May 2001

NASA/TM-2001-210863

Equivalent-Continuum Modeling of

Nano-Structured Materials

Gregory M. Odegard and Thomas S. Gates

Langley Research Center, Hampton, Virginia

Lee M. Nicholson

ICASE

Langley Research Center, Hampton, Virginia

Kristopher E. Wise

Langley Research Center, Hampton, Virginia
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May 2001

NASA/TM-2001-210863

Equivalent-Continuum Modeling of

Nano-Structured Materials

Gregory M. Odegard and Thomas S. Gates

Langley Research Center, Hampton, Virginia

Lee M. Nicholson

ICASE

Langley Research Center, Hampton, Virginia

Kristopher E. Wise

Langley Research Center, Hampton, Virginia
Available from:

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(301) 621-0390 (703) 605-6000
 1

Equivalent-Continuum Modeling of Nano-Structured Materials 

Gregory M. Odegard*, Thomas S. Gates, Lee M. Nicholson§, and Kristopher E. Wise* 

NASA Langley Research Center, Hampton, Virginia 

Abstract

A method has been developed for modeling structure-property relationships of nano-structured 

materials. This method serves as a link between computational chemistry and solid mechanics 

by substituting discrete molecular structures with an equivalent-continuum model. It has been 

shown that this substitution may be accomplished by equating the vibrational potential energy of 

a nano-structured material with the strain energy of representative truss and continuum models. 

As an important example with direct application to the development and characterization of 

single-walled carbon nanotubes, the model has been applied to determine the effective 

continuum geometry of a graphene sheet. A representative volume element of the equivalentcontinuum model has been developed with an effective thickness. This effective thickness has 

been shown to be similar to, but slightly smaller than, the interatomic spacing of graphite. 

KEY WORDS: nanotechnology, nanotubes, continuum mechanics, molecular mechanics, finite 

elements 

Nomenclature 

Graphite orientation

aβ - Unit-cell base axes 

D - Graphite orientation indicator 

nβ - Orientation integers 

Force field

Eel - Non-bonded electrostatic potential energy 

Eg

 - Vibrational potential energy of graphene sheet 

Em - Potential energy of nano-structured material 

EvdW - Non-bonded van der Waals potential energy 

Eθ

 - Bond-angle variation potential energy 

Eρ - Bond stretching potential energy 

Eτ

 - Bond torsion potential energy 

Eω - Bond inversion potential energy 

Ki

ρ - Constant associated with C-C bond stretching in graphite 

Ki

θ

 - Constant associated with C-C-C bond-angle variation in graphite 

θι - Deformed bond-angle of C-C-C bonds in graphite 

 

*

 National Research Council, NASA Langley Research Center 

§

 ICASE, Structures and Materials Division, NASA Langley Research Center 
 2

Θi - Equilibrium bond-angle of C-C-C bonds in graphite 

ρi - Deformed bond length of C-C bonds in graphite 

Ρι - Equilibrium bond length of C-C bonds in graphite 

Truss and continuum models

a - Elastic rod type of outer portion of truss representative volume element 

j Ai - Cross-sectional area of rod i of truss member type j 

Ant - Cross sectional area of carbon nanotube 

b - Elastic rod type of inner portion of truss representative volume element 

j

ir - Deformed distance between joints of rod i of truss member type j

j Ri - Undeformed distance between joints of rod i of truss member type j 

t - Wall thickness of continuum nanotube 

Yg - Young’s modulus of graphene sheet 

j Yi - Young’s modulus of rod i of truss member type j 

Ynt - Young’s modulus of carbon nanotube 

r′ - Mid-plane radius of continuum nanotube wall

r′′ - Inner radius of continuum nanotube wall 

r′′′ - Outer radius of continuum nanotoube wall 

Λc

 - Mechanical strain energy of the continuum model 

Λt

 - Mechanical strain energy of the truss model 

νg - Poison’s ratio of graphene sheet 

Boundary conditions

B1..B10 - Displacement field constants 

( ) dk

α - Vector of periocity of representative volume element face α

fk - RVE and macroscopic coordinate system offset vector 

h - Height of macroscopic graphene plate 

( )

k u α - Displacement vector of representative volume element face α 

ˆuk - Global displacement vector 

w - Width of macroscopic graphene plate 

xk - Cartesian coordinate system of the representative volume element 

ˆk x - Global coordinate system

α - Representative volume element face number 

∆ - Prescribed displacement applied to macroscopic plate 

εkl - Strain tensor 

σkl - Stress tensor 
 3

1. Introduction 

Nano-structured materials have excited considerable interest in the materials research community 

in the last few years partly due to their potentially remarkable mechanical properties [1]. In 

particular, materials such as carbon nanotubes, nanotube and nanoparticle-reinforced polymers 

and metals, and nano-layered materials have shown considerable promise. For example, carbon 

nanotubes could potentially have a Young’s modulus as high as 1 TPa and a tensile strength 

approaching 100 GPa. The design and fabrication of these materials are performed on the 

nanometer scale with the ultimate goal to obtain highly desirable macroscopic properties. 

