Abstract
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.
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References
Green, A.E., Naghdi, P.M.: Re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. A 432, 1171–1194 (1991)
Green, A.E., Naghdi, P.M.: On undamped heat wave in elastic solids. J. Therm. Stress 15(2), 253–264 (1992)
Green, A.E., Naghdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 9, 1–8 (1993)
Choudhuri, S.K.R.: On a thermoelastic three-phase-lag model. J. Therm. Stress 30(3), 231–238 (2007)
Eringen, A.C.: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 28, 1291–1301 (1990)
Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999)
Iesan, D., Ciarletta, M.: Non-Classical Elastic Solids. Longman Scientific and Technical, Harlow (1993)
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)
Marin, M.: A domain of influence theorem for microstretch elastic materials. Nonlinear Anal. RWA 11(5), 3446–3452 (2010)
Marin, M.: Harmonic vibrations in thermoelasticity of microstretch materials. ASME J. Vibr. Acoust. 132(4), 044501-044501-6 (2010). doi:10.1115/1.4000971
Straughan, B.: Heat waves. In: Applied Mathematical Sciences, vol. 177. Springer, New York (2011)
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)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964)
Fried, E., Gurtin, M.E.: Thermomechanics of the interface between a body and its environment. Contin. Mech. Thermodyn. 19(5), 253–271 (2007)
John, F.: Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Commun. Pure Appl. Math. 13, 551–585 (1960)
Quintanilla, R.: Structural stability and continuous dependence of solutions of thermoelasticity of type III. Discrete Contin. Dyn. Syst. Ser. B 1, 463–470 (2001)
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)
Ames, K.A., Straughan, B.: Continuous dependence results for initially prestressed thermoelastic bodies. Int. J. Eng. Sci. 30, 7–13 (1992)
Wilkes, N.S.: Continuous dependence and instability in linear thermoelasticity. SIAM J. Math. Anal. 11, 292–299 (1980)
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)
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)
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Communicated by Andreas Öchsner.
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Marin, M., Öchsner, A. The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity. Continuum Mech. Thermodyn. 29, 1365–1374 (2017). https://doi.org/10.1007/s00161-017-0585-7
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DOI: https://doi.org/10.1007/s00161-017-0585-7