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Reflection and transmission of plane wave at the interface between two distinct nonlocal triclinic micropolar generalized thermoelastic half spaces under DPL and LS theory

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Abstract

In this model, we have examined the problem of plane wave reflection and transmission occurring at the interface between two distinct nonlocal triclinic micropolar generalized thermoelastic half-spaces. It is evident that when an incident quasi-P (\(q_{P}\)) wave encounters the interface between two dissimilar half-spaces, distinct waves emerges in the form of reflected and transmitted coupled quasi-P \((C_{qP})\), coupled quasi-SV \((C_{qSV})\), coupled quasi-SH \((C_{qSH})\), coupled quasi-thermal \((C_{qT})\), and coupled quasi-transverse micro-rotational \((C_{qTM})\) waves, each possessing unique phase speed. Through the imposition of appropriate boundary conditions, we have derived the expressions for phase speed, reflection/transmission coefficients and energy ratios associated with the reflected and transmitted waves which depends on the nonlocal and micro-polar parameter, elastic properties of material and incident angle. To gain the behaviour of these parameters on reflection and transmission phenomena, we have conducted numerical simulations using MATLAB programming. The results have been presented graphically which provides a visual representation of nonlocal and micropolar parameters on phase speed and reflection/transmission coefficients as well as on the energy ratios of different waves under dual phase lag (DPL) and Lord–Shulman (LS) theory.

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Acknowledgements

The corresponding author is grateful to Guru Ghasidas Vishwavidyalaya, Bilaspur, Chhattisgarh, India for support under "Research Seed Money Grant Scheme" and is also thankful to the Department of Science and Technology, New Delhi, India for providing support under FIST program (Ref. No. SR/FST/MS-I/2022/122 dated 19 December 2022).

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Authors and Affiliations

  1. Department of Mathematics, Guru Ghasidas Vishwavidyalaya Bilaspur, Chhattisgarh, 495009, India

    Deepak Kumar & Brijendra Paswan

  2. Department of Mathematics, Buddha Post-Graduate College (Affiliation-Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur), Kushinagar, Uttar Pradesh, 274403, India

    Pooja Singh

  3. Department of Mathematics, Indian Institute of Technology (ISM) Dhanbad, Jharkhand, 826004, India

    Amares Chattopadhyay

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  1. Deepak Kumar
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Correspondence to Brijendra Paswan.

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Appendices

Appendix A

$$\begin{aligned}{} & {} \frac{C_{12}}{C_{11}}=a_{1}, ~\frac{C_{13}}{C_{11}}=a_{2}, \frac{C_{14}}{C_{11}}=a_{3}, ~\frac{C_{15}}{C_{11}}=a_{4}, ~\frac{C_{16}}{C_{11}}=a_{5}, ~\frac{C_{26}}{C_{11}}=a_{6}, ~\frac{C_{36}}{C_{11}}=a_{7}, ~\frac{C_{46}}{C_{11}}=a_{8}, ~\frac{C_{56}}{C_{11}}=a_{9},\\{} & {} \frac{C_{66}}{C_{11}}=a_{10}, ~\frac{C_{25}}{C_{11}}=a_{11}, ~\frac{C_{35}}{C_{11}}=a_{12}, ~\frac{C_{45}}{C_{11}}=a_{13}, ~\frac{C_{55}}{C_{11}}=a_{14}, ~\frac{C_{22}}{C_{11}}=a_{15}, ~\frac{C_{23}}{C_{11}}=a_{16}, ~\frac{C_{24}}{C_{11}}=a_{17}, \\{} & {} \frac{C_{34}}{C_{11}}=a_{18}, ~\frac{C_{44}}{C_{11}}=a_{19}, ~\frac{C_{33}}{C_{11}}=a_{20}. \\{} & {} L_{1}=a_{5}\left( p_{1}^{(n)}\right) ^{2}+a_{13}\left( p_{3}^{(n)}\right) ^{2}+\left( a_{3}+a_{9}\right) p_{1}^{(n)}p_{3}^{(n)},~~D_{1}=\left( p_{1}^{(n)}\right) ^{2}+a_{14}\left( p_{3}^{(n)}\right) ^{2}+2a_{14}p_{1}^{(n)}p_{3}^{(n)},\\{} & {} L_{2}=a_{4}\left( p_{1}^{(n)}\right) ^{2}+a_{12}\left( p_{3}^{(n)}\right) ^{2}+\left( a_{2}+a_{14}\right) p_{1}^{(n)}p_{3}^{(n)},~~L_{4}=b_{2}p_{1}^{(n)}\vartheta ^{*}\left( \tau ''+\left( c^{2}\eta \tau _{q}^{2}\right) /2\right) \\{} & {} D_{2}=a_{10}\left( p_{1}^{(n)}\right) ^{2}+a_{19}\left( p_{3}^{(n)}\right) ^{2}+2a_{8}p_{1}^{(n)}p_{3}^{(n)},~~D_{3}=a_{14}\left( p_{1}^{(n)}\right) ^{2}+a_{20}\left( p_{3}^{(n)}\right) ^{2}+2a_{12}p_{1}^{(n)}p_{3}^{(n)},\\{} & {} L_{3}=a_{9}\left( p_{1}^{(n)}\right) ^{2}+a_{18}\left( p_{3}^{(n)}\right) ^{2}+\left( a_{7}+a_{13}\right) p_{1}^{(n)}p_{3}^{(n)},~~\vartheta ^{*}=\left( 1-\epsilon ^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \\{} & {} D_{4}=B_{77}\left( p_{1}^{(n)}\right) ^{2}+B_{66}\left( p_{3}^{(n)}\right) ^{2}, ~~D_{5}=\left( \left( p_{1}^{(n)}\right) ^{2}+b_{1}\left( p_{3}^{(n)}\right) ^{2}\right) \tau ',\\{} & {} L_{5}=b_{3}p_{3}^{(n)}\vartheta ^{*}\left( \tau ''+\left( c^{2}\eta \tau _{q}^{2}\right) /2\right) \\{} & {} \Delta _{0}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =\Gamma _{22}(n)\Gamma _{33}(n)\Gamma _{44}(n)\Gamma _{55}(n)+\Gamma _{22}(n)\Gamma _{35}(n)\Gamma _{44}(n)\Gamma _{53}(n)-\Gamma _{32}(n)\Gamma _{23}(n)\Gamma _{44}(n)\Gamma _{55}(n),\\{} & {} \Delta _{1}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =\Gamma _{44}(n)\Gamma _{21}(n)\Gamma _{33}(n)\Gamma _{55}(n)+\Gamma _{44}(n)\Gamma _{35}(n)\Gamma _{53}(n)\Gamma _{21}(n)-\Gamma _{44}(n)\Gamma _{23}(n)\Gamma _{31}(n)\Gamma _{55}(n)\\{} & {} \qquad \qquad \qquad \qquad +\Gamma _{44}(n)\Gamma _{23}(n)\Gamma _{51}(n)\Gamma _{35}(n),\\{} & {} \Delta _{2}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =\Gamma _{21}(n)\Gamma _{32}(n)\Gamma _{44}(n)\Gamma _{55}(n)-\Gamma _{22}(n)\Gamma _{31}(n)\Gamma _{44}(n)\Gamma _{55}(n)+\Gamma _{22}(n)\Gamma _{51}(n)\Gamma _{35}(n)\Gamma _{44}(n),\\{} & {} \Delta _{3}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =0,\\ {}{} & {} \Delta _{4}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =-\Gamma _{21}(n)\Gamma _{32}(n)\Gamma _{44}(n)\Gamma _{53}(n)+\Gamma _{31}(n)\Gamma _{22}(n)\Gamma _{44}(n)\Gamma _{53}(n)+\Gamma _{51}(n)\Gamma _{44}(n)\Gamma _{22}(n)\Gamma _{33}(n)\\{} & {} \qquad \qquad \qquad \qquad -\Gamma _{51}(n)\Gamma _{44}(n)\Gamma _{23}(n)\Gamma _{32}(n) \end{aligned}$$
$$\begin{aligned}{} & {} \Gamma _{11}(n)=\left( p_{1}^{(n)}\right) ^{2}+a_{14}\left( p_{3}^{(n)}\right) ^{2}+2a_{14}p_{1}^{(n)}p_{3}^{(n)}-\rho c_{n}^{2}\left( 1-\epsilon ^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) ,~~\Gamma _{23}(n)=\Gamma _{32}(n),\\{} & {} \Gamma _{31}(n)=\Gamma _{13}(n),~~\Gamma _{12}(n)=a_{5}\left( p_{1}^{(n)}\right) ^{2}+a_{13}\left( p_{3}^{(n)}\right) ^{2}+\left( a_{3}+a_{9}\right) p_{1}^{(n)}p_{3}^{(n)},\\{} & {} \Gamma _{13}(n)=a_{4}\left( p_{1}^{(n)}\right) ^{2}+a_{12}\left( p_{3}^{(n)}\right) ^{2}+\left( a_{2}+a_{14}\right) p_{1}^{(n)}p_{3}^{(n)},\\{} & {} \Gamma _{15}(n)=\beta _{1}p_{1}^{(n)}, ~~\Gamma _{21}(n)=\Gamma _{12}(n),\\{} & {} \Gamma _{22}(n)=a_{10}\left( p_{1}^{(n)}\right) ^{2}+a_{19}\left( p_{3}^{(n)}\right) ^{2}+2a_{8}p_{1}^{(n)}p_{3}^{(n)}-\rho c_{n}^{2}\left( 1-\epsilon ^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) ,\\{} & {} \Gamma _{31}(n)=a_{4}\left( p_{1}^{(n)}\right) ^{2}+a_{12}\left( p_{3}^{(n)}\right) ^{2}+\left( a_{2}+a_{14}\right) p_{1}^{(n)}p_{3}^{(n)}\\{} & {} \Gamma _{33}(n)=a_{14}\left( p_{1}^{(n)}\right) ^{2}+a_{20}\left( p_{3}^{(n)}\right) ^{2}+2a_{12}p_{1}^{(n)}p_{3}^{(n)}-\rho c_{n}^{2}\left( 1-\epsilon ^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \\{} & {} \Gamma _{35}(n)=\beta _{3}p_{3}^{(n)},~~\Gamma _{44}(n)=\left( B_{77}\left( p_{1}^{(n)}\right) ^{2}+B_{66}\left( p_{3}^{(n)}\right) ^{2}-J\rho c_{n}^{2}\left( 1-\epsilon ^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \right) \\{} & {} \Gamma _{55}(n)=\left( \left( p_{1}^{(n)}\right) ^{2}+b_{1}\left( p_{3}^{(n)}\right) ^{2}\right) \tau '-ik\zeta \rho \left( 1-\epsilon ^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \\{} & {} \Gamma _{51}(n)=b_{2}p_{1}^{(n)}\left( 1-\epsilon ^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \left( \tau ''+c_{n}^{2}\tau _{q}^{2}/2\right) \\{} & {} \Gamma _{53}(n)=b_{3}p_{3}^{(n)}\left( 1-\epsilon ^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \left( \tau ''+c_{n}^{2}\tau _{q}^{2}/2\right) \end{aligned}$$

