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Reflection and refraction of plane waves in layered nonlocal elastic and anisotropic thermoelastic medium

  • Lata, Parveen (Department of Basic and Applied Sciences, Punjabi University)
  • Received : 2018.01.24
  • Accepted : 2018.02.19
  • Published : 2018.04.10

Abstract

In the present paper, we have considered a layered medium of two semi-infinite nonlocal elastic solids with intermediate transversely isotropic magnetothermoelastic solid. The intermediate slab is of uniform thickness with the effects of two temperature, rotation and Hall current and with and without energy dissipation. A plane longitudinal or transverse wave propagating through one of the nonlocal elastic solid half spaces, is made incident upon transversely isotropic slab and it results into various reflected and refracted waves. The amplitude ratios of various reflected and refracted waves are obtained by using appropriate boundary conditions. The effect of nonlocal parameter on the variation of various amplitude ratios with angle of incidence are depicted graphically. Some cases of interest are also deduced.

Keywords

References

  1. Achenbach, J.D. (1973), Wave Propagation in Elastic Solids, Elsevier, North-Holland, Amsterdam.
  2. Atwa, S.Y. and Jahangir, A. (2014), "Two temperature effects on plane waves in generalized thermo-microstretch elastic solid", J. Thermophys., 35(1), 175-193. https://doi.org/10.1007/s10765-013-1541-9
  3. Boley, B.A. and Tolins, I.S. (1962), "Transient coupled thermoelastic boundary value problem in the half space", J. Appl. Mech., 29(4), 637-646. https://doi.org/10.1115/1.3640647
  4. Chandrasekharaiah, D.S. (1998), "Hyperbolic thermoelasticity: A review of recent literature", Appl. Mech. Rev., 51(12), 705-729. https://doi.org/10.1115/1.3098984
  5. Chaudhary, S., Kaushik, V.P. and Tomar, S.K. (2010), "Transmission of plane SH-waves through a monoclinic layer embedded between two different self-reinforced elastic solid half-spaces", J. Appl. Math. Mech., 6(19), 22-43.
  6. Chen, P.J. and Gurtin, M.E. (1968), "On a theory of heat conduction involving two parameters", Zeitschrift fur Angewandte Mathematik und Physik (ZAMP), 19, 614-627. https://doi.org/10.1007/BF01594969
  7. Chen, P.J., Gurtin, M.E. and Williams, W.O. (1968), "A note on simple heat conduction", J. Appl. Math. Phys. (ZAMP), 19, 969-970. https://doi.org/10.1007/BF01602278
  8. Chen, P.J., Gurtin, M.E. and Williams, W.O. (1969), "On the thermodynamics of non-simple elastic materials with two temperatures", J. Appl. Math. Phys. (ZAMP), 20(1), 107-112. https://doi.org/10.1007/BF01591120
  9. Das, P. and Kanoria, M. (2014), "Study of finite thermal waves in a magnetothermoelastic rotating medium", J. Therm. Stress., 37(4), 405-428. https://doi.org/10.1080/01495739.2013.870847
  10. Deshpande, V.S. and Fleck, N.A. (2005), "One-dimensional response of sandwich plates to underwater shock loading", J. Mech. Phys. Sol., 53(11), 2347-2383. https://doi.org/10.1016/j.jmps.2005.06.006
  11. Dhaliwal, R.S. and Singh, A. (1980), Dynamic Coupled Thermoelasticity, Hindustance Publisher Corp., New Delhi, India.
  12. Edelen, D.G.B. and Laws, N. (1971), "On the thermodynamics of systems with non-locality", Arch. Rat. Mech. Analy., 43(1), 36-44. https://doi.org/10.1007/BF00251544
  13. Elphinstone, M.J. and Lakhtakia, A. (1994), "Plane wave response of an elastic chiral solid slab sandwiched between achiral solid half spaces", J. Acoust. Soc. Am., 95(2), 617-627. https://doi.org/10.1121/1.408422
