Impact of Initial Stress on Reflection and Transmission of SV-Wave between Two Orthotropic Thermoelastic Half-Spaces

Subject Areas : Engineering

B  Prasad ¹ (Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, India)
S  Kundu ² (Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, India)
P  Chandra Pal ³ (Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, India)

Received: 2019-07-18 Accepted : 2019-09-19 Published : 2019-12-30

Keywords: Thermal wave, Anisotropic, Transmission coefficient, Reflection coefficient, Initial stress,

Abstract :

Reflection and transmission of plane waves between two initially stressed thermoelastic half-spaces with orthotropic type of anisotropy is studied. Incidence of a SV-type wave from the lower half-space is considered and the amplitude ratios of the reflected and transmitted SV-wave, P-wave and thermal wave are obtained by using appropriate boundary conditions. Numerical computation for a particular model is performed and graphs are plotted to study the effect of angle of incidence of the wave and the initial stress parameters of the half-spaces. From the graphical results, it is found that the modulus of reflection and transmission coefficients of the thermal wave is very less in comparison to reflection and transmission coefficients of P- and SV-waves. It is also observed that for vertical incidence of SV-wave we have only reflected and refracted SV-waves and there is no reflected or refracted P and thermal waves, whereas for horizontal incidence of SV-wave there exists only reflected SV-wave and no other reflected or transmitted wave exists. Moreover, it is found that all the reflection and transmission coefficients are strongly affected by the initial stress parameters of the both half-spaces.

References:

