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Scattering of cylindrical inclusions in half space with inhomogeneous shear modulus due to SH wave

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Abstract

Dynamic responses around cylindrical inclusion in inhomogeneous medium are discussed. A mathematical model of inhomogeneous half space is established. The shear modulus of the medium is assumed to change in two dimensions. Based on complex function theory, the governing equations are derived. Meanwhile, the auxiliary function is introduced. By solving the governing equation, the analytical expressions of the displacement field and stress field formed by Bessel function and Hankel function are obtained. The unknown coefficients can be obtained by boundary conditions. According to numerical examples, the results of this paper are compared with published results to verify the validity of the method. Meanwhile, the effects of inhomogeneous parameters, reference wave number and burial location on the dynamic stress concentration factor (DSCF) around a cylindrical inclusion are discussed.

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Funding

This work is supported by the National Natural Science Foundation of China (No. 11872156) and the Research Team Project of Heilongjiang Natural Science Foundation (Grant No.TD2020A001) and the Fundamental Research Funds for the Central Universities (Grant No. 3072020CFT0202) 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

    Zailin Yang, Jinlai Bian, Yunqiu Song, Yong Yang & Menghan Sun

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

    Zailin Yang, Yong Yang & Menghan Sun

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  1. Zailin Yang
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  2. Jinlai Bian
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Appendix

Appendix

1.1 Expression for stress fields in half space

Incident waves in half-space

$$\begin{aligned} \tau_{rz}^{i} = & \frac{1}{2}\mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta {\text{e}}^{ - i\alpha } + \overline{\zeta }{\text{e}}^{i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \left[ {\left( {\beta \overline{z} + \gamma } \right)\left( {ik_{T} {\text{e}}^{ - i\alpha } - 1} \right){\text{e}}^{i\theta } + \left( {\beta z + \gamma } \right)\left( {ik_{T} {\text{e}}^{i\alpha } - 1} \right){\text{e}}^{ - i\theta } } \right] \\ \end{aligned}$$
(40)
$$\begin{aligned} \tau_{\theta z}^{i} = & \frac{1}{2}i\mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta e^{ - i\alpha } + \overline{\zeta }e^{i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \left[ {\left( {\beta \overline{z} + \gamma } \right)\left( {ik_{T} e^{ - i\alpha } - 1} \right)e^{i\theta } - \left( {\beta z + \gamma } \right)\left( {ik_{T} e^{i\alpha } - 1} \right)e^{ - i\theta } } \right] \\ \end{aligned}$$
(41)

Reflected waves in half-space

$$\begin{aligned} \tau_{rz}^{r} = & \frac{1}{2}\mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta e^{i\alpha } + \overline{\zeta }e^{ - i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \left[ {\left( {\beta \overline{z} + \gamma } \right)\left( {ik_{T} e^{i\alpha } - 1} \right)e^{i\theta } + \left( {\beta z + \gamma } \right)\left( {ik_{T} e^{ - i\alpha } - 1} \right)e^{ - i\theta } } \right] \\ \end{aligned}$$
(42)
$$\begin{aligned} \tau_{\theta z}^{r} = & \frac{1}{2}i\mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta e^{i\alpha } + \overline{\zeta }e^{ - i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \left[ {\left( {\beta \overline{z} + \gamma } \right)\left( {ik_{T} e^{i\alpha } - 1} \right)e^{i\theta } - \left( {\beta z + \gamma } \right)\left( {ik_{T} e^{ - i\alpha } - 1} \right)e^{ - i\theta } } \right] \\ \end{aligned}$$
(43)

