# Study on Damage Constitutive Model of Rock under Freeze-Thaw-Confining Pressure-Acid Erosion
### Determination of Force Damage Variables  
Because rock is a non-uniform material with a large number of randomly distributed pores, the damage process of rock is the process of the initiation, development, and accumulation of internal pores under external load action, which is directly related to the number of damaged cells. The destruction of micro-elements is generally random. The damage variable \(D\) is defined as the ratio of the number of damaged cells \(n\) under a certain stress level to the total number of cells \(N\) in the initial state, namely  
\[D=\frac{n}{N} \tag{6}\]  
Under the action of external load, the failure of the rock element is random. According to the Krajicinovic model, the damage variable is the failure probability \(P\) of the element [43]. If the probability density function of the element failure is \(\phi(x)\), then \(P\) is the cumulative distribution function of \(F\), namely  
\[D=P=\int\limits_{0}^{F}\phi(x)dx \tag{7}\]  
Studies have shown that when establishing a statistical constitutive model of rock damage, the study of the damage evolution characteristics during rock failure is based on the assumption that the strength of the rock's microelement obeys the normal or Weibull distribution. Compared with the normal distribution, the Weibull distribution is a less conservative type of distribution, which reflects the distribution of random variables more truly to a certain extent. So, this article assumes that the strength of the rock element obeys the Weibull distribution, and its probability density function is:  
\[\phi(F)=\frac{m}{F_{0}}(\frac{F}{F_{0}})^{m-1}\exp\left[-(\frac{F}{F_{0}})^{m }\right] \tag{8}\]  
where: \(F\) represents the intensity of the infinitesimal element, \(m\) and \(F_{0}\) are the parameters of the Weibull distribution. From Equation (7), the damage variable of the rock under load is  
\[D=\int\limits_{0}^{F}\phi(x)dx=1-\exp\left[-(\frac{F}{F_{0}})^{m}\right] \tag{9}\]  
Determination of the Total Damage Variable of Acid-Etched Rock under the Coupling Effect of Freezing-Thawing and Confining Pressure  
Because rocks exhibit different damage characteristics under chemical corrosion, freeze-thaw cycles, and loads, acid-corroded rocks' total damage under the coupling of freeze-thaw and confining pressure is not a simple superposition of their respective damages. Suppose that the volume of the defective material expresses the damage of the rock under the two states of freeze-thaw and load, that is, the ratio of the defect volume to the total volume of the material, assuming that the effective volumes of undamaged rock, freeze-thaw damaged rock and freeze-thaw-confining pressure coupling effect damage rock are \(V_{0}\), \(V_{1}\), and \(V_{2}\), respectively, then  
\[D_{\text{n}}=1-\frac{V_{1}}{V_{0}} \tag{10}\]  
\[D_{\text{m}}=1-\frac{V_{2}}{V_{1}} \tag{11}\]  
Combining Equations (10) and (11) can obtain the total rock damage variable under the coupling of freeze-thaw and confining pressure  
\[D_{S}=D_{\text{m}}+D_{\text{n}}-D_{\text{m}}D_{\text{n}} \tag{12}\]  
where: \(D_{S}\) is the total rock damage variable under the coupling effect of freeze-thaw and confining pressure. The essence of rock freeze-thaw damage is that the volume expansion force generated by volume expansion acts on the initial microvoids and pores in the rock and brings certain tensile stress to the pores, causing the microvoids and pores to expand and even crack. Rock will also cause grain slip and dislocation under load. The damage induced by grain slip and dislocation under load and the damage caused by freezing and thawing interact and couple with each other, causing more complicated changes in the physical and mechanical properties of the rock and the damage mechanism. Equation (12) shows that under the combined action of freeze-thaw and load, the total damage of the rock will be aggravated and show obvious nonlinear characteristics, but the coupling effect of freeze-thaw and load will weaken the total damage of the rock.