One of the fundamental issues that needs to be addressed in modeling macroscopic mechanical 

behavior of nano-structured materials based on molecular structure is the large difference in time 

and length scales. On the opposite ends of the time and length scale spectrum are computational 

chemistry and solid mechanics, each of which consists of highly developed and reliable 

modeling methods. Computational chemistry models predict molecular properties based on 

known quantum interactions, and computational solid mechanics models predict the macroscopic 

mechanical behavior of materials idealized as continuous media based on known bulk material 

properties. However, a corresponding model does not exist in the intermediate time and length 

scale range. If a hierarchical approach is used to model the macroscopic behavior of nanostructured materials, then a methodology must be developed to link the molecular structure and 

macroscopic properties. Even though there is a long history of modeling bulk properties of 

materials based on molecular properties, a simple link between the firmly established disciplines 

of computational chemistry and solid mechanics has not been established. 

In this paper, a methodology for linking computational chemistry and solid mechanics models 

has been developed. This tool allows molecular properties of nano-structured materials obtained 

through molecular mechanics models to be directly used in determining the corresponding bulk 

properties of the material at the macroscopic scale. The advantages of the proposed method are 

its simplicity and direct connection with computational chemistry and solid mechanics. 

This approach consists of two major steps. First, the chemical structure and vibrational potential 

energy function of the nano-structured material, which is determined from computational 

chemistry, is used to obtain an equivalent mechanical pin-jointed truss model. Second, the truss 

representation of the nano-structured material is substituted with an equivalent-continuum 

model. The mechanical behavior of the continuum model closely approximates that of the nanostructured material. 

As an important example, with direct application to the development and characterization of 

single-walled carbon nanotubes (SWNT), the proposed method has been used to determine the 

effective geometry of a graphene sheet. A representative volume element (RVE) of the graphene 

layer has been modeled as a continuous plate with an effective thickness that has been 

determined from the nano-structured properties of graphite by using the proposed method. 
 4

2. Background and basic concepts 

A discussion of some of the necessary background and concepts associated with characterization 

and modeling of carbon nanotubes is required for the development of the proposed method. 

2.1 Carbon nanotubes 

Carbon nanotubes have become a primary focus in nanotechnology research due to their 

exceptionally high stiffness and strength. One of the fundamental issues scientists are 

confronting is the characterization of the mechanical behavior and properties of individual 

carbon nanotubes. Many experimental [2-6] and theoretical [7-11] studies have been performed 

on single- and multi-walled carbon nanotubes. In particular, deformation modes and overall tube 

stiffnesses have been closely examined. 

In order experimentally characterize the mechanical behavior of nanotubes, a thorough 

understanding of the physical properties, such as effective cross-sectional area and moment of 

inertia, and materials properties, such as Young’s modulus and Poisson’s ratio, is necessary. 

These quantities are traditionally associated with the macroscopic scale, where the characteristic 

dimensions of a continuum solid are well defined. The determination of these properties has 

been attempted in many of the studies cited above without proper regard to an acceptable 

definition of the nanotube geometry. Accurate values of macroscopic physical and mechanical 

properties are crucial in establishing a meaningful link between nanotube properties and the 

properties of larger structures, such as nanotube-reinforced polymer composites. Therefore, 

caution should be used when applying continuum-type properties to nano-structured materials. 

In many studies, it has been assumed that the nanotube “wall thickness” is merely the interatomic 

spacing of two or more graphene sheets [3, 6-11], which is about 0.34 nm in single-crystal 

graphite. While this simple idealization appears to have intuitive merit, it does not necessarily 

reflect the effective thickness that is representative of continuum properties. In order to avoid 

this problem, Hernandez et al. [8] proposed the use of a specific Young’s modulus, i.e., Young’s 

modulus per unit thickness. Even though this approach is convenient for studies concerned with 

the relative stiffnesses of nanotubes, it is of little use when modeling a nanotube as a continuum 

structure. Another proposed solution to this dilemma is to assume that the nanotube is a solid 

cylinder [12-14]. This method is certainly convenient, however, significant inconsistencies arise 

when comparing moduli data of single wall nanotubes (SWNT) and multi-walled nanotubes 

(MWNT) when both are assumed to be a solid cylinder. 

A continuum modeling approach, which relies on an effective wall thickness, was proposed by 

Yakobson et al. [15]. They developed an elastic continuous shell model to represent a carbon 

nanotube. In their model, a wall thickness (0.066 nm) was calculated based on an assumed value 

of Young’s modulus of the nanotube (5.5 TPa). This approach is extremely useful in 

deformation analysis of nanotubes (e.g., buckling), however, it is not convenient when 

attempting to calculate the modulus of the nanotube based on empirical or theoretical data where 

the thickness is not known a priori. Govindjee and Sackman [14] utilized the Bernoulli-Euler 

beam theory to show that for a MWNT in bending, a large number of atomic layers are necessary 

in order to be able to assume that the cross-section of the nanotube is a continuum. This type of 
 5

approach assumes that the nanotube is a solid cylinder instead of using a more realistic model of 

single or multiple hollow cylinders with inter-wall interactions. Ru [16] used an elastic shell 

continuum model to investigate buckling of double-walled nanotubes subjected to axial 

compression. It was shown that the van der Waals forces between the layers did not increase the 

critical axial strains that correspond to buckling. Specific thicknesses for the inner and outer 

walls were not discussed in detail. Ru [17] proposed that the effective bending stiffness of 

SWNT should be regarded as an independent material parameter not related to the bending 

stiffness associated with the Bernoulli-Euler beam bending theory. This simplification was 

suggested in order to explain inconsistencies associated with the use of the classical bending 

formula for SWNT using a wall thickness of 0.34 nm. 

