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

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.

[1]  F. Josse,et al.  Reflection of bulk waves at a piezoelectric crystal-viscous conductive liquid interface , 1992 .

[2]  J. Sharma,et al.  Effect of rotation and thermal relaxation on Rayleigh waves in piezothermoelastic half space , 2008 .

[3]  Gérard A. Maugin,et al.  Electrodynamics of Continua I: Foundations and Solid Media , 1989 .

[4]  G. Maugin,et al.  Intermodulation and generation of elastic and piezoelectric waves in anisotropic solids , 1989 .

[5]  G. Maugin,et al.  Wave motions in unbounded poroelastic solids infused with compressible fluids , 2002 .

[6]  Baljeet Singh,et al.  On the theory of generalized thermoelasticity for piezoelectric materials , 2005, Appl. Math. Comput..

[7]  G. Rosi,et al.  The effect of fluid streams in porous media on acoustic compression wave propagation, transmission, and reflection , 2013 .

[8]  S. M. Abo-Dahab,et al.  On the reflection of the generalized magneto-thermo-viscoelastic plane waves , 2003 .

[9]  A. Abd-alla,et al.  REFLECTION OF LONGITUDINAL WAVES UNDER INITIAL STRESSES AT AN INTERFACE IN PIEZOELECTRIC MEDIA , 2007 .

[10]  Khaled A. Elsibai,et al.  REFLECTION OF THERMOELASTIC WAVES AT A SOLID HALF-SPACE WITH TWO RELAXATION TIMES , 1996 .

[11]  Francesco dell’Isola,et al.  Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua , 2012 .

[12]  H. Lord,et al.  A GENERALIZED DYNAMICAL THEORY OF THERMOELASTICITY , 1967 .

[13]  M. Othman,et al.  Reflection of magneto-thermo-elastic waves from a rotating elastic half-space in generalized thermoelasticity under three theories , 2008 .

[14]  G. Maugin,et al.  THERMOELASTIC WAVE AND FRONT PROPAGATION , 2002 .

[15]  G. Maugin,et al.  Reprint of: Wave momentum in models of generalized continua , 2013 .

[16]  Abo-el-nour N. Abd-alla,et al.  THE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM , 2000 .

[17]  Rajneesh Kumar,et al.  Reflection/transmission of plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic half-spaces , 2008 .

[18]  Francesco dell’Isola,et al.  Variational formulation of pre-stressed solid-fluid mixture theory, with an application to wave phenomena , 2008 .

[19]  Vlado A. Lubarda,et al.  On the elastic moduli and compliances of transversely isotropic and orthotropic materials , 2008 .

[20]  J. Achenbach.  Wave propagation in elastic solids , 1962 .

[21]  Zhen-Bang Kuang,et al.  Reflection and transmission of waves in pyroelectric and piezoelectric materials , 2011 .

[22]  Jiashi Yang,et al.  The mechanics of piezoelectric structures , 2006 .

[23]  Xiaodong Wang.  On the dynamic behaviour of interacting interfacial cracks in piezoelectric media , 2001 .

[24]  Ivan Giorgio,et al.  Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials , 2014 .

[25]  L. Nappa,et al.  THERMOELASTIC DEFORMATION OF MICROSTRETCH ELASTIC BEAMS , 1996 .

[26]  A. C. Eringen,et al.  Electrodynamics of continua. Volume 1. Foundations and solid media. Volume 2 - Fluids and complex media , 1990 .

[27]  Jean-Louis Guyader,et al.  Switch between fast and slow Biot compression waves induced by ''second gradient microstructure{''} at material discontinuity surfaces in porous media , 2013 .

[28]  S. Gavrilyuk,et al.  Propagation of acoustic waves in porous media and their reflection and transmission at a pure-fluid/porous-medium permeable interface , 2010 .

[29]  A. Shuvalov,et al.  Theory of reflection of acoustoelectric waves in a semiinfinite piezoelectric medium. III : Resonance reflection in the neighborhood of a branch of outflowing waves , 1991 .

[30]  M. Othman,et al.  Reflection of magneto-thermoelastic waves with two relaxation times and temperature dependent elastic moduli , 2008 .

[31]  Zuo-Guang Ye,et al.  Handbook of Advanced Dielectric, Piezoelectric and Ferroelectric Materials : Synthesis, Properties and Applications , 2008 .

[32]  A. Abd-alla.  Nonlinear constitutive equations for thermo-electroelastic materials , 1999 .

[33]  Jiashi Yang,et al.  An Introduction to the Theory of Piezoelectricity , 2004 .

[34]  S. Crampin.  Distinctive Particle Motion of Surface Waves as a Diagnostic of Anisotropic Layering , 1975 .

[35]  Francesco dell’Isola,et al.  Boundary Conditions at Fluid-Permeable Interfaces in Porous Media: a Variational Approach , 2009 .

[36]  J. N. Sharma,et al.  Reflection of piezothermoelastic waves from the charge and stress free boundary of a transversely isotropic half space , 2008 .

[37]  H. Kim,et al.  Particle dynamics calculations of wall stresses and slip velocities for couette flow of smooth inelastic spheres , 1994 .

[38]  Khaled A. Elsibai,et al.  Reflection and refraction of thermoelastic waves at an interface of two semi-infinite media with two relaxation times , 1997 .

[39]  A. Chattopadhyay,et al.  Reflection and refraction of plane quasi-P waves at a corrugated interface between distinct triclinic elastic half spaces , 2009 .

[40]  A. Abd-alla,et al.  Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezoelectric media under initial stresses , 2009 .

[41]  K. S. Aleksandrov,et al.  Reflection and refraction of bulk acoustic waves in piezoelectrics under uniaxial stress , 2009 .