Reflection and Refraction of Plane Harmonic Waves at an Interface Between Elastic Solid and Magneto-thermoelastic Diffusion Solid with Voids
Kumar Rajneesh 1, Kansal Tarun 2
1 Department of Mathematics, Kurukshetra University Kurukshetra-136119, India
E-mail: rajneesh_kuk@rediffmail.com2 Department of Mathematics, M.N.College, Shahabad (M.)-136135, India
E-mail: tarun1_kansal@yahoo.co.in – corresponding author
Received:
Received: 21 July 2016; revised: 23 January 2017; accepted: 24 January 2017; published online: 16 March 2017
DOI: 10.12921/cmst.2016.0000036
Abstract:
The problem of the reflection and refraction phenomenon due to longitudinal and transverse waves incident obliquely at a plane interface between uniform elastic solid half-space and magneto-thermoelastic diffusive solid half-space with voids has been studied. It is found that the amplitude ratios of various reflected and refracted waves are functions of the angle of incidence and frequency of the incident wave. The amplitude ratios and energy ratios have been computed numerically for a particular model. The variations of energy ratios with angle of incidence are shown graphically.
Key words:
amplitude ratios, elastic waves, energy ratios, magneto-thermoelastic diffusive solid, reflection, refraction
References:
[1] H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15, 299-309 (1967).
[2] A.E. Green, K.A. Lindsay, Thermoelasticity, Journal of Elas- ticity 2, 1-7 (1972).
[3] R.S. Dhaliwal, H.H. Sherief, Generalized thermoelasticity for anisotropic media, Quarterly of Applied Mathematics 38, 1-8 (1980).
[4] D.S. Chanderashekhariah, Thermoelasticity with second sound: a review, Applied Mechanics Review 39, 355-376 (1986).
[5] R.B. Hetnarski, J. Ignaczak, Generalized thermoelasticity, Journal of Thermal Stresses 22, 451-476 (1999).
[6] J.W. Nunziato, S.C. Cowin, A non-linear theory of elastic material with voids, Archive for Rational Mechanics and Analysis 72, 175-201 (1979).
[7] S.C. Cowin, J.W. Nunziato,Linear theory of elastic materials with voids, Journal of Elasticity 13, 125-147 (1983).
[8] D. Iesan, A theory of thermoelastic materials with voids, Acta Mechanica 60, 67-89 (1986).
[9] M. Ciarletta, A. Scalia, On the nonlinear theory of nonsimple thermoelastic materials with voids, ZAMM 73, 7-75 (1993).
[10] W. Nowacki, Dynamical problems of thermodiffusion in solids-I, Bulletin of Polish Academy of Sciences Series, Science and Technology 22, 55-64 (1974a).
[11] W. Nowacki, Dynamical problems of thermodiffusion in solids-II, Bulletin of Polish Academy of Sciences Series, Science and Technology 22, 205-211 (1974b).
[12] W. Nowacki, Dynamical problems of thermodiffusion in solids-III, Bulletin of Polish Academy of Sciences Series, Science and Technology 22, 257-266 (1974c).
[13] W.Nowacki,Dynamicalproblemsofdiffusioninsolids,Engineering Fracture Mechanics 8, 261-266 (1976).
[14] W. Dudziak, S.J. Kowalski, Theory of thermodiffusion for solids, International Journal of Heat and Mass Transfer 32, 2005-2013 (1989).
[15] Z.S. Olesiak, Y.A. Pyryev, A coupled quasi-stationary problem of thermodiffusion for an elastic cylinder, International Journal of Engineering Science 33, 773-780 (1995).
[16] J.A. Gawinecki, P. Kacprzyk, P. Bar-Yoseph, Initial boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermoelastic diffusion in solid body, Journal for Analysis and its Applications 19, 121-130 (2000).
[17] J. A. Gawinecki, A. Szymaniec, Global solution of the Cauchy problem in nonlinear thermoelastic diffusion in solid body, Proceedings in Applied Mathematics and Mechanics 1, 446-447 (2002).
[18] H.H. Sherief, F.A. Hamza and H.A. Saleh, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42, 591-608 (2004).
[19] M. Aouadi, Uniqueness and reciprocity theorems in the the- ory of generalized thermoelastic diffusion, Journal of Thermal Stresses 30, 665-678 (2007).
[20] H.H. Sherief, H.A. Saleh, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42, 4484-4493 (2005).
