Cyclic constitutive equations of rock with coupled damage induced by compaction and cracking

In this paper, the cyclic constitutive equations were proposed to describe the constitutive behavior of cyclic loading and unloading. Firstly, a coupled damage variable was derived, which contains two parts, i.e., the compaction-induced damage and the cracking-induced damage. The compaction-induced damage variable was derived from a nonlinear stress–strain relation of the initial compaction stage, and the cracking-induced damage variable was established based on the statistical damage theory. Secondly, based on the total damage variable, a damage constitutive equation was proposed to describe the constitutive relation of rock under the monotonic uniaxial compression conditions, whereafter, the application of this model is extended to cyclic loading and unloading conditions. To validate the proposed monotonic and cyclic constitutive equations, a series of mechanical tests for marble specimens were carried out, which contained the monotonic uniaxial compression (MUC) experiment, cyclic uniaxial compression experiments under the variable amplitude (CUC-VA) and constant amplitude (CUC-CA) conditions. The results show that the proposed total damage variable comprehensively reflects the damage evolution characteristic, i.e., the damage variable firstly decreases, then increases no matter under the conditions of MUC, CUC-VA or CUC-CA. Then a reasonable consistency is observed between the experimental and theoretical curves. The proposed cyclic constitutive equations can simulate the whole cyclic loading and unloading behaviors, such as the initial compaction, the strain hardening and the strain softening. Furthermore, the shapes of the theoretical curves are controlled by the modified coefficient, compaction sensitivity coefficient and two Weibull distributed parameters.