On a connection between certain semi-infinite and finite crack problems

Abstract

Two models of steady state brittle crack motion are commonly used. They are the models of Yoff6 (1) and of Craggs (2) (similar models have also been used by Barenblatt etc. of the Russian school). As is well known, Yoffr's model consists of a constant finite length crack moving with uniform velocity (i.e. the material closes up behind the crack as the crack advances) whereas Craggs's model consists of a semi-infinite crack with a finite region near its tip acted upon by a non-zero internal stress. Both these models have also been used with a Dugdale zone as an attempt to take into account the effect of plasticity at the moving crack tip. The two models cited above give a similar r -¢ stress-singularity at the leading crack tip, the stress intensity factor in each case having the same velocity dependence. Although the two stress intensity factors are not identical the similarity between them is sufficient to make any predictions based on them qualitatively the same. There is a physical peculiarity about each of these models however; in the Yoff6 model there is the closing up of the material behind the crack and in the Craggs model the finite region of internal stress behind the crack tip leads to difficulties when one thinks of a stress applied at infinity. In this note we wish to compare the semi-infinite crack model with the finite crack model not restricting ourselves to loadings such as considered by Yoff6 and Craggs. It turns out that a semi-infinite crack can be directly related to a problem involving a dislocation pile up of finite length rather than to a finite length crack. This is not surprising really since a semi-infinite crack has only one stress-singularity at its single tip, whereas a finite length crack has two stress singularities (one at each end) and a dislocation pile up has only one. The analysis serves as a simple example of the use of singular integral equations and we hope should be of some pedagogic interest. We consider shear cracks in plane strain moving along the plane y = 0 with speed V in the positive x direction (the case of cracks under tension is mathematically very similar). Since we are considering shear cracks the only non-zero stress acting on the y --0 plane will be the shear stress axy. To solve the problem we take as the fundamental solution the stress field of an edge dislocation moving with velocity V along the x axis in the positive x direction. This stress field is well known (see for example Hirth and L~Sthe (3)) and the stress axy on y = 0 can be written as ~bC2t [ +y2)2--4ye~tt , 2 t 0 j (1)