Scattering waves generated by inclusion

$$\begin{aligned} \tau_{rz}^{s} = & - \frac{1}{2}\mu_{0} \beta \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \sum\limits_{n = - \infty }^{\infty } {A_{n} } \left[ {\left( {\left( {\beta \overline{z} + \gamma } \right)\left( {H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n} + H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{{2}} }}{{\left| {\zeta_{{2}} } \right|}}} \right)^{ - n} } \right)} \right.} \right. \\ & + \left. {k_{T} \left( {\beta z + \gamma } \right)\left( {\beta \overline{z} + \gamma } \right)\left( \begin{gathered} \frac{1}{{\beta z_{2} + \gamma }}H_{n + 1}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{{2}} }}{{\left| {\zeta_{{2}} } \right|}}} \right)^{ - n - 1} \hfill \\ - \frac{1}{{\beta z_{1} + \gamma }}H_{n - 1}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n - 1} \hfill \\ \end{gathered} \right)} \right){\text{e}}^{i\theta } \\ & + \left( {\left( {\beta z + \gamma } \right)\left[ {H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n} + H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{{2}} }}{{\left| {\zeta_{{2}} } \right|}}} \right)^{ - n} } \right]} \right. \\ & \left. { + \left. {k_{T} \left( {\beta z + \gamma } \right)\left( {\beta \overline{z} + \gamma } \right)\left( \begin{gathered} \frac{1}{{\beta \overline{z}_{1} + \gamma }}H_{n + 1}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n + 1} \hfill \\ - \frac{1}{{\beta \overline{z}_{2} + \gamma }}H_{n - 1}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{2} }}{{\left| {\zeta_{2} } \right|}}} \right)^{ - n + 1} \hfill \\ \end{gathered} \right)} \right){\text{e}}^{ - i\theta } } \right] \\ \end{aligned}$$
(44)
$$\begin{aligned} \tau_{\theta z}^{s} = & - \frac{1}{2}i\mu_{0} \beta \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \sum\limits_{n = - \infty }^{\infty } {A_{n} } \left[ {\left( {\left( {\beta \overline{z} + \gamma } \right)(H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n} + H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{{2}} }}{{\left| {\zeta_{{2}} } \right|}}} \right)^{ - n} } \right.} \right. \\ & + \left. {k_{T} \left( {\beta z + \gamma } \right)\left( {\beta \overline{z} + \gamma } \right)\left( \begin{gathered} \frac{1}{{\beta z_{2} + \gamma }}H_{n + 1}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{{2}} }}{{\left| {\zeta_{{2}} } \right|}}} \right)^{ - n - 1} \hfill \\ - \frac{1}{{\beta z_{1} + \gamma }}H_{n - 1}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n - 1} \hfill \\ \end{gathered} \right)} \right){\text{e}}^{i\theta } \\ & - \left( {\left( {\beta z + \gamma } \right)\left[ {H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n} + H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{{2}} }}{{\left| {\zeta_{{2}} } \right|}}} \right)^{ - n} } \right]} \right. \\ & \left. { + \left. {k_{T} \left( {\beta z + \gamma } \right)\left( {\beta \overline{z} + \gamma } \right)\left( \begin{gathered} \frac{1}{{\beta \overline{z}_{1} + \gamma }}H_{n + 1}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n + 1} \hfill \\ - \frac{1}{{\beta \overline{z}_{2} + \gamma }}H_{n - 1}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{2} }}{{\left| {\zeta_{2} } \right|}}} \right)^{ - n + 1} \hfill \\ \end{gathered} \right)} \right){\text{e}}^{ - i\theta } } \right] \\ \end{aligned}$$
(45)

Standing waves in inclusion

$$\tau_{rz}^{t} = \frac{1}{2}\mu_{2} k_{2} \sum\limits_{n = - \infty }^{\infty } {B_{n} \left[ {J_{n - 1} \left( {k_{2} \left| {z_{1} } \right|} \right)\left( {\frac{{z_{1} }}{{\left| {z_{1} } \right|}}} \right)^{n - 1} {\text{e}}^{i\theta } - J_{n + 1} \left( {k_{2} \left| {z_{1} } \right|} \right)\left( {\frac{{z_{1} }}{{\left| {z_{1} } \right|}}} \right)^{n + 1} {\text{e}}^{ - i\theta } } \right]}$$
(46)
$$\tau_{\theta z}^{t} = \frac{1}{2}i\mu_{2} k_{2} \sum\limits_{n = - \infty }^{\infty } {B_{n} \left[ {J_{n - 1} \left( {k_{2} \left| {z_{1} } \right|} \right)\left( {\frac{{z_{1} }}{{\left| {z_{1} } \right|}}} \right)^{n - 1} {\text{e}}^{i\theta } + J_{n + 1} \left( {k_{2} \left| {z_{1} } \right|} \right)\left( {\frac{{z_{1} }}{{\left| {z_{1} } \right|}}} \right)^{n + 1} {\text{e}}^{ - i\theta } } \right]}$$
(47)

