The Micromorphic Approach to Generalized Heat Equations

References

[1] J. Fourier, Theorie analytique de la chaleur, par M. Fourier. Chez Firmin Didot, père et fils, 1822.Suche in Google Scholar

[2] J. Fourier, The Analytical Theory of Heat. Translated, with notes, by Alexander Freeman. University Press, Cambridge, 1878.Suche in Google Scholar

[3] C. Cattaneo, Sulla conduzione del calore, in: A. Pignedoli, (ed.) Some Aspects of Diffusion Theory, Vol. 42, Springer-Verlag, Berlin, (1948).Suche in Google Scholar

[4] C. Cattaneo, Sur une forme de lequation de la chaleur eliminant le paradoxe dune propagation instantanee, C. R. Hebd. Seances Acad. Sci. 247 (1958), no. 4, 431–433.Suche in Google Scholar

[5] P. Vernotte, Les paradoxes de la théorie continue de léquation de la chaleur, C. R. Hebd. Seances Acad. Sci. 246 (1958), no. 22, 3154–3155.Suche in Google Scholar

[6] B. D. Coleman, M. Fabrizio and D. R. Owen, On the thermodynamics of second sound in dielectric crystals, Arch. Rational Mech. Anal. 80 (1982), no. 2, 135–158.10.1007/BF00250739Suche in Google Scholar

[7] B. D. Coleman, M. Fabrizio and D. R. Owen, Thermodynamics and the constitutive relations for second sound in crystals, in: G. Grioli, (ed.) Thermodynamics and Constitutive Equations, Vol. 228, Springer-Verlag, Berlin, 1985, 20–43.10.1007/BFb0017953Suche in Google Scholar

[8] D. Jou, J. Casas-Vazquez and G. Lebon, Extended irreversible thermodynamics, Rep. Prog. Phys. 51 (1988), no. 8, 1105–1179.10.1088/0034-4885/51/8/002Suche in Google Scholar

[9] I. Müller and T. Ruggeri, Extended Thermodynamics, Vol. 37. Springer Tracts in Natural Philosophy. Springer-Verlag, New York, 1993.10.1007/978-1-4684-0447-0Suche in Google Scholar

[10] C. Bai and A. S. Lavine, On hyperbolic heat conduction and the second law of thermodynamics, J. Heat Transfer117 (1995), no. 2, 256–263.10.1115/1.2822514Suche in Google Scholar

[11] A. Barletta and E. Zanchini, Hyperbolic heat conduction and local equilibrium: A second law analysis, Int. J. Heat. Mass Transf. 40 (1997), no. 5, 1007–1016.10.1016/0017-9310(96)00211-6Suche in Google Scholar

[12] E. Zanchini, Hyperbolic-heat-conduction theories and nondecreasing entropy, Phys. Rev. B60 (1999), no. 2, 991–997.10.1103/PhysRevB.60.991Suche in Google Scholar

[13] R. J. Swenson, Heat conduction - finite or infinite propagation, J. Non-Equilib. Thermodyn. 3 (1978), no. 1, 39–48.10.1515/jnet.1978.3.1.39Suche in Google Scholar

[14] V. Peshkov, Second sound in helium II, J. Phys. USSR8 (1944), 381–386.Suche in Google Scholar

[15] V. Peshkov, Determination of the velocity of propagation of the second sound in helium II, J. Phys. USSR10 (1946), 389–398.10.1016/B978-0-08-015816-7.50016-XSuche in Google Scholar

[16] H. E. Jackson and C. T. Walker, Thermal conductivity, second sound, and phonon-phonon interactions in NaF, Phys. Rev. B3 (1971), no. 4, 1428–1439.10.1103/PhysRevB.3.1428Suche in Google Scholar

[17] V. Narayanamurti and R. C. Dynes, Observation of second sound in bismuth, Phys. Rev. Lett. 28 (1972), no. 22, 1461–1465.10.1103/PhysRevLett.28.1461Suche in Google Scholar

[18] T. Da Yu, Shock wave formation around a moving heat source in a solid with finite speed of heat propagation, Int. J. Heat. Mass Transf. 32 (1989), no. 10, 1979–1987.10.1016/0017-9310(89)90166-XSuche in Google Scholar

