Journal of Elasticity

, Volume 124, Issue 1, pp 81–106 | Cite as

A Helical Cauchy-Born Rule for Special Cosserat Rod Modeling of Nano and Continuum Rods

  • Ajeet Kumar
  • Siddhant Kumar
  • Prakhar Gupta


We present a novel scheme 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. The uniformity in strain then enables us to deduce the constitutive laws by just solving the warping of the helical rod’s cross-section (smallest repeating cell for nanorods) but under certain constraints. The constraints are shown to be critical in the absence of which, the 6-parameter family reduces to a well known 2-parameter family of uniform helical equilibria. 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. A scheme for the passage from nano to continuum scale is also presented to derive the constitutive laws of a continuum rod via atomistic calculations of nanorods. The bending, twisting, stretching and shearing stiffnesses of diamond nanorods and carbon nanotubes are computed to demonstrate our theory. We show that our scheme is more general and accurate than existing schemes allowing us to deduce shearing stiffness and several coupling stiffnesses of a nanorod for the first time.


Cauchy-Born rule Special Cosserat rod Elastic constitutive modeling Helical symmetry Molecular mechanics 

Mathematics Subject Classification

74B20 74A25 74Q15 



We thank the anonymous reviewers for their several useful suggestions. P. Gupta acknowledges support from the DST-INSPIRE fellowship.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Applied MechanicsIIT DelhiNew DelhiIndia
  2. 2.Department of Mechanical EngineeringIIT DelhiNew DelhiIndia

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