Understanding Crystal Dislocations: A Mathematical Approach

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In summary, the conversation revolved around the need for a reference on the theory of dislocations in crystals with a mathematical emphasis. The suggested references included Callister, a free manuscript by Professor Sir Bhadesia, and a paper by Hirth. Landau/Lifshitz was also mentioned as a potential resource. The conversation concluded with the mention of classic texts such as "Elementary Dislocation Theory" by Johannes Weertman and Julia R. Weertman, and the paper "Theory of Steady-State Creep Based on Dislocation Climb" by J. Weertman.
  • #1
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!
 
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  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
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  • #6
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 😍
 
  • #8
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
 
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1. What is a crystal dislocation?

A crystal dislocation, also known as a lattice defect, is a deviation from the regular atomic arrangement in a crystal lattice. It can be thought of as a "line" or "plane" of atoms that is not aligned with the rest of the crystal's lattice structure.

2. How do dislocations affect the properties of crystals?

Dislocations can significantly impact the mechanical, electrical, and thermal properties of crystals. They can act as barriers to dislocation motion, leading to increased strength and hardness in materials. Dislocations can also influence the electrical conductivity and thermal conductivity of crystals, making them important in the design and development of materials for various applications.

3. What are the different types of dislocations?

There are two main types of dislocations: edge dislocations and screw dislocations. An edge dislocation is a linear defect where the atoms on one side of the dislocation line are shifted relative to the atoms on the other side. A screw dislocation is a spiral defect where the atoms are displaced in a helical manner around the dislocation line.

4. How are dislocations studied mathematically?

Dislocations can be described and studied using mathematical models and equations. These models take into account the crystal's atomic structure, the type and location of the dislocation, and the forces acting on the dislocation. The mathematical approach to understanding dislocations allows for the prediction of their behavior and properties, which can aid in the development of new materials.

5. Why is understanding crystal dislocations important?

Understanding crystal dislocations is crucial for designing and developing materials with desired properties. Dislocations can significantly affect the mechanical, electrical, and thermal properties of materials, making them important in various industries such as aerospace, electronics, and construction. Additionally, dislocations play a key role in the deformation and failure of materials, making them essential to study for improving the durability and reliability of materials.

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