Generalized macroscale model for Cosserat elasticity using Generalized Multiscale Finite Element Method

Abstract

In this paper, we develop a multiscale computational approach for heterogeneous Cosserat media without scale separation and high contrast. Cosserat medium has been used in many applications to describe materials whose particles possess rotational degrees of freedom. Many previous findings investigate Cosserat medium in the presence of heterogeneities. Approaches that include homogenization and numerical homogenization have been developed for problems with scale separation, such as periodicity. In these approaches, macroscopic models are derived that show the resulting equations can be Cauchy medium or Cosserat medium depending on the Cosserat intrinsic length scale. In our paper, we consider computational macroscopic models for Cosserat medium without scale separation. We use Generalized Multiscale Finite Element Method (GMsFEM) to derive a macroscopic model on a coarse computational grid, that does not resolve the small scales and the contrast. In this approach, multiple macroscopic basis functions are used to describe the small scales. The resulting macroscopic model is similar to multicontinua model as it contains multiple macroscopic parameters at each macroscopic point (or grid). We present numerical results for different heterogeneous media types. Our numerical results show that using a few multiscale basis functions (i.e., macroscopic parameters), we can achieve a good accuracy.