Dislocation Screening at a Crack Tip

2 Dislocation Screening at a Crack Tip From the standpoint of the linear theory of elasticity, the problem of crack tip shielding may be seen as a question of solving the boundary value problem of a crack in the presence of a dislocation in its vicinity. As indicated schematically in fig. 11.15, the perspective we will adopt here is that of the geometrically sterilized two-dimensional problem in which both the crack front and the dislocation line are infinite in extent and perfectly straight. As we will see, even this case places mathematical demands of some sophistication. The basic question we 

To be concrete, we note that we seek a potential function r] (z) such that u (z) = (2//r) Im[r)(z)]. Note that the factor of 2 here is consonant with the convention 

Note that in this case, the function x (to) is presumed to be known (usually by virtue of the symmetric geometry of the transformed problem), and is a geometric 

For the case of interest here, namely, the fields that arise for a crack in the presence of a dislocation, the procedure described above is disarmingly simple. Through a suitable choice of transformation, the problem of a crack and dislocation can be mapped onto the problem of a single subsurface dislocation beneath a traction-free surface already solved by the method of images in section 8.5.2. 

Before writing down the solution we must first broach the subject of the geometric transformation relating the wedge crack problem to the allied problem of a free surface. If we consider a wedge crack with opening angle a, all points on the line z = must be mapped onto the line z = re. The transformation that 

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