On an Exact Mapping and a Higher Order Born Rule for 

Use in Analyzing Graphene Carbon Nanotubes

by Michael J. Leamy, Peter W. Chung, and Raju Namburu

ARL-TR-3117 December 2003

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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5067

ARL-TR-3117 December 2003

On an Exact Mapping and a Higher Order Born Rule for 

Use in Analyzing Graphene Carbon Nanotubes

Michael J. Leamy

U.S. Military Academy

West Point, NY

Peter W. Chung and Raju Namburu

Computational and Information Sciences Directorate, ARL

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On an Exact Mapping and a Higher Order Born Rule for Use in Analyzing 

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14. ABSTRACT

As part of an effort to model single-walled graphene carbon nanotubes, a transformation is first described which maps atom 

locations originally on a planar sheet to atom locations on a cylindrical nanotube. The mapping is parametrized by the two 

components describing the chiral vector, and thus is valid for generating the atomic locations of arbitrary chirality nanotubes. 

Attention is then turned towards quantifying the dimensions of the unit cell, again parametrized by the chiral vector’s two 

components. Taken together, the mapping plus the unit cell dimensions are used to generate unique atomic locations of 

arbitrarily long carbon nanotubes. A second mapping is described which generalizes the commonly used Born rule to higher 

orders. The Born rule is an effective tool for linking bulk material deformations to atomic displacements, particularly for 

homogenous deformations. For nonhomogenous deformations, such as those involving nonzero curvature, the standard Born 

rule can inaccurately link the two scales. The higher order mapping described herein allows nonhomogenous deformations to be 

mapped down to the atomic scale to an arbitrary degree of precision.

15. SUBJECT TERMS

Born rule, nanotube, atomistic/continuum, kinematics

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19a. NAME OF RESPONSIBLE PERSON

Peter W. Chung

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410-278-6027

Standard Form 298 (Rev. 8/98)

Prescribed by ANSI Std. Z39.18
iii

Contents

List of Figures iv

Acknowledgments v

1. Background 1

2. Introduction 1

3. General Chirality Graphene Sheet to Nanotube Mapping 4

3.1 Mapping...........................................................................................................................4

3.2 Unit Cell ..........................................................................................................................8

4. On a Higher Order Born Rule 9

5. Conclusions 12

6. References 13

Distribution List 15
iv

List of Figures

Figure 1. Lattice geometry for the graphene sheet..........................................................................5

Figure 2. Mapping Φ taking planar manifold P to cylindrical manifold C . ...............................6

Figure 3. Cylinder (blue) generated using mapping 2 3 Y :(n, m; N, M )∈ Z a ( x, y,z)∈R , where 

(N = 9, M = 5) chirality has been chosen and where the range of indices used to generate 

the figure are − 5 ≤ n ≤ 5, 0 ≤ m ≤ 8. The red plane indicates the locus of (X, Y, Z) 

locations using the given indices and before applying the map.................................................8

Figure 4. Results of procedure for determining the indices in the unit cell (left column) and 

subsequent mapping (right column) for chiral vectors (a) N = 5, M = 5, (b) N = 6, M = 3, 

and (c) N = 9, M = 0.................................................................................................................10
v

Acknowledgments

The authors gratefully acknowledge partial support through the U.S. Army Research Laboratory 

(ARL)-sponsored Mathematical Sciences Center of Excellence within the Department of 

Mathematical Sciences at the U.S. Military Academy, LTC Tyge Rugenstein, the ARL Director's 

Research Initiative (DRI-FY02-CIS-01), and additional support through the Computational 

Sciences & Engineering Branch, High Performance Computing Division of ARL.
vi

INTENTIONALLY LEFT BLANK.
1

1. Background

As part of an effort to model single-walled graphene carbon nanotubes, a transformation is first 

described which maps atom locations originally on a planar sheet to atom locations on a 

cylindrical nanotube. The mapping is parametrized by the two components describing the chiral 

vector, and thus is valid for generating the atomic locations of arbitrary chirality nanotubes. 

