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Nonlocal analysis of Rayleigh-type wave propagating in a gradient layered structure

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Abstract

The present article aims to study the propagation behavior of Rayleigh-type waves using the nonlocal theory of elasticity in a layered structure constituted of a gradient transversely isotropic stratum perfectly bonded with a gradient monoclinic substrate. At first a constitutive relation is established for the assumed layered structure. Thereafter in view of suitable boundary conditions dispersion relation for the propagation of Rayleigh-type wave is obtained by considering a complex quantity wavenumber. The obtained result well agrees with the classical result and therefore validates the present study. The phase velocities and the attenuation coefficient for the Rayleigh-type wave propagation are numerically computed for the materials CdSe and LiNbO3; and the same are illustrated graphically. A significant effect of the affecting parameters on the propagation and the attenuation curves are depicted against the wavenumber. Comparative analysis of the influence of these parameters on the propagation and attenuation of Rayleigh-type waves is marked distinctly which serves as a salient feature of the present study. The techniques utilised the present problem and the obtained results may find potential application in various aspects.

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No data associated in the manuscript.

References

  1. A.C. Eringen, Nonlocal Continuum Field Theories (Springer Verlag, New York, 2001)

    MATH  Google Scholar 

  2. G.Z. Voyiadjis, Handbook of Nonlocal Continuum Mechanics for Materials and Structures (Springer Nature, Berlin, 2019)

    Book  Google Scholar 

  3. A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54(9), 4703–4710 (1983)

    Article  ADS  Google Scholar 

  4. A.C. Eringen, Theory of Nonlocal Elasticity and Some Applications. Princeton Univ NJ Dept of Civil Engineering (1984)

  5. E. Kroner, Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3(5), 731–742 (1967)

    Article  MATH  Google Scholar 

  6. A.C. Eringen, Nonlocal Continuum Field Theories (Springer, Berlin, 2002)

    MATH  Google Scholar 

  7. A.C. Eringen, D. Edelen, On nonlocal elasticity. Int. J. Eng. Sci. 10(3), 233–248 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. M.N.L. Narasimhan, B.M. McCay, Dispersion of surface waves in nonlocal dielectric fluids. Arch. Mech 33(3), 385–400 (1981)

    MATH  Google Scholar 

  9. L.L. Ke, Y.S. Wang, Z.D. Wang, Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory. Compos. Struct. 94(6), 2038–2047 (2012)

    Article  Google Scholar 

  10. L.H. Tong, Y.S. Liu, D.X. Geng, S.K. Lai, Nonlinear wave propagation in porous materials based on the Biot theory. J. Acoust. Soc. Am. 142(2), 756–770 (2017)

    Article  ADS  Google Scholar 

  11. C. Liu, J. Yu, X. Wang, B. Zhang, X. Zhang, H. Zhou, Reflection and transmission of elastic waves through nonlocal piezoelectric plates sandwiched in two solid half-spaces. Thin-Walled Struct. 168, 108306 (2021)

    Article  Google Scholar 

  12. J. Yu, C. Liu, C. Yang, B. Zhang, X. Zhang, Y. Zhang, Elastic wave attenuation in a functionally graded viscoelastic couple stress plate, sandwiched between two elastic half-spaces. Appl. Math. Model. 108, 670–684 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Roy, D.P. Acharya, S. Acharya, Rayleigh wave in a rotating nonlocal magnetoelastic half-plane. J. Theor. Appl. Mech. 45(4), 61 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Tong, Y. Yu, W. Hu, Y. Shi, C. Xu, On wave propagation characteristics in fluid saturated porous materials by a nonlocal Biot theory. J. Sound Vib. 379, 106–118 (2016)

    Article  ADS  Google Scholar 

  15. V.T.N. Anh, P.C. Vinh, The incompressible limit method and Rayleigh waves in incompressible layered nonlocal orthotropic elastic media. Acta Mech. 1-19 (2022)

  16. S. Kazemirad, L. Mongeau, Rayleigh wave propagation method for the characterization of a thin layer of biomaterials. J. Acoust. Soc. Am. 133(6), 4332–4342 (2013)

    Article  ADS  Google Scholar 

  17. S. Biswas, Surface waves in piezothermoelastic transversely isotropic layer lying over piezothermoelastic transversely isotropic half-space. Acta Mech (2020). https://doi.org/10.1007/s00707-020-02848-8