It has been postulated by Cornwell and Wille [12] and Robertson et al. [18] that the strain energy 

induced by bending a flat graphene sheet into a circular nanotube causes an increase in the 

apparent stiffness of the tube as a function of the nanotube radius. Conversely, Halicioglu [19] 

has shown that significant radial and circumferential stresses are present in the nanotube that are 

also dependent on the tube radius. Therefore, it follows that in order to properly model the 

mechanical behavior of a SWNT using continuum mechanics, the effective geometry must be 

known. If the nanotube is modeled as a continuous hollow cylinder with an effective wall 

thickness, then a simple first step is to model the flat graphene sheet in order to determine the 

effective wall thickness. In the current study, the effective thickness will be calculated as an 

example of the proposed approach. 

2.2 Notation for graphite orientation 

In order to clarify some of the notation used in this paper, a brief description of the commonly 

used notation for graphite is necessary. A graphene sheet is composed of bonded carbon atoms 

arranged in a repeating array of regular hexagons (Figure 1). The simplest way of specifying the 

orientation of a graphite structure is in terms of a vector D that joins two equivalent points 

(atoms) of the graphene sheet. The vector may be expressed as: 

D n a = β β (1) 

where the index β = 1,2; aβ are the unit-cell base vectors of the graphene sheet (Figure 1); nβ is a 

set of integers where n1 ≥ n2; and the summation convention is used for repeated indices. It is 

common to express equation (1) in terms of the integers as the ordered pair (n1,n2). 

2.3 Representative volume element 

To reduce the time and number of computations associated with modeling the graphene sheet, a 

representative volume element (RVE) for graphene was used in this study (Figure 2). The 

selected RVE allows each degree of freedom of the carbon atom associated with bond stretching 

and bond-angle variation in the hexagonal ring to be completely modeled by truss and continuum 

finite element model nodal-displacement degrees of freedom. Also, this RVE allows the 

displacements on the boundary of the proposed chemical, truss, and continuum models to 

correspond exactly. Furthermore, macroscopic loading conditions applied to a continuous 
 6

graphene plate can be easily reduced to periodic boundary conditions that are applied to the 

RVE. 

3. Modeling procedure 

The proposed method of modeling nano-structured materials with an equivalent-continuum is 

outlined below. The approach uses the energy terms that are used in molecular mechanics 

modeling for the development of a continuum solid. Therefore, a brief description of molecular 

mechanics is given first followed by an outline of the equivalent-truss and equivalent-continuum 

model development. 

3.1 Molecular mechanics

An important component in molecular mechanics calculations of the nano-structure of a material 

is the description of the forces between individual atoms. This description is characterized by a 

force field. In the most general form, the total potential energy of the force field for a nanostructured material is described by the sum of many individual energy contributions: 

m vdW el E EEEEE E ρθτω =++++ + (2) 

where Eρ, Eθ, Eτ, and Eω are the energies associated with bond stretching, angle variation, 

torsion, and inversion, respectively. The non-bonded interaction energies consist of van der 

Waals, EvdW, and electrostatic, Eel, terms. Various functional forms may be used for these energy 

terms depending on the particular material and loading conditions considered [20]. Obtaining 

accurate parameters for a force field amounts to fitting a set of experimental or calculated data to 

the assumed functional form, specifically, the force constants and equilibrium structure. In 

situations where experimental data are either unavailable or very difficult to measure, quantum 

mechanical calculations can be a critical source of information for defining the force field. 

3.2 Truss model 

Due to the nature of the molecular force field, a pin-jointed truss model may be used to represent 

the energies given by equation (2) where each truss member represents the forces between two 

atoms. Therefore, a truss model allows the mechanical behavior of the nano-structured system to 

be accurately modeled in terms of displacements of the atoms. This mechanical representation 

of the lattice behavior then serves as an intermediate step between linking the vibrational 

potential with an equivalent-continuum model. In the truss model, each truss element 

corresponds to a chemical bond or a significant non-bonded interaction. The stretching potential 

of each bond corresponds with the stretching of the corresponding truss element. Traditionally, 

atoms in a lattice have been viewed as masses that are held in place with atomic forces that 

resemble elastic springs [21]. Therefore, bending of truss elements is not needed to simulate the 

chemical bonds, and it is assumed that each truss joint is pinned, not fixed. 
 7

The mechanical strain energy, Λt, of the truss model is expressed in the form: 

( )2 t

2

j j

i i j j

i i j

j i i

A Y r R

R

Λ= −  (3) 

where j Ai and j Yi are the cross-sectional area and Young’s modulus of rod i of truss member type 

j, respectively. The term j j

i i r R− is the stretching of rod i of truss member type j, where j Ri and 

j

ir are the undeformed and deformed lengths of the truss elements, respectively. For equation 

(3), as well as all equations up through section 4.2, the summation convention for repeated 

indices is not used. 

In order to represent the chemical behavior with the truss model, equation (3) must be equated 

with equation (2) in a physically meaningful manner. Each of the two equations are sums of 

energies for particular degrees of freedom. The main difficulty in the substitution is specifying 

equation (3), which has stretching terms only, for equation (2), which also has bond-angle 

variance and torsion terms. No generalization can be made for overcoming this difficulty for 

every nano-structured system. A feasible solution must be determined for a specific nanostructured material depending on the geometry, loading conditions, and degree of accuracy 

sought in the model. 

