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A general approximation of the exponential Cauchy–Born hypothesis to model arbitrarily shaped shell‐like nanostructures within continuum mechanics

In this paper, a continuum membrane theory and its subsequent finite element approximation for the description of arbitrary shell‐like nanostructures such as graphene‐based nanostructures is presented. This is carried out by applying a multiscale approach where the continuum membrane is linked to the underlying atomistic lattice. This linkage is performed by the exponential generalization of the Cauchy–Born hypothesis, because the classical Cauchy–Born hypothesis is restricted to three‐dimensional bulk structures and is thus not applicable to shell‐like structures. However, the approximations of the exponential Cauchy–Born hypothesis published so far are limited to structures with a planar reference configuration. In this paper, we present an extended approximation, which does not require the reference configuration to be planar and is thus applicable to arbitrarily shaped shell‐like nanostructures. A detailed elaboration of the related finite element implementation with important computational aspects is presented. Finally, the accuracy of the proposed method and its implementation is verified with several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.