Barenblatts Theory of Brittle Cracks

For simplicity, we will restrict ourselves to the case of purely normal forces. Let the crack occupy [-c(0, (0] We denote the cohesive forces as This function will be zero except in a small region near each tip, which in fact may be regarded as beyond the tip itself. Let the crack plus cohesive region occupy [-a t),a t)] where a(t) c(t). For convenience, let us regard p(x,t) as given over the small regions c x a, even though they are off the crack face. 

In practice, f x, t) will be very large over these regions so the value of p(x, t) is irrelevant. We now impose the condition that no stress singularity exists, due to the presence of f(x, t). We can express this condition by simply saying that the combination p x,t)-f x,t) has stress intensity factor, as given by (4.2.1 p), equal to zero, or 

In the first integral, we can replace a t) by c t) with negligible error since p(x, t) is a smooth function. Using (4.2.1 p) again allows us to write (4.7.1) in the form 

It is worth noting that exactly this type of reasoning has been used to incorporate regions of plastic flow at crack tips, for ductile materials [Dugdale (1960), Bilby et al. (1963), Lardner (1974), Golden and Graham (1984)]. 

Let us first write down an approximate expression for the displacement derivative near x = a(t). By virtue of condition (4.7.1), the relation (4.2.9), in this context, can be put in the form 

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