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Dynamic Response of Tunnel Structures in Inhomogeneous Medium Under SH Wave: Shear Modulus in Quadratic Functional Form

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Abstract

The task of this work is to study the scattering of SH waves by homogeneous tunnel structures in an unbounded inhomogeneous medium. The shear modulus is assumed to be a function of coordinates (x,y). A two-dimensional scattering model is established. Selecting different inhomogeneous parameters, the medium has different properties, expressed as a rigid variation. The stress concentration phenomenon of the structure is analyzed for material design. Based on the complex function theory, the expressions of wave field in the tunnel are derived. The stress concentration phenomenon on the tunnel is discussed with numerical examples. The distribution of dynamic stress concentration factor on the inner and outer boundaries is analyzed under different influencing factors. Finally, it is found that the distribution of dynamic stress concentration factor is significantly affected by the inhomogeneous parameters and reference wave numbers of the medium.

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All data generated or analyzed to support the findings of this study are included within the article.

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Funding

This work is supported by the National Natural Science Foundation of China (No. 12002143) and Research Team Project of Heilongjiang Natural Science Foundation (No.TD2020A001) and the program for Innovative Research Team in China Earthquake Administration.

Author information

Authors and Affiliations

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

    Zailin Yang, **lai Bian & Yong Yang

  2. School of Resource Engineering, Longyan University, Longyan, 364012, China

    Huanan Xu

  3. Key Laboratory of Advanced Material of Ship and Mechanics, Ministry of Industry and Information Technology, Harbin Engineering University, Harbin, 150001, China

    Zailin Yang & Yong Yang

Authors
  1. Zailin Yang
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  2. **lai Bian
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  3. Huanan Xu
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  4. Yong Yang
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Contributions

ZY and YY conceived the idea. JB and HX carried out formula derivation and example analysis. All authors contributed to the writing and revision.

Corresponding authors

Correspondence to Huanan Xu or Yong Yang.

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No conflict of interest exists in the submission of this manuscript, and the manuscript is approved by all authors for publication.

Appendices

Appendix

Stress Fields in an Unbounded Inhomogeneous Medium

Incident waves in an unbounded medium

$$ \begin{gathered} \tau_{rz}^{{\text{i}}} = \frac{1}{2}\mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta {\text{e}}^{ - i\alpha } + \overline{\zeta } {\text{e}}^{i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta } } \right)} \right] \hfill \\ \left[ {\left( {\beta \overline{z} + \eta } \right)\left( {ik_{T} {\text{e}}^{ - i\alpha } - 1} \right){\text{e}}^{i\theta } + \left( {\beta z + \eta } \right)\left( {ik_{T} {\text{e}}^{i\alpha } - 1} \right){\text{e}}^{ - i\theta } } \right] \hfill \\ \end{gathered} $$
(A1)
$$ \begin{gathered} \tau_{\theta z}^{{\text{i}}} = \frac{1}{2}i\mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta e^{ - i\alpha } + \overline{\zeta } e^{i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta } } \right)} \right] \hfill \\ \left[ {\left( {\beta \overline{z} + \eta } \right)\left( {ik_{T} {\text{e}}^{ - i\alpha } - 1} \right){\text{e}}^{i\theta } - \left( {\beta z + \eta } \right)\left( {ik_{T} {\text{e}}^{i\alpha } - 1} \right){\text{e}}^{ - i\theta } } \right] \hfill \\ \end{gathered} $$
(A2)

