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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.

Literature:

Kröner, E.:

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

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

Differential geometry und tensor algebra:

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

General relativity:

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

Alexander Unzicker, 1996-03-21