Second gradient thermoelasticity with microtemperatures

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This research was concerned with a linear theory of thermoelasticity with microtempera- tures where the second thermal displacement gradient and the second gradient of microtemperatures are included in the classical set of independent constitutive variables. The master balance laws of micromorphic continua, the theory of the strain gradient of elasticity, and Green-Naghdi thermome- chanics were used to derive a second gradient theory. The semigroup theory of linear operators al- lowed us to prove that the problem of the second gradient thermoelasticity with microtemperatures is well-posed. For the equations of isotropic rigids, we presented a natural extension of the Cauchy- Kovalevski-Somigliana solution of isothermal theory. In the case of stationary vibrations, the funda- mental solutions of the basic equations were obtained. Uniqueness and instability of the solutions were obtained in the case of antiplane shear deformations.

Keywords

elastic solids with microtemperatures / solids with microstructure / second gradient theory / constitutive equations / well-posed problems

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] A. E. Green, P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. R. Soc. A, 432 (1991), 171-194. https://doi.org/10.1098/rspa.1991.0012

[2] A. E. Green, P. M. Naghdi, A demonstration of consistency of an entropy balance with balance energy, ZAMP, 42 (1991), 159-168. https://doi.org/10.1007/BF00945790

[3] A. E. Green, P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elast., 31 (1993), 189-208. https://doi.org/10.1007/BF00044969

[4] M. Fabrizio, F. Franchi, R. Nibbi, Second gradient Green-Naghdi type ther- moelasticity and viscoelasticity, Mech. Res. Commun., 126 (2022), 104014. https://doi.org/10.1016/j.mechrescom.2022.104014

[5] D. Ieşan, Thermal stresses that depend on temperature gradients, Z. Angew. Math. Phys., 74 (2023), 138. https://doi.org/10.1007/s00033-023-02034-5

[6] D. Ieşan, R. Quintanilla, A second gradient theory of thermoelasticity, J. Elast., 154 (2023), 629-643. https://doi.org/10.1007/s10659-023-10020-1

[7] A. C. Eringen, Mechanics of micromorphic continua, in Mechanics of Generalized Con- tinua (eds. E. Kröner), IUTAM Symposia, Springer, Berlin, Heidelberg, (1968), 18-35. https://doi.org/10.1007/978-3-662-30257-62

[8] R. Grot, Thermodynamics of a continuum with microstructure, Int. J. Eng. Sci., 7 (1969), 801-814. https://doi.org/10.1016/0020-7225(69)90062-7

[9] A. C. Eringen, C. B. Kafadar, Polar field theories, in Continuum Physics , (eds. A. C. Eringen ), Academic Press, New York, (1976), 1-73. https://doi.org/10.1016/B978-0-12-240804-5.50007-5

[10] D. Ieşan, R. Quintanilla, Qualitative properties in strain gradient thermoelasticity with microtem- peratures, Math. Mech. Solids, 23 (2018), 240-258. https://doi.org/10.1177/10812865166808

[11] R. A. Toupin, Theories of elasticity with couple-stress, Arch. Ration. Mech. Anal., 17 (1964), 85-112. https://doi.org/10.1007/BF00253050

[12] R. D. Mindlin, Microstructure in linear elasticity, Arch. Ration. Mech. Anal., 16 (1964), 51-77. https://doi.org/10.1007/BF00248490

[13] R. D. Mindlin, N. N. Eshel, On first strain gradient theories in linear elasticity, Int. J. Solids Struct., 4 (1968), 109-124. https://doi.org/10.1016/0020-7683(68)90036-X

[14] A. E. Green, R. S. Rivlin, Multipolar continuum mechanics, Arch. Ration. Mech. Anal., 17 (1964), 113-147. https://doi.org/10.1007/BF00253051

[15] S. Forest, M. Amestoy, Hypertemperature in thermoelastic solids, Comptes Rendus. Mécanique, 336 (2008), 347-353. https://doi.org/10.1016/j.crme.2008.01.007

[16] J. Goldstein, Semigroups of Linear Operators and Applications , Oxford University Press, New York and Oxford, 1985.

[17] D. Gilbarg, N. S. Trudinger, Elliptic Partial Diff erential Equations of Second Order , Springer, Berlin Heidelberg New York, 2001. https://doi.org/10.1007/978-3-642-61798-0

[18] V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, T. V. Burchuladze, Three-Dimensional Prob- lems of the Mathematical Theory of Elasticity and Thermoelasticity , North-Holland, Amsterdam, New York, Oxford, 1979.

[19] K. A. Ames, B. Straughan, Non-standard and Improperly Posed Problems , Academic Press, San Diego, 1997.