A new transversely isotropic nonlinear creep model for layered phyllite and its application

Abstract

Phyllite, which is a low-grade metamorphic rock with well-developed foliation planes, is encountered frequently during tunnel construction in western China. Its creep behavior is affected significantly by the foliation planes and has a crucial influence on the long-term safety of tunnel structures. Uniaxial compressive creep testing was conducted to analyze the time-dependent features of phyllite obtained from the Zhegu mountain tunnel on the Wenma expressway, China. A new creep model that connects a Maxwell body, a Kelvin body, and a nonlinear visco-plastic body was proposed to describe both the full creep process (including the transient, steady, and accelerated creep stages) and the transversely isotropic characteristics of phyllite. The creep model was also applied to investigate the long-term safety of a cracked tunnel lining in phyllite bedrock. The results showed that the creep strength and corresponding axial strain of phyllite exhibited maximum and minimum values at θ (the angle between the loading direction and the weak planes) = 90° and 30°, respectively. Good agreement was found between the calculated and experimental creep curves, indicating that the creep model replicates the physical creep process of phyllite well. The safety of the cracked lining was affected mainly by the damage degree of cracks and the creep behavior of the surrounding rock. Uncracked sections, because of their greater stiffness, were more sensitive to creep load than cracked ones. The inclination angle of foliation planes influenced the location of unsafe sections (those with a safety factor less than one), and this effect was weakened as the number of pre-existing cracks increased.

Appendices

Appendix 1: Implementation of ubhm model in FLAC 3D

The three-dimensional finite difference formulas of ubhm is derived as as follows.

• (1) Basic formulas

The incremental expression of Eq. 3 has the form:

$$ \varDelta {e}_{ij}=\varDelta {e}_{ij}^1+\varDelta {e}_{ij}^2+\varDelta {e}_{ij}^3 $$

(27)

$$ \varDelta {e}_{ij}^1=\frac{\varDelta {S}_{ij}}{2{G}_1}+\frac{{\overline{S}}_{ij}}{2{H}_1}\varDelta t $$

(28)

$$ {\overline{S}}_{ij}=2{H}_2\varDelta {e}_{ij}^2+2{G}_2{\overline{e}}_{ij}^2\varDelta t $$

(29)

$$ {\dot{\varepsilon}}_{ij}^3=\frac{H(f)}{2{H}_3}\frac{\partial g}{\partial {\sigma}_{ij}}\varDelta t $$

(30)

where

$$ {\overline{S}}_{ij}=\frac{{\overline{S}}_{ij}^N+{\overline{S}}_{ij}^O}{2} $$

(31)

$$ {\overline{e}}_{ij}^2=\frac{e_{ij}^{2,N}+{e}_{ij}^{2,O}}{2} $$

(32)

The superscripts N and O denote new and old values during a time step, respectively. ${\overline{S}}_{ij}^o$ and ${\overline{S}}_{ij}^N$ are the new and old deviatoric stress tensors, respectively. ${e}_{ij}^{2,N}$ and ${e}_{ij}^{2,O}$ are the new and old deviatoric strain tensors, respectively.

Substituting Eqs. (31) and (32) into Eq. (29) yields:

$$ {e}_{ij}^{2,N}=\frac{1}{A}\left[B{e}_{ij}^{2,O}+\frac{\varDelta t}{4{H}_2}\left({S}_{ij}^N+{S}_{ij}^O\right)\right] $$

(33)

where

$$ A=1+\frac{G_2\varDelta t}{2{H}_2} $$

(34)

$$ B=1-\frac{G_2\varDelta t}{2{H}_2} $$

(35)

Substituting Eqs. (28) and (33) into Eq. (27) yields:

$$ {S}_{ij}^N=\frac{1}{a}\left[\varDelta {e}_{ij}-\varDelta {e}_{ij}^{3\mathrm{J}}-\varDelta {e}_{ij}^{3S}+b{S}_{ij}^O-\left(\frac{B}{A}-1\right){e}_{ij}^{2,O}\right] $$

(36)

where

$$ a=\frac{1}{2{G}_1}+\frac{\varDelta t}{4}\left(\frac{1}{H_1}+\frac{1}{A{H}_2}\right) $$

(37)

$$ b=\frac{1}{2{G}_1}-\frac{\varDelta t}{4}\left(\frac{1}{H_1}+\frac{1}{A{H}_2}\right) $$

(38)

$\varDelta {e}_{ij}^{3\mathrm{J}}$, $\varDelta {e}_{ij}^{3S}$are the plastic strain of weak planes and rock matrix, respectively.

