Abstract
We study the swelling of a gel annulus attached to a rigid core when it is immersed in a solvent. For equilibrium states, the free-energy function of the gel can be converted into a strain energy function, and as a result the gel can be treated as a compressible hyperelastic material. Asymptotic methods are used to study the inhomogeneous swelling in order to obtain the leading-order solution. Some analytical insights are then deduced. Because of the compressive hoop stress in this state, at some stage instability can occur, leading to wrinkles in the gel. An incremental deformation theory in nonlinear elasticity is used to conduct a linear bifurcation analysis for understanding such instability. More specifically, the critical loading for the onset of a wrinkled state is obtained. Detailed discussions on the behaviors of various physical quantities in this critical state are given. It is found that the critical mode number, while insensitive to the material parameters, is greatly influenced by the ratio of the inner and outer radii of the gel. Also, an interesting finding is that the critical swelling ratio is an increasing function of this geometrical parameter, which implies that a thin annulus is more likely to be unstable than a thick one.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Eddington, T.D., Beebe, J.D.: Flow control with hydrogels. Adv. Drug Deliv. Rev. 56, 199–210 (2004)
Sun, J.-Y., Keplinger, C., Whitesides, G.M., et al.: Ionic skin. Adv. Mater. (2014). doi:10.1002/adma.201403441
Kuhn, W., Hargitay, B.: Reversible dilation and contraction by changing the state of ionization of high-polymer acid networks. Nature 165, 514–516 (1950)
Zwieniecki, M.A., Melcher, P.J., Holbrook, N.M.: Hydrogel controls of xylem hydraulic reisstence in plants. Science 291, 1059–1062 (2001)
Beebe, D.J., Moore, J., Bauer, J., et al.: Functional hydrogel structures for autonomous flow control inside microfluidic channels. Nature 404, 588–590 (2000)
Bush, A.W.: Perturbation Methods for Engineers and Scientists. CRC Press, Boca Raton (1992)
Holmes, M.H.: Introduction to Perturbation Methods. Springer, New York (1995)
Zhao, X., Hong, W., Suo, Z.: Inhomogeneous and anisotropic equilibrium state of a swollen hydrogel containing a hard core. Appl. Phys. Lett. 92, 051904 (2008)
Hong, W., Liu, Z., Suo, Z.: Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load. Int. J. Solids Struct. 46, 3282–3289 (2009)
Flory, P.J., Rehner Jr, J.: Statistical mechanics of cross-linked polymer networks II. Swelling. J. Chem. Phys. 11, 521 (1943)
Dai, H.-H., Song, Z.: Some analytical formulas for the equilibrium states of a swollen hydrogel shell. Soft Matter 7, 8473 (2011)
Chen, X., Dai, H.-H.: Asymptotic solutions and new insights for cylinder and core-shell polymer gel in a solvent. Soft Matter 9, 8664 (2013)
Wu, Z., Zhong, Z.: Inhomogeneous equilibrium swelling of core-shell-coating gels. Soft Mater. 11, 215–220 (2013)
Roxburgh, D.G., Ogden, R.W.: Stability and vibration of pre-stressed compressible elastic plates. Int. J. Eng. Sci. 32, 427–454 (1994)
Huang, R., Suo, Z.: Wrinkling of a compressible elastic film on a viscous layer. J. Appl. Phys. 91, 1135 (2002)
Timoshenko, S., Woinowsky-Krieger, S.: Theory of plates and shells, 2nd edn. McGraw-Hill, New York (1987)
Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, pp. 57–60. Pergamon, London (1959)
Destrade, M., Murphy, J.G., Ogden, R.W.: On deforming a sector of a circular cylindrical tube into an intact tube: existence, uniqueness, and stability. http://arxiv.org/abs/1301.5117 (2013)
Kang, M.K., Huang, R.: Swell-induced surface instability of confined hydrogel layers on substrates. J. Mech. Phys. Solids 58, 1582–1598 (2010)
Xiao, Z., Li, M., Zhou, J.: Surface instability of a swollen cylinder hydrogel. Acta Mech. Solida Sin. 25, 550–556 (2013)
Dervaux, J., Couder, Y., Guedeau-Boudeville, M.A., et al.: Shape transition in artificial tumors: from smooth buckles to singular creases. Phys. Rev. Lett. 107, 018103 (2011)
Pence, T.J., Tsai, H.: Swelling-induced microchannel formation in nonlinear elasticity. IMA J. Appl. Math. 70, 173–189 (2005)
Treolar, L.R.G.: The Physics of Rubber Elasticity. Oxford University Press, Oxford (1975)
Cai, S., Suo, Z.: Equations of state for ideal elastomeric gels. EPL 97, 34009 (2012)
Li, J., Hu, Y., Vlassak, J.J., et al.: Experimental determination of equations of state for ideal elastomeric gels. Soft Matter 8, 8121 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, X., Dai, HH. Swelling and instability of a gel annulus. Acta Mech. Sin. 31, 627–636 (2015). https://doi.org/10.1007/s10409-015-0496-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-015-0496-4