  14. Eringen, A.C. (1972), "Nonlocal polar elastic continua", J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  15. Eringen, A.C. and Edelen, D.G.B. (1972), "On nonlocal elasticity", J. Eng. Sci., 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0
  16. Ezzat, M.A. and AI-Bary, A.A. (2017), "Fractional magnetothermoelastic materials with phase lag green-naghdi theories", Steel Compos. Struct., 24(3), 297-307. https://doi.org/10.12989/SCS.2017.24.3.297
  17. Ezzat, M.A., EI-Karamany, A.S. and EI-Bary, A.A. (2016), "Modelling of memory dependent derivative in generalized thermoelasticity", Eur. Phys. J. Plus, 131(10), 372. https://doi.org/10.1140/epjp/i2016-16372-3
  18. Green, A.E. and Naghdi, P.M. (1991), "A re-examination of the basic postulates of thermomechanics", Proceedings of the Royal Society of London Series A, 432(1885), 171-194. https://doi.org/10.1098/rspa.1991.0012
  19. Green, A.E. and Naghdi, P.M. (1992), "On undamped heat waves in an elastic solid", J. Therm. Stress., 15(2), 253-264. https://doi.org/10.1080/01495739208946136
  20. Green, A.E. and Naghdi, P.M. (1993), "Thermoelasticity without energy dissipation", J. Elast., 31(3), 189-208. https://doi.org/10.1007/BF00044969
  21. Jahangir, A. (2012), "Propagation of plane magneto-thermo-elastic waves in a rotating, electrically conducting and transversely isotropic medium", Sci. Res. Essays, 7(10), 1148-1155.
  22. Kaushal, S., Kumar, R. and Miglani, A. (2011), "Wave propagation in temperature rate dependent thermoelasticity with two temperatures", Math. Sci., 5, 125-146.
  23. Keith, C.M. and Crampin. (1977), "Seismic body waves in anisotropic media, reflection and refraction at a plane interface", Geophys. J. Int., 49(1), 181-208. https://doi.org/10.1111/j.1365-246X.1977.tb03708.x
  24. Khurana, A. and Tomar, S.K. (2009), "Longitudinal wave response of a chiral slab interposed between micropolar halfspaces", J. Sol. Struct., 46(1), 135-150. https://doi.org/10.1016/j.ijsolstr.2008.08.018
  25. Kroner, E. (1967), "Elasticity theory of materials with long range cohesive forces", J. Sol. Struct., 3(5), 731-742. https://doi.org/10.1016/0020-7683(67)90049-2
  26. Kumar, R., Sharma, N. and Lata, P. (2017), "Effects of hall current and two temperatures in transversely isotropic magnetothermoelastic with and without energy dissipation due to ramp type heat", Mech. Adv. Mater. Struct., 24(8), 625-635. https://doi.org/10.1080/15376494.2016.1196769
  27. Kumar, R., Sharma, N. and Lata, P. (2016b), "Thermomechanical interactions due to hall current in transversely isotropic thermoelastic with and without energy dissipation with two temperature and rotation", J. Sol. Mech., 8(4), 840-858.
  28. Kumar, R. and Gupta, V. (2013), "Plane wave propagation in an anisotropic thermoelastic medium with fractional order derivative and void", J. Thermoelast., 1(1), 21-34.
  29. Kumar, R., Sharma, N. and Lata, P. (2016a), "Thermomechanical interactions in transversely isotropic magneto thermoelastic medium with vacuum and with and without energy dissipation with combined effects of rotation, vacuum and two temperature", Appl. Math. Model., 40(13-14), 6560-6575. https://doi.org/10.1016/j.apm.2016.01.061
  30. Kumar, R., Sharma, N. and Lata, P. (2016), "Effects of hall current in a transversely isotropic magnetothermoelastic with and without energy dissipation due to normal force", Struct. Eng. Mech., 57(1), 91-103. https://doi.org/10.12989/sem.2016.57.1.091
  31. Kumar, R. and Mukhopdhyay, S. (2010), "Effects of thermal relaxation times on plane wave propagation under two temperature thermoelasticity", J. Eng. Sci., 48(2), 128-139. https://doi.org/10.1016/j.ijengsci.2009.07.001
