Locking-free finite elements for large deformations

Abstract

This thesis deals with locking and hourglassing issues that arise in nonlinear finite element analyses of problems in mechanics. The major focus lies on the analysis of these numerical deficiencies, the design of suitable benchmarks and the development of novel remedies. A new nonlinear locking phenomenon is described. It is caused by parasitic nonlinear strain terms and it is particularly pronounced for large element deformations in combination with higher-order integration and a critical parameter, such as the element aspect ratio or the Poisson's ratio. To avoid this problem within the popular class of enhanced assumed strain formulations, novel strain enhancements are presented. An analytical solution of a tailored finite bending problem is used to benchmark the newly proposed element formulations. Further, the problem of hourglassing in both compression and tension of solid bodies is analysed. It is shown that the underlying causes of hourglassing can be explained by geometry-induced and material-induced trigger mechanisms of structural instabilities. Crucial for understanding as well as benchmarking is the analytical in-depth analysis of a large strain bifurcation problem. Based on these insights, an obvious remedy for the geometric hourglassing phenomenon is presented. The last part of this thesis is devoted to the efficient algorithmic treatment of the computation of instability points. The difficulties in choosing a suitable load-stepping approach with methods from the literature are discussed and a methodological idea of an adaptive load-stepping scheme is presented. Efficiency and practicability are demonstrated for several benchmarks.