## 4 Elastoplastic Damage Constitutive Model Considering High-Temperature Damage  
### Incremental Iterative Algorithm for Z-P Damage Constitutive Model  
The FLAC\({}^{\text{3D}}\) 5.0 software is applicable for numerical analysis that utilizes a three-dimensional explicit finite difference approach. This algorithm provides accurate simulation of the yield, plastic flow, softening, and large deformation behavior of geotechnical materials, particularly in elastoplastic analysis and analysis of materials undergoing significant deformation, and has unique advantages. Meanwhile, the simple process control language and modular design of FLAC\({}^{\text{3D}}\) provide users with a convenient secondary development interface, which enables users to develop corresponding calculation models as necessary and embed self-defined constitutive models into FLAC\({}^{\text{3D}}\). All constitutive models in FLAC\({}^{\text{3D}}\) basically follow the same algorithm. Specifically, by substituting parameters such as the stress state at a given time t and the strain increment at the \(\Delta t\) time step into the corresponding constitutive equation in FLAC\({}^{\text{3D}}\), the stress increment and stress state at time \(t+\Delta t\) can be obtained. If all the deformations that occur are elastic, the stress increment can be directly derived from it, but if plastic strain occurs, the assumed stress increment must be corrected to obtain a new stress increment and stress state. According to the incremental numerical algorithm in Flac\({}^{\text{3D}}\)[33], this section presents the incremental iterative scheme for the high-temperature-load coupled damage elastoplastic constitutive model, which is based on the Z-P yield criterion. The Z-P yield criterion is written in the following form after introduction of coupling damage:  
\[f=\tan\overline{\varphi}p+\frac{q}{g(\theta)}-(1-D_{c})\overline{c}=0 \tag{14}\]  
The plastic potential function using uncorrelated flow can be expressed as  
\[g=\tan\overline{\psi}p+\frac{q}{g*(\theta)}-(1-D_{c})\overline{c*}=0 \tag{15}\]  
where \(\tan\overline{\psi}\), \(g*(\theta)\), \(\overline{c*}\) are obtained by replacing internal friction angle \(\varphi\) in the expressions of \(\tan\overline{\varphi}\), \(g(\theta)\), \(\overline{c}\) with expansion angle \(\psi\), respectively. The generalized strain increment \(\Delta\varepsilon\) corresponding to p and q consists of two parts: the shear strain increment \(\Delta\gamma\) and the volumetric strain increment \(\Delta\varepsilon_{v}\):  
\[\Delta\gamma=\sqrt{2\Delta\varepsilon_{ij}\Delta\varepsilon_{ij}},\Delta \varepsilon_{v}=\Delta\varepsilon_{kk} \tag{16}\]  
where \(\Delta\varepsilon_{ij}=\Delta\varepsilon_{ij}-\frac{\Delta\varepsilon_{kk}}{3}\) denotes the deviatoric strain increment. The [PERSON]'s law equation, relating the elastic shear strain increment and the elastic volumetric strain increment, can be described as follows:  
\[\Delta q=G(D_{c})\Delta\gamma^{e},\Delta p=K(D_{c})\Delta\varepsilon_{v}^{e} \tag{17}\]  
The non-destructive shear modulus \(G_{0}\) and bulk modulus \(K_{0}\) can be employed to express the damage shear modulus \(G(D_{c})\) and bulk modulus \(K(D_{c})\):  
\[G(D_{c})=(1-D_{c})G_{0},K(D_{c})=(1-D_{c})K_{0} \tag{18}\]According to the plastic flow law, when considering shear failure, the increase in plastic shear strain \(\Delta\gamma^{p}\) and the increase in plastic volumetric strain \(\Delta\varepsilon_{v}{}^{p}\) can be expressed as  
\[\Delta\gamma^{p}=\lambda\frac{\partial g}{\partial q}=\frac{\lambda}{g*(\theta)},\Delta\varepsilon_{v}^{p}=\lambda\frac{\partial g}{\partial p}=\lambda\tan \overline{\psi} \tag{19}\]  
where \(\lambda\) is the plastic factor.