We present two atomistic models for the energy of a one-dimensional elastic crystal. We assume that the macroscopic displacement equals the microscopic one. The energy of the ﬁrst model is given by a two-body interaction potential, and we assume that the atoms follow a continuous and piecewise smooth macroscopic (continuum) deformation. We calculate the ﬁrst terms of the Taylor expansion (with respect to the parameter representing the interatomic distance) of the atomistic energy, and deduce that the coefﬁcients of that Taylor expansion represent, respectively, an elastic energy, a sharp-interface energy, and a smooth-interface energy. The second atomistic model is a variant of the ﬁrst one, and its Taylor expansion predicts, in addition, a new term that accounts for the repulsion force between two sharp interfaces.
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Carlos Mora-Corral, Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity. Interfaces Free Bound. 11 (2009), no. 3, pp. 421–446DOI 10.4171/IFB/217