1.2 Wave fields expression (Sect. 5.1) in boundary conditions

$$\begin{aligned} E^{1} & = \varphi_{0} \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right]\exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta {\text{e}}^{ - i\alpha } + \overline{\zeta }{\text{e}}^{i\alpha } } \right)} \right] \hfill \\ & + \varphi_{0} \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right]\exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta {\text{e}}^{i\alpha } + \overline{\zeta }{\text{e}}^{ - i\alpha } } \right)} \right] \hfill \\ \end{aligned}$$
(48)
$$\begin{aligned} E^{2} &= \mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta {\text{e}}^{ - i\alpha } + \overline{\zeta }{\text{e}}^{i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \quad \left[ {\left( {\beta \overline{z} + \gamma } \right)\left( {ik_{T} {\text{e}}^{ - i\alpha } - 1} \right){\text{e}}^{i\theta } + \left( {\beta z + \gamma } \right)\left( {ik_{T} {\text{e}}^{i\alpha } - 1} \right){\text{e}}^{ - i\theta } } \right] \\ & \quad + \mu_{0} \varphi_{0} \beta \exp \left[ {\frac{{ik_{T} }}{2}\left( {\zeta {\text{e}}^{i\alpha } + \overline{\zeta }{\text{e}}^{ - i\alpha } } \right)} \right]\exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \quad \left[ {\left( {\beta \overline{z} + \gamma } \right)\left( {ik_{T} {\text{e}}^{i\alpha } - 1} \right){\text{e}}^{i\theta } + \left( {\beta z + \gamma } \right)\left( {ik_{T} {\text{e}}^{ - i\alpha } - 1} \right){\text{e}}^{ - i\theta } } \right] \\ \end{aligned}$$
(49)
$$E_{n}^{11} = - \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right]\left[ {H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{1} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n} + H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{2} }}{{\left| {\zeta_{2} } \right|}}} \right)^{ - n} } \right]$$
(50)
$$E_{n}^{12} = J_{n} \left( {k_{2} \left| {z_{1} } \right|} \right)\left( {\frac{{z_{1} }}{{\left| {z_{1} } \right|}}} \right)^{n}$$
(51)
$$\begin{aligned} E_{n}^{21} = & \mu_{0} \beta \exp \left[ { - \frac{1}{2}\left( {\zeta + \overline{\zeta }} \right)} \right] \\ & \left[ {\left( {\left( {\beta \overline{z} + \gamma } \right)\left( {H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{1} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n} + H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{2} }}{{\left| {\zeta_{2} } \right|}}} \right)^{ - n} } \right)} \right.} \right. \\ & \left. { + k_{T} \left( {\beta z + \gamma } \right)\left( {\beta \overline{z} + \gamma } \right)\left( \begin{gathered} \frac{1}{{\beta z_{2} + \gamma }}H_{n + 1}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{{2}} }}{{\left| {\zeta_{{2}} } \right|}}} \right)^{ - n - 1} \hfill \\ - \frac{1}{{\beta z_{1} + \gamma }}H_{n - 1}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n - 1} \hfill \\ \end{gathered} \right)} \right){\text{e}}^{i\theta } \\ & + \left( {\left( {\beta z + \gamma } \right)\left( {H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{1} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n} + H_{n}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{2} }}{{\left| {\zeta_{2} } \right|}}} \right)^{ - n} } \right)} \right. \\ & \left. {\left. { + k_{T} \left( {\beta z + \gamma } \right)\left( {\beta \overline{z} + \gamma } \right)\left( \begin{gathered} \frac{1}{{\beta \overline{z}_{1} + \gamma }}H_{n + 1}^{(1)} \left( {k_{T} \left| {\zeta_{{1}} } \right|} \right)\left( {\frac{{\zeta_{1} }}{{\left| {\zeta_{1} } \right|}}} \right)^{n + 1} \hfill \\ - \frac{1}{{\beta \overline{z}_{2} + \gamma }}H_{n - 1}^{(1)} \left( {k_{T} \left| {\zeta_{2} } \right|} \right)\left( {\frac{{\zeta_{2} }}{{\left| {\zeta_{2} } \right|}}} \right)^{ - n + 1} \hfill \\ \end{gathered} \right)} \right){\text{e}}^{ - i\theta } } \right] \\ \end{aligned}$$
(52)
$$E_{n}^{22} = \mu_{2} k_{2} \left[ {J_{n - 1} \left( {k_{2} \left| {z_{1} } \right|} \right)\left( {\frac{{z_{1} }}{{\left| {z_{1} } \right|}}} \right)^{n - 1} {\text{e}}^{i\theta } - J_{n + 1} \left( {k_{2} \left| {z_{1} } \right|} \right)\left( {\frac{{z_{1} }}{{\left| {z_{1} } \right|}}} \right)^{n + 1} {\text{e}}^{ - i\theta } } \right]$$
(53)

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Yang, Z., Bian, J., Song, Y. et al. Scattering of cylindrical inclusions in half space with inhomogeneous shear modulus due to SH wave. Arch Appl Mech 91, 3449–3461 (2021). https://doi.org/10.1007/s00419-021-01975-5

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