## 3 CUH Model: Framework and Theory
### Third-Order Flexibility Matrix of the Model  
Based on Equation (14), the incremental relationship is expressed in the form of a third-order flexibility matrix:  
\[\{\mathrm{d}\epsilon\}_{3\times 1}=\bigg{\{}\left[C^{\mathrm{e}}\right]_{3 \times 3}+\left[C^{\mathrm{p}}\right]_{3\times 3}\bigg{\}}\{\mathrm{d} \sigma\}_{3\times 1} \tag{34}\]  
For isotropic materials:  
\[\left[C^{\mathrm{e}}\right]_{3\times 3}=\frac{1}{E} \begin{bmatrix}1&-\
u&-\
u\\ -\
u&1&-\
u\\ -\
u&-\
u&1\end{bmatrix}  \tag{35}\]  
where \(E\) is the elastic modulus and \(\
u\) is Poisson's ratio. In Equation (34), the third-order plastic flexibility matrix \([C^{\mathrm{e}}]_{3\times 3}\) is given as follows:  
\[\left[C^{\mathrm{p}}\right]_{3\times 3}=-\frac{\left\{\frac{\partial\delta}{ \partial\delta}\right\}_{3\times 1}\left\{\frac{\partial\tilde{H}}{\partial\
u} \right\}_{1\times 3}^{T}}{\frac{\partial\tilde{H}}{\partial\tilde{H}}\left[\left\{ \frac{\partial\tilde{H}}{\partial\tilde{H}}\right\}_{1\times 3}^{T}\left\{\frac{ \partial\tilde{H}}{\partial\tilde{H}}\right\}_{3\times 1}\right]} \tag{36}\]  
where the yield surface equation, \(\tilde{f}\), is differentiated with respect to the true stress, \(\sigma\), and the hardening parameter, \(\tilde{H}\), the plastic potential surface equation, \(\tilde{g}\), is differentiated with respect to the transformed stress, \(\tilde{\sigma}\), and the hardening parameter, \(\tilde{H}\), is differentiated with respect to the true strain, \(\epsilon^{p}\). These differentiation processes require the application of the chain rule. The yield surface equation, \(\tilde{f}\), and plastic potential surface equation, \(\tilde{g}\), are generally expressed as functions of \(\bar{p}\) and \(\tilde{q}\), while \(\bar{p}\) and \(\tilde{q}\) are functions of stress, \(\sigma_{i}\)Based on the SMP strength criterion and the transformed stress (TS) method, we obtain the following:  
\[\left\{ \begin{array}{l}\tilde{p}=p=(\sigma_{1}+\sigma_{2}+\sigma_{3})/3\\ \tilde{q}=q_{\rm c}=\\ \frac{I_{1}I_{2}-9I_{3}+3\sqrt{(I_{1}I_{2}-I_{3})(I_{1}I_{2}-9I_{3})}}{4I_{2}} \end{array} \right. \tag{37}\]  
where \(I_{1}\), \(I_{2}\), and \(I_{3}\) are the first, second, and third invariants of the principal stress tensor, respectively, and \(q_{\rm c}\) is the transformed generalized shear stress based on the SMP strength criterion. The derivation results for Equation (36) of the CSUH model are provided in [11] and will not be reiterated here. Equation (8) can also be expressed as follows:  
\[\left\{\mathrm{d}\varepsilon_{V}^{\rm P}\right\}=\bar{\Lambda}\left\{\frac{ \partial\tilde{g}}{\partial\tilde{p}}\right\} \tag{38}\]  
The derivative of the hardening parameter with respect to plastic strain can also be expressed as the derivative with respect to plastic volumetric strain. Therefore, for the CSUH model, Equation (36) can be rewritten as follows:  
\[\left[C_{ij}^{\rm P}\right]=c_{\rm P}\frac{\bar{M}_{\rm c}^{4}-\tilde{\eta}^ {4}}{\bar{M}_{\rm Y}^{4}-\tilde{\eta}^{4}}\frac{\frac{\partial\tilde{f}}{ \partial\tilde{\gamma}}}{\frac{\partial\tilde{\gamma}}{\partial\tilde{\gamma }}}\frac{\partial\tilde{g}}{\partial\tilde{\gamma}_{i}},i=1,2,3,\ j=1,2,3 \tag{39}\]  
The loading and unloading criteria are defined as follows:  
\[\mathrm{d}f_{trial}=\frac{\partial\tilde{f}}{\partial\sigma_{1}}\mathrm{d} \sigma_{1}+\frac{\partial\tilde{f}}{\partial\sigma_{2}}\mathrm{d}\sigma_{2}+ \frac{\partial\tilde{f}}{\partial\sigma_{3}}\mathrm{d}\sigma_{3} \tag{40}\]  
If the strain increment \(\mathrm{d}\varepsilon\mathrm{d}\backslash\mathrm{varepsilom}\mathrm{d}\varepsilon\) calculated using the elastic flexibility matrix in Equation (34) indicates loading, the total flexibility matrix, which includes both the elastic and plastic flexibility matrices, should be used to recalculate the stress-strain increment in Equation (34). If the strain increment calculated using only the elastic flexibility matrix indicates unloading, the calculation is considered correct, and the next step of elastic prediction can proceed.