Reflection of Plane Waves at Micropolar Piezothermoelastic Half-space
Kumar Rajneesh 1, Sharma Nidhi 2, Lata Parveen 3*, Marin Marin 4
1Department of Mathematics, Kurukshetra University, Kurukshetra 136119, India
2Department of Mathematics, MM University, Mullana, Ambala, Haryana, India
3Department of Basic and Applied Sciences, Punjabi University, Patiala, Punjab, India
4Department of Mathematics, University of Brasov, Romania
*E-mail: parveenlata@pbi.ac.in
Received:
Received: 30 December 2016; revised: 15 December 2017; accepted: 18 December 2017; published online: 30 January 2018
DOI: 10.12921/cmst.2016.0000069
Abstract:
A problem of reflection at a free surface of micropolar orthotropic piezothermoelastic medium is discussed in the present paper. It is found that there exist five type plane waves in micropolar orthotropic piezothermoelastic medium, namely quasi longitudinal displacement wave (quasi LD wave), quasi thermal wave (quasi T wave), quasi CD-I, quasi CD-II wave and electric potential wave (PE wave). The amplitude ratios corresponding to reflected waves are obtained numerically. The effect of angle of incidence and thermopiezoelectric interactions on the reflected waves are studied for a specific model. Some particular cases of interest are also discussed.
Key words:
angle of incidence, micropolar, orthotropic, piezothermoelastic, reflection coefficients
References:
[1] A.C.Eringen,E.S.Suhubi,Non-lineartheoryofmicro-elastic solids, International Journal of Engineering Science 2, 189– 203 (1964).
[2] A.C.Eringen,Lineartheoryofmicropolarelasticity,Journal of Math and Mechanics 15, 909–923 (1996).
[3] W. Nowacki, Couple Stress in the Theory of Thermoelasticity, Proc. ITUAM Symposia,Vienna, Editors H. Parkus and L. I. Sedov, Springer-Verlag, 259–278 (1966).
[4] A.C. Eringen, Foundations of micropolar thermoelasticity, Course of lectures No. 23, CSI H. Udline Springer, 1970.
[5] A.C. Eringen, Microcontinuum field theory I, Foundations and Solids (Springer, New York), 1992.
[6] D. Iesan, The plane micropolar strain of orthotropic elas- tic solids, Archiwum Mechaniki Stosowanej 25, 547–561 (1973).
[7] D.Iesan,Torsionofanisotropicmicropolarelasticcylinders, Zeitschrift für Angewandte Mathematik und Mechanik 54, 773–779 (1974).
[8] D. Iesan, Bending of orthotropic micropolar elastic beams by terminal couples, Analele Stiintifice Ale Uni. IASI 20, 411–418 (1974).
[9] S.Nakamura,R.Benedict,R.Lakes,Finiteelementmethod for orthotropic micropolar elasticity, International, Journal of Engineering Science 22, 319–330 (1984).
[10] R.Kumar,S.Choudhary,Mechanicalsourcesinorthotropic micropolar continua, Proceedings of the Indian Academy of Sciences (EarthandPlanetary Sciences) 111, 133–141 (2002).
[11] R. Kumar, S. Choudhary, Influence of Green’s function for orthotropic micropolar continua, Archive of Mechanics 54, 185–198 (2002).
[12] R.Kumar,S.Choudhary,Dynamicalbehavioroforthotropic micropolar elastic medium, Journal of Vibration and Control 8, 1053–1069 (2002).
[13] R.Kumar,S.Choudhary,Responseoforthotropicmicroploar elastic medium due to various Sources, Meccanica 38, 349– 368 (2003).
[14] R.Kumar,S.Choudhary,Responseoforthotropicmicropolar elastic medium due to time harmonic sources, Sadhana 29, 83–92 (2004).
[15] R.D. Mindlin, On the equations of motion of piezoelectric crystals, in: N.I. Muskilishivili, Problems of continuum Me- chanics, 70th Birthday Volume, SIAM, Philadelphia, 282–290 (1961).
[16] W. Nowacki, Some general theorems of thermo- piezoelectricity, Journal of Thermal Stresses 1, 171–182 (1978).
[17] W. Nowacki, Foundations of linear piezoelectricity, in: H. Parkus (Ed.), Electromagnetic Interactions in Elastic Solids, Springer, Wein, Chapter 1(1979).
