New insight into large deformation analysis of stretch-based and invariant-based rubber-like hyperelastic elastomers

A nonlinear finite element formulation for large deformation analysis of hyperelastic membranes in Total Lagrangian framework is developed. In many cases, it is convenient to utilize the invariant-based strain energy density function to analyze the structures when employing the finite element method. However, in this contribution, both invariant-based and stretch-based formulations are developed. Formulations are completely general and not restricted to axisymmetric loading and geometry of membrane. Furthermore, initially flat and non-flat membrane can be investigated by this method. Incompressible as well as compressible materials can be modeled. A method for imposing the plane stress condition to the formulation for compressible materials is presented. Moreover, isotropic and anisotropic materials can be considered. Some problems are analyzed and results compared to the available ones in the literature. The comparison between solution based on invariant-based method and stretch-based method with exact fourth-order elasticity tensor obtained in this work demonstrates that the stretch-based method derived in this paper works very well. The presented examples demonstrate that the developed formulations have no restriction or simplification with respect to the modeled object geometry and inherent anisotropy.