# Creep Behavior of Rocks and Its Application to the Long-Term Stability of Deep Rock Tunnels
## Abbreviations  
VOFCM Variable-order fractional creep model  
IEFV Improved elastic element Fractional variable  
BDT Brazilian disk test  
FENLCM Five-element non-linear creep damage model  
HFD Hadamard fractional derivative  
CTVDE Coupled thermo-visco-plastic damage element  
NLVEDDM Non-linear visco-elastic-plastic damage model  
VEPCM Visco-elasto-plastic creep model  
GKM Generalized Kelvin model  
NLVPCM Non-linear viscoelastic-plastic creep model  
EVPD Visco-elastic visco-plastic with isotropic damage  
UCC Ubiquitous corrosion Coulomb  
FDCD Fractional derivative creep damage  
**List of symbols**  
\(P_{w}\) Pore water pressure  
\(\sigma_{0}\) Initial stress; constant stress; deviatoric stress  
\begin{tabular}{l l} \(\sigma;\sigma_{1}\,;\,\sigma_{3}\) & Applied stress \\ \(\sigma_{1}\,;\,\sigma_{2}\); \(\sigma_{3}\) & Respectively 1 st, 2 nd, and 3 rd principal stress \\ \(\tilde{\sigma};\tilde{\sigma}\) & Respectively second and first partial derivatives of the stress \\ \(\Delta\sigma\) & Stress increment \\ \(\tau_{i}\,\tau_{t_{0}}^{\beta_{c}}\) & Shear stress; initial shear stress at time \(t_{0}\) \\ \(\tau_{t_{i}}^{\beta_{c}}\) & Critical shear stress at critical time \(t_{c}\) \\ \(V_{t}^{\beta_{c}}\) & Increase rate of the shear stress \\ \(\sigma_{s};\,\sigma_{s}^{\prime}\) & Yield stress; creep strength \\ \(S_{ij};e_{ij}\) & Deviatoric stress tensor \\ \(\left(\hat{S}_{ij}\right)_{0}\) & Constant value of the deviatoric stress tensor \\ \(S_{be}\) & Three-dimensional creep threshold \\ \(S_{s};\,S_{\gamma}\) & Long-term strength \\ \(f_{2}^{\
u p}\) & Second invariant of the deviatoric visco-plastic strain tensor \\ \(f_{2}^{\
u p}\) & Onset of tertiary creep expressed by the second invariant \\ \(\sigma_{m}\) & Main stress tensor; spherical stress tensor \\ \(\delta_{ij}\) & Kronecker function \\ \(F\) & Yield function \\ \(\varepsilon_{0}\) & Strain before accelerating creep \\ \(\varepsilon(t)\); \(\varepsilon_{11}(t)\); \(u\); \(\varepsilon_{n}\) & Creep strain \\ \(\varepsilon_{ij}(t)\) & 3D creep strain \\ \(\tilde{\varepsilon}\); \(\dot{\varepsilon}\) & Respectively second and first partial derivatives of the creep strain \\ \(t\) & Creep time \\ \(t_{1}\) & Creep time; unloading time \\ \(t_{n}\) & n-th level charge start time \\ \(\eta_{1}\); \(\eta_{2}\); \(\eta_{3}\); \(\eta\); \(n_{K}\); \(\eta_{m}\); \(\eta_{sc}\) & Viscosity coefficient of rocks \\ \(\eta_{1}(T)\); \(\eta_{2}(T)\) & Viscosity coefficient at temperature \(T\) \\ \(G_{0}\) & Bulk modulus \\ \(G_{1}\) & Elastic shear modulus, viscoelastic shear modulus \\ \(G_{2}\) & Viscoelastic shear modulus \\ \(G_{K}\) & Shear modulus \\ \(E_{0}\) & Elastic modulus, viscoelasticity modulus; shear modulus \\ \(E_{1}\); \(E\); \(E_{k}\); \(E_{M}\) & Elastic modulus of rocks \\ \(E_{0}\) & Elastic, or viscoelasticity modulus of rocks \\ \(E_{0}(T)\); \(E_{1}(T)\) & Elastic modulus at temperature \(T\) \\ \(n_{0}\) & Accelerating creep exponent with no water pressure \\ \(a\); \(b\); \(\varphi\); \(B\); \(A_{sc}\) & fitting parameters \\ C & Fitting parameter that represents the duration of stress; \\  & internal frictional angle \\ \(t_{0}\) & Transition time between the 2 nd and 3 rd creep phases \\ \(\Gamma\) & Gamma function \\ \(\alpha\) & Parameter controlling the damage evolution with time, \\ \(\mu\); \(\gamma_{t}\); \(\gamma_{1}\); \(\gamma_{2}\) & Fractional order derivative \\ \(m\); \(n\); \(\beta\) & Constants related to the materials \\ \(n\) & Tertiary creep factor \((n\geq 1)\) \\ \(C(\sigma_{1}\,\sigma_{3}\), \(\phi\) ) & Additional criterion for tertiary creep \\ \(\lambda\); \(H_{1}\); \(H_{2}\); \(H_{3}\) & Viscous parameters \\ \(D_{ini}\) & Initial damage variable \\ \(e^{*}\) & Initial strain value of the tertiary creep \\ \(K\); \(\sigma_{a}\); \(\sigma_{b}\); \(A\) & Creep parameters \\ \(h\); \(p\); \(q\) & Constants \\ \(\theta\) & Relaxation time of materials \\ \end{tabular}