A VDQ‐based multifield approach to the 2D compressible nonlinear elasticity

Abstract

SummaryA numerical multifield methodology is developed to address the large deformation problems of hyperelastic solids based on the 2D nonlinear elasticity in the compressible and nearly incompressible regimes. The governing equations are derived using the Hu‐Washizu principle, considering displacement, displacement gradient, and the first Piola‐Kirchhoff stress tensor as independent unknowns. In the formulation, the tensor form of equations is replaced by a novel matrix‐vector format for computational purposes. In the solution strategy, based on the variational differential quadrature (VDQ) technique and a transformation procedure, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing derivative and integral matrix operators. The present method can be regarded as a viable alternative to mixed finite element methods because it is locking free and does not involve complexities related to considering several DOFs for each element in the finite element exterior calculus. Simple implementation is another advantage of this VDQ‐based approach. Some well‐known examples are solved to demonstrate the reliability and effectiveness of the approach. The results reveal that it has good performance in the large deformation problems of hyperelastic solids in compressible and nearly incompressible regimes.