[19] J. Wang and J.-S. Wang, Carbon nanotube thermal transport: Ballistic to diffusive, Appl. Phys. Lett. 88 (2006), no. 11, 111909.10.1063/1.2185727Suche in Google Scholar

[20] D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, et al. Nanoscale thermal transport, J. Appl. Phys. 93 (2003), no. 2, 793–818.10.1063/1.1524305Suche in Google Scholar

[21] D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E. Goodson, et al. Nanoscale thermal transport. II. 2003–2012, Appl. Phys. Rev. 1 (2014), no. 1, 011305.10.1063/1.4832615Suche in Google Scholar

[22] D. Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, 2nd ed., John Wiley & Sons, Chichester, 2014.10.1002/9781118818275Suche in Google Scholar

[23] D. Jou and A. Cimmelli Vito, Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: An overview, Commun. Appl. Ind. Math. 7 (2016), no. 2, 196–226.10.1515/caim-2016-0014Suche in Google Scholar

[24] B. C. Larson, J. Z. Tischler and D. M. Mills, Nanosecond resolution time-resolved x-ray study of silicon during pulsed-laser irradiation, J. Mater. Res. 1 (1986), no. 1, 144–154.10.1557/JMR.1986.0144Suche in Google Scholar

[25] A. Sellitto, V. A. Cimmelli and D. Jou, Mesoscopic Theories of Heat Transport in Nanosystems. SEMA SIMAI Springer Series, vol. 6. Springer International Publishing, Switzerland, 2016.10.1007/978-3-319-27206-1Suche in Google Scholar

[26] J. Kaiser, T. Feng, J. Maassen, X. Wang, X. Ruan and M. Lundstrom, Thermal transport at the nanoscale: A Fourier’s law vs. phonon Boltzmann equation study, J. Appl. Phys. 121 (2017), no. 4, 044302.10.1063/1.4974872Suche in Google Scholar

[27] F. X. Alvarez and D. Jou, Memory and nonlocal effects in heat transport: From diffusive to ballistic regimes, Appl. Phys. Lett. 90 (2007), no. 8, 083109.10.1063/1.2645110Suche in Google Scholar

[28] Y. Guo and M. Wang, Phonon hydrodynamics and its applications in nanoscale heat transport, Phys. Rep. 595 (2015), 1–44.10.1016/j.physrep.2015.07.003Suche in Google Scholar

[29] C. Körner and H. W. Bergmann, The physical defects of the hyperbolic heat conduction equation, Appl. Phys. A67 (1998), no. 4, 397–401.10.1007/s003390050792Suche in Google Scholar

[30] I. Müeller, Zur Ausbreitungsgeschwindigkeit Von Störungen in Kontinuierlichen Medien, Technische Hochschule, Aachen, 1966.Suche in Google Scholar

[31] I. Muller, Zum Paradoxon der Warmeleitungstheorie, Z. Phys. 198 (1967), 329–344.10.1007/BF01326412Suche in Google Scholar

[32] D. Jou, J. Casas-Vázquez and G. Lebon, Extended Irreversible Thermodynamics, Springer Netherlands, Dordrecht, 2010.10.1007/978-90-481-3074-0Suche in Google Scholar

[33] İ. Temizer and P. Wriggers, A micromechanically motivated higher-order continuum formulation of linear thermal conduction, ZAMM J. Appl. Math. Mech. 90 (2010), no. 10–11, 768–782.10.1002/zamm.201000009Suche in Google Scholar

[34] R. A. Guyer and J. A. Krumhansl, Solution of the linearized phonon Boltzmann equation, Phys. Rev. 148 (1966), no. 2, 766–778.10.1103/PhysRev.148.766Suche in Google Scholar

[35] S. Both, B. Czél, T. Fülöp, G. Gróf, Á. Gyenis, R. Kovács, et al. Deviation from the Fourier law in room-temperature heat pulse experiments, J. Non-Equilib. Thermodyn. 41 (2016), no. 1, 41–48.10.1515/jnet-2015-0035Suche in Google Scholar