3.3 Equivalent-continuum model 

For many years, researchers have developed methods of modeling large-area truss structures 

with equivalent-continuum models [22-27]. These studies indicate that various methods and 

assumptions have been employed in which equivalent-continuum models have been developed 

that adequately represent truss structures. In this study, the truss and continuum models are 

assumed to be equivalent under the following conditions: 

1. Truss lattices with pinned joints can be modeled accurately with an equivalent-continuum 

model that is based on classical continuum mechanics. For this case, micropolar [28] 

continuum assumptions are not necessary.

2. The models have the same degrees of freedom.

3. The displacements along the edges of the RVE are identical for the two models subjected 

to the same static loading conditions.

4. The same amount of thermoelastic strain energy is stored in the two models when 

deformed by identical static loading conditions.

If these criteria are satisfied, then a particular continuum model, such as a beam, plate, shell, or 

three-dimensional solid, may be directly substituted for a discrete truss lattice. The parameters 

of the solid, such as the elastic properties and geometry, are determined based on the above 

criteria. In some cases the strain energy of the continuum, Λc

, can be easily formulated 

analytically and compared directly with equation (3) to obtain the equivalent-continuum 
 8

properties. In other cases, especially with complex geometries and deformations, numerical 

tools need to be used to determine the continuum parameters. 

Once an equivalent-continuum model has been determined, the mechanical behavior of larger 

structures made of the nano-structured material may be predicted using the standard methods of 

continuum mechanics.

4. Example: effective geometry of a graphene sheet 

In this section, a graphene sheet is modeled as a continuous plate with a finite thickness that 

represents the effective thickness for the determination of continuum-type mechanical and 

physical properties. By using the methodology described above, the molecular mechanics model 

is substituted with a truss model and subsequently an equivalent-plate model. The continuum 

model may then be used in further solid mechanics-based analyses of SWNT. 

4.1 Molecular mechanics model 

The force constants used in this example were taken from the MM3 force field of Allinger and 

coworkers [29-31]. Due to the nature of the material and loading conditions in the present study, 

only the bond stretching and bond-angle variation parameters were used. Torsion, inversion, and 

non-bonded interactions were assumed to be negligible for the case of a graphene lattice 

subjected to small deformations. For this example, the vibrational potential energy of a graphene 

sheet with carbon-to-carbon bonds is expressed as a sum of simple harmonic functions: 

() ( ) 2 2 g

ii i i i i

i i

EK K ρ θ =   ρ −Ρ + θ −Θ (4)

where the terms Ρi andΘ i refer to the undeformed interatomic distance of bond i and the 

undeformed bond-angle i, respectively. The quantities ρi and θi are the distance and bond-angle 

after stretching and angle variance, respectively (see Figure 3). Ki

ρ and Kiθ are the force 

constants associated with the stretching and angle variance, respectively, of the chemical bonds. 

Using the parameters for the MM3 force field [29-31], the force constants used in this example 

are: 

7

2 2

10

2 2

46900 3.26 10

63 4.38 10

i

i

kcal nJ K

mole nm bond nm

kcal nJ K

mole rad angle rad

ρ −

θ −

= =⋅ ⋅ ⋅

= =⋅ ⋅ ⋅

 (5) 

and the equilibrium bond length, Ρi , is 0.140 nm. 
 9

4.2 Truss model 

Since it is difficult to express the mechanical strain energy, Λt

, of the truss model in terms of the 

variable truss joint angles that are specified in molecular mechanics (θi-Θi), the RVE has been 

instead modeled with extra rods between nearly adjacent joints to represent the interaction 

between the corresponding carbon atoms (Figure 3). In order to represent the chemical model, 

which has bond stretching and variable angles as degrees of freedom, with a truss model that has 

stretching degrees of freedom only, two types of elastic rods, a and b, are incorporated into the 

truss RVE. 

The mechanical strain energy, Λt

, of the discrete truss system in Figure 3 is expressed in the form 

of equation (3): 

( ) ( ) 2 2 t

2 2

aa bb

ii ii aa bb

a b ii ii

i i i i

AY AY rR rR

R R

Λ= − + −   (6) 

where the superscripts correspond to rod types a and b, respectively. Comparing equations (4) 

and (6), it is clear that the bond stretching term in the equation (4) can be related to the first term 

of equation (6) for the rods of type a: 

2

a a

i i

a

i

A Y K

R

ρ = (7) 

where it is assumed that a

i i ρ = r and a Ρ =i i R . However, the second terms in equations (4) and 

(6) cannot be related directly. In order to equate the constants, the chemical bond-angle variation 

must be expressed in terms of the elastic stretching of the truss elements of type b. For 

simplicity, it may be assumed that the prescribed loading conditions consist of small, elastic 

deformations only. This assumption is not an over-simplification for the graphene sheet since the 

deformations for highly stiff linear-elastic materials subjected to many practical loading 

conditions are quite small. 

In order to express the Young’s modulus of the rods of type b in terms of the bond-angle 

constant, a simple analysis of the deformation was performed (Figure 4). For small deformations 

of the hexagonal RVE, changes in the bond-angles are small. Therefore, in the second terms of 

equations (4) and (6), it can be assumed that a a

i i r R = and that small angle approximations are 

valid. With these assumptions it can be shown that: 

2

b b

i i

i i a

i

r R

R

− θ −Θ ≈ (8) 

Substitution of equation (8) into equations (4) and (6) results in the following approximation: 
 10

2

3

bbb Ki i ii R AY θ = (9) 

Therefore, the Young’s moduli of the two rod types are: 

2 3

2

a

a b ii i

i i a bb

i ii

K R K Y Y

A RA

ρ θ

= = (10) 

The strain energy of the truss model may be expressed as: 

( ) ( ) 2 2

2

3

4( )

t aa bb

ii ii b

i i i

K K rR rR

R

θ

ρ Λ= − + −   (11) 

Thus, the strain energy of the truss model is expressed terms of the vibrational potential energy 

constants. 

4.3 Equivalent-plate model 

Working with the assumptions discussed herein, the next step in linking the molecular and 

continuum models is to replace the equivalent-truss model with an equivalent-continuous plate 

with a finite thickness (Figure 3). This replacement is accomplished by determining and 

equating the strain energies of the truss and continuum models for a specific set of applied loads. 

For the case of the RVE shown in Figure 2, a direct solution of strain energy of the equivalent 

plate in terms of displacements is quite difficult to obtain in closed form. A simple and accurate 

way of calculating the strain energy of the truss and continuum RVE representations is to model 

them numerically by using the finite element method [32]. For a given set of loading conditions, 

the strain energies of the two models may then be quickly calculated and compared. Once this is 

accomplished, then the finite thickness of the equivalent plate is determined, and therefore, the 

effective thickness of a graphene sheet is known. 

In order to satisfy the first condition of the criteria specified in section 3.3, it was assumed that 

the equivalent plate may be modeled using plate-like finite elements. To satisfy the second 

condition, it was assumed that the nodes of the two models are located at the same points, with 

the same degrees of freedom. A simple configuration that satisfies this requirement is shown in 

Figure 3, where the finite element nodes are located at each truss joint on the RVE edges (the 

intersections of the type b rods are not joined) and at the corners of the 4-noded equivalent-plate 

elements. These nodes are allowed to translate in the x1 and x2 directions only (The origin of the 

coordinates is indicated in Figure 3 and located at the centroid of the RVE). Therefore, the 

degrees of freedom of the two models are identical. For condition 3, the displacement gradients 

of the truss elements and the edges of the continuum elements were assumed to be linear. 

Finally, to satisfy condition 4, the two finite element models were subjected to identical loading 

conditions with the total strain energies calculated based on the nodal displacements. This step 

was performed iteratively while optimizing the plate thickness, which was the only available 

adjustable parameter. The corresponding plate thickness is the assumed effective thickness of 

the graphene sheet. 
 11

While the mechanical properties of the truss elements have been determined as described above, 

those of the graphene sheet were taken from the literature. Reliable values of the in-plane 

mechanical properties of graphite have been known for quite some time since they can be 

measured macroscopically, i.e., without any assumptions regarding the graphene sheet thickness. 

For this example, typical values of the mechanical properties were used [33-35]: 

1030 GPa

0.170

g

g

Y =

ν =

 (12) 

where Yg

 and νg are Young’s modulus and Poisson’s ratio of graphite, respectively. 

4.4 Boundary conditions 

In order to determine an effective plate thickness, the truss and continuum models were 

subjected to three sets of identical loading conditions for each strain energy calculation. For 

each set of loading conditions, a corresponding effective thickness was determined. The loading 

conditions correspond to the three fundamental in-plane deformations of a plate, that is, uniform 

axial tension along x1 and x2 and pure shear loading. It was assumed that these loading 

conditions were applied to a macroscopic graphene sheet, i.e. a graphite sheet that has 

dimensions on the macroscopic scale so that there are a very large number of RVEs contained 

within the entire sheet. In order to numerically calculate the strain energy of the RVE under 

these loading conditions, the corresponding periodic boundary conditions for the these 

fundamental loadings were determined and applied to the macroscopic graphene sheet. 

Using the method developed by Whitcomb et al. [36] and Chapman and Whitcomb [37] for 

woven composites, the boundary conditions for the graphene RVE (Figure 2) was determined in 

terms of the overall macroscopic displacement field. The periodic displacement conditions for 

the graphene RVE are given by: 

( ) ( ) () () () () ˆk

k mm k m l

l

u u xd u x d

x

α αα α ∂

+= +

∂ (13) 

where ( ) uk

α is the displacement of a specific RVE face; ( ) dkα is the vector of periodicity; ˆuk is the 

macroscopic, volume-averaged displacement; l x is the RVE coordinate system located at the 

centroid of the RVE (Figure 3); the superscript α is the RVE face under consideration; and the 

subscripts k,l,m = 1,2, where repeated indices imply summation over the range of the index. The 

vector of periodicity is any vector that connects two equivalent points in a periodic array. The 

vectors of periodicity of particular interest here are those that connect points in adjacent RVEs. 

