  A translational gauge approach of the Einstein type is proposed for obtaining
the stresses that are due to non-singular screw dislocation. The stress
distribution of second order around the screw dislocation is 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 us to
continue the classically known quadratic stresses inside the core. The gauge
equation is chosen in the Hilbert--Einstein form, and it plays the role of
non-conventional incompatibility law. The stress function method is used, and
it leads to the modified stress potential given by two constituents: the
conventional one, say, the `background' and a short-ranged gauge contribution.
The latter just causes additional stresses, which are localized. The asymptotic
properties of the resulting stresses are studied. Since the gauge contributions
are short-ranged, the background stress field dominates sufficiently far from
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 does not occur. Expressions
for the asymptotically far stresses provide self-consistently new length scales
dependent on the elastic parameters. These lengths could characterize an
exteriority of the dislocation core region.
