Dynamic stress concentration of a non-circular cavern in a viscoelastic medium subjected to a plane P-wave

Abstract

An analytical method for solving the dynamic response of an underground cavern with arbitrary shape in a viscoelastic rock mass under a plane P-wave is deduced based on the theory of stress wave and the method of complex function. The seismic quality factors are used to represent the viscoelastic behavior of the rock mass. The dynamic stress concentration factors (DSCF) of an arch-shaped cavern as example are discussed with considering different influencing parameters, e.g., seismic quality factors (under two frequencies) of surrounding rock mass, incident angle of incident plane P-wave, arch radius, and span height ratio of the cavern. The research result shows that the seismic quality factors of the surrounding rock mass are of significant importance for the scattering of the plane P-wave. The seismic quality factors of the surrounding rock mass change the distribution characteristics and scale of the DSCF compared with an elastic rock mass. Under the coupling effect of seismic quality factor and other effects, the change of DSCF around the cavern is more complex.

Appendices

Appendix A

Coefficients of $E_{n}^{(11)}$, $E_{n}^{(12)}$, $E_{n}^{(21)}$, $E_{n}^{(22)}$, $E_{{}}^{(1)}$, and $E_{{}}^{(2)}$ as presented in Eqs. (32) and (33)

$$E_{n}^{(11)} = - \alpha^{2} (\overline{\lambda } + \overline{\mu })H_{n}^{(1)} (\alpha \left| {w(\eta )} \right|)\left[ {\frac{w(\eta )}{{\left| {w(\eta )} \right|}}} \right]^{n} + \frac{{\overline{\mu }\alpha^{2} \eta^{2} w^{\prime}(\eta )}}{{\overline{w^{\prime}(\eta )} }}H_{n - 2}^{(1)} (\alpha \left| {w(\eta )} \right|)\left[ {\frac{w(\eta )}{{\left| {w(\eta )} \right|}}} \right]^{n - 2}, $$

(43)

$$E_{n}^{(12)} = \frac{{i\overline{\mu }\beta^{2} \eta^{2} w^{\prime}(\eta )}}{{\overline{w^{\prime}(\eta )} }}H_{n - 2}^{(1)} (\beta \left| {w(\eta )} \right|)\left[ {\frac{w(\eta )}{{\left| {w(\eta )} \right|}}} \right]^{n - 2}, $$

(44)

$$E_{n}^{(21)} = - \alpha^{2} (\overline{\lambda } + \overline{\mu })H_{n}^{(1)} (\alpha \left| {w(\eta )} \right|)\left[ {\frac{w(\eta )}{{\left| {w(\eta )} \right|}}} \right]^{n} + \frac{{\overline{\mu }\alpha^{2} \overline{\eta }^{2} \overline{w^{\prime}(\eta )} }}{w^{\prime}(\eta )}H_{n + 2}^{(1)} (\alpha \left| {w(\eta )} \right|)\left[ {\frac{w(\eta )}{{\left| {w(\eta )} \right|}}} \right]^{n + 2}, $$

(45)

$$E_{n}^{(22)} = - \frac{{i\overline{\mu }\beta^{2} \overline{\eta }^{2} \overline{w^{\prime}(\eta )} }}{w^{\prime}(\eta )}H_{n + 2}^{(1)} (\beta \left| {w(\eta )} \right|)\left[ {\frac{w(\eta )}{{\left| {w(\eta )} \right|}}} \right]^{n + 2}, $$

(46)

$$E_{{}}^{(1)} = \left( {(\overline{\lambda } + \overline{\mu }) + \frac{{\mu \eta^{2} w^{\prime}(\eta )}}{{\overline{w^{\prime}(\eta )} }}\exp ( - 2i\theta_{0} )} \right)\alpha^{2} \varphi_{0} \cdot \exp \left[ {\frac{i\alpha }{2}(w(\eta )\exp ( - i\theta_{0} ) + \overline{w(\eta )} \exp (i\theta_{0} ))} \right], $$

(47)

$$E_{{}}^{(2)} = \left( {(\overline{\lambda } + \overline{\mu }) + \frac{{\mu \overline{\eta }^{2} \overline{w^{\prime}(\eta )} }}{w^{\prime}(\eta )}\exp (2i\theta_{0} )} \right)\alpha^{2} \varphi_{0} \cdot \exp \left[ {\frac{i\alpha }{2}(w(\eta )\exp ( - i\theta_{0} ) + \overline{w(\eta )} \exp (i\theta_{0} ))} \right]. $$

