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.
References
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)
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)
Bi, Z., Sekerka, R.F.: Phase-field model of solidification of a binary alloy. Physica A 261, 95–106 (1998)
Bowen, R.M.: Toward a thermodynamics and mechanics of mixtures. Arch. Ration. Mech. Anal. 24, 370–403 (1967)
Bowen, R.M.: The thermochemistry of a reacting mixture of elastic materials with diffusion. Arch. Ration. Mech. Anal. 34, 97–110 (1969)
Bowen, R.M.: Theory of mixtures. In: Eringen, A.C. (ed.) Continuum Physics 3. Academic Press, New York (1976)
Bowen, R.M., Wiese, J.C.: Diffusion in mixtures of elastic materials. Int. J. Eng. Sci. 7, 689–722 (1969)
Buonsanti, M., Fosdick, R., Royer-Carfagni, G.: Chemomechanical equilibrium of bars. J. Elast. 84, 167–188 (2006)
Cahn, J.C.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)
Carlson, D.E.: Linear thermoelasticity. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VIa/2. Springer, New York (1972)
De Groot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. Dover, New York (1984)
Echebarria, B., Folch, R., Karma, A., Plapp, M.: Quantitative phase-field model of alloy solidification. Phys. Rev. E 70, 061604 (2004)
Fried, E., Gurtin, M.E.: Coherent solid-state phase transitions with atomic diffusion: a thermomechanical treatment. J. Stat. Phys. 95, 1361–1427 (1999)
Fried, E., Sellers, S.: Theory for atomic diffusion on fixed and deformable crystal lattices. J. Elast. 59, 67–81 (2000)
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)
Green, A.E., Naghdi, P.M.: On basic equations for mixtures. Q. J. Mech. Appl. Math. 22, 427–438 (1969)
Gurtin, M.E.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92, 178–192 (1992)
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)
Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular Theory of Gases and Liquids. Wiley, New York (1954)
Iesan, D.: On the theory of mixtures of elastic solids. J. Elast. 35, 251–268 (1994)
Jabbour, M.E., Bhattacharya, K.: A continuum theory of multispecies thin solid film growth by chemical vapor deposition. J. Elast. 73, 13–74 (2003)
Klisch, S.M.: A mixture of elastic materials with different constituent temperatures and internal constraints. Int. J. Eng. Sci. 40, 805–828 (2002)
Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. Lond. Ser. A 454, 2617–2654 (1998)
Mielke, A.: Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete Contin. Dyn. Syst., Ser. S 6, 479–499 (2013)
Morro, A.: Phase-field models for fluid mixtures. Math. Comput. Model. 45, 1042–1052 (2007)
Morro, A.: Governing equations in non-isothermal diffusion. Int. J. Non-Linear Mech. 55, 90–97 (2013)
Morro, A.: Balance and constitutive equations for diffusion in mixtures of fluids. Meccanica 49, 2109–2123 (2014)
Müller, I.: A thermodynamic theory of mixtures of fluids. Arch. Ration. Mech. Anal. 28, 1–39 (1968)
Müller, I.: Thermodynamics of mixtures of fluids. J. Méc. 14, 267–303 (1975)
Müller, I.: Thermodynamics of mixtures and phase field theory. Int. J. Solids Struct. 38, 1105–1113 (2001)
Passman, S.L., Nunziato, J.W.: A theory of multiphase mixtures. In: Truesdell, C. (ed.) Rational Thermodynamics. Springer, New-York (1984)
Rajagopal, K.R., Tao, L.: Mechanics of Mixtures. World Scientific, Singapore (1996)
Sekerka, R.F.: Similarity solutions for a binary diffusion couple with diffusivity and density dependent on composition. Prog. Mater. Sci. 49, 511–536 (2004)
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)
Truesdell, C.: Rational Thermodynamics. Springer, New York (1984), Chap. 5
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)
Volokh, K.Y.: Stresses in growing soft tissues. Acta Biomater. 2, 493–504 (2006)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Morro, A. Diffusion in Mixtures of Reacting Thermoelastic Solids. J Elast 123, 59–84 (2016). https://doi.org/10.1007/s10659-015-9547-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-015-9547-0