Abstract
In this short note we consider a recent modification of the Green–Lindsay thermoelastic theory proposed at Yu et al. (Meccanica 53:2543–2554, 2018). We consider a functional defined on the solutions of the problem. It allows us to obtain the continuous dependence of the solutions with respect to the initial conditions and to the supply terms, the time exponential decay of solutions and an alternative of Phragmén–Lindelöf type for the spatial behaviour.
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Notes
Here \(\theta \) is the relative temperature and \(q_i\) is the heat flux vector
It is worth noting that the best value for the \(K_{\omega }\) involves the study of a very cumbersome system of nonlinear equations.
References
Bofill F, Quintanilla R (1995) Some qualitative results for the theory of thermo-microstretch elastic solids. Int J Eng Sci 33:2115–2125
Cattaneo C (1948) Sulla conduzioini del calore. Atti Semin Mat Fis Univ Modena 3:83–101
Chirita S (1995) Saint-Venant’s principle in linear elasticity. J Thermal Stress 18:485–496
Flavin JN, Knops RJ, Payne LE (1989) Decay estimates for the constrained elastic cylinder of variable cross-section. Q Appl Math 47:325–350
Green AE, Lindsay K (1972) Thermoelasticity. J Elast 2:1–7
Green AE, Naghdi PM (1992) On undamped heat waves in elastic solids. J Thermal Stress 15:252–264
Green AE, Naghdi PM (1993) Thermoelasticity without energy dissipation. J Elast 31:189–208
Horgan CO, Quintanilla R (2005) Spatial behaviour of solutions of the dual-phase-lag heat equation. Math Methods Appl Sci 28:43–57
Iesan D, Quintanilla R (2000) On a theory of thermoelasticity with microtemperatures. J Thermal Stress 23:199–216
Jun Yu Y, Xue Z-N, Tian X-G (2018) A modified Green–Lindsay thermoelastidcity with strain rate to eliminate discontinuity. Meccanica 53:2543–2554
Leseduarte MC, Magaña A, Quintanilla R (2010) On the time decay of solutions in porous-thermo-elasticity of type II. Discrete Contin Dyn Syst Ser B 13:375–391
Leseduarte MC, Quintanilla R (2014) On the spatial behavior in type III thermoelastodynamics. J Appl Math Phys (ZAMP) 65:165–177
Leseduarte MC, Quintanilla R (2018) Spatial behavior in high order partial differential equations. Math Methods Appl Sci 41:2480–2493
Lord H, Shulman Y (1967) A generalized dynamic theory of thermoelasticity. J Mech Phys Solids 15:299–309
Navarro CB, Quintanilla R (1984) On existence and uniquenes in incremental thermoelasticity. J Appl Math Phys (ZAMP) 35:206–215
Quintanilla R (2001) End effects in thermoelasticity. Math Methods Appl Sci 24:93–102
Quintanilla R, Racke R (2003) Stability in thermoelasticity of type III. Discrete Contin Dyn Syst B 3:383–400
Quintanilla R, Straughan B (2000) Growth and uniqueness in thermoelasticity. Proc R Soc Lond Ser A 456:1419–1429
Saint-Venant AJCB (1853) Mémoire sur la torsion des prismes. In: Mémoires présentés pour divers Savants a Ácadémie des Sciences de l’Institut Impérial de France, vol 14, pp 233–560
Saint-Venant AJCB (1856) Mémoire sur la flexion des prismes. J Math Pures Appl 1(Ser. 2):89–189
Acknowledgements
The investigation reported in this paper is supported by the project project “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P)(AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness. The author thanks to the anonymous referee his useful suggestions concerning this submission.
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Quintanilla, R. Some qualitative results for a modification of the Green–Lindsay thermoelasticity. Meccanica 53, 3607–3613 (2018). https://doi.org/10.1007/s11012-018-0889-0
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DOI: https://doi.org/10.1007/s11012-018-0889-0