[36] P. Ván and T. Fülöp, Universality in heat conduction theory: Weakly nonlocal thermodynamics, Ann. Phys. 524 (2012), no. 8, 470–478.10.1002/andp.201200042Suche in Google Scholar

[37] G. Lebon, Heat conduction at micro and nanoscales: A review through the prism of extended irreversible thermodynamics, J. Non-Equilib. Thermodyn. 39 (2014), no. 1, 35–59.10.1515/jnetdy-2013-0029Suche in Google Scholar

[38] S. L. Sobolev, Two-temperature discrete model for nonlocal heat conduction, J. Phys. III France3 (1993), no. 12, 2261–2269.10.1051/jp3:1993273Suche in Google Scholar

[39] S. L. Sobolev, Two-temperature Stefan problem, Phys. Lett. 197 (1995), no. 3, 243–246.10.1016/0375-9601(94)00939-MSuche in Google Scholar

[40] L. S. Sergei, Local non-equilibrium transport models, Physics-Uspekhi40 (1997), no. 10, 1043.10.1070/PU1997v040n10ABEH000292Suche in Google Scholar

[41] S. L. Sobolev, Nonlocal diffusion models: Application to rapid solidification of binary mixtures, Int. J. Heat. Mass Transf. 71 (2014), 295–302.10.1016/j.ijheatmasstransfer.2013.12.048Suche in Google Scholar

[42] S. L. Sobolev, Nonlocal two-temperature model: Application to heat transport in metals irradiated by ultrashort laser pulses, Int. J. Heat. Mass Transf. 94 (2016), 138–144.10.1016/j.ijheatmasstransfer.2015.11.075Suche in Google Scholar

[43] D. Y. Tzou, A unified field approach for heat conduction from macro- to micro-scales, J. Heat Transfer117 (1995), no. 1, 8–16.10.1115/1.2822329Suche in Google Scholar

[44] G. Chen, Ballistic-diffusive heat-conduction equations, Phys. Rev. Lett. 86 (2001), no. 11, 2297–2300.10.1103/PhysRevLett.86.2297Suche in Google Scholar

[45] A. V. Luikov, V. A. Bubnov and I. A. Soloviev, On wave solutions of the heat-conduction equation, Int. J. Heat. Mass Transf. 19 (1976), no. 3, 245–248.10.1016/0017-9310(76)90027-2Suche in Google Scholar

[46] V. A. Bubnov, Wave concepts in the theory of heat, Int. J. Heat. Mass Transf. 19 (1976), no. 2, 175–184.10.1016/0017-9310(76)90110-1Suche in Google Scholar

[47] P. Ireman and Q.-S. Nguyen, Using the gradients of temperature and internal parameters in continuum thermodynamics, C. R. MeC. 332 (2004), no. 4, 249–255.10.1016/j.crme.2004.01.012Suche in Google Scholar

[48] G. A. Maugin, Internal variables and dissipative structures, J. Non-Equilib. Thermodyn. 15 (1990), no. 2, 173–192.Suche in Google Scholar

[49] G. A. Maugin and W. Muschik, Thermodynamics with internal variables. Part I. General concepts, J. Non-Equilib. Thermodyn. 19 (1994), no. 3, 217–249.10.1515/jnet.1994.19.3.217Suche in Google Scholar

[50] G. A. Maugin and W. Muschik, Thermodynamics with internal variables. Part II. Applications, J. Non-Equilib. Thermodyn. 19 (1994), no. 3, 250–289.10.1515/jnet.1994.19.3.250Suche in Google Scholar

[51] S. Forest, J.-M. Cardona and R. Sievert, Thermoelasticity of second-grade media, in: Maugin G. A., Drouot R., Sidoroff F. (eds.), Continuum Thermomechanics, Springer Netherlands, Dordrecht, (2000), 163–176.10.1007/0-306-46946-4_12Suche in Google Scholar

[52] S. Forest and M. Amestoy, Hypertemperature in thermoelastic solids, C. R. MeC. 336 (2008), no. 4, 347–353.10.1016/j.crme.2008.01.007Suche in Google Scholar

