The Volterra Formula

The reciprocal theorem points the way to an idea that will allow for the calculation of the displacements associated with a dislocation loop of arbitrary shape. As discussed in chap. 2, the reciprocal theorem asserts that two sets of equilibrium fields (or and f ), associated with the same 

Within the context of the elastic Green function, the reciprocal theorem serves as a jumping off point for the construction of fundamental solutions to a number of different problems. For example, we will first show how the reciprocal theorem may be used to construct the solution for an arbitrary dislocation loop via consideration of a distribution of point forces. Later, the fundamental dislocation solution will be bootstrapped to construct solutions associated with the problem of a cracked solid. 

We consider the generic problem of a dislocated solid with a loop such as that shown in fig. 8.10. Here we imagine two different configurations associated with 

The basic idea is to exploit the fact that the displacements associated with the point force are already known, and correspond to the elastic Green function. Further, the displacements associated with the dislocation are prescribed on the slip plane. We can rewrite the surface integral that spans the slipped region as 

In this case f )(+) refers to the traction associated with the dislocation just above the slip plane while t (—) refers to the value of the traction just below the slip plane. However, since the displacement fields (u ) associated with the point force are continuous across the slip plane, Similarly, the tractions 

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This problem is a bit nasty, but will illustrate the way in which the Volterra formula can be used to derive dislocation displacement fields, (a) Imitate the derivation we performed for the screw dislocation, this time to obtain the displacement fields for an edge dislocation. Use the Volterra formula (as we did for the screw dislocation) to obtain these displacements. Make sure you are careful to explain the logic of your various steps. In particular, make sure at the outset that you make a decent sketch that explains what surface integral you will perform and why. 

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