[1] Love A.E.H., 1944, A Treatise on Mathematical Theory of Elasticity, Dover Publication, New York.
[2] Biot M.A., 1940, The influence of initial stress on elastic waves, Journal of Applied Physics 11(8): 522-530.
[3] Biot M.A., 1965, Mechanics of Incremental Deformations, John Wiley and Sons, New York.
[4] Dey S., Addy S.K., 1977, Reflection of plane waves under initial stresses at a free surface, International Journal of Non-Linear Mechanics 12(6): 371-381.
[5] Chattopadhyay A., Bose S., Chakraborty M., 1982, Reflection of elastic waves under initial stress at a free surface: P and SV motion, The Journal of Acoustical Society of America 72(1): 255-263.
[6] Dey S., Roy N., Dutta A., 1985, Reflection and refraction of P-waves under initial stresses at an interface, Indian Journal of Pure and Applied Mathematics 16(9): 1051-1071.
[7] Sinha A.N., Sinha S.B., 1974, Reflection of thermoelastic waves at a solid half-space with thermal relaxation, Journal of Physics of the Earth 22(2): 237-244.
[8] Sinha S.B., Elsibai K.A., 1996, Reflection of thermoelastic waves at a solid half-space with two relaxation times, Journal of Thermal Stresses 19(8): 749-762.
[9] Sinha S.B., Elsibai K.A., 1997, Reflection and refraction of thermoelastic waves at an interface of two semi-infinite media with two relaxation times, Journal of Thermal Stresses 20(2): 129-145.
[10] Abd-Ala A.N., Al-Dawy A.A.S, 2000, The reflection phenomena of SV-waves in a generalized thermoelastic medium, International Journal of Mathematics and Mathematical Sciences 23(8): 529-546.
[11] Kumar R., Singh M., 2008, Reflection/transmission of plane waves at an imperfectly bounded interface of two orthotropic generalized thermoelastic half-spaces, Material Science and Engineering 472(1-2): 83-96.
[12] Singh B., Singh L., Deswal S., 2011, Propagation of waves at an interface between two thermoelastic half-spaces with diffusion, Journal of Theoretical and Applied Mechanics, Sofia 41(1): 61-80.
[13] Montanaro A., 1999, On singular surfaces in isotropic linear thermoelasticity with initial stress, The Journal of Acoustical Society of America 106(3): 1586-1588.
[14] Singh B., Kumar A., Singh J., 2006, Reflection of generalized thermoelastic waves from a solid half-space under hydrostatic initial stress, Applied Mathematics and Computation 177(1): 170-177.
[15] Othman M.I.A., Song Y., 2007, Reflection of plane waves from an elastic solid half-space under hydrostatic initial stress without energy dissipation, International Journal of Solids and Structures 44(17): 5651-5664.
[16] Singh B., Singh J., Ailawalia, P., 2010, Thermoelastic waves at an interface between two solid half-spaces under hydrostatic initial stress, Archive of Mechanics 62(3): 177-193.
[17] Chakraborty N., Singh M.C., 2011, Reflection and refraction of a plane thermoelastic wave at a solid-solid interface under perfect boundary condition, in presence of normal initial stress, Applied Mathematical Modelling 35(11): 5286-5301.
[18] Singh M.C., Chakraborty N., 2013, Reflection and refraction of P-, SV- and thermal wave, at an initially stressed solid-liquid interface in generalized thermoelasticity, Applied Mathematical Modelling 37(1-2): 463-475.
[19] Singh M.C., Chakraborty N., 2017, Reflection and refraction of plane waves at the interface of two visco-thermoelastic liquid layers in presence of magnetic field and compressional stress, with the outer core inside the earth as a model, International Journal of Applied and Computational Mathematics 3(3): 2107-2124.
[20] Kumar R., Kumar R., 2011, Wave propagation in orthotropic generalized thermoelastic half-space with voids under initial stress, International Journal of Applied Mathematics and Mechanics 7(13): 17-44.
[21] Abd-Alla A.M., Abo-Dahab S.M., Hammad H.H.A, 2011, Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field, Applied Mathematical Modelling 35(6): 2981-3000.
[22] Ahmed S.M., Abo-Dahab S.M., 2012, Influence of initial stress and gravity field on propagation of Rayleigh and Stoneley waves in a thermoelastic orthotropic granular medium, Mathematical Problems in Engineering 2012: 245965.
[23] Pal P.C., Kumar S., Mandal D., 2014, Wave propagation in an inhomogeneous anisotropic generalized thermoelastic solid, Journal of Thermal Stresses 37(7): 817-831.
[24] Sharma M.D., 2010, Wave propagation in a pre-stressed anisotropic generalized thermoelastic medium, Earth, Planet and Space 62(4): 381-390.
[25] Sharma M.D., 2014, Propagation and attenuation of Rayleigh waves in generalized thermoelastic media, Journal of Seismology 18(1): 61-79.
[26] Gupta R.R., Gupta R.R., 2014, Reflection of waves in a rotating transversely isotropic thermoelastic half-space under initial stress, Journal of Solid Mechanics 6(2): 229-239.
[27] Sharma J.N., Kaur R., 2014, Study of reflection and transmission of plane waves at thermoelastic-diffusive solid/liquid interface, Latin American Journal of Solids and Structures 11(12): 2141-2170.
[28] Kakar R., Kakar S., 2015, Modelling of magneto-thermoelastic plane waves at the interface of two prestressed solid half-spaces without energy dissipation, Earthquakes and Structures 8(6): 1299-1323.
[29] Kumar R., Garg S.K., Ahuja S., 2015, Wave propagation in fibre-reinforced transversely isotropic thermoelastic media with initial stress at the boundary, Journal of Solid Mechanics 7(2): 223-238.
[30] Sur A., Kanoria M., 2014, Finite thermal wave propagation in a half-space due to variable thermal loading, Applications and Applied Mathematics: An International Journal 9(1): 94-121.
[31] Sur A., Kanoria M., 2015, Propagation of thermal waves in a functionally graded thick plate, Mathematics and Mechanics of Solids 22(4): 718-736.
[32] Sur A., Kanoria M., 2017, Modeling of fibre-reinforced magneto-thermoelastic plate with heat sources, Procedia Engineering 173: 875-882.
[33] Sur A., Kanoria M., 2017, Three-dimensional thermoelastic problem under two-temperature theory, International Journal of Computational Methods 14(3): 1750030-1-17.
[34] Karmakar R., Sur A., Kanoria M., 2016, Generalized thermoelastic problem of an infinite body with a spherical cavity under dual-phase-lags, Journal of Applied Mechanics and Technical Physics 57(4): 652-665.
[35] Prasad B., Pal P.C., Kundu S., Prasad N., 2017, Effect of inhomogeneity due to temperature on the propagation of shear waves in an anisotropic layer, AIP Conference Proceedings 1860: 020053-1-7.
[36] Kumar R., Kaur M., 2017, Reflection and transmission of plane waves at micropolar piezothermoelastic solids, Journal of Solid Mechanics 9(3): 508-526.
[37] Sur A., Pal P., Kanoria M., 2018, Modeling of memory-dependent derivative in a fiber-reinforced plate under gravitational effect, Journal of Thermal Stresses 41(8): 973-992.
[38] Sur A., Paul S., Kanoria M., 2019, Modeling of memory-dependent derivative in a functionally graded plate, Waves in Random and Complex Media, doi:10.1080/17455030.2019.1606962.
[39] Purkait P., Sur A., Kanoria M., 2019, Elasto-thermodiffusive response in a spherical shell subjected to memory-dependent heat transfer, Waves in Random and Complex Media, doi:10.1080/17455030.2019.1599464.