Variation on a Theme of Irwin

Cracks as a Superposition of Dislocations. A scheme that will suit our aim of building a synthetic description of cracks and any allied dislocations is to think of a crack as an array of dislocations. Indeed, the majority of our work has already been done earlier in the context of our consideration of dislocation pile-ups in section 11.4.2. In fact, our present analysis will do little more than demonstrate that the solutions written down there are relevant in the crack context as well. The more fundamental significance of the perspective to be offered here is that we will soon want to build up solutions in which cracks and dislocations are equal partners. 

One of the key points to be made in the present discussion is the substantiation of the claim that a continuous distribution of infinitesimal dislocations can be assembled in a way that respects the traction-free boundary conditions associated with crack faces. In particular, we consider a crack which runs from —a to a, with a remotely applied stress of the form o y = tq. In this mode II setting (see fig. 11.16), we insist that on the crack faces Oyy = o y = 0. Our claim is that such a crack can be represented by a distribution of infinitesimal dislocations with a density of the form 

This distribution is determined by insisting that the traction-free condition, 

We denote the stress field due to a single Volterra dislocation at position (x, y ) by olp x — x, y — y ). We also note that the original interpretation of the elastic energy release rate offered in section 2.4.4 can be reinterpreted in terms of the driving forces on the various dislocations making up the crack. Now that we have seen how the crack itself may be written in terms of dislocations, we turn to the question of how to think about such a crack when there are other dislocations in its vicinity. 

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