On Linear Theory of Thermoelasticity for an Anisotropic Medium Under a Recent Exact Heat Conduction Model

  • Manushi Gupta
  • Santwana Mukhopadhyay
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 834)


The aim of this paper is to discuss about a new thermoelasticity theory for a homogeneous and anisotropic medium in the context of a recent heat conduction model proposed by Quintanilla (2011). The coupled thermoelasticity being the branch of science that deals with the mutual interactions between temperature and strain in an elastic medium had become the interest of researchers since 1956. Quintanilla (2011) have introduced a new model of heat conduction in order to reformulate the heat conduction law with three phase-lags and established mathematical consistency in this new model as compared to the three phase-lag model. This model has also been extended to thermoelasticity theory. Various Taylor’s expansion of this model has gained the interest of many researchers in recent times. Hence, we considered the model’s backward time expansion of Taylor’s series upto second-order and establish some important theorems. Firstly, uniqueness theorem of a mixed type boundary and initial value problem is proved using specific internal energy function. Later, we give the alternative formulation of the problem using convolution which incorporates the initial conditions into the field equations. Using this formulation, the convolution type variational theorem is proved. Further, we establish a reciprocal relation for the model.


Non-Fourier heat conduction model Generalized thermoelasticiy Uniqueness Variational principle Reciprocity theorem 


