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The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials

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Abstract

In this paper, the basic equations of motion, of Gauss and of heat conduction, together with constitutive relations for pyro- and piezoelectric media, are presented. Three thermoelastic theories are considered: classical dynamical coupled theory, the Lord–Shulman theory with one relaxation time and Green and Lindsay theory with two relaxation times. For incident elastic longitudinal, potential electric and thermal waves, referred to as qP, φ-mode and T-mode waves, which impinge upon the interface between two different transversal isotropic media, reflection and refraction coefficients are obtained by solving a set of linear algebraic equations. A case study is investigated: a system formed by two semi-infinite, hexagonal symmetric, pyroelectric–piezoelectric media, namely Cadmium Selenide (CdSe) and Barium Titanate (BaTiO3). Numerical results for the reflection and refraction coefficients are obtained, and their behavior versus the incidence angle is analyzed. The interaction with the interface give rises to different kinds of reflected and refracted waves: (i) two reflected elastic waves in the first medium, one longitudinal (qP-wave) and the other transversal (qSV-wave), and a similar situation for the refracted waves in the second medium; (ii) two reflected potential electric waves and a similar situation for the refracted waves; (iii) two reflected thermal waves and a similar situation for the refracted waves. The amplitudes of the reflected and refracted waves are functions of the incident angle, of the thermal relaxation times and of the media elastic, electric, thermal constants. This study is relevant to signal processing, sound systems, wireless communications, surface acoustic wave devices and military defense equipment.

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References

  1. Achenbach J.D.: Wave Propagation in Elastic Solids. North-Holland, New York (1994)

    Google Scholar 

  2. Auld B.A.: Acoustic Fields and Waves in Solids, vols 1 and 2, 2nd edn.. Kreiger, Malabar (1990)

    Google Scholar 

  3. Royer D., Dieulesaint E.: Elastic Waves in Solids I, Free and Guided Propagation. Springer, Berlin (2000)

    MATH  Google Scholar 

  4. Yang J.: An Introduction to the Theory of Piezoelectricity. Springer, Boston (2005)

    MATH  Google Scholar 

  5. Yang J.: The Mechanics of Piezoelectric Structures. World Scientific Publishing Co, Singapore (2008)

    Google Scholar 

  6. Ye Z.G.: Handbook of Dielectric, Piezoelectric and Ferroelectric Materials Synthesis, Properties and Applications. Woodhead Publishing Limited, CRC Press, New York (2008)

  7. Nayfeh A.H.: Wave Propagation in Layered Anisotropic Media. North-Holland, Amsterdam (1995)

    MATH  Google Scholar 

  8. Bardzokas, D.I., Kudryavtsev, B.A., Senik, N.A.: Wave Propagation in Electromagnetoelastic Media. Editorial URSS (2005)

  9. Crampin S.: Distinctive particle motion of surface waves as a diagnostic of anisotropic layering. Geophs. J. R. Astron. Soc. 40, 177–186 (1975)

    Article  Google Scholar 

  10. Alshits V.I., Darinskii A.N., Shuvalov A.L.: Theory of reflection of acoustoelectric waves in semi-infinite piezoelectric medium. III. Resonance reflection in the neighborhood of a branch of outflowing waves. Sov. Phys. Crystallogr. 36, 145–153 (1991)

    Google Scholar 

  11. Shana Z., Josse F.: Reflection of bulk waves at a piezoelectric crystal-viscous conductive liquid interface. J. Acoust. Soc. Am. 91, 854–860 (1992)

    Article  Google Scholar 

  12. Abd-alla A.N., Al-sheikh F.A.: Reflection and transmission of longitudinal waves under initial stresses at an interface in piezoelectric media. Arch. Appl. Mech. 79(9), 843–857 (2008)

    Article  Google Scholar 

  13. Burkov S.I., Sorokin B.P., Glushkov D.A., Aleksandrov K.S.: Reflection and refraction of bulk acoustic waves in piezoelectrics under uniaxial stress. Acoust. Phys. 55(2), 178–185 (2009)

    Article  Google Scholar 

  14. Sinha S.B., Elsibai K.A.: Reflexion of thermoelastic waves at a solid half-space with two relaxation times. J. Thermal Stresses 19, 763–777 (1996)

    Article  Google Scholar 

  15. Sinha S.B., Elsibai K.A.: Reflexion and refraction of thermoelastic waves at an interface of two semi-infinite media with two relaxation times. J. Thermal Stresses 20, 129–146 (1997)

    Article  MathSciNet  Google Scholar 

  16. Abd-alla A.N., Al-dawy A.A.S.: The reflection phenomena of SV-wave in a generalized thermoelastic medium. Int. J. Math. Math. Sci. 23(8), 529–546 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kumar R., Singh M.: Reflection/transmission of plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic half-spaces. Mater. Sci. Eng. A 472, 83–96 (2008)

    Article  Google Scholar 

  18. Singh B.: On the theory of generalized thermoelasticity for piezoelectric materials. Appl. Math. Comput. 171(1,1), 398–405 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sharma J.N., Walia V., Gupta S.K.: Reflection of piezothermoelastic waves from the charge and stress free boundary of a transversely isotropic half space. Int. J. Eng. Sci. 46(2), 131–146 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang X.D.: On the dynamic behavior of interacting interfacial cracks in piezoelectric media. Int. J. Solids Struct. 38((5), 815–831 (2001)

