Skip to main content
SpringerLink
Log in

Microdeformation and Microtemperature

  • Chapter
  • First Online:
Internal Variables in Thermoelasticity

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 243))

Abstract

The introduction of double dual internal variables provides the complete extension of the classical thermoelasticity theory onto the case of microstructured solids. This extension keeps the structure of canonical balances of momentum and energy and provides the thermodynamically consistent evolution equations for microdeformation and microtemperature. Evolution equations in the case of dual internal variables are hyperbolic and coupled with the equations of macromotion.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Baczyński ZF (2003) Dynamic thermoelastic processes in microperiodic composites. J Thermal Stress 26(1):55–66

    Article  Google Scholar 

  2. Berezovski A, Berezovski M (2013) Influence of microstructure on thermoelastic wave propagation. Acta Mech 224(11):2623–2633

    Article  MathSciNet  MATH  Google Scholar 

  3. Berezovski A, Engelbrecht J (2012) Waves in microstructured solids: dispersion and thermal effects. In: Proceedings of the 23rd international congress of theoretical and applied mechanics, Beijing, China, pp SM07–005

    Google Scholar 

  4. Berezovski A, Engelbrecht J (2013) Thermoelastic waves in microstructured solids: dual internal variables approach. J Coupled Syst Multiscale Dyn 1(1):112–119

    Article  Google Scholar 

  5. Berezovski A, Engelbrecht J, Maugin GA (2009) Internal variables and generalized continuum theories. In: IUTAM symposium on progress in the theory and numerics of configurational mechanics. Springer, pp 149–158

    Google Scholar 

  6. Berezovski A, Engelbrecht J, Maugin GA (2011) Generalized thermomechanics with dual internal variables. Arch Appl Mech 81(2):229–240

    Google Scholar 

  7. Berezovski A, Engelbrecht J, Maugin GA (2011) Thermoelasticity with dual internal variables. J Thermal Stress 34(5–6):413–430

    Google Scholar 

  8. Berezovski A, Engelbrecht J, Salupere A, Tamm K, Peets T, Berezovski M (2013) Dispersive waves in microstructured solids. Int J Solids Struct 50(11):1981–1990

    Article  Google Scholar 

  9. Capriz G (1989) Continua with microstructure. Springer, Berlin

    Google Scholar 

  10. Cardona J, Forest S, Sievert R (1999) Towards a theory of second grade thermoelasticity. Extracta Math 14(2):127–140

    MathSciNet  MATH  Google Scholar 

  11. Carlson DE (1972) Linear thermoelasticity. Handbuch der Physik. Springer, Berlin, pp 297–345

    Google Scholar 

  12. Chandrasekharaiah D (1998) Hyperbolic thermoelasticity: a review of recent literature. Appl Mech Rev 51(12):705–729

    Google Scholar 

  13. Chen Y, Lee JD (2003) Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables. Phys A: Stat Mech Appl 322:359–376

    Article  MATH  Google Scholar 

  14. Chen Y, Lee JD, Eskandarian A (2003) Examining the physical foundation of continuum theories from the viewpoint of phonon dispersion relation. Int J Eng Sci 41(1):61–83

    Article  MathSciNet  MATH  Google Scholar 

  15. Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys 47(2):597–613

    Article  Google Scholar 

  16. Dell’Isola F, Gavrilyuk S (2012) Variational models and methods in solid and fluid mechanics. Springer Science & Business Media, Berlin

    Book  Google Scholar 

  17. Dell’Isola F, Rosa L, Woźniak C (1998) A micro-structured continuum modelling compacting fluid-saturated grounds: the effects of pore-size scale parameter. Acta Mech 127(1–4):165–182

    Article  MathSciNet  MATH  Google Scholar 

  18. Eringen AC (1999) Microcontinuum field theories: I. Foundations and solids. Springer, Berlin

    Book  MATH  Google Scholar 

  19. Eringen AC, Suhubi ES (1964) Nonlinear theory of simple micro-elastic solids -I. Int J Eng Sci 2(2):189–203

    Article  MathSciNet  MATH  Google Scholar 

  20. Fish J, Chen W (2001) Higher-order homogenization of initial/boundary-value problem. J Eng Mech 127(12):1223–1230

    Article  Google Scholar 

  21. Fish J, Filonova V, Kuznetsov S (2012) Micro-inertia effects in nonlinear heterogeneous media. Int J Numer Methods Eng 91(13):1406–1426

    Article  MathSciNet  Google Scholar 

  22. Forest S (2013) Micromorphic media. Generalized continua from the theory to engineering applications. Springer, Berlin, pp 249–300

    Google Scholar 

  23. Forest S, Amestoy M (2008) Hypertemperature in thermoelastic solids. Comptes Rendus Mécanique 336(4):347–353

    Article  MATH  Google Scholar 

  24. Geers MG, Kouznetsova VG, Brekelmans W (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175–2182

    Article  MATH  Google Scholar 

  25. Grot RA (1969) Thermodynamics of a continuum with microstructure. Int J Eng Sci 7(8):801–814

    Article  MATH  Google Scholar 

  26. Gyarmati I (1970) Nonequilibrium thermodynamics (field theory and variational principles). Springer, Berlin

    Google Scholar 

  27. Hetnarski RB, Eslami MR, Gladwell G (2009) Thermal stresses: advanced theory and applications, vol 41. Springer, Berlin