  1. 1.
    Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253 (1956)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chandrasekharaiah, D.S.: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39(3), 355–376 (1986)CrossRefGoogle Scholar
  3. 3.
    Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51(12), 705–729 (1998)CrossRefGoogle Scholar
  4. 4.
    Joseph, D.D., Preziosi, L.: Heat waves. Rev. Mod. Phys. 61, 41–73 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hetnarski, R.B., Ignaczak, J.: Generalized thermoelasticity. J. Therm. Stresses 22, 451–476 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dreyer, W., Struchtrup, H.: Heat pulse experiments revisited. Continuum Mech. Therm. 5, 3–50 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ozisik, M.N., Tzou, D.Y.: On the wave theory of heat conduction. ASME J. Heat Transfer 116, 526–535 (1994)CrossRefGoogle Scholar
  8. 8.
    Ignaczak, J., Ostoja-Starzewski, M.: Thermoelasticity With Finite Wave Speeds. Oxford University Press, New York (2010)zbMATHGoogle Scholar
  9. 9.
    Muller, I., Ruggeri, T.: Extended Thermodynamics. Springer Tracts on Natural Philosophy. Springer, New York (1993). Scholar
  10. 10.
    Marín, E.: Does Fourier’s law of heat conduction contradict the theory of relativity? Latin-American J. Phys. Edu. 5, 402–405 (2011)Google Scholar
  11. 11.
    Lord, H.W., Shulman, Y.A.: Generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (1967)CrossRefGoogle Scholar
  12. 12.
    Green, A.E., Lindasy, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)CrossRefGoogle Scholar
  13. 13.
    Cattaneo, C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Compte Rendus 247, 431–433 (1958)zbMATHGoogle Scholar
  14. 14.
    Vernotte, P.: Les paradoxes de la theorie continue de l’equation de la chaleur. Compte Rendus 246, 3154–3155 (1958)zbMATHGoogle Scholar
  15. 15.
    Vernotte, P.: Some possible complications in the phenomena of thermal conduction. Compte Rendus 252, 2190–2191 (1961)Google Scholar
  16. 16.
    Green, A.E., Naghdi, P.M.: A re-examination of the base postulates of thermoemechanics. Proc. R. Soc. Lond. A 432, 171–194 (1991)CrossRefGoogle Scholar
  17. 17.
    Green, A.E., Naghdi, P.M.: On undamped heat waves in an elastic solid. J. Therm. Stresses 15, 253–264 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tzou, D.Y.: A unified field approach for heat conduction from macro to micro scales. ASME J. Heat Transfer 117, 8–16 (1995)CrossRefGoogle Scholar
  20. 20.
    Tzou, D.Y.: The generalized lagging response in small-scale and high-rate heating. Int. J. Heat Mass Transfer 38(17), 3231–3240 (1995)CrossRefGoogle Scholar
  21. 21.
    Roychoudhuri, S.K.: On a thermoelastic three-phase-lag model. J. Therm. Stresses 30, 231–238 (2007)CrossRefGoogle Scholar
  22. 22.
    Dreher, M., Quintanilla, R., Racke, R.: Ill-posed problems in thermomechanics. Appl. Math. Lett. 22, 1374–1379 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Quintanilla, R.: Exponential stability in the dual-phase-lag heat conduction theory. J. Non-Equilib. Thermodyn. 27, 217–227 (2002)CrossRefGoogle Scholar
  24. 24.
    Horgan, C.O., Quintanilla, R.: Spatial behaviour of solutions of the dual-phase-lag heat equation. Math. Methods Appl. Sci. 28, 43–57 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kumar, R., Mukhopadhyay, S.: Analysis of the effects of phase-lags on propagation of harmonic plane waves in thermoelastic media. Comput. Methods Sci. Tech. 16(1), 19–28 (2010)CrossRefGoogle Scholar
  26. 26.
    Mukhopadhyay, S., Kumar, R.: Analysis of phase-lag effects on wave propagation in a thick plate under axisymmetric temperature distribution. Acta Mech. 210, 331–344 (2010)CrossRefGoogle Scholar
  27. 27.
    Mukhopadhyay, S., Kothari, S., Kumar, R.: On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags. Acta Mech. 214, 305–314 (2010)CrossRefGoogle Scholar
  28. 28.
    Quintanilla, R.: A condition on the delay parameters in the one-dimensional dual-phase-lag thermoelastic theory. J. Therm. Stresses 26, 713–721 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Quintanilla, R., Racke, R.: A note on stability of dual-phase-lag heat conduction. Int. J. Heat Mass Transfer 49, 1209–1213 (2006)CrossRefGoogle Scholar
  30. 30.
    Quintanilla, R., Racke, R.: Qualitative aspects in dual-phase-lag thermoelasticity. SIAM J. Appl. Math. 66, 977–1001 (2006)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Quintanilla, R., Racke, R.: Qualitative aspects in dual-phase-lag heat conduction. Proc. R. Soc. Lond. A 463, 659–674 (2007)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Quintanilla, R., Racke, R.: A note on stability in three-phase-lag heat conduction. Int. J. Heat Mass Transfer 51, 24–29 (2008)CrossRefGoogle Scholar
  33. 33.
    Quintanilla, R.: Some solutions for a family of exact phase-lag heat conduction problems. Mech. Res. Commun. 38, 355–360 (2011)CrossRefGoogle Scholar
  34. 34.
    Leseduarte, M.C., Quintanilla, R.: Phragman-Lindelof alternative for an exact heat conduction equation with delay. Commun. Pure Appl. Math. 12(3), 1221–1235 (2013)zbMATHGoogle Scholar
  35. 35.
    Quintanilla, R.: On uniqueness and stability for a thermoelastic theory. Math. Mech. Solids 22(6), 1387–1396 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ignaczak, J.: A completeness problem for stress equations of motion in the linear elasticity theory. Arch. Mech. Stos 15, 225–234 (1963)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Gurtin, M.E.: Variational principles for linear Elastodynamics. Arch. Ration. Mech. Anal. 16, 34–50 (1964)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Iesan, D.: Principes variationnels dans la theorie de la thermoelasticite couplee. Ann. Sci. Univ. ‘Al. I. Cuza’ Iasi Mathematica 12, 439–456 (1966)zbMATHGoogle Scholar
  39. 39.
    Iesan, D.: On some reciprocity theorems and variational theorems in linear dynamic theories of continuum mechanics. Memorie dell’Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. Ser. 4(17), 17–37 (1974)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Nickell, R., Sackman, J.: Variational principles for linear coupled thermoelasticity. Quart. Appl. Math. 26, 11–26 (1968)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Iesan, D.: Sur la théorie de la thermoélasticité micropolaire couplée. C. Rend. Acad. Sci. Paris 265, 271–275 (1967)zbMATHGoogle Scholar
  42. 42.
    Nowacki, W.: Fundamental relations and equations of thermoelasticity. In: Francis, P.H., Hetnarski, R.B. (eds.) Dynamic Problems of Thermoelasticity (English Edition). Noordhoff Internationa Publishing, Leyden (1975)Google Scholar
  43. 43.
    Maysel, V.M.: The Temperature Problem of the Theory of Elasticity. Kiev (1951). (in Russian)Google Scholar
  44. 44.
    Predeleanu, P.M.: On thermal stresses in viscoelastic bodies. Bull. Math. Soc. Sci. Math. Phys. 3(51), 223–228 (1959)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Ionescu-Cazimir, V.: Problem of linear thermoelasticity: theorems on reciprocity I. Bull. Acad. Polon. Sci. Ser. Sci. Tech. 12, 473–480 (1964)zbMATHGoogle Scholar
  46. 46.
    Scalia, A.: On some theorems in the theory of micropolar thermoelasticity. Int. J. Eng. Sci. 28, 181–189 (1990)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Lebon, G.: Variational Principles in Thermomechanics. Springer-Wien, New York (1980). Scholar
  48. 48.
    Carlson, D.E.: Linear thermoelasticity. In: Truesdell, C. (ed.) Flugge’s Handbuch der Physik, vol. VI a/2, pp. 297–345. Springer, Heidelberg (1973). Scholar
  49. 49.
    Hetnarski, R.B., Ignaczak, J.: Mathematical Theory of Elasticity. Taylor and Francis, New York (2004)zbMATHGoogle Scholar
  50. 50.
    Hetnarski, R.B., Eslami, M.R.: Thermal Stresses: Advanced Theory and Applications. In: Gladwell, G.M.L. (ed.) Solid Mechanics and Its Applications, vol. 158. Springer, Dordrecht (2010).
  51. 51.
    Chirita, S., Ciarletta, M.: Reciprocal and variational principles in linear thermoelasticity without energy dissipation. Mech. Res. Commun. 37, 271–275 (2010)CrossRefGoogle Scholar
  52. 52.
    Mukhopadhyay, S., Prasad, R.: Variational and reciprocal principles in linear theory of type-III thermoelasticity. Math. Mech. Solids 16, 435–444 (2011)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Kothari, S., Mukhopadhyay, S.: Some theorems in linear thermoelasticity with dual phase-lags for an Anisotropic Medium. J. Therm. Stresses 36, 985–1000 (2013)CrossRefGoogle Scholar
  54. 54.
    Kumari, B., Mukhopadhyay, S.: Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay. Math. Mech. Solids 22(5), 1177–1189 (2016, 2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndian Institute of Technology (BHU)VaranasiIndia

Personalised recommendations