A New Extension of Cauchy–Born Rule for Monolayer Crystal Films
 Sheng Lu^{1} and
 Chongdu Cho^{1}Email author
DOI: 10.1007/s1167101095763
© The Author(s) 2010
Received: 23 February 2010
Accepted: 5 March 2010
Published: 23 March 2010
Abstract
By combining with the physical concept of inscribed surface, the standard Cauchy–Born rule (CBR) is straightly extended to have a rigorous and accurate atomistic continuum theory for the monolayer crystal films. Resorting to using Tersoff–Brenner potential, the present theory to graphite sheet and singlewalled carbon nanotubes (SWCNTs) is applied to evaluate the mechanical properties. The results are validated by the comparison with previously reported studies.
Keywords
Carbon nanotubes Cauchy–Born rule Monolayer crystalline film Inscribed surface Mechanical propertiesIntroduction
Based on atomistic simulations, a series of values of mechanical properties of CNTs are given and agreed qualitatively with experimental studies, but such methods quickly become computationally extremely demanding as the number of atoms increases. In other words, there exists a rigorous limitation on atomistic simulation by its time and length scales. So, a direct link between the continuum analysis and atomistic simulation is needed to investigate the mechanical properties of CNTs more effectively. In general, the socalled CBR [1, 2] is viewed as the fundamental assumption for linkage between the deformation descriptions of crystal configurations and the continuum theories of crystal mechanics. Without considerations of diffusions, phase transitions, lattice defects, slips, or other nonhomogeneities, it is quite suitable for the spacefilling materials. In a spacefilling material, the crystal deformation is homogenous and continuous at the atomic scale, and the lattice vector and its tangent are coincident. Thus, CBR is widely accepted as the form of a = F·A, where F denotes a twopoint deformation gradient tensor, A and a denote one lattice vector on the respective undeformed and deformed crystals.
However, the standard CBR fails to extend directly to the case of monolayer crystal films. In short, if the monolayer crystal film is treated as a surface, the deformation gradient F maps the tangent space of the surface and the lattice vectors are regarded as chords of the surface. Obviously, the finite length lattice vector does not fall into the tangent space of infinitesimal material vectors, so the deformation gradient F cannot give an accurate description of relationship between the undeformed and the deformed lattice vectors.
To generalize the standard CBR in the monolayer crystal films, two main type modifications are developed until now. In the study of finite crystal elasticity for curved single layer lattices, Arroyo and Belytschko [3, 4] developed the exponential CBR. First, the undeformed lattice vector A is mapped into the tangent space by the exponential inverse mapping to get an undeformed finite line element. After the deformation calculations of finite line element on the tangent space, the deformed finite line element on the tangent space is pulled back to the deformed surface by the exponential mapping to determine the deformed lattice vector a. In practice, it requires the knowledge of the geodesic curves of the surface. That means much computational cost is needed to pay to solve a set of nonlinear partial differential equations. Besides, on the deformation gradients in atomistic continuum modeling, Sunyk and Steinmann took into account the second quadratic term in the Taylor’s series expansion of the deformation field [5]. Then, Guo et al. [6] and Wang et al. [7] performed the higher order CBR for predicating the mechanical properties of SWCNTs. For the monolayer films, they illuminated that higher order term of the deformation gradient can pull the tangent vector close to the deformed manifold. Sometimes, this method has to be in the face of convergence problem. Both of two modified CBRs mentioned in the above are approximate methods from mathematics. However, based on the physical concept of inscribed surface, the present study discovered that the standard CBR can be extended straightly to describe monolayer crystal films accurately.
New Extension of CBR
Interestingly, the chord AB of circumcircle C_{1} exactly is the tangent of the inscribed circle C_{2} at any time of deforming. That is to say, the deformation gradient of the lattice vector AB maps the tangent space of the inscribed circle C_{2} at tangent point m. In other words, the variation of the bond (the chord of the circumcircle C_{1}) can be investigated by the deformation gradient (the tangents) of the inscribed circle C_{2} at the tangent point.
The extensions of CBR for different types of materials
Monolayer crystal film  Spacefilling material  

Noncentrosymmetric  Centrosymmetric  Noncentrosymmetric  Centrosymmetric 

 a = F·(A + )  a = F·A 
Applications to Graphite Sheets and SWCNTs
From the outer dashed line (the atomic surface without considering the inner shift) to the outer solid line (the atomic surface with considering the inner shift), there is no change except that the radius of circle becomes a little larger. This is so called the phenomenon of relaxation. However, by comparing with the inner dashed curve (the inscribed surface without considering the inner shift), the inner solid curve pqr (the inscribed surface with considering the inner shift) has obvious changes in shape to average the curvature distribution. From the view of energy, the curve with small curvature will be much steadier than with large curvature. So it is not difficult to say that the inner shift gives the deformation gradient a selfadjustment to uniformly distribute the energy. This adjustment of energy distribution of the inscribed surface can be comprehended as the physical origin of relaxation.
where a is the equilibrium bond length of graphite sheet, r _{ a } and r _{ z } denote the respective radii of armchair CNTs and zigzag CNTs. Obviously, the radii of SWCNTs depend on a pair of parameters (n,m) as well as the inner shift .
Conclusions
In summary, the present study has discovered that the mechanical phenomena of monolayer crystal films are governed by their inscribed surface, not atomic surface. Based on this congenital advantage of inscribed surface, a new extension of CBR, called inscribed CBR, is proposed to build a rigorous and accurate atomistic continuum theory. It straightly connects the continuum mechanics with monolayer crystal films at nanoscale. Applications of this theory to the graphite sheet and SWCNTs are validated by previously reported empirical studies.
Declarations
Acknowledgment
This work was supported by Inha University.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Authors’ Affiliations
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