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L. N. McCartney, 'Analytical model for debonded interfaces associated with fibre fractures or matrix cracks', Proc. ICCM-12, Paris, July 1999. 9. A. G. Metcalfe, M. E. Gulden and G. K. Schmitz, Glass technology, 12, (1971), 15-23. 10. McCartney, 'Failure by stress corrosion of bundles of fibres', Proc. Roy. Soc. Lond. A374, 475 (1981). 11. R. F. Churchhouse, 'Handbook of Applicable Mathematics', Vol. 3 on 'Numerical Methods', series edited by W Ledermann, John Wiley and Sons, Chichester - New York -Brisbane Toronto, 1981, pp.

As will be seen next, the zero-toughness solution is incompatible with the Linear Elastic Fracture Mechanics (LEFM) tip asymptote (8) for small but non-zero IC and, consequently, is not uniformly valid along the fracture for IC « 1. The latter suggests that a boundary layer possessing the LEFM tip behavior has to exist near the tip of the moving fracture for arbitrary small non-zero IC. Thus, the expansion (10) actually corresponds to the outer small toughness solution (valid everywhere except at the tip boundary layer).

Numerical simulations The numerical procedure proposed in the previous section has been used to evaluate the flexural behaviour of the cross-section reinforced with two different levels of fibers. The three dimensionless numbers, N~l), N;;) and wf affect the shape of the curves. N~l) takes into account the contribution of the primary reinforcement. N~2) describes the fiber percentage and the fiber strength, wf their length, once that the geometrical and mechanical parameters of the element have been fixed.