On the theoretical average crack spacing in brittle matrix composites containing continuous aligned fibres

Abstract

A brittle matrix composite containing aligned continuous fibres (or fibres which are long relative to the critical length) under tension will have a final crack spacing of between x and 2x, where x is the length over which the additional load sustained by the fibres at the crack is transferred back to the matrix. Notional values of the stress transfer at the fibre m a t r i x interface can be evaluated if x can be determined from experimental measurements of average crack spacing. The average crack spacing in a length of composite has been quoted by Aveston et aL [1] as (1.364-+ 0.002)x, from work by Gale based on the analogous problem of minimum average spacing between cars of length x parked at random in a given space. Beeby [2], in a study of cracking in reinforced concrete, has noted that, while a mean value of 1.5x has commonly been assumed, there are theoretical reasons for believing that a value of 1.33x is more correct. We set out below a solution to the problem of average crack spacing which yields, as the length of composite considered approaches infinity (or, practically, becomes large compared to x), an average spacing of 1.337x. Let m(l) be the average number of cracks in a brittle matrix composite of length l. Without loss of generality we take x = 1. I f the first crack appears at a point t units along the composite, it is clear that m(l/ t ) , the average number of cracks given that the first crack appears at t, is