An efficient numerical method to solve the problems of 2D incompressible nonlinear elasticity

Abstract

Presented herein is a numerical variational approach to the two-dimensional (2D) incompressible nonlinear elasticity. The governing equations are derived based upon the minimum total energy principle by considering the displacement and a pressure-like field as the two independent unknowns. The tensor equations are replaced by equations in a novel matrix-vector form. The proposed solution method is based upon the variational differential quadrature (VDQ) method and a transformation procedure. Using the introduced VDQ-based approach, the energy functional is precisely discretized in a direct way. Being locking-free, simple implementation and computational efficiency are the main features of this method. Also, it is free from numerical artifacts and instabilities. Some important problems of 2D incompressible elasticity are addressed to test the method. It is revealed that it can be efficiently utilized to capture the large strains of incompressible solids. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.