A compressible approach in finite element analysis of rubber-elastic materials

Abstract

This paper presents a finite element procedure for rubber-elastic materials based on a strain energy function of the compressible type. A brief review on the material's description and the related numerical methods is given. A compressible strain energy function is developed by adding a bulk (dilatational) term to the Ogden-Tschoegl model and is used for finite element formulations. An ideal dilatation test is used to show that the penalty method is equivalent to modifying a strain energy function from an incompressible form to a compressible form and that the penalty parameter is related to the Lame's constant λ. The numerical approach to near incompressibility in this work is a natural penalty method which does not use an artificial penalty function. A method for finite element implementation of a strain energy function in terms of principal stretches is proposed, in which the stresses and the material moduli are initially calculated in principal directions and are then transformed to the axial directions of an active coordinate system. Two finite element formulations, the total Lagrangian (TL) and the updated Lagrangian (UL), are given explicitly. Finite element codes for plane stress, plane strain, axisymmetric and three-dimensional problems are developed. The TL formulation is used in programming except for the three-dimensional code which uses the UL formulation. Numerical experiments are conducted on benchmark problems to show the effectiveness of the proposed procedure for the analyses of rubber-elastic materials.