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Journal of Elasticity

, Volume 123, Issue 1, pp 59–84 | Cite as

Diffusion in Mixtures of Reacting Thermoelastic Solids

  • A. Morro
Article

Abstract

A chemically reacting mixture of elastic solids is considered. As a constitutive assumption, the peculiar functions (such as the free energy, the entropy, and the stress) of a constituent are taken to be functions of a set of variables pertaining to that constituent. The interaction terms, namely the growth of mass, linear momentum, and energy, are allowed to depend on the set of variables pertaining to all of the constituents. While the dependence on the mass density is usually disregarded, the paper accounts also for such a dependence, which seems to be in order especially in connection with reacting mixtures where the mass densities change also in the reference configuration. The thermodynamic restrictions are derived by starting from the non-negative value of the sum of entropy growths and involving the properties of the peculiar functions. The results so obtained for stresses and chemical potentials are examined in connection with similar schemes (swelling solids). While the correct relations for the mass diffusion flux arise from balance equations, an analysis is given of whether and how Fick-type models are acceptable possibly depending on the fluid or solid character of the mixture.

Keywords

Thermoelastic solids Reactive solids Diffusion fluxes 

Mathematics Subject Classification (2010)

74A30 74A65 74F20 

Notes

Acknowledgements

The research leading to this paper has been developed under the auspices of INDAM (Italy). The author is grateful to an anonymous reviewer for helpful comments on an earlier version.