    Article  MATH  Google Scholar 

  18. S. Kumar, P.C. Pal, S. Bose, Propagation of Rayleigh waves in anisotropic layer overlying a semi-infinite sandy medium. Ain Shams Eng. J. 6, 621–627 (2015)

    Article  Google Scholar 

  19. S. Mana, T.C. Anjali, Rayleigh type wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer. Appl. Math. Model. 83, 590–613 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Gupta, S. Pramanik, S.K. Das, S. Saha, Dynamic analysis of wave propagation and buckling phenomena in carbon nanotubes (CNTs). Wave Motion 104, 102730 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. K.H. Lee, Characteristics of a crack propagating along the gradient in functionally gradient materials. Int. J. Solids Struct. 41(11–12), 2879–2898 (2004)

    Article  MATH  Google Scholar 

  22. N. Mawassy, H. Reda, J.F. Ganghoffer, H. Lakiss, Wave propagation analysis in non-local flexoelectric composite materials. Compos. Struct. 278, 114696 (2021)

    Article  Google Scholar 

  23. J. Xiao, J. Wang, Nonlinear Vibration of FGM Sandwich Nanoplates with Surface Effects. Acta Mech.. Solida Sin. 1–8 (2022)

  24. A.S. Sattari, Z.H. Rizvi, H.D. Aji, F. Wuttke, Study of wave propagation in discontinuous and heterogeneous media with the dynamic lattice method. Sci. Rep. 12(1), 1–16 (2022)

    Article  Google Scholar 

  25. N. Pradhan, S.K. Samal, Surface waves propagation in a homogeneous liquid layer overlying a monoclinic half-space. Appl. Math. Comput. 414, 126655 (2022)

    MathSciNet  MATH  Google Scholar 

  26. B. Kaur, B. Singh, Rayleigh-type surface wave in nonlocal isotropic diffusive materials. Acta Mech. 232(9), 3407–3416 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. P.C. Vinh, R.W. Ogden, On the Rayleigh wave speed in orthotropic elastic solids. Meccanica 40(2), 147–161 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Chattopadhyay, A. Keshri, On the generation of G type seismic waves. Acta Geol. Pol. 26(2), 131–138 (1978)

    Google Scholar 

  29. S. Saha, A.K. Singh, M.S. Chaki, Analysis of generated shear wave due to stress discontinuity in a monoclinic layered structure. Waves Random Complex Media 1–29 (2021)

  30. S. Saha, A.K. Singh, A. Chattopadhyay, Analysis of reflection and refraction of plane wave at the separating interface of two functionally graded incompressible monoclinic media under initial stress and gravity. Eur. Phys. J. Plus 135(2), 1–31 (2020)

    Article  Google Scholar 

  31. J. Achenbach, Wave Propagation in Elastic Solids (Elsevier, Hoboken, 2012)

    MATH  Google Scholar 

  32. M.A. Biot, Mechanics of incremental deformations (1965)

  33. J.N. Sharma, M. Pal, D. Chand, Propagation characteristics of Rayleigh waves in transversely isotropic piezothermoelastic materials. J. Sound Vib. 284(1–2), 227–248 (2005)

    Article  ADS  Google Scholar 

  34. B. Paswan, S.A. Sahu, A. Chattopadhyay, Reflection and transmission of plane wave through fluid layer of finite width sandwiched between two monoclinic elastic half-spaces. Acta Mech. 227(12), 3687–3701 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. V.A. Sveklo, Plane waves and Rayleigh waves in anisotropic media. In Dokl. Akad. Nauk SSSR (Vol. 59, pp. 871–874) (1948)

  36. Y. Pang, J.X. Liu, Y.S. Wang, X.F. Zhao, Propagation of Rayleigh-type surface waves in a transversely isotropic piezoelectric layer on a piezomagnetic half-space. J. Appl. Phys. 103(7), 074901 (2008)

    Article  ADS  Google Scholar 

  37. L. Gold, Rayleigh wave propagation on anisotropic (cubic) media. Phys. Rev. 104(6), 1532 (1956)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. W.W. Johnson, The propagation of Stoneley and Rayleigh waves in anisotropic elastic media. Bull. Seismol. Soc. Am. 60(4), 1105–1122 (1970)

    Google Scholar 

  39. D. Royer, E. Dieulesaint, Rayleigh wave velocity and displacement in orthorhombic, tetragonal, hexagonal, and cubic crystals. J. Acoust. Soc. Am. 76(5), 1438–1444 (1984)

    Article  ADS  Google Scholar 

  40. M. Destrade, M. Destrade, Rayleigh waves in anisotropic crystals rotating about the normal to a symmetry plane. J. Appl. Mech. 71(4), 516–520 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author Mr. Nirakara Pradhan conveys his sincere thanks to Kalinga Institute of Industrial Technology Deemed to be University for providing fellowship and facilitating us with its best research facility.