Attention is then turned towards quantifying the dimensions of the unit cell, again parametrized 

by the chiral vector’s two components. Taken together, the mapping plus the unit cell 

dimensions are used to generate unique atomic locations of arbitrarily long carbon nanotubes.

A second mapping is described which generalizes the commonly used Born rule to higher orders. 

The Born rule is an effective tool for linking bulk material deformations to atomic 

displacements, particularly for homogeneous deformations. For nonhomogeneous deformations, 

such as those involving nonzero curvature, the standard Born rule can inaccurately link the two 

scales. The higher order mapping described herein allows nonhomogeneous deformations to be 

mapped down to the atomic scale to an arbitrary degree of precision. 

The two mappings described herein are anticipated to be important developments leading to 

accurate deformation modeling of graphene carbon nanotubes of arbitrary chirality: the first 

mapping will allow for the generation of the initial atomic locations of the undeformed carbon 

nanotube, and the second mapping will allow subsequent deformations of the nanotube to be 

accurately modeled at the atomic level. 

2. Introduction

The Born rule is an important component in the development of continuum models of atomic 

lattices. It serves as an elementary kinematic assumption that establishes a connection between 

atomic motion and continuum gradients, and its origins can be found in the work of Cauchy

(1828a, 1828b, 1829a, 1829b), who hypothesized that the gradients of the deformation agree 

with the motion of molecules in a deforming solid. Later, in his first book on dynamics of 

crystal lattices, Born generalized the hypothesis of Cauchy by allowing for the separate 

relaxation of internal microstructure (Born, 1915). This text was later translated into English 

(Born and Huang, 1954). Accounting for internal relaxation has important implications for many 

types of crystal structures and crystal/continuum mechanics, e.g., in elastic properties of crystals 

in the absence or upon removal of centrosymmetry(Cousins, 1978a, 1978b, 1979, 1982; Cousins 

and Martin, 1978), in deformations involving phase transformations (Ericksen, 1984; Zanzotto, 

1992, 1996), in contributions to the elastic properties from n-body interactions (Martin, 1975a,

1975b, 1975c), local instabilities in the deforming Bravais lattice (Friesecke and Thiel, 2002),
2

and in generalized schemes for transitioning atom mechanics to continuum mechanics (Friesecke 

and James, 2000), among others. 

In general, a crystalline lattice L in its reference (unloaded) configuration is composed of a 

number of interpenetrating Bravais lattices whose points are given by

i k

i X = M a + p , with i = 1, 2, 3, Mi ∈ Z, (1)

where X are the lattice points, ai are the linearly independent lattice vectors or Bravais base 

vectors, and pk are the shift vectors for the inner atoms. For N + 1 atoms in the basis, the index 

k runs from 0 to N. The base vectors ai generate a so-called skeletal Bravais lattice S⊂L

where each point in S along with the k-shifted atoms compose a cluster, or the so-called 

microstructural motif (see Zanzotto, 1992). When the crystal is undeformed, the base vectors are 

maximal or, namely, they represent the smallest possible periodic unit of the crystal. When the 

crystal is subjected to a deformation, the base vectors ai are no longer maximal in general. For 

this reason, let us call { M M Z}

i

i

i R = X = a , ∈ the maximal skeletal lattice of L . Let D be 

the set of all allowable deformation gradient tensors F and let R ’ be the maximal skeletal 

lattice in the deformed configuration generated by the set of lattice vectors i a' . Then the formal 

definition of the Born rule is

FR = R′ , i.e., i

j

a′

i = mi Fa with (m ) GL( Z )

j

i ∈ 3, , for all F∈D. (2)

The notation GL(3, Z) comes from group theory referring to the classical linear group of 

invertible 3 × 3 matrices with integer coefficients, which effectively offers the ability to select 

larger sublattices that are maximal in their representation of crystal periodicity in the context of 

deformations. 