    Google Scholar 

  28. Ignaczak J, Ostoja-Starzewski M (2009) Thermoelasticity with finite wave speeds. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

  29. Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61(1):41–73

    Article  MathSciNet  MATH  Google Scholar 

  30. Kouznetsova V, Geers M, Brekelmans W (2004) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 193(48):5525–5550

    Article  MATH  Google Scholar 

  31. Mariano PM (2001) Multifield theories in mechanics of solids. Adv Appl Mech 38:1–93

    Article  Google Scholar 

  32. Mariano PM, Stazi FL (2005) Computational aspects of the mechanics of complex materials. Arch Comput Methods Eng 12(4):391–478

    Article  MathSciNet  MATH  Google Scholar 

  33. Matolcsi T, Ván P, Verhás J (2005) Fundamental problems of variational principles: objectivity, symmetries and construction. Variational and extremum principles in macroscopic problems. Elsevier, Amsterdam, pp 57–74

    Google Scholar 

  34. Maugin GA (1990) Internal variables and dissipative structures. J Non-Equilib Thermodyn 15(2):173–192

    Article  Google Scholar 

  35. Maugin GA (1993) Material inhomogeneities in elasticity. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  36. Maugin GA (2006) On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Arch Appl Mech 75(10–12):723–738

    Article  MATH  Google Scholar 

  37. Maugin GA, Metrikine AV (2010) Mechanics of generalized continua: one hundred years after the Cosserats. Springer, Berlin

    Book  MATH  Google Scholar 

  38. Maugin GA, Muschik W (1994) Thermodynamics with internal variables. Part I. General concepts. J Non Equilib Thermodyn 19:217–249

    MATH  Google Scholar 

  39. Mindlin RD (1964) Micro-structure in linear elasticity. Arch Rational Mech Anal 16(1):51–78

    Article  MathSciNet  MATH  Google Scholar 

  40. Nemat-Nasser S, Hori M (1993) Micromechanics: overall properties of heterogeneous materials. Elsevier, Amsterdam

    MATH  Google Scholar 

  41. Nowacki W (1986) Thermoelasticity. Pergamon, New York

    MATH  Google Scholar 

  42. Özdemir I, Brekelmans W, Geers MG (2008) FE2 computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput Methods Appl Mech Eng 198(3):602–613

    Article  MATH  Google Scholar 

  43. Parnell WJ (2006) Coupled thermoelasticity in a composite half-space. J Eng Math 56(1):1–21

    Article  MathSciNet  MATH  Google Scholar 

  44. Pindera MJ, Khatam H, Drago AS, Bansal Y (2009) Micromechanics of spatially uniform heterogeneous media: a critical review and emerging approaches. Compos Part B: Eng 40(5):349–378

    Article  Google Scholar 

  45. Rice JR (1971) Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J Mech Phys Solids 19(6):433–455

    Article  MATH  Google Scholar 

  46. Straughan B (2011) Heat waves. Springer, New York

    Book  MATH  Google Scholar 

  47. Suhubi ES (1975) Thermoelastic solids. Continuum physics, vol 2. Academic Press, San Diego, pp 173–265

    Google Scholar 

  48. Tamma KK, Zhou X (1998) Macroscale and microscale thermal transport and thermo-mechanical interactions: some noteworthy perspectives. J Thermal Stress 21(3–4):405–449

    Google Scholar 

  49. Tzou DY (2014) Macro-to microscale heat transfer: the lagging behavior. Wiley, New York

    Book  Google Scholar 

  50. Ván P (2005) Exploiting the second law in weakly non-local continuum physics. Period Polytech Mech Eng 49(1):79–94

    MathSciNet  Google Scholar 

  51. Ván P (2009) Weakly nonlocal non-equilibrium thermodynamics–variational principles and second law. Applied wave mathematics. Springer, Berlin, pp 153–186

    Google Scholar 

  52. Ván P, Berezovski A, Engelbrecht J (2008) Internal variables and dynamic degrees of freedom. J Non-Equilib Thermodyn 33(3):235–254

    Article  MATH  Google Scholar 

  53. Ván P, Berezovski A, Papenfuss C (2014) Thermodynamic approach to generalized continua. Contin Mech Thermodyn 26(3):403–420

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This chapter is derived in part from the article published in Arch. Appl. Mech. (2014) 84:1249–1261. Copyright\(\copyright \) Springer-Verlag, available online: https://link.springer.com/article/10.1007/s00419-014-0858-6

Author information

Authors and Affiliations

  1. Department of Cybernetics, School of Science, Tallinn University of Technology, Tallinn, Estonia

    Arkadi Berezovski

  2. Institute of Particle and Nuclear Physics, MTA WIGNER Research Centre for Physics, Budapest, Hungary

    Peter Ván

  3. Department of Energy Engineering, Montavid Thermodynamic Research Group, Budapest University of Technology and Economics, Budapest, Hungary

    Peter Ván

Authors
  1. Arkadi Berezovski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Ván
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arkadi Berezovski .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Berezovski, A., Ván, P. (2017). Microdeformation and Microtemperature. In: Internal Variables in Thermoelasticity. Solid Mechanics and Its Applications, vol 243. Springer, Cham. https://doi.org/10.1007/978-3-319-56934-5_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-56934-5_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-56933-8

  • Online ISBN: 978-3-319-56934-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation