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The reflection and transmission of a quasi-longitudinal displacement wave at an imperfect interface between two nonlocal orthotropic micropolar half-spaces

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Abstract

This work is concerned with the reflection and transmission of quasi-longitudinal displacement wave incident at an imperfect interface between two nonlocal orthotropic micropolar half-spaces. The linear spring model is used to describe the imperfection of bonding behavior at the interface. Reflection, transmission coefficients and energy ratios of waves have been derived analytically for when a longitudinal displacement wave strikes for both imperfect and perfect interfaces. Finally, numerical examples are provided to show the effect of the imperfect interface, nonlocal parameter and incident angle on the reflection and transmission coefficients.

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  1. Faculty of Civil Engineering, Hanoi Architectural University, Km 10 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

    Do Xuan Tung

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  1. Do Xuan Tung
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Correspondence to Do Xuan Tung.

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Appendices

Appendix

The coefficients of characteristic equation

$$\begin{aligned} t_{6}= & {} k^2 (-Q_{22} +\rho c^2 \epsilon ^2 k^2) (-Q_{88} +\rho c^2 \epsilon ^2 k^2) (B_{44} -\rho c^2 \epsilon ^2 j k^2).\\ t_{4}= & {} Q_{22} \bigg ((B_{66} k^2 + \kappa _{12}) Q_{88} - \kappa _{21} (\kappa _{21} + Q_{88})\bigg ) + c^2 k^2 \bigg [\epsilon ^2 \bigg (\kappa _{21}^2 - (B_{66} k^2 + \kappa _{12}) Q_{22}\\&\quad + \kappa _{21} Q_{22}+j k^2 (-Q_{11}Q_{22} + (Q_{12} + Q_{78})^2)\bigg ) - \bigg (j Q_{22}+ \epsilon ^2 \big (\kappa _{12}-\kappa _{21}+ k^2 (B_{66}\\&\quad + j (Q_{22}+Q_{77}))\big )\bigg ) Q_{88}\bigg ] \rho + c^4 \epsilon ^2 k^4 \bigg [B_{66} \epsilon ^2 k^2 + 2 j (Q_{22} + Q_{88}) + \epsilon ^2 \big (\kappa _{12} - \kappa _{21}\\&\quad + j k^2 \big (Q_{11}+2Q_{22}+Q_{77}+2Q_{88}\big )\big )\bigg ] \rho ^2-3 c^6 \epsilon ^4 j k^6 (1+ \epsilon ^2 k^2) \rho ^3+ B_{44} k^2 \bigg (Q_{11} Q_{22}\\&\quad - (Q_{12}+Q_{78})^2 +Q_{77}Q_{88}-c^2 \big (Q_{22}+Q_{88}+\epsilon ^2 k^2 (Q_{11}+Q_{22}+Q_{77}+Q_{88})\big ) \rho \\&\quad +2 c^4 \epsilon ^2 k^2 (1+ \epsilon ^2 k^2) \rho ^2\bigg ).\\ t_{2}= & {} 2\kappa _{12}\kappa _{21}Q_{12}-B_{66}k^2 Q_{12}^2-\kappa _{12}Q_{12}^2+\kappa _{21}Q_{12}^2+B_{66}k^2Q_{11}Q_{22}+\kappa _{12}Q_{11} Q_{22}-\kappa _{21}Q_{11}Q_{22}\\&\quad - \kappa _{21}^2 Q_{77} + B_{44} k^2 Q_{11} Q_{77} + 2 \kappa _{12} \kappa _{21} Q_{78} - 2 B_{66} k^2 Q_{12} Q_{78} - 2\kappa _{12} Q_{12} Q_{78} + 2 \kappa _{21} Q_{12} Q_{78}\\&\quad -B_{66}k^2Q_{78}^2-\kappa _{12}Q_{78}^2+\kappa _{21}Q_{78}^2-\kappa _{12}^2Q_{88}+B_{66}k^2Q_{77}Q_{88}+\kappa _{12}Q_{77}Q_{88}-\kappa _{21} Q_{77}Q_{88}\\&\quad -c^2 \bigg [-\kappa _{21}^2+k^2 (B_{44} Q_{11} - j Q_{12}^2) + (B_{66} k^2 + \kappa _{12}) Q_{22} + k^2 \bigg (B_{44} Q_{77} + j \big (Q_{11} Q_{22}\\&\quad -Q_{78} (2 Q_{12}+Q_{78})\big )\bigg )+\big (\kappa _{12} + k^2 (B_{66}+jQ_{77})\big ) Q_{88}-\kappa _{21} (Q_{22}+Q_{88})\\&\quad +\epsilon ^2 k^2 \bigg (-\kappa _{12}^2 - \kappa _{21}^2 + k^2 \big (B_{44} Q_{11} - j Q_{12}^2+ B_{66} (Q_{11} + Q_{22})\big ) + \kappa _{12} (Q_{11} + Q_{22} + Q_{77}\\&\quad +Q_{88})-\kappa _{21} (Q_{11}+Q_{22}+Q_{77}+Q_{88})+k^2 \big ((B_{44}+B_{66}) Q_{77}+B_{66}Q_{88}+ j (Q_{11} (Q_{22}+ Q_{77})\\&\quad -Q_{78}(2 Q_{12}+Q_{78})+Q_{77}Q_{88})\big )\bigg )\bigg ] \rho + c^4 k^2 (1 + \epsilon ^2 k^2) (B_{44} + B_{44} \epsilon ^2 k^2 + 2 B_{66} \epsilon ^2 k^2 + 2 \epsilon ^2 \kappa _{12}\\&\quad -2 \epsilon ^2 \kappa _{21}+2\epsilon ^2 j k^2 Q_{11}+j Q_{22}+\epsilon ^2 j k^2 Q_{22}+2 \epsilon ^2 j k^2 Q_{77} + j Q_{88}\\&\quad +\epsilon ^2 j k^2 Q_{88}) \rho ^2 - 3 c^6 j k^4 (\epsilon + \epsilon ^3 k^2)^2 \rho ^3.\\ t_{0}= & {} (-Q_{11} +\rho c^2 (1 + \epsilon ^2 k^2)) \bigg (\kappa _{12}^2 - \kappa _{12} Q_{77} +\rho c^2 (1 + \epsilon ^2 k^2) \kappa _{12}\\&\quad + (Q_{77} -\rho c^2 (1 + \epsilon ^2 k^2)) \big (\kappa _{21} + k^2 (-B_{66} +\rho c^2 j (1 + \epsilon ^2 k^2))\big )\bigg ) \end{aligned}$$