Appendix B

$$\begin{aligned}{} & {} \frac{C'_{12}}{C'_{11}}=a'_{1}, ~\frac{C'_{13}}{C'_{11}}=a'_{2}, ~\frac{C'_{14}}{C'_{11}}=a'_{3}, ~\frac{C'_{15}}{C'_{11}}=a'_{4}, ~\frac{C'_{16}}{C'_{11}}=a'_{5},\frac{C'_{26}}{C'_{11}}=a'_{6},\frac{C'_{36}}{C'_{11}}=a'_{7}, ~\frac{C'_{46}}{C'_{11}}=a'_{8}, ~\frac{C'_{56}}{C'_{11}}=a'_{9},\\{} & {} \frac{C'_{66}}{C_{11}}=a'_{10}, ~\frac{C'_{25}}{C'_{11}}=a'_{11}, ~\frac{C'_{35}}{C'_{11}}=a'_{12}, ~\frac{C'_{45}}{C'_{11}}=a'_{13}, ~\frac{C'_{55}}{C'_{11}}=a'_{14}, ~\frac{C'_{22}}{C'_{11}}=a'_{15}, ~\frac{C'_{23}}{C'_{11}}=a'_{16}, ~\frac{C'_{24}}{C'_{11}}=a'_{17}, \\{} & {} \frac{C'_{34}}{C'_{11}}=a'_{18}, ~\frac{C'_{44}}{C'_{11}}=a'_{19}, ~\frac{C'_{33}}{C'_{11}}=a'_{20} \\{} & {} L_{6}=a'_{5}\left( p_{1}^{(n)}\right) ^{2}+a'_{13}\left( p_{3}^{(n)}\right) ^{2}+\left( a'_{3}+a'_{9}\right) p_{1}^{(n)}p_{3}^{(n)},~~D_{6}=\left( p_{1}^{(n)}\right) ^{2}+a'_{14}\left( p_{3}^{(n)}\right) ^{2}+2a'_{14}p_{1}^{(n)}p_{3}^{(n)},\\{} & {} L_{7}=a'_{4}\left( p_{1}^{(n)}\right) ^{2}+a'_{12}\left( p_{3}^{(n)}\right) ^{2}+\left( a'_{2}+a'_{14}\right) p_{1}^{(n)}p_{3}^{(n)},~~L_{9}=b'_{2}p_{1}^{(n)}\vartheta ^{*}\left( \tau ''+\left( c^{2}\eta ' {\tau '_{q}}^{2}\right) /2\right) \\{} & {} D_{7}=a'_{10}\left( p_{1}^{(n)}\right) ^{2}+a'_{19}\left( p_{3}^{(n)}\right) ^{2}+2a'_{8}p_{1}^{(n)}p_{3}^{(n)},~~D_{8}=a'_{14}\left( p_{1}^{(n)}\right) ^{2}+a'_{20}\left( p_{3}^{(n)}\right) ^{2}+2a'_{12}p_{1}^{(n)}p_{3}^{(n)},\\{} & {} L_{8}=a'_{9}\left( p_{1}^{(n)}\right) ^{2}+a'_{18}\left( p_{3}^{(n)}\right) ^{2}+\left( a'_{7}+a'_{13}\right) p_{1}^{(n)}p_{3}^{(n)},~~\vartheta '^{*}=\left( 1-\epsilon '^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \\{} & {} {D_{9}}={B'_{77}}\left( p_{1}^{(n)}\right) ^{2}+{B'_{66}}\left( p_{3}^{(n)}\right) ^{2}, ~~{D_{10}}=\left( \left( p_{1}^{(n)}\right) ^{2}+b'_{1} \left( p_{3}^{(n)}\right) ^{2}\right) \tau ',\\{} & {} L_{10}=b'_{3}\vartheta '^{*}p_{3}^{(n)}\left( \tau '' +\left( c'^{2} \eta ' {\tau '_{q}}^{2}\right) /2\right) \\{} & {} \Delta _{5}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =\Gamma '_{44}(n)\Gamma '_{21}(n)\Gamma '_{33}(n)\Gamma '_{55}(n)+\Gamma '_{44}(n)\Gamma '_{35}(n)\Gamma '_{53}(n)\Gamma '_{21}(n)-\Gamma '_{44}(n)\Gamma '_{23}(n)\Gamma '_{31}(n)\Gamma '_{55}(n)\\{} & {} \qquad \qquad \qquad \qquad \quad +\Gamma '_{44}(n)\Gamma '_{23}(n)\Gamma '_{51}(n)\Gamma '_{35}(n),\\{} & {} \Delta _{6}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =\Gamma '_{21}(n)\Gamma '_{32}(n)\Gamma '_{44}(n)\Gamma '_{55}(n)-\Gamma '_{22}(n)\Gamma '_{31}(n)\Gamma '_{44}(n)\Gamma '_{55}(n)+\Gamma '_{22}(n)\Gamma '_{51}(n)\Gamma '_{35}(n)\Gamma '_{44}(n),\\{} & {} \Delta _{7}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =0,\\ {}{} & {} \Delta _{8}\left( p_{1}^{(n)},p_{3}^{(n)}\right) =-\Gamma '_{21}(n)\Gamma '_{32}(n)\Gamma '_{44}(n)\Gamma '_{53}(n)+\Gamma '_{31}(n)\Gamma '_{22}(n)\Gamma '_{44}(n)\Gamma '_{53}(n)+\Gamma '_{51}(n)\Gamma '_{44}(n)\Gamma '_{22}(n)\Gamma '_{33}(n)\\{} & {} \qquad \qquad \qquad \qquad \quad -\Gamma '_{51}(n)\Gamma '_{44}(n)\Gamma '_{23}(n)\Gamma '_{32}(n) \end{aligned}$$
$$\begin{aligned}{} & {} \Gamma '_{11}(n)=\left( p_{1}^{(n)}\right) ^{2}+a'_{14}\left( p_{3}^{(n)}\right) ^{2}+2a'_{14}p_{1}^{(n)}p_{3}^{(n)}-\rho ' c_{n}^{2}\left( 1-\epsilon '^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) ,\\{} & {} \Gamma '_{12}(n)=a'_{5}\left( p_{1}^{(n)}\right) ^{2}+a'_{13}\left( p_{3}^{(n)}\right) ^{2}+\left( a'_{3}+a'_{9}\right) p_{1}^{(n)}p_{3}^{(n)},\\{} & {} \Gamma '_{13}(n)=a'_{4}\left( p_{1}^{(n)}\right) ^{2}+a'_{12}\left( p_{3}^{(n)}\right) ^{2}+\left( a'_{2}+a'_{14}\right) p_{1}^{(n)}p_{3}^{(n)},~~\Gamma '_{15}(n)=\beta '_{1}p_{1}^{(n)}, ~~\Gamma '_{21}(n)=\Gamma '_{12}(n),\\{} & {} \Gamma '_{22}(n)=a'_{10}\left( p_{1}^{(n)}\right) ^{2}+a'_{19}\left( p_{3}^{(n)}\right) ^{2}+2a'_{8}p_{1}^{(n)}p_{3}^{(n)}-\rho ' c_{n}^{2}\left( 1-\epsilon '^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) ,\\{} & {} \Gamma '_{31}(n)=a'_{4}\left( p_{1}^{(n)}\right) ^{2}+a'_{12}\left( p_{3}^{(n)}\right) ^{2}+\left( a'_{2}+a'_{14}\right) p_{1}^{(n)}p_{3}^{(n)}\\{} & {} \Gamma '_{23}(n)=\Gamma '_{32}(n),~\Gamma '_{31}(n)=\Gamma '_{13}(n), \\{} & {} \Gamma '_{33}(n)=a'_{14}\left( p_{1}^{(n)}\right) ^{2}+a'_{20}\left( p_{3}^{(n)}\right) ^{2}+2a'_{12}p_{1}^{(n)}p_{3}^{(n)}-\rho ' c_{n}^{2}\left( 1-\epsilon '^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \\{} & {} \Gamma '_{35}(n)=\beta '_{3}p_{3}^{(n)},\\{} & {} \Gamma '_{44}(n)=\left( B'_{77}\left( p_{1}^{(n)}\right) ^{2}+B'_{66}\left( p_{3}^{(n)}\right) ^{2}-J' \rho ' c_{n}^{2}\left( 1-\epsilon '^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \right) \\{} & {} \Gamma '_{55}(n)=\left( \left( \left( p_{1}^{(n)}\right) ^{2}+b'_{1}\left( p_{3}^{(n)}\right) ^{2}\right) \tau '-ik \rho ' \left( 1-\epsilon '^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \right) \\{} & {} \Gamma '_{51}(n)=b'_{2} p_{1}^{(n)}\left( 1-\epsilon '^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \left( \tau ''+c_{n}^{2}{\tau '_{q}}^{2}/2\right) \\{} & {} \Gamma '_{53}(n)=b'_{3}p_{3}^{(n)}\left( 1-\epsilon '^{2}\left( \left( p_{1}^{(n)}\right) ^{2}+\left( p_{3}^{(n)}\right) ^{2}\right) \right) \left( \tau ''+c_{n}^{2}{\tau '_{q}}^{2}/2\right) \end{aligned}$$
$$\begin{aligned} {\left\{ \begin{array}{ll} &{}g_{1j}=1 ~(j=1,2,3,4,5); ~g_{1j}=-1 ~\left( j=6,7,8,9,10\right) ;\\ &{}g_{2j}=\frac{V_{j}}{V_{0}} ~(j=1,2,3,4,5); ~g_{2j}=-\frac{V_{j}}{V_{0}} ~\left( j=6,7,8,9,10\right) \\ &{}g_{3j}=\frac{W_{j}}{W_{0}} ~(j=1,2,3,4,5); ~g_{3j}=-\frac{W_{j}}{W_{0}} ~\left( j=6,7,8,9,10\right) ; ~g_{4j}=\frac{F_{j}}{F_{0}} ~~(j=1,2,3,4,5); \\ &{}g_{4j}=-\frac{F_{j}}{F_{0}} ~\left( j=6,7,8,9,10\right) ; ~~g_{5j}=\frac{G_{j}}{G_{0}} ~(j=1,2,3,4,5); ~g_{5j}=-\frac{G_{j}}{G_{0}} ~~\left( j=6,7,8,9,10\right) ;\\ &{}g_{6j}=\left\{ a_{4}p_{1}^{(n)}+a_{9}\left( p_{1}^{(n)}+a_{13}p_{3}^{(n)}\right) V_{j}+\left\{ \left( a_{12}+a_{14}\right) p_{3}^{(n)}+p_{1}^{(n)}\right\} W_{j}+\kappa _{1}F_{j}\right\} ~~(j,n=1,2,3,4,5);\\ &{}g_{6j}=-\left\{ a'_{4}p_{1}^{(n)}+a'_{9}\left( p_{1}^{(n)}+a'_{13}p_{3}^{(n)}\right) V_{j}+\left\{ \left( a'_{12}+a'_{14}\right) p_{3}^{(n)}+p_{1}^{(n)}\right\} W_{j}+\kappa '_{1}F_{j}\right\} ~~(j,n=6,7,8,9,10);\\ &{}g_{7j}=\left\{ a_{3}p_{1}^{(n)}+\left( a_{8}p_{1}^{(n)}+a_{19}p_{3}^{(n)}\right) V_{j}+\left\{ \left( a_{13}+a_{18}\right) p_{3}^{(n)}+p_{1}^{(n)}\right\} W_{j}+\kappa _{3}F_{j}\right\} ~~(j,n=1,2,3,4,5);\\ &{}g_{7j}=-\left\{ a'_{3}p_{1}^{(n)}+\left( a'_{8}p_{1}^{(n)}+a'_{19}p_{3}^{(n)}\right) V_{j}+\left\{ \left( a'_{13}+a'_{18}\right) p_{3}^{(n)}+p_{1}^{(n)}\right\} W_{j}+\kappa '_{3}F_{j}\right\} ~~(j,n=6,7,8,9,10);\\ &{}g_{8j}=\left\{ a_{2}p_{1}^{(n)}+\left( a_{7}p_{1}^{(n)}+a_{18}p_{3}^{(n)}\right) V_{j}+\left\{ \left( a_{12}+a_{20}\right) p_{3}^{(n)}+p_{1}^{(n)}\right\} W_{j}-\beta _{33}G_{j}\right\} ~(j,n=1,2,3,4,5);\\ &{}g_{8j}=-\left\{ a'_{2}p_{1}^{(n)}+\left( a'_{7}p_{1}^{(n)}+a'_{18}p_{3}^{(n)}\right) V_{j}+\left\{ \left( a'_{12}+a'_{20}\right) p_{3}^{(n)}+p_{1}^{(n)}\right\} W_{j}-\beta '_{33}G_{j}\right\} ~(j,n=6,7,8,9,10);\\ &{}g_{9j}=B_{66}p_{3}^{(n)}F_{j} ~~(j,n=1,2,3,4,5); ~g_{9j}=-B'_{66}p_{3}^{(n)}F_{j} ~~(j,n=6,7,8,9,10);\\ &{}g_{10j}=K_{3}p_{3}^{(n)}G_{j} ~~(j,n=1,2,3,4,5); ~g_{10j}=-K'_{3}p_{3}^{(n)}G_{j} ~~(j,n=6,7,8,9,10);\\ &{}f_{1}= f_{2}= f_{3}= f_{4}= f_{5}=-1,\\ &{}f_{6}= -\left\{ a_{4}p_{1}^{(0)}+a_{9}\left( p_{1}^{(0)}+a_{13}p_{3}^{(0)}\right) V_{0}+\left\{ \left( a_{12}+a_{14}\right) p_{3}^{(0)}+p_{1}^{(0)}\right\} W_{0}+\kappa _{1}F_{0}\right\} , \\ &{}f_{7}=-\left\{ a_{3}p_{1}^{(0)}+\left( a_{8}p_{1}^{(0)}+a_{19}p_{3}^{(0)}\right) V_{0}+\left\{ \left( a_{13}+a_{18}\right) p_{3}^{(0)}+p_{1}^{(0)}\right\} W_{0}+\kappa _{3}F_{0}\right\} , ~f_{9}=-B_{66}p_{3}^{(0)}F_{0},\\ &{}f_{8}=-\left\{ a_{2}p_{1}^{(0)}+\left( a_{7}p_{1}^{(0)}+a_{18}p_{3}^{(0)}\right) V_{0}+\left\{ \left( a_{12}+a_{20}\right) p_{3}^{(0)}+p_{1}^{(0)}\right\} W_{0}-\beta _{33}G_{0}\right\} , ~f_{10}=-K_{3}p_{3}^{(0)}G_{0}\\ &{}P_{j}^{*}= \left\{ g_{6j}+g_{7j}V_{j}+g_{8j}W_{j}+g_{9j}F_{j}+K_{3}\right\} ; ~(j=1,2,3,4,5)\\ &{}P_{j}^{*}= \left\{ g_{6j}+g_{7j}V_{j}+g_{8j}W_{j}+g_{9j}F_{j}+K'_{3}\right\} ; ~\left( j=6,7,8,9,10\right) \end{array}\right. } \end{aligned}$$

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Kumar, D., Paswan, B., Singh, P. et al. Reflection and transmission of plane wave at the interface between two distinct nonlocal triclinic micropolar generalized thermoelastic half spaces under DPL and LS theory. Acta Mech 235, 3245–3270 (2024). https://doi.org/10.1007/s00707-024-03893-3

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