2. Atomistic and continuum modeling of nanomaterials

2.1 Atomistic-continuum bridging methods for elastic nanomaterials

  • Standard Cauchy-Born rule is used to evaluate nonlinear constitutive relations for crystalline materials which have translational symmetry in them.

  • However, translational periodicity cannot be retained upon uniform twisting or arbitratly straining the nanorod/nanotubes

  • We proposed "Helical Cauchy-Born" rule to derive nonlinearly elastic constitutive laws for special Cosserat rod modeling of nano and continuum rods.

  • We first construct a 6-parameter (corresponding to the six strains in the theory of special Cosserat rods) family of helical rod configurations subjected to uniform strain along their arc-length. An explicit formula for the 6-parameter helical map is derived which maps atoms in the repeating cell of a nanorod to their images for the purpose of repeating cell energy minimization.

  • The bending, twisting, stretching, and shearing stiffnesses of diamond nanorods and carbon nanotubes are computed to demonstrate our theory.

A helical Cauchy-Born Rule for special Cosserat rods

2.2 Effect of material nonlinearity

  • We show the importance of incorporating material nonlinearity for accurate determination of spatial buckling of nanorods and nanotubes.

  • Both the nanorods and nanotubes are modeled as a special Cosserat rod whose nonlinear material laws are obtained using the recently proposed helical Cauchy-Born rule.

  • We present Euler buckling of solid diamond nanorods whose normalized buckling load, obtained from fully atomistic calculations, exhibits an interesting trend. The buckling load starts from unity at large aspect ratio of the nanorod, then as the aspect ratio is decreased, the buckling load increases slowly and finally decreases rapidly. We attribute this trend to material nonlinearity of the nanorod's core at large compressive strain. We also discuss how surface stress affects buckling in nanorods.

  • We then present the effect of compression and twist on buckling of single-walled carbon nanotubes. Interestingly, for highly twisted nanotubes, fully atomistic calculations show the first buckled mode to be different from a typical Euler buckling mode.

  • Both the observations about nanorods and nanotubes are accurately replicated in the finite element special Cosserat rod simulation when the material nonlinearity is also incorporated. However, the simulation results exhibit completely different trend when only linear material laws are incorporated.

Fully atomistic buckling simulations of Carbon Nanotubes

Reference Publication:
Gupta, P. , Kumar, A., 2017. Effect of material nonlinearity on spatial buckling of nanorods and nanotubes. Journal of Elasticity,126, 155-171.

2.2 Analytical study of surface stress effects on nanorods

  • We present a continuum formulation to obtain the effects of surface residual stress and surface elastic constants on extensional and torsional stiffnesses of isotropic circular nanorods.

  • Analytical expressions of axial force, twisting moment, extensional and torsional stiffnesses are obtained.

  • Unlike the case of rectangular nanorods, we show that the stiffnesses of circular nanorods also depend on surface residual stress components. This is attributed to non-zero surface curvature inherent in circular nanorods.

  • We further normalize these expressions and analyze their asymptotic limits in the limit of the nanorod's radius approaching both zero and infinity which correspond to surface dominated and bulk dominated regimes respectively.

  • Finally, we use the recently proposed Helical Cauchy-Born rule and perform molecular statics calculations to obtain axial force, twisting moment and stiffnesses of the tungsten nanorod. The tungsten material is selected since its bulk crystal exhibits isotropy in the stress-free state.

  • The results from molecular statics calculations are shown to match accurately with the derived continuum formulae.

Normalized extensional stiffness of tungsten nanorod

Normalized twisting stiffness of tungsten nanorod

Reference Publication:
 Gupta, P. , Kumar, A., 2018. Effect of surface elasticity on extensional and torsional stiffnesses of isotropic circular nanorods. Mathematics and Mechanics of Solids, 24(6), 1613-1629.

2.3 Phonons in chiral nanorods and nanotubes

  • A Cosserat rod based continuum approach is presented to obtain phonon dispersion curves of flexural, torsional, longitudinal, shearing and radial breathing modes in chiral nanorods and nanotubes.

  • Upon substituting the continuum wave form in the linearized dynamic equations of stretched and twisted Cosserat rods, we obtain analytical expression of a coefficient matrix (in terms of the rod’s stiffnesses, induced axial force and twisting moment) whose eigenvalues and eigenvectors give us frequencies and mode shapes, respectively, for each of the above phonon modes.

  • We show that, unlike the case of achiral tubes, these phonon modes are intricately coupled in chiral tubes due to extension-torsion-inflation and bending-shear couplings inherent in them. This coupling renders the conventional approach of obtaining stiffnesses from the long wavelength limit slope of dispersion curves redundant.

  • However, upon substituting the frequencies and mode shapes (obtained independently from phonon dispersion molecu lar data) in the eigenvalue-eigenvector equation of the above mentioned coefficient matrix, we are able to obtain all the stiffnesses (bending, twisting, stretching, shearing and all cou pling stiffnesses corresponding to extension-torsion, extension-inflation, torsion-inflation and bending-shear couplings) of chiral nanotubes.

  • Finally, we show unusual effects of the single-walled carbon nanotube’s chirality as well as stretching and twisting of the nanotube on its phonon dispersion curves obtained from the molecular approach. These unusual effects are accurately reproduced in our continuum formulation.

Longitudinal, torsional and radial breathing modes of a (10,10) SWCNT.


Bending frequency in the stress-free, axially compressed and twisted state.

Reference Publication:

Gupta, P. , Kumar, A., 2019. Phonons in chiral nanorods and nanotubes: a Cosserat rod based continuum approach. Mathematics and Mechanics of Solids, 24(12), 3897-3919.