Abstract
In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a quasi-longitudinal Primary (qP) wave and a quasi-transverse Secondary (qS) wave; and (2) a mainly thermal (qT) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.
Similar content being viewed by others
References
Abd-alla A.N., Yahia A., Abo-dahab S.: On the reflection of the generalized magneto–thermo–viscoelastic plane waves. Chaos Solitons Fractals 16, 211–231 (2003)
Abd-Alla A.N., Abo-dahab S.M.: The influence of the viscosity on reflection and refraction of plane shear elastic waves in two magnetized semi-infinite media is investigated. Meccanica 43(4), 437–448 (2008)
Abd-alla A.N., Al-sheikh F.A., Al-Hossain A.Y.: The reflection phenomena of quasi-vertical transverse waves in piezoelectric medium under initial stresses. Meccanica 47(3), 731–744 (2012)
Abd-alla A.N., Al-Hossain A.Y., Elhaes H., Ibrahim M.: Reflection and refraction of waves in nano-smart materials: anisotropic thermo-piezoelectric materials. J. Comput. Theor. Nanosci. 11(3), 715–726 (2014)
Abd-alla A.N., Hamdan A.M., Giorgio I., Del Vescovo D.: The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials. Arch. Appl. Mech. (2014). doi:10.1007/s00419-014-0852-z
Achenbach J.D.: Wave Propagation in Elastic Solids. North-Holland Series in Applied Mathematics and Mechanics 16. North-Holland, New York (1973)
Andreaus U., Chiaia B., Placidi L.: Soft-impact dynamics of deformable bodies. Contin. Mech. Thermodyn. 25, 375–398 (2013). doi:10.1007/s00161-012-0266-5
Andreaus U., dell’Isola F., Porfiri M.: Piezoelectric passive distributed controllers for beam flexural vibrations. J. Vib. Control 10(5), 625–659 (2004)
Auld B.: Acoustic Field and Waves in Solids. R. E. Krieger Pub. Com., Malabar (1990)
Berezovski A., Maugin G.A.: Thermoelastic wave and front propagation. J. Therm. Stress. 25(8), 719–743 (2002)
Boehler J.P.: Representations for isotropic and anisotropic non-polynomial tensor functions. In: Boehler, J.P. (ed.) Applications of Tensor Functions in Solid Mechanics, CISM Courses and Lectures, pp. 31–53. Springer, Wien (1987)
Carcaterra A., Roveri N., Pepe G.: Fractional dissipation generated by hidden wave-fields. Math. Mech. Solids (2014). doi:10.1177/1081286513518941
Carcaterra A., Roveri N.: Energy distribution in impulsively excited structures. Shock Vib. 19(5), 1143–1163 (2012)
Cazzani A., Lovadina C.: On some mixed finite element methods for plane membrane problems. Comput. Mech. 20(6), 560–572 (1997)
Cuomo, M., Greco, L.: Isogeometric analysis of space rods: Considerations on stress locking. ECCOMAS 2012—European congress on computational methods in applied sciences and engineering, e-Book Full Papers, 5094–5112 (2012)
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)
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)
dell’Isola F., Sciarra G., Vidoli S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. A: Math. Phys. Eng. Sci. 465(2107), 2177–2196 (2009)
dell’Isola F., Vidoli S.: Continuum modelling of piezoelectromechanical truss beams: an application to vibration damping. Arch. Appl. Mech. 68(1), 1–19 (1998)
Del Vescovo D., Giorgio I.: Dynamic problems for metamaterials: review of existing models and ideas for further research. Int. J. Eng. Sci. (2014). doi:10.1016/j.ijengsci.2014.02.022
Eremeyev V.A.: Acceleration waves in micropolar elastic media. Dokl. Phys. 50(4), 204–206 (2005)
Greco L., Cuomo M.: B-Spline interpolation of Kirchhoff–Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)
Greco L., Cuomo M.: An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)
Green A., Lindsay K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)
Guo S.H.: The thermo-electromagnetic waves in piezoelectric solids. Acta Mech. 219, 231–240 (2011)
Huang H.T., Zhou L.M.: Micromechanics approach to the magnetoelectric properties of laminate and fibrous piezoelectric/magnetostrictive composites. J. Phys. D: Appl. Phys. 37, 3361–3366 (2004)
Keith C.M., Crampin S.: Seismic body waves in anisotropic media: reflection and refraction at a plane interface. Geophys. J. R. Astron. Soc. 49(1), 181–208 (1977)
Kumar R., Kumar R.: Wave propagation at the boundary surface of elastic and initially stressed visco-thermoelastic diffusion with voids media. Meccanica 48(9), 2173–2188 (2013)
Kuang Z.B., Yuan X.G.: Reflection and transmission of waves in pyroelectric and piezoelectric materials. J. Sound Vib. 330(6), 1111–1120 (2011)
Lord H.W., Shulman Y.: A generalized theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)
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)
Luongo A., Paolone A., Piccardo G.: Postcritical behavior of cables undergoing two simultaneous galloping modes. Meccanica 33(3), 229–242 (1998)
Luongo A., Piccardo G.: Linear instability mechanisms for coupled translational galloping. J. Sound Vib. 288(4–5), 1027–1047 (2005)
Madeo A., Neff P., Ghiba I.D., Placidi L., Rosi G.: Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Contin. Mech. Thermodyn. (2013). doi:10.1007/s00161-013-0329-2
Maurini C., Pouget J., dell’Isola F.: On a model of layered piezoelectric beams including transverse stress effect. Int. J. Solids Struct. 41(16–17), 4473–4502 (2004)
Maurini C., Pouget J., dell’Isola F.: Extension of the Euler–Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach. Comput. Struct. 84(22–23), 1438–1458 (2006)
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. Contin. Mech. Thermodyn. 6, 1–24 (2012)
Madeo A., dell’Isola F., Darve F.: A continuum model for deformable, second gradient porous media partially saturated with compressible fluids. J. Mech. Phys. Solids 61(11), 2196–2211 (2013)
Misra A., Marangos O.: Parametric studies of wave propagation through imperfect interfaces using micromechanics based effective stiffness. Rev. Prog. Quant. Nondestruct. Eval. 27, 1074–1081 (2008)
Misra A., Marangos O.: Effect of contact viscosity and roughness on interface stiffness and wave propagation. Rev. Prog. Quant. Nondestruct. Eval. 28A, 105–112 (2009)
Nayfeh A.H.: Wave Propagation in Layered Anisotropic Media. North-Holland, Amsterdam (1995)
Neff P., Ghiba I.D., Madeo A., Placidi L., Rosi G.: A unifying perspective: the relaxed linear micromorphic continuum. Contin. Mech. Thermodyn. (2013). doi:10.1007/s00161-013-0322-9
Othman M.I.A., Song Y.Q.: Reflection of magneto–thermo–elastic waves from a rotating elastic half-space. Int. J. Eng. Sci. 46(5), 459–474 (2008)
Parton V., Kudryavtsev B.: Electromagnetoelasticity: Piezoelectrics and Electrically Conductive Solids. Gordon and Breach, NY (1988)
Placidi L., Hutter K.: An anisotropic flow law for incompressible polycrystalline materials. Zeitschrift Für Angewandte Mathematik und Physik - ZAMP 57, 160–181 (2006). doi:10.1007/s00033-005-0008-7
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)
Placidi L., Greve R., Seddik H., Faria S.H.: Continuum-mechanical, anisotropic flow model, based on an anisotropic flow enhancement factor. Contin. Mech. Thermodyn. 22, 221–237 (2010). doi:10.1007/s00161-009-0126-0
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 19(5), 555–578 (2014)
Placidi L.: A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Contin. Mech. Thermodyn. (2014). doi:10.1007/s00161-14-0338-9
Quiligotti S., Maugin G.A., dell’Isola F.: Wave motions in unbounded poroelastic solids infused with compressible fluids. Zeitschrift Für Angewandte Mathematik und Physik - ZAMP 53(6), 1110–1113 (2002)
