Continuum Mechanics and Thermodynamics

, Volume 29, Issue 6, pp 1365–1374 | Cite as

The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity

  • Marin MarinEmail author
  • Andreas Öchsner
Original Article


This study is concerned with the mixed initial boundary value problem for a dipolar body in the context of the thermoelastic theory proposed by Green and Naghdi. For the solutions of this problem we prove a result of Hölder’s-type stability on the supply terms. We impose middle restrictions on the thermoelastic coefficients, which are common in continuum mechanics. For the same conditions we propose a continuous dependence result with regard to the initial data.


Dipolar bodies Green–Naghdi thermoelasticity G–N-type III Hölder stability Continuous dependence 


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  1. 1.
    Green, A.E., Naghdi, P.M.: Re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A 432, 1171–1194 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Green, A.E., Naghdi, P.M.: On undamped heat wave in elastic solids. J. Therm. Stress 15(2), 253–264 (1992)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 9, 1–8 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Choudhuri, S.K.R.: On a thermoelastic three-phase-lag model. J. Therm. Stress 30(3), 231–238 (2007)CrossRefGoogle Scholar
  5. 5.
    Eringen, A.C.: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 28, 1291–1301 (1990)CrossRefzbMATHGoogle Scholar
  6. 6.
    Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Iesan, D., Ciarletta, M.: Non-Classical Elastic Solids. Longman Scientific and Technical, Harlow (1993)zbMATHGoogle Scholar
  8. 8.
    Sharma, K., Marin, M.: Effect of distinct conductive and thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space. U.P.B. Sci. Bull. Ser. A Appl. Math. Phys. 75(2), 121–132 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Marin, M.: A domain of influence theorem for microstretch elastic materials. Nonlinear Anal. RWA 11(5), 3446–3452 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Marin, M.: Harmonic vibrations in thermoelasticity of microstretch materials. ASME J. Vibr. Acoust. 132(4), 044501-044501-6 (2010). doi: 10.1115/1.4000971 CrossRefGoogle Scholar
  11. 11.
    Straughan, B.: Heat waves. In: Applied Mathematical Sciences, vol. 177. Springer, New York (2011)Google Scholar
  12. 12.
    Sharma, K., Marin, M.: Reflection and transmission of waves from imperfect boundary between two heat conducting micropolar thermoelastic solids. An. Sti. Univ. Ovidius Constanta 22(2), 151–175 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fried, E., Gurtin, M.E.: Thermomechanics of the interface between a body and its environment. Contin. Mech. Thermodyn. 19(5), 253–271 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    John, F.: Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Commun. Pure Appl. Math. 13, 551–585 (1960)CrossRefzbMATHGoogle Scholar
  17. 17.
    Quintanilla, R.: Structural stability and continuous dependence of solutions of thermoelasticity of type III. Discrete Contin. Dyn. Syst. Ser. B 1, 463–470 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ames, K.A., Payne, L.E.: Continuous dependence on initial-time geometry for a thermoelastic system with sign-indefinite elasticities. J. Math. Anal. Appl. 189, 693–714 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ames, K.A., Straughan, B.: Continuous dependence results for initially prestressed thermoelastic bodies. Int. J. Eng. Sci. 30, 7–13 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wilkes, N.S.: Continuous dependence and instability in linear thermoelasticity. SIAM J. Math. Anal. 11, 292–299 (1980)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Majeed, A., Zeeshan, A., Ellahi, R.: Unsteady ferromagnetic liquid flow and heat transfer analysis over a stretching sheet with the effect of dipole and prescribed heat flux. J. Mol. Liq. 223, 528–533 (2016)CrossRefGoogle Scholar
  22. 22.
    Zeeshan, A., Majeed, A., Ellahi, R.: Effect of magnetic dipole on viscous ferro-fluid past a stretching surface with thermal radiation. J. Mol. Liq. 215, 549–554 (2016)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceTransilvania University of BrasovBrasovRomania
  2. 2.Griffith School of EngineeringGriffith UniversityNathanAustralia

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