[21] R. Kumar, T. Kansal, Variational principle, uniqueness and reciprocity theorems in the theory of generalized thermoelas- tic diffusion material, QScience Connect 2013, 1-18 (2013).
[22] P. Borejko, Reflection and transmission coefficients for three- dimensional plane waves in elastic media, Wave Motion 24, 371-393 (1996).
[23] C.M. Wu, B. Lundberg, Reflection and transmission of the energy of harmonic elastic waves in a bent bar, Journal of Sound and Vibration 190, 645-659 (1996).
[24] S.B. Sinha, K.A. Elsibai, Reflection and refraction of ther- moelastic waves at an interface of two semi-infinite media with two relaxation times, Journal of Thermal Stresses 20, 129-145 (1997).
[25] M.D. Sharma, M.L. Gogna, Reflection and refraction of plane harmonic waves at an interface between elastic solid and porous solid saturated by viscous liquid, PAGEOPH 138, 249-266 (1992).
[26] S.K. Tomar, A. Arora, Reflection and transmission of elas- tic waves at an elastic/porous solid saturated by immisci- ble fluids, International Journal of Solids and Structures 43, 1991-2013 (2006).
[27] R. Kumar, P. Sarthi, Reflection and refraction of thermoelas- tic plane waves at an interface of two thermoelastic media without energy dissipation, Archives of Mechanics 58, 155- 185 (2006).
[28] A.N. Abd-alla, A.A.S. Al-dawy, The reflection phenomena of SV-waves in a generalized thermoelastic medium, International Journal of Mathematics and Mathematical Sciences 23, 529-546 (2000).
[29] M. Ciarletta, M.A. Sumbatyan, Reflection of plane waves by the free boundary of a porous elastic half-space, Journal of Sound and Vibration 259, 253-264 (2003).
[30] J. Singh, S.K. Tomar, Plane waves in thermo-elastic material with voids, Mechanics of Materials 39, 932-940 (2007).
[31] B. Singh, Wave propagation in a generalized thermoelastic material with voids, Applied Mathematics and Computation 189, 698-709 (2007).
[32] J. Singh, S.K. Tomar, Reflection and transmission of trans- verse waves at a plane interface between two different porous elastic half-spaces, Applied Mathematics and Computation 176, 364-378 (2006).
[33] N.C. Das, A. Lahiri, S. Sarkar and B. Basu, Reflection of gen- eralized thermoelastic waves from isothermal and insulated boundaries of a half-space, Computers and Mathematics with Applications 56, 2795-2805 (2008).
[34] S.S. Singh, Reflection and transmission of couple longitudi- nal waves at a plane interface between two dissimilar half- spaces of thermo-elastic materials with voids, Applied Math- ematics and Computation 218, 3359-3371 (2011).
[35] R. Kumar, R. Kumar, Reflection and transmission at the plane boundary of elastic half-space and initially stressed thermoe- lastic diffusion with voids half-space, MMMS 9, 185-213 (2013).
[36] A.N. Abd-alla, Relaxation effects on reflection of generalized magneto-thermoelastic waves, Mechanics Research Commu- nications 27, 591-600 (2000).
[37] M.I.A. Othman, Y. Song, Reflection of magneto-thermo- elastic waves with two relaxation times and temperature dependent elastic moduli, Applied Mathematics and Mod- elling 32, 483-500 (2008).
[38] M.I.A. Othman, R. Kumar, Reflection of magneto-thermoelas- ticity waves with temperature dependent properties in gen- eralized thermoelasticity, International Communications in Heat and Mass Transfer 36, 513-520 (2009).
[39] S.M. Abo-Dahab, Propagation of P waves from stress-free surface elastic half-space with voids under thermal relax- ation and magnetic field, Applied Mathematics and Mod- elling 34, 1798-1806 (2010).
[40] M.I.A. Othman, Generalized electro-magneto-thermoelas- ticity in case of thermal shock plane waves for a finite con- ducting half-Space with two relaxation times, Mechanics and Mechanical Engineering 14, 5-30 (2010).
[41] B. Singh, L. Singh and S. Deswal, Reflection of plane waves from a free surface of a generalized magneto-thermoelastic solid half-space with diffusion, Journal of Theoretical and Applied Mechanics 52, 385-394 (2014).
[42] R.D. Borcherdt, Reflection-refraction of general P and type-I S waves in elastic and anelastic solids, Geophysical Journal of Royal Astronomical Society 70, 621-638 (1982).
[43] J.D. Achenbach, Wave propagation in elastic solids, North- Holland, Amsterdam (1973).
[44] R. Kumar, T. Kansal, Reflection and refraction of plane waves at an interface of an elastic solid half-space and a thermoelastic diffusive solidhalf-space, Archives of Mechanics 64, 293-317 (2012).
[45] K.E. Bullen, An introduction to the theory of Seismology, Cambridge University press, England (1963).
The problem of the reflection and refraction phenomenon due to longitudinal and transverse waves incident obliquely at a plane interface between uniform elastic solid half-space and magneto-thermoelastic diffusive solid half-space with voids has been studied. It is found that the amplitude ratios of various reflected and refracted waves are functions of the angle of incidence and frequency of the incident wave. The amplitude ratios and energy ratios have been computed numerically for a particular model. The variations of energy ratios with angle of incidence are shown graphically.
Key words:
amplitude ratios, elastic waves, energy ratios, magneto-thermoelastic diffusive solid, reflection, refraction
References:
[1] H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15, 299-309 (1967).
[2] A.E. Green, K.A. Lindsay, Thermoelasticity, Journal of Elas- ticity 2, 1-7 (1972).
[3] R.S. Dhaliwal, H.H. Sherief, Generalized thermoelasticity for anisotropic media, Quarterly of Applied Mathematics 38, 1-8 (1980).
[4] D.S. Chanderashekhariah, Thermoelasticity with second sound: a review, Applied Mechanics Review 39, 355-376 (1986).
[5] R.B. Hetnarski, J. Ignaczak, Generalized thermoelasticity, Journal of Thermal Stresses 22, 451-476 (1999).
[6] J.W. Nunziato, S.C. Cowin, A non-linear theory of elastic material with voids, Archive for Rational Mechanics and Analysis 72, 175-201 (1979).
[7] S.C. Cowin, J.W. Nunziato,Linear theory of elastic materials with voids, Journal of Elasticity 13, 125-147 (1983).
[8] D. Iesan, A theory of thermoelastic materials with voids, Acta Mechanica 60, 67-89 (1986).
[9] M. Ciarletta, A. Scalia, On the nonlinear theory of nonsimple thermoelastic materials with voids, ZAMM 73, 7-75 (1993).
[10] W. Nowacki, Dynamical problems of thermodiffusion in solids-I, Bulletin of Polish Academy of Sciences Series, Science and Technology 22, 55-64 (1974a).
[11] W. Nowacki, Dynamical problems of thermodiffusion in solids-II, Bulletin of Polish Academy of Sciences Series, Science and Technology 22, 205-211 (1974b).
[12] W. Nowacki, Dynamical problems of thermodiffusion in solids-III, Bulletin of Polish Academy of Sciences Series, Science and Technology 22, 257-266 (1974c).
[13] W.Nowacki,Dynamicalproblemsofdiffusioninsolids,Engineering Fracture Mechanics 8, 261-266 (1976).
[14] W. Dudziak, S.J. Kowalski, Theory of thermodiffusion for solids, International Journal of Heat and Mass Transfer 32, 2005-2013 (1989).
[15] Z.S. Olesiak, Y.A. Pyryev, A coupled quasi-stationary problem of thermodiffusion for an elastic cylinder, International Journal of Engineering Science 33, 773-780 (1995).
[16] J.A. Gawinecki, P. Kacprzyk, P. Bar-Yoseph, Initial boundary value problem for some coupled nonlinear parabolic system of partial differential equations appearing in thermoelastic diffusion in solid body, Journal for Analysis and its Applications 19, 121-130 (2000).
[17] J. A. Gawinecki, A. Szymaniec, Global solution of the Cauchy problem in nonlinear thermoelastic diffusion in solid body, Proceedings in Applied Mathematics and Mechanics 1, 446-447 (2002).
[18] H.H. Sherief, F.A. Hamza and H.A. Saleh, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42, 591-608 (2004).
[19] M. Aouadi, Uniqueness and reciprocity theorems in the the- ory of generalized thermoelastic diffusion, Journal of Thermal Stresses 30, 665-678 (2007).
[20] H.H. Sherief, H.A. Saleh, A half space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42, 4484-4493 (2005).
[21] R. Kumar, T. Kansal, Variational principle, uniqueness and reciprocity theorems in the theory of generalized thermoelas- tic diffusion material, QScience Connect 2013, 1-18 (2013).
[22] P. Borejko, Reflection and transmission coefficients for three- dimensional plane waves in elastic media, Wave Motion 24, 371-393 (1996).
