Abstract
This note is concerned with the linear (and linearized) Type III thermoelastodynamic theory proposed by Green and Naghdi. We investigate the spatial behavior of the solutions when we assume the positivity of the elasticity tensor, the thermal conductivity tensor, the mass density and the heat capacity. However, we do not assume (a priori) the positivity of the internal energy. We first obtain a Phragmén-Lindelöf alternative of exponential type for the solutions. Later, we prove that the decay can be controlled by the exponential of a second-degree polynomial. This is similar to other thermoelastic situations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Maxwell J.C.: Theory of Heat. Dover, Mineola (2001)
Hetnarski R.B., Ignaczak J.: Generalized thermoelasticity. J. Therm. Stress. 22, 451–470 (1999)
Hetnarski R.B., Ignaczak J.: Nonclassical dynamical thermoelasticity. Int. J. Solids Struct. 37, 215–224 (2000)
Ignaczak J., Ostoja-Starzewski M.: Thermoelasticity with Finite Wave Speeds. Oxford Mathematical Monographs, Oxford (2010)
Straughan B.: Heat Waves. Springer, Berlin (2011)
Green A.E., Naghdi P.M.: On undamped heat waves in an elastic solid. J. Therm. Stress. 15, 253–264 (1992)
Green A.E., Naghdi P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)
Green, A.E., Naghdi, P.M.: A unified procedure for construction of theories of deformable media, I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua, Proc. R. Soc. Lond. A 448, 335–356 (357–377, 379–388) (1995)
Iesan D.: On the theory of thermoelasticity without energy dissipation. J. Therm. Stress. 21, 295–307 (1998)
Iesan D.: Thermopiezoelectricity without energy dissipation. Proc. R. Soc. Lond. A 464, 631–656 (2008)
Iesan D., Quintanilla R.: On the thermoelastic bodies with inner structure and microtemperatures. J. Math. Anal. Appl. 354, 12–23 (2009)
Lazzari B., Nibbi R.: On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary. J. Math. Anal. Appl. 338, 317–329 (2008)
Leseduarte M.C., Magaña A., Quintanilla R.: On the time decay of solutions in porous-thermo-elasticity of type II. Discret. Cont. Dyn. Syst. Ser. B 13, 375–391 (2010)
Leseduarte M.C., Quintanilla R.: On uniqueness and continuous dependence in type III thermoelasticity. J. Math. Anal. Appl. 395, 429–436 (2012)
Liu Z., Quintanilla R.: Analiticity of solutions in the type III thermoelastic plates. IMA J. Appl. Math. 75, 356–365 (2010)
Liu Z., Quintanilla R.: Energy decay rates of mixed type II and type III thermoelastic system. Discret. Cont. Dyn. Syst. Ser. B 14, 1433–1444 (2010)
Liu Y., Lin C.: Phragmén-Lindelöf alternative and continuous dependence-type results for the thermoelasticity of type III. Appl. Anal. Int. J. 83, 431–449 (2008)
Messaoudi S.A., Soufyane A.: Boundary stabilization of memory type in thermoelasticity of type III. Appl. Anal. 87, 13–28 (2008)
Puri P., Jordan P.M.: On the propagation of plane waves in type-III thermoelastic media. Proc. R. Soc. Lond. A 460, 3203–3221 (2004)
Qin Y., Deng S., Huang L., Ma Z., Su X.: Global existence for the three-dimensional thermoelastic equations of Type II. Q. Appl. Math. 68, 333–348 (2010)
Quintanilla R.: On the spatial behaviour in thermoelasticity without energy dissipation. J. Therm. Stress. 21, 213–224 (1999)
Quintanilla R.: Damping of end effects in a thermoelastic theory. Appl. Math. Lett. 14, 137–141 (2001)
Quintanilla R.: Some remarks on growth and uniqueness in thermoelasticity. IJMMS 10, 617–623 (2003)
Quintanilla R.: Convergence and structural stability in thermoelasticity. Appl. Math. Comput. 135, 287–300 (2003)
Quintanilla R.: Impossibility of localization in linear thermoelasticity. Proc. R. Soc. Lond. A 463, 3311–3322 (2007)
Quintanilla R., Racke R.: Stability in thermoelasticity of type III. Discret. Cont. Dyn. Syst. Ser. B 3, 383–400 (2003)
Quintanilla R., Straughan B.: Growth and uniqueness in thermoelasticity. Proc. R. Soc. Lond. A 456, 1419–1429 (2000)
Quintanilla R., Straughan B.: A note on discontinuity waves in type III thermoelasticity. Proc. R. Soc. Lond. A 460, 1169–1175 (2004)
Quintanilla R., Straughan B.: Energy bounds for some non-standard problems in thermoelasticity. Proc. R. Soc. Lond. A 461, 1147–1162 (2005)
Yang L., Wang Y.G.: Well-posedness and decay estimates for Cauchy problems of linear thermoelastic systems of type III in 3-D. Indiana Univ. Math. J. 55, 1333–1361 (2006)
Flavin J.N., Knops R.J., Payne L.E.: Decay estimates for the constrained elastic cylinder of variable cross-section. Q. Appl. Math. 47, 325–350 (1989)
Horgan C.O., Payne L.E., Wheeler L.T.: Spatial decay estimates in transient heat conduction. Q. Appl. Math. 42, 119–127 (1984)
Horgan C.O., Quintanilla R.: Spatial decay of transient end effects in functionally graded heat conducting materials. Q. Appl. Math. 59, 529–542 (2001)
Flavin J.N., Knops R.J., Payne L.E.: Energy bounds in dynamical problems for a semi-infinite elastic beam. In: Eason, G., Ogden, R.W. (eds.) Elasticity: Mathematical Methods and Applications, pp. 101–111. Ellis Horwood, Chichester (1989)
Quintanilla R.: On existence in thermoelasticity without energy dissipation. J. Therm. Stress. 25, 195–202 (2002)
Leseduarte M.C., Quintanilla R.: Phragmén-Lindelöf alternative for an exact heat conduction equation with delay. Commun. Pure Appl. Anal. 12, 1221–1235 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leseduarte, M.C., Quintanilla, R. On the spatial behavior in Type III thermoelastodynamics. Z. Angew. Math. Phys. 65, 165–177 (2014). https://doi.org/10.1007/s00033-013-0321-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-013-0321-5