    MATH  Google Scholar 

  21. Kuang Z.B., Yuan X.G.: Reflection and transmission of waves in pyroelectric and piezoelectric materials. J. Sound Vib. 330(6), 1111–1120 (2011)

    Article  Google Scholar 

  22. Abd-alla A.N., Yahia A.A., Abo-Dahab S.M.: On reflection of the generalized magneto-thermo-viscoelastic plane waves. Chaos Solitons Fractals 16, 211–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Othman M.I.A., Song Y.: Reflection of magneto-thermoelastic waves with two relaxation times and temperature dependent elastic moduli. Appl. Math. Model. 32(4), 483–500 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Othman M.I.A., Song Y.: Reflection of magneto-thermo-elastic waves from a rotating elastic half-space. Int. J. Eng. Sci. 46(5), 459–474 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chattopadhyay A., Gupa S., Sharma V.K., Kumari P.: Reflection and refraction of plane quasi P waves at a corrugated interface between distinct triclinic elastic half spaces. Int. J. Solids Struct. 46, 3241–3256 (2009)

    Article  MATH  Google Scholar 

  26. Daher N., Maugin G.A.: Intermodulation and generation of elastic and piezoelectric waves in anisotropic solids. J. Acoust. Soc. Am. 85(6), 2338–2345 (1989)

    Article  MathSciNet  Google Scholar 

  27. Quiligotti S., Maugin G.A., dell’Isola F.: Wave motions in unbounded poroelastic solids infused with compressible fluids. ZAMP 53((6), 1110–1113 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Berezovski A., Maugin G.A.: Thermoelastic wave and front propagation. J. Therm. Stresses 25(8), 719–743 (2002)

    Article  MathSciNet  Google Scholar 

  29. Placidi L., dell’Isola F., Ianiro N., Sciarra G.: Variational formulation of pre-stressed solid-fluid mixture theory, with an application to wave phenomena. Eur. J. Mech. A/Solids 27, 582–606 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. dell’Isola F., Madeo A., Seppecher P.: Boundary conditions at fluid-permeable interfaces in porous media: A variational approach. Int. J. Solids Struct. 46(17), 3150–3164 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Madeo A., Gavrilyuk S.: Propagation of acoustic waves in porous media and their reflection and transmission at a pure-fluid/porous-medium permeable interface. Eur. J. Mech. A/Solids 29(5), 897–910 (2010)

    Article  MathSciNet  Google Scholar 

  32. dell’Isola F., Madeo A., Placidi L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D Continua. ZAMM J. Appl. Math. Mech. 92(1), 52–71 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Madeo A., Djeran-Maigre I., Rosi G., Silvani C.: The effect of fluid streams in porous media on acoustic compression wave propagation, transmission, and reflection. Continuum Mech. Thermodyn. 6, 1–24 (2012)

    Google Scholar 

  34. Rosi G., Madeo A., Guyader J.L.: Switch between fast and slow Biot compression waves induced by second gradient microstructure at material discontinuity surfaces in porous media. Int. J. Solids Struct. 50(10), 1721–1746 (2013)

    Article  Google Scholar 

  35. Placidi, L., Rosi, G., Giorgio, I., Madeo, A.: Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math. Mech. Solids (2013). doi:10.1177/1081286512474016

  36. Rousseau M., Maugin G.A.: Reprint of: wave momentum in models of generalized continua. Wave Motion 50(8), 1251–1261 (2013)

    Article  MathSciNet  Google Scholar 

  37. Maugin G.A., Eringen A.C.: Electrodynamics of continua, foundations and solid media. Springer, New York (1989)

    Google Scholar 

  38. Abd-alla A.N.: Nonlinear constitutive equations for thermo-electroelastic materials. Mech. Res. Commun. 24((3), 335–346 (1999)

    Article  MathSciNet  Google Scholar 

  39. Lubarda V.A., Chen M.C.: On the elastic moduli and compliances of transversely isotropic and orthotropic materials. J. Mech. Mater. Struct. 3(1), 153–171 (2008)

    Article  Google Scholar 

  40. Sharma J.N., Walia V., Gupta S.K.: Effect of rotation and thermal relaxation on Rayleigh waves in piezothermoelastic half space. Int. J. Mech. Sci. 50(3), 433–444 (2008)

    Article  MATH  Google Scholar 

  41. Lord H., Shulman Y.: A generalized dynamical theory of thermoelasaticity. J. Mech. Phys. Solid 15, 299–309 (1967)

    Article  MATH  Google Scholar 

  42. Green A.E, Lindsay K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

    Abo-el-nour N. Abd-alla & Abdelmonam M. Hamdan

  2. Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

    Abo-el-nour N. Abd-alla

  3. Department of Mechanical and Aerospace Engineering, Sapienza University, Rome, Italy

    Ivan Giorgio & Dionisio Del Vescovo

  4. International Research Center for the Mathematics and Mechanics of Complex Systems, University of L’Aquila, L’Aquila, Italy

    Ivan Giorgio & Dionisio Del Vescovo

Authors
  1. Abo-el-nour N. Abd-alla
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  2. Abdelmonam M. Hamdan
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  3. Ivan Giorgio
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  4. Dionisio Del Vescovo
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Correspondence to Abo-el-nour N. Abd-alla.

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Abd-alla, Aen.N., Hamdan, A.M., Giorgio, I. et al. The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials. Arch Appl Mech 84, 1229–1248 (2014). https://doi.org/10.1007/s00419-014-0852-z

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  • DOI: https://doi.org/10.1007/s00419-014-0852-z

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