The volumetric strain is:

$$ {\sigma}_0^N={\sigma}_0^O+K\left(\varDelta {e}_{vol}-\varDelta {e}_{vol}^{3\mathrm{J}}-\varDelta {e}_{vol}^{3\mathrm{S}}\right) $$

(39)

The Mohr-Coulomb criterion of rock matrix are:

$$ {f}^{\mathrm{s}}={\sigma}_1-{\sigma}_3{N}_{\varphi }+2c\sqrt{N_{\varphi }} $$

(40)

$$ {f}^{\mathrm{t}}={\sigma}_{\mathrm{t}}-{\sigma}_3 $$

(41)

where f^s is shear yield criterion, f^t is tensile yield criterion, N_φ = (1 + sinφ)/(1-sinφ), c, φ, σ_t are cohesion, friction angle, and tensile strength of rock matrix, respectively.

The corresponding potential functions are:

$$ {g}^{\mathrm{s}}={\sigma}_1-{\sigma}_3{N}_{\psi } $$

(42)

$$ {g}^{\mathrm{t}}=-{\sigma}_3 $$

(43)

where g^s is shear potential function and g^t is tensile potential function, N_ψ = (1 + sinψ)/(1-sinψ), ψ is dilation angle.

The function h(σ₁, σ₃) = 0, which is represented by the diagonal between the strength envelope of f^s = 0 and f^t = 0 in the principal stress plane (see Fig. 23), is defined to determine the yield type of rock matrix:

$$ h={\sigma}_3-{\sigma}_t+{\alpha}^p\left({\sigma}_1-{\sigma}^p\right) $$

(44)

where

$$ {\alpha}^p=\sqrt{1+{N}_{\varphi}^2}+{N}_{\varphi } $$

(45)

$$ {\sigma}^p={\sigma}_{\mathrm{t}}{N}_{\varphi }-2c\sqrt{N_{\varphi }} $$

(46)

Fig. 23

Definition of h and domains used in determining yield type of rock matrix

If the stress falls within domain 1, then shear failure occurs, and the new stress is revised using the flow rule derived from g^s. If the stress falls within domain 2, then tensile failure occurs, and the new stress is re-calculated adopting the flow rule derived from g^t.

• (2) Strain increment of rock matrix

Expressing Eqs. (33) and (39) in principal axes, the definition of trial stresses can be written as follows:

$$ {S}_i^N={\widehat{S}}_i^N-\frac{1}{a}\varDelta {e}_i^3 $$

(47)

$$ {\sigma}_0^N={\widehat{\sigma}}_0^N- K\varDelta {\varepsilon}_{vol}^3 $$

(48)

Adding (47) and (48) and expressing the result in principal axes yields:

$$ \Big\{{\displaystyle \begin{array}{l}{\sigma}_1^N={\widehat{\sigma}}_1^N-\left[{\alpha}_1\varDelta {\varepsilon}_1^3+{\alpha}_2\left(\varDelta {\varepsilon}_2^3+\varDelta {\varepsilon}_3^3\right)\right]\\ {}{\sigma}_2^N={\widehat{\sigma}}_2^N-\left[{\alpha}_1\varDelta {\varepsilon}_2^3+{\alpha}_2\left(\varDelta {\varepsilon}_1^3+\varDelta {\varepsilon}_3^3\right)\right]\\ {}{\sigma}_3^N={\widehat{\sigma}}_3^N-\left[{\alpha}_1\varDelta {\varepsilon}_3^3+{\alpha}_2\left(\varDelta {\varepsilon}_1^3+\varDelta {\varepsilon}_2^3\right)\right]\end{array}} $$

