## 3 CUH Model: Framework and Theory  
### Three-Dimensional Constitutive Model Framework  
[PERSON] et al. [11] proposed a unified hardening (CUH) model based on the Modified Cam-Clay (MCC) model, which can describe both overconsolidated clay and sand. The CUH model effectively captures the stress-strain behavior of various soil types, including clay, sand, and rockfill, under different initial conditions and stress paths. It represents a theoretical advancement and innovative development of the MCC model [12]. To apply the constitutive model to true triaxial laboratory test predictions or engineering practice, it is necessary to extend the model to three dimensions. The Transformed Stress (TS) method, proposed by [PERSON] et al. [18], is a three-dimensional extension approach for constitutive models. Its principal stress formulation is expressed as follows:  
\[\tilde{\sigma_{i}}=p+\frac{q_{c}}{q}(\sigma_{i}-p),i=1,2,3 \tag{26}\]  
where \(\sigma_{i}\), \(p\), and \(q\) represent the principal stresses, mean principal stress, and generalized shear stress in the real stress space, respectively; \(\tilde{\sigma_{i}}\) and \(q_{c}\) denote the principal stresses and generalized shear stress in the transformed stress space, respectively. The \(q_{c}\) varies based on different strength criteria. The mean principal stress PPP in the real stress space is equal to that in the transformed stress space. Since this method can be formulated based on different strength criteria (such as the SMP criterion, Mohr-Coulomb criterion, Lade criterion, and generalized nonlinear strength criteria), it is applicable to materials with varying strength characteristics. Figure 1 illustrates the transformed stress method based on the SMP strength criterion. Figure 1a shows the SMP strength criterion and the generalized Mises strength criterion in the real stress space. The SMP criterion in the real stress space is transformed into the generalized Mises strength criterion in the transformed stress space, assuming that the constitutive model is established in the transformed stress space. The real stress-strain relationship is obtained by differentiating the yield surface in the transformed stress space with respect to the real stress space. Figure 1b presents the transformation illustration in the \(\pi\)-plane. Figure 1c depicts the yield surface of the CSUH model in both the real principal stress space and the transformed principal stress space. Figure 1d shows the \(p\)-\(q\) cross-section of Figure 1c, where \(M_{\text{e}}\) represents the critical state stress ratio under triaxial extension. When applying the TS method to extend the CSUH model into three dimensions, it is assumed that the CSUH model is formulated in the transformed stress space. The plastic potential surface equation, \(g\), of the CSUH model in the transformed stress space is expressed as follows:  
\[\tilde{g}=\ln\hat{p}^{2}+\ln\biggl{(}1+\frac{\tilde{q}^{2}}{M_{\rm c}^{2}\hat{p }^{2}}\biggr{)}=0 \tag{27}\]  
where \(\hat{p}\) and \(\tilde{q}\) denote the effective mean principal stress and shear stress in the transformed stress space, respectively. The tilde symbol (-) indicates variables in the transformed stress space. The parameter \(p_{y}\) represents the horizontal coordinate of the plastic potential surface's intersection with the \(p\)-axis. The characteristic state stress ratio, \(\tilde{M}_{c}\), has a geometric interpretation as the stress ratio at the upper quadrant point of the ellipse. The yield surface equation, \(f\), of the CSUH model in the transformed stress space is given by the following:  
\[\tilde{f}=\ln\biggl{[}\biggl{(}1+\frac{(1+\chi)\tilde{q}^{2}}{M^{2}\hat{p}^{2} -\chi\tilde{q}^{2}}\biggr{)}\hat{p}+\hat{p}_{\rm s}\biggr{]}-\ln(\hat{p}_{x0}+ \hat{p}_{\rm s})-\frac{\hat{H}}{c_{\rm P}}=0 \tag{28}\]  
where \(M\) is the critical state stress ratio, representing the slope of the conventional triaxial compression critical state line (CSL) in the \(p\)-\(q\) space. The critical state parameter, \(\chi\), adjusts the shape of the yield surface.