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Scattering of SH waves by lined tunnel in inhomogeneous right-angle space with variable shear modulus

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Abstract

Based on the theory of complex function, the seismic response of tunnel structure in inhomogeneous elastic right-angle space under different stiffness is solved. The inhomogeneous form of shear modulus is used to reflect the variation regular of stiffness. The governing equations are obtained by means of displacement auxiliary function and mapping function. At the same time, the calculation model is established by means of the image method, and the series solution of the wave field is constructed and mainly analyses the distribution of displacement amplitude on the surface and dynamic stress concentration factor on the lining.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 12072085) and the Natural Science Foundation of Heilongjiang Province of China (No. ZD2021A001) and Research Team Project of Heilongjiang Natural Science Foundation (No. TD2020A001) and the program for Innovative Research Team in China Earthquake Administration.

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Authors and Affiliations

  1. College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin, 150001, China

    Jin-lai Bian, Zai-lin Yang, Yong Yang & Meng-han Sun

  2. Key Laboratory of Advanced Material of Ship and Mechanics, Ministry of Industry and Information Technology, Harbin Engineering University, Harbin, China

    Zai-lin Yang, Yong Yang & Meng-han Sun

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  1. Jin-lai Bian
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  2. Zai-lin Yang
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Each author contributed to this paper. JL-B and ZL-Y wrote this paper, YY and MH-S analyzed the results, JL-B and YY processed the data, and ZL-Y provided methodological guidance.

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Correspondence to Meng-han Sun.

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Appendix

Appendix

Stress expressions for incident and reflected waves

$$\begin{aligned}{} & {} \begin{array}{l} \tau _{rz}^{1\text{ i }} =-\frac{\mu _{0} \varphi _{0} \beta }{2\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\exp \left( {\frac{ik_{t} }{2}\left( {\zeta \text{ e}^{-i\alpha }+\bar{{\zeta }}\text{ e}^{i\alpha }} \right) } \right) \\ \left[ {\left( {\beta \bar{{Z}}+\eta } \right) \left( {1-ik_{t} \text{ e}^{-i\alpha }} \right) \text{ e}^{i\theta }+\left( {\beta Z+\eta } \right) \left( {1-ik_{t} \text{ e}^{i\alpha }} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.1)
$$\begin{aligned}{} & {} \begin{array}{l} \tau _{\theta z}^{1\text{ i }} =-\frac{i\mu _{0} \varphi _{0} \beta }{2\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\exp \left( {\frac{ik_{t} }{2}\left( {\zeta \text{ e}^{-i\alpha }+\bar{{\zeta }}\text{ e}^{i\alpha }} \right) } \right) \\ \left[ {\left( {\beta \bar{{Z}}+\eta } \right) \left( {1-ik_{t} \text{ e}^{-i\alpha }} \right) \text{ e}^{i\theta }-\left( {\beta Z+\eta } \right) \left( {1-ik_{t} \text{ e}^{i\alpha }} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.2)
$$\begin{aligned}{} & {} \begin{array}{l} \tau _{rz}^{2\text{ r }} =-\frac{\mu _{0} \varphi _{0} \beta }{2\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\exp \left( {\frac{ik_{t} }{2}\left( {\zeta \text{ e}^{i\alpha }+\bar{{\zeta }}\text{ e}^{-i\alpha }} \right) } \right) \\ \left[ {\left( {\beta \bar{{Z}}+\eta } \right) \left( {1-ik_{t} \text{ e}^{i\alpha }} \right) \text{ e}^{i\theta }+\left( {\beta Z+\eta } \right) \left( {1-ik_{t} \text{ e}^{-i\alpha }} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.3)
$$\begin{aligned}{} & {} \begin{array}{l} \tau _{\theta z}^{2\text{ r }} =-\frac{i\mu _{0} \varphi _{0} \beta }{2\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\exp \left( {\frac{ik_{t} }{2}\left( {\zeta \text{ e}^{i\alpha }+\bar{{\zeta }}\text{ e}^{-i\alpha }} \right) } \right) \\ \left[ {\left( {\beta \bar{{Z}}+\eta } \right) \left( {1-ik_{t} \text{ e}^{i\alpha }} \right) \text{ e}^{i\theta }-\left( {\beta Z+\eta } \right) \left( {1-ik_{t} \text{ e}^{-i\alpha }} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.4)
$$\begin{aligned}{} & {} \begin{array}{l} \tau _{rz}^{3\text{ r }} =-\frac{\mu _{0} \varphi _{0} \beta }{2\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\exp \left( {-\frac{ik_{t} }{2}\left( {\zeta \text{ e}^{-i\alpha }+\bar{{\zeta }}\text{ e}^{i\alpha }} \right) } \right) \\ \left[ {\left( {\beta \bar{{Z}}+\eta } \right) \left( {1+ik_{t} \text{ e}^{-i\alpha }} \right) \text{ e}^{i\theta }+\left( {\beta Z+\eta } \right) \left( {1+ik_{t} \text{ e}^{i\alpha }} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.5)
$$\begin{aligned}{} & {} \begin{array}{l} \tau _{\theta z}^{3\text{ r }} =-\frac{i\mu _{0} \varphi _{0} \beta }{2\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\exp \left( {-\frac{ik_{t} }{2}\left( {\zeta \text{ e}^{-i\alpha }+\bar{{\zeta }}\text{ e}^{i\alpha }} \right) } \right) \\ \left[ {\left( {\beta \bar{{Z}}+\eta } \right) \left( {1+ik_{t} \text{ e}^{-i\alpha }} \right) \text{ e}^{i\theta }-\left( {\beta Z+\eta } \right) \left( {1+ik_{t} \text{ e}^{i\alpha }} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.6)
$$\begin{aligned}{} & {} \begin{array}{l} \tau _{rz}^{4\text{ r }} =-\frac{\mu _{0} \varphi _{0} \beta }{2\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\exp \left( {-\frac{ik_{t} }{2}\left( {\zeta \text{ e}^{i\alpha }+\bar{{\zeta }}\text{ e}^{-i\alpha }} \right) } \right) \\ \left[ {\left( {\beta \bar{{Z}}+\eta } \right) \left( {1+ik_{t} \text{ e}^{i\alpha }} \right) \text{ e}^{i\theta }+\left( {\beta Z+\eta } \right) \left( {1+ik_{t} \text{ e}^{-i\alpha }} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.7)
$$\begin{aligned}{} & {} \begin{array}{l} \tau _{\theta z}^{4\text{ r }} =-\frac{i\mu _{0} \varphi _{0} \beta }{2\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\exp \left( {-\frac{ik_{t} }{2}\left( {\zeta \text{ e}^{i\alpha }+\bar{{\zeta }}\text{ e}^{-i\alpha }} \right) } \right) \\ \left[ {\left( {\beta \bar{{Z}}+\eta } \right) \left( {1+ik_{t} \text{ e}^{i\alpha }} \right) \text{ e}^{i\theta }-\left( {\beta Z+\eta } \right) \left( {1+ik_{t} \text{ e}^{-i\alpha }} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.8)

