Elastic fields of an edge dislocation

It is well known that the algebra associated with edge dislocations can be forbidding. As Prof. Frank (of the Frank-Read source fame) noted once,

• I found all that elasticity mathematics rather difficult, but I found it easier to concentrate on the screw dislocation, with only one displacement variable, instead of two for the edge. So I became particularly fond of the screw dislocation. Mott and Nabarro liked to work with edge dislocations., because they liked two-dimensional diagrams. I was less afraid than they were of the third dimension, and more afraid of algebra.

Even the great Eshelby called the displacement field expressions of an edge dislocation field "rather forbidding expressions" in a pedagogical paper that he wrote in 1966.

This paper, published in the British Journal of Applied Physics (the abstract of which is given below), describes a process to obtain the elastic stress fields of the edge dislocation using what Eshelby calls a wedge dislocation:

• The elastic field of an edge dislocation is found in a simple manner by making use of the relation between an edge dislocation and a `wedge’ dislocation made by inserting or removing a narrow wedge of material.

As is his wont, Eshelby uses some elegant cutting and welding operations to derive some very interesting results; further, these gedanken experiments also give a nice physical feel to the problem at hand.

What I found the most interesting about the paper is the connection that Eshelby makes to the stress fields associated with a tilt boundary in obtaining the elastic stress fields of an edge dislocation.

It is well known that a tilt boundary can be understood as an array of edge dislocations; and, the stress fields and energy associated with the tilt boundaries are obtained using that of the edge dislocations.

Eshelby, cunningly, stands the problem on its head. What he calls a wedge is a terminating tilt boundary; and, using some rather simple physical arguments, Eshelby obtains the field for such a wedge, and derives the fields for the edge dislocations from them.

On the whole, this five page pedagogical piece is a real gem for its simplicity and the understanding it brings to the usually intractable problem.

Here are a few posts of mine in my blog giving the details of the derivation: post 1, post 2, and post 3. These posts are but notes that I wrote for myself while reading the paper. There is nothing as pleasurable as learning from the master himself. Have fun!