A modified approximation of the exponential Cauchy-Born rule for arbitrary shell-like nanostructures

Image: Julian Ochs

C. Findeisen, J. Ochs, J. Wackerfuß

The mechanical behavior of shell-like nanostructures, such as carbon nanotubes (CNT) can be described within the continuum mechanics by a generalization of the Cauchy-Born (CB) rule, introduced by Arroyo and Belytschko. However, the finite element approximation presented in the literature requires the reference configuration to be planar. In case the configuration of interest is initially not planar, a preprocessing step is required that isometrically maps the planar reference configuration into the initial configuration. In order to calculate more complex structures, in this project an extended finite element approximation has been developed which is applicable to shell-like atomic structures, whose reference configuration does not need to be planar and consequently the afore-mentioned steps are no longer necessary. The accuracy of the proposed model has been verified by different numerical examples.

Concept of the exponential Born hypothesis
Deformation of a junction consisting of 3the carbon nanotubes under axial loading (top view)
(side view)
Comparison of the numerical results (continuum model vs. full atomistic model)


Publications

    • Ochs, J., Wackerfuß, J., 2023. Continuous modelling of single layer 2D carbon nanostructures based on a modified Cauchy–Born rule. PAMM Proceedings in Applied Mathematics and Mechanics 23, e202300197. https://doi.org/10.1002/pamm.202300197
    • Findeisen, C., Wackerfuß, J., 2016. A general approximation of the exponential Cauchy-Born hypothesis to model arbitrarily shaped shell-like nanostructures within continuum mechanics. International Journal for Numerical Methods in Engineering 105, 747-780. https://doi.org/10.1002/nme.4992
    • Findeisen, C., Wackerfuß, J., 2014. A modified approximation of the exponential Cauchy-Born rule forarbitrary shell-like nanostructures. PAMM Proceedings in Applied Mathematics and Mechanics 14, 565-566. https://doi.org/10.1002/pamm.201410270