From Generalized Theories of Media with Fields of Defects to Closed Variational Models of the Coupled Gradient Thermoelasticity and Thermal Conductivity

  • Sergey LurieEmail author
  • Petr Belov
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 120)


A gradient theory of the coupled theory of elasticity, thermoelasticity and thermal conductivity based on a generalized model of media with fields of defects is developed. Defectiveness is defined only by free dilatation and the tensor of incompatible distortions is determined by the spherical tensor. In the general case, the unified model includes scale parameters that are responsible for mechanical and temperature scale effects. In the particular case the proposed model describes the gradient thermoelasticity, in which effects are controlled by a mechanical scale parameter, and in the limit case, it describes the classical thermoelasticity, when this parameter tends to zero. The analysis of boundary problems of the general model is given. Particular cases are considered, and it is shown that gradient thermal conductivity and thermoelasticity make it possible to simulate thermal resistance and size effects.


Generalized Mindlin’s medium Free dilatations Gradient thermoelasticity Reversible thermal conductivity Cohesive interactions Scale effects 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors are deeply grateful to Professor Holm Altenbach for professional and useful discussions of our research. This work was carried out with support from the Russian Government Foundation of Institute of Applied Mechanics of RAS, Project -AAAA-A-19-119012290177-0 and particularly supported by the Russian Foundation for Basic Research grant 18-01-00553-1.