[18] W. Nowacki, Mathematical models of phenomenological piezo-electricity, New Problems in Mechanics of Continua, University of Waterloo Press, Waterloo, Ontario, 1983.
[19] D.S.Chandrasekharaiah,Atemperature-rate-dependentthe- ory of thermopiezoelectricity, Journal of Thermal Stresses 7, 293–306 (1984).
[20] D.S.Chandrasekharaiah,Ageneralizedlinearthermoelastic- ity theory for piezoelectric media, Acta Mechanica 71, 39–49 (1988).
[21] W.Q. Chen, On the general solution for piezothermoelastic for transverse isotropy with application, ASME, Journal of Applied Mechanics 67, 705–711 (2000).
[22] P.F. Hou, W. Luo, Y.T. Leung, A point heat source on the surface of a semi-infinite transverse isotropic piezothermoe- lastic material, SME Journal of Applied Mechanics 75, 1–8 (2008).
[23] A.N. Abd-Alla, F.A. Alshaikh, Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezo- electric media under initial stresses, Archive of Applied Mechanics 79(9), 843–857 (2009).
[24] A.N. Abd-Alla, F.A. Alshaikh, The effect of the initial stresses on the reflection and transmission of plane quasi-vertical transverse waves in piezoelectric materials, Proceedings of World Academy of Science, Engineering and Technology 38, 660–668 (2009).
[25] Y. Pang, Y.S.Wang, J.X. Liu, D.N.Fang, Reflection and refrac- tion of plane waves at the interface between piezoelectricand piezomagnetic media, International Journal of Engineering Science 46, 1098–1110 (2008).
[26] J.N. Sharma, V. Walia, S.K. Gupta, Reflection of piezo- thermoelastic waves from the charge and stress free boundary of a transversely isotropic half space, International Journal of Engineering Science, 46(2), 131–146 (2008).
[27] Z.B. Kuang , X.G. Yuan, Reflection and transmission of waves in pyroelectric and piezoelectric materials, Journal of Sound and Vibration 330 (6), 1111–1120 (2011).
[28] A.N. Abd-Alla, F.A. Alshaikh, A.Y. Al-Hossain, The reflec- tion phenomena of quasi-vertical transverse waves in piezo- electric medium under initial stresses, Meccanica 47(3), 731– 744 (2012).
[29] F.A. Alshaikh, The mathematical modelling for studying the influence of the initial stresses and relaxation times on reflec- tion and refraction waves in piezothermoelastic half-space, Applied Mathematics 3(8), 819–832 (2012).
[30] F.A. Alshaikh, Reflection of Quasi Vertical Transverse Waves in the Thermo-Piezoelectric Material under Initial Stress (Green- Lindsay Model), International Journal of Pure and Applied Sciences and Technology 13, 27–39 (2012).
[31] M.M. Meerschaert, R.J. Mc Gough, Attenuated Fractional Wave Equations With Anisotropy, Journal of Vibration and Acoustics 136,051004–1 to 051004–5 (2014).
[32] A. Sur, M. Kanoria, Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell, Lat. Am. J. Solids Struct. 11(7), 1132–1162 (2014).
[33] S.M. Abo-Dahab, Magnetic field effect on three plane waves propagation at interface between solid-liquid media placed under initial stress in the context of GL model, Applied Math- ematics and Information Sciences 9(6), 3119–3131 (2015).
[34] A.M. Abd-Alla, S.M. Abo-Dahab, Effect of initial stress, ro- tation and gravity on propagation of the surface waves in fibre-reinforced anisotropic solid elastic media, Journal of Computational and Theoretical Nanoscience 12(2), 305–315 (2015).
[35] A.K. Vashishth, H. Sukhija, Reflection and transmission of plane waves from fluid-piezothermoelastic solid interface, Applied Mathematics and Mechanics 36(1), 11–36 (2015).
[36] R. Kumar, A. Kumar, Elastodynamic response of thermal laser pulse in micropolar thermoelastic diffusion medium, Journal of Thermodynamics, ?? (2016).
[37] S.R. Mahmoud, An analytical solution for the effect of initial stress, rotation, magnetic field and a periodic loading in a thermoviscoelastic medium with a spherical cavity, Mechan- ics of Advanced Materials and Structures 23(1), 1–7 (2016).
