## 2 Non-Equilibrium Thermodynamic-Based Model Framework
### Energy Dissipation at a Mesoscopic Level  
Entropy is similar to conserved quantities and has superposition and non-zero properties. However, the difference is that it can be produced, also known as entropy production. It is found that the behavior of entropy is also effective when deviating from thermodynamic equilibrium [30]. Thus, the total entropy balance equation can be written as:  
\[\sum_{a=S,L,G}\left(\rho^{a}\dot{\vartheta}_{a}\right)=R/T+\
abla_{k}f_{k}-\rho ^{L}v_{k}^{LS}\
abla_{k}\vartheta_{L}-\rho^{G}v_{k}^{GS}\
abla_{k}\vartheta_{G} \tag{4}\]  
where \(\vartheta=S/\rho\) is specific entropy. Different to the irregular motion of micro molecules, the interaction between particles (fluctuation motion) is generally non-elastic, which will lead to energy dissipation [34]. If there is no sustained excitation, the fluctuation motion of particles will decay in the form of macro energy dissipation until the fluctuation disappears and reaches equilibrium again. As shown in Figure 1, the dissipation process of geotechnical materials will produce granular entropy \(S_{g}\) and thermal entropy \(S\). Since the interaction between particles is non-elastic, entropy production will be generated along with it and finally granular entropy \(S_{g}\) will decay to thermal entropy \(S\). The granular entropy balance equation can be obtained by analogy in Equation (4)  
\[\rho^{S}\dot{\beta}_{g}=R_{g}/T_{g}-I_{g} \tag{5}\]  
where \(\delta_{g}=S_{g}/\rho^{S}\) is specific granular entropy, \(R_{g}\) is granular entropy production, and \(I_{g}\) is the granular entropy decay rate, which is the entropy production of the system caused by the granular temperature. For unsaturated geotechnical materials containing water and gas, the change of saturation will change the attraction force and cementation between particles, which will cause the recombination movement between particles [17; 35], such as the wetting collapsibility phenomenon. Therefore, the change rate of saturation is also a driving force of particle fluctuation, which should be attributed to the dissipation mechanism at the mesoscopic level. Thus, the change rate of saturation \(\dot{s}_{r}\) can be regarded as a dissipative force, and the suction \(s\) is the corresponding dissipative flow. Considering the viscosity at the mesoscopic level and the dissipation caused by saturation change, the granular entropy production rate \(R_{g}\) is expressed as:  
\[R_{g}=\sigma_{ij}^{pg}\dot{s}_{ij}+\dot{s}T_{g}\dot{S}_{r} \tag{6}\]  
where \(\dot{s}T_{g}\dot{S}_{r}\) is the dissipation at the mesoscopic level due to saturation changes, and when \(T_{g}\)=0, that is, no dissipation occurs at the mesoscopic level, \(\dot{s}T_{g}\dot{S}_{r}\) also disappears (multiplied by \(T_{g}\) to reflect this phenomenon). Similarly, the viscous dissipation relationship at the mesoscopic level can also be expressed as:  
\[\sigma_{ij}^{pg}=T_{g}\lambda_{ijkl}^{S}\dot{\varepsilon}_{kl} \tag{7}\]  
where \(\sigma_{ij}^{pg}\) is the viscous stress and \(\lambda_{ijkl}^{S}\) is the migration coefficient at the mesoscopic level. Additionally, when \(T_{g}=0\), there is no dissipation at the mesoscopic level and the viscous dissipation disappears. The energy density associated with the fluctuation motion of the particles \(w_{g}\) is expressed as \(w_{g}=b\rho^{S}\big{(}T_{g}\big{)}^{2}/2\), where \(b\) is the material parameter [29]. According to Equation (1), there is  
\[\rho^{S}\delta_{g}=\frac{\partial w_{g}}{\partial T_{g}}=\rho^{S}bT_{g} \tag{8}\]  
Figure 1: The two-stage irreversible dissipation processes of the granular system. Thus, \(\delta_{g}=bT_{g}\) can be obtained. Similarly, defining the tensor \(\lambda_{ijkl}^{g}\) as  
\[\lambda_{ijkl}^{g}=\lambda_{sg}\delta_{il}\delta_{jk}+\left(\lambda_{eg}-\frac{ \lambda_{sg}}{3}\right)\delta_{ij}\delta_{lk} \tag{9}\]  
where \(\lambda_{sg}\) and \(\lambda_{eg}\) are the migration coefficients at the mesoscopic level.