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An asymptotic analysis of the buckling of a highly shear-resistant vesicle

Published online by Cambridge University Press:  25 June 2009

SYLVAIN REBOUX ,
GILES RICHARDSON  and
OLIVER E. JENSEN
SYLVAIN REBOUX
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: [email protected]
GILES RICHARDSON
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: [email protected]
OLIVER E. JENSEN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: [email protected]
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Abstract

The static compression between two smooth plates of an axisymmetric capsule or vesicle is investigated by means of asymptotic analysis. The governing equations of the vesicle are derived from thin-shell theory and involve a bending stiffness B, a shear modulus H, the unstressed vesicle radius a and a constant surface-area constraint. The sixth-order free-boundary problem obtained by a balance-of-forces approach is addressed in the limit when the dimensionless parameter C = Ha2/B is large and the plate displacements are small. When the plate displacement is of order aC−1/2, the vesicle undergoes a sub-critical buckling instability which is captured by leading-order asymptotics. Asymptotic linear and quadratic force–displacement relations for the pre- and post-buckled solutions are determined. The leading-order post-buckled solution is described by a simple fourth-order problem, exhibiting stress-focusing with stretching and bending confined to a narrow boundary layer. In contrast, in the pre-buckled state, stretching occurs over a larger length scale than bending. The results are in good qualitative agreement with numerical simulations for finite values of C.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 20 , Issue 6 , December 2009 , pp. 479 - 518
Copyright
Copyright © Cambridge University Press 2009

