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Coupling Analysis for Rock Mass Supported with CMC or CFC Rockbolts Based on Viscoelastic Method

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Abstract

Rockbolts have been widely used in rock reinforcement for high-stress conditions in mining and civil engineering. However, the interaction mechanism between the rockbolt and the rock mass is still unclear. To fully understand the coupling mechanism of a rock mass supported with rockbolts, this article studied the coupling effect and the time-dependent behavior of a rock mass supported with continuously mechanically coupled (CMC) or continuously frictionally coupled (CFC) rockbolts. The elastic solutions of the interaction model were obtained in the coupled state. In addition, viscoelastic analytical solutions were used to describe the rheological properties of the coupling model, and the solutions were acquired by setting the constitutive models of the rockbolt and rock mass to a one-dimensional Kelvin model and a three-dimensional Maxwell model based on the material properties. According to the proposed coupling model, the rock mass stress and displacement fields, and the rockbolt axial force strongly depend on the relative deformation modulus of the rock mass and rockbolt. In addition, a lower viscosity coefficient of the rockbolt or rock mass produces a larger rock mass displacement. Moreover, as the relative deformation modulus increases, the distance to the neutral point beyond the rockbolt head increases. Furthermore, the position of the neutral point is independent of time.

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Abbreviations

A b :

Cross-sectional area of the rockbolt

K :

Bulk modulus of the rock mass

ρ :

Radial coordinate

r :

Tunnel radius

d :

Rockbolt diameter

σ 0 :

Initial rock mass stress

\(u_{{\rho_{0} }}\) :

Rock mass displacement under the initial rock mass stress

\(\varepsilon_{{\theta_{0} }}\) :

Rock mass tangential strain under the initial rock mass stress

\(u_{{\rho_{2} }}^{\prime }\) :

Change of displacement in the unreinforced zone

\(\varepsilon_{{\rho_{2} }}^{\prime }\) :

Change of radial strain in the unreinforced zone

\(\sigma_{{\theta_{2} }}^{\prime }\) :

Change of tangential strain in the unreinforced zone

\(\sigma_{{\rho_{1} }}\) :

Radial stress in the reinforced zone

\(\sigma_{{\theta_{1} }}\) :

Tangential stress in the reinforced zone

\(\varepsilon_{{\rho_{1} }}\) :

Radial strain in the reinforced zone

\(\varepsilon_{{\theta_{1} }}\) :

Tangential strain in the reinforced zone

\(u_{{\rho_{1} }}\) :

Rock mass displacement in the reinforced zone

ε :

Axial strain of the rockbolt

S θ :

Rockbolt spacing in the tangential direction

t :

Time

G r :

Shear modulus of the rock mass

σ ij :

Stress tensor

T N :

Axial force of the rockbolt in the neutral point

T 1 :

Rockbolt axial force in front of the neutral point

\(\overline{{P_{{{\text{a}}K}} }}^{\prime } (s), \, \overline{{Q_{{{\text{a}}K}} }}^{\prime } (s)\) :

Operator function of the rockbolt viscoelastic constitutive model after Laplace transformation

τ a :

Additional shear stress beyond the neutral point

\(\tau_{{{\text{B}}_{ 1} }}\) :

Shear stress before the neutral point

p k, q k :

Constant parameters of the rockbolt material

E r :

Deformation modulus of the rock mass

E b :

Deformation modulus of the rockbolt

Θ :

Angular coordinate

R :

Radius of the reinforced zone

L :

The length of the rockbolt

μ r :

Poisson’s ratio of the rock mass

\(\varepsilon_{{\rho_{0} }}\) :

Rock mass radial strain under the initial rock mass stress

\(u_{{\rho_{1} }}^{\prime }\) :

Change of displacement in the reinforced zone

\(\varepsilon_{{\rho_{1} }}^{\prime }\) :

Change of radial strain in the reinforced zone

\(\varepsilon_{{\theta_{1} }}^{\prime }\) :

Change of tangential strain in the reinforced zone

c :

Distance from the concentrated force to the rockbolt end

\(\sigma_{{\rho_{2} }}\) :

Radial stress in the unreinforced zone

\(\sigma_{{\theta_{2} }}\) :

Tangential stress in the unreinforced zone

\(\varepsilon_{{\rho_{2} }}\) :

Radial strain in the unreinforced zone

\(\varepsilon_{{\theta_{2} }}\) :

