## 1 Introduction  
Coal is still the largest and most reliable source of energy in the world. Since the majority of China's coal resources are located in the western region, the coal mining operations are being gradually shifted to this region [1, 2, 3, 4]. During the construction of vertical shafts in the western region, the Cretaceous water-rich bedrock section contains a large number of sandstones with low strength and weak cementation, which often leads to the fracture of frozen pipe due to the instability or excessive deformation of surrounding rocks, causing serious engineering accidents [5, 6]. Hence, it is of great significance to examine the mechanical properties and constitutive relationship of Cretaceous water-rich sandstone under the effect of surrounding rocks. To study the mechanical properties of ultra-deep sandstone reservoir rocks, [PERSON] et al. [7] used a variety of laboratory tests to examine the relationships between the compressive strength, internal friction angle, shear strength, and other mechanical parameters of sandstone and the confining pressure and temperature and compared them with those of shallow sandstone. [PERSON] et al. [8] studied the mechanical properties of granite under uniaxial compression by using computerized tomography (CT) scanning technology because of the significant influence of joints and fissures and deduced the constitutive relationship closer to the actual stress state of the sample. [PERSON] et al. [9] studied the impact of lateral pressure on porosity, permeability, deformation, the mechanical properties, and failure mechanism and obtained the fitting equation for the relation between the physical and mechanical properties of sandstone and confining pressure. Although several studies have been conducted on the mechanical properties of rocks under different conditions [10; 11; 12; 13], the mechanical properties of the Cretaceous water-rich sandstone in the western region of China have been rarely examined, and the stress state of sandstone under complex stress conditions needs to be further studied. The stress state of rock under the crustal stress is very complex, and the rock mechanics basically involves studying the complete stress-strain relationship to characterize the deformation and failure of rocks, i.e., establishing the constitutive relationship of rock [14]. The early studies on the constitutive relationship of rocks were mainly based on the classical continuum mechanics theory. For example, [PERSON] et al. [15] characterized the post-peak elastic modulus obtained during the compressive failure of rock as a function of strain and considered the internal friction angle as the intermediate variable to establish the post-peak nonlinear constitutive relationship on the basis of the Mohr-Coulomb strength criterion. The obtained model accurately described the post-peak mechanical behavior of marble under different confining pressures. [PERSON] et al. [16] introduced the post-peak strength decline index to describe the influence of confining pressure on the post-peak residual strength and secant modulus and established a post-peak strain softening mechanical model of rock by considering the influence of confining pressure. The rationality of the established model was verified by comparing the experimental and numerical simulation results. However, rock is a coherent aggregate of one or more minerals and has complex mechanical properties. On the basis of the continuum theory, the precision of the constitutive model is restricted to the actual working conditions [17]. Subsequently, it was found that the damage constitutive model established by introducing damage mechanics can more accurately describe the stress state of rocks. [PERSON] et al. [18; 19] derived a three-dimensional (3D) damage evolution equation and a damage-softening constitutive equation for rock considering that the micro-element strength of rock obeyed Weibull distribution and normal distribution.