Motivated by the mechanical analysis of multiphase or damaged materials, a general approach relating fabric tensors characterizing microstructure to the fourth rank elasticity tensor is proposed. Using a Fourier expansion in spherical harmonics, the orientation distribution function of a positive, radially symmetric microstructural property is approximated by a scalar and a symmetric, traceless second rank tensor. Following this approximation, a general expression of the elastic free energy potential is derived from representation theorems for anisotropic scalar functions. Based on a homogeneity assumption for the elastic constitutive law with respect to the microstructural property, a particular elasticity model is developed that involves three independent constants beside the fabric tensors. Strict positive definiteness of the corresponding elasticity tensor is ensured under explicit conditions on the independent constants for arbitrary fabric tensors.
Keywords: Anisotropy; Continuum mechanics; Elasticity; Fabric; Microstructure