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The Quarterly Journal of Mechanics and Applied Mathematics 1993 46(2):331-350; doi:10.1093/qjmam/46.2.331
© 1993 by Oxford University Press
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ASYMPTOTIC ANALYSIS OF THE REINFORCEMENT OF A BRITTLE CRACK BY BRIDGING FIBRES

A. B. MOVCHAN{dagger} and J. R. WILLIS

( School of Mathematical Sciences, University of Bath Bath BA2 7AY )

{dagger} On leave from: Institute of Precision Mechanies and Optics. St Petersburg. Russia.

The subject of the paper is the cracking of a brittle matrix reinforced by unidirectional fibres. A crack is introduced into the matrix, transverse to the fibres, which bridge the crack and tend to tie the faces together. The fibres are treated in an ‘averaged’ manner. the uncracked composite material is modelled as homogeneous linearly elastic material and the effect of the bridging fibres is modelled by a nonlinear law of cohesion between the crack faces. The problem is formulated as a nonlinear, highly singular, integral equation. The ‘long-crack’ limit is addressed, which corresponds to the bridging fibres carrying most of the load across the crack. The equation then contains a small parameter and is solved by a matched expansion technique. This has only recently been expressed in a form suitable for the singular perturbation of integral equations and is likely to have applicability beyond the specific example considered here.


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