Tangential strain in the unreinforced zone

\(u_{{\rho_{2} }}\) :

Radial displacement in the unreinforced zone

\(\varepsilon_{{\rho_{0} }}\) :

Initial strain of the rock mass under the initial stress

S z :

Rockbolt spacing in the longitudinal direction

σ b :

Axial stress of the rockbolt

u :

Displacement in the semi-infinite plane under the Mindlin solution

ε ij :

Strain tensor

ρ n :

The position of the neutral point

T 2 :

Rockbolt axial force beyond the neutral point

\(\begin{aligned} \overline{P}^{\prime } (s),\;\;\overline{Q}^{\prime } (s) \hfill \\ \overline{P}^{\prime \prime } (s),\;\;\overline{Q}^{\prime \prime } (s) \hfill \\ \end{aligned}\) :

Operator function of the rock mass viscoelastic constitutive model after Laplace transformation

τ b :

Shear stress caused by the rock mass deformation

\(\tau_{{{\text{B}}_{ 2} }}\) :

Shear stress beyond the neutral point

D :

Differential operator

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Acknowledgements

This study was supported by the National Natural Science Foundation of China (no. 51479108) and the Taishan Scholar Talent Team Support Plan for Advantaged & Unique Discipline Areas.

Author information

Author notes

  1. Gang Wang and Wei Han contributed to the work equally and should be regarded as co-first authors.

Authors and Affiliations

  1. Shandong Provincial Key Laboratory of Civil Engineering Disaster Prevention and Mitigation, Shandong University of Science and Technology, Qingdao, 266590, China

    Gang Wang, Wei Han & Ke Wang

  2. State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao, 266590, China

    Gang Wang, Yujing Jiang & Hengjie Luan

Authors
  1. Gang Wang
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  2. Wei Han
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Appendix

Appendix

The elastic solutions in the reinforced zone of the rock mass supported with CFC or CMC rockbolts are shown below.

The radial stress can be expressed as

$$\sigma_{{\rho_{1} }} = C_{2} \cdot \ln \rho + C_{3} \cdot \rho^{{\frac{{ - 2\xi \mu_{\text{r}}^{2} + ( - \xi + 2E_{\text{r}} )\mu_{\text{r}} + \xi - 2E_{\text{r}} }}{{2\xi \mu_{\text{r}}^{2} + (\xi - E_{\text{r}} )\mu_{\text{r}} - \xi + E_{\text{r}} }}}} \; + \;C_{1} .$$
(45)

The tangential stress can be written as

$$\begin{aligned} \sigma_{{\theta_{1} }} & = \frac{1}{{\left( {\left( {1 - \mu_{\text{r}} } \right)E_{\text{r}} + \xi \mu_{\text{r}} \left( {1 + \mu_{\text{r}} } \right)} \right)E_{\text{r}} }}C_{3} \cdot E_{\text{r}} \left( {1 - \mu_{\text{r}} } \right)\left( { - E_{\text{r}} + \xi \left( {1 + \mu_{\text{r}} } \right)} \right)\rho^{{\frac{{ - 2\xi \mu_{\text{r}}^{2} + ( - \xi + 2E_{\text{r}} )\mu_{\text{r}} + \xi - 2E_{\text{r}} }}{{2\xi \mu_{\text{r}}^{2} + (\xi - E_{\text{r}} )\mu_{\text{r}} - \xi + E_{\text{r}} }}}} \\ \, & \;\;\;{ + }\,\,2\left( {\frac{1}{2}C_{2} \cdot E_{\text{r}} \ln \rho + \frac{1}{2}\left( {C_{1} + C_{2} } \right)E_{\text{r}} + \xi \left( {\mu_{\text{r}} - \frac{1}{2}} \right)C_{2} \left( {1 + \mu_{\text{r}} } \right)} \right) \cdot (1 - \mu_{\text{r}} )E_{\text{r}} + \xi \mu_{\text{r}} (1 + \mu_{\text{r}} ). \\ \end{aligned}$$
(46)

