Variational asymptotic homogenization of finitely deformed heterogeneous elastomers

The objective of this paper is to develop a micromechanics approach to the homogenization and macroscopic stability analysis of finitely deformed heterogeneous elastomers. An Euler-Newton predictor-corrector method is developed for homogenization. It consists of an Euler predictor and a Newton corrector step. Each step involves formulating a variational statement using the variational asymptotic method, discretizing the statement in a finite-dimensional space, and solving the problem using an Euler/multilevel Newton method. An explicit expression for the effective tangent stiffness is obtained in the Euler predictor step and then used for macroscopic stability analysis. The present approach is validated (1) by homogenizing long fiber- and short fiber-reinforced elastomers undergoing uniaxial, biaxial, or shear deformation and (2) by predicting the onset-of-failure curves of porous elastomers with square and hexagonal arrangements of cylindrical voids, undergoing transverse biaxial compression. The present approach is found to be capable of handling various microstructures and complex loading conditions. Different failure modes are found to compete for material failure. More sophisticated hyperelastic material models can be implemented in the present approach.