[53] Q.-S. Nguyen, Gradient thermodynamics and heat equations, C. R. MeC. 338 (2010), no. 6, 321–326.10.1016/j.crme.2010.07.010Suche in Google Scholar

[54] E. C. Aifantis, Further comments on the problem of heat extraction from hot dry rocks, Mech. Res. Commun. 7 (1980), no. 4, 219–226.10.1016/0093-6413(80)90042-7Suche in Google Scholar

[55] E. C. Aifantis, On the problem of diffusion in solids, Acta Mech. 37 (1980), no. 3, 265–296.10.1007/BF01202949Suche in Google Scholar

[56] S. Forest and E. C. Aifantis, Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua, Int. J. Solids Struct. 47 (2010), no. 25–26, 3367–3376.10.1016/j.ijsolstr.2010.07.009Suche in Google Scholar

[57] I. Müller, Thermodynamics of mixtures and phase field theory, Int. J. Solids Struct. 38 (2001), no. 6–7, 1105–1113.10.1016/S0020-7683(00)00076-7Suche in Google Scholar

[58] C. Wozniak, Thermoelasticity of non-simple oriented materials, Int. J. Eng. Sci. 5 (1967), no. 8, 605–612.10.1016/0020-7225(67)90059-6Suche in Google Scholar

[59] A. C. Eringen and E. S. Suhubi, Nonlinear theory of simple micro-elastic solids—I, Int. J. Eng. Sci. 2 (1964), no. 2, 189–203.10.1016/0020-7225(64)90004-7Suche in Google Scholar

[60] E. S. Suhubl and A. C. Eringen, Nonlinear theory of micro-elastic solids—II, Int. J. Eng. Sci. 2 (1964), no. 4, 389–404.10.1016/0020-7225(64)90017-5Suche in Google Scholar

[61] R. A. Grot, Thermodynamics of a continuum with microstructure, Int. J. Eng. Sci. 7 (1969), no. 8, 801–814.10.1016/0020-7225(69)90062-7Suche in Google Scholar

[62] D. Ieşan, On the theory of heat conduction in micromorphic continua, Int. J. Eng. Sci. 40 (2002), no. 16, 1859–1878.10.1016/S0020-7225(02)00066-6Suche in Google Scholar

[63] A. C. Eringen, Mechanics of micromorphic materials, in: Görtler H. (ed.), Applied Mechanics: Proceedings of the Eleventh International Congress of Applied Mechanics Munich (Germany) 1964, Springer, Berlin, Heidelberg (1966), 131–138.10.1007/978-3-662-29364-5_12Suche in Google Scholar

[64] A. C. Eringen, Balance laws of micromorphic mechanics, Int. J. Eng. Sci. 8 (1970), no. 10, 819–828.10.1016/0020-7225(70)90084-4Suche in Google Scholar

[65] A. C. Eringen, Balance laws of micromorphic continua revisited, Int. J. Eng. Sci. 30 (1992), no. 6, 805–810.10.1016/0020-7225(92)90109-TSuche in Google Scholar

[66] A. C. Eringen, Microcontinuum Field Theories: I. Foundations and Solids, Springer Verlag, New York, 1999.10.1007/978-1-4612-0555-5Suche in Google Scholar

[67] S. L. Koh, Theory of a second-order microfluid, Rheol. Acta12 (1973), no. 3, 418–424.10.1007/BF01502994Suche in Google Scholar

[68] P. Říha, On the theory of heat-conducting micropolar fluids with microtemperatures, Acta Mech. 23 (1975), no. 1, 1–8.10.1007/BF01177664Suche in Google Scholar

[69] P. Riha, Poiseuille flow of microthermopolar fluids, Acta Tech. 22 (1977), 602–613.Suche in Google Scholar

[70] P. Verma, D. Singh and K. Singh, Hagen-Poiseuille flow of microthermopolar fluids in a circular pipe, Acta Tech. 24 (1979), 402–412.Suche in Google Scholar

[71] P. Říha, On the microcontinuum model of heat conduction in materials with inner structure, Int. J. Eng. Sci. 14 (1976), no. 6, 529–535.10.1016/0020-7225(76)90017-3Suche in Google Scholar

