Consistent finite element procedures for nonlinear rubber elasticity with a higher order strain energy function

Abstract

The use of higher order terms in the Rivlin's polynomial strain energy density function is necessary to describe the elastic behavior of rubber undergoing very large and complex deformation. In this paper, the material response tensor for general Rivlin's strain energy density function is derived in a consistent manner such that both major and minor symmetries are retained. Lack of minor symmetry in the material response tensor will lead to numerical convergence difficulties, especially in shear dominant problems. The projection method is used to avoid volumetric locking due to the nearly incompressible nature of rubber. The relation between the numerical penalty number and the material bulk modulus is characterized. The importance of this relation is demonstrated in the study of the apparent Young's modulus of bonded rubber units. The need to include higher order terms in the strain energy density function is presented in the numerical examples. Several classical elasticity problems are also analyzed.