## 4 Damage Degree Analysis  
The strength of expansive soils is significantly influenced by the action of two factors: freeze-thaw cycles and weathered sand content. In order to investigate the change pattern of the strength of expansive soils under the combined effect of both conditions, a model is developed here to predict the degree of soil damage with the initial dynamic elastic modulus as the reference index.  
Taking the maximum dynamic elastic modulus of soil samples that have not undergone freeze-thaw cycles as a nondestructive criterion, the degree of the damage caused by freeze-thaw cycles to \(E_{\text{dmax}}\) is defined as \(D_{n}^{w}\) according to the strain equivalence assumption proposed by [PERSON] [23; 24; 25], which represents the ratio of the strength lost to the original strength after n freeze-thaw cycles times for soils mixed with w content of weathered sand, as the following equation shows:  
\[D_{n}^{w}=1-\frac{E_{\text{dmax}}(n,w)}{E_{\text{dmax}}(0,w)} \tag{7}\]  
where \(E_{\text{dmax}}(n,w)\) denotes the maximum dynamic elastic modulus of a modified expansive soil with weathered sand content of \(w\) after \(n\) times of freezing-thawing.  
According to the calculation results, making the damage degree versus the freeze-thaw cycle curve as shown in Figure 11, the figure for the damage degree of the soil under the \(n\)-th freeze-thaw can be found using the following equation to fit:  
\[D_{n}^{w}=\frac{n}{\alpha+\beta n} \tag{8}\]  
where \(\alpha\) and \(\beta\) are the fitting parameters, the fitting parameters of the four sand blending rates are averaged to obtain the fitting parameters under the influence of the number of freeze-thaw cycles, and the results are as follows:  
\[\alpha=3.56653,\beta=7.02598\]  
Figure 11: Curve of damage degree with the number of freezing and thawing.  
The effect factor of weathered sand admixture \(w\) on strength under \(n\) freeze-thaw cycles is defined as \(\gamma_{w}^{n}\), and can be calculated from the following equation:  
\[\lambda_{w}^{n}=\frac{E_{\text{dmax}}(w,n)}{E_{\text{dmax}}(0,n)} \tag{9}\]  
\(\lambda_{w}^{n}>1\) indicates the increase in strength and \(\lambda_{w}^{n}>1\) indicates the decrease in strength. The relationship with the sand mixing rate is shown in Figure 12.  
It can be seen that the two satisfy a parabolic relationship, and a quadratic polynomial fit of these data results in the following equation:  
\[\lambda_{w}^{n}=1+\eta w+\xi w^{2} \tag{10}\]  
The \(R^{2}\) values after the fit was completed were all greater than 0.99, and the fitting effect was good. The parameters under the 5 freeze-thaw conditions were taken as the average values, and the fitted parameters of the effect of sand blending rate on strength were finally obtained, i.e.,  
\[\eta=-2.73674,\xi=11.636894\]  
In summary, under the joint influence of freeze-thaw action and weathered sand improvement, the equation of calculating the initial dynamic elastic modulus damage degree of the soil is as follows:  
\[D=\lambda_{w}^{n}(1-D_{n}^{w}) \tag{11}\]  
Substituting the parameter \(\alpha\), \(\beta\), \(\eta\), \(\zeta\), into the equation, i.e.,  
\[D=(1-\frac{n}{356653+7.02598n}) \tag{12}\] \[\times(1-2.73674w+11.63689w^{2})\]  
\[E_{\text{dmax}}(n,w)=DE_{\text{dmax}}(0,0) \tag{13}\]  
To verify the fitting accuracy of the damage calculation formula, the value of the initial dynamic elastic modulus was calculated by Equation (13), and then compared with the real value of the initial elastic modulus in Table 2; the average error of 20 sets of data was found to be 1.59%, and the maximum error appeared in the soil sample with 20% weathered sand content which had undergone freeze-thaw once. The calculated value is 361.0108 MPa and the actual value is 346.4378 MPa, so the error reached was 4.21%. Taken  
Figure 12: Variation curve of influence factor with the number of freezing and thawing.  
together, the accuracy of the predicted damage degree value of Equation (12) meets the engineering requirements.