  We calculate the dislocation glide mobility in solid $^4$He within a model
that assumes the existence of a superfluid field associated with dislocation
lines. Prompted by the results of this mobility calculation, we study within
this model the role that such a superfluid field may play in the motion of the
dislocation line when a stress is applied to the crystal. To do this, we relate
the damping of dislocation motion, calculated in the presence of the assumed
superfluid field, to the shear modulus of the crystal. As the temperature
increases, we find that a sharp drop in the shear modulus will occur at the
temperature where the superfluid field disappears. We compare the drop in shear
modulus of the crystal arising from the temperature dependence of the damping
contribution due to the superfluid field, to the experimental observation of
the same phenomena in solid $^4$He and find quantitative agreement. Our results
indicate that such a superfluid field plays an important role in dislocation
pinning in a clean solid $^4$He at low temperatures and in this regime may
provide an alternative source for the unusual elastic phenomena observed in
solid $^4$He.

Other Condensed Matter;Soft Condensed Matter