References

  1. 1.
    Araujo, R.P., McElwain, D.L.S.: A mixture theory for the genesis of residual stresses in growing tissues I: a general formulation. SIAM J. Appl. Math. 65, 1261–1284 (2005) CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Baek, S., Srinivasa, A.R.: Diffusion of a fluid through an elastic solid undergoing large deformation. Int. J. Non-Linear Mech. 39, 201–218 (2004) CrossRefzbMATHGoogle Scholar
  3. 3.
    Bi, Z., Sekerka, R.F.: Phase-field model of solidification of a binary alloy. Physica A 261, 95–106 (1998) CrossRefADSGoogle Scholar
  4. 4.
    Bowen, R.M.: Toward a thermodynamics and mechanics of mixtures. Arch. Ration. Mech. Anal. 24, 370–403 (1967) CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Bowen, R.M.: The thermochemistry of a reacting mixture of elastic materials with diffusion. Arch. Ration. Mech. Anal. 34, 97–110 (1969) CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bowen, R.M.: Theory of mixtures. In: Eringen, A.C. (ed.) Continuum Physics 3. Academic Press, New York (1976) Google Scholar
  7. 7.
    Bowen, R.M., Wiese, J.C.: Diffusion in mixtures of elastic materials. Int. J. Eng. Sci. 7, 689–722 (1969) CrossRefzbMATHGoogle Scholar
  8. 8.
    Buonsanti, M., Fosdick, R., Royer-Carfagni, G.: Chemomechanical equilibrium of bars. J. Elast. 84, 167–188 (2006) CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Cahn, J.C.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961) CrossRefGoogle Scholar
  10. 10.
    Carlson, D.E.: Linear thermoelasticity. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VIa/2. Springer, New York (1972) Google Scholar
  11. 11.
    De Groot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. Dover, New York (1984) Google Scholar
  12. 12.
    Echebarria, B., Folch, R., Karma, A., Plapp, M.: Quantitative phase-field model of alloy solidification. Phys. Rev. E 70, 061604 (2004) CrossRefADSGoogle Scholar
  13. 13.
    Fried, E., Gurtin, M.E.: Coherent solid-state phase transitions with atomic diffusion: a thermomechanical treatment. J. Stat. Phys. 95, 1361–1427 (1999) CrossRefADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Fried, E., Sellers, S.: Theory for atomic diffusion on fixed and deformable crystal lattices. J. Elast. 59, 67–81 (2000) CrossRefzbMATHGoogle Scholar
  15. 15.
    Gandhi, M.V., Rajagopal, K.R., Wineman, A.S.: Some nonlinear diffusion problems within the context of the theory of interacting continua. Int. J. Eng. Sci. 25, 1441–1457 (1987) CrossRefzbMATHGoogle Scholar
  16. 16.
    Green, A.E., Naghdi, P.M.: On basic equations for mixtures. Q. J. Mech. Appl. Math. 22, 427–438 (1969) CrossRefzbMATHGoogle Scholar
  17. 17.
    Gurtin, M.E.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92, 178–192 (1992) ADSMathSciNetGoogle Scholar
  18. 18.
    Heida, M., Málek, J., Rajagopal, K.R.: On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework. Z. Angew. Math. Phys. 63, 145–169 (2012) CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular Theory of Gases and Liquids. Wiley, New York (1954) zbMATHGoogle Scholar
  20. 20.
    Iesan, D.: On the theory of mixtures of elastic solids. J. Elast. 35, 251–268 (1994) CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Jabbour, M.E., Bhattacharya, K.: A continuum theory of multispecies thin solid film growth by chemical vapor deposition. J. Elast. 73, 13–74 (2003) CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Klisch, S.M.: A mixture of elastic materials with different constituent temperatures and internal constraints. Int. J. Eng. Sci. 40, 805–828 (2002) CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. Ser. A 454, 2617–2654 (1998) CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Mielke, A.: Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete Contin. Dyn. Syst., Ser. S 6, 479–499 (2013) CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Morro, A.: Phase-field models for fluid mixtures. Math. Comput. Model. 45, 1042–1052 (2007) CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Morro, A.: Governing equations in non-isothermal diffusion. Int. J. Non-Linear Mech. 55, 90–97 (2013) CrossRefADSGoogle Scholar
  27. 27.
    Morro, A.: Balance and constitutive equations for diffusion in mixtures of fluids. Meccanica 49, 2109–2123 (2014) CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Müller, I.: A thermodynamic theory of mixtures of fluids. Arch. Ration. Mech. Anal. 28, 1–39 (1968) CrossRefzbMATHGoogle Scholar
  29. 29.
    Müller, I.: Thermodynamics of mixtures of fluids. J. Méc. 14, 267–303 (1975) zbMATHGoogle Scholar
  30. 30.
    Müller, I.: Thermodynamics of mixtures and phase field theory. Int. J. Solids Struct. 38, 1105–1113 (2001) CrossRefzbMATHGoogle Scholar
  31. 31.
    Passman, S.L., Nunziato, J.W.: A theory of multiphase mixtures. In: Truesdell, C. (ed.) Rational Thermodynamics. Springer, New-York (1984) Google Scholar
  32. 32.
    Rajagopal, K.R., Tao, L.: Mechanics of Mixtures. World Scientific, Singapore (1996) Google Scholar
  33. 33.
    Sekerka, R.F.: Similarity solutions for a binary diffusion couple with diffusivity and density dependent on composition. Prog. Mater. Sci. 49, 511–536 (2004) CrossRefGoogle Scholar
  34. 34.
    Shi, J.J.-J., Rajagopal, K.R., Wineman, A.S.: Applications of the theory of interacting continua to the diffusion of a fluid through a non-linear elastic media. Int. J. Eng. Sci. 19, 871–889 (1981) CrossRefzbMATHGoogle Scholar
  35. 35.
    Truesdell, C.: Rational Thermodynamics. Springer, New York (1984), Chap. 5 CrossRefzbMATHGoogle Scholar
  36. 36.
    Truesdell, C., Noll, W.: The non-linear field theory of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/3, pp. 17–19. Springer, Berlin (1965) Google Scholar
  37. 37.
    Volokh, K.Y.: Stresses in growing soft tissues. Acta Biomater. 2, 493–504 (2006) CrossRefGoogle Scholar
  38. 38.
    Wheeler, A.A., Boettinger, W.J., McFadden, G.B.: Phase-field model for isothermal phase transitions in binary alloys. Phys. Rev. A 45, 7424–7439 (1992) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.DIBRISGenovaItaly

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