The radial strain can be expressed as

$$\begin{aligned} \varepsilon_{{\rho_{1} }} & = - \frac{1}{{\left( {\xi \mu_{\text{r}}^{2} + (\xi + E_{\text{r}} )\mu_{\text{r}} - E_{\text{r}} } \right)E_{\text{r}}^{2} }}\left( {2\left( { - \frac{1}{2}C_{3} E_{\text{r}}^{2} \left( {\xi \mu_{\text{r}} - 1} \right)\rho^{{\frac{{ - 2\xi \mu_{\text{r}}^{2} + ( - \xi - 2E_{\text{r}} )\mu_{\text{r}} + \xi + 2E_{\text{r}} }}{{2\xi \mu_{\text{r}}^{2} + (\xi + E_{\text{r}} )\mu_{\text{r}} - \xi - E_{\text{r}} }}}} } \right)} \right. \\ & \;\; + \left( {\xi \mu_{\text{r}}^{2} + (\xi + E_{\text{r}} )\mu_{\text{r}} - E_{\text{r}} } \right)\left( {\left( {\mu_{\text{r}} - \frac{1}{2}} \right)E_{\text{r}} C_{2} \ln \rho - C_{2} \xi \mu_{\text{r}}^{3} + \frac{1}{2}C_{2} \xi \mu_{\text{r}}^{2} } \right. \\ & \;\; + \left( {\left( {\frac{3}{2}C_{2} + C_{1} } \right)E_{\text{r}} + C_{2} \xi } \right)\mu_{\text{r}} + \left( { - \frac{1}{2}C_{1} - C_{2} } \right)\left. {\left. {E_{\text{r}} - \frac{1}{2}C_{2} \xi } \right)} \right)\left. {\left( {1 + \mu_{r} } \right)} \right). \\ \end{aligned}$$
(47)

The tangential strain can be expressed as

$$\begin{aligned} \varepsilon_{{\theta_{1} }} & = - \frac{1}{{(\xi \mu_{\text{r}}^{2} + (\xi + E_{\text{r}} )\mu_{\text{r}} - E_{\text{r}} )E_{\text{r}}^{2} }}2(1 + \mu_{\text{r}} )\left( {E_{\text{r}} C_{3} \left( {\xi \mu_{\text{r}}^{2} + \frac{1}{2}\mu_{\text{r}} \left( {\xi + E_{\text{r}} } \right)} \right.} \right. \\ & \;\; - \frac{1}{2}\left( {E_{\text{r}} + \xi } \right)\rho^{{\frac{{ - 2\xi \mu_{\text{r}}^{2} + ( - \xi - 2E_{\text{r}} )\mu_{\text{r}} + \xi + 2E_{\text{r}} }}{{2\xi \mu_{\text{r}}^{2} + (\xi + E_{\text{r}} )\mu_{\text{r}} - \xi - E_{\text{r}} }}}} + \left( {\xi \mu_{\text{r}}^{2} + (\xi + E_{\text{r}} )\mu_{\text{r}} - E_{\text{r}} } \right)\left( {\left( {\mu_{\text{r}} - \frac{1}{2}} \right)E_{\text{r}} C_{2} \ln \rho } \right. \\ & \;\;\left. { - C_{2} \xi \mu_{\text{r}}^{3} + \frac{1}{2}C_{2} \xi \mu_{\text{r}}^{2} + \left( {\left( {\frac{1}{2}C_{2} + C_{1} } \right)E_{\text{r}} + C_{2} \xi } \right)\mu_{\text{r}} - \frac{1}{2}(C_{1} + C_{2} )E_{\text{r}} - \frac{1}{2}C_{2} \xi } \right). \\ \end{aligned}$$
(48)

The displacement can be written as

$$\begin{aligned} u_{{\rho_{1} }} & = - \frac{1}{{((\mu_{r} - 1)E_{\text{r}} + \xi \mu_{\text{r}} (1 + \mu_{\text{r}} ))E_{\text{r}}^{2} }}2\rho (E_{\text{r}} C_{3} ((\frac{1}{2}\mu_{\text{r}} - 1)E_{\text{r}} + \xi \left( {\mu_{\text{r}} - \frac{1}{2}} \right)(1 + \mu_{\text{r}} )) \\ & \quad \rho^{{\frac{{ - 2\xi \mu_{\text{r}}^{2} + ( - \xi - 2E_{\text{r}} )\mu_{\text{r}} + \xi + 2E_{\text{r}} }}{{2\xi \mu_{\text{r}}^{2} + (\xi + E_{\text{r}} )\mu_{r} - \xi - E_{\text{r}} }}}} + ((\mu_{\text{r}} - 1)E_{\text{r}} + \xi \mu_{\text{r}} (1 + \mu_{\text{r}} ))(\left( {\mu_{\text{r}} - \frac{1}{2}} \right)E_{\text{r}} C_{2} \ln \rho + \left( {\left( {\frac{1}{2}C_{2} + C_{1} } \right)\mu_{\text{r}} } \right. \\ & \quad \left. { - \frac{1}{2}C_{1} - \frac{1}{2}C_{2} } \right)E_{\text{r}} - \xi \left( {\mu_{\text{r}} - \frac{1}{2}} \right)(\mu_{\text{r}} - 1)(1 + \mu_{\text{r}} )C_{2} ))(1 + \mu_{\text{r}} ). \\ \end{aligned}$$
(49)