(49)

where

$$ {\alpha}_1=K+\frac{2}{3a} $$

(50)

$$ {\alpha}_2=K-\frac{1}{3a} $$

(51)

For shear failure, partial differentiation of Eq. (42):

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\mathrm{g}}^{\mathrm{s}}}{\partial {\sigma}_1}=1\\ {}\frac{\partial {\mathrm{g}}^{\mathrm{s}}}{\partial {\sigma}_2}=0\\ {}\frac{\partial {\mathrm{g}}^{\mathrm{s}}}{\partial {\sigma}_3}=-{N}_{\psi}\end{array}} $$

(52)

Substituting Eq. (52) into Eq. (49) yields:

$$ \Big\{{\displaystyle \begin{array}{l}{\sigma}_1^N={\widehat{\sigma}}_1^N-{\lambda}^{\ast}\left[{\alpha}_1-{\alpha}_2{N}_{\psi}\right]\\ {}{\sigma}_2^N={\widehat{\sigma}}_2^N-{\lambda}^{\ast }{\alpha}_2\left(1-{N}_{\psi}\right)\\ {}{\sigma}_3^N={\widehat{\sigma}}_3^N-{\lambda}^{\ast}\left({\alpha}_2-{\alpha}_1{N}_{\psi}\right)\end{array}} $$

(53)

where

$$ {\lambda}^{\ast }=\frac{f^s}{2{H}_3}\left[\left(n-1\right)a{t}^{n-2}-\frac{b}{t^2}\right]\varDelta t $$

(54)

For tensile failure, partial differentiation of Eq. (43):

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\mathrm{g}}^{\mathrm{t}}}{\partial {\sigma}_1}=0\\ {}\frac{\partial {\mathrm{g}}^{\mathrm{t}}}{\partial {\sigma}_2}=0\\ {}\frac{\partial {\mathrm{g}}^{\mathrm{t}}}{\partial {\sigma}_3}=-1\end{array}} $$

(55)

Substituting Eq. (55) into Eq. (49) yields:

$$ \Big\{{\displaystyle \begin{array}{l}{\sigma}_1^N={\widehat{\sigma}}_1^N-{\lambda}^{\ast }{\alpha}_2\\ {}{\sigma}_2^N={\widehat{\sigma}}_2^N-{\lambda}^{\ast }{\alpha}_2\\ {}{\sigma}_3^N={\widehat{\sigma}}_3^N-{\lambda}^{\ast }{\alpha}_1\end{array}} $$

(56)

where

$$ {\lambda}^{\ast }=\frac{f^{\mathrm{t}}}{2{H}_3}\left[\left(n-1\right)a{t}^{n-2}-\frac{b}{t^2}\right]\varDelta t $$

(57)

• (3) Strain increment of weak planes

The relationship between the global coordinate system and local system is shown in Fig. 24. The stress tensors in the local system are computed adopting Eq. (58):

$$ {\sigma}^{\hbox{'}}={C}^T\sigma C $$

(58)

where C is transformation matrix (Manh et al. 2015).

Fig. 24

Coordinate systems of weak planes

Fig. 25

Yield criterion of weak planes

The Mohr-Coulomb criteria of weak planes are:

$$ {f}_{\mathrm{j}}^{\mathrm{s}}=\tau -{\sigma}_{\mathrm{n}}\tan {\varphi}_{\mathrm{j}}-{c}_{\mathrm{j}} $$

(59)

$$ {f}_{\mathrm{j}}^{\mathrm{t}}={\sigma}_{\mathrm{n}}-{\sigma}_{\mathrm{j}}^{\mathrm{t}};{\sigma}_{\mathrm{j}\mathrm{max}}^{\mathrm{t}}=\raisebox{1ex}{${c}_{\mathrm{j}}$}\!\left/ \!\raisebox{-1ex}{$\tan {\varphi}_{\mathrm{j}}$}\right. $$

(60)

where ${f}_{\mathrm{j}}^{\mathrm{s}}$ is shear yield criterion, ${f}_{\mathrm{j}}^{\mathrm{t}}$ is tensile yield criterion, c_j, φ_j, and σ_j^t are cohesion, friction angle, and tensile strength of weak planes, respectively; σ^t_jmax is the maximum tensile strength for a weak plane with nonzero friction angle; σ_n is normal stress.

