Visco-plastic response of deep tunnels based on a fractional damage creep constitutive model

Abstract

Hard rock tunnels under high geo-stresses, and weak, soft rock tunnels show evident continued deformation after excavation, which is closely associated with the time-dependent behavior of rocks. In this paper, a novel fractional damage visco-plastic model was put forward to describe the creep response of rocks with the following elements: (1) an Abel dashpot, (2) a damaged Abel dashpot coupled with damage formulation that is based on a statistical distribution of microfractures, (3) elastic spring, and (4) Hoek–Brown plastic element. Firstly, the creep equation of the visco-plastic model was derived and validated against experimental data. Secondly, a closed-form analytical solution for the creep deformation of the surrounding rock around deep, circular tunnels was obtained by adopting the proposed model. Then, parametric studies were conducted to reveal the influence of the time-dependent parameters on the deformation of surrounding rocks. Finally, laboratory tests were conducted to provide data to validate the model. The auxiliary tunnel of the **** II hydropower station was chosen to demonstrate the analytical solution’s applicability to real-world problems. The results showed that: (1) the proposed constitutive model can adequately reflect the primary, secondary and tertiary creep stages of rocks; (2) the tunnel deformation increases as the Geological Strength Index (GSI) value in the Hoek–Brown model decreases, and for each time-dependent parameter, its influence on the tunnel deformation is more evident in weak rock mass than that in rock mass with higher GSI; (3) all features regarding the relationship between the tunnel deformation and the parameters of surrounding rocks agree well with physical meanings of each parameter; and (4) the deformation curves of the analytical solution and laboratory and field tests are consistent with each other with respect to curve shape and the magnitude, indicating that the proposed analytical solution can be reliably used to predict and study the creep deformation of tunnels.

Appendix: Elasto-plastic stress field in circular tunnels with Hoek–Brown failure criterion for the rock mass

The derivation of the elastic and plastic stress fields of a circular tunnel with the surrounding rock satisfying the Hoek–Brown criterion has already been given by former researchers (Cai et al. [7]) as follows:

In the plastic zone, the stress equilibrium equation is:

$$ \frac{{{\text{d}}\sigma_{r}^{{\text{p}}} }}{{{\text{d}}r}} + \frac{{\sigma_{r}^{{\text{p}}} - \sigma_{\theta }^{{\text{p}}} }}{r} = 0 $$

(45)

where σ_r^p, σ_θ^p are the radial stress and tangential stress in the plastic zone, respectively; r is the distance to the central point of the tunnel section.

Substituting Eq. 27 into Eq. (45) and $\sigma_{1} = \sigma_{\theta }$, $\sigma_{3} = \sigma_{r}$, $\sigma_{{r|r = R_{0} }}^{{\text{p}}} = 0$:

$$ \int_{0}^{{{\sigma_{r}{\text{p}}} }} {\frac{{{\text{d}}\sigma_{r}^{{\text{p}}} }}{{\sqrt {m\sigma_{{\text{c}}} \sigma_{r}^{{\text{p}}} + s\sigma_{{\text{c}}}^{2} } }}} = \int_{{R_{0} }}^{r} {\frac{{{\text{d}}r}}{r}} $$

(46)

Thus,

$$ \sigma_{r}^{{\text{p}}} = \frac{{m\sigma_{{\text{c}}} }}{4}\left( {{\text{In}}\frac{r}{{R_{0} }}} \right)^{2} + \sqrt s \sigma_{{\text{c}}} {\text{In}}\frac{r}{{R_{0} }} $$

(47)

$$ \sigma_{\theta }^{{\text{p}}} = \frac{{m\sigma_{{\text{c}}} }}{4}\left( {{\text{In}}\frac{r}{{R_{0} }}} \right)^{2} + \left( {\sqrt s \sigma_{{\text{c}}} { + }\frac{{m\sigma_{{\text{c}}} }}{2}} \right){\text{In}}\frac{r}{{R_{0} }} + \sqrt s \sigma_{{\text{c}}} $$

(48)

where R₀ is the radius of the tunnel, σ_c is the uniaxial compression strength of rock, m and s are material parameters related to the rock features.

Considering the influence of the change in plastic volume, the axial stress under plane strain conditions based on the plastic potential function (Yu et al. [40]) is:

$$ \sigma_{z}^{{\text{p}}} = \frac{{(1 + \sin \psi )\sigma_{\theta }^{{\text{p}}} + (1 - \sin \psi )\sigma_{r}^{{\text{p}}} }}{2} $$

(49)

where ψ is the dilation angle.