[72] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Soc. London Ser. 432 (1991), no. 1885, 171–194.10.1098/rspa.1991.0012Suche in Google Scholar

[73] A. E. Green and P. M. Naghdi, On thermodynamics and the nature of the second law, Proc. Royal Soc. London Ser. 357 (1977), no. 1690, 253–270.10.1098/rspa.1977.0166Suche in Google Scholar

[74] A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elast. 31 (1993), no. 3, 189–208.10.1007/BF00044969Suche in Google Scholar

[75] D. Ieşan and L. Nappa, On the theory of heat for micromorphic bodies, Int. J. Eng. Sci. 43 (2005), no. 1–2, 17–32.10.1016/j.ijengsci.2004.09.003Suche in Google Scholar

[76] S. Forest, Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech. 135 (2009), no. 3, 117–131.10.1061/(ASCE)0733-9399(2009)135:3(117)Suche in Google Scholar

[77] K. Saanouni and M. Hamed, Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage: Formulation and computational aspects, Int. J. Solids Struct. 50 (2013), no. 14–15, 2289–2309.10.1016/j.ijsolstr.2013.03.027Suche in Google Scholar

[78] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics. The Non-Linear Field Theories of Mechanics, Springer, Berlin, Heidelberg, 2004.10.1007/978-3-662-10388-3Suche in Google Scholar

[79] W. Kaminski, Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure, J. Heat Transfer112 (1990), no. 3, 555–560.10.1115/1.2910422Suche in Google Scholar

[80] Q.-S. Nguyen and S. Andrieux, The non-local generalized standard approach: A consistent gradient theory, C. R. MeC. 333 (2005), no. 2, 139–145.10.1016/j.crme.2004.09.010Suche in Google Scholar

[81] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), no. 2, 258–267.10.1063/1.1744102Suche in Google Scholar

[82] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D92 (1996), no. 3–4, 178–192.10.1016/0167-2789(95)00173-5Suche in Google Scholar

[83] W. Dreyer and H. Struchtrup, Heat pulse experiments revisited., Continuum Mech. Thermodyn. 5 (1993), no. 1, 3–50.10.1007/BF01135371Suche in Google Scholar

[84] D. Ieşan, Thermoelasticity of bodies with microstructure and microtemperatures, Int. J. Solids Struct. 44 (2007), no. 25–26, 8648–8662.10.1016/j.ijsolstr.2007.06.027Suche in Google Scholar

[85] D. Ieşan and R. Quintanilla, On thermoelastic bodies with inner structure and microtemperatures, J. Math. Anal. Appl. 354 (2009), no. 1, 12–23.10.1016/j.jmaa.2008.12.017Suche in Google Scholar

[86] D. Ieşan and A. Scalia, Plane deformation of elastic bodies with microtemperatures, Mech. Res. Commun. 37 (2010), no. 7, 617–621.10.1016/j.mechrescom.2010.09.005Suche in Google Scholar

[87] P. Germain, The method of virtual power in continuum mechanics. Part 2: Microstructure, SIAM J. Appl. Math. 25 (1973), no. 3, 556–575.10.1137/0125053Suche in Google Scholar

[88] F. Mandl, Statistical Physics, 2nd ed., The Manchester Physics. John Wiley & Sons, Chichester, 2008.Suche in Google Scholar

[89] R. H. Swendsen, Statistical mechanics of colloids and Boltzmann’s definition of the entropy, Am. J. Phys. 74 (2006), no. 3, 187–190.10.1119/1.2174962Suche in Google Scholar

[90] J. Šesták and P. Holba, Heat inertia and temperature gradient in the treatment of DTA peaks, J. Therm. Anal. Calorim. 113 (2013), no. 3, 1633–1643.10.1007/s10973-013-3025-3Suche in Google Scholar

[91] P. Holba and J. Šesták, Heat inertia and its role in thermal analysis, J. Therm. Anal. Calorim. 121 (2015), no. 1, 303–307.10.1007/s10973-015-4486-3Suche in Google Scholar