## 4 Damage Constitutive Model  
After decades of development, damage constitutive models for rock and soil under loading have been formulated by combining elastic and viscoplastic mesoscopic damage during deformation [28]. Particularly, the random distribution of micro-cracks in the specimen and the evolutionary damage in later stages are correlated with the initial damage [29]. Therefore, it is assumed herein that the axial strain of the specimen satisfies the following Equation (1) [30]:  
\[P(\varepsilon)=\frac{m}{F}(\frac{\varepsilon}{F})^{m-1}\exp[-(\frac{ \varepsilon}{F})^{m}] \tag{1}\]  
where m and F are fitting constants, \(\varepsilon\) refers to the axial strain, and \(P(\varepsilon)\) denotes the distribution function of probability. As demonstrated in Figure 7, if \(N_{\text{D}}\) is designed to represent the number of microelements that have been destroyed inside a damaged specimen under a certain level of loading, the total number of microelements inside the specimen is signified by \(N_{\text{T}}\), and the their ratio can be defined as the damage variable \(D\), as follows [30]:  
\[D=\frac{N_{D}}{N_{T}} \tag{2}\]  
The number of destroyed microelements \(N_{\text{t}}\) in the specimen is defined as follows:  
\[N_{t}(\varepsilon)=\int_{0}^{\varepsilon}NP(x)dx=N\Big{\{}1-\exp\Bigl{[}-( \frac{\varepsilon}{F})\Bigr{]}^{m}\Big{\}} \tag{3}\]  
The damage variable can be expressed by the following Equation (4). \[D=1-\exp\Bigl{[}-(\frac{\varepsilon}{F})\Bigr{]}^{m} \tag{4}\]  
According to the principle of effective stress, the axial stress \(\sigma\) is defined as Equation (5):  
\[\sigma_{i}=\frac{\sigma}{1-D}=E\varepsilon(1-D) \tag{5}\]  
where \(\sigma_{i}\) is the effective stress, and \(E\) is elastic modulus. The mechanical constitutive model plays a pivotal role in comprehending the mechanical behavior of reactive materials. Given that the stress-strain curves of stabilized soft clay exhibit linear rises and strain softening as their two most prominent features, a piecewise constitutive model based on mesoscopic damage mechanics is employed to accurately fit  
Figure 7: Schematic diagram of microelements under external loading. and address these curves. Drawing from previous research [31], the constitutive model equation is represented by Equation (6). The fitting parameters \(a\) and \(b\) of the constitutive model are determined by analyzing the specific UCS experimental data for five different fiber contents in the 5 mm short-fiber group using Equation (6), and these results are presented in Table 5. The correlation coefficient \(\mathbb{R}^{2}\) predicted by the constitutive model exceeds 0.95. These fitting outcomes affirm that the aforementioned constitutive equation adeptly captures the stress-strain relationship of stabilized soft clay samples. \[\frac{\sigma}{s_{u}}= \begin{cases}a\frac{\varepsilon}{t_{u}}+(3-2a)(\varepsilon/ \varepsilon_{u})^{2}+(a-1.5)(\varepsilon/\varepsilon_{u})^{3}&\varepsilon\leq \varepsilon_{u}\\ \frac{\varepsilon/\varepsilon_{u}}{b(\varepsilon/\varepsilon_{u}-1)^{2}+ \varepsilon/\varepsilon_{u}}&\varepsilon>\varepsilon_{u}\end{cases}  \tag{6}\]  
where \(s_{u}\) is the peak stress, and \(\varepsilon_{u}\) represents the strain at the peak point. The fitting parameters \(a\) and \(b\) in the proposed constitutive model carry specific physical significance [32]. Specifically, \(a\) in the rising section of the stress-strain curves reflects the ratio of the initial elastic modulus to the peak modulus of the materials, providing insights into the ductility characteristics of the materials. On the other hand, \(b\) in the descending section of the curves reveals the britteness of the materials. As indicated in Table 6, with the increase in fiber content, \(a\) exhibits an upward trend, signifying an improved ductility of the fiber-reinforced cement-stabilized soil, which aligns with the test results. Simultaneously, \(b\) demonstrates a decrease followed by an increase, with the final value being lower than that of cement soil without fibers.