Reccia E., Cazzani A., Cecchi A.: FEM–DEM modeling for out-of-plane loaded masonry panels: a limit analysis approach. Open Civ. Eng. J. 6(1), 231–238 (2012)
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)
Rosi G., Giorgio I., Eremeyev V.A.: Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids. ZAMM-J. Appl. Math. Mech. 93(12), 914–927 (2013). doi:10.1002/zamm.201200285
Royer D., Dieulesaint E.: Elastic Waves in Solids I, Free and Guided Propagation. Springer, Berlin (2000)
Rupender K.R.: Propagation of waves in an electro-microstretch generalized thermoelastic semi-space. Acta Mech. Sin. 25(5), 619–628 (2009)
Ryu J., Shashank P., Kenji U., Kim H.: Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials. J. Electroceram. 8, 107–119 (2002)
Sharma J.N., Kumar V., Chand D.: Reflection of generalized thermoelastic waves from boundary of a half-space. J. Therm. Stress. 26, 925–942 (2003)
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)
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, 433–444 (2008)
Singh B.: On the theory of generalized thermoelasticity for piezoelectric materials. Appl. Math. Comput. 171(1), 398–405 (2005)
Singh B.: Wave propagation in a prestressed piezoelectric half-space. Acta Mech. 211(3–4), 337–344 (2010)
Singh S.S.: Transverse wave at a plane interface in thermo-elastic materials with voids. Meccanica 48(3), 617–630 (2013)
Song Y.Q., Bai J.T., Ren Z.Y.: Study on the reflection of photothermal waves in a semi-conducting medium under generalized thermoelastic theory. Acta Mech. 223, 1545–1557 (2012)
Srinivasan G., Laletsin V.M.: Giant magnetoelectric effects in layered composites of nicked zinc ferrite and lead zirconate titanate. Solid State Commun. 124, 373–378 (2002)
Sciarra G., dell’Isola F., Ianiro N., Madeo A.: A variational deduction of second gradient poroelasticity part I: general theory. J. Mech. Mater. Struct. 3(3), 507–526 (2008)
Spencer A.J.M.: The formulation of constitutive equation for anisotropic solids. In: Boehler, J.P. (ed.) Mechanical Behavior of Anisotropic Solids, pp. 2–26. Martinus Nijhoff Publishers, The Hague (1982)
Tomar S.K., Khurana A.: Elastic waves in an electro-microelastic solid. Int. J. Solids Struct. 45, 278–302 (2008)
Van Suchtelen J.: Product properties: a new application of composite materials. Philips Res. Rep. 27, 28–37 (1972)
Vidoli S., dell’Isola F.: Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks. Eur. J. Mech. A/Solids 20(3), 435–456 (2001)
Wu, L.: Piezoelectric Thin Film: Fabrication, Properties and Applications. Proceeding of The Second Symposium on Piezoelectricity, Acoustic Waves, and Device Applications 5 (2007)
Yang J.: The Mechanics of Piezoelectric Structures. World Scientific Publishing Co., Singapore (2008)
Yang J.: An Introduction to the Theory of Piezoelectricity. Springer, Boston (2005)
Yuan X., Kuang Z.: Waves in pyroelectrics. J. Therm. Stress. 31(12), 1190–1211 (2008). doi:10.1080/01495730802508046
Ye Z.G.: Handbook of Dielectric, Piezoelectric and Ferroelectric Materials Synthesis, Properties and Applications. Woodhead Publishing Limited, CRC Press, New York (2008)
Yuan X., Zhu Z.H.: Reflection and refraction of plane waves at interface between two piezoelectric media. Acta Mech. 223, 2509–2521 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.
Rights and permissions
About this article
Cite this article
Abd-alla, Aen., Giorgio, I., Galantucci, L. et al. Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity. Continuum Mech. Thermodyn. 28, 67–84 (2016). https://doi.org/10.1007/s00161-014-0400-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-014-0400-7