[23] C.M. Wu, B. Lundberg, Reflection and transmission of the energy of harmonic elastic waves in a bent bar, Journal of Sound and Vibration 190, 645-659 (1996).
[24] S.B. Sinha, K.A. Elsibai, Reflection and refraction of ther- moelastic waves at an interface of two semi-infinite media with two relaxation times, Journal of Thermal Stresses 20, 129-145 (1997).
[25] M.D. Sharma, M.L. Gogna, Reflection and refraction of plane harmonic waves at an interface between elastic solid and porous solid saturated by viscous liquid, PAGEOPH 138, 249-266 (1992).
[26] S.K. Tomar, A. Arora, Reflection and transmission of elas- tic waves at an elastic/porous solid saturated by immisci- ble fluids, International Journal of Solids and Structures 43, 1991-2013 (2006).
[27] R. Kumar, P. Sarthi, Reflection and refraction of thermoelas- tic plane waves at an interface of two thermoelastic media without energy dissipation, Archives of Mechanics 58, 155- 185 (2006).
[28] A.N. Abd-alla, A.A.S. Al-dawy, The reflection phenomena of SV-waves in a generalized thermoelastic medium, International Journal of Mathematics and Mathematical Sciences 23, 529-546 (2000).
[29] M. Ciarletta, M.A. Sumbatyan, Reflection of plane waves by the free boundary of a porous elastic half-space, Journal of Sound and Vibration 259, 253-264 (2003).
[30] J. Singh, S.K. Tomar, Plane waves in thermo-elastic material with voids, Mechanics of Materials 39, 932-940 (2007).
[31] B. Singh, Wave propagation in a generalized thermoelastic material with voids, Applied Mathematics and Computation 189, 698-709 (2007).
[32] J. Singh, S.K. Tomar, Reflection and transmission of trans- verse waves at a plane interface between two different porous elastic half-spaces, Applied Mathematics and Computation 176, 364-378 (2006).
[33] N.C. Das, A. Lahiri, S. Sarkar and B. Basu, Reflection of gen- eralized thermoelastic waves from isothermal and insulated boundaries of a half-space, Computers and Mathematics with Applications 56, 2795-2805 (2008).
[34] S.S. Singh, Reflection and transmission of couple longitudi- nal waves at a plane interface between two dissimilar half- spaces of thermo-elastic materials with voids, Applied Math- ematics and Computation 218, 3359-3371 (2011).
[35] R. Kumar, R. Kumar, Reflection and transmission at the plane boundary of elastic half-space and initially stressed thermoe- lastic diffusion with voids half-space, MMMS 9, 185-213 (2013).
[36] A.N. Abd-alla, Relaxation effects on reflection of generalized magneto-thermoelastic waves, Mechanics Research Commu- nications 27, 591-600 (2000).
[37] M.I.A. Othman, Y. Song, Reflection of magneto-thermo- elastic waves with two relaxation times and temperature dependent elastic moduli, Applied Mathematics and Mod- elling 32, 483-500 (2008).
[38] M.I.A. Othman, R. Kumar, Reflection of magneto-thermoelas- ticity waves with temperature dependent properties in gen- eralized thermoelasticity, International Communications in Heat and Mass Transfer 36, 513-520 (2009).
[39] S.M. Abo-Dahab, Propagation of P waves from stress-free surface elastic half-space with voids under thermal relax- ation and magnetic field, Applied Mathematics and Mod- elling 34, 1798-1806 (2010).
[40] M.I.A. Othman, Generalized electro-magneto-thermoelas- ticity in case of thermal shock plane waves for a finite con- ducting half-Space with two relaxation times, Mechanics and Mechanical Engineering 14, 5-30 (2010).
[41] B. Singh, L. Singh and S. Deswal, Reflection of plane waves from a free surface of a generalized magneto-thermoelastic solid half-space with diffusion, Journal of Theoretical and Applied Mechanics 52, 385-394 (2014).
[42] R.D. Borcherdt, Reflection-refraction of general P and type-I S waves in elastic and anelastic solids, Geophysical Journal of Royal Astronomical Society 70, 621-638 (1982).
[43] J.D. Achenbach, Wave propagation in elastic solids, North- Holland, Amsterdam (1973).
[44] R. Kumar, T. Kansal, Reflection and refraction of plane waves at an interface of an elastic solid half-space and a thermoelastic diffusive solidhalf-space, Archives of Mechanics 64, 293-317 (2012).
[45] K.E. Bullen, An introduction to the theory of Seismology, Cambridge University press, England (1963).
