Coupled macro–meso damage constitutive model for fractured rocks based on logistic growth theory

Rock mass is a complex system contains large amounts of macroscopic and mesoscopic damage. The evolution of this internal macroscopic and mesoscopic damage under stress conditions has a significant impact on the stability of rock masses. However, most of the current damage constitutive models do not consider the coupled effects of damage at different scales, including mesoscopic and macroscopic damage in fractured rock masses, also unable to reflect the nonlinear characteristics shown in the process of rock damage. In order to investigate the effect of the nonlinear process of the macro- and mesoscopic damage evolution on the mechanical behavior of fractured rock masses, this study first divided the deformation and damage processes of fractured sandstone under uniaxial compression into three phases based on acoustic emission test data: the compaction phase, elastic phase, and damage phase. Subsequently, the mesoscopic damage variables for fractured sandstone in the compaction phase were derived from damage mechanics and nonlinear dynamics; the mesoscopic damage variables for fractured sandstone in the damage phase were derived from acoustic emission theory, micro-unit strength theory, and nonlinear dynamics; and the macroscale damage variables for fractured sandstone were derived from damage mechanics and fracture mechanics. The Lemaitre strain equivalence hypothesis was used to couple the mesoscopic and macroscopic damage variables, thereby constructing a coupled macro–meso formulation for the damage variables and a constitutive model for fractured sandstone under uniaxial compression. The results of comparisons with several sets of experimental data indicated that the damage constitutive model proposed in this study could accurately reflect the stress–strain relationship and damage evolution of fractured sandstone during uniaxial compression.