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No funding was received for conducting this study.

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Authors and Affiliations

  1. Department of Mathematics, School of Applied Sciences, KIIT Deemed to be University, Bhubaneswar, 751024, India

    Nirakara Pradhan, Shalini Saha & Sapan Kumar Samal

  2. Christ (Deemed to be University), Bangalore, 560029, India

    Snehamoy Pramanik

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  1. Nirakara Pradhan
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  2. Shalini Saha
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Correspondence to Shalini Saha.

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Appendices

Appendix A

$$\begin{aligned} F_1&= B'_{22} B'_{33} k^2H^2 - 2 B'_{23} B'_{33} k^2H^2 - 2 B'^2_{22} k^2H^2 (c/\beta _1)^2 \epsilon ^2_1 + 2 B'_{22} B'_{23} k^2H^2 (c/\beta _1)^2 \epsilon ^2_1 \\& \quad - 4 B'_{22} B'_{33} k^2H^2 (c/\beta _1)^2 \epsilon ^2_1 + 4 B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^4_1,\\ F_2&= 4 a B'_{22} B'_{33} kH - 4 a B'_{23} B'_{33} kH - 2 a B'^2_{22} kH (c/\beta _1)^2 \epsilon ^2_1 \\&\quad + 2 a B'_{22} B'_{23} kH (c/\beta _1)^2 \epsilon ^2_1 - 4aB'_{22} B'_{33} kH (c/\beta _1)^2 \epsilon ^2_1,\\ F_3&= 2 a^2 B'_{22}B'_{33} - 2 a^2 B'_{23}B'_{33} + 4 B'_{22}B'_{23} k^2H^2 - 4 B'_{22}B'_{22} k^2H^2 + 2 B'^2_{22} k^2H^2 (c/\beta _1)^2 \\& \quad - 2 B'_{22}B^2_{33} k^2H^2 (c/\beta _1)^2 + 4 B'_{22}B'_{33} k^2H^2 (c/\beta _1)^2 + 8 B'^2_{22} k^2H^2 (c/\beta _1)^2 \epsilon ^2_1\\& \quad- 4 B'_{22}B'_{23} k^2H^2 (c/\beta _1)^2 \epsilon ^2_1 + 4 B'_{22}B'_{33} k^2H^2 (c/\beta _1)^2 \epsilon ^2_1 - 8 B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^2_1 - 8 B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^4_1,\\ F_4&= 4 a B'_{22} B'_{23} kH - 4 a B'_{22} B'_{33} kH + 2 a B'^2_{22} kH (c/\beta _1)^2 - 2 a B'_{22} B'_{23} kH (c/\beta _1)^2 \\& \quad+4 a B'_{22} B'_{33} kH (c/\beta _1)^2 + 2 a B'^2_{22} kH (c/\beta _1)^2 \epsilon ^2_1 - 2 a B'_{22} B'_{23} kH (c/\beta _1)^2 \epsilon ^2_1 + 4 a B'_{22} B'_{33} kH (c/\beta _1)^2 \epsilon ^2_1,\\ F_5&= 2 a^2 B'_{22} B'_{23} - 2 a^2 B^2_{23} + 2 B'^2_{22} k^2H^2 - 2 B'_{22} B'_{23} k^2H^2 - 6 B'^2_{22} k^2H^2 (c/\beta _1)^2 + 2 B'_{22} B'_{23} k^2H^2 (c/\beta _1)^2 \\&\quad + 4 B'^2_{22} k^2H^2 (c/\beta _1)^4 - 6 B'^2_{22} k^2H^2 (c/\beta _1)^2 \epsilon ^2_1 + 2 B'_{22} B'_{23} k^2H^2 (c/\beta _1)^2 \epsilon ^2_1 + 8 B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^2_1 + 4 B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^4_1,\\ J_1&= C'^2_{34} k^2H^2 \rho '^2_1 - C'_{33} C'_{44} k^2H^2 \rho '^2_1 + B'_{22} C'_{33} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 \\& \quad+ B'_{22} C'_{44} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 - B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^4_2 \rho '^2_2, \\ J_2&= -2 i C'_{24} C'_{33} k^2H^2 \rho '^2_1 + 2 i C'_{23} C'_{34} k^2H^2 \rho '^2_1 + 2 i B'_{22} C'_{24} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2\\& \quad+ 2 i B'_{22} C'_{34} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 + 2 C'^2_{34} kH \rho '^2_1 log\,b - 2 C'_{33} C'_{44} kH \rho '^2_1 log\,b \\&\quad + B'_{22} C'_{33} kH (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 log\,b + B'_{22} C'_{44} kH (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 log\,b,\\ J_3&= -C^2_{23} k^2H^2 \rho '^2_1 + C22 C'_{33} k^2H^2 \rho '^2_1 + 2 C'_{24} C'_{34} k^2H^2 \rho '^2_1 \\&\quad - 2 C'_{23} C'_{44} k^2H^2 \rho '^2_1 - B'_{22} C'_{33} k^2H^2 (c/\beta _1)^2 \rho '_1 \rho '_2 - B'_{22} C'_{44} k^2H^2 (c/\beta _1)^2 \rho '_1 \rho '_2 \\ &\quad - B'_{22} C'_{22} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 - B'_{22} C'_{33} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 \\&\quad - 2 B'_{22} C'_{44} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 + 2 B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^2_2 \rho '^2_2 + 2 B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^4_2 \rho '^2_2 - 3 i C'_{24} C'_{33} kH \rho '^2_1 log\,b \\& \quad+ 3 i C'_{23} C'_{34} kH \rho '^2_1 log\,b + i B'_{22} C'_{24} kH (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 log\,b \\&\quad + i B'_{22} C'_{34} kH (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 log\,b + 2 C'^2_{34} \rho '^2_1 log\,b -2 C'_{33} C'_{44} \rho '_1 log\,b,\\ J_4&= -2 i C'_{23} C'_{24} k^2H^2 \rho '^2_1 + 2 i C'_{22} C'_{34} k^2H^2 \rho '^2_1 - 2 i B'_{22} C'_{24} k^2H^2 (c/\beta _1)^2 \rho '_1 \rho '_2\\&\quad - 2 i B'_{22} C'_{34} k^2H^2 (c/\beta _1)^2 \rho '_1 \rho '_2\\& \quad- 2 i B'_{22} C'_{24} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 - 2 i B'_{22} C'_{34} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 - C^2_{23} kH \rho '^2_1 log\,b + C'_{22} C'_{33} kH \rho '^2_1 log\,b\\&\quad + 2 C'_{24} C'_{34} kH \rho '^2_1 log\,b - 2 C'_{23} C'_{44} kH \rho '^2_1 log\,b - B'_{22} C'_{33} kH (c/\beta _1)^2 \rho '_1 \rho '_2 log\,b - B'_{22} C'_{44} kH (c/\beta _1)^2 \rho '_1 \rho '_2 log\,b\\&\quad - B'_{22} C'_{33} kH (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 log\,b - B'_{22} C'_{44} kH (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 log\,b\\&\quad - 2 i C'_{24} C'_{33} \rho '^2_1 log\,b + 2 i C'_{23} C'_{34} \rho '^2_1 log\,b,\\ J_5&= C^2_{24} k^2H^2 \rho '^2_1 - C'_{22} C'_{44} k^2H^2 \rho '^2_1 + B'_{22} C'_{22} k^2H^2 (c/\beta _1)^2 \rho '_1 \rho '_2 \\&\quad + B'_{22} C'_{44} k^2H^2 (c/\beta _1)^2 \rho '_1 \rho '_2 + B'_{22} C'_{22} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 \\&\quad + B'_{22} C'_{44} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 - B'^2_{22} k^2H^2 (c/\beta _1)^4 \rho '^2_2 - 2 B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^2_2 \rho '^2_2 \\ &\quad - B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^4_2 \rho '^2_2 + q^4 (C'^2_{34} k^2H^2 \rho '^2_1 - C'_{33} C'_{44} k^2H^2 \rho '^2_1 + B'_{22} C'_{33} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2\\&\quad + B'_{22} C'_{44} k^2H^2 (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 - B'^2_{22} k^2H^2 (c/\beta _1)^4 \epsilon ^4_2 \rho '^2_2) - i C'_{23} C'_{24} kH \rho '^2_1 log\,b \\&\quad + i C'_{22} C'_{34} kH \rho '^2_1 log\,b - i B'_{22} C'_{24} kH (c/\beta _1)^2 \rho '_1 \rho '_2 log\,b - i B'_{22} C'_{34} kH (c/\beta _1)^2 \rho '_1 \rho '_2 log\,b \\&\quad - i B'_{22} C'_{24} kH (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 log\,b - i B'_{22} C'_{34} kH (c/\beta _1)^2 \epsilon ^2_2 \rho '_1 \rho '_2 log\,b \\&\quad + 2 C'_{24} C'_{34} \rho '^2_1 log\,b - C'_{23} C'_{44} \rho '^2_1 log\,b. \end{aligned}$$
$$\begin{aligned} \varDelta _{11}&= (s_1\gamma _1+i)e^{-ks_1H},\,\varDelta _{12}=(s_2\gamma _2+i)e^{-ks_2H},\,\varDelta _{13}=(i-s_1\gamma _3)e^{ks_1H},\,\varDelta _{14}=(i-s_2\gamma _4)e^{ks_2H},\\ \varDelta _{15}&= \varDelta _{16}=0, \varDelta _{21}=(B'_{23}\gamma _1i+B'_{33}s_1)e^{-ks_1H},\,\,\,\varDelta _{22}=(B'_{23}\gamma _2i+B'_{33}s_2)e^{-ks_2H},\varDelta _{23}=(B'_{23}\gamma _3i+B'_{33}s_1)e^{ks_1H},\\ \varDelta _{24}&= (B'_{23}\gamma _4i+B'_{33}s_2)e^{ks_2H},\varDelta _{31}=(B'_{22}-B'_{23})(s_1\gamma _1+i),\varDelta _{32}=(B'_{22}-B'_{23})(s_2\gamma _2+i),\varDelta _{25}=\varDelta _{26}=0,\\ \varDelta _{33}&= (B'_{22}-B'_{23})(i-s_1\gamma _3),\,\,\,\varDelta _{34}=(B'_{22}-B'_{23})(i-s_2\gamma _4), \varDelta _{35}=2[C'_{34}q_1-(C'_{24}i-C'_{44}q_1)\eta _3-C'_{44}i],\\ \varDelta _{36}&= 2[C'_{34}q_2-(C'_{24}i-C'_{44}q_2)\eta _4-C'_{44}i], \varDelta _{41}=B'_{23}i\gamma _1+B'_{33}s_1,\,\varDelta _{42}=B'_{23}i\gamma _2+B'_{33}s_2,\,\varDelta _{43}=B'_{23}i\gamma _3+B'_{33}s_1,\\ \varDelta _{44}&= B'_{23}i\gamma _4+B'_{33}s_2,\varDelta _{45}=C'_{33}q_1-(C'_{23}i-C'_{34}q_1)\eta _3-C'_{34}i,\,\,\,\varDelta _{46}=C'_{33}q_2-(C'_{23}i-C'_{34}q_2)\eta _4-C'_{34}i, \varDelta _{51}=\gamma _1,\\ \varDelta _{52}&= \gamma _2, \varDelta _{53}=\gamma _3, \varDelta _{54}=\gamma _4, \varDelta _{55}=-\eta _3, \varDelta _{56}=-\eta _4, \varDelta _{61}=\varDelta _{62}=\varDelta _{63}=\varDelta _{64}=1, \varDelta _{65}=\varDelta _{66}=-1. \end{aligned}$$