Scattering waves generated by outer boundary

$$\begin{aligned}{} & {} \begin{array}{l} \tau _{rz}^{\text{ s }} =\frac{\mu _{0} }{\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\sum \limits _{n=-\infty }^{n=+\infty } {A_{n} } \\ \left[ {\left( {\begin{array}{l} \left( {\frac{\partial S_{n}^{\left( 1 \right) } }{\partial Z_{1} }+\frac{\partial S_{n}^{\left( 2 \right) } }{\partial Z_{1} }+\frac{\partial S_{n}^{\left( 3 \right) } }{\partial Z_{1} }+\frac{\partial S_{n}^{\left( 4 \right) } }{\partial Z_{1} }} \right) \left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) \\ -\frac{1}{2}\beta \left( {\beta \bar{{Z}}+\eta } \right) \left( {S_{n}^{\left( 1 \right) } +S_{n}^{\left( 2 \right) } +S_{n}^{\left( 3 \right) } +S_{n}^{\left( 4 \right) } } \right) \\ \end{array}} \right) } \right. \text{ e}^{i\theta }+ \\ \left. {\left( {\begin{array}{l} \left( {\frac{\partial S_{n}^{\left( 1 \right) } }{\partial \bar{{Z}}_{1} }+\frac{\partial S_{n}^{\left( 2 \right) } }{\partial \bar{{Z}}_{1} }+\frac{\partial S_{n}^{\left( 3 \right) } }{\partial \bar{{Z}}_{1} }+\frac{\partial S_{n}^{\left( 4 \right) } }{\partial \bar{{Z}}_{1} }} \right) \left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) \\ -\frac{1}{2}\beta \left( {\beta Z+\eta } \right) \left( {S_{n}^{\left( 1 \right) } +S_{n}^{\left( 2 \right) } +S_{n}^{\left( 3 \right) } +S_{n}^{\left( 4 \right) } } \right) \\ \end{array}} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.9)
$$\begin{aligned}{} & {} \begin{array}{l} \tau _{\theta z}^{\text{ s }} =\frac{i\mu _{0} }{\sqrt{\left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) } }\sum \limits _{n=-\infty }^{n=+\infty } {A_{n} } \\ \left[ {\left( {\begin{array}{l} \left( {\frac{\partial S_{n}^{\left( 1 \right) } }{\partial Z_{1} }+\frac{\partial S_{n}^{\left( 2 \right) } }{\partial Z_{1} }+\frac{\partial S_{n}^{\left( 3 \right) } }{\partial Z_{1} }+\frac{\partial S_{n}^{\left( 4 \right) } }{\partial Z_{1} }} \right) \left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) \\ -\frac{1}{2}\beta \left( {\beta \bar{{Z}}+\eta } \right) \left( {S_{n}^{\left( 1 \right) } +S_{n}^{\left( 2 \right) } +S_{n}^{\left( 3 \right) } +S_{n}^{\left( 4 \right) } } \right) \\ \end{array}} \right) } \right. \text{ e}^{i\theta }- \\ \left. {\left( {\begin{array}{l} \left( {\frac{\partial S_{n}^{\left( 1 \right) } }{\partial \bar{{Z}}_{1} }+\frac{\partial S_{n}^{\left( 2 \right) } }{\partial \bar{{Z}}_{1} }+\frac{\partial S_{n}^{\left( 3 \right) } }{\partial \bar{{Z}}_{1} }+\frac{\partial S_{n}^{\left( 4 \right) } }{\partial \bar{{Z}}_{1} }} \right) \left( {\beta Z+\eta } \right) \left( {\beta \bar{{Z}}+\eta } \right) \\ -\frac{1}{2}\beta \left( {\beta Z+\eta } \right) \left( {S_{n}^{\left( 1 \right) } +S_{n}^{\left( 2 \right) } +S_{n}^{\left( 3 \right) } +S_{n}^{\left( 4 \right) } } \right) \\ \end{array}} \right) \text{ e}^{-i\theta }} \right] \\ \end{array} \end{aligned}$$
(A.10)