[38] M.A. Ezzat, A.S. EI-Karamany, A.A. EI-Bary, Generalized thermoelasticity with memory dependent derivatives involv- ing two temperatures, Mechanics of Advanced materials and Structures 23(5), 545–553 (2016).
[39] W.S. Slaughter, The Linearized Theory of Elasticity, Birkhauser, Basel (2002).
A problem of reflection at a free surface of micropolar orthotropic piezothermoelastic medium is discussed in the present paper. It is found that there exist five type plane waves in micropolar orthotropic piezothermoelastic medium, namely quasi longitudinal displacement wave (quasi LD wave), quasi thermal wave (quasi T wave), quasi CD-I, quasi CD-II wave and electric potential wave (PE wave). The amplitude ratios corresponding to reflected waves are obtained numerically. The effect of angle of incidence and thermopiezoelectric interactions on the reflected waves are studied for a specific model. Some particular cases of interest are also discussed.
Key words:
angle of incidence, micropolar, orthotropic, piezothermoelastic, reflection coefficients
References:
[1] A.C.Eringen,E.S.Suhubi,Non-lineartheoryofmicro-elastic solids, International Journal of Engineering Science 2, 189– 203 (1964).
[2] A.C.Eringen,Lineartheoryofmicropolarelasticity,Journal of Math and Mechanics 15, 909–923 (1996).
[3] W. Nowacki, Couple Stress in the Theory of Thermoelasticity, Proc. ITUAM Symposia,Vienna, Editors H. Parkus and L. I. Sedov, Springer-Verlag, 259–278 (1966).
[4] A.C. Eringen, Foundations of micropolar thermoelasticity, Course of lectures No. 23, CSI H. Udline Springer, 1970.
[5] A.C. Eringen, Microcontinuum field theory I, Foundations and Solids (Springer, New York), 1992.
[6] D. Iesan, The plane micropolar strain of orthotropic elas- tic solids, Archiwum Mechaniki Stosowanej 25, 547–561 (1973).
[7] D.Iesan,Torsionofanisotropicmicropolarelasticcylinders, Zeitschrift für Angewandte Mathematik und Mechanik 54, 773–779 (1974).
[8] D. Iesan, Bending of orthotropic micropolar elastic beams by terminal couples, Analele Stiintifice Ale Uni. IASI 20, 411–418 (1974).
[9] S.Nakamura,R.Benedict,R.Lakes,Finiteelementmethod for orthotropic micropolar elasticity, International, Journal of Engineering Science 22, 319–330 (1984).
[10] R.Kumar,S.Choudhary,Mechanicalsourcesinorthotropic micropolar continua, Proceedings of the Indian Academy of Sciences (EarthandPlanetary Sciences) 111, 133–141 (2002).
[11] R. Kumar, S. Choudhary, Influence of Green’s function for orthotropic micropolar continua, Archive of Mechanics 54, 185–198 (2002).
[12] R.Kumar,S.Choudhary,Dynamicalbehavioroforthotropic micropolar elastic medium, Journal of Vibration and Control 8, 1053–1069 (2002).
[13] R.Kumar,S.Choudhary,Responseoforthotropicmicroploar elastic medium due to various Sources, Meccanica 38, 349– 368 (2003).
[14] R.Kumar,S.Choudhary,Responseoforthotropicmicropolar elastic medium due to time harmonic sources, Sadhana 29, 83–92 (2004).
[15] R.D. Mindlin, On the equations of motion of piezoelectric crystals, in: N.I. Muskilishivili, Problems of continuum Me- chanics, 70th Birthday Volume, SIAM, Philadelphia, 282–290 (1961).
[16] W. Nowacki, Some general theorems of thermo- piezoelectricity, Journal of Thermal Stresses 1, 171–182 (1978).
[17] W. Nowacki, Foundations of linear piezoelectricity, in: H. Parkus (Ed.), Electromagnetic Interactions in Elastic Solids, Springer, Wein, Chapter 1(1979).
[18] W. Nowacki, Mathematical models of phenomenological piezo-electricity, New Problems in Mechanics of Continua, University of Waterloo Press, Waterloo, Ontario, 1983.
[19] D.S.Chandrasekharaiah,Atemperature-rate-dependentthe- ory of thermopiezoelectricity, Journal of Thermal Stresses 7, 293–306 (1984).