Appendix B

$$\begin{aligned} \gamma '_1&= -\dfrac{\left( B'_{22}+B'_{23}\right) is_1+\left( B'_{22}-B'_{23}\right) \dfrac{a}{k}i}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_1+\left( B'_{22}-B'_{23}\right) \dfrac{a}{k}s_1+2\rho '_1c^2},\\ \gamma '_2&= -\dfrac{\left( B'_{22}+B'_{23}\right) is_2+\left( B'_{22}-B'_{23}\right) \dfrac{a}{k}i}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_2+\left( B'_{22}-B'_{23}\right) \dfrac{a}{k}s_2+2\rho '_1c^2},\\ \gamma '_{3}&= -\dfrac{-\left( B'_{22}+B'_{23}\right) is_1+\left( B'_{22}-B'_{23}\right) \dfrac{a}{k}i}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_1-\left( B'_{22}-B'_{23}\right) \dfrac{a}{k}s_1+2\rho '_1c^2},\\ \gamma '_{4}&= -\dfrac{-\left( B'_{22}+B'_{23}\right) is_2+\left( B'_{22}-B'_{23}\right) \dfrac{a}{k}i}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_2-\left( B'_{22}-B'_{23}\right) \dfrac{a}{k}s_2+2\rho '_1c^2},\\ \eta '_3&= -\dfrac{-C'_{24}-\left( C'_{23}+C'_{44}\right) iq_1+C'_{34}q^2_1}{-C'_{22}-2C'_{24}iq_1+C'_{44}q^2_1+\rho '_2c^2},\,\,\,\eta '_4=-\dfrac{-C'_{24}-\left( C'_{23}+C'_{44}\right) iq_2+C'_{34}q^2_2}{-C'_{22}-2C'_{24}iq_2+C'_{44}q^2_2+\rho '_2c^2}.\\ \varDelta '_{11}&= (s_1\gamma '_1+i)e^{-ks_1H},\,\,\, \varDelta '_{12}=(s_2\gamma '_2+i)e^{-ks_2H},\,\,\,\varDelta '_{13}=(i-s_1\gamma '_3)e^{ks_1H},\,\,\,\varDelta '_{14}=(i-s_2\gamma '_4)e^{ks_2H}, \varDelta '_{15}=\varDelta '_{16}=0,\\ \varDelta _{21}&= (B'_{23}\gamma '_1i+B'_{33}s_1)e^{-ks_1H},\,\,\,\varDelta '_{22}=(B'_{23}\gamma '_2i+B'_{33}s_2)e^{-ks_2H},\varDelta '_{23}=(B'_{23}\gamma '_3i+B'_{33}s_1)e^{ks_1H},\\ \varDelta '_{24}&= (B'_{23}\gamma '_4i+B'_{33}s_2)e^{ks_2H}, \varDelta '_{25}=\varDelta '_{26}=0, \varDelta '_{31}=(B'_{22}-B'_{23})(s_1\gamma '_1+i),\,\,\,\varDelta '_{32}=(B'_{22}-B'_{23})(s_2\gamma '_2+i),\\ \varDelta '_{33}&= (B'_{22}-B'_{23})(i-s_1\gamma '_3),\,\,\,\varDelta '_{34}=(B'_{22}-B'_{23})(i-s_2\gamma '_4),\\ \varDelta '_{35}&= 2[C'_{34}q_1-(C'_{24}i-C'_{44}q_1)\eta '_3-C'_{44}i],\,\,\,\varDelta '_{36}=2[C'_{34}q_2-(C'_{24}i-C'_{44}q_2)\eta '_4-C'_{44}i],\\ \varDelta '_{41}&= B'_{23}i\gamma '_1+B'_{33}s_1,\,\varDelta '_{42}=B'_{23}i\gamma '_2+B'_{33}s_2,\,\varDelta '_{43}=B'_{23}i\gamma '_3+B'_{33}s_1,\,\varDelta '_{44}=B'_{23}i\gamma '_4+B'_{33}s_2,\\ \varDelta '_{45}&= C'_{33}q_1-(C'_{23}i-C'_{34}q_1)\eta '_3-C'_{34}i,\,\,\,\varDelta '_{46}=C'_{33}q_2-(C'_{23}i-C'_{34}q_2)\eta '_4-C'_{34}i.\,\,\varDelta '_{51}=\gamma '_1,\\ \varDelta '_{52}&= \gamma '_2, \varDelta '_{53}=\gamma '_3, \varDelta '_{54}=\gamma '_4, \varDelta '_{55}=-\eta '_3, \varDelta '_{56}=-\eta '_4, \varDelta '_{61}=\varDelta '_{62}=\varDelta '_{63}=\varDelta '_{64}=1, \varDelta '_{65}=\varDelta '_{66}=-1. \end{aligned}$$