Scattering waves generated by the outer boundary

$$ \begin{gathered} \tau_{rz}^{{\text{s}}} = \frac{1}{2}\mu_{0} \beta \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta } } \right)} \right] \hfill \\ \sum\limits_{n = - \infty }^{n = + \infty } {A_{n} } \left[ {\left( {k_{T} H_{n - 1}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n - 1} - H_{n}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n} } \right)} \right.\left( {\beta \overline{z} + \eta } \right){\text{e}}^{i\theta } \hfill \\ \left. { - \left( {H_{n}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n} + k_{T} H_{n + 1}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n + 1} } \right)\left( {\beta z + \eta } \right){\text{e}}^{ - i\theta } } \right] \hfill \\ \end{gathered} $$
(A3)
$$ \begin{aligned} & \tau_{\theta z}^{{\text{s}}} = \frac{1}{2}i\mu_{0} \beta \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta } } \right)} \right] \\ & \sum\limits_{n = - \infty }^{n = + \infty } {A_{n} } \left[ {\left( {k_{T} H_{n - 1}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n - 1} - H_{n}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n} } \right)} \right.\left( {\beta \overline{z} + \eta } \right){\text{e}}^{i\theta } \\ &\qquad \left. {{ + }\left( {H_{n}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n} + k_{T} H_{n + 1}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n + 1} } \right)\left( {\beta z + \eta } \right){\text{e}}^{ - i\theta } } \right] \end{aligned} $$
(A4)

Refracted waves generated by the outer boundary

$$\begin{aligned} & \tau_{rz}^{{\text{f}}} = \frac{1}{2}\mu_{2} k_{2} \sum\limits_{n = - \infty }^{\infty }\\ & {B_{n} \left[ {H_{n - 1}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{n - 1} {\text{e}}^{i\theta } - H_{n + 1}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{n + 1} {\text{e}}^{ - i\theta } } \right]}\end{aligned} $$
(A5)
$$\begin{aligned} & \tau_{\theta z}^{{\text{f}}} = \frac{1}{2}i\mu_{2} k_{2} \sum\limits_{n = - \infty }^{\infty }\\ & {B_{n} \left[ {H_{n - 1}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{n - 1} {\text{e}}^{i\theta } + H_{n + 1}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{n + 1} {\text{e}}^{ - i\theta } } \right]}\end{aligned} $$
(A6)

Scattering waves generated by the inner boundary

$$\begin{aligned} \tau_{rz}^{{{\text{s}}R_{{1}} }} & = \frac{1}{2}\mu_{2} k_{2} \sum\limits_{n = - \infty }^{\infty } C_{n} \left[ H_{n - 1}^{(1)} \left( {k_{2} \left| {z_{{1}} } \right|} \right)\left( {\frac{{z_{{1}} }}{{\left| {z_{{1}} } \right|}}} \right)^{n - 1} {\text{e}}^{i\theta }\right.\\ &\quad \left. - H_{n + 1}^{(1)} \left( {k_{2} \left| {z_{{1}} } \right|} \right)\left( {\frac{{z_{{1}} }}{{\left| {z_{{1}} } \right|}}} \right)^{n + 1} {\text{e}}^{ - i\theta } \right]\end{aligned} $$
(A7)
$$\begin{aligned} \tau_{\theta z}^{{{\text{s}}R_{{1}} }} & = \frac{1}{2}i\mu_{2} k_{2} \sum\limits_{n = - \infty }^{\infty } C_{n} \left[ H_{n - 1}^{(1)} \left( {k_{2} \left| {z_{{1}} } \right|} \right)\left( {\frac{{z_{{1}} }}{{\left| {z_{{1}} } \right|}}} \right)^{n - 1} {\text{e}}^{i\theta }\right.\\ & \quad \left. + H_{n + 1}^{(1)} \left( {k_{2} \left| {z_{{1}} } \right|} \right)\left( {\frac{{z_{{1}} }}{{\left| {z_{{1}} } \right|}}} \right)^{n + 1} {\text{e}}^{ - i\theta } \right]\end{aligned} $$
(A8)

