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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on July 21, 2009
The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(4):481-494; doi:10.1093/qjmam/hbp017
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© The author 2009. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Junction Conditions for Cracked Elastic Thin Solids Under Bending and Shear

M. Gei{dagger}

( Department of Mechanical and Structural Engineering, University of Trento, Via Mesiano 77, I-38123 Trento, Italy )

A. B. Movchan{ddagger}

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

I. S. Jones§

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

{dagger} <massimiliano.gei{at}unitn.it>

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

§ Correspondence author. <I.S.Jones{at}ljmu.ac.uk>

Received 20 February 2009. Revise 22 June 2009. Accepted 23 June 2009.


  Abstract

The paper presents an asymptotic model of the spring-like behaviour of small elastic junctions within two-dimensional thin solids subjected to a flexural or shearing load. The approach employs the method of compound asymptotic expansions. It is shown that the flexural stiffness increases quadratically with the increase of the thin ligament length, whereas the shear stiffness is relatively high and it is less sensitive to the change of the thickness of the thin ligament within the junction region. The expressions for the effective junction conditions agree well with the independent numerical simulations for test configurations. In addition to the junction conditions, our model also provides a uniform asymptotic approximation of elastic displacements within a thin solid containing surface breaking cracks separated by a thin bridge.


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