A higher-order porous thermoelastic problem with microtemperatures

doi:10.1007/s10483-023-3049-8

Abstract

Abstract: In this paper, we study a porous thermoelastic problem with microtemperatures assuming parabolic higher order in time derivatives for the thermal variables. The model is derived and written as a coupled linear system. Then, a uniqueness result is proved by using the logarithmic convexity method in the case that we do not assume that the mechanical energy is positive definite. Finally, the existence of the solution is obtained by introducing an energy function and applying the theory of linear semigroups.

Key words: higher order, thermoelasticity, microtemperature, logarithmic convexity method, existence and uniqueness

2010 MSC Number:

J. R. FERNÁNDEZ, R. QUINTANILLA. A higher-order porous thermoelastic problem with microtemperatures. Applied Mathematics and Mechanics (English Edition), 2023, 44(11): 1911-1926.

TrendMD

References

[1] CATTANEO, C. On a form of heat equation which eliminates the paradox of instantaneous propagation. Comptes Rendus de l’Académie des Sciences de Paris, 247, 431–433(1958)
[2] CHEN, P. J. and GURTIN, M. E. On a theory of heat involving two temperatures. Zeitschrift für Angewandte Mathematik und Physik, 19, 614–627(1968)
[3] CHEN, P. J., GURTIN, M. E., and WILLIAMS, W. O. A note on non-simple heat conduction. Zeitschrift für Angewandte Mathematik und Physik, 19, 969–970(1968)
[4] CHEN, P. J., GURTIN, M. E., and WILLIAMS, W. O. On the thermodynamics of nonsimple materials with two temperatures. Zeitschrift für Angewandte Mathematik und Physik, 20, 107–112(1968)
[5] GREEN, A. E. and NAGHDI, P. M. On undamped heat waves in an elastic solid. Journal of Thermal Stresses, 15, 253–264(1992)
[6] GREEN, A. E. and NAGHDI, P. M. Thermoelasticity without energy dissipation. Journal of Elasticity, 31, 189–208(1993)
[7] GREEN, A. E. and NAGHDI, P. M. A unified procedure for construction of theories of deformable media. I. classical continuum physics. Proceedings of the Royal Society of London A, 448, 335–356(1995)
[8] GREEN, A. E. and NAGHDI, P. M. A unified procedure for construction of theories of deformable media. II. generalized continua. Proceedings of the Royal Society of London A, 448, 357–377(1995)
[9] GREEN, A. E. and NAGHDI, P. M. A unified procedure for construction of theories of deformable media. III. mixtures of interacting continua. Proceedings of the Royal Society of London A, 448, 379–388(1995)
[10] TZOU, D. Y. The generalized lagging response in small-scale and high-rate heating. International Journal of Heat and Mass Transfer, 38, 3231–3240(1995)
[11] CHOUDHURI, S. K. R. On a thermoelastic three-phase-lag model. Journal of Thermal Stresses, 30, 231–239(2007)
[12] QUINTANILLA, R. Moore-Gibson-Thompson thermoelasticity. Mathematics and Mechanics of Solids, 24, 4020–4031(2019)
[13] ERINGEN, A. C. Microcontinuum Field Theories. I. Foundations and Solids, Springer, New York (1999)
[14] IEŞAN, D. Thermoelastic models of continua. Solid Mechanics and Its Applications, Kluwer Academic Publisher, Dordrecht (2004)
[15] COWIN, S. C. and NUNZIATO, J. W. Linear elastic materials with voids. Journal of Elasticity, 13, 125–147(1983)
[16] GROT, R. Thermodynamics of a continuum with microstructure. International Journal of Engineering Science, 7, 801–814(1969)
[17] RIHA, P. On the theory of heat-conducting micropolar fluids with microtemperatures. Acta Mechanica, 23, 1–8(1975)
[18] RIHA, P. On the microcontinuum model of heat conduction in materials with inner structure. International Journal of Engineering Science, 14, 529–535(1976)
[19] IEŞAN, D. Thermoelasticity of bodies with microstructure and microtemperatures. International Journal of Solids and Structures, 44, 8648–8653(2007)
