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We consider the one-dimensional shearing motion of a material exhibiting elastic behaviour when the stress is below some threshold. The threshold represents a limit to the deformability, i.e. no further deformation can occur on increasing the stress. The mathematical formulation leads to a free boundary problem for the wave equation, whose structure depends on whether the stress (and the velocity) are continuous across the propagating interface for the strain threshold .<br />Local existence and uniqueness are proved for the continuous case (in which the interface propagation is subsonic). Some explicit solutions are calculated for another case (with a supersonic interface). It is shown that the model with strain threshold is never the limit of hyperelastic systems.