Progressive failure constitutive model for softening behavior of rocks based on maximum entropy theory

Considering that rock failure is a gradual process when subjected to triaxial stress conditions, a new statistical damage constitutive model is proposed to describe the progressive failure of rocks. The model is based on continuous damage mechanics and maximum entropy theory while the commonly used statistical damage model is based on continuous damage mechanics and the conventional Weibull distribution, which is used to describe the strength of mesoscopic rock elements. Weibull distribution is a distribution function with a specific assumption that the nth central moment and the geometric mean of the statistical variable are constant. The maximum entropy distribution is the only unbiased distribution and the Weibull distribution is a special case of the maximum entropy distribution. According to the maximum entropy theory, the damage variable is defined without any prior assumptions of the theoretical distributions. The rock is hypothesized to be divided into two parts: the damaged portion and undamaged portion. The bearing capacity of the damaged part is also considered in the new model so that it is more in accordance with the actual situation. The mesoscopic rock elemental strength is calculated based on energy release rate principles to avoid the deficiencies in using the conventional stress or strain criteria approaches, and the effect of rock initial fissures is emphasized. A new method is presented to determine the unknown parameters in the constitutive equations. The applicability of the new statistical damage constitutive model is verified by experimental data. It is shown that the theoretical model is in good agreement with the test data trend and can simulate the softening behavior of rock well. Admittedly, the proposed model proposed in this paper is a basic model without considering some important aspects of rock deformation mechanics, such as the absences of the complex stress conditions. The purpose of this paper was to illustrate that the constitutive model can be established in the framework of continuous damage mechanics and maximum entropy theory.