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On the thermomechanics of continuous media with diffusion and/or weak nonlocality

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Abstract

Working in parallel on the energy equation in a special form and the associated canonical equation of momentum, we focus attention on the case of deformable media which are basically finitely elastic, but which also admit the existence of thermodynamicallyirreversible phenomena by means of a diffusive internal variable of state, or alternately an additional degree of freedom, in anycase presenting some weak nonlocality (gradient effects). Two descriptions follow thereof, one that can be called standard according to rational thermomechanics (there exists a generalized internal force or thermodynamically conjugated force for a variable and its gradient separately, and the entropy flux has its classical definition) or the so-called field-theoretic viewpoint in which only one generalized force (based on a variational derivative of the energy) is used. In the latter one, the entropy flux deviates form its classical definition but, simultaneously, by virtue of the space–time consistency, the Eshelby stress tensor has to be altered. Simple examples with diffusion of an internal variable illustrate these formulations that may be valuable in the description of some bio- and geomaterials.

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References

  1. Bowen R.M. (1967) Towards a thermodynamics and mechanics of mixtures. Arch. Rat. Mech. Anal. 24, 370–403

    Article  MATH  MathSciNet  Google Scholar 

  2. Bowen, R.M.: Theory of Mixtures, in: Eringen A.C., Continuum Physics, Vol. III, pp.1–127. Academic, New York (1976)

  3. Bui H.D. (1978) Mécanique de la Rupture Fragile. Masson, Paris

    Google Scholar 

  4. de Groot S.R., Mazur P. (1962) Non-equilibrium Thermodynamics. North-Holland, Amsterdam

    Google Scholar 

  5. Epstein M. Maugin G.A. (1995) Thermoelastic material forces: definition and geometric aspects. C. R. Acad. Sci. Paris II-320, 63–68

    Google Scholar 

  6. Epstein M. Maugin G.A. (2000) Thermomechanics of volumetric growth in uniform bodies. Int. J. Plasticity 16, 951–978

    Article  Google Scholar 

  7. Forest, S., Cordona, J.-M.: Thermoelasticity of second-grade materials, In: Maugin, G.A., Drouot, R. Sidoroff F., (eds.) 163–176, Kluwer, Continuum Thermoechanics: The Art and Science of Modeling Materials’ Behaviour Dordrecht (2000)

  8. Forest S. Sievert R.(2003) Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160, 71–111

    Article  Google Scholar 

  9. Frémond M., Nedjar B., (1993) Endommagement et principe des puissances virtuelles. C. R. Acad. Sci. Paris. 317-II, 857–864

    MATH  Google Scholar 

  10. Frémond M., Nedjar B. (1996) Damage, Gradient of damage and principle of virtual power. Int. J. Solids Struct. 33, 1083–1103

    Article  MATH  Google Scholar 

  11. Germain P. (1973a) La méthode des puissances virtuelles en mécanique des milieux continus- I: Théorie du second gradient. J. Mécanique (Paris) 12, 135–274

    Google Scholar 

  12. Germain P. (1973b) The method of virtual power in continuum mechanics-II: microstructure. SIAM J. Appl. Math. 25, 556–575

    Article  MATH  Google Scholar 

  13. Goldstein H. (1950) Classical Mechanics. Addison–Wesley, Reading

    MATH  Google Scholar 

  14. Grinfeld M.A. (1991) Thermodynamic Methods in the Theory of Heterogeneous Systems. Longman, Harlow

    MATH  Google Scholar 

  15. Ireman P., Nguyen Q.-S. (2004) Using the gradients of the temperature and internal parameters in continuum thermodynamics. C. R. Mécanique 333, 249–255