(48)

Appendix B

Coefficients of R, b₂, b₃, b₄, and b₅ in Eq. (42):

$$R = \frac{b}{{1 - b_{2} - b_{4} /3}}, $$

(49)

$$b_{2} = c_{12} + c_{22} + c_{32} - c_{11} c_{21} - (c_{11} + c_{21} )c_{31}^{{}}, $$

(50)

$$b_{3} = c_{13} + c_{23} + c_{33} + c_{11} c_{22} + c_{12} c_{21} + (c_{12} - c_{11} c_{21} + c_{22} )c_{31} + (c_{11} + c_{21} )c_{32}, $$

(51)

$$\begin{gathered} b_{4} = c_{14} + c_{24} + c_{34} { - }c_{11} c_{23} + c_{12} c_{22} { - }c_{13} c_{21} - (c_{13} + c_{23} + c_{11} c_{22} + c_{12} c_{21} )c_{31} \\ + (c_{21} - c_{11} c_{21} + c_{22} )c_{32} - (c_{11} + c_{21} )c_{33} \\ \end{gathered}, $$

(52)

$$\begin{gathered} b_{5} = c_{15} + c_{25} + c_{35} + c_{11} c_{24} + c_{12} c_{23} + c_{13} c_{22} + c_{14} c_{21} + (c_{14} + c_{24} - c_{11} c_{23} + c_{12} c_{22} - c_{13} c_{21} )c_{31} \\ + (c_{13} + c_{23} + c_{11} c_{22} + c_{12} c_{21} )c_{32} + (c_{12} - c_{11} c_{21} + c_{22} )c_{33} + (c_{11} + c_{21} )c_{34} \\ \end{gathered}$$

(53)

where

$c_{l1} = - \lambda_{l} \Phi_{l}$, $c_{l2} = - \Phi_{l} \left[ {1 + \frac{{\lambda_{l}^{2} }}{2}\left( {\Phi_{l} - 1} \right)} \right]$, $c_{l3} = \lambda_{l} \Phi_{l} \left( {\Phi_{l} - 1} \right)\left[ {1 + \frac{{\lambda_{l}^{2} }}{6}\left( {\Phi_{l} - 2} \right)} \right]$,

$c_{l4} = \Phi_{l} \left( {\Phi_{l} - 1} \right)\left[ {\frac{1}{2} + \frac{{\lambda_{l}^{2} }}{2}\left( {\Phi_{l} - 2} \right) + \frac{{\lambda_{l}^{4} }}{24}\left( {\Phi_{l} - 2} \right)\left( {\Phi_{l} - 3} \right)} \right]$,

$c_{l5} = { - }\lambda_{l} \Phi_{l} \left( {\Phi_{l} - 1} \right)\left( {\Phi_{l} - 2} \right)\left[ {\frac{1}{2} + \frac{{\lambda_{l}^{2} }}{6}\left( {\Phi_{l} - 3} \right) + \frac{{\lambda_{l}^{4} }}{1204}\left( {\Phi_{l} - 3} \right)\left( {\Phi_{l} - 4} \right)} \right]$, $l$ = 1, 2, 3,

$\lambda_{1} = 2\cos (k_{1} \pi )$, $\lambda_{2} = 2\cos (k_{2} \pi )$, $\lambda_{3} = 2\cos (k_{3} \pi )$,

$\Phi_{1} = \frac{{v_{1} }}{\pi }{ - }1$, $\Phi_{2} = \frac{{v_{2} }}{\pi }{ - }1$,$\Phi_{3} = \frac{{v_{3} }}{\pi }{ - }1$,

$v_{1} = \frac{3\pi }{2}{\text{ - arctan}}\frac{b - a}{{2h - H}}$, $v_{2} = \pi + {\text{arctan}}\frac{b - a}{{2h - H}}$,$v_{3} = \frac{3\pi }{2}$, and the value of $k_{1}$,$k_{2}$, $k_{3}$ can be obtained by trial method according to the following equation:

$$\Phi_{1} \cos \left( {k_{1} \pi } \right) + \Phi_{2} \cos \left( {k_{2} \pi } \right) + \Phi_{3} \cos \left( {k_{3} \pi } \right) = 0. $$