Shales have become increasingly important because they play key roles in modern energy and environmental geomechanics applications, such as nuclear waste storage, non-conventional oil and gas operations and CO2 geological storage. Shale behaves in a quasi-brittle manner, often exhibiting linear elasticity before reaching its peak stress. Furthermore, softening of the material leads to a residual state in which pure plastic flow is observed under constant values of deviatoric stress. Degradation of the elastic moduli and the accumulation of irreversible strains are believed to be primarily caused by the debonding and decohesion mechanisms in the structure, as well as the growth of microcracks. To capture these features, a constitutive model that couples elastic, plastic and damage theories is developed. The isotropic damage model is based on the Weibull probabilistic theory and describes the failure of brittle materials. This model is coupled with a modified version of the Lade-Duncan criterion to account for non-linear dependency of shear resistance with a mean stress typical of geomaterials. The two surfaces of damage and plasticity and the rate equations for the internal variables are postulated and thermodynamic consistency is subsequently investigated. The coupled plastic-damage constitutive model is integrated with an implicit stress return algorithm for the plastic part, while the damage part can be integrated implicitly. Numerical back-calculations of experimental results from two quasi-brittle shales demonstrate the ability of the model to reproduce the primary features of their mechanical behavior.
Parisio, F., Samat, S., & Laloui, L. (2015). Constitutive analysis of shale: A coupled damage plasticity approach. International Journal of Solids and Structures, 75–76, 88–98. https://doi.org/10.1016/j.ijsolstr.2015.08.003