Multi-Scale Analysis and Homogenization of Geological Materials with Inclusion

"Rock masses comprised of stiff inclusions in softer matrix are common in various geological environments. The heterogeneity and multi-scale nature of these materials make the evaluation of their mechanical properties a challenging task, which cannot be accomplished using conventional laboratory tests and composite mixture rules. We employ the homogenization method to estimate the mechanical properties of geological materials with inclusions. The homogenization method solves the equations of elasticity on the material (microscopic) scale for a unit cell subjected to unit strains under periodic boundary condition. The micro-scale elasticity equations are solved using the Finite Elements Method, and are in turn used to obtain the structure scale (macroscopic) material matrix.In this research, Finite Element models for unit cells with spherical and ellipsoidal inclusions were developed. Using the unit cells, a parametric study of the geometrical and statistical microscopic properties of the material composites was performed. The following parameters were studied: volumetric ratio, Young’s modulus ratio, size-distribution of inclusion and inclination angle of inclusions. The main purpose of the parametric study was to study the deviation from isotropy as a function of the aforementioned parameters.It was found that for volumetric ratio of 5%-40%, the deviation from isotropy increases, as the volumetric ratio of inclusions increases. For spherical inclusions, most of the cases studied exhibit quasiisotropic behaviour; whereas ellipsoidal inclusions exhibit considerable deviation from isotropy. The inclination of the ellipsoidal inclusions enlarges the deviation from isotropy (9% at random orientation and 26% at 0°)."