  1. Anisimov SI, Kapeliovich BL, Perel’man TL (1974) Electron emission from metal surfaces exposed to ultrashort laser pulses. Sov Phys-JETP 39(2):375–377Google Scholar
  2. Antoci S, Mihich L (1999) A four-dimensional Hooke’s law can encompass linear elasticity and inertia. Il Nuovo Cimento B 8:873–880Google Scholar
  3. Bahar LY, Hetnarski RB (1978) State space approach to thermoelasticity. J Therm Stress 1:135–145Google Scholar
  4. Barone M, Selleri F (eds) (2012) Frontiers of Fundamental Physics. Springer Science & Business Media, New YorkGoogle Scholar
  5. Belov PA, Lurie SA (2012) Ideal nonsymmetric 4D-medium as a model of invertible dynamic thermoelasticity. Mech Solids 47(5):580–590CrossRefGoogle Scholar
  6. Belov PA, Gorshkov AG, Lurie SA (2006) Variational model of nonholonomic 4D-media. Mech Solids 41(6):22–35Google Scholar
  7. Biot MA (1956) Thermoelasticity and irreversible thermo-dynamics. J Appl Phys 27:240–253CrossRefGoogle Scholar
  8. Cardona JM, Sievert R (2000) Thermoelasticity of second-grade media. In: Maugin GA, Drouot R, Sidoroff F (eds) Continuum hermomechanics, the Art and Science of Modelling Material Behaviour, Kluwer Academic Publishers, Dordrecht, pp 163–176Google Scholar
  9. Carlson DE (1972) Linear thermoelasticity. In: Flügge S (ed) Handbuch der Physik, Springer, Berlin, vol VIa/2: Linear Theories of Elasticity and Thermoelasticity (ed. by C. Truesdell), pp 297–345CrossRefGoogle Scholar
  10. Cattaneo C, Hebd CR (1958) Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée. C R Acad Sci 247:431–432Google Scholar
  11. Challamel N, Grazide C, Picandet V, Perrot A, Zhang Y (2016) A nonlocal fourier’s law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices. Comptes Rendus Mécanique 344(4):388–401CrossRefGoogle Scholar
  12. Chen J, Zhang G, Li B (2012) Thermal contact resistance across nanoscale silicon dioxide and silicon interface. Journal of Applied Physics 112(6):064,319CrossRefGoogle Scholar
  13. Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13:167–178CrossRefGoogle Scholar
  14. Dhar A (2008) Heat transport in low-dimensional systems. Adv Phys 57:457CrossRefGoogle Scholar
  15. Dong H, Wen R B Melnik (2014) Relative importance of grain boundaries and size effects in thermal conductivity of nanocrystalline materials. Scientific Reports 4(7037)Google Scholar
  16. Filopoulos SP, Papathanasiou TK, Markolefas SI, Tsamaphyros GJ (2014) Generalized thermoelastic models for linear elastic materials with micro-structure Part II: Enhanced Lord–Shulman model. J Therm Stress 37:642–659CrossRefGoogle Scholar
  17. Forest S, Aifantis EC (2010) Some links between recent gradient thermo – elasto – plasticity theories and the thermomechanics of generalized continua. Int J Solids Struct 47:3367–3376CrossRefGoogle Scholar
  18. Gaughey AJH, Kaviany M (2006) Phonon transport in molecular dynamics simulations: Formulation and thermal conductivity prediction. Advances in Heat Transfer 39:169–254Google Scholar
  19. Gendelman OV, Savin AV (2010) Nonstationary heat conduction in one-dimensional chains with conserved momentum. Phys Rev E 81:0201,103Google Scholar
  20. Green AE, Lindsay KA (1972) Thermoelasticity. J Elast 2:1–7Google Scholar
  21. Gudlur P (2010) Thermoelastic Properties of Particle Reinforced Composites at the Micro and Macro Scales. Texas, A & M UniversityGoogle Scholar
  22. Gusev A, Lurie S (2013) Wave-relaxation duality of heat propagation in fermi–pasta–ulam chains.Modern Phys Lett B 26(22):1250,145CrossRefGoogle Scholar
  23. Harry G, Bodiya TP, DeSalvo R (eds) (2012) Optical Coatings and Thermal Noise in Precision Measurement. Cambridge University Press, CambridgeGoogle Scholar
  24. Hopkins PE, Kassebaum JL, Norris PM (2009) Effects of electron scattering at metal-nonmetal interfaces on electron-phonon equilibration in gold films. J Appl Phys 105(2):023,710CrossRefGoogle Scholar
  25. Iesan D (1983) Thermoelasticity of nonsimple materials. J Therm Stress 6:385–397Google Scholar
  26. Iesan D (1991) On the theory of mixtures of thermoelastic solids. J Therm Stress 13(4):389–408CrossRefGoogle Scholar
  27. Iesan D (2004) Thermoelastic Models of Continua. Kluwer Acad. Publ., DordrechtCrossRefGoogle Scholar
  28. Iesan D (2009) Classical and Generalized Models of Elastic Rods. Chapman & Hall CRC Press, New YorkGoogle Scholar
  29. Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61:41–73CrossRefGoogle Scholar
  30. Jou D, Casa-Vásquez J, Lebon G (2010) Extended Irreversible Thermodynamics. Springer, New YorkCrossRefGoogle Scholar
  31. Kaganov MI, Lifshitz IM, Tanatarov LV (1957) Relaxation between electrons and crystalline lattices. Sov Phys JETP 4(2):173–178Google Scholar
  32. Kapitza PL (1971) The study of heat transfer in Helium II. In: Helium 4 - The Commonwealth and International Library: Selected Readings in Physics, Elsevier, pp 114–153CrossRefGoogle Scholar
  33. Kavner A, Panero W (2004) Temperature gradients and evaluation of thermoelastic properties in the synchrotron-based laser-heated diamond cell. Physics of the Earth and Planetary Interiors 143–144:527–539CrossRefGoogle Scholar
  34. Kienzler R, Maugin GA (eds) (2001) Configurational Mechanics of Materials, CISM International Centre for Mechanical Sciences, vol 427. Springer, ViennaGoogle Scholar
  35. Knyazeva AG, Evstigneev NK (2010) Interrelations between heat and mechanical processes during solid phase chemical conversion under loading. Procedia Computer Science 1(1):2613–2622CrossRefGoogle Scholar
  36. Landau LD, Lifshitz EM (1986) Course of Theoretical Theory of Elasticity. Theory of Elasticity. Vol. 7. Pergamon Press, OxfordGoogle Scholar
