A generalized micromorphic approach accounting for variation and dispersion of preferred material directions

Abstract Materials exhibiting a heterogeneous and non-uniform composition in terms of elastic and anisotropic properties such as biological tissues require special efforts to accurately describe their constitutive behavior. In contrast to classical models, micromorphic formulations can predict the macroscopically observable material response as originated from distinct scale-dependent micro-structural deformation mechanisms. This is facilitated by additional independent degrees of freedom and associated additional strain and stress quantities. Here, a generalized continuum is mathematically constructed from a macro-continuum and a micro-continuum which are both adequately coupled on kinematics and constitutive levels as well as by micro-boundary conditions. In view of biomechanical modeling, the potential of the formulation is studied for a number of academic examples characterized by an anisotropic material composition to elucidate the micromorphic material response as compared with the one obtained using a classical continuum mechanics approach. The results demonstrate the ability of the generalized continuum approach to address non-affine elastic reorientation of the preferred material direction in the macro-space and its dispersion in the micro-space as affecting deformation, strain and stress on the macroscopic level. In particular, if the anisotropy in the micromorphic formulation is solely linked to the extra degrees of freedom and associated strain and stress measures, the deformation for small and large strains is shown to be distinctly different to the classical response. Together with the ability to implicitly account for scale-dependent higher-order deformation effects in the constitutive law the proposed generalized micromorphic formulation provides an advanced description, especially for fibrous biological materials.