The shear mode associated with the interplanar elastic parameter C 44 of graphite is anomalously soft in the low frequency region. An analysis of dislocation dynamics has been made and several experiments are described which explore the temperature, pressure and magnetic field dependence of C 44 . It appears that the anomalous elastic behaviour can be accounted for quantitatively from the properties of glissile dislocations lying in the basal plane, and the degree of softening is not coupled to the electronic states. A similar mode-softening observed in hexagonal BN suggests that this type of elastic anomaly may be common to layer materials with weak interplanar coupling. It is proposed that a study of the specific heat at very low temperature could give valuable information about the dislocation dynamics.