For completeness, it is noteworthy to mention the hypothesis of Cauchy, which is more 

restrictive than Born’s and states that for all F∈D, L ‘ is the deformed configuration 

corresponding to the undeformed L, such that FL = L ‘. The so-called Cauchy rule, or CauchyBorn rule, says that the atomic motion agrees with the gross deformation of the crystal, whereas 

the Born rule says that only the motion of the skeletal lattice agrees with the gross deformation 

and the microstructural motif is free to relax into its appropriate equilibrium configuration. It 

follows then that when the base vector of the skeletal lattice corresponds to the vector connecting 

nearest neighbors in a single species crystal, the Cauchy and Born hypotheses are equivalent.

In general deforming crystals, the Born rule for a maximal lattice R can be surprisingly 

restrictive despite the freedom to choose a maximal skeletal lattice in L. In particular, and as is 

discussed further, the linear nature of the rule prevents it from being applied to certain types of 

reduced dimension problems undergoing nonhomogeneous deformations, phenomena involving 

diffusion, lattice shuffles, or local instabilities (Cousins and Martin, 1978; Zanzotto, 1996; 

Friesecke and Thiel, 2002; Arroyo and Belytschko, 2002). Another possible approach for 
3

connecting atomic motion and continuum deformations is described by Zanzotto (1992) through

the so-called weak Born rule, which allows greater flexibility in the selection of the sublattices 

but is still subject to similar requirements as the Born rule.* 

In spite of some of its limitations, the Born rule continues to be applied in advanced 

computational methods with success (see Tadmor et al., 1996; Shilkrot et al., 2002). In general, 

the rule has been found to yield excellent results in three-dimensiona l (3-D) bulk or twodimensional (2-D) planar materials. The concern in this report, however, is in its application to 

reduced dimensions such as thin films embedded in 3-D space. The motivation for such 

problems arises naturally in contemporary problems of atomistic nanotubes and thin films, where 

the nanotube geometry can be constructed from a planar graphene sheet. The key issue in the

modeling of such applications is the kinematic hypothesis inherent in the Born rule, which 

assumes that some level of homogeneity exists in the deformation map. While the concept of 

homogeneous deformation is quite standard for a 3-D body, it is not the same for generalized 

geometric surfaces. These objections were noted by Arroyo and Belytschko (2002) who 

recognized a fundamental distinction between (1) tangents of the undeformed and deformed 

bodies that are mapped from one to the other via the deformation gradient F and (2) the chords 

created by lines connecting two atoms in a curved manifold. The chords are of critical 

importance because they determine the relative energy of an atom to its neighbor. For 

generalized geometric surfaces, chords are not mapped correctly by the deformation gradient.

Arroyo and Belytschko recognized the need to map chords correctly for generalized geometric 

surfaces and therefore proposed an alternative, but approximate, method to handle Born rule-like

kinematics on curved manifolds using the so-called exponential Born rule. At its core is an 

exponential map that maps the tangent of a manifold back into the manifold itself, thereby 

intrinsically parameterizing the surface with the local tangents. The scheme serves as a 

correction step following the application of the traditional Born rule. Although effective, it 

requires knowledge of the surface via geodesics which, in general, involves the solution of a 

nonlinear system of ordinary differential equations whose unknowns are the parametric 

coordinates of the geodesic. For the special case of a surface that forms a uniform cylinder, the 

calculation is straightforward and can be done analytically when accompanied by a geometric 

approximation regarding atom placement on a curved manifold (Arroyo and Belytschko, 2003). 

The end result is a membrane theory based only on the geodesic curvature of the cylinder, in the 

form of a correction that accurately maps one generalized cylindrical surface into another. It 

would be useful for nanotube applications to explicitly consider the nanotube chiral indices 

(Harris, 1999) into the mapping scheme. 