Figure 5 shows the vectors of periodicity for each set of faces of the RVE. It should be noted 

that dk

(1) is parallel to a2 and dk(3) is parallel to a1 (see Figures 1 and 5). The vectors of periodicity 

for each face are defined with respect to the xk coordinate system by: 
 12

( )

(1)

(2)

(3)

3 3, 2 2

3 3, 2 2

3,0

a a

i i

k

a a

i i

k

a

k i

R R d

R R d

d R

 

=  

 

 

= − 

 

=

 (14) 

Substitution of equation (14) and the geometry of the RVE into equation (13) gives the following 

constraints that represent the periodic boundary conditions for each group of opposing faces in 

the graphene RVE: 

( )

( )

(1) (1)

2 2 22

1 2

(2) (2)

2 2 22

1 2

(3) (3

2

33 3 3 ˆ ˆ 3 , 3 3, 22 2 2

33 3 3 ˆ ˆ 3 , 3 3, 22 2 2

3 , 2

aa a a

i i i k ik a

k ki

aa a a

i i i k ik a

k ki

a

i

k k

R R R u Ru u x x u x Rx

x x

R R R u Ru u x x u x Rx

x x

R

u xu

 ∂ ∂   − − + =− − + +   ∂ ∂    

 ∂ ∂   + += + − +   ∂ ∂    

   =

 

)

2

1

3 ˆ , 3 2

a

i k a

i

R u x R

x

  ∂   − +   ∂    

 (15) 

Three different sets of displacement boundary conditions were applied to the RVE in this study. 

Each set is discussed below. 

4.4.1 Case I: uniaxial displacements along (n1,0) 

A macroscopic graphene sheet is subjected to uniaxial loading conditions as shown in Figure 6. 

An axial displacement ∆ is prescribed on the upper edge while the lower edge is constrained. 

The displacement is relatively small and the h×w plate is perfectly elastic. If it is assumed that 

the global displacements 1 uˆ and 2 uˆ vary linearly over the global coordinates 1 xˆ and 2 xˆ , then these 

displacements may be written as: 

( )

( )

1 11 2

2 32 4

ˆˆ ˆ

ˆˆ ˆ

k

k

u x Bx B

u x Bx B

= +

= +

 (16) 

where B1, B2, B3, and B4 are constants which depend on the geometry and applied global 

displacements. The boundary conditions for this case are: 
 13

( )

2 1

2 1

1 2

ˆ ˆ ,

2

ˆ ˆ , 0 2

ˆ ˆ 0, 0

h

u x

h

u x

u x

   = ∆  

   − =  

=

 (17) 

Since small elastic displacements are assumed, the strains and constitutive equations are [38]: 

( ) ( )

1 2

1 2

1 2

11 22 22 22 11

ˆ ˆ

ˆ ˆ

1 1 g g

g g

u u

x x

Y Y 11

∂ ∂

ε= ε= ∂ ∂

ε = σ −ν σ ε = σ −ν σ

 (18) 

where ε11 and ε22 are normal strains, and σ11, and σ22 are normal stresses in the plate. 

Substitution of equations (17) and (18) into equation (16) and solving for the constants B1, B2, 

B3, and B4 results in the following displacement functions: 

( )

( )

1 1

2 2

ˆˆ ˆ

ˆˆ ˆ 2

g

k

k

ux x

h

ux x

h

ν ∆ = −

∆ ∆

= +

 (19) 

The global and RVE coordinates are related by: 

ˆkkk x = + x f (20) 

where fk is a vector indicating the relative position of the two coordinate systems with respect to 

each other. Using equations (15), (19), and (20), the RVE boundary displacements are 

determined to be: 
 14

( )

( )

( )

(1) (1)

1 2 2 12 2

(1) (1)

2 2 2 22 2

(2) (2)

1 2 2 12 2

(2)

22 2

33 3 3 , 3 3, 22 2

33 3 3 , 3 3, 22 2

33 3 3 , 3 3, 22 2

3 3 3 , 2 2

a a ga

ii i a

i

aa a

ii i a

i

a a ga

ii i a

i

a a

i i

RR R u x x u x Rx

h

RR R u x x u x Rx

h

RR R u x x u x Rx

h

R R

ux x

 ν ∆   − − + =− − −

 

 ∆   − − + =− − +

 

 ν ∆   + += + +

 



+ + ( ) (2)

22 2

(3) (3)

1 21 2

(3) (3)

2 22 2

3 3 3, 2

3 33 , , 2 2

3 3 , , 2 2

a

a i

i

a a ga

i ii

a a

i i

R u x Rx

h

R RR

u xu x

h

R R

u xu x

∆   =++

 

  ν ∆    =− −

  

     = −

  

 (21) 

4.4.2 Case II: uniaxial displacements along (0,n1) 

The same macroscopic graphene sheet studied in Case I is now subjected to a different set of 

displacements, as shown in Figure 7. The boundary conditions for this case are: 

( )

1 2

1 2

2 1

ˆ ˆ ,

2

ˆ ˆ , 0

2

ˆ ˆ ,0 0

w

u x

w

u x

u x

   = ∆  

   − =  

=

 (22) 

If the same assumptions are made for this case as for Case I, then equations (16), (18), and (22) 

may be used to determine the following global displacements: 

( )

( )

1 1

2 2

ˆˆ ˆ 2

ˆˆ ˆ

k

g

k

ux x

w

ux x

w

∆ ∆

= +

ν ∆ = −

 (23) 

Using equations (15), (20), and (23), the RVE boundary displacements are determined to be: 
 15

( )

( )

( )

(1) (1)

1 2 2 12 2

(1) (1)

2 2 2 22 2

(2) (2)

1 2 2 12 2

(2)

22 2

33 3 3 , 3 3, 22 2

33 3 3 , 3 3, 22 2

33 3 3 , 3 3, 22 2

3 3 3 , 2 2

aa a

ii i a

i

a a ga

ii i a

i

aa a

ii i a

i

a a

i i

RR R u x x u x Rx

w

R R R u x x u x Rx

w

RR R u x x u x Rx

w

R R

ux x

 ∆   − − + =− − +

 