Appendix C

$$\begin{aligned} \gamma ''_1&= -\dfrac{\left( B'_{22}+B'_{23}\right) is_1}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_1+2\rho '_1c^2\left\{ 1+k^2\epsilon ^2_1(1-s^2_1)\right\} }, \\ \gamma ''_2&= -\dfrac{\left( B'_{22}+B'_{23}\right) is_2}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_2+2\rho '_1c^2\left\{ 1+k^2\epsilon ^2_1(1-s^2_2)\right\} }, \\ \gamma ''_3&= -\dfrac{-\left( B'_{22}+B'_{23}\right) is_1}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_1+2\rho '_1c^2\left\{ 1+k^2\epsilon ^2_1(1-s^2_1)\right\} },\\ \gamma ''_4&= -\dfrac{-\left( B'_{22}+B'_{23}\right) is_2}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_2+2\rho '_1c^2\left\{ 1+k^2\epsilon ^2_1(1-s^2_2)\right\} },\\ \eta ''_3&= -\dfrac{-C'_{24}-\left( C'_{23}+C'_{44}\right) iq_1+C'_{34}q^2_1}{-C'_{22}-2C'_{24}iq_1+C'_{44}q^2_1+\rho '_2c^2\left\{ 1+\epsilon ^2_2k^2\left( 1-q^2_1\right) \right\} },\\ \eta ''_4&= -\dfrac{-C'_{24}-\left( C'_{23}+C'_{44}\right) iq_2+C'_{34}q^2_2}{-C'_{22}-2C'_{24}iq_2+C'_{44}q^2_2+\rho '_2c^2\left\{ 1+\epsilon ^2_2k^2\left( 1-q^2_2\right) \right\} },\\ \varDelta ''_{11}&= (s_1\gamma ''_1+i)e^{-ks_1H},\,\,\, \varDelta ''_{12}=(s_2\gamma ''_2+i)e^{-ks_2H},\,\,\,\varDelta ''_{13}=(i-s_1\gamma ''_3)e^{ks_1H},\,\,\,\varDelta ''_{14}=(i-s_2\gamma ''_4)e^{ks_2H},\\ \varDelta ''_{15}&= \varDelta ''_{16}=0, \varDelta ''_{21}=(B'_{23}\gamma ''_1i+B'_{33}s_1)e^{-ks_1H},\,\,\,\varDelta ''_{22}=(B'_{23}\gamma ''_2i+B'_{33}s_2)e^{-ks_2H},\\ \varDelta ''_{23}&= (B'_{23}\gamma ''_3i+B'_{33}s_1)e^{ks_1H},\,\,\,\varDelta ''_{24}=(B'_{23}\gamma ''_4i+B'_{33}s_2)e^{ks_2H}, \varDelta ''_{25}=\varDelta ''_{26}=0,\\ \varDelta ''_{31}&= (B'_{22}-B'_{23})(s_1\gamma ''_1+i),\,\,\,\varDelta ''_{32}=(B'_{22}-B'_{23})(s_2\gamma ''_2+i), \varDelta ''_{33}=(B'_{22}-B'_{23})(i-s_1\gamma ''_3),\,\,\,\varDelta ''_{34}=(B'_{22}-B'_{23})(i-s_2\gamma ''_4),\\ \varDelta ''_{35}&= 2[C'_{34}q_1-(C'_{24}i-C'_{44}q_1)\eta ''_3-C'_{44}i],\,\,\,\varDelta ''_{36}=2[C'_{34}q_2-(C'_{24}i-C'_{44}q_2)\eta ''_4-C'_{44}i],\\ \varDelta ''_{41}&= B'_{23}i\gamma ''_1+B'_{33}s_1,\,\varDelta ''_{42}=B'_{23}i\gamma ''_2+B'_{33}s_2,\,\varDelta ''_{43}=B'_{23}i\gamma ''_3+B'_{33}s_1,\,\varDelta ''_{44}=B'_{23}i\gamma ''_4+B'_{33}s_2,\\ \varDelta ''_{45}&= C'_{33}q_1-(C'_{23}i-C'_{34}q_1)\eta ''_3-C'_{34}i,\,\,\,\varDelta ''_{46}=C'_{33}q_2-(C'_{23}i-C'_{34}q_2)\eta ''_4-C'_{34}i,\,\,\varDelta ''_{51}=\gamma _1,\\ \varDelta ''_{52}&= \gamma ''_2, \varDelta ''_{53}=\gamma ''_3, \varDelta ''_{54}=\gamma ''_4, \varDelta ''_{55}=-\eta ''_3, \varDelta ''_{56}=-\eta ''_4, \varDelta ''_{61}=\varDelta ''_{62}=\varDelta ''_{63}=\varDelta ''_{64}=1, \varDelta ''_{65}=\varDelta ''_{66}=-1. \end{aligned}$$

