Continuum Mechanics and Thermodynamics

, Volume 30, Issue 2, pp 319–345 | Cite as

Dual and mixed nonsymmetric stress-based variational formulations for coupled thermoelastodynamics with second sound effect

  • Balázs Tóth
Original Article


Some new dual and mixed variational formulations based on a priori nonsymmetric stresses will be developed for linearly coupled irreversible thermoelastodynamic problems associated with second sound effect according to the Lord–Shulman theory. Having introduced the entropy flux vector instead of the entropy field and defining the dissipation and the relaxation potential as the function of the entropy flux, a seven-field dual and mixed variational formulation will be derived from the complementary Biot–Hamilton-type variational principle, using the Lagrange multiplier method. The momentum-, the displacement- and the infinitesimal rotation vector, and the a priori nonsymmetric stress tensor, the temperature change, the entropy field and its flux vector are considered as the independent field variables of this formulation. In order to handle appropriately the six different groups of temporal prescriptions in the relaxed- and/or the strong form, two variational integrals will be incorporated into the seven-field functional. Then, eliminating the entropy from this formulation through the strong fulfillment of the constitutive relation for the temperature change with the use of the Legendre transformation between the enthalpy and Gibbs potential, a six-field dual and mixed action functional is obtained. As a further development, the elimination of the momentum- and the velocity vector from the six-field principle through the a priori satisfaction of the kinematic equation and the constitutive relation for the momentum vector leads to a five-field variational formulation. These principles are suitable for the transient analyses of the structures exposed to a thermal shock of short temporal domain or a large heat flux.


Nonsymmetric stresses Thermoelasticity Two-way coupling Second sound effect Lord–Shulman theory Dual–mixed variational formulations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ackerman, C.C., Bertman, B., Fairbank, H.A., Guyer, R.A.: Second sound in solid helium. Phys. Rev. Lett. 16, 789–791 (1966)ADSCrossRefGoogle Scholar
  2. 2.
    Anthony, K.: Hamilton’s action principle and thermodynamics of irreversible processes a unifying procedure for reversible and irreversible processes. J. Nonnewton Fluid Mech. 96(1–2), 291–339 (2001)CrossRefMATHGoogle Scholar
  3. 3.
    Aouadi, M.: Generalized theory of thermoelastic diffusion for anisotropic media. J. Therm. Stresses 31(3), 270–285 (2008)CrossRefGoogle Scholar
  4. 4.
    Apostolakis, G., Dargush, G.F.: Mixed variational principles for dynamic response of thermoelastic and poroelastic continua. Int. J. Solids Struct. 50(5), 642–650 (2013)CrossRefGoogle Scholar
  5. 5.
    Apostolakis, G., Dargush, G.F.: Variational methods in irreversible thermoelasticity: theoretical developments and minimum principles for the discrete form. Acta Mech. 224(9), 2065–2088 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Askar Altay, G., Cengiz Dökmeci, M.: Some variational principles for linear coupled thermoelasticity. Int. J. Solids Struct. 33(26), 3937–3948 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bailey, C.D.: Hamilton’s principle and calculus of variations. Acta Mech. 44(1), 49–57 (1982)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bargmann, H.: Recent developments in the field of thermally induced waves and vibrations. Nucl. Eng. Des. 27(3), 372–381 (1974)CrossRefGoogle Scholar
  9. 9.
    Baruch, M., Riff, R.: Hamilton’s principle, Hamilton’s law—6\(^n\) correct formulations. AIAA J. 20(5), 687–692 (1982)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Batra, G.: On a principle of virtual work for thermo-elastic bodies. J. Elast. 21(2), 131–146 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bem, Z.: Existence of a generalized solution in thermoelasticity with one relaxation time. J. Therm. Stresses 5(2), 195–206 (1982)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ben-Amoz, M.: On a variational theorem in coupled thermoelasticity. J. Appl. Mech. 32(4), 943–945 (1965)CrossRefGoogle Scholar
  13. 13.
    Berdichevsky, V.L.: Variational Principles of Continuum Mechanics: I. Fundamentals. Springer, Berlin (2009)MATHGoogle Scholar
  14. 14.
    Beris, A.N., Edwards, B.J.: Thermodynamics of Flowing Systems with Internal Microstructures. Oxford University Press, New York (1994)Google Scholar
  15. 15.
