Correlation structures of statistically isotropic stiffness and compliance TRFs through upscaling

This paper reports a procedure to develop random fields of material properties on a mesoscale level, coarser than the microscale level of heterogeneous material microstructure. Since the anisotropy of properties at the mesoscale level is unavoidable, tensor-valued random fields (TRFs) need to be constructed. The construction satisfies three criteria: (i) the passage from the micro to mesoscale must be conducted according to micromechanics, (ii) any anisotropic properties must be grasped, and (iii) full (spatial) correlation structure must be accounted for. The construction is illustrated in the example of a linear elastic microstructure modeling a planar interpenetrating phase composite (IPC), where each phase is interconnected throughout the microstructure. The most general representation of a statistically homogeneous and isotropic correlation structure of the TRFs of stiffness and compliance is employed. Four material functions of the representation are determined through scale-dependent homogenization, which grasps any auto- and cross-correlations within/among different components of TRFs. The Gaussianity of the TRFs is also assessed with a finding that, overall, the smaller is the mesoscale and the more pronounced is the microstructural randomness, the more non-Gaussian are the mesoscale TRFs.