Wave Fields (Sect. 5.1) in Boundary Conditions

$$ E_{n}^{11} = - \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right]H_{n}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n} $$
(A9)
$$ E_{n}^{12} = H_{n}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{n} $$
(A10)
$$ E_{n}^{13} = H_{n}^{(1)} \left( {k_{2} \left| {z_{1} } \right|} \right)\left( {\frac{{z_{1} }}{{\left| {z_{1} } \right|}}} \right)^{n} $$
(A11)
$$ \begin{aligned} & E_{n}^{21} = - \mu_{0} \beta \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right]\\ & \left[ {\left( {k_{T} H_{n - 1}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n - 1} - H_{n}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n} } \right)} \right. \\ & \left. \left( {\beta \overline{z} + \eta } \right){\text{e}}^{i\theta } - \left( H_{n}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n}\right.\right. \\ & \left.\left. + k_{T} H_{n + 1}^{(1)} \left( {k_{T} \left| \zeta \right|} \right)\left( {\frac{\zeta }{\left| \zeta \right|}} \right)^{n + 1} \right)\left( {\beta z + \eta } \right){\text{e}}^{ - i\theta } \right] \end{aligned} $$
(A12)
$$\begin{aligned} & E_{n}^{22} = \mu_{2} k_{2} \left[ H_{n - 1}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{n - 1} {\text{e}}^{i\theta }\right.\\ & \left. - H_{n + 1}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{{n{ + }1}} {\text{e}}^{ - i\theta } \right]\end{aligned} $$
(A13)
$$\begin{aligned} & E_{n}^{23} = \mu_{2} k_{2} \left[ H_{n - 1}^{(1)} \left( {k_{2} \left| {z_{{1}} } \right|} \right)\left( {\frac{{z_{{1}} }}{{\left| {z_{{1}} } \right|}}} \right)^{n - 1} {\text{e}}^{i\theta }\right.\\ & \left. - H_{n + 1}^{(1)} \left( {k_{2} \left| {z_{{1}} } \right|} \right)\left( {\frac{{z_{{1}} }}{{\left| {z_{{1}} } \right|}}} \right)^{{n{ + }1}} {\text{e}}^{ - i\theta } \right]\end{aligned} $$
(A14)
$$ E_{n}^{32} = H_{n - 1}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{n - 1} {\text{e}}^{i\theta } - H_{n + 1}^{(2)} \left( {k_{2} \left| z \right|} \right)\left( {\frac{z}{\left| z \right|}} \right)^{{n{ + }1}} {\text{e}}^{ - i\theta } $$
(A15)
$$ E_{n}^{33} = H_{n - 1}^{(1)} \left( {k_{2} \left| {z_{{1}} } \right|} \right)\left( {\frac{{z_{{1}} }}{{\left| {z_{{1}} } \right|}}} \right)^{n - 1} {\text{e}}^{i\theta } - H_{n + 1}^{(1)} \left( {k_{2} \left| {z_{{1}} } \right|} \right)\left( {\frac{{z_{{1}} }}{{\left| {z_{{1}} } \right|}}} \right)^{{n{ + }1}} {\text{e}}^{ - i\theta } $$
(A16)
$$ E^{1} = \varphi_{0} \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta {\text{e}}^{ - i\alpha } + \overline{\zeta }{\text{e}}^{i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] $$
(A17)
$$ \begin{gathered} E^{2} = \mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta {\text{e}}^{ - i\alpha } + \overline{\zeta }{\text{e}}^{i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \hfill \\ \left[ {\left( {\beta \overline{z} + \eta } \right)\left( {ik_{T} {\text{e}}^{ - i\alpha } - 1} \right){\text{e}}^{i\theta } + \left( {\beta z + \eta } \right)\left( {ik_{T} {\text{e}}^{i\alpha } - 1} \right){\text{e}}^{ - i\theta } } \right] \hfill \\ \end{gathered} $$
(A18)

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Yang, Z., Bian, J., Xu, H. et al. Dynamic Response of Tunnel Structures in Inhomogeneous Medium Under SH Wave: Shear Modulus in Quadratic Functional Form. Acta Mech. Solida Sin. 36, 457–468 (2023). https://doi.org/10.1007/s10338-023-00395-y

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