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References

Blyth, M. G. & Pozrikidis, C. (2004) Solution space of axisymmetric capsules enclosed by elastic membranes. Eur. J. Mech. – A/Solids 23, 877–892.CrossRefGoogle Scholar
Canham, P. B. (1970) The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26, 61–81.CrossRefGoogle ScholarPubMed
Evans, E. A. & Skalak, R. (1980) Mechanics and Thermodynamics of Biomembranes, CRC Press. Boca Raton, Florida, 97–112.Google Scholar
Evkin, A. Y. & Kalamkarov, A. L. (2001) Analysis of large deflection equilibrium states of composite shells of revolution. Part 1. General model and singular perturbation analysis. Int. J. Solids Struct. 38, 8961–8974.CrossRefGoogle Scholar
Fery, A. & Weinkamer, R. (2007) Mechanical properties of micro- and nanocapsules: Single-capsule measurements. Polymer 48, 7221–7235.CrossRefGoogle Scholar
Graff, M., Scheidl, R., Troger, H. & Weinmuller, E. (1985) An investigation of the complete postbuckling behavior of axisymmetrical spherical-shells. ZAMP 36, 803–821.Google Scholar
Gregory, R. D., Milac, T. I. & Wan, F. Y. M. (1998) A thick hollow sphere compressed by equal and opposite concentrated axial loads: An asymptotic solution. SIAM J. Appl. Math. 59, 1080–1097.Google Scholar
Gupta, N. K., Sheriff, N. M. & Velmurugan, R. (2008) Experimental and theoretical studies on buckling of thin spherical shells under axial loads. Int. J. Mech. Sci. 50, 422–432.CrossRefGoogle Scholar
Helfer, E., Harlepp, S., Bourdieu, L., Robert, J., MacKintosh, F. C. & Chatenay, D. (2001) Buckling of actin-coated membranes under application of a local force. Phys. Rev. Lett. 87, 088103.CrossRefGoogle ScholarPubMed
Huang, Y. Q., Doerschuk, C. M. & Kamm, R. D. (2001) Computational modeling of RBC and neutrophil transit through the pulmonary capillaries. J. Appl. Physiol. 90, 545–564.CrossRefGoogle ScholarPubMed
Komura, S., Tamura, K. & Kato, T. (2005) Buckling of spherical shells adhering onto a rigid substrate. Eur. Phys. J. E – Soft Matter 18, 343–358.CrossRefGoogle ScholarPubMed
Kriegsmann, G. A. & Lange, C. G. (1980) On large axisymmetrical deflection states of spherical shells. J. Elast. 10, 179–192.CrossRefGoogle Scholar
Kumar, M. N. R. (2000) Nano and microparticles as controlled drug delivery devices. J. Pharm. Sci. 3, 234–258.Google Scholar
Landau, L. D. & Lifshitz, E. M. (1986) Theory of Elasticity, 3rd ed., Pergamon, New York7, 54–57.Google Scholar
Lidmar, J., Mirny, L. & Nelson, D. R. (2003) Virus shapes and buckling transitions in spherical shells. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 68, 051910.CrossRefGoogle ScholarPubMed
Lobkovsky, A. E. & Witten, T. A. (1997) Properties of ridges in elastic membranes. Phys. Rev. E 55, 1577–1589.CrossRefGoogle Scholar
Monllor, P., Bonet, M. A. & Cases, F. (2007) Characterization of the behaviour of flavour microcapsules in cotton fabrics. Eur. Polym. J. 43, 2481–2490.CrossRefGoogle Scholar
Murray, F. J. & Wright, F. W. (1961) The buckling of thin spherical shells. J. Aerosp. Sci. 28, 223–236.CrossRefGoogle Scholar
Noguchi, H. & Gompper, G. (2005) Shape transitions of fluid vesicles and red blood cells in capillary flows. Proc. Natl. Acad. Sci. USA 102, 14159–14164.CrossRefGoogle ScholarPubMed
Pamplona, D. C. & Calladine, C. R. (1993) The mechanics of axially symmetric liposomes. J. Biomech. Eng. 115, 149–159.CrossRefGoogle ScholarPubMed
Pamplona, D. C., Greenwood, J. A. & Calladine, C. R. (2005) The buckling of spherical liposomes. J. Biomech. Eng. 127, 1062–1069.CrossRefGoogle ScholarPubMed
Parker, D. F. & Wan, F. Y. M. (1984) Finite polar dimpling of shallow caps under sub-buckling axisymmetric pressure distributions. SIAM J. Appl. Math. 44, 301–326.CrossRefGoogle Scholar
Parker, K. H. & Winlove, C. P. (1999) The deformation of spherical vesicles with permeable, constant-area membranes: Application to the red blood cell. Biophys. J. 77, 3096–107.CrossRefGoogle Scholar
Pauchard, L. & Rica, S. (1998) Contact and compression of elastic spherical shells: The physics of a ping-pong ball. Phil. Mag. B 78 (9), 225–233.CrossRefGoogle Scholar
Pogoroelov, A. V. (1986) Bendings of Surfaces and Stability of Shells. Translations of Mathematical Monographs Vol. 72, American Mathematical Society, Providence, RI.Google Scholar
Pozrikidis, C. (2003) Shell theory for capsules and cells. Ch. 2 (pages 35–101) of Modelling and Simulation of Capsules and Biological Cells, ed. Pozrikidis, C., CRC Press, Boca Raton, Florida.CrossRefGoogle Scholar
Preston, S. P., Jensen, O. E. & Richardson, G. (2008) Buckling of an axisymmetric vesicle under compression: The effects of resistance to shear. Q. J. Mech. Appl. Math. 61, 1–24.CrossRefGoogle Scholar
Risso, F. & Carin, M. (1987) Compression of a capsule: Mechanical laws of membranes with negligible bending stiffness. Phys. Rev. E 69, 061601.CrossRefGoogle Scholar
Scheidl, R. & Troger, H. (1987) A comparison of the postbuckling behavior of plates and shells. Comput. Struct. 27, 157 – 163.CrossRefGoogle Scholar
Schrooyen, P. M., van der Meer, R. & Kruif, C. G. D. (2001) Microencapsulation: Its application in nutrition. Proc. Nutr. Soc. 60, 475–479.CrossRefGoogle ScholarPubMed
Schwarz, U. (2007) Soft matters in cell adhesion: Rigidity sensing on soft elastic substrates. Soft Matter 3, 263–266.CrossRefGoogle ScholarPubMed
Smith, A. E., Moxham, K. E. & Middelberg, A. P. J. (1998) On uniquely determining cell-wall material properties with the compression experiment. Chem. Eng. Sci. 53, 3913–3922.CrossRefGoogle Scholar
Vaziri, A. & Mahadevan, L. (2008) Localized and extended deformations of elastic shells. Proc. Natl. Acad. Sci. 105, 7913–7918.CrossRefGoogle ScholarPubMed
Wan, F. Y. M. (1980) The dimpling of spherical caps. Mech. Today 5, 495–508.CrossRefGoogle Scholar
Wan, K., Chan, V. & Dillard, D. A. (2003) Constitutive equation for elastic indentation of a thin-walled bio-mimetic microcapsule by an atomic force microscope tip. Colloids Surf. B: Biointerf. 27, 241–248.CrossRefGoogle Scholar
Witten, T. A. (2007) Stress focusing in elastic sheets. Rev. Mod. Phys. 79, 643–33.CrossRefGoogle Scholar
Zarda, P. R., Chien, S. & Skalak, R. (1977) Elastic deformations of red blood cells. J. Biomech. 10, 211–21.CrossRefGoogle ScholarPubMed