  Superfluidity, liquid flow without friction, is familiar in helium. The first
evidence for "supersolidity", its analogue in quantum solids, came from recent
torsional oscillator (TO) measurements involving 4-He. At temperatures below
200 mK, TO frequencies increased, suggesting that some of the solid decoupled
from the oscillator. This behavior has been replicated by several groups but
solid 4-He does not respond to pressure differences and persistent currents and
other signatures of superflow have not been seen. Both experiments and theory
indicate that defects are involved. These should also affect the solid's
mechanical behavior and so we have measured the shear modulus of solid 4-He at
low frequencies and strains. We observe large increases below 200 mK, with the
same dependence on measurement amplitude, 3-He impurity concentration and
annealing as the decoupling seen in TO experiments. This unusual elastic
behavior is explained in terms of a dislocation network which is pinned by 3-He
at the lowest temperatures but becomes mobile above 100 mK. The frequency
changes in TO experiments appear to be related to the motion of these
dislocations, perhaps by disrupting a possible supersolid state.
