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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on April 16, 2008
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbn009
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© The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Asymptotic Results For Bifurcations In Pure Bending Of Rubber Blocks

Ciprian D. Coman{dagger}

( Department of Mathematics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, United Kingdom )

Michel Destrade

( Institut Jean Le Rond d'Alembert, Centre National de la Recherche Scientifique (CNRS)/Université Pierré et Marie Curie, Case 162, 75252 Paris Cedex 05, France )

{dagger} cdc{at}maths.gla.ac.uk

Received 28 November 2007. Revise 22 February 2008. Accepted 7 March 2008.


  Abstract

The bifurcation of an incompressible neo-Hookean thick block with a ratio of thickness to length {eta}, subject to pure bending, is considered. The two incremental equilibrium equations corresponding to a nonlinear pre-buckling state of strain are reduced to a fourth-order linear eigenproblem that displays a multiple turning point. It is found that for 0 < {eta} < {infty}, the block experiences an Euler-type buckling instability which in the limit {eta} -> {infty} degenerates into a surface instability. Singular perturbation methods enable us to capture this transition, while direct numerical simulations corroborate the analytical results.


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