[20] BAZARRA, N., FERNÁNDEZ, J. R., and QUINTANILLA, R. Lord-Shulman thermoelasticity with microtemperatures. Applied Mathematics and Optimization, 84, 1667–1685(2021)
[21] LIU, Z., QUINTANILLA, R., and WANG, Y. Dual-phase-lag heat conduction with microtemperatures. Zeitschrift für Angewandte Physik, 101, e202000167(2021)
[22] IEŞAN, D. Incremental equations in thermoelasticity. Journal of Thermal Stresses, 3, 41–56(1980)
[23] KNOPS, R. J. and WILKES, E. W. Theory of elastic stability. Handbuch der Physic, Springer-Verlag, Berlin (1973)
[24] AMES, B. and STRAUGHAN, B. Continuous dependence results for initially prestressed thermoelastic bodies. International Journal of Engineering Science, 30, 7–13(1992)
[25] KNOPS, R. J. Instability and the ill-posed Cauchy problem in elasticity. Mechanics of Solids, The Rodney Hill 60th Anniversary, Elsevier, Berlin (1982)
[26] KNOPS, R. J. and PAYNE, L. E. Growth estimates for solutions of evolutionary equations in Hilbert spaces with applications to elastodynamics. Archive of Rational Mechanics and Analysis, 41, 363–398(1971)
[27] PELLICER, M. and QUINTANILLA, R. On uniqueness and instability for some thermomechanical problems involving the Moore-Gibson-Thompson equation. Zeitschrift für Angewandte Physik, 71, 84(2020)
[28] IEŞAN, D. and QUINTANILLA, R. On the theory of thermoelasticity with microtemperatures. Journal of Thermal Stresses, 23, 199–215(2000)
[29] IEŞAN, D. and QUINTANILLA, R. On thermoelastic bodies with inner structure and microtemperatures. Journal of Mathematical Analysis and Applications, 354, 12–23(2009)
[30] BEZEROVSKI, A., ENGELBRETCH, J., and MAUGIN, G. A. Thermoelasticity with dual internal variables. Journal of Thermal Stresses, 34, 413–430(2011)
[31] BORGMEYER, K., QUINTANILLA, R., and RACKE, R. Phase-lag heat conduction: decay rates for limit problems and well-posedness. Journal of Evolution Equations, 14, 863–884(2014)
[32] CHIRIŢA, S., D’APICE, C., and ZAMPOLI, V. The time differential three-phase-lag heat conduction model: thermodynamic compatibility and continuous dependence. International Journal of Heat and Mass Transfer, 102, 226–232(2016)
[33] FABRIZIO, M. and LAZZARI, B. Stability and second law of thermodynamics in dual-phase-lag heat conduction. International Journal of Heat and Mass Transfer, 74, 484–489(2014)
[34] QUINTANILLA, R. and RACKE, R. A note on stability in dual-phase-lag heat conduction. International Journal of Heat and Mass Transfer, 49, 1209–1213(2006)
[35] QUINTANILLA, R. and RACKE. R. A note on stability in three-phase-lag heat conduction. International Journal of Heat and Mass Transfer, 51, 24–29(2008)
[36] DELL’ORO, F. and PATA, V. A hierarchy of heat conduction laws. Discrete of Continuous Dynamical Systems Series S, 16, 2613–2635(2023)
[37] FERNÁNDEZ, J. R. and QUINTANILLA, R. Uniqueness for a high order ill posed problem. Proceedings of the Royal Society of Edinburgh, 153, 1425–1438(2023)
[38] MAGAÑA, A. and QUINTANILLA, R. On the existence and uniqueness in phase-lag thermoe-lasticity. Meccanica, 53, 125–134(2018)
[39] QUINTANILLA, R. and RACKE. R. Spatial behavior in phase-lag heat conduction. Differential and Integral Equations, 28, 291–308(2015)
[40] AMES, B. and STRAUGHAN, B. Non-standard and improperly posed problems. Mathematics in Science and Engineering, 194, Springer, Berlin (1997)
[41] FLAVIN, J. N. and RIONERO, S. Qualitative Estimates for Partial Differential Equations: an Introduction, CRC Press, Boca Raton (1995)
[42] ZAMPOLI, V. Uniqueness theorems about high-order time differential thermoelastic models. Ricerche di Matematica, 67, 929–950(2018)
[43] GOLDSTEIN, J. A. Semigroups of linear operators and applications. Oxford Mathematical Monographs, Oxford University Press, Oxford (1985)