Field-Boundary Element Method for Nonlinear Solid Mechanics

Abstract

T HE boundary integral equation approach for the solution of initial and boundary value problems in linear and nonlinear continuum mechanics has proved to be extremely effective. This approach has received considerable attention from researchers in the last two decades. The method was first developed for linear problems. Extensions have been made to include problems that encompass both geometric and material nonlinearities. The essence of all integral equation approaches lies in the use of a fundamental solution to the governing differential equation. These fundamental solutions are sought in infinite space, to the highest order operator in the governing differential equation. In nonlinear problems, difficulties arise with the fact that the fundamental solution to the nonlinear equation is unknown, even if it exists. Thus, alternative schemes must be used. One way to circumvent this difficulty is to use the fundamental solution in infinite space of just the linear portion of the governing equation. As the result, the integral equation will contain domain integrals in addition to the boundary integrals. These domain integrals will primarily be associated with the nonlinearities in the system, as shall be seen. The apparent advantage in reducing the dimensionality is lost in the case of nonlinear problems. However, the integral equation approach has a number of advantages over other available numerical methods. Interelement continuity requirements for interior unknowns may be significantly less in the case of the integral equation approach than in the finite element approach. An incremental procedure is most likely to be used, and in such a scheme the interior unknowns may be evaluated in terms of the previous increment using the integral equation, in the case of the integral equation approach. Thus, the only unknowns in this instance would be