nonlinear waves in solids
Nonlinear Hyperbolic Equations — Theory, Computation Methods, and Applications, volume 24 of Notes on Numerical Fluid Mechanics, Vieweg+Teubner Verlag, Wiesbaden, (1989)
DOI: 10.1007/978-3-322-87869-4_56
Abstract
The nonlinear scalar constitutive equations of gases lead to a change in sound speed from point to point as would be found in linear inhomogeneous (and time dependent) media. The nonlinear tensor constitutive equations of solids introduce the additional local effect of solution dependent anisotropy. The speed of a wave passing through a point changes with propagation direction and its rays are inclined to the front. It is an open question whether the widely used operator splitting techniques achieve a dimensional splitting with physically reasonable results for these multi-dimensional problems. Maybe this is the main reason why the theoretical and numerical investigations of multi-dimensional wave propagation in nonlinear solids are so far behind gas dynamics. We hope to promote the subject a little by a discussion of some fundamental aspects of the solution of the equations of nonlinear elastodynamics. We use methods of characteristics because they only integrate mathematically exact equations which have a direct physical interpretation.
