## 5 Freeze-Thaw Strength Decay Model
### Parameter Determination  
First, based on the above generalization idea of the exponential decay model, the magnitude of the unloading residual confining pressure value is determined. The exponential function model is selected to fit the unloaded-damaged samples of groups A, B, C, and D under the action of freeze-thaw cycles in terms of the peak strength versus the number of freeze-thaw cycles. The results are shown in Figure 12, all of which indicate a good correlation between the unloaded-damaged samples reload peak strengths after freeze-thawing (both uniaxial and triaxial) and the number of freeze-thaw cycles.  
Table 3 lists the specific values of the four groups of unloaded-damaged sample fitting parameters. \(F_{0}\) is the uniaxial and triaxial peak strength before freeze-thaw of the unloaded-damaged samples, and its value can be directly obtained from the mechanical strength test of the unloaded-damaged samples. To correct the decay constant, the exponential function model is used to fit the decay constant and the value of the unloading residual confining pressure, as shown in Figure 13. The correlation constant \(R^{2}\) = 0.999, and the correction results are shown in Equation (9).  
\[\lambda=0.01952\times\exp(-0.08683\times\sigma_{3}^{\prime}) \tag{9}\]  
Figure 12: Peak strength of unloaded-damaged samples after freeze-thaw. (**a**) Uniaxial reloading. (**b**) Triaxial reloading.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fitting Parameter** & **Group A** & **Group B** & **Group C** & **Group D** \\ \hline \(F_{0}\) & 110.35915 & 125.06573 & 145.77302 & 36.59078 \\ \(\lambda\) & 0.01171 & 0.00931 & 0.00677 & 0.01957 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fitting parameter values of freeze-thaw number and peak strength of the unloaded-damaged samples.  
Figure 13: The relationship between the unloading residual confining pressure and the decay constant.  
The average change in porosity of the unloaded-damaged samples with the number of freeze-thaw cycles at different initial confining pressures is largely the same, and the values are approximately equal. Therefore, the average change in porosity of the unloaded-damaged samples in the three groups can be linearly fitted with the number of freeze-thaw cycles according to Equation (5). The results are shown in Figure 14, with the correlation constant \(R^{2}=0.98\), which is a good fit. The value of \(k=0.08266\), which is the microstructural deterioration constant after freeze-thaw cycling for the unloaded-damaged samples; the results are shown in Equation (10).  
\[\Delta P_{n}=0.08266\times n \tag{10}\]  
Substituting Equations (9) and (10) into Equation (8), the relationship between the peak uniaxial and triaxial reload strength and the average change in porosity after freezing and thawing of the unloaded-damaged samples is obtained, as shown in Equation (11).  
\[F_{n}=F_{0}e^{-\frac{0.01952\times\text{exp}(-0.08683\times\sigma_{3}^{\prime} )}{0.08266}\Delta P_{n}} \tag{11}\]  
where \(F_{n}\) and \(F_{0}\) are the peak uniaxial and triaxial reload strengths of the unloaded damaged sample after \(n\) freeze-thaw cycles and before freeze-thawing, respectively, \(\sigma_{3}^{\prime}\) is the unloading residual confining pressure value, and \(\Delta P_{n}\) is the average change in porosity.