    Biot, M.A.: Theory of stress–strain relations in anisotropic viscoelasticity and relaxation phenomena. J. Appl. Phys. 25, 1385–1391 (1954)ADSCrossRefMATHGoogle Scholar
  16. 16.
    Biot, M.A.: Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys. Rev. 97, 1463–1469 (1955)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Biot, M.A.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253 (1956)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Boley, B.A., Weiner, J.H.: Theory of Thermal Stresses. Dover Publications, Mineola, New York (1960). Originally published by Wiley, Inc, New York, in 1960Google Scholar
  19. 19.
    Cannarozzi, A.A., Ubertini, F.: Mixed variational method for linear coupled thermoelastic analysis. Int. J. Solids Struct. 38(4), 717–739 (2001)CrossRefMATHGoogle Scholar
  20. 20.
    Carini, A., Genna, F.: Some variational formulations for continuum nonlinear dynamics. J. Mech. Phys. Solids 46(7), 1253–1277 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Cengiz Dökmeci, M.: Hamilton’s principle and associated variational principles in polar thermopiezoelectricity. Phys. A 389, 2966–2974 (2010)CrossRefGoogle Scholar
  22. 22.
    Chandrasekharaiah, D.S.: Thermoelasticity with second sound: a review. Appl. Mech. Rev. 39(3), 355–376 (1986)ADSCrossRefMATHGoogle Scholar
  23. 23.
    Chandrasekharaiah, D.S.: Variational and reciprocal principles in micropolar thermoelasticity. Int. J. Eng. Sci. 25(1), 55–63 (1987)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Chandrasekharaiah, D.S.: A uniqueness theorem in the theory of thermoelasticity without energy dissipation. J. Therm. Stresses 19(3), 262–272 (1996)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Chester, M.: Second sound in solids. Phys. Rev. 131, 2013–2015 (1963)ADSCrossRefGoogle Scholar
  26. 26.
    Chiriţă, S., Ciarletta, M.: Reciprocal and variational principles in linear thermoelasticity without energy dissipation. Mech. Res. Commun. 37, 271–275 (2010)CrossRefMATHGoogle Scholar
  27. 27.
    de Groot, S.R.: Thermodynamics of Irreversible Processes. North-Holland Publishing Company, Amsterdam (1951)Google Scholar
  28. 28.
    Dhaliwal, R.S., Sherief, H.H.: Generalized thermoelasticity for anisotropic media. Q. Appl. Math. 33, 1–8 (1980)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Dym, C.L., Shames, I.H.: Solid Mechanics. A Variational Approach, Augmented Edition. Springer, New York (2013)MATHGoogle Scholar
  30. 30.
    Ebrahimzadeh, Z., Leok, M., Mahzoon, M.: A novel variational formulation for thermoelastic problems. Commun. Nonlinear Sci. Numer. Simul. 22, 263–268 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Eslami, M.R., Hetnarski, R.B., Ignaczak, J., Noda, N., Sumi, N., Tanigawa, Y.: Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications, vol. 197. Springer, Netherland (2013)CrossRefMATHGoogle Scholar
  32. 32.
    Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2(1), 1–7 (1972)CrossRefMATHGoogle Scholar
  33. 33.
    Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31(3), 189–208 (1993)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Grmela, M., Öttinger, H.C.: Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E 56, 6620–6632 (1997)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C (ed.) Mechanics of Solids, Encyclopedia of Physics, vol. 6a/2, pp. 1–295. Springer, Berlin (1972)Google Scholar
  36. 36.
    Gyarmati, I.: Non-equilibrium Thermodynamics: Field Theory and Variational Principles. Springer, Berlin (1970)CrossRefGoogle Scholar
  37. 37.
    Hamilton, W.R.: On a general method in dynamics. Philos. Trans. R. Soc. Lond. 124, 247–308 (1834)CrossRefGoogle Scholar
  38. 38.
    Hamilton, W.R.: Second essay on a general method in dynamics. Philos. Trans. R. Soc. Lond. 125, 95–144 (1835)CrossRefGoogle Scholar
  39. 39.
    Har, J., Tamma, K.: Advances in Computational Dynamics of Particles, Materials and Structures. Wiley, West Sussex (2002)MATHGoogle Scholar
  40. 40.