 ν ∆   − − + =− − −

 

 ∆   + += + −

 



 + +

 ( ) (2)

22 2

(3) (3)

1 21 2

(3) (3)

2 22 2

3 3 3, 2

3 33 , , 2 2

3 3 , , 2 2

g a

a i

i

a aa

i ii

a a

i i

R u x Rx

w

R RR

u xu x

w

R R

u xu x

ν ∆  =+−  

  ∆    =− +

  

     = −

  

 (24)

4.4.3 Case III: (pure-shear displacements) 

The macroscopic graphene sheet is now subjected to pure shear, as indicated in Figure 8. The 

constraints imposed for in-plane shear loading assume that the shearing faces normal to the 

direction of the applied shear are displaced by identical amounts. The displacement is relatively 

small and the h×w plate is perfectly elastic. If it is assumed that the global displacements 1 uˆ and 

2 uˆ vary linearly over 1 xˆ and 2 xˆ , then these displacements may be written as: 

( )

( )

1 51 62 7

2 8 1 9 2 10

ˆˆ ˆ ˆ

ˆˆ ˆ ˆ

k

k

u x Bx Bx B

u x Bx Bx B

=++

=++

 (25) 

The boundary conditions are: 

1 2 12

2 2 22

11 11

21 21

ˆ ˆ ˆˆ ,0 ,0

2 2

ˆ ˆ ˆˆ ,0 ,

2 2

ˆˆ ˆˆ , ,

2 2

ˆˆ ˆˆ , ,

2 2

w w

u x ux

w w

u x ux

h h

ux ux

h h

ux ux

    −= =   

    − = =∆   

     = −   

   = −  

 (26) 

Substitution of equation (26) into equation (25) results in: 
 16

( )

( )

1

2 1

ˆ ˆ 0

ˆˆ ˆ 2

k

k

u x

ux x

w

=

∆ ∆

= +

 (27) 

Using equations (15), (20), and (27), the RVE boundary displacements may be determined: 

( )

( )

( )

(1) (1)

1 2 2 12 2

(1) (1)

2 2 2 22 2

(2) (2)

1 2 2 12 2

(2) (2)

2 2 2 22

3 3 3 , 3 3, 2 2

33 3 3 , 3 3, 22 2

3 3 3 , 3 3, 2 2

3 3 3 , 33 2 2

a a

i i a

i

aa a

ii i a

i

a a

i i a

i

a a

i i

i

R R u x x u x Rx

RR R u x x u x Rx

w

R R u x x u x Rx

R R u x x u xR

   − − + =− −

 

 ∆   − − + =− − +

 

   + += +

 

   + += +

  ( ) 2

(3) (3)

1 21 2

(3) (3)

2 22 2

3 , 2

3 3 , , 2 2

3 33 , , 2 2

a

a i

a a

i i

a aa

i ii

R

x

w

R R

u xu x

R RR

u xu x

w

∆ −

     = −

  

  ∆    =− +

  

 (28) 

To simplify the application of equations (21), (24), and (28) to finite element boundary 

conditions, without loss of generality, the following assumption can be made: 

( ) ux d ux ˆ ˆ km m km ( ) ( ) α + =− (29) 

4.5 Finite element modeling 

The truss and continuum representations of the RVE were modeled using ANSYS® 5.4 [39]. In 

the truss model, each pin-jointed extensible rod was modeled using a finite truss element 

(LINK1) with two degrees of freedom at each node (displacements parallel to x1 and x2). Rod 

types a and b were assumed to have the same cross-sectional area and different Young’s moduli. 

The entire RVE was modeled with the cross-sectional areas of the type a rods divided by a factor 

of 2, since these rods are sharing their total area with adjacent RVEs. The equivalent-plate RVE 

was modeled by using two plane-stress 4-noded quadrilateral elements (PLANE42) with linear 

displacement fields on the edges (Figure 3). The boundary conditions described above were 

applied to each node for a macroscopic graphene sheet height, width, and displacement of 1x106

nm, 1x106

 nm, and 100 nm (corresponding to global uniaxial and shear strains of 0.01%), 

respectively. For both models, the total strain energy of each element was calculated for the 

given boundary conditions, then summed to obtain the strain energy of the entire RVE model. A 

systematic variation of thickness was used to calculate the strain energy of the equivalent plate, 
 17

and an iterative process was used to find the critical thickness (accurate to 0.01 nm) that resulted 

in the equivalence of the strain energies of the truss and equivalent-plate models. 

5. Discussion

Using the method described herein, effective thicknesses were determined for the three loading 

cases. For loading cases I and II, the resulting effective thickness was calculated to be 0.28 nm. 

For load case III, the effective thickness was 0.24 nm. These values are close to, but smaller 

than, the widely accepted value of the graphitic interatomic spacing, 0.34 nm, and much larger 

than the value suggested by Yakobson et al. [15], 0.066 nm. Even though there is a difference in 

the effective plate thickness based on different loading conditions, the difference is very small 

compared to the range of values currently used. 

