The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on July 6, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2007 60(3):319-336; doi:10.1093/qjmam/hbm009
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Singular behaviour in a generalized boundary eigenvalue problem for annular plates in tension
( Department of Mathematics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW )
( School of Mathematics and Statistics, University of Western Australia, Crawley 6009, Western Australia )
Corresponding author cdc{at}maths.gla.ac.uk
Received 17 December 2006. Accepted 23 March 2007.
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Abstract |
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A novel application of boundary-layer asymptotic techniques to a generalized linear eigenvalue problem is presented. Our investigation is concerned with a bifurcation equation that governs the formation of wrinkles in thin annular plates subjected to in-plane tensile loading on the inner boundary. If 
denotes the ratio of the inner and outer radii of the annulus, then the critical wrinkling load satisfies 
= C(), where the function C is available only numerically. It is known that there is a critical value 
such that 
as 
but, until now, little has been understood about this singular behaviour. Asymptotic methods enable us to capture accurately and describe the nature of this blow-up phenomenon which we show is sensitive to the forms of the boundary conditions imposed at the edges of the annular plate. Our analytical findings are complemented by a series of comparisons with direct numerical simulations that shed further light on the singular behaviour.