## A Numerical Scheme for Axisymmetric Elastic Waves in Solids

### BibTeX

@MISC{And_anumerical,

author = {Xiao Lin And and Xiao Lin and Josef Ballmann},

title = {A Numerical Scheme for Axisymmetric Elastic Waves in Solids},

year = {}

}

### OpenURL

### Abstract

This paper is dedicated to Professor Alan Jeffrey, University of Newcastle Upon Tyne, on the occasion of his 65th birthday. 1 Some analytical solutions were obtained, e.g. by Laturelle [1,2] for a half space using Laplace and Hankel transformations, and by Miklowitz [3] for a rod using the approximate MindlinHerrmann theory. These show that the source term complicates the analytical solving of the system in a way that does not occur in the plane problem. Therefore a numerical method giving a good approximate solution will be very useful for practical applications. Hirose and Achenbach [4] developed a time-domain boundary element method to study the elastic wave interaction in an axisymmetric body. For plane problems without a source term, Lin and Ballmann [5-8] have extended the Godunov-type characteristic-based finite difference methods of gasdynamics for stress waves in elastic-plastic solids and have obtained a great number of results. Problems with cylindrical symmetry can be treated in a similar way. Nevertheless, the existing numerical schemes dealing with hyperbolic systems with a source term seem not yet as well-developed as those for the systems without a source term. A widely used method is the time splitting technique which alternately solves a system of conservation laws without any source term and a system of ordinary differential equations modeling the source effect. However, it seems that this technique can produce misleading results, see Westenberger and Ballmann [9]. For one-dimensional problems, some promising efforts were made by Glimm et al [10], Glaz and Liu [11] and Roe [12]. In this paper, we first propose an explicit finite difference scheme for the numerical integration of hyperbolic PDEs with a source term. Then, this scheme is applied to so...