The values of the parameters A, C1, C2 and C3 in 45–49 are given below:

$$A = \frac{{\left( \begin{aligned} & 32(\frac{1}{2}r^{{\frac{{2E_{r} + \xi }}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} (\mu _{r} - \frac{1}{2})E_{r} ((\mu _{r}^{3} + \frac{1}{2}\mu _{r}^{2} - \frac{1}{2}\mu _{r} )\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} ) \\ & ((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} )\mu _{r}^{2} R^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} + r^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} R^{{\frac{{2E_{r} + \xi }}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} ((\mu _{r} - \frac{1}{2}) \\ & ((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} )\mu _{r}^{2} \ln \left( R \right) \\ & - (\mu _{r} - \frac{1}{2})((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} )\mu _{r}^{2} \ln \left( r \right) \\ & - \frac{1}{2}E_{r} ((\mu _{r} - \frac{1}{2})^{2} (1 + \mu _{r} )^{2} (\mu _{r}^{4} - \frac{1}{4}\mu _{r}^{3} - \frac{1}{4}\mu _{r}^{2} - \frac{1}{4}\mu _{r} + \frac{1}{4})\xi ^{2} + \frac{1}{2}(\mu _{r} - \frac{1}{2})(1 + \mu _{r} )^{2} (\mu _{r} - 1)E_{r} (\mu _{r}^{3} + \frac{1}{2}\mu _{r}^{2} - 2\mu _{r} + 1)\xi \\ & + \frac{1}{2}(\mu _{r} - 1)^{2} E_{r}^{2} (\mu _{r}^{3} - \frac{3}{4}\mu _{r}^{2} - \frac{1}{2}\mu _{r} + \frac{1}{2}))))\sigma _{0} R^{2} \\ \end{aligned} \right)}}{\begin{aligned} & 8r^{{\frac{{2E_{r} + \xi }}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} E_{r} ((\mu _{r}^{5} - \frac{3}{4}\mu _{r}^{3} + \frac{3}{4}\mu _{r}^{2} + \frac{1}{4}\mu _{r} - \frac{1}{4})\xi + (\mu _{r}^{3} - \frac{1}{2}\mu _{r}^{2} - \mu _{r} + 1)(\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} )R^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} \\ & + 8((\mu _{r}^{2} + 1)(\mu _{r} - \frac{1}{2})(1 + \mu _{r} )\xi + \mu _{r}^{2} E_{r} (\mu _{r} - 1))(((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )\ln \left( R \right) - ((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi \\ & + (\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )\ln \left( r \right) - \frac{1}{2}E_{r} ((\mu _{r}^{3} + \frac{1}{2}\mu _{r}^{2} - \frac{1}{2}\mu _{r} )\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} ))r^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} R^{{\frac{{2E_{r} + \xi }}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} \\ \end{aligned} },$$
$$C_{1} = \frac{{\left( \begin{gathered} 8( - 2r^{{\frac{{^{{2E_{{r + \xi }} }} }}{{2\xi \mu _{r} ^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} \left( { - \left( {\mu _{r} - \frac{1}{2}} \right)E_{r} \left( {\left( {\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2}} \right)\xi + (\mu _{r} - 1)E_{r} } \right)\mu _{r} ^{2} \ln (R) + (\mu _{{\text{r}}} - \frac{1}{2}} \right)^{2} (1 + \mu _{{\text{r}}} )^{2} (\mu _{{\text{r}}} - 1)\mu _{{\text{r}}} ^{2} \xi ^{2} \hfill \\ + \left( {\mu _{{\text{r}}} - \frac{1}{2}} \right)\left( {\mu _{r} ^{4} - \frac{7}{2}\mu _{r} ^{3} + \frac{7}{4}\mu _{r} ^{2} - \frac{1}{4}} \right)(1 + \mu _{r} )E_{r} \xi - (\mu _{r} - 1)E_{r} ^{2} \left( {\mu _{r} ^{3} - \frac{1}{2}\mu _{r} ^{2} - \frac{1}{2}\mu _{r} + \frac{1}{2}} \right))((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \frac{1}{2}(\mu _{{\text{r}}} - 1)E_{{\text{r}}} )R^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r} ^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} + \hfill \\ ((\mu _{r} ^{2} + 1)(\mu _{r} - \frac{1}{2})(1 + \mu _{r} )\xi + \mu _{r} ^{2} E_{r} (\mu _{r} - 1))(\xi (\mu _{r} - \frac{1}{2})(1 + \mu _{r} )((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} )\ln (R) - ((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} ) \hfill \\ ((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )\ln (r) - \frac{1}{2}\xi (1 + \mu _{r} )(\mu _{r} - 1)((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + 2E_{r} (\mu _{r} - \frac{3}{4})))r^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r} ^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} R^{{\frac{{^{{2E_{r} }} + \xi }}{{2\xi \mu _{r} ^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} )\sigma _{0} \hfill \\ \end{gathered} \right)}}{{\left( \begin{gathered} 8r^{{\frac{{^{{2E_{r} + \xi }} }}{{2\xi \mu _{r} ^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} E_{r} ((\mu _{r} ^{5} - \frac{3}{4}\mu _{r} ^{3} + \frac{3}{4}\mu _{r} ^{2} + \frac{1}{4}\mu _{r} - \frac{1}{4})\xi + (\mu _{r} ^{3} - \frac{1}{2}\mu _{r} ^{2} - \mu _{r} + 1)(\mu _{r} - 1)E_{r} )((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} )R^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r} ^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} + \hfill \\ 8((\mu _{r} ^{2} + 1)(\mu _{r} - \frac{1}{2})(1 + \mu _{r} )\xi + \mu _{r} ^{2} E_{r} (\mu _{r} - 1))(((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} )((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )\ln (R) - ((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} ) \hfill \\ ((\mu _{r} ^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )\ln (r) - \frac{1}{2}E_{r} ((\mu _{r} ^{3} + \frac{1}{2}\mu _{r} ^{2} - \frac{1}{2}\mu _{r} )\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} ))r^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r} ^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} R^{{\frac{{^{{2E_{r} + \xi }} }}{{2\xi \mu _{r} ^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} ) \hfill \\ \end{gathered} \right)}},$$
$$C_{2} = \frac{{ - \left( \begin{aligned} & 4E_{r} \sigma _{0} ((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} )(4r^{{\frac{{2E_{r} + \xi }}{{2\xi \mu _{r}^{2} + \left( {\xi + E_{r} } \right)\mu _{r} - \xi - E_{r} }}}} (\mu _{r} - \frac{1}{2})((\mu _{r} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} ) \\ & \mu _{r}^{2} R^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} + ((\mu _{r}^{2} + 1)(\mu _{r} - \frac{1}{2})(1 + \mu _{r} )\xi + \mu _{r}^{2} E_{r} (\mu _{r} - 1))r^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} R^{{\frac{{2E_{r} + \xi }}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} ) \\ \end{aligned} \right)}}{{\left( \begin{aligned} & 8r^{{\frac{{2E_{r} + \xi }}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} E_{r} ((\mu _{r}^{5} - \frac{3}{4}\mu _{r}^{3} + \frac{3}{4}\mu _{r}^{2} + \frac{1}{4}\mu _{r} - \frac{1}{4})\xi + (\mu _{r}^{3} - \frac{1}{2}\mu _{r}^{2} - \mu _{r} + 1)(\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2}) \\ & \xi + \frac{1}{2}(\mu _{r} - 1)E_{r} )R^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} + 8((\mu _{r}^{2} + 1)(\mu _{r} - \frac{1}{2})(1 + \mu _{r} )\xi + \mu _{r}^{2} E_{r} (\mu _{r} - 1))(((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + \\ & (\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )\ln \left( R \right) - ((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi + (\mu _{r} - 1)E_{r} )((\mu _{r}^{2} + \frac{1}{2}\mu _{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} ) \\ & \ln \left( r \right) - \frac{1}{2}E_{r} ((\mu _{r}^{3} + \frac{1}{2}\mu _{r}^{2} - \frac{1}{2}\mu _{r} )\xi + \frac{1}{2}(\mu _{r} - 1)E_{r} ))r^{{\frac{{\mu _{r} (2\mu _{r} \xi + 2E_{r} + \xi )}}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} R^{{\frac{{2E_{r} + \xi }}{{2\xi \mu _{r}^{2} + (\xi + E_{r} )\mu _{r} - \xi - E_{r} }}}} \\ \end{aligned} \right)}},$$
$$C_{3} = \frac{{\left( \begin{aligned} 16\sigma_{0} ((\mu_{r}^{2} + \mu_{r} )\xi + (\mu_{r} - 1)E_{r} )((\mu_{r} - \frac{1}{2})((\mu_{r}^{2} + \frac{1}{2}\mu_{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} ))((\mu_{r}^{2} + \frac{1}{2}\mu_{r} - \frac{1}{2})\xi \hfill \\ + \left( {\mu_{r} - 1} \right)E_{r} )\mu_{r}^{2} \ln R - (\mu_{r} - \frac{1}{2})((\mu_{r}^{2} + \frac{1}{2}\mu_{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )((\mu_{r}^{2} + \frac{1}{2}\mu_{r} - \frac{1}{2})\xi + \left( {\mu_{r} - 1} \right)E_{r} ) \hfill \\ \mu_{r}^{2} \ln r - \frac{1}{2}E_{r} ((\mu_{r} - \frac{1}{2})(1 + \mu_{r} ))(\mu_{r}^{4} - \frac{1}{4}\mu_{r}^{2} + \frac{1}{4})\xi + (\mu_{r}^{3} - \frac{1}{2}\mu_{r}^{2} - \frac{1}{2}\mu_{r} + \frac{1}{2})(\mu_{r} - 1) \hfill \\ \end{aligned} \right)}}{{\left( \begin{aligned} 8(((\mu_{r}^{2} + \frac{1}{2}\mu_{r} - \frac{1}{2})\xi + \left( {\mu_{r} - 1} \right)E_{r} )((\mu_{r}^{2} + \frac{1}{2}\mu_{r} - \frac{1}{2})\xi - \frac{1}{2}E_{r} )\ln R - ((\mu_{r}^{2} + \frac{1}{2}\mu_{r} - \frac{1}{2})\xi + \left( {\mu_{r} - 1} \right)E_{r} )((\mu_{r}^{2} + \frac{1}{2}\mu_{r} - \frac{1}{2}) \hfill \\ \xi - \frac{1}{2}E_{r} ))((\mu_{r}^{2} + 1)(\mu_{r} - \frac{1}{2})(1 + \mu_{r} )\xi + \mu_{r}^{2} E_{r} (\mu_{r} - 1))R^{{\frac{{ - 2\xi \mu_{r}^{2} + \left( { - \xi - 2E_{r} } \right)\mu_{r} + \xi + 2E_{r} }}{{2\xi \mu_{r}^{2} + \left( {\xi + E_{r} } \right)\mu_{r} - \xi - E_{r} }}}} + r^{{\frac{{ - 2\xi \mu_{r}^{2} + \left( { - \xi - 2E_{r} } \right)\mu_{r} + \xi + 2E_{r} }}{{2\xi \mu_{r}^{2} + \left( {\xi + E_{r} } \right)\mu_{r} - \xi - E_{r} }}}} E_{r} (8(\mu_{r} - \frac{1}{2})^{2} (1 + \mu_{r} )^{2} \hfill \\ (\mu_{r}^{3} - \frac{1}{2}\mu_{r}^{2} + \frac{1}{2})\xi^{2} ) + 12(\mu_{r} - \frac{1}{2})(1 + \mu_{r} )^{2} (\mu_{r}^{2} - \frac{3}{2}\mu_{r} + \frac{5}{6})(\mu_{r} - 1)E_{r} \xi + 4\left( {\mu_{r} - 1} \right)^{2} E_{r}^{2} (\mu_{r}^{3} + \frac{1}{2}\mu_{r}^{2} - \mu_{r} + 1) \hfill \\ \end{aligned} \right)}}.$$

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Wang, G., Han, W., Jiang, Y. et al. Coupling Analysis for Rock Mass Supported with CMC or CFC Rockbolts Based on Viscoelastic Method. Rock Mech Rock Eng 52, 4565–4588 (2019). https://doi.org/10.1007/s00603-019-01840-6

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