The coefficients of equation system for imperfect interface

$$\begin{aligned} \begin{aligned}&I_{1k}=-(Q_{88}^{+}\alpha _{k}\xi _{k}+Q_{78}^{+}\beta _{k}-\kappa _{21}^{+}); I_{2k}=-(Q_{12}^{+}\alpha _{k}+Q_{22}^{+}\beta _{k}\xi _{k}); I_{3k}=-B_{44}^{+}\xi _{k}\\&I_{4k}=\left( Q_{88}^{+}\alpha _{k}\xi _{k}+Q_{78}^{+}\beta _{k}-\kappa _{21}^{+}+\dfrac{k_\mathrm{T}}{k}i\alpha _{k}\right) ;\quad I_{5k}=\left( Q_{12}^{+}\alpha _{k}+Q_{22}^{+}\beta _{k}\xi _{k}+\dfrac{k_\mathrm{N}}{k}i\beta _{k}\right) \\&I_{6k}=\left( B_{44}^{+}\dfrac{k}{k_\mathrm{C}}i\xi _{k}-1\right) \quad {(k=1, 2, 3 )}\\&I_{1k}=(Q_{88}^{-}\alpha _{k}\xi _{k}+Q_{78}^{-}\beta _{k}-\kappa _{21}^{-}); I_{2k}=(Q_{12}^{-}\alpha _{k}+Q_{22}^{-}\beta _{k}\xi _{k}); I_{3k}=B_{44}^{-}\xi _{k}\\&I_{4k}=-\dfrac{k_\mathrm{T}}{k}i\alpha _{k}; I_{5k}=-\dfrac{k_\mathrm{N}}{k}i\beta _{k}; I_{6k}=1\quad {(k=4, 5, 6 )}\\&I_{10}=(Q_{88}^{+}\alpha _{0}\xi _{0}+Q_{78}^{+}\beta _{0}-\kappa _{21}^{+}); I_{20}=(Q_{12}^{+}\alpha _{0}+Q_{22}^{+}\beta _{0}\xi _{0}); I_{30}=-B_{44}^{+}\xi _{0}\\&I_{40}=-\left( Q_{88}^{+}\alpha _{0}\xi _{0}+Q_{78}^{+}\beta _{0}-\kappa _{21}^{+}+\dfrac{k_\mathrm{T}}{k}i\alpha _{0}\right) ; I_{50}=-\left( Q_{12}^{+}\alpha _{0}+Q_{22}^{+}\beta _{0}\xi _{0}+\dfrac{k_\mathrm{N}}{k}i\beta _{0}\right) \\&I_{60}=-\left( B_{44}^{+}\dfrac{k}{k_\mathrm{C}}i\xi _{1}-1\right) \\ \end{aligned} \end{aligned}$$

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Tung, D.X. The reflection and transmission of a quasi-longitudinal displacement wave at an imperfect interface between two nonlocal orthotropic micropolar half-spaces. Arch Appl Mech 91, 4313–4328 (2021). https://doi.org/10.1007/s00419-021-02011-2

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