Skip to main content
SpringerLink
Log in

Discrete and Continuum Thermomechanics

  • Living reference work entry
  • First Online:
Encyclopedia of Continuum Mechanics

Synonyms

Discrete and continuum thermoelasticity; Mechanics of discrete media: thermoelasticity; Thermoelasticity: from particle dynamics to continuum mechanics

Definitions

An approach for transition from discrete to continuum description of thermomechanical behavior of solids is discussed. The transition is carried out for several perfect anharmonic crystals with pair force interactions: one-dimensional crystal, quasi-one-dimensional crystal (a chain possessing longitudinal and transverse motions), two- and three-dimensional crystals with simple lattice. Macroscopic balance equations are derived from equations of motion for particles using continualization. Macroscopic parameters, such as stress, heat flux, deformation, thermal energy, etc., are represented via parameters of the discrete system. An approach for derivation of equations of state relating thermal pressure, thermal energy, and specific volume is presented. Derivation of constitutive equations for heat transfer is...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Barton MA, Stacey FD (1985) The Gruneisen parameter at high pressure: a molecular dynamical study. Phys Earth Planet Inter 39:167

    Google Scholar 

  • Dove MT, Fang H (2016) Negative thermal expansion and associated anomalous physical properties: review of the lattice dynamics theoretical foundation. Rep Prog Phys 79:066503

    Google Scholar 

  • Dudnikova TV, Komech AI, Spohn H (2003) On the convergence to statistical equilibrium for harmonic crystals. J Math Phys 44:2596

    Google Scholar 

  • Gendelman OV, Savin AV (2010) Nonstationary heat conduction in one-dimensional chains with conserved momentum. Phys Rev E 81:020103

    Google Scholar 

  • Grüneisen E (1912) Theorie des festen Zustandes einatomiger Elemente. Annalen der Physik 344(12):257

    Google Scholar 

  • Harris L, Lukkarinen J, Teufel S, Theil F (2008) Energy transport by acoustic modes of harmonic lattices. SIAM J Math Anal 40(4):1392

    Google Scholar 

  • Hoover WG (1991) Computational statistical mechanics. Studies in modern thermodynamics. Elsevier Science, Amsterdam, p 314

    Google Scholar 

  • Hoover WG, Hoover CG (2013) Time reversibility, computer simulation, algorithms, chaos. World Scientific, Hackensack

    Google Scholar 

  • Hoover WG, Holian BL, Posch HA (1993) Comment i on “Possible experiment to check the reality of a nonequilibrium temperature”. Phys Rev E 48:3196

    Google Scholar 

  • Indeitsev DA, Naumov VN, Semenov BN, Belyaev AK (2009) Thermoelastic waves in a continuum with complex structure. ZAMM 89(4):279

    Google Scholar 

  • Inogamov NA, Petrov Yu V, Zhakhovsky VV, Khokhlov VA, Demaske BJ, Ashitkov SI, Khishchenko KV, Migdal KP, Agranat MB, Anisimov SI, Fortov VE, Oleynik II (2012) Two-temperature thermodynamic and kinetic properties of transition metals irradiated by femtosecond lasers. AIP Conf Proc 1464:593

    Google Scholar 

  • Irvine RD, Stacey FD (1975) Pressure dependence of the thermal gruneisen parameter, with application to the Earth’s lower mantle and outer core. Phys Earth Planet Inter 11:157

    Google Scholar 

  • Kannan V, Dhar A, Lebowitz JL (2012) Nonequilibrium stationary state of a harmonic crystal with alternating masses. Phys Rev E 85:041118

    Google Scholar 

  • Krivtsov AM (1999) Constitutive equations of the nonlinear crystal lattice. ZAMM 79(S2):419

    Google Scholar 

  • Krivtsov AM (2003) From nonlinear oscillations to equation of state in simple discrete systems. Chaos, Solitons Fractals 17(1):79

    Google Scholar 

  • Krivtsov AM (2007a) Dynamics of energy characteristics in one-dimensional crystal. Proceeding of XXXIV summer school “Advanced Problems in Mechanics”. St.-Petersburg

    Google Scholar 

  • Krivtsov AM (2007b, in Russian) Deformation and fracture of solids with microstructure. Fizmatlit, Moscow

    Google Scholar 

  • Krivtsov AM (2014) Energy oscillations in a one-dimensional crystal. Dokl Phys 59:427

    Google Scholar 

  • Krivtsov AM (2015) Heat transfer in infinite harmonic one-dimensional crystals. Dokl Phys 60407

