Large scale computations of effective elastic properties of rubber with carbon black fillers

Abstract

A general method, based on a multi-scale approach, is proposed to derive the effective elastic shear modulus of a rubber with 14\% of carbon black fillers from finite element (FE) and fast Fourier transform (FFT) methods. The complex multi-scale microstructure of such material was generated numerically from a mathematical model of its morphology which was identified from statistical moments out of TEM images. For FE computations, the simulated microstructures were meshed from three-dimensional reconstruction of isosurface using Marching Cubes algorithm with a special attention to the quality of the topology and the geometry of the mesh. To compute the shear modulus and to determine the representative volume element, homogeneous boundary conditions were prescribed on meshes and combined with a domain decomposition method. Regarding parallel computing, specific difficulties related to the highly heterogeneous microstructures and complex geometry are pointed out. Hereby, the experimental shear modulus (1.8MPa) obtained from DMA (dynamic mechanical analysis) was estimated by the Hashin-Shtrikman lower bound (1.4MPa) and the computations on simulated microstructures (2.4MPa). The shear modulus was determined for two materials with the same volume fraction but different distribution of fillers. The current model of microstructures is capable of estimating the relative effect of the mixing time in processing associated with change in morphology on the elastic behavior. The computations also provide the local fields of stress/strain in the elastomeric matrix.