Substituting Eqs. (47) and (48) into Eq. (49) yields:

$$ \sigma_{z}^{{\text{p}}} = \frac{{m\sigma_{{\text{c}}} }}{4}\left( {\ln \frac{r}{{R_{0} }}} \right)^{2} + \left[ {s\sqrt {\sigma_{{\text{c}}} } + \frac{{m\sigma_{{\text{c}}} }}{4}(1 + \sin \psi )} \right]\ln \frac{r}{{R_{0} }} + \frac{1 + \sin \psi }{2}s\sqrt {\sigma_{{\text{c}}} } $$

(50)

In the elastic zone, it meets two boundary conditions, that is,

$$ \sigma_{{r|r = R_{{\text{p}}} }}^{{\text{e}}} = \sigma_{{r|r = R_{{\text{p}}} }}^{{\text{p}}} $$

(51)

$$ \sigma_{r|r \to \infty }^{{\text{e}}} = \sigma_{\theta |r \to \infty }^{{\text{e}}} = p_{0} $$

(52)

where R_p is the radius of the plastic zone.

The elastic stresses are:

$$ \sigma_{r}^{{\text{e}}} = p_{0} - \frac{{R_{{\text{p}}}^{2} }}{{r^{2} }}\left[ {p_{0} - \frac{{m\sigma_{{\text{c}}} }}{4}\left( {{\text{In}}\frac{{R_{{\text{p}}} }}{{R_{0} }}} \right)^{2} - \sqrt s \sigma_{{\text{c}}} {\text{In}}\frac{{R_{{\text{p}}} }}{{R_{0} }}} \right] $$

(53)

$$ \sigma_{\theta }^{{\text{e}}} = p_{0} + \frac{{R_{{\text{p}}}^{2} }}{{r^{2} }}\left[ {p_{0} - \frac{{m\sigma_{{\text{c}}} }}{4}\left( {{\text{In}}\frac{{R_{{\text{p}}} }}{{R_{0} }}} \right)^{2} - \sqrt s \sigma_{{\text{c}}} {\text{In}}\frac{{R_{{\text{p}}} }}{{R_{0} }}} \right] $$

(54)

where σ_r^e and σ_θ^e are the radial stress and tangential stress, respectively;

Based on the boundary condition: ${\sigma }_{\theta |r={R}_{p}}^{\mathrm{e}}={\sigma }_{\theta |r={R}_{p}}^{\mathrm{p}}$, the radius of the plastic zone can be obtained:

$$ R_{{\text{p}}} = R_{0} {\text{e}}^{{\text{M}}} $$

(55)

$$ M = \frac{{\sqrt {m^{2} + 16mp_{0} /\sigma_{{\text{c}}} + 16s} - 4\sqrt s - m}}{2m} $$

(56)

Substituting Eqs. (55) and (53) into Eq. (54) yields:

$$ \sigma_{r}^{{\text{e}}} = p_{0} - \frac{{R_{0}^{2} }}{{r^{2} }}N_{1} {\text{e}}^{{{\text{2M}}}} $$

(57)

$$ \sigma_{\theta }^{{\text{e}}} = p_{0} + \frac{{R_{0}^{2} }}{{r^{2} }}N_{1} {\text{e}}^{{{\text{2M}}}} $$

(58)

$$ N_{1} = p_{0} - \sqrt s \sigma_{{\text{c}}} M - \frac{{m\sigma_{{\text{c}}} }}{4}M^{2} $$

(59)

According to the generalized Hooke’s law (Hou and Niu [18]):

$$ \varepsilon_{z}^{{\text{e}}} = [\sigma_{z}^{{\text{e}}} - \nu (\sigma_{r}^{{\text{e}}} + \sigma_{\theta }^{{\text{e}}} )]/E $$

(60)

where ε_z^e is the axial strain, E is the elastic modulus and ν is the Poisson’s ratio.

Thus, in the original geo-stress field:

$$ p_{0} = \nu (p_{0} + p_{0} ) + E\varepsilon_{z}^{{\text{e}}} $$

(61)

In the elastic zone:

$$ \sigma_{z}^{{\text{e}}} = \nu (\sigma_{r}^{{\text{e}}} + \sigma_{\theta }^{{\text{e}}} ) + E\varepsilon_{z}^{{\text{e}}} $$

(62)

The difference between Eqs. (62) and (61) is:

$$ \sigma_{z}^{{\text{e}}} = \nu (\sigma_{r}^{{\text{e}}} + \sigma_{\theta }^{{\text{e}}} - 2p_{0} ) + p_{0} $$

(63)

Substituting Eqs. (57) and (48) into Eq. (63) yields (Hou and Niu [18]:

$$ \sigma_{z}^{{\text{e}}} = p_{0} $$

(64)