The corresponding potential functions are:

$$ {g}_{\mathrm{j}}^{\mathrm{s}}=\tau -{\sigma}_{\mathrm{n}}\tan {\psi}_{\mathrm{j}} $$

(61)

$$ {g}_{\mathrm{j}}^{\mathrm{t}}=-{\sigma}_{\mathrm{n}} $$

(62)

where g_j^s is shear potential function, g_j^t is tensile potential function, and Ψ_j is dilation angle of weak planes.

The function h_j(σ₁, σ₃) = 0, which is represented by the diagonal between the strength envelope of f_j^s = 0 and f_j^t = 0 in the principal stress plane (see Fig. 26), is defined:

$$ {h}_{\mathrm{j}}=\tau -{\tau}_{\mathrm{j}}^{\mathrm{p}}-{\alpha}_{\mathrm{j}}^{\mathrm{p}}\left({\sigma}_{\mathrm{n}}-{\sigma}_{\mathrm{j}}^{\mathrm{t}}\right) $$

(63)

where

$$ {\tau}_{\mathrm{j}}^{\mathrm{p}}={c}_{\mathrm{j}}-\tan {\varphi}_{\mathrm{j}}{\sigma}_{\mathrm{j}}^{\mathrm{t}} $$

(64)

$$ {\alpha}_{\mathrm{j}}^{\mathrm{p}}=\sqrt{1+\tan {\varphi}_{\mathrm{j}}^2}-\tan {\varphi}_{\mathrm{j}} $$

(65)

Fig. 26

Definition of h_j and domains used in determining yield type of weak planes

Fig. 27

Flow chart of the implementation of ubhm in FLAC3D

If the stress falls within domain 1, then shear failure occurs, and the new stress is revised using the flow rule derived from g_j^s. If the stress falls within domain 2, then tensile failure occurs, and the new stress is re-calculated adopting the flow rule derived from g_j^t.

For shear failure, partial differentiation of Eq. (61):

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\mathrm{g}}_{\mathrm{j}}^{\mathrm{s}}}{\partial {\sigma}_{1^{\hbox{'}}{1}^{\hbox{'}}}}=0\\ {}\frac{\partial {\mathrm{g}}_{\mathrm{j}}^{\mathrm{s}}}{\partial {\sigma}_{2^{\hbox{'}}{2}^{\hbox{'}}}}=0\\ {}\frac{\partial {\mathrm{g}}_{\mathrm{j}}^{\mathrm{s}}}{\partial {\sigma}_{3^{\hbox{'}}{3}^{\hbox{'}}}}=\tan {\psi}_j\\ {}\frac{\partial {\mathrm{g}}_{\mathrm{j}}^{\mathrm{s}}}{\partial \tau }=1\end{array}} $$

(66)

Substituting Eq. (66) into Eq. (49) yields:

$$ \Big\{{\displaystyle \begin{array}{l}{\sigma}_{1^{\hbox{'}}{1}^{\hbox{'}}}^N={\sigma}_{1^{\hbox{'}}{1}^{\hbox{'}}}^O-{\lambda}^s{\alpha}_2\tan {\psi}_{\mathrm{j}}\\ {}{\sigma}_{2^{\hbox{'}}{2}^{\hbox{'}}}^N={\sigma}_{2^{\hbox{'}}{2}^{\hbox{'}}}^O-{\lambda}^s{\alpha}_2\tan {\psi}_{\mathrm{j}}\\ {}{\sigma}_{3^{\hbox{'}}{3}^{\hbox{'}}}^N={\sigma}_{3^{\hbox{'}}{3}^{\hbox{'}}}^O-{\lambda}^s{\alpha}_1\tan {\psi}_{\mathrm{j}}\\ {}{\tau}^N={\tau}^O-2{\lambda}^s{G}_1\end{array}} $$

