Some biological tissues exhibit a sharp reduction of deformability beyond some stress threshold, below which they may be considered elastic. The simplest constitutive equation used to describe the mechanical behavior of tendons and ligaments is a stress-strain relation in which the stress is a sharply increasing function of the strain (for instance exponential, [Fung, 1993]). The system can be thus described by classical hyperelasticity. A limit case of such a behavior is a material that beyond

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... a certain stress is no longer deformable. Here we investigate the mechanical response of a layer of material characterized by a no-deformation stress threshold below which the system is modeled as a neo-Hookean nonlinear elastic solid. In particular, we consider the case in which the bottom face of the layer is kept fixed while on the top a stress eventually exceeding threshold is applied. This problem reveals surprisingly complicated. The corresponding mathematical model is a free boundary problem for the wave equation, in which the free boundary conditions may be of two different types, according to whether the stress is continuous or discontinuous across the interface. The first class, corresponding to a rather artificial set of initial data, produces a subsonic motion of the interface. The second, corresponding to the case in which the system is initially at rest and the applied load is or becomes larger than the threshold value, exhibits instead a supersonic interface. Both situations have been studied, obtaining an existence and uniqueness theorem for the former case and studying the latter case for some specific data. The mathematical model We consider a homogeneous slab of thickness h loaded on the top surface with a known shear stress σ(t) > τ o , where τ o is the stress threshold. Let x = X +f (y, t) be a pure shear motion, f (y, t) being the unknown displacement. The elastic and fully stretched regions are divided by a sharp interface 0 ≤ s(t) ≤ h. In the deformable region 0 < y < s(t)