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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on January 7, 2009
The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(1):1-18; doi:10.1093/qjmam/hbn023
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© The author 2009. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Asymptotic model of fields in a thin-walled structure with crack-like defects

V. V. Zalipaev{dagger} and 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} <v.zalipaev{at}liv.ac.uk>

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

§ Correspondingauthor <i.s.jones{at}ljmu.ac.uk>

Received 29 March 2008. Revise 23 September 2008. Revise 17 November 2008. Accepted 18 November 2008.


  Abstract

The transition from two-dimensional (2D) wave propagation through the square periodic structure in anti-plane shear time-harmonic case to a discretised model of a 2D lattice with masses connected by springs is considered. A model of a defect in the middle part of the thin-walled bridges is presented. As a first part of the asymptotic model, the effective transmission condition in the vicinity of the transverse cut of the thin-walled bridges is discussed. Then, a boundary layer determining the asymptotic expansion of the field near the tip of the crack is constructed. Stress intensity factors are evaluated for deep cracks in the junction regions. The corresponding boundary layer analysis is non-trivial and has not been attempted elsewhere.


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