(67)

where

$$ {\lambda}^{\mathrm{s}}=\frac{f_{\mathrm{j}}^{\mathrm{s}}}{2{G}_1}+\frac{f_{\mathrm{j}}^{\mathrm{s}}}{2{H}_3}\left[\left(n-1\right)a{t}^{n-2}-\frac{b}{t^2}\right]\varDelta t $$

(68)

For tensile failure, partial differentiation of Eq. (62):

$$ \Big\{{\displaystyle \begin{array}{l}\frac{\partial {\mathrm{g}}_{\mathrm{j}}^{\mathrm{t}}}{\partial {\sigma}_{1^{\hbox{'}}{1}^{\hbox{'}}}}=0\\ {}\frac{\partial {\mathrm{g}}_{\mathrm{j}}^{\mathrm{t}}}{\partial {\sigma}_{2^{\hbox{'}}{2}^{\hbox{'}}}}=0\\ {}\frac{\partial {\mathrm{g}}_{\mathrm{j}}^{\mathrm{t}}}{\partial {\sigma}_{3^{\hbox{'}}{3}^{\hbox{'}}}}=1\\ {}\frac{\partial {\mathrm{g}}_{\mathrm{j}}^{\mathrm{t}}}{\partial \tau }=0\end{array}} $$

(69)

Substituting Eq. (69) into Eq. (49) yields:

$$ \Big\{{\displaystyle \begin{array}{l}{\sigma}_{1^{\hbox{'}}{1}^{\hbox{'}}}^N={\sigma}_{1^{\hbox{'}}{1}^{\hbox{'}}}^O-{\lambda}^t{\alpha}_2\\ {}{\sigma}_{2^{\hbox{'}}{2}^{\hbox{'}}}^N={\sigma}_{2^{\hbox{'}}{2}^{\hbox{'}}}^O-{\lambda}^t{\alpha}_2\\ {}{\sigma}_{3^{\hbox{'}}{3}^{\hbox{'}}}^N={\sigma}_{3^{\hbox{'}}{3}^{\hbox{'}}}^O-{\lambda}^t{\alpha}_1\end{array}} $$

(70)

where

$$ {\lambda}^t=\frac{f_{\mathrm{j}}^{\mathrm{t}}}{2{H}_3}\left[\left(n-1\right)a{t}^{n-2}-\frac{b}{t^2}\right]\varDelta t $$

(71)

Appendix 2: Safety factor of secondary lining

The safety factor is the most intuitive index to evaluate the safety of lining. We adopted the calculation method recommended in the design specification of highway tunnels (Ministry of Transport of PRC 2004):

• (1) For e₀ ≤ 0.2 h, the bearing capacity of lining is controlled by the compressive strength of concrete. Thus, its safety factor can be calculated using Eq. (72)

$$ K=\frac{\phi \alpha {R}_a bh}{N} $$

(72)

where K is the safety factor, R_a is the compressive strength, N is axial force, b is the width of section, h is the height of section, φ is the longitudinal bending coefficient of lining, and α is the influential coefficient of eccentricity:

$$ \alpha =1+0.648\left({e}_0/h\right)-12.569{\left({e}_0/h\right)}^2+15.444{\left({e}_0/h\right)}^3 $$

(73)

where

$$ {\mathrm{e}}_0=\mathrm{N}/\mathrm{M} $$

(74)

e₀ is eccentricity of the axial force, and M is the bending moment.

• (2) For e₀ > 0.2 h, the bearing capacity of lining is controlled by the tensile strength of concrete. Thus, its safety factor can be calculated using Eq. (75):

$$ K=\varphi \frac{1.75{R}_l bh}{\frac{6{e}_0}{h}-1}\times \frac{1}{N} $$

(75)

where R_l is the tensile strength of concrete.