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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on October 31, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2007 60(4):457-471; doi:10.1093/qjmam/hbm019
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Q. Jl Mech. Appl. Math, Vol. 60. No. 4 © The author 2007. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Two-parameter asymptotic approximations in the analysis of a thin solid fixed on a small part of its boundary

V. V. Zalipaev{dagger} and A. B. Movchan{ddagger}

( Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK )

I. S. Jones§

( School of Engineering, John Moores University, Liverpool L3 3AF, UK )

{dagger} < v.zalipaev{at}liverpool.ac.uk>

{ddagger} < abm{at}liv.ac.uk>

§ < i.s.jones{at}ljmu.ac.uk>

Received 13 April 2007. Revise 12 July 2007. Accepted 14 July 2007.


  Abstract

Planar elasticity problems are considered for thin domains fixed along a small part of the end region boundary. The analysis involves two small parameters: the normalized thickness of the body and the normalized length of the fixed part of the boundary. The aim of the paper is to derive an asymptotic approximation of the solution to a boundary-value problem in such a domain and, in particular, analyze the ‘effective boundary conditions’, which occur for the leading-order terms of the asymptotics. We include applications for problems of both anti-plane shear and plane strain elasticity.


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