Reflection of plane waves at the initially stressed surface of a fiber-reinforced thermoelastic half space with temperature dependent properties

  • Sunita Deswal
  • Baljit Singh Punia
  • Kapil Kumar KalkalEmail author


In this paper, a model of two dimensional problem of generalized thermoelasticity for a fiber-reinforced anisotropic elastic medium under the effect of temperature dependent properties is established. Reflection phenomena of plane waves in an initially stressed thermoelastic medium is studied in the context of two theories proposed by Lord–Shulman and Green–Lindsay. Using proper boundary conditions, the amplitude ratios and energy ratios for various reflected waves are presented. The phase speeds, reflection coefficients and energy ratios are computed numerically with the help of MATLAB programming and are depicted graphically to show the effect of initial stress and temperature dependent properties. It is found that there is no dissipation of energy at the boundary surface during reflection. A comparison between the two theories is also depicted in the present investigation.


Reflection Anisotropic material Fiber-reinforced Initial stress Temperature dependent elastic modulus 

Mathematics Subject Classification

74A15 80A20 


  1. Abbas, I.A., Abd-Alla, A.N.: Effect of initial stress on a fiber-reinforced anisotropic thermoelastic thick plate. Int. J. Thermophys. 32, 1098–1110 (2011)CrossRefGoogle Scholar
  2. Achenbach, J.D.: Wave Propagation in Elastic Solids. North Holland, Amsterdam (1973)zbMATHGoogle Scholar
  3. Aouadi, M.: Temperature dependence of an elastic modulus in generalized linear micropolar thermoelasticity. Z. Angew. Math. Phys. 57, 1057–1074 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Belfield, A.J., Rogers, T.G., Spencer, A.J.M.: Stress in elastic plates reinforced by fiber lying in concentric circles. J. Mech. Phys. Solids 31, 25–54 (1983)CrossRefzbMATHGoogle Scholar
  5. Biot, M.A.: Mechanics of Incremental Deformation. Wiley, New York (1965)CrossRefGoogle Scholar
  6. Darabseh, T., Yilmaz, N., Bataineh, M.: Transient thermoelasticity analysis of functionally graded thick hollow cylinder based on Green–Lindsay model. Int. J. Mech. Mater. Des. 8, 247–255 (2012)CrossRefGoogle Scholar
  7. Deswal, S., Kalkal, K.K., Sheoran, S.S.: Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction. Physica B 496, 57–68 (2016)CrossRefGoogle Scholar
  8. Deswal, S., Yadav, R., Kalkal, K.K.: Propagation of waves in an initially stressed generalized electro-microstrectch thermoelastic medium with temperature dependent properties under the effect of rotation. J. Thermal Stress. 40, 281–301 (2017)CrossRefGoogle Scholar
  9. Elsagheer, M., Abo-Dahab, S.M.: Reflection of thermoelastic waves from insulated boundary fiber-reinforced half-space under influence of rotation and magnetic field. Appl. Math. Inf. Sci. 10, 1129–1140 (2016)CrossRefGoogle Scholar
  10. Ezzat, M.A., Othman, M.I.A., El-Karamany, A.S.: The dependence of the modulus of elasticity on the reference temperature in generalized thermoelasticity. J. Thermal Stress. 24, 1159–1176 (2001)CrossRefGoogle Scholar
  11. Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)CrossRefzbMATHGoogle Scholar
  12. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. Royal Soc. Lond. A 432, 171–194 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Thermal Stress. 15, 253–264 (1992)MathSciNetCrossRefGoogle Scholar
  14. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–209 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hashin, Z., Rosen, W.B.: The elastic moduli of fiber-reinforced materials. J. Appl. Mech. 31, 223–232 (1964)CrossRefGoogle Scholar
  16. Kalkal, K.K., Deswal, S.: Effect of phase lags on three-dimensional wave propagation with temperature-dependent properties. Int. J. Thermophys. 35, 952–969 (2014)CrossRefGoogle Scholar
  17. Kumar, R., Garg, S.K., Ahuja, S.: Wave propagation in fiber-reinforced transversely isotropic thermoelastic media with initial stress at the boundary surface. J. Solid Mech. 7, 223–238 (2015)Google Scholar
  18. Lomakin, V.A.: The Theory of Elasticity of Non-Homogeneous Bodies. Moscow (1976)Google Scholar
  19. Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)CrossRefzbMATHGoogle Scholar
  20. Montanaro, A.: On singular surfaces in isotropic linear thermoelasticity with initial stress. J. Acoust. Soc. Am. 106, 1586–1588 (1999)CrossRefGoogle Scholar
  21. Othman, M.I.A., Kumar, R.: Reflection of magneto-thermoelasticity waves with temperature dependent properties in generalized thermoelasticity. Int. Commun. Heat Mass Transf. 36, 513–520 (2009)CrossRefGoogle Scholar
  22. Othman, M.I.A., Atwa, S.Y.: Generalized magneto-thermoelasticity in a fiber-reinforced anisotropic half-space with energy dissipation. Int. J. Thermophys. 33, 1126–1142 (2012)CrossRefGoogle Scholar
  23. Othman, M.I.A., Said, S.M.: The effect of rotation on two-dimensional problem of a fiber-reinforced thermoelastic with one relaxation time. Int. J. Thermophys. 33, 160–171 (2012)CrossRefGoogle Scholar
  24. Othman, M.I.A., Said, S.M.: 2D problem of magneto-thermoelasticity fiber-reinforced medium under temperature dependent properties with three-phase-lag model. Meccanica 49, 1225–1241 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Othman, M.I.A., Said, S.M.: The effect of rotation on a fiber-reinforced medium under generalized magneto-thermoelasticity with internal heat source. Mech. Adv. Mater. Struct. 22, 168–183 (2015)CrossRefGoogle Scholar
  26. Panda, S.P., Panda, S.: Micromechanical finite element analysis of effective properties of a unidirectional short piezoelectric fiber reinforced composite. Int. J. Mech. Mater. Des. 11, 41–57 (2015)CrossRefGoogle Scholar
  27. Pipkin, A.C.: Finite deformation of ideal fiber-reinforced composites. In: Sendeckyj, G.P. (ed.) Composites Materials, 2nd edn, pp. 251–308. Academic, New York (1973)Google Scholar
  28. Rogers, T.G.: Finite deformations of strongly anisotropic materials. In: Hutton, J.F., Pearson, J.R.A., Walters, K. (eds.) Theoretical Rheology, pp. 141–168. Applied Science Publication, London (1975)Google Scholar
  29. Sengupta, P.R., Nath, S.: Surface waves in fiber-reinforced anisotropic elastic media. Sadhana 26, 363–370 (2001)CrossRefGoogle Scholar
  30. Singh, B., Singh, S.J.: Reflection of plane waves at free surface of a fiber-reinforced elastic half space. Sadhana 29, 249–257 (2004)CrossRefzbMATHGoogle Scholar
  31. Singh, B.: Wave propagation in thermally conducting linear fiber-reinforced composite materials. Arch. Appl. Mech. 75, 513–520 (2006)CrossRefzbMATHGoogle Scholar
  32. Xiong, Q., Tian, X.: Effect of initial stress on a fiber-reinforced thermoelastic porous media without energy dissipation. Trans. Porous Media 111, 81–95 (2016)MathSciNetCrossRefGoogle Scholar
  33. Yadav, R., Kalkal, K.K., Deswal, S.: Two temperature theory of initially stressed electro-microstretch medium without energy dissipation. Microsyst. Tech. 23, 4931–4940 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Sunita Deswal
    • 1
  • Baljit Singh Punia
    • 1
  • Kapil Kumar Kalkal
    • 1
    Email author
  1. 1.Department of MathematicsGuru Jambheshwar University of Science and TechnologyHisarIndia

Personalised recommendations