[20] D.S.Chandrasekharaiah,Ageneralizedlinearthermoelastic- ity theory for piezoelectric media, Acta Mechanica 71, 39–49 (1988).
[21] W.Q. Chen, On the general solution for piezothermoelastic for transverse isotropy with application, ASME, Journal of Applied Mechanics 67, 705–711 (2000).
[22] P.F. Hou, W. Luo, Y.T. Leung, A point heat source on the surface of a semi-infinite transverse isotropic piezothermoe- lastic material, SME Journal of Applied Mechanics 75, 1–8 (2008).
[23] A.N. Abd-Alla, F.A. Alshaikh, Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezo- electric media under initial stresses, Archive of Applied Mechanics 79(9), 843–857 (2009).
[24] A.N. Abd-Alla, F.A. Alshaikh, The effect of the initial stresses on the reflection and transmission of plane quasi-vertical transverse waves in piezoelectric materials, Proceedings of World Academy of Science, Engineering and Technology 38, 660–668 (2009).
[25] Y. Pang, Y.S.Wang, J.X. Liu, D.N.Fang, Reflection and refrac- tion of plane waves at the interface between piezoelectricand piezomagnetic media, International Journal of Engineering Science 46, 1098–1110 (2008).
[26] J.N. Sharma, V. Walia, S.K. Gupta, Reflection of piezo- thermoelastic waves from the charge and stress free boundary of a transversely isotropic half space, International Journal of Engineering Science, 46(2), 131–146 (2008).
[27] Z.B. Kuang , X.G. Yuan, Reflection and transmission of waves in pyroelectric and piezoelectric materials, Journal of Sound and Vibration 330 (6), 1111–1120 (2011).
[28] A.N. Abd-Alla, F.A. Alshaikh, A.Y. Al-Hossain, The reflec- tion phenomena of quasi-vertical transverse waves in piezo- electric medium under initial stresses, Meccanica 47(3), 731– 744 (2012).
[29] F.A. Alshaikh, The mathematical modelling for studying the influence of the initial stresses and relaxation times on reflec- tion and refraction waves in piezothermoelastic half-space, Applied Mathematics 3(8), 819–832 (2012).
[30] F.A. Alshaikh, Reflection of Quasi Vertical Transverse Waves in the Thermo-Piezoelectric Material under Initial Stress (Green- Lindsay Model), International Journal of Pure and Applied Sciences and Technology 13, 27–39 (2012).
[31] M.M. Meerschaert, R.J. Mc Gough, Attenuated Fractional Wave Equations With Anisotropy, Journal of Vibration and Acoustics 136,051004–1 to 051004–5 (2014).
[32] A. Sur, M. Kanoria, Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell, Lat. Am. J. Solids Struct. 11(7), 1132–1162 (2014).
[33] S.M. Abo-Dahab, Magnetic field effect on three plane waves propagation at interface between solid-liquid media placed under initial stress in the context of GL model, Applied Math- ematics and Information Sciences 9(6), 3119–3131 (2015).
[34] A.M. Abd-Alla, S.M. Abo-Dahab, Effect of initial stress, ro- tation and gravity on propagation of the surface waves in fibre-reinforced anisotropic solid elastic media, Journal of Computational and Theoretical Nanoscience 12(2), 305–315 (2015).
[35] A.K. Vashishth, H. Sukhija, Reflection and transmission of plane waves from fluid-piezothermoelastic solid interface, Applied Mathematics and Mechanics 36(1), 11–36 (2015).
[36] R. Kumar, A. Kumar, Elastodynamic response of thermal laser pulse in micropolar thermoelastic diffusion medium, Journal of Thermodynamics, ?? (2016).
[37] S.R. Mahmoud, An analytical solution for the effect of initial stress, rotation, magnetic field and a periodic loading in a thermoviscoelastic medium with a spherical cavity, Mechan- ics of Advanced Materials and Structures 23(1), 1–7 (2016).
[38] M.A. Ezzat, A.S. EI-Karamany, A.A. EI-Bary, Generalized thermoelasticity with memory dependent derivatives involv- ing two temperatures, Mechanics of Advanced materials and Structures 23(5), 545–553 (2016).
[39] W.S. Slaughter, The Linearized Theory of Elasticity, Birkhauser, Basel (2002).
