Reflection of Plane Waves from Surface of a Generalized Thermo-viscoelastic Porous Solid Half-space with Impedance Boundary Conditions
Department of Mathematics
Post Graduate Government College
Sector-11, Chandigarh – 160 011, India
E-mail: bsinghgc11@gmail.com
Received:
Received: 23 January 2018; revised: 25 December 2018; accepted: 27 December 2018; published online: 30 December 2018
DOI: 10.12921/cmst.2018.0000012
Abstract:
A phenomenon of reflection of plane waves from a thermally insulated surface of a solid half-space is studied in the context of Lord-Shulman theory of generalized thermo-viscoelasticity with voids. The governing equations of generalized thermo-viscoelastic medium with voids are specialized in x-z plane. The plane wave solution of these equations shows the existence of three coupled longitudinal waves and a shear vertical wave in a generalized thermo-viscoelastic medium with voids. For incident plane wave (longitudinal or shear), three coupled longitudinal waves and a shear vertical wave reflect back in the medium. The mechanical boundary conditions on the free surface of solid half-space are consid- ered as impedance boundary conditions, in which the shear force tractions are assumed to vary linearly with the tangential displacement components multiplied by the frequency. The impedance corresponds to the constant of proportionality. The appropriate potentials of incident and reflected waves in the half-space will satisfy the required impedance boundary condi- tions. A non-homogeneous system of four equations in the amplitude ratios of reflected waves is obtained. These amplitude ratios are functions of material parameters, impedance parameter, angle of incidence, thermal relaxation and speeds of plane waves. Using relevant material parameters for medium, the amplitude ratios are computed numerically and plotted against certain ranges of the impedance parameter and the angle of incidence.
Key words:
amplitude ratios, generalized thermo-viscoelasticity, plane waves, reflection, thermal relaxation, voids
References:
[1] S.C. Cowin, J.W. Nunziato, Linear theory of elastic materials with voids, J. Elast. 13, 125–147 (1983).
[2] D. Iesan, A theory of thermoelastic materials with voids, Acta Mech. 60, 67–89 (1986).
[3] D. Iesan, Thermoelastic Models of Cortinua, Kluwer Academic Publishers, Boston, Dordrecht, London, 2004.
[4] D.Iesan,Sometheoremsinthetheoryofelasticmaterialswith voids, J. Elast. 15, 215–224 (1985).
[5] M. Ciarletta, A. Scalia, On uniqueness and reciprocity in linear thermoelasticity of materials with voids, J. Elast. 32, 1–17 (1993).
[6] S. Ciarletta, A. Scalia, On the spatial and temporal behaviour in linear thermoelasticity of materials with voids, J. Therm. Stresses 24, 433–455 (2001).
[7] S. Ciarletta, M. Ciarletta, V. Tibullo, Rayleigh surface waves on a Kelvin-Voigt viscoelastic half-space, J. Elast. 115, 61–76 (2014).
[8] D.Iesan,L.Nappa,Thermalstressesinplanestrainofporous elastic bodies, Meccanica 39, 125–138 (2004).
[9] S. Chirita, C. D’Apice, On Saint-Venant’s principle for a linear poroelastic material in plane strain, J. Math. Anal. Appl. 363, 454–467 (2010).
[10] S.Chirita,C.D’Apice,OnSaint-Venant’sprincipleinaporoelastic arch-like region, Math. Methods Appl. Sci. 33, 1743– 1754 (2010).
[11] M. Ciarletta, F. Passarella, M. Svanadze, Planes waves and uniqueness theorems in the coupled linear theory of elasticity for solids with double porosity, J. Elast. 114, 55–68 (2014).
[12] P. Puri, S.C. Cowin, Plane waves in linear elastic materials with voids, J. Elast. 15, 167–183 (1985).
[13] D.S. Chandrasekharaiah, Surface waves in an elastic halfspace with voids, Acta Mech. 62, 77–85 (1986).
[14] D.S. Chandrasekharaiah, Rayleigh Lamb waves in an elastic plate with voids, J. Appl. Mech. 54, 509–512 (1987).
[15] B. Singh, Wave propagation in a generalized thermoelastic material with voids, Appl. Math. Comp. 189, 698–709 (2007).
[16] M. Ciarletta, B. Straughan, Thermo-poroacoustic acceleration
[18] A.V. Bucur, F. Passarella, V. Tibullo, Rayleigh surface waves in the theory of thermoelastic materials with voids, Meccanica 49, 2069–2078 (2014).
[19] D. Iesan, On a theory of thermoviscoelastic materials with voids, J. Elast. 104, 369–384 (2011).
