  The elastic properties of hcp $^4$He samples have been shown to display
various anomalies. The elastic shear modulus stiffens and the moment of
rotational inertia drops when the temperature is lowered below $\sim$ 0.2 K.
The relation between these two quantities is studied within the framework of
classical deformable-body mechanics. A model based on the formation by plastic
flow of extremely soft two-dimensional layers of dislocations is studied
analytically and numerically. This model accounts quantitatively for a number
of experiments. Other situations, in which it may seem less relevant, are
discussed.