  32. Kumar, R. and Kansal, T. (2011), "Reflection of plane waves at the free surface of a transversely isotropic thermoelastic diffusive solid half-space", J. Appl. Math. Mech., 7(14), 57-78.
  33. Lata, P., Kumar, R. and Sharma, N. (2016), "Plane waves in anisotropic thermoelastic medium", Steel Compos. Struct., 22(3), 567-587. https://doi.org/10.12989/scs.2016.22.3.567
  34. Liu, L. and Bhattacharya, K. (2009), "Wave propagation in a sandwich structure", J. Sol. Struct., 46(17), 3290-3300. https://doi.org/10.1016/j.ijsolstr.2009.04.023
  35. Marin, M. and Baleanu, D. (2016), "On vibrations in thermoelasticity without energy dissipation for micropolar bodies", Bound. Val. Prob., 1-19.
  36. Marin, M. (1997), "On weak solutions in elasticity of dipolar bodies with voids", J. Comput. Appl. Math., 82(1-2), 291-297. https://doi.org/10.1016/S0377-0427(97)00047-2
  37. Marin, M. (2008), "Weak solutions in elasticity of dipolar porous materials", Math. Prob. Eng., 1-8.
  38. Marin, M. (2016), "An approach of a heat flux dependent theory for micropolar porous media", Meccan., 51(5), 1127-1133. https://doi.org/10.1007/s11012-015-0265-2
  39. Othman, M.I.A. and Abd-Elaziz, E.M. (2017), "Plane waves in a magneto-thermoelastic solids with voids and microtemperatures due to hall current and rotation", Result. Phys., 7, 4253-4263. https://doi.org/10.1016/j.rinp.2017.10.053
  40. Othman, M.I.A. and Jahangir, A. (2015), "Plane waves on rotating microstretch elastic solid with temperature dependent elastic properties", Appl. Math. Informat. Sci., 9(6), 2963-2972.
  41. Othman, M.I.A. (2010), "Generalized electro-magnetothermoelasticity in case of thermal shock waves for a finite conducting half-space with two relaxation times", Mech. Mech. Eng., 14(1), 5-30.
  42. Polizzotto, C. (2001), "Nonlocal elasticity and related variational principles", J. Sol. Struct., 38, 7359-7380. https://doi.org/10.1016/S0020-7683(01)00039-7
  43. Said, M.S. and Othman, M.I.A. (2016), "Wave propagation in a two temperature fibre-reinforced magneto-thermoelastic medium with three phase lag models", Struct. Eng. Mech., 57(2), 201-220. https://doi.org/10.12989/sem.2016.57.2.201
  44. Sharma, K. and Marin, M. (2013), "Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space", Univ. Politeh. Bucharest Sci. Bullet. Ser., 75(2), 121-132.
  45. Sharma, K. and Bhargava, R.R. (2014), "Propagation of thermoelastic plane waves at an imperfect boundary of thermal conducting viscous liquid/generalized thermolastic solid", Afrika Mathematika, 25, 81-102. https://doi.org/10.1007/s13370-012-0099-1
  46. Singh, D., Kaur, G. and Tomar, S.K. (2017), "Waves in nonlocal elastic solids with voids", J. Elast.
  47. Slaughter, W.S. (2002), The Linearised Theory of Elasticity, Birkhausar.
  48. Vasiliew, V.V. and Lurie, S.A. (2016), "On correct nonlocal generalized theories of elasticity", Phys. Mesomech., 19(3), 269-281. https://doi.org/10.1134/S102995991603005X
  49. Warren, W.E. and Chen, P.J. (1973), "Wave propagation in the two temperature theory of thermoelasticity", J. Acta Mech., 16, 21-33. https://doi.org/10.1007/BF01177123
  50. Youssef, H.M. (2006), "Theory of two temperature generalized thermoelasticity", IMA J. Appl. Math., 71(3), 383-390. https://doi.org/10.1093/imamat/hxh101
  51. Youssef, H.M. (2011), "Theory of two-temperature thermoelasticity without energy dissipation", J. Therm. Stress., 34(2), 138-146. https://doi.org/10.1080/01495739.2010.511941

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