Arc-length procedures with BEM in physically nonlinear problems

Geometrically or physically nonlinear problems are often characterised by the presence of critical points with snapping behaviour in the structural response. These structural or material instabilities usually lead to inefficiency of standard numerical solution techniques. Special numerical procedures are, therefore, required to pass critical points. The authors mean to present an arc-length procedure combined with the Boundary Element Method (BEM). The arc-length methods are intended to enable solution algorithms to pass limit points. Particularly for snap-back behaviour, the arc-length methods are the only procedures, which enable to follow the equilibrium path. The interest is mainly devoted to softening models where snap-backs and snap-throughs usually occur. No BEM applications have been possible so far due to the lack of a procedure enforcing the arc-length constraint. This paper intends to overcome such a difficulty. The analysis beyond the collapse point is mainly justified by two facts: (1) the investigation concerns a structural component and, therefore, it may be desirable to incorporate the load/deflection response of this component within a further analysis of the complete structure; (2) it may be important to know not just the collapse load but whether or not this collapse is of a ductile or brittle form. The procedure can be easily applied both to plasticity and damage, but numerical results will be presented for 2D elastoplasticity. Accurate results will be obtained in the case of both hardening and softening plasticity. The procedure can be used in BEM applications on nonlocal continuum models of the integral type and it can be easily extended to elasto-plasto-dynamics and to buckling in elastoplasticity. © 2004 Elsevier Ltd. All rights reserved.