    Google Scholar 

  • Krivtsov AM (2015) On unsteady heat conduction in a harmonic crystal. ArXiv:1509.02506

    Google Scholar 

  • Krivtsov AM, Kuzkin VA (2011) Derivation of equations of state for ideal crystals of simple structure. Mech Solids 46(3):387

    Article  Google Scholar 

  • Kuzkin VA (2010) Interatomic force in systems with multibody interactions. Phys Rev E 82:016704

    Article  Google Scholar 

  • Kuzkin VA (2014) Comment on “Negative thermal expansion in single-component systems with isotropic interactions”. J Phys Chem 118(41):9793

    Article  Google Scholar 

  • Kuzkin VA, Krivtsov AM (2011) Equivalent thermo-mechanical parameters for perfect crystals. In: IUTAM symposium on the vibration analysis of structures with uncertainties. IUTAM bookseries. Springer.

    Book  Google Scholar 

  • Kuzkin VA, Krivtsov AM (2015) Nonlinear positive/negative thermal expansion and equations of state of a chain with longitudinal and transverse vibrations. Phys Stat Sol b 252:1664

    Article  Google Scholar 

  • Kuzkin VA, Krivtsov AM (2017a) An analytical description of transient thermal processes in harmonic crystals. Phys Sol State 59(5):1051

    Article  Google Scholar 

  • Kuzkin VA, Krivtsov AM (2017b) Fast and slow thermal processes in harmonic scalar lattices. J Phys: Condens Matter 29: 505401

    Google Scholar 

  • Kuzkin VA, Krivtsov AM, Jones RE, Zimmerman JA (2015) Material stress representation of equivalent stress tensor for discrete solids. Phys Mesomech 18(1):13

    Article  Google Scholar 

  • Lepri S (ed) (2016) Thermal transport in low dimensions. From statistical physics to nanoscale heat transfer. Springer, Cham/Heidelberg/New York/Dordrecht/London

    Google Scholar 

  • Lepri S, Livi R, Politi A (2003) Thermal conduction in classical low-dimensional lattices. Phys. Rep. 377:1

    Article  MathSciNet  Google Scholar 

  • Mie G (1903) Zur kinetischen theorie der einatomigen korper. Annalen der Physik 316(8):657

    Article  MATH  Google Scholar 

  • Panchenko AYu, Podolskaya EA, Krivtsov AM (2017) Analysis of equations of state and determination of the Gruneisen function for two-dimensional crystal lattices. Dokl Phys 62(3):141

    Article  Google Scholar 

  • Rieder Z, Lebowitz JL, Lieb E (1967) Properties of a harmonic crystal in a stationary nonequilibrium state. J Math Phys 8:1073

    Article  Google Scholar 

  • Weinberger CR, Tucker GJ (ed) (2016) Multiscale materials modeling for nanomechanics. Springer, Cham

    Google Scholar 

  • Zhilin PA, Altenbach H, Ivanova EA, Krivtsov AM (2013) Material strain tensor. In: Altenbach H et al (ed) Generalized continua as models for materials. Advanced structured materials, vol 22. Springer, Berlin/Heidelberg, pp 234–295

    Chapter  Google Scholar 

Download references

Acknowledgements

This work was supported by the Russian Science Foundation (RSCF grant No. 17-71-10213).

Author information

Authors and Affiliations

  1. Department of Theoretical Mechanics, Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, Russia

    Anton M. Krivtsov & Vitaly A. Kuzkin

  2. Laboratory for Discrete Models in Mechanics, Institute for Problems in Mechanical Engineering RAS, Saint Petersburg, Russia

    Anton M. Krivtsov & Vitaly A. Kuzkin

Authors
  1. Anton M. Krivtsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vitaly A. Kuzkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton M. Krivtsov .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Institut für Mechanik, Magdeburg, Germany

    Holm Altenbach

  2. Griffith University (Gold Coast Campus), Griffith School of Engineering Griffith University (Gold Coast Campus), Southport, Queensland, Australia

    Andreas Öchsner

Section Editor information

  1. Department of Theoretical Mechanics, Institute of Applied Mathematics and Mechanics, Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya 29, 195251, St. Petersburg, Russia

    Elena Ivanova

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Germany

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Krivtsov, A.M., Kuzkin, V.A. (2018). Discrete and Continuum Thermomechanics. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_67-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53605-6_67-1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53605-6

  • Online ISBN: 978-3-662-53605-6

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation