Understanding Crystal Dislocations: A Mathematical Approach

[etotheipi]

I'm in not too urgent (but a little pressing, i.e. I have an assignment on this due Friday... ) need of some reference that treats the theory of dislocations in crystal with a mathematical emphasis (i.e. tensors); specifically, pertaining to Burgers' vectors and the strain response to applied stress in crystals containing dislocations. Does such a reference exist?

The lecturer recommended Callister but it's general purpose and doesn't go into enough detail on this section.

Thanks!

 

[Frabjous]

Callister the materials science book? There are several good texts on dislocations (older versions are fine)
https://www.amazon.com/dp/0080966721/?tag=pfamazon01-20
https://www.amazon.com/gp/product/0521864364/?tag=pfamazon01-20

or are you looking for something truly scary

 

[etotheipi]

Or even an internet reference, to be honest. I don't think I'd consider buying a book unless I could use it for other stuff too.

A few minutes ago I did find a free manuscript by Prof Sir Bhadesia
https://www.phase-trans.msm.cam.ac.uk/2001/crystal.html

think that's got a few bits in it but a little strange notation (Mackenzie & Bowles?). Maybe, worth a shot.

 

[Frabjous]

Your going to be disappointed. The problem is that the number density of dislocations is high and that mechanisms abound. So the focus has not been on formalism.

 

[Frabjous]

This paper
Hirth, J.P. A brief history of dislocation theory. Metall Mater Trans A 16, 2085–2090 (1985). https://doi.org/10.1007/BF02670413

has this paragraph

Brown,52while considering magnetic properties of dis- locations, originated the concept of smearing discrete dislocations into a continuous array of infinitesimal dis- locations. This method has resulted in connections with powerful methods of mathematics but describes properties of the net dislocation density and has some problems in uniqueness and the description of arrays of dislocations of opposite sign. In early work, Nye53described the connec- tion between the net dislocation density tensor and the lat- tice curvature. Kondo54and Bilby, Bullough, and Smith55 showed that the Cartan torsion of space is the continuum equivalent of the dislocation, with the Cartan circuit closely related to the Burgers circuit.56The latter authors used the continuum description to derive the geometric properties of grain boundaries. Kr/Sners7 developed the concept of the incompatibility, proportional to derivatives of the dislocation density, and descriptions of the elastic fields in terms of it. Further advances are discussed in several reviews. 57.58,59

and refs 52-59
52. W.E Brown: Phys. Rev., 1941, vol. 60, p. 139.
53. J. E Nye: Acta MetaU., 1953, vol. 1, p. 153.
54. K. Kondo: RAAG Memoirs of the Unifying Study of the Basic Prob-
lems in Engineering Sciences by Means of Geometry, Gakujutsu Buuken Fukyu-Kai, Tokyo, 1955, vol. I, p. 453; also see Ref. 46, p. 761.
55. B.A. Bilby, R. Bullough, and E. Smith: Proc. Roy. Soc. London, 1955, vol. A231, p. 263.
56. E. KriJner: in Dislocation Modeling of Physical Systems, M.F. Ashby, R. Bullough, C. S. Hartley, and J. P. Hirth, eds., Pergamon, Oxford, 1981, p. 285.
57. E. Kr/Sner: Ergeb. angew. Math, 1958, vol. 5, p. 1.
58. E. Cosserat and F. Cosserat: Theorie des Corps Deformables,
Herman, Pads, 1909.
59. Mechanics of Generalized Media, E. KriSner, ed., Springer, Berlin,
1968.

 

[etotheipi]

thanks, I'll take a look and see which of those I can access! should be some useful stuff in there.

was also going to say that I discovered that Landau/Lifshitz wrote a little section on dislocations near the end of vol. 7 on elasticity theory... think I hit the jackpot, there

 

[Frabjous]

In case anybody laughs at you for looking at dislocations

https://royalsocietypublishing.org/toc/rspa1969/371/1744

 

[Astronuc]

FUNDAMENTAL ASPECTS OF DISLOCATION THEORY, Conference Proceedings, April 1969
https://www.govinfo.gov/content/pkg...VPUB-C13-3e85db87f8d45249963643f05e447bd7.pdf

A classic text - Elementary Dislocation Theory
Johannes Weertman and Julia R. Weertman, Published: 25 June 1992
https://global.oup.com/academic/product/elementary-dislocation-theory-9780195069006?cc=us&lang=en&#

J. Weertman, Theory of Steady‐State Creep Based on Dislocation Climb Journal of Applied Physics 26, 1213 (1955); https://doi.org/10.1063/1.1721875 - see the references of folks like Sherby, Dorn and Mott
https://aip.scitation.org/doi/pdf/10.1063/1.1721875