    He, J.H.: Asymptotic methods for solitary solutions and compactons. In: Abstract and Applied Analysis, pp. 1–130 (2012) 916793Google Scholar
  41. 41.
    He, J.H.: Hamilton’s principle for dynamical elasticity. Appl. Math. Lett. 72, 65–69 (2017)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Herrmann, G.: On variational principles in thermoelasticity and heat conduction. Q. Appl. Math. 21(2), 151–155 (1963)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Ignaczak, J.: Linear dynamic thermoelasticity: a survey. Shock Vib. Dig. 13(9), 3–8 (1981)CrossRefGoogle Scholar
  44. 44.
    Ignaczak, J.: A note on uniqueness in thermoelasticity with one relaxation time. J. Therm. Stress. 5(3–4), 257–264 (1982)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Kaminski, W.: Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure. J. Heat Transf. 112(3), 555–560 (1990)CrossRefGoogle Scholar
  46. 46.
    Kaufman, A.N.: Dissipative Hamiltonian systems: a unifying principle. Phys. Lett. A 100(8), 419–422 (1984)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Keramidas, G.A., Ting, E.C.: A finite element formulation for thermal stress analysis. Part I: variational formulation. Nucl. Eng. Des. 39, 267–275 (1976)CrossRefGoogle Scholar
  48. 48.
    Kim, J.: Extended framework of Hamilton’s principle for thermoelastic continua. Comput. Math. Appl. 73(7), 1505–1523 (2017)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Kim, J., Dargush, G.F., Ju, Y.K.: Extended framework of Hamilton’s principle for continuum dynamics. Int. J. Solids Struct. 50(20–21), 3418–3429 (2013)CrossRefGoogle Scholar
  50. 50.
    Kim, J., Dargush, G.F., Lee, H.S.: Extended framework of Hamilton’s principle in heat diffusion. Int. J. Mech. Sci. 114, 166–176 (2016)CrossRefGoogle Scholar
  51. 51.
    Kline, K.A., DeSilva, C.N.: Variational principles for linear coupled thermoelasticity with microstructure. Int. J. Solids Struct. 7, 129–142 (1971)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Kotowski, R.: Hamilton’s principle in thermodynamics. Arch. Mech. 44, 203–215 (1992)MathSciNetMATHGoogle Scholar
  53. 53.
    Lanczos, C.: The Variational Principles of Mechanics, 4th edn. Dover Publications, New York (1970)MATHGoogle Scholar
  54. 54.
    Leitmann, G.: Some remarks on Hamilton’s principle. J. Appl. Mech. 30(4), 623–625 (1963)ADSMathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Li, X.: A generalized theory of thermoelasticity for an anisotropic medium. Int. J. Eng. Sci. 30(5), 571–577 (1992)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Liu, G.L.: A vital innovation in Hamilton principle and its extension to initial-value problems. In: Proceedings of the 4th International Conference on Nonlinear Mechanics, pp. 90–97. Shanghai University Press, Shanghai, China (2002)Google Scholar
  57. 57.
    Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)ADSCrossRefMATHGoogle Scholar
  58. 58.
    Lubarda, V.A.: On thermodynamic potentials in linear thermoelasticity. Int. J. Solids Struct. 41(26), 7377–7398 (2004)CrossRefMATHGoogle Scholar
  59. 59.
    Lucia, U.: Macroscopic irreversibility and microscopic paradox: a constructal law analysis of atoms as open systems. Sci. Rep. 6(35796) (2016).
  60. 60.
    Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, vol. 17, 2nd edn. Springer, New York (1999)Google Scholar
  61. 61.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover Publications, New York (1983)MATHGoogle Scholar
  62. 62.
    Maugin, G.A., Kalpakides, V.K.: A Hamiltonian formulation for elasticity and thermoelasticity. J. Phys. A: Math. Gen. 35(50), 10775–10788 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Maxwell, J.: On the dynamical theory of gases. Philos. Trans. R. Soc. 175, 49–88 (1867)CrossRefGoogle Scholar
  64. 64.
    Morrison, P.J.: Bracket formulation for irreversible classical fields. Phys. Lett. A 100(8), 423–427 (1984)ADSMathSciNetCrossRefGoogle Scholar
  65. 65.