It may be assumed that the Young’s modulus of a carbon nanotube can be calculated using: 

1 ( ) nt nt Y A − ∝ (30) 

where Ant is the cross-sectional area of the hollow continuum cylinder with a constant mid-plane 

radius, r′ . The inner radius, r′′, and outer radius, r′′′ , of the tube are: 

2

2

t

r r

t

r r

′′ ′ = −

′′′ ′ = +

 (31)

where t is the wall thickness. The cross-sectional area of the hollow continuum cylinder is: 

2 nt A tr = π ′ (32) 

The calculated cross-sectional areas are shown in Table 1. The percentage difference in the 

calculated Young’s modulus based on the effective thickness obtained for the three load cases 

with respect to the interatomic spacing is also shown in Table 1. The results suggest that 

measured and calculated values of physical and mechanical properties of carbon nanotubes that 

are dependent on the dimensions of the continuum tube may differ significantly based on the 

assumed geometry. 

The proposed method for continuum modeling may be applied to other nano-structured materials 

such as nanotube-reinforced polymer composites and will yield satisfactory results if the right 

assumptions are made. If the material is subjected to large deformations such that non-linear 

deformation and failure are key issues, then the particular approximations made in this study 

may not be appropriate. 
 18

6. Conclusions 

A method has been presented for modeling structure-property relationships of nano-structured 

materials. This method serves to link computational chemistry, which is used to predict 

molecular properties, and solid mechanics, which describes macroscopic mechanical behavior 

based on macroscopic material properties. This link is established by replacing discrete 

molecular structures with equivalent-continuum models. It has been shown that this replacement 

may be accomplished by equating the vibrational potential energy of nano-structured materials 

with the mechanical strain energy of representative truss and continuum models. 

As an important example with direct applications to the development and characterization of 

carbon nanotubes, the model has been applied to determine the effective geometry of a graphene 

sheet. A representative volume element (RVE) of the chemical structure of graphene has been 

substituted with RVEs of equivalent-truss and equivalent-continuum models. As a result, an 

effective thickness of the continuum model has been determined. This effective thickness has 

been shown to be similar to, but slightly smaller than, the interatomic spacing of graphite. 

Acknowledgements 

This work was performed while Dr. Odegard and Dr. Wise held National Research Council 

Research Associateship Awards at NASA Langley Research Center. 

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Properties of Nanotubes. Solid State Communications. Vol. 110, 1999, pp. 227-230. 

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2511-2514. 

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California: University Science Books, 1997. 
 20

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Atlanta, Georgia: 1981. 

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McGraw-Hill Publishing Company Limited, 1971. 

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1981. 
 21

[34] Pierson, H.O.: Handbook of Carbon, Graphite, Diamond and Fullerenes: Properties, 

Processing and Applications. New Jersey: Noyes Publications, 1993. 

[35] Harris, P.J.F.: Carbon Nanotubes and Related Structures: New Materials for the Twentyfirst Century. Cambridge: Cambridge University Press, 1999. 

[36] Whitcomb, J.D.; Chapman, C.D., and Tang, X.: Derivation of Boundary Conditions for 

Micromechanics Analyses of Plain and Satin Weave Composites. Journal of Composite 

Materials. Vol. 34, no. 9, 2000, pp. 724-747. 

[37] Chapman, C.D. and Whitcomb, J.D.: Thermally Induced Damage Initiation and Growth 

in Plain and Satin Weave Carbon-Carbon Composites. Mechanics of Composite 

Materials and Structures. Vol. 7, no. 2, 2000, pp. 177-194. 

[38] Timoshenko, S. and Goodier, J.N.: Theory of Elasticity. New York: McGraw-Hill Book 

Company, 1951. 

[39] SAS IP, Inc.: Canonsburg, PA, 1997. 
 22

continuum continuum % difference of

wall thickness cross-sectional area Young's modulus

[nm] [nm2

] [%]

load cases I and II 0.28 1.76r' 21

load case III 0.24 1.51r' 42

average of I, II, and III 0.27 1.70r' 26

interatomic spacing 0.34 2.14r' -

Table 1 – Percent difference in calculated Young’s modulus for different wall thicknesses with 

respect to the Young’s modulus calculated with the interatomic spacing. 
 23

Figure 1 – Unit vector notation for a graphene sheet 
 24

Figure 2 – Representative volume element of a graphene sheet 
 25

Figure 3 – Representative volume elements for the chemical, truss, and continuum models 
 26

Figure 4 – Schematic of the geometry of deformation of the representative volume element 
 27

Figure 5 – Vectors of periodicity of the RVE 
 28

Figure 6 – Loading case I: uniaxial displacements along (n1,0) 
 29

Figure 7 – Loading case II: uniaxial displacements along (0,n2) 
 30

Figure 8 – Loading case III: pure shear displacements 
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Equivalent-Continuum Modeling of Nano-Structured Materials

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6. AUTHOR(S)

Gregory M. Odegard, Thomas S. Gates, Lee M. Nicholson and Kristopher

E. Wise

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NASA Langley Research Center

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13. ABSTRACT (Maximum 200 words)

A method has been developed for modeling structure-property relationships of nano-structured materials. This

method serves as a link between computational chemistry and solid mechanics by substituting discrete molecular

structures with an equivalent-continuum model. It has been shown that this substitution may be accomplished

by equating the vibrational potential energy of a nano-structured material with the strain energy of representative

truss and continuum models. As an important example with direct application to the development and

characterization of single-walled carbon nanotubes, the model has been applied to determine the effective

continuum geometry of a graphene sheet. A representative volume element of the equivalent-continuum model

has been developed with an effective thickness. This effective thickness has been shown to be similar to, but

slightly smaller than, the interatomic spacing of graphite.

14. SUBJECT TERMS

nanotechnology, nanotubes, continuum mechanics, molecular mechanics,

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