To account for surface relaxation in ultra-thin films, we consider the simplest one-dimensional discrete chain with harmonic interactions of up to second nearest neighbors. We assume that the springs, describing interactions of the nearest neighbors (NN) and next to nearest neighbors (NNN) have incompatible reference lengths, which introduce a hyper-pre-stress and results in a formation of the exponential surface boundary layers. For a finite body loaded by a system of (double) forces at the boundary, we explicitly find the displacement field and compute the energies of the inhomogeneous stressed and reference configurations. We then obtain a simple expression for the hyper-pre-stress related contribution to the surface energy and show an unusual scaling of the total energy with the film thickness. For ultra-thin films we report an anomalous stiffness increase due to the overlapping of the surface boundary layers. Implications of the micro level hyper-pre-stress in fracture mechanics and in the theory of non-Bravais lattices are also discussed.

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Considering a one-dimensional problem of debonding of a thin film in the context of Griffith's theory, we show that the dynamical solution converges, when the speed of loading goes down to 0, to a quasi-static solution including an unstable phase of propagation. In particular, the jump of the debonding induced by this instability is governed by a principle of conservation of the total quasi-static energy, the kinetic energy being negligible.