 

*Our definition of the Born rule differs slightly from that of Zanzotto (1992). We presume that the Born rule pertains only to 

the mapping of the skeletal Bravais lattice and not to the selection of the sublattice itself. The “requirements” mentioned here are 

in reference to the assumptions of homogeneous deformations in the absence of instabilities, diffusion, and preservation of a 

Euclidean metric.
4

In this report, we propose an alternative method to the Born rule and exponential Born rule for 

general manifolds. In doing so, we explicitly consider nanotube indices for specific application 

to carbon nanotubes and develop the closed-form expressions for mapping an arbitrary graphene 

sheet into a cylindrical configuration. The larger aim is to determine the two-axis bending 

rigidity and torsional rigidity of a graphene carbon nanotube of arbitrary chirality. As a first step

toward realizing this aim, attention is initially focused on developing a mapping for graphene 

sheets of arbitrary chirality which take Bravais indices (n, m) and locate atomic positions

(x, y, z) on the surface of a cylindrical carbon nanotube. In order to ensure a mapping which is 

one-to-one, bounds on (n, m) are developed which define the domain of the so-called unit cell. 

The second part of this report details the proposed mapping, termed herein the higher order Born 

rule, intended to replace the standard Born rule when linking bulk material deformations to

atomic deformations.* In contrast to the exponential Born rule, the higher order Born rule 

eliminates the need for a correction step and is considerably quicker (in a computational sense) 

and easier to implement. Depending on the order to which the rule is taken, it can be used to 

yield a scale-linking rule of arbitrary precision. 

3. General Chirality Graphene Sheet to Nanotube Mapping

3.1 Mapping

In this section, a mapping 2 3 Ψ :(n,m) ∈ Z a (x, y,z) ∈ R is developed, taking integers n and m, 

indicating the usual Bravais indices for a graphene sheet, to position (x, y, z) on a manifold N,

representing the surface of a graphene carbon nanotube. Specifically, consider the planar 

graphene sheet of figure 1 in which a1 and a2 denote Bravais base vectors, C = Na1 + Ma2

denotes the chiral vector, T N a1 M a2 = ′ + ′ denotes the translation vector normal to C, and q

indicates the angle C makes with the reference Cartesian frame (X, Y, Z) with unit vectors i

and j . The length of the chiral vector can be found in any standard reference on lattice 

geometry,

2 2 C = C = l

0 N + NM + M , (3)

where 0l denotes the length of the basis vectors a1 and a2, which for graphene is 0.246

o

A . The 

length of T determines the unit cell, defined to be the smallest rectangle defined by C and a

translate of C such that all four corners of the unit cell coincide with an atomic lattice point. 

 

*We should also note that at the time of writing, another paper (Sunyk and Steinmann, 2003) on a higher order gradient 

concept appeared that discussed the application of the Cauchy-Born rule to inhomogeneous in-plane (2-D) deformations of 

atomistic/continuum problems. The work presented herein, however, differs in the generalization of the Born rule for arbitrary 

precision and its application for reduced dimensions problems in a full 3-D sense.
5

Figure 1. Lattice geometry for the graphene sheet.

This length is also well documented, see for example Harris (1999), and is given by,

( ) 





− =

− ≠

= = ′ + ′ ′+ ′ =

H H

H H

C d N M zd

C d N M zd

T l N N M M

3 / 3 3

2 2 3 / 3

T 0

, (4)

where d H denotes the highest common divisor of N and M and z denotes any integer. Any 

lattice point on the graphene sheet can be located with a position vector r and lattice indices n

and m, or chiral and translational coordinates c and t, 

c t

r = na + ma = ce + te 1 2, (5)

where unit vectors along C and T are denoted by ce and te , respectively.

