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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 17/2006

C. MALYSHEV##
THE EINSTEINIAN GAUGE APPROACH AND
THE STRESS FIELD OF
NON-SINGULAR SCREW DISLOCATION
IN THE SECOND ORDER APPROXIMATION

This preprint was accepted October 30, 2006

ABSTRACT:
The stress field of non-singular screw dislocation is investigated
in the second order approximation by means of the translational
gauge approach of the Einstein-type. The stresses of second order
around the screw dislocation are classically known for the hollow
circular cylinder with traction-free external and internal
boundaries. The inner boundary surrounds the dislocation's core,
which is not captured by the conventional solution. The present
gauge approach enables the classically known quadratic stresses be
continued inside the core. The corresponding gauge equation is
chosen in the Hilbert--Einstein form. The gauge equation plays a
role of non-conventional incompatibility law, and the stress
function method is used for its solution. The modified stress
potential is obtained as a sum of two parts: conventional, say,
`background' and a short-ranged gauge contribution. The latter
just causes additional stresses localized within the dislocation's
core. The asymptotic properties of the resulting stresses are
studied. Since the gauge contributions are short-ranged, the
background stress field dominates at the distances sufficiently
large with respect to the core. The outer cylinder's boundary is
traction-free. At sufficiently moderate distances, the second
order stresses acquire regular continuation within the core
region, and the cut-off at the core is absent. Expressions for the
asymptotically far stresses provide us self-consistently with new
length parameters of the unconventional origin. These lengths are
given by means of the elastic moduli and could characterize an
exteriority of the dislocation core.

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