A simplified multiscale damage model for the transversely isotropic shale rocks under tensile loading

A simplified multiscale damage model is proposed for the transversely isotropic shale rocks under tensile loading. In this framework, the multiscale representations for the shale rocks are presented by introducing the microcrack-weakened equivalent solid with hierarchical microstructures, whose transversely isotropic properties are obtained by performing multilevel homogenization procedures. To simplify the calculation process for the damage-induced properties, the equivalent isotropic medium is attained by applying the Voigt–Reuss–Hill averaging process to the transversely isotropic solid. Subsequently, the microcrack-induced inelastic compliances are approximately derived in terms of microcrack opening displacements in the equivalent isotropic medium of the shale rock under tensile loading. The sizes and orientations of microcracks are taken as random variables. Both stationary and evolutionary damage models are considered. Microcrack kinetic equations are characterized through the use of a fracture mechanics-based stability criterion and microcrack geometry within a representative volume element. Numerical examples including experimental validations and comparisons with existing micromechanical models are presented to verify the proposed multiscale damage model. Finally, the influences of the silt inclusions and porosity on the material intrinsic and damage-induced properties are discussed.