    Article  Google Scholar 

  16. Kats E.I., Lebedev V.V. Dynamics of liquid crystals (in Russian) Nauka, Moscow (1988)

  17. Leslie F.M. (1968) Constitutive equations for liquid crystals. Arch. Rat. Mech. Anal. 28, 265–283

    Article  MATH  MathSciNet  Google Scholar 

  18. Lorentz E., Andrieux S. (1999) A variational formulation of nonlocal damage models. Int. J. Plasticity 15, 119–198

    Article  MATH  Google Scholar 

  19. Maugin G.A. (1972) Remarks on dissipative processes in the continuum theory of micromagnetics. J. Phys. (UK) A5, 1550–1562

    Google Scholar 

  20. Maugin G.A. (1974a) Sur la dynamique des milieux déformables avec spin magnétique – Théorie classique J. Mécanique (Paris), 13, 75–96

    MathSciNet  MATH  Google Scholar 

  21. Maugin G.A. (1974b) Quasi-electrostatics of electrically polarized continua. Lett. Appl. Eng. Sci. 2, 293–306

    Google Scholar 

  22. Maugin G.A. (1976) On the foundations of the electrodynamics of deformable media with interactions. Lett. Appl. Eng. Sci. 4, 3–17

    Google Scholar 

  23. Maugin G.A. (1977) Deformable dielectrics II – Voigt’s intramolecular force balance in elastic dielectrics. Arch. Mech. 29, 143–159

    MATH  Google Scholar 

  24. Maugin G.A. (1979) Nonlocal theories or gradient-type theories: a matter of convenience? Arch. Mech. 31, 1–26

    MathSciNet  Google Scholar 

  25. Maugin G.A. (1980) The method of virtual power in continuum mechanics: application to coupled fields. Acta Mech. 35, 1–80

    Article  MATH  MathSciNet  Google Scholar 

  26. Maugin G.A. (1990) Internal variables and dissipative structures. J. Non-Eq. Thermodyn. 15, 173–192

    Article  Google Scholar 

  27. Maugin G.A. (1993) Material Inhomogeneities in Elasticity. Chapman and Hall, London

    MATH  Google Scholar 

  28. Maugin G.A. (1995) Material forces: concepts and applications. ASME Appl. Mech. Rev. 48, 213–245

    Article  MathSciNet  Google Scholar 

  29. Maugin G.A. (1996) Canonical momentum and energy in elastic systems with additional state variables. C. R. Acad. Sci. Paris IIb-323, 407–412

    MATH  Google Scholar 

  30. Maugin G.A. (1997) Thermomechanics of inhomogeneous-heterogeneous systems: application to the irreversible progress of two- and three-dimensional defects. ARI 50, 41–56

    Google Scholar 

  31. Maugin G.A. Remarks on the thermomechanics of weakly nonlocal theories In: Brillard, A., Ganghoffer, J.-F., Nonlocal Aspects in Solid Mechanics (Proc. Euromech Coll.378) pp. 2–9. Univ. Mulhouse, France (1998)

  32. Maugin G.A. (1999) The Thermomechanics of Nonlinear Irreversible Behaviors. World Scientific, Singapore

    MATH  Google Scholar 

  33. Maugin G.A. (2000) On the universality of the thermomechanics of forces driving singular sets. Arch. Appl. Mech. 70, 31–45

    Article  MATH  Google Scholar 

  34. Maugin G.A. (2003) Pseudo-plasticity and pseudo-inhomogenity effects in materials mechanics. J. Elast. (Special issue in Memory of C. A. Truesdell) 71, 81–103

    MATH  MathSciNet  Google Scholar 

  35. Truesdell C.A. (1984) Rational Thermodynamics, 2nd enlarged edition. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  36. Truskinovskii L.M. (1983) The chemical tensor. Geokhimiya 12, 1730–1744

    Google Scholar 

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  1. Laboratoire de Modélisation en Mécanique, UMR 7607 C NRS, Université Pierre et Marie Curie, Case 162, 4 place Jussieu, 75252, Paris Cedex 05, France

    Gérard A. Maugin

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Maugin, G.A. On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Arch Appl Mech 75, 723–738 (2006). https://doi.org/10.1007/s00419-006-0062-4

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  • DOI: https://doi.org/10.1007/s00419-006-0062-4

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