A nonincremental approach for large displacement problems

Abstract

In structural mechanics, nearly all the current computations for time dependent nonlinear problems (e.g. plasticity, viscoplasticity or damage) use step-by-step methods. In contrast, for small displacement problems, the large time incremental (LATIN) method, introduced by Ladevèze [ C.r. Acad. Sci. Paris Ser. II 300, 41–44 (1985).], is an iterative method which accounts for the whole loading process in a single time increment which is not a priori limited. To give an idea of the step length, several loading cycles (or even several thousand) can be simulated in a single time increment. The performance of the method is excellent in problems with many degrees of freedom or complicated loads [Ph. Boisse, P. Bussy and P. Ladevèze, Int. J. numer. Meth. Engng 29, 632–647 (1990); P. Ladevèze, In: New Advances in Computational Structural Mechanics (Edited by P. Ladevèze and O. C. Zienkiewick), pp. 3–21. Elsevier, Oxford (1992)]. A preliminary extension to large displacement problems has been presented and applied to deep drawing simulation in [P. Bussy, P. Rougée and P. Vauchez, In: Proc. Numerical Methods in Engineering, pp. 102–109. Elsevier, Oxford (1990)]. The present work concerns another extension, suitable for material models with internal variables, as described by Ladevèze [ C.r. Acad. Sci. Paris Ser. II 309, 1095–1099 (1989)]; more details concerning this extension can be found in [P. Ladevèze, Sur une théorie des grandes transformations: modelisation et calcul, Rapport interne du LMT no. 125 (1991)]. The objective herein is to describe the main ideas of the method, and to show with simple beam buckling problems how the pre and post buckling response of a structure may be obtained simultaneously without any continuation technique and with less than 10 iterations.