Abstract
This chapter introduces a geometrically nonlinear, continuum thermomechanical framework for case II diffusion: a type of non-Fickian diffusion characterized by the wave-like propagation of a low-molecular weight solvent in a polymeric solid. The key objective of this contribution is to derive the coupled system of governing equations describing case II diffusion from fundamental balance principles. A general form for the Helmholtz energy is proposed and the resulting constitutive laws are derived from logical, thermodynamically consistent argumentation. The chapter concludes by comparing the model developed here to various others in the literature. The approach adopted to derive the governing equations is not specific to case II diffusion, rather it encompasses a wide range of applications wherein heat conduction, species diffusion and finite inelastic effects are coupled. The presentation is thus applicable to the generality of models for non-Fickian diffusion: an area of increasing research interest.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aifantis, E.C.: On the problem of diffusion in solids. Acta Mechanica 37, 265–296 (1980)
Alfrey, T., Gurnee, E.F., Lloyd, W.G.: Diffusion in glassy polymers. Journal of Polymer Science Part C 12, 249–261 (1966)
Bargmann, S., McBride, A.T., Steinmann, P.: Models of solvent penetration in glassy polymers with an emphasis on case II diffusion. A comparative review. Applied Mechanics Reviews, doi:10.1115/1.4003955 (in press)
Cohen, D.S.: Theoretical models for diffusion in glassy polymers. Journal of Polymer Science: Polymer Physics Edition 2(22), 1001–1009 (1984)
Cohen, D.S., White, A.B.: Sharp fronts due to diffusion and stress at the glass transition in polymers. Journal of Polymer Science: Part B: Polymer Physics 27, 1731–1747 (1989)
Cottrell, A.H.: Effect of solute atoms on the behavior of dislocations. In: Wills, H.H. (ed.) Report of a Conference on Strength of Solids, pp. 30–36. The Physical society, University of Bristol, London (1948)
El Afif, A., Grmela, M.: Non-Fickian mass transport in polymers. Journal of Rheology 46(3), 591–628 (2002)
Fick, A.: Über Diffusion. Poggendorff’s Annalen der Physik und Chemie 94, 58–86 (1855)
Flory, P.J.: Thermodynamics of high polymer solutions. Journal of Chemical Physics 9(8), 66–661 (1941)
Govindjee, S., Simo, J.C.: Coupled stress–diffusion: Case II. Journal of the Mechanics and Physics of Solids 41(5), 863–887 (1993)
Holzapfel, G.A., Simo, J.C.: Entropy elasticity of isotropic rubber-like solids at finite strains. Computer Methods in Applied Mechanics and Engineering 132(1-2), 17–44 (1996)
Holzapfel, G.A., Simo, J.C.: A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. International Journal of Solids and Structures 33(20-22), 3019–3034 (1996)
Huggins, M.L.: Solutions of long chain compounds. Journal of Chemical Physics 9, 440 (1941)
Javili, A., Steinmann, P.: On thermomechanical solids with boundary structures. International Journal of Solids and Structures 47(24), 3245–3253 (2010)
Kalospiros, N.S., Ocone, R., Astarita, G., Meldon, J.H.: Analysis of anomalous diffusion and relaxation in solid polymers. Industrial & Engineering Chemistry Research 30, 851–864 (1991)
Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice–Hall Inc., Englewood Cliffs (1969)
Maxwell, J.C.: On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London 157, 49–88 (1867)
McBride, A.T., Javili, A., Steinmann, P., Bargmann, S.: Geometrically nonlinear continuum thermomechanics with surface energies coupled to diffusion (2011) (review)
Simo, J.C., Hughes, T.J.R.: Computational inelasticity. In: Marsden, J.E., Wiggins, S., Sirovich, L. (eds.) Interdisciplinary Applied Mathematics, vol. 7. Springer, Heidelberg (1998)
Steinmann, P.: On boundary potential energies in deformational and configurational mechanics. Journal of the Mechanics and Physics of Solids 56, 772–800 (2008)
Steinmann, P., McBride, A., Bargmann, S., Javili, A.: A deformational and configurational framework for geometrically nonlinear continuum thermomechanics coupled to diffusion. International Journal of Non-Linear Mechanics, doi:10.1016/j.ijnonlinmec.2011.05.009 (in press)
Thomas, N.L., Windle, A.H.: A deformation model for case ii diffusion. Polymer 21(6), 613–619 (1980)
Thomas, N.L., Windle, A.H.: Diffusion mechanics of the system pmma-methanol. Polymer 22(5), 627–639 (1981)
Thomas, N.L., Windle, A.H.: A theory of case II diffusion. Polymer 23(4), 529–542 (1982)
Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Springer, New York (2004)
Vesely, D.: Diffusion of liquids in polymers. International Materials Reviews 53(5), 299–315 (2008)
Vijalapura, P.K., Govindjee, S.: Numerical simulation of coupled-stress case II diffusion in one dimension. Journal of Polymer Science: Part B: Polymer Physics 41, 2091–2108 (2003)
Vijalapura, P.K., Govindjee, S.: An adaptive hybrid time-stepping scheme for highly non-linear strongly coupled problems. International Journal for Numerical Methods In Engineering 64, 819–848 (2005)
Wu, J.C., Peppas, N.A.: Modeling of penetrant diffusion in glassy polymers with an integral sorption Deborah number. Journal of Polymer Science: Part B: Polymer Physics 31, 1503–1518 (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
McBride, A.T., Bargmann, S., Steinmann, P. (2011). Geometrically Nonlinear Continuum Thermomechanics Coupled to Diffusion: A Framework for Case II Diffusion. In: Markert, B. (eds) Advances in Extended and Multifield Theories for Continua. Lecture Notes in Applied and Computational Mechanics, vol 59. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22738-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-22738-7_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22737-0
Online ISBN: 978-3-642-22738-7
eBook Packages: EngineeringEngineering (R0)