    Nappa, L.: Variational principles in micromorphic thermoelasticity. Mech. Res. Commun. 28(4), 405–412 (2001)CrossRefMATHGoogle Scholar
  66. 66.
    Nickell, R.E., Sackman, J.L.: Variational principles for linear coupled thermoelasticity. Q. Appl. Math. 26, 11–26 (1968)MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Nowacki, W.: Thermoelasticity, 2nd edn. Pergamon Press, Oxford (1986)MATHGoogle Scholar
  68. 68.
    Oden, J.T., Reddy, J.N.: Variational Methods in Theoretical Mechanics, 2nd edn. Springer, Berlin (1983)CrossRefMATHGoogle Scholar
  69. 69.
    Onsager, L.: Reciprocal relations in irreversible processes I. Phys. Rev. 37, 405–426 (1931)ADSCrossRefMATHGoogle Scholar
  70. 70.
    Onsager, L.: Reciprocal relations in irreversible processes II. Phys. Rev. 38, 2265–2279 (1931)ADSCrossRefMATHGoogle Scholar
  71. 71.
    Prigogine, I.: Introduction to Thermodynamics of Irreversible Processes, 3rd edn. Wiley, New York (1968)MATHGoogle Scholar
  72. 72.
    Rafalski, P.: A variational principle for the coupled thermoelastic problem. Int. J. Eng. Sci. 6(8), 465–471 (1968)CrossRefMATHGoogle Scholar
  73. 73.
    Rayleigh, J.W.S.: Theory of Sound. I. & II., 2nd edn. Dover Publications, New York (1887). Reprint in 1945Google Scholar
  74. 74.
    Reddy, J.N.: Variational principles for linear coupled dynamic theory of thermoviscoelasticity. Int. J. Eng. Sci. 14(7), 605–616 (1976)MathSciNetCrossRefMATHGoogle Scholar
  75. 75.
    Serra, E., Bonaldi, M.: A finite element formulation for thermoelastic damping analysis. Int. J. Numer. Methods Eng. 78(6), 671–691 (2009)MathSciNetCrossRefMATHGoogle Scholar
  76. 76.
    Sherief, H.H.: On uniqueness and stability in generalized thermoelasticity. Q. Appl. Math. 45, 773–778 (1987)MathSciNetCrossRefMATHGoogle Scholar
  77. 77.
    Sherief, H.H., Dhaliwal, R.S.: A uniqueness theorem and a variational principle for generalized thermoelasticity. J. Therm. Stresses 3, 223–230 (1980)CrossRefGoogle Scholar
  78. 78.
    Smith Jr., D.R., Smith, C.V.: When is Hamilton’s principle an extremum principle? AIAA J. 12(11), 1573–1576 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar
  79. 79.
    Tabarrok, B.: Complementary variational principles in elastodynamics. Comput. Struct. 19(1–2), 239–246 (1984)CrossRefMATHGoogle Scholar
  80. 80.
    Tabarrok, B., Rimrott, F.P.J.: Variational Methods and Complementary Formulations in Dynamics. Kluwer, The Netherlands (1994)CrossRefMATHGoogle Scholar
  81. 81.
    Tonti, E.: Variational formulation for every nonlinear problem. Int. J. Eng. Sci. 22(11), 1343–1371 (1984)MathSciNetCrossRefMATHGoogle Scholar
  82. 82.
    Tóth, B.: Multi-field dual-mixed variational principles using non-symmetric stress field in linear elastodynamics. J. Elast. 122, 113–130 (2016)MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    Vujanovic, B., Djukic, D.S.: On one variational principle of Hamilton’s type for nonlinear heat transfer problem. Int. J. Heat Mass Transf. 15(5), 1111–1123 (1972)CrossRefGoogle Scholar
  84. 84.
    Weinstock, R.: Calculus of Variations. With Applications to Physics and Engineering. Dover Publications, New York (1974)MATHGoogle Scholar
  85. 85.
    Youssef, H.M., Al-Lehaibi, E.A.: Variational principle of fractional order generalized thermoelasticity. Appl. Math. Lett. 23(10), 1183–1187 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute of Applied MechanicsUniversity of MiskolcMiskolcHungary

Personalised recommendations