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Dislocations in crystalls and non-riemannian geometry

Das gleiche auf Deutsch
Dislocations, one-dimensional lattice defects in crystalline
bodies, were first used in the 30's to describe the plastic
deforamtions of metals. In the early 50's, there was growing
interest in this theory, after the discovery of its relation
to non-riemannian geometry. This field theory, mainly developed
by Kröner [1],[2], shows certain analogies to electrodynamics.
However, until the present day there is no complete theory
of the dynamics of dislocations.
We will visualize the different types of dislocations (edge d.,
screw d.) and discuss some related phenomena ("forces" between
dislocations, stress fields, "pair creation", dynamic effects,
velocity of sound).
We will briefly mention the related topics of tensor analysis
(div, inc, curl, similar to div and curl in vector analysis).
Then we discuss some topics of nonriemannian geometry with
torsion (the connection is not necessarily symmetric) like
metric, connection, curvature and torsion tensor. We will
not give a mathematical rigid approach, but rather focus on
visualization of terms by showing the relations to dislocation
theory and general relativity. Some knowledge in the latter
and in vector analysis will be useful.

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Literature:

[1] Kontinuumstheorie der Versetzungen und Eigenspannungen, Springer 1958, and

[2] Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Arch. Rat. Mech. Anal. IV (1960), 273-313.

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Differential geometry und tensor algebra:

[3] Schouten,J.,A.: Ricci-Calculus (Springer), chap.3 "Linear connexions".
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General relativity:

[4] Landau-Lifschitz II, §81 ff, §91 ff.

Alexander Unzicker, 1996-03-21