# Waterproof Performance of Sealing Gasket in Shield Tunnel: A Review
## 3 Rubber Material Characteristics
### Constitutive Model and Aging Mechanism  
The rubber material of gaskets is hyper-elastic, which means that its mechanical constitutive model cannot be described by the typical stress-strain curve. In continuum mechanics, for materials such as hyper-elastic rubber, the phenomenological theory can be used to determine the constitutive model of materials by obtaining the relationship between strain energy density and strain tensor invariants. Commonly used constitutive models include the Arruda-Boyce model, Mooney-Rivlin model (Equation (1)), neo-Hookean model, etc. \[\mathrm{W}=\mathrm{C}_{10}(\mathrm{I}_{1}-3)+\mathrm{C}_{01}(\mathrm{I}_{2}-3) \tag{1}\]  
where W is the strain energy density, \(\mathrm{C}_{10}\) and \(\mathrm{C}_{01}\) are constants, and \(\mathrm{I}_{1}\) and \(\mathrm{I}_{2}\) are the strain invariants. Due to its simple mathematical form and high simulation accuracy, the Mooney-Rivlin model is often implemented in gasket performance [37]. A constitutive model of hyperelastic rubber considering thermal-viscoelastic coupling (Equation(2)) and a rheological model based on non-Newtonian fluid (Equation(3)) were established based on different states of the rubber [38]. In terms of the thermal-mechanical aging model, [PERSON] [39] has been devoted to exploring the thermal-mechanical aging process of rubber and its relationship with property changes since the 1990s. Based on the Arrhenius formula (Equation (4)) and the Dakin prediction equation (Equation(5)), a P-T-tag model was proposed, considering the influence of temperature and time on performance deterioration (Equation(6)). \[\mathrm{\Psi}=-\frac{\mathrm{G}_{8}}{2}\frac{\mathrm{T}}{\mathrm{T}_{0}} \lambda_{\mathrm{m}}\left[1+\mathrm{b}\left(\frac{\mathrm{T}-\mathrm{T}_{0}}{ \mathrm{T}_{0}}\right)\right]\ln\left\{1-\frac{\mathrm{I}_{1}-3}{\lambda_{ \mathrm{m}}[1+\mathrm{b}(\mathrm{T}-\mathrm{T}_{0})/\mathrm{T}_{0}]}\right\}+ \mathrm{d}\left(\mathrm{T}-\mathrm{T}_{0}-\mathrm{T}\mathrm{ln}\frac{\mathrm{T }}{\mathrm{T}_{0}}\right) \tag{2}\]  
where \(\mathrm{G}_{8}\) is the isothermal infinitesimal shear modulus, \(\lambda_{\mathrm{m}}\) is the isothermal average maximum elongation, \(\mathrm{d}\) is the material constant, and \(\mathrm{b}\) is the temperature correlation factor of the maximum elongation. \[\sigma_{12}=\mathrm{Y}+\frac{\mathrm{A}\gamma}{\mathrm{B}\gamma^{1-\mathrm{n}}} \tag{3}\]  
where Y and B are constant, and \(\gamma\) stands for the shear distance of the non-Newtonian liquid. \[\mathrm{K}=\mathrm{A}_{0}\mathrm{e}^{-\frac{\mathrm{E}}{\mathrm{kT}}} \tag{4}\]  
where K is the chemical reaction rate, \(\mathrm{A}_{0}\) is the pre-exponential factor, E is the activation energy, R is the molar gas constant, and T is the thermodynamic temperature. \[\mathrm{f}_{\mathrm{t}}(\mathrm{P})=\mathrm{kt} \tag{5}\]  
where \(\mathrm{f}_{\mathrm{t}}(\mathrm{P})\) is the reaction rate function, \(k\) is the coefficient, and \(t\) is the reaction time. \[\mathrm{log}\left[-\mathrm{log}\frac{\mathrm{Y}}{\mathrm{B}}\right]=\mathrm{B} _{0}+\mathrm{B}_{1}/\mathrm{T}+\mathrm{B}_{2}\times\mathrm{logt} \tag{6}\]  
where \(\mathrm{y}=\mathrm{f}(\mathrm{P})\), \(\mathrm{B}_{0}=\mathrm{log}(\mathrm{A}_{0}/2.303)\), \(\mathrm{B}_{1}=-\mathrm{E}/2.303\), \(\mathrm{B}_{2}\) is a constant. Based on thermodynamics, the mathematical relationship among the prevailing predicting models: the kinetic curve method, linear relation method, variable reduction method, and response function method, were analyzed, and the advantages of the proposed three-element aging mathematical model (Equation (6)) in rubber service life prediction were proved. [PERSON] et al. [40] calculated the activation energy of the thermal-oxygen aging reaction process of phenyl ether silicone rubber by taking the critical value of rubber as the key index and establishing the relationship between the temperature and rubber aging prediction. [PERSON] et al.