Probabilistic optimization of a continuum mechanics model to predict differential stress-induced damage in claystone

• A model of anisotropic damage is optimized by using the maximum likelihood method.
• Only one damage parameter is needed to predict differential stress induced damage in claystone.
• The damage model is not sensitive to the initial damage threshold C0 when C0 is below 1 MPa.

Phenomenological modeling of anisotropic damage in rock raises many fundamental thermodynamic and mechanical issues. In this paper, the maximum likelihood method is used to analyze the performance of the Differential Stress Induced Damage (DSID) model recently formulated by Xu and Arson [1]. The stress/strain relationship is a nonlinear function of parameters including unknown constants (i.e., damage constitutive parameters) and known variables (e.g., elastic parameters and controlled stress state). Logarithmic transformation, normalization and forward deletion are employed, in order to find the optimum number of constitutive parameters, as a trade off between accuracy and simplicity. For Eastern France claystone subject to deviatoric stress loading (e.g., triaxial and proportional compression loading), it is found that (1) only one damage parameter (a2) is needed in the expression of the free energy to predict stress/strain curves; (2) a2 controls the deviation of the current principal directions of stress to the principal directions of damage; (3) model parameters involved in the damage criterion can be related to a2. As a result, a2 is the only parameter needed to model differential-stress induced damage in Eastern France claystone. It is also shown that within the set of assumptions made in this study, the DSID model is not sensitive to the initial damage threshold C0, except for C0>106Pa, a range of values in which only one constitutive parameter becomes insufficient to predict the stress/strain curves of damaged claystone. Coupling probabilistic calibration and optimization methods to numerical codes promises to allow adapting the complexity of anisotropic damage models to different rocks and stress paths.