  37. Lepri S, Livi R, Politi A (2003) Thermal conduction in classical lowdimensional lattices. Phys Rep 377:1–80CrossRefGoogle Scholar
  38. Lord HW, Shulman YA (1967) Generalized dynamical theory of thermoelasticity. J Mech Phys Solids 15:299–309CrossRefGoogle Scholar
  39. Lurie SA, Belov PA (2000) Mathematical Models of Continuum Mechanics and Physical Fields. Izd-vo VTs, MoscowGoogle Scholar
  40. Lurie SA, Belov PA (2001) A variational model for nonholonomic media. Mekh Komp Mater Konstr [J Comp Mech Design (Engl Transl)] 7(2):266–276Google Scholar
  41. Lurie SA, Belov PA (2013) Theory of space–time dissipative elasticity and scale effects. Nanoscale Systems MMTA 2(1):66–178Google Scholar
  42. Lurie SA, Belov PA (2018) On the nature of the relaxation time, the Maxwell–Cattaneo and Fourier law in the thermodynamics of a continuous medium and the scale effects in thermal conductivity. Continuum Mechanics and Thermodynamics DOI Scholar
  43. Lurie SA, Belov PA, Solyaev YO (2019) Mechanistic model of generalized non-antisymmetrical electrodynamics. In: Altenbach H, Belyaev A, Eremeyev V, Krivtsov A, Porubov A (eds) Dynamical Processes in Generalized Continua and Structures, Springer, Cham, Advanced Structured Materials, vol 103Google Scholar
  44. Mahan GD (1988) Nonlocal theory of thermal conductivity. Phys Rev B 38(3):1963–1969CrossRefGoogle Scholar
  45. Majumdar A, Reddy P (2004) Role of electron–phonon coupling in thermal conductance of metal–nonmetal interfaces. Appl Phys Lett 84(23):4768–4770CrossRefGoogle Scholar
  46. Martinez F, Quintanilla R (1998) On the incremental problem in thermoelasticity of nonsimple materials. Z Angew Math Mech 78:703–710Google Scholar
  47. Maxwell JC (1865) A dynamical theory of the electromagnetic field. Philosophical Transactions of the Royal Society of London 155:459–512Google Scholar
  48. Molaro JL, Byrne S, Langer SA (2015) Grain-scale thermoelastic stresses and spatiotemporal temperature gradients on airless bodies, implications for rock breakdown. J Geophys Res: Planets 120(2):255–277Google Scholar
  49. Müller I, Ruggeri T (1993) Extended Thermodynamics, Springer Tracts in Natural Philosophy, vol 37. Springer, BerlinCrossRefGoogle Scholar
  50. Nowacki W (1986) Thermoelasticity. Pergamon Press, OxfordCrossRefGoogle Scholar
  51. Ordonez-Miranda J, Alvarado-Gil JJ, Yang R (2011) The effect of the electron-phonon coupling on the effective thermal conductivity of metal-nonmetal multilayers. J Appl Phys 109(9):094,310CrossRefGoogle Scholar
  52. Povstenko Y (2015) Fractional Thermoelasticity. SpringerGoogle Scholar
  53. Qiu B, Bao H, Zhang G, Wu Y, Ruan X (2012) Molecular dynamics simulations of lattice thermal conductivity and spectral phonon mean free path of PbTe: Bulk and nanostructures. Computat Mater Sci 53(1):278–285CrossRefGoogle Scholar
  54. Rawat V, Sands TD (2006) Growth of TiN/GaN metal/semiconductor multilayers by reactive pulsed laser deposition. J Appl Phys 100(6):064,901CrossRefGoogle Scholar
  55. Schelling PK, Phillpot SR, Keblinski P (2002) Comparison of atomic-level simulation methods for computing thermal conductivity. Phys Rev B 65:144,306Google Scholar
  56. Sellan DP, Turney JE, McGaughey AJH, Amon CH (2010) Cross-plane phonon transport in thin films. J Appl Phys 108(11):113,524CrossRefGoogle Scholar
  57. Sherief HH (1993) State space formulation for generalized thermoelasticity with one relaxation time including heat sources. J Therm Stress 16:163–180CrossRefGoogle Scholar
  58. Sherief HH, Helmy K (1999) A two dimensional generalized thermoelasticity problem for a halfspace. J Therm Stresses 22:897–910Google Scholar
  59. Sobolev SL (1991) Transport processes and traveling waves in systems with local nonequilibrium. Sov Phys Usp 34(3):217–229CrossRefGoogle Scholar
  60. Stewart DA, Norris PM (2000) Size effects on the thermal conductivity of thin metallic wires: microscale implications. Microscale Thermophysical Engineering 4(2):89–101Google Scholar
  61. Turney JE, Landry ES, McGaughey AJH, Amon CH (2009) Predicting phonon properties and thermal conductivity from anharmonic lattice dynamics calculations and molecular dynamics simulations. Phys Rev B 79(6):064,301Google Scholar
  62. Tzou DY (2014) Macro- to Microscale Heat Transfer: the Lagging Behavior, 2nd edn. Series in Chemical and Mechanical Engineering, Taylor & Francis, Washington, D.C.Google Scholar
  63. Vernotte MP (1958) Les paradoxes de la théorie continue de l’équation de la chaleur. C R de l’Académie des Sci 246(3):154–3155Google Scholar
  64. Williams D (1989) The elastic energy-momentum tensor in special relativity. Ann of Phys 196(2):345–360CrossRefGoogle Scholar
  65. Xu Y, Li G (2009) Strain effect analysis on phonon thermal conductivity of two-dimensional nanocomposites. J Appl Phys 106(11):114,302CrossRefGoogle Scholar
  66. Zenkour AM, Abouelregal AE, Alnefaie KA, Zhang X, Aifantis EC (2015) Nonlocal thermoelasticity theory for thermal-shock nanobeams with temperature-dependent thermal conductivity. J Therm Stress 38(9):1049–1067CrossRefGoogle Scholar
  67. Zhang G, Li BW (2010) Impacts of doping on thermal and thermoelectric properties of nanomaterials. NanoScale 2:1058–1068CrossRefGoogle Scholar
  68. Zhou Y, Anglin B, Strachan A (2007) Phonon thermal conductivity in nanolaminated composite metals via molecular dynamics. J Chem Phys 127(8):184,702–11CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Applied Mechanics of Russian Academy of SciencesMoscowRussia

Personalised recommendations