Appendix D

$$\begin{aligned} \gamma '''_1&= -\dfrac{\left( B'_{22}+B'_{23}\right) is_1}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_1+2\rho '_1c^2},\,\,\,\gamma '''_2=-\dfrac{\left( B'_{22}+B'_{23}\right) is_2}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_2+2\rho '_1c^2}, \\ \gamma '''_3&= -\dfrac{-\left( B'_{22}+B'_{23}\right) is_1}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_1+2\rho '_1c^2},\,\,\,\gamma '''_4=-\dfrac{-\left( B'_{22}+B'_{23}\right) is_2}{-2B'_{22}+\left( B'_{22}-B'_{23}\right) s^2_2+2\rho '_1c^2},\\ \eta '''_3&= -\dfrac{-C'_{24}-\left( C'_{23}+C'_{44}\right) iq_1+C'_{34}q^2_1}{-C'_{22}-2C'_{24}iq_1+C'_{44}q^2_1+\rho '_2c^2},\,\,\,\eta '''_4=-\dfrac{-C'_{24}-\left( C'_{23}+C'_{44}\right) iq_2+C'_{34}q^2_2}{-C'_{22}-2C'_{24}iq_2+C'_{44}q^2_2+\rho '_2c^2},\\ \varDelta '''_{11}&= (s_1\gamma '''_1+i)e^{-ks_1H},\, \varDelta '''_{12}=(s_2\gamma '''_2+i)e^{-ks_2H},\,\varDelta '''_{13}=(i-s_1\gamma '''_3)e^{ks_1H},\\ \varDelta '''_{14}&= (i-s_2\gamma '''_4)e^{ks_2H}, \varDelta '''_{15}=\varDelta '''_{16}=0,\\ \varDelta '''_{21}&= (B'_{23}\gamma '''_1i+B'_{33}s_1)e^{-ks_1H},\,\,\,\varDelta '''_{22}=(B'_{23}\gamma '''_2i+B'_{33}s_2)e^{-ks_2H},\\ \varDelta '''_{23}&= (B'_{23}\gamma '''_3i+B'_{33}s_1)e^{ks_1H},\,\,\,\varDelta '''_{24}=(B'_{23}\gamma '''_4i+B'_{33}s_2)e^{ks_2H}, \varDelta '''_{25}=\varDelta '''_{26}=0,\\ \varDelta '''_{31}&= (B'_{22}-B'_{23})(s_1\gamma '''_1+i),\,\,\,\varDelta '''_{32}=(B'_{22}-B'_{23})(s_2\gamma '''_2+i),\\ \varDelta '''_{33}&= (B'_{22}-B'_{23})(i-s_1\gamma '''_3),\,\,\,\varDelta '''_{34}=(B'_{22}-B'_{23})(i-s_2\gamma '''_4),\\ \varDelta '''_{35}&= 2[C'_{34}q_1-(C'_{24}i-C'_{44}q_1)\eta '''_3-C'_{44}i],\,\,\,\varDelta '_{36}=2[C'_{34}q_2-(C'_{24}i-C'_{44}q_2)\eta '''_4-C'_{44}i],\\ \varDelta '''_{41}&= B'_{23}i\gamma '''_1+B'_{33}s_1,\,\varDelta '''_{42}=B'_{23}i\gamma '''_2+B'_{33}s_2,\,\varDelta '''_{43}=B'_{23}i\gamma '''_3+B'_{33}s_1,\,\varDelta '''_{44}=B'_{23}i\gamma '''_4+B'_{33}s_2,\\ \varDelta '''_{45}&= C'_{33}q_1-(C'_{23}i-C'_{34}q_1)\eta '''_3-C'_{34}i,\,\,\,\varDelta '''_{46}=C'_{33}q_2-(C'_{23}i-C'_{34}q_2)\eta '''_4-C'_{34}i,\,\,\varDelta '''_{51}=\gamma '''_1,\\ \varDelta '''_{52}&= \gamma '''_2, \varDelta '''_{53}=\gamma '''_3, \varDelta '''_{54}=\gamma '''_4, \varDelta '''_{55}=-\eta '''_3, \varDelta '''_{56}=-\eta '''_4, \varDelta '''_{61}=\varDelta '''_{62}=\varDelta '''_{63}=\varDelta '''_{64}=1, \varDelta '''_{65}=\varDelta '''_{66}=-1. \end{aligned}$$

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Pradhan, N., Saha, S., Samal, S.K. et al. Nonlocal analysis of Rayleigh-type wave propagating in a gradient layered structure. Eur. Phys. J. Plus 138, 410 (2023). https://doi.org/10.1140/epjp/s13360-023-04012-2

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