where

$$\begin{aligned} \left\{ {{\begin{array}{*{20}c} {{\begin{array}{*{20}c} {\frac{\partial S_{n}^{\left( 1 \right) } }{\partial Z_{1} }=\frac{\beta k_{t} }{2\left( {\beta Z_{1} +\eta } \right) }H_{n-1}^{\left( 1 \right) } \left( {k_{t} \left| {\zeta _{1} } \right| } \right) \left( {\frac{\zeta _{1} }{\left| {\zeta _{1} } \right| }} \right) ^{n-1}} \\ {\frac{\partial S_{n}^{\left( 2 \right) } }{\partial Z_{1} }=-\frac{\beta k_{t} }{2\left( {\beta Z_{2} +\eta } \right) }H_{n+1}^{\left( 1 \right) } \left( {k_{t} \left| {\zeta _{2} } \right| } \right) \left( {\frac{\zeta _{2} }{\left| {\zeta _{2} } \right| }} \right) ^{-n-1}} \\ {\frac{\partial S_{n}^{\left( 3 \right) } }{\partial Z_{1} }=\left( {-1} \right) ^{n}\frac{\beta k_{t} }{2\left( {\beta Z_{3} +\eta } \right) }H_{n-1}^{\left( 1 \right) } \left( {k_{t} \left| {\zeta _{3} } \right| } \right) \left( {\frac{\zeta _{3} }{\left| {\zeta _{3} } \right| }} \right) ^{-n-1}} \\ {\frac{\partial S_{n}^{\left( 4 \right) } }{\partial Z_{1} }=\left( {-1} \right) ^{n+1}\frac{\beta k_{t} }{2\left( {\beta Z_{4} +\eta } \right) }H_{n+1}^{\left( 1 \right) } \left( {k_{t} \left| {\zeta _{4} } \right| } \right) \left( {\frac{\zeta _{4} }{\left| {\zeta _{4} } \right| }} \right) ^{-n-1}} \\ \end{array} }} &{} {{\begin{array}{*{20}c} {\frac{\partial S_{n}^{\left( 1 \right) } }{\partial \bar{{Z}}_{1} }=-\frac{\beta k_{t} }{2\left( {\beta \bar{{Z}}_{1} +\eta } \right) }H_{n+1}^{\left( 1 \right) } \left( {k_{t} \left| {\zeta _{1} } \right| } \right) \left( {\frac{\zeta _{1} }{\left| {\zeta _{1} } \right| }} \right) ^{n+1}} \\ {\frac{\partial S_{n}^{\left( 2 \right) } }{\partial \bar{{Z}}_{1} }=\frac{\beta k_{t} }{2\left( {\beta \bar{{Z}}_{2} +\eta } \right) }H_{n-1}^{\left( 1 \right) } \left( {k_{t} \left| {\zeta _{2} } \right| } \right) \left( {\frac{\zeta _{2} }{\left| {\zeta _{2} } \right| }} \right) ^{-n+1}} \\ {\frac{\partial S_{n}^{\left( 3 \right) } }{\partial \bar{{Z}}_{1} }=\left( {-1} \right) ^{n+1}\frac{\beta k_{t} }{2\left( {\beta \bar{{Z}}_{3} +\eta } \right) }H_{n+1}^{\left( 1 \right) } \left( {k_{t} \left| {\zeta _{3} } \right| } \right) \left( {\frac{\zeta _{3} }{\left| {\zeta _{3} } \right| }} \right) ^{n+1}} \\ {\frac{\partial S_{n}^{\left( 4 \right) } }{\partial \bar{{Z}}_{1} }=\left( {-1} \right) ^{n}\frac{\beta k_{t} }{2\left( {\beta \bar{{Z}}_{4} +\eta } \right) }H_{n-1}^{\left( 1 \right) } \left( {k_{t} \left| {\zeta _{4} } \right| } \right) \left( {\frac{\zeta _{4} }{\left| {\zeta _{4} } \right| }} \right) ^{-n+1}} \\ \end{array} }} \\ \end{array} }} \right. \end{aligned}$$