Based on this description of the graphene sheet, a mapping can be developed which describes the 

positions of atoms on a carbon nanotube formed by rolling-up the graphene sheet along C, such 

that the two end-points of C occupy the same point on the nanotube, and such that T indicates 

the tubular direction. The development of this mapping starts first with the simple map Φ

taking points 3(X ,Y, Z) ∈ R to 3(x, y,z) ∈ R ,

x (X,Y, Z) rsin(X /r) = Φ1 = , y = Φ2( X,Y, Z) = Y , z = Φ (X,Y ,Z) = r cos(X /r)− r 3, (6)

which, when applied to points on the planar manifold P, P = {( X,Y,0) | 0 ≤ X ≤ r,0 ≤ Y ≤ l}, 

yields points on a new cylindrical manifold C with radius r and axis {(0, y,r)| 0 ≤ y ≤ l}, as 

shown in figure 2. 

X

Y

C

T

a1

a2

(N, M)

(1,0) (2,0) (3,0)

(0,1) (1,1)

(-1,2) (0,2) (1,2) (2,2)

(N′, M′)

Unit Cell
6

Figure 2. Mapping Φ taking planar manifold P to cylindrical manifold C .

The mapping just described is that required to roll-up a graphene sheet along chiral vectors given 

by (N, 0). To roll-up the sheet along any general chiral vector (N, M), Φ must be applied first to 

points referenced to a coordinate system aligned with the chiral and translational directions, i.e., 

3

(c,t,Z) ∈ R , thereby rolling up an angled sheet Pq (angle to the horizontal measured by q) into 

an angled cylinder C q. Following this initial mapping, a second mapping C Φ (corresponding to 

a coordinate transformation) must be applied to transform the new points on C q referenced to 

the chiral and translational axes to points referenced to the (X, Y, Z) system. The composition 

mapping is denoted by 3 3: (c,t, Z) R (x, y,z) R

R C Ψ = Φ o Φ ∈ a ∈ and is given as 

( , , ) cos(q )sin ( / ) sin (q ) 1x c t Z r c r t

R

= Ψ = − ,

( , , ) sin (q )sin ( / ) cos(q ) 2y c t Z r c r t

R

= Ψ = + ,

and

z c t Z r (c r) r

R

= Ψ3( , , ) = cos / − , (7)

where : (c,t, Z) (x, y,z) Φ

C a is represented in matrix form by







































 −

=





















Z

t

c

z

y

x

0 0 1

sin cos 0

cos sin 0

q q

q q

, (8)

Y

X

(r,l)

Z

X

Z

Φ

P C

Y
7

and where a superscript R indicates this mapping holds for points in 3 R and not in 2Z , as 

desired.

The desired mapping Ψ follows closely from R Ψ . Noting that the Bravais base vectors for 

graphene are given as

a i

1 0 = l , a i j2 0 0

2

3

2

1

= l + l , (9)

expressions for coordinates (c, t) can be developed using equations 5, 8, and 9,

sin( )

2

3

cos( )

2

0 q 0 q

m

l

n m

c l +

+

= , cos( )

2

3

sin( )

2

0 q 0 q

m

l

n m

t l +

+

= − , (10)

while expressions for q and r can be developed in terms of chiral indices N and M, 

















+ +

=

−

2 2

1 3

2

1

sin

N NM M

M

q ,

p

2 2

0

2

1 l N NM M

r

+ +

= . (11)

The desired map 2 3 Ψ :(n, m;N,M ) ∈ Z a (x, y,z) ∈ R is then arrived at after further 

manipulation,

( )

( )

























+ +

 + + − +











+

= = + + − 2 2

2 2 2

2 2 2

1 0

2

3

4 3

2

4 3 sin

4

1

( , ; , )

N NM M

N NM M M mM mn

x n m N M l N NM M M

p p

p

Y

( ) ( )

2 2

2 2 2 2

0

3 2 3 4 3

8 N NM M

l n m M mM N NM M M

+ +

+ − + + −

+ ,

( )

























+ +

 + + − +











+

= = 2 2

2 2 2

2 0

2

3

4 3

2

3 sin

4

1

( , ; , )

N NM M

N NM M M mM mn

y n m N M l M

p p

p

Y

( ) ( )

( )

2 2 2

2 2

2 2 2

0 4 3

2 3 3 4 3

8

N NM M M

N NM M

l n m M m M N NM M M

+ + −

+ +

+ − + + −

− ,

( )

























+ +

 + + − +











+

= = + + 2 2

2 2 2

2 2

3 0

2

3

4 3

2

cos

2

1

( , ; , )

N NM M

N NM M M mM mn

z n m N M l N NM M

p p

p

Y

0 2 2

2

N NM M

l

− + +

p

, (12)
8

where arguments after the ; indicate chiral parameters. An example of the map is shown in 

figure 3 for a (9, 5) chiral vector. 

