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Elastic fields of an edge dislocation

Mogadalai Gururajan's picture

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!



Henry Tan's picture

Eshelby's work on the elastic fields of an edge dislocation is a nice example in describing the atomistic scale deformation analytically using concepts of continuum mechanics.

Pradeep Sharma's picture


I quite agree, this is very nice paper but hard to find. I was not aware of it until I  happened to buy the collected works of Eshelby compiled recently by Markenscoff and Gupta. There are many other papers by him which are also not quite as widely known but are true gems.

Mogadalai Gururajan's picture

Dear Tan:

What you say is true; however, one should always remember that the fields are not properly defined inside of what is called the "core"; that is why, in his paper, Eshelby keeps insisting that the expressions are valid at a few Burgers vectors from the dislocation.

Dear Pradeep:

I agree with you; as the old poem goes, "Full many a gem of purest ray serene" that the collected works of Eshlby bears; and, from that point of view, the Merkenscoff and Gupta collection is indeed a boon. For those who do not have access to the book, however, a complete listing of all the 56 papers of Eshelby is available in the memoir that Bilby wrote in the Biographical memoirs of the Fellows of the Royal Society. Further, from the memoir, I also learnt that

  • Eshelby taught himself theory of elasticity for his thesis on ‘Stationary and moving dislocations’; was well versed in Sanskrit (among other classical languages); was not keen on doing experimental work; was an avid second-hand book buyer; though not active in politics, was typically anti-establishment; was clear and amusing as a lecturer, and prepared his lectures with great care.

There is also a signed photograph of Eshelby in the memoir to boot!

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