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In this paper we present two atomistic models for the energy of aone-dimensional elastic crystal. We assume that the macroscopic displacementequals the microscopic one. The energy of the first model is given by atwo-body interaction potential, and we assume that the atoms follow acontinuous and piecewise smooth macroscopic (continuum) deformation. Wecalculate the first terms of the Taylor expansion (with respect to theparameter representing the interatomic distance) of the atomistic energy, andobtain that the coefficients of that Taylor expansion represent, respectively,an elastic energy, a sharp-interface energy, and a smooth-interface energy. Thesecond atomistic model is a variant of the first one, and its Taylor expansionpredicts, in addition, a new term that accounts for the repulsion force betweentwo sharp interfaces.

Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity