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Nonlinear elastic models are widely used to describe the elastic response of crystalline solids, for example, the well-known Cauchy-Born model. While the Cauchy-Born model only depends on the strain, effects of higher order strain gradients are significant and higher order continuum models are preferred, in various applications such as defect dynamics and modeling of carbon nanotubes. In this paper, we rigorously derive a higher order nonlinear elasticity model for crystals from its atomistic description in one dimension. We show that, compared to the second order accuracy of the Cauchy-Born model, the higher order continuum model in this paper is of fourth oder accuracy with respect to the interatomic spacing in the thermal dynamic limit. In addition, we discuss the key issues for the derivation of higher order continuum models in more general cases. The theoretical convergence results are demonstrated by numerical experiments.

A priori analysis of a higher-order nonlinear elasticity model for an atomistic chain with periodic boundary condition