Establishing rock stress–strain relationship through analysis of damage growth rate

Establishing constitutive relationships that describe the deformation behaviour of rocks remains a central focus in rock mechanics. With the introduction of growth laws, adoption of distinct assumptions for the damage evolution rate under loading, and incorporation of a new parameter termed the maximum damage degree to represent the practical characteristic that the load-bearing capacity of rocks does not vanish completely with progressive deformation, three damage constitutive relations are derived, namely, linear, logarithmic and quadratic forms. The proposed models include two damage evolution parameters. These parameters can be determined analytically through closed-form expressions based on conventional mechanical properties, thereby ensuring the practicality and applicability of the models. Validation is performed using existing compression test data for granite and marble. Additionally, the effect of damage evolution parameters on the stress–strain response is examined. The results show that for identical strain conditions, the damage degree predicted by the quadratic damage constitutive relationship is greater than that obtained from the linear and logarithmic relationships in sequence. Error statistics show that the quadratic damage constitutive relationship exhibits the best performance, followed by the linear and logarithmic models. Furthermore, both the quadratic and linear relationships outperform the existing damage relationship employing the Weibull function. In the proposed models, the two damage evolution parameters respectively govern the size and shape of deformation curve. By incorporating the variation of conventional mechanical parameters with confining pressure, the proposed model can predict the deformation response of rocks under various confining pressures.