Figure 3. Cylinder (blue) generated using mapping 

2 3 Y :(n, m; N, M )∈ Z a ( x, y,z)∈R , where 

(N = 9, M = 5) chirality has been chosen and 

where the range of indices used to generate the 

figure are − 5 ≤ n ≤ 5, 0 ≤ m ≤ 8. The red plane 

indicates the locus of (X, Y, Z) locations using 

the given indices and before applying the map.

3.2 Unit Cell

Application of Ψ maps points from a plane to a cylinder, but without proper choice of the 

indices ranges for (n, m), results in an overdetermination of the cylindrical manifold—i.e., the 

map is not one-to-one, as more than one point on the plane is mapped to the same point on the 

cylinder. To prevent this occurrence, bounds on the indices are developed next which ensure a 

domain on the plane which is mapped one-to-one onto the cylinder.

Attention is focused on the unit cell of figure 1 in order to develop the indices ranges. For chiral 

vectors with associated q < 60°, the minimum index mmin for which a lattice point lies in the 

unit cell is 0, while the minimum index nmin is determined by N′ . Expressions for N′ and M ′

can be developed using the translation vector and its magnitude, 

( ) 1 2 T = T − sin q i + cosq j = N′a + M ′a , (13)

with equation 9. The desired expressions are













= ′ = − q − cosq

3

1

sin

0

min l

T

n N , cosq

3

2

0

l

T

M ′ = . (14)
9

The maximum index nmax is simply N, while the maximum index mmax is determined by the m

index of a vector formed by the addition of T and C—i.e., the top-right corner of the unit cell. In 

order to determine this index, it is convenient to introduce the dual basis vectors 1a and 2a such 

that j

i

j

i

a ⋅ a = d where the center dot denotes the Cartesian inner product, 

















a = i − j

3

1 3

0

1

l

,

















a = j

3

1 2 3

0

2

l

. (15)

Then,

2

max m = (T +C) ⋅ a

M

l

T

= cosq +

3

2

0

. (16)

Note that use of equations 4 and 11 in equations 14 and 16 yields indices bounds 

(nmin ,nmax ,mmin ,mmax ) which are only functions of the chiral vector’s N and M. 

With bounds identified for the indices m and n, two conditions must still be met on any pair of 

indices ( ) min max min max n ≤ n ≤ n ,m ≤ m ≤ m to ensure their existence inside of the unit cell: 

0 ≤ c ≤ C and 0 ≤ t ≤ T where it is recalled that c and t are chiral and translational coordinates 

corresponding to (n, m) and are given by equation 10 while C and T are the chiral and translation 

vector magnitudes given byequations 3 and 4. Given a chiral vector (N, M), a single nested loop 

can be easily programmed using the minimum and maximum bounds on n and m plus the 

conditional checks to generate a list of index pairs of atoms located in the unit cell. The mapping 

Ψ given in equation 12 can then be applied to locate the atoms on the nanotube manifold N. 

Results of this procedure are shown in figure 4 where unit cells and nanotubes for several 

common chiralities are shown.

