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A Second Gradient Theory of Thermoelasticity

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Abstract

This paper is concerned with a linear theory of thermoelasticity without energy dissipation, where the second gradient of displacement and the second gradient of the thermal displacement are included in the set of independent constitutive variables. In particular, in the case of rigid heat conductors the present theory leads to a fourth order equation for temperature. First, the basic equations of the second gradient theory of thermoelasticity are presented. The boundary conditions for thermal displacement are derived. The field equations for homogeneous and isotropic solids are established. Then, a uniqueness result for the basic boundary-initial-value problems is presented. An existence theorem is established for the first boundary value problem. The problem of a concentrated heat source is investigated using a solution of Cauchy-Kowalewski-Somigliana type.

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References

  1. Green, A.E., Naghdi, P.M.: A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. A 432, 171 (1991)

    MathSciNet  Google Scholar 

  2. Green, A.E., Naghdi, P.M.: A demonstration of consistency of an entropy balance with balanced energy. Z. Angew. Math. Phys. 42, 159 (1991)

    Article  MathSciNet  Google Scholar 

  3. Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189 (1993)

    Article  MathSciNet  Google Scholar 

  4. Quintanilla, R.: Thermoelasticity without energy dissipation of nonsimple materials. Z. Angew. Math. Mech. 83, 172 (2003)

    Article  MathSciNet  Google Scholar 

  5. Bargmann, S., Steinmann, P.: Theoretical and compuational aspects of non-classical thermoelasticity. Comput. Methods Appl. Mech. Eng. 196, 516 (2006)

    Article  Google Scholar 

  6. Fabrizio, M., Franchi, F., Nibbi, R.: Second gradient Green-Naghdi type thermoelasticity and viscoelasticity. Mech. Res. Commun. 126, 104014 (2022)

    Article  Google Scholar 

  7. Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385 (1962)

    Article  MathSciNet  Google Scholar 

  8. Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17, 85 (1964)

    Article  MathSciNet  Google Scholar 

  9. Mindlin, R.D.: Microstructure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51 (1964)

    Article  Google Scholar 

  10. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109 (1968)

    Article  Google Scholar 

  11. Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementation and new results. Int. J. Solids Struct. 48, 192 (2011)

    Article  Google Scholar 

  12. Forest, S., Amestoy, M.: Hypertemperature in thermoelastic solids. C. R., Méc. 336, 347 (2008)

    Article  Google Scholar 

  13. Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113 (1964)

    Article  MathSciNet  Google Scholar 

  14. Goldstein, J.A.: Semigroup of Linear Operators and Applications. Oxford Mathematical Monographs, New York (1985)

    Google Scholar 

  15. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)

    Google Scholar 

  16. Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Handbuch der Physik, Vol. VI a/2, pp. 1–296. Springer, Berlin (1972)

    Google Scholar 

  17. Ciarletta, M., Ieşan, D.: Non-classical Elastic Solids. Longman Scientific & Technical, London (1993)

    Google Scholar 

  18. Nowacki, W.: Thermoelasticity, 2nd edn. Pergamon, Elmsford (1986)

    Google Scholar 

  19. Ieşan, D., Scalia, A.: Thermoelastic Deformations. Kluwer, Dordrecht (1996)

    Book  Google Scholar 

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Acknowledgements

We express our gratitude to the referees for their useful suggestions. The work of R. Quintanilla has been funded by the research project PID2019-105118GBI00, funded by the Spanish Ministry of Science, Innovation and Universities and FEDER “A way to make Europe”.

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Authors and Affiliations

  1. Institute of Mathematics, Romanian Academy, Iaşi, Romania

    D. Ieşan

  2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, C. Colom 11, 08222, Terrassa, Barcelona, Spain

    R. Quintanilla

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  1. D. Ieşan
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  2. R. Quintanilla
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D. Iesan and R. Quintanilla wrote the main manuscript. All authors reviewed the manuscript.

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Correspondence to R. Quintanilla.

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Ieşan, D., Quintanilla, R. A Second Gradient Theory of Thermoelasticity. J Elast 154, 629–643 (2023). https://doi.org/10.1007/s10659-023-10020-1

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