[20] D. Iesan, On the Nonlinear Theory of Thermoviscoelastic Materials with Voids, J. Elast. 128, 1–16 (2017).
[21] K. Sharma, P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, J. Therm. Stresses 36, 94–111 (2013).
[22] M.M. Svanadze, Potential method in the linear theory of viscoelastic materials with voids, J. Elast. 114, 101–126 (2014).
[23] S.K. Tomar, J. Bhagwan, H. Steeb, Time harmonic waves in a thermo-viscoelastic materials with voids, J. Vib. Control 20, 1119–1136 (2014).
[24] S. Chirita, On the spatial behaviour of the steady-state vibrations in thermoviscoelastic porous materials, J. Therm. Stresses 38, 96–109 (2015).
[25] S. Chirita and A. Danescu, Surface waves problem in a thermoviscoelastic porous half-space, Wave Motion 54, 100–114 (2015).
[26] C.D’Apice,S.Chirita,Planeharmonicwavesinthetheoryof thermoviscoelastic materials with voids, J. Therm. Stresses 39, 142–155 (2016).
[27] A.V. Bucur, Rayleigh surface waves problem in linear thermoviscoelasticity with voids, Acta Mechanica 227, 1199–1212 (2016).
[28] R.S. Lakes, Viscoelastic Materials, Cambridge University Press, Cambridge, 2009.
[29] E. Godoy, M. Durn, J.-C. Ndlec, On the existence of the surface waves in an elastic half-space with impedance boundary conditions, Wave Motion 49, 585–594 (2012).
[30] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15, 299–309 (1967).
[31] A.Hobbs,S.Murakami,T.Hosoda,S.Ohtsuka,M.Miyajima, S. Sugatani, T. Nakamura, Evolution of grain and micro-void structure in electroplated copper interconnects, Mater. Trans. 43, 1629–1632 (2002).
[32] T.C. Wang, Y.L. Wang, T.E. Hsieh, S.C. Chang, Y.L. Cheng,
Copper voids improvements for the copper dual damascene in-
waves in elastic materials with voids, J. Math. Anal. Appl. 333,
142–150 (2007). 571
[17] M. Ciarletta, M. Svanadze, L. Buonanno, Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids, Eur. J. Mech., (A-Solids) 28, 897–903 (2009).
[33] P. Gondcharton, B. Imbert, L. Benaissa, M. Verdier, Voiding phenomena in copper-copper bonded structures: Role of creep, ECS J. Solid State Sci. Tech. 4, P77-P82 (2015).
A phenomenon of reflection of plane waves from a thermally insulated surface of a solid half-space is studied in the context of Lord-Shulman theory of generalized thermo-viscoelasticity with voids. The governing equations of generalized thermo-viscoelastic medium with voids are specialized in x-z plane. The plane wave solution of these equations shows the existence of three coupled longitudinal waves and a shear vertical wave in a generalized thermo-viscoelastic medium with voids. For incident plane wave (longitudinal or shear), three coupled longitudinal waves and a shear vertical wave reflect back in the medium. The mechanical boundary conditions on the free surface of solid half-space are consid- ered as impedance boundary conditions, in which the shear force tractions are assumed to vary linearly with the tangential displacement components multiplied by the frequency. The impedance corresponds to the constant of proportionality. The appropriate potentials of incident and reflected waves in the half-space will satisfy the required impedance boundary condi- tions. A non-homogeneous system of four equations in the amplitude ratios of reflected waves is obtained. These amplitude ratios are functions of material parameters, impedance parameter, angle of incidence, thermal relaxation and speeds of plane waves. Using relevant material parameters for medium, the amplitude ratios are computed numerically and plotted against certain ranges of the impedance parameter and the angle of incidence.
Key words:
amplitude ratios, generalized thermo-viscoelasticity, plane waves, reflection, thermal relaxation, voids
References:
[1] S.C. Cowin, J.W. Nunziato, Linear theory of elastic materials with voids, J. Elast. 13, 125–147 (1983).
[2] D. Iesan, A theory of thermoelastic materials with voids, Acta Mech. 60, 67–89 (1986).
[3] D. Iesan, Thermoelastic Models of Cortinua, Kluwer Academic Publishers, Boston, Dordrecht, London, 2004.
[4] D.Iesan,Sometheoremsinthetheoryofelasticmaterialswith voids, J. Elast. 15, 215–224 (1985).
[5] M. Ciarletta, A. Scalia, On uniqueness and reciprocity in linear thermoelasticity of materials with voids, J. Elast. 32, 1–17 (1993).
[6] S. Ciarletta, A. Scalia, On the spatial and temporal behaviour in linear thermoelasticity of materials with voids, J. Therm. Stresses 24, 433–455 (2001).
[7] S. Ciarletta, M. Ciarletta, V. Tibullo, Rayleigh surface waves on a Kelvin-Voigt viscoelastic half-space, J. Elast. 115, 61–76 (2014).
[8] D.Iesan,L.Nappa,Thermalstressesinplanestrainofporous elastic bodies, Meccanica 39, 125–138 (2004).
[9] S. Chirita, C. D’Apice, On Saint-Venant’s principle for a linear poroelastic material in plane strain, J. Math. Anal. Appl. 363, 454–467 (2010).
[10] S.Chirita,C.D’Apice,OnSaint-Venant’sprincipleinaporoelastic arch-like region, Math. Methods Appl. Sci. 33, 1743– 1754 (2010).
[11] M. Ciarletta, F. Passarella, M. Svanadze, Planes waves and uniqueness theorems in the coupled linear theory of elasticity for solids with double porosity, J. Elast. 114, 55–68 (2014).
[12] P. Puri, S.C. Cowin, Plane waves in linear elastic materials with voids, J. Elast. 15, 167–183 (1985).
[13] D.S. Chandrasekharaiah, Surface waves in an elastic halfspace with voids, Acta Mech. 62, 77–85 (1986).
[14] D.S. Chandrasekharaiah, Rayleigh Lamb waves in an elastic plate with voids, J. Appl. Mech. 54, 509–512 (1987).
[15] B. Singh, Wave propagation in a generalized thermoelastic material with voids, Appl. Math. Comp. 189, 698–709 (2007).
[16] M. Ciarletta, B. Straughan, Thermo-poroacoustic acceleration
[18] A.V. Bucur, F. Passarella, V. Tibullo, Rayleigh surface waves in the theory of thermoelastic materials with voids, Meccanica 49, 2069–2078 (2014).
[19] D. Iesan, On a theory of thermoviscoelastic materials with voids, J. Elast. 104, 369–384 (2011).
[20] D. Iesan, On the Nonlinear Theory of Thermoviscoelastic Materials with Voids, J. Elast. 128, 1–16 (2017).
[21] K. Sharma, P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids, J. Therm. Stresses 36, 94–111 (2013).
[22] M.M. Svanadze, Potential method in the linear theory of viscoelastic materials with voids, J. Elast. 114, 101–126 (2014).
[23] S.K. Tomar, J. Bhagwan, H. Steeb, Time harmonic waves in a thermo-viscoelastic materials with voids, J. Vib. Control 20, 1119–1136 (2014).
[24] S. Chirita, On the spatial behaviour of the steady-state vibrations in thermoviscoelastic porous materials, J. Therm. Stresses 38, 96–109 (2015).
[25] S. Chirita and A. Danescu, Surface waves problem in a thermoviscoelastic porous half-space, Wave Motion 54, 100–114 (2015).
[26] C.D’Apice,S.Chirita,Planeharmonicwavesinthetheoryof thermoviscoelastic materials with voids, J. Therm. Stresses 39, 142–155 (2016).
[27] A.V. Bucur, Rayleigh surface waves problem in linear thermoviscoelasticity with voids, Acta Mechanica 227, 1199–1212 (2016).
[28] R.S. Lakes, Viscoelastic Materials, Cambridge University Press, Cambridge, 2009.
[29] E. Godoy, M. Durn, J.-C. Ndlec, On the existence of the surface waves in an elastic half-space with impedance boundary conditions, Wave Motion 49, 585–594 (2012).
[30] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids 15, 299–309 (1967).
[31] A.Hobbs,S.Murakami,T.Hosoda,S.Ohtsuka,M.Miyajima, S. Sugatani, T. Nakamura, Evolution of grain and micro-void structure in electroplated copper interconnects, Mater. Trans. 43, 1629–1632 (2002).
[32] T.C. Wang, Y.L. Wang, T.E. Hsieh, S.C. Chang, Y.L. Cheng,
Copper voids improvements for the copper dual damascene in-
waves in elastic materials with voids, J. Math. Anal. Appl. 333,
142–150 (2007). 571
[17] M. Ciarletta, M. Svanadze, L. Buonanno, Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids, Eur. J. Mech., (A-Solids) 28, 897–903 (2009).
[33] P. Gondcharton, B. Imbert, L. Benaissa, M. Verdier, Voiding phenomena in copper-copper bonded structures: Role of creep, ECS J. Solid State Sci. Tech. 4, P77-P82 (2015).