Refraction waves generated by outer boundary

$$\begin{aligned} \tau _{rz}^{\text{ f }}= & {} \frac{\mu _{2} k_{2} }{2}\sum \limits _{n=-\infty }^{n=+\infty } {B_{n} \left[ {H_{n-1}^{\left( 2 \right) } \left( {k_{2} \left| {Z_{1} } \right| } \right) \left( {\frac{Z_{1} }{\left| {Z_{1} } \right| }} \right) ^{n-1}\text{ e}^{i\theta }-H_{n+1}^{\left( 2 \right) } \left( {k_{2} \left| {Z_{1} } \right| } \right) \left( {\frac{Z_{1} }{\left| {Z_{1} } \right| }} \right) ^{n+1}\text{ e}^{-i\theta }} \right] } \end{aligned}$$
(A.11)
$$\begin{aligned} \tau _{\theta z}^{\text{ f }}= & {} \frac{i\mu _{2} k_{2} }{2}\sum \limits _{n=-\infty }^{n=+\infty } {B_{n} \left[ {H_{n-1}^{\left( 2 \right) } \left( {k_{2} \left| {Z_{1} } \right| } \right) \left( {\frac{Z_{1} }{\left| {Z_{1} } \right| }} \right) ^{n-1}\text{ e}^{i\theta }+H_{n+1}^{\left( 2 \right) } \left( {k_{2} \left| {Z_{1} } \right| } \right) \left( {\frac{Z_{1} }{\left| {Z_{1} } \right| }} \right) ^{n+1}\text{ e}^{-i\theta }} \right] } \end{aligned}$$
(A.12)

Scattering waves generated by inner boundary

$$\begin{aligned} \tau _{rz}^{\text{ s}_{r_{c} } }= & {} \frac{\mu _{2} k_{2} }{2}\sum \limits _{n=-\infty }^{n=+\infty } {C_{n} \left[ {H_{n-1}^{\left( 1 \right) } \left( {k_{2} \left| {Z_{c} } \right| } \right) \left( {\frac{Z_{c} }{\left| {Z_{c} } \right| }} \right) ^{n-1}\text{ e}^{i\theta }-H_{n+1}^{\left( 1 \right) } \left( {k_{2} \left| {Z_{c} } \right| } \right) \left( {\frac{Z_{c} }{\left| {Z_{c} } \right| }} \right) ^{n+1}\text{ e}^{-i\theta }} \right] } \end{aligned}$$
(A.13)
$$\begin{aligned} \tau _{\theta z}^{\text{ s}_{r_{c} } }= & {} \frac{i\mu _{2} k_{2} }{2}\sum \limits _{n=-\infty }^{n=+\infty } {C_{n} \left[ {H_{n-1}^{\left( 1 \right) } \left( {k_{2} \left| {Z_{c} } \right| } \right) \left( {\frac{Z_{c} }{\left| {Z_{c} } \right| }} \right) ^{n-1}\text{ e}^{i\theta }+H_{n+1}^{\left( 1 \right) } \left( {k_{2} \left| {Z_{c} } \right| } \right) \left( {\frac{Z_{c} }{\left| {Z_{c} } \right| }} \right) ^{n+1}\text{ e}^{-i\theta }} \right] } \end{aligned}$$
(A.14)

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Bian, Jl., Yang, Zl., Yang, Y. et al. Scattering of SH waves by lined tunnel in inhomogeneous right-angle space with variable shear modulus. Z. Angew. Math. Phys. 74, 161 (2023). https://doi.org/10.1007/s00033-023-02047-0

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