4. On a HigherOrder Born Rule

The Born rule is commonly employed to determine atomic lattice positions from the bulk 

material deformation gradient F. Specifically, letting Ri denote the position vector of the i

th atom 

in the continuum before deformation, and letting ri denote the position of the same atom after 

deformation; the Born rule relates the two,

i F Ri

r = ⋅ , (17)

where F referenced to Cartesian coordinates (X1, X2, X3) in the undeformed configuration and

(x1, x2, x3) in the deformed configuration can be stated as
10

y 

(

o

A )

(a)

y 

(

o

A )

(b)

y 

(

o

A )

x ( o

A )

(c)

Figure 4. Results of procedure for determining the indices in the unit cell (left column) and subsequent mapping 

(right column) for chiral vectors (a) N = 5, M = 5, (b) N = 6, M = 3, and (c) N = 9, M = 0. 
11

i J

J

i

iJ i J

X

x

F e e e e

R

r

F ⊗

∂

∂

= ⊗ =

∂

∂

= . (18) 

As discussed by Arroyo and Belytschko (2002), the deformation gradient maps points on the 

tangent space associated with the undeformed configuration to points on the tangent space 

associated with the deformed configuration, and therefore inadequately maps deformations with 

nonzero curvature. They address this deficiency by applying the Born rule as stated in 

equation 17, and then correcting ri through the exponential mapping. Note that this correction 

follows the application of the Born rule. Furthermore, by their own account, the implementation 

of the exponential map is “not straight forward” and requires additional simplifications. Herein, 

a higher order Born rule is proposed which improves upon the standard Born rule by improving 

the initial mapping from Ri to ri, obviating the need for a follow-up correction, in contrast to the 

approach of (Arroyo and Belytschko, 2002).

As opposed to equation 17, the differential element dR maps to dr exactly through the rule

dr = F ⋅ dR , (19) 

which can be integrated once to obtain,

∫

r = F⋅ dR . (20)

Further integration of the right-hand side requires an expression for F = F(R) . It is chosen to 

expand F in a Taylor series about R = 0 without loss of generality,

( ) ( ) ( )

2 R R

R

F

F R F 0

R 0

⋅ + O

∂

∂

= +

=

. (21)

Substitution of equation 21 into 20, and denoting the constant third-order tensor

R R 0

F

=

∂

∂

as K ,

yields the desired higher order Born rule,

( ) ( )

2

r = F 0 ⋅R + ⋅ R ⋅dR + O R ∫K . (22) 

Comparison of equation 22 with the standard Born rule (equation 17) reveals the underlying 

assumption of the Born rule—that the deformation gradient is constant over all R in which it is 

applied. This is an adequate approximation for homogeneous or small-gradient deformations 

like expansion, contraction, or shear, but may be inadequate for high-gradient deformations or 

for bending-like deformations with nonzero curvature. 

Next, an example deformation mapping is exercised in order to demonstrate the utility of the 

higher order Born rule. Consider again the mapping Φ illustrated in figure 2, taking points 

3

1 2 3

(X , X , X ) ∈ R to 3

1 2 3

(x , x , x )∈ R ,
12

x (X , X , X ) rsin (X /r) 1 = Φ1 1 2 3 = 1, 2 2 1 2 3 2x = Φ (X , X , X ) = X , 

x = Φ (X , X , X ) = r cos(X / r) − r 3 3 1 2 3 1. (23)

The deformation gradient for this map is

( )

( ) 



















−

=

sin / 0 0

0 1 0

cos / 0 0

1

1

X r

X r

F , (24)

while the only nonzero component of K is 1/r K 311 = − . Note that this indicates negative 

curvature in the z-x plane with radius of curvature r for all ( , , ) X1 X2 X3, as expected. Note 

further that the higher order Born rule (equation 22) for this example evaluates as

( )

2

1 2

3

2

1

r 1e1 2e2e O R

X

r

= X + X − + , (25)

where the nonzero 3e coefficient demonstrates the ability of the higher order Born rule to more 

accurately map points from the plane to the cylinder. If desired, further accuracy can be obtained 

by taking more terms in the expansion (equation 21). In contrast, the standard Born rule for this 

mapping is inadequate and would only map the plane onto itself. 

