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Linear elastic chain with a hyper-pre-stress. (English) Zbl 1035.74005
Summary: 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 and next to nearest neighbors have incompatible reference lengths, which introduce a hyper-pre-stress and results in a formation of 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 inhomogeneous stressed and reference configurations. We then obtain a simple expression for hyper-pre-stress-related contribution to the surface energy, and show an unusual scaling of total energy with film thickness. For ultra-thin films, we report an anomalous stiffness increase due to the overlapping of surface boundary layers. Implications for the micro level hyper-pre-stress in fracture mechanics and for the theory of non-Bravais lattices are also discussed.

MSC:
74A60 Micromechanical theories
74B99 Elastic materials
74K35 Thin films
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