5. Conclusions

Two important mappings have been developed in an effort to accomplish the larger goal of 

determining the two-axis bending and torsional rigidity of graphene carbon nanotubes. The first 

mapping allows atomic lattice coordinates to be generated for graphene carbon nanotubes of 

arbitrary chirality. In addition to the mapping, bounds on the Bravais indices have been 

developed (thereby describing a unit cell) which ensure the mapping is one-to-one. The second 

mapping developed allows subsequent deformations of the nanotube to be linked to atomic 

displacements using the newly termed higher order Born rule. Although the new rule’s use has 

been demonstrated in the context of graphene carbon nanotubes, it is expected to be equally 

valuable in the study of mechanics associated with any generalized geometric surface of reduced 

dimension, which is embedded in 3-D space.
13

6. References

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Arroyo, M.; Belytschko, T. A Finite Deformation Membrane Based on Inter-Atomic Potentials 

for the Transverse Mechanics of Nanotubes. Mechanics of Materials 2003, 35, 193–215. 

Born, M. Dynamik der Kristallgitter; Teubner: Leipzig, Berlin, 1915.

Born, M.; Huang, K. Dynamical Theory of Crystal Lattices; Clarendon Press: Oxford, 1954.

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Mathematiques. Available in Cauchy, Augustin Louis Oeuvres Complètes, Tome 20, 1828a; 

pp 253–277.

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de Points Matériels Sollicités par des Forces D’attraction ou de Répulsion Mutuelle. 

Exercices De Mathematiques. Available in Cauchy, Augustin Louis Oeuvres Complètes, 

Tome 21, 1829b; pp 162–173.

Cousins, C. S. G.; Martin, J. W. Extended Cauchy Discrepancies: Measures of Non-Central, 

Non-Isotropic Interactions in Crystals. J. Phys. F: Metal Phys. 1978, 8 (11), 2279–2291.

Cousins, C. S. G. Inner Elasticity. J. Phys. C: Solid State Phys. 1978a, 11, 4867–4879.

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1978b, 11, 4881–4900.

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Ericksen, J. L. The Cauchy and Born Hypotheses for Crystals. In Phase Transformations and 

Material Instabilities in Solids. Proceedings of a Conference Conducted by the Mathematics 

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pp 61–77.

Friesecke, G.; James, R. D. A Scheme for the Passage From Atomic to Continuum Theory for 

Thin Films, Nanotubes and Nanorods. J. Mech. Phys. Solids 2000, 48, 1519–1540.

Friesecke, G.; Theil, F. Validity and Failure of the Cauchy-Born Hypothesis in a TwoDimensional Mass-Spring Lattice. Journal of Nonlinear Science 2002, 12 (5), 445–478.

Harris, Peter J. F. Carbon Nanotubes and Related Structures. Cambridge University Press: 

Cambridge, UK, 1999. 

Martin, J. W. Many-Body Forces in Metals and the Brugger Elastic Constants. J. Phys. C: Solid 

State Phys. 1975a, 8, 2837–2857.

Martin, J. W. Many-Body Forces in Metals and the Brugger Elastic Constants: II. Inner Elastic 

Constants. J. Phys. C: Solid State Phys. 1975b, 8, 2858–2868.

Martin, J. W. Many-Body Forces in Solids: Elastic Constants of Diamond-Type Crystals. 

J. Phys. C: Solid State Phys. 1975c, 8, 2869–2888.

Shilkrot, L. E.; Curtin, W. A.; Miller, R. E. A Coupled Atomistic/Continuum Model of Defects 

in Solids. J. Mech. Phys. Solids 2002, 50, 2085–2106.

Sunyk, R.; Steinmann, P. On Higher Gradients in Continuum-Atomistic Modeling.

International Journal of Solids and Structures 2003, 40, 6877–6896.

Tadmor, E. B.; Ortiz, M.; Phillips, R. Quasicontinuum Analysis of Defects in Solids. Phil. Mag. 

A 1996, 73 (6), 1529–1563.

Zanzotto, G. On the Material Symmetry Group of Elastic Crystals and the Born Rule. Arch.

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