Composite Structures 93 (2011) 1809–1818

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Composite Structures

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A continuum mechanics framework and a constitutive model for predictingthe orthotropic elastic properties of microtubules

K.M. Liew a,⇑, Ping Xiang a, Yuzhou Sun b

a Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kongb Department of Civil Engineering and Architecture, Zhongyuan University of Technology, Zhengzhou 450007, China

a r t i c l e i n f o

Article history:Available online 19 February 2011

Keywords:MicrotubulesContinuum constitutive modelElastic modulusCauchy–Born rule

0263-8223/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruct.2011.01.017

⇑ Corresponding author. Tel.: +852 34426581; fax:E-mail address: kmliew@cityu.edu.hk (K.M. Liew).

a b s t r a c t

This paper presents a theoretical framework for modeling the orthotropic elastic properties of microtu-bules. We propose a constitutive model to describe the detailed microscale information and continuumproperties of the microtubules. The microtubule is viewed as being transformed from an equivalent pla-nar structure, a fictitious-bond is introduced to evaluate the system energy and related with deformationgradients, and a representative unit cell is considered to bridge the microscale energy and the continuumstrain energy. In a representative unit cell, the tubulin monomers and guanosine molecules are treated asspheroids, and the fictitious-bond vectors are evaluated through the higher-order Cauchy–Born rule. Todeal with this polyatomic bio-composite structure that has large quantities of different types of chemicalelements, a homogenization technique is performed to calculate the fictitious-bond energy. The structureof the microtubule is thus determined by minimizing the potential of the representative unit cell. Withthe established model, the fictitious-bond lengths between adjacent molecules are evaluated and the lon-gitudinal and circumferential moduli are calculated.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction support and maintain the shape of cells, and take charge of cell

The cytoskeleton contains microtubules, microfilaments, andintermediate filaments. Microtubules are regarded as typical poly-atomic bio-composite structures, which have intricate buildup ofatomic ingredients and spread throughout the cytoskeleton. Theyplay essential roles in maintaining its stability, rigidity, and integ-rity. They are the largest filaments, frequently spanning wholecells, and have an outer diameter of around 25 nm and a lengththat varies from tens of nanometers to tens or even hundreds ofmicrometers [1]. The formation of microtubules is considered tobe a self-assembly process that composed primarily of a single re-peated and well-arranged macromolecular: tubulin. Tubulin is aheterodimeric protein composed of two similar subunits: a-tubu-lin and b-tubulin. In a material view, tubulins are polyatomic com-posites that made up of thousands of different types of atoms. Thea-tubulin and b-tubulin subunits are assembled end-to-end toinitially form a tubulin heterodimer. These heterodimers thenpolymerize head to tail in an energy-dependant manner and formprotofinaments as they grow helically. Electron microscope imagesof microtubules in in vivo cells show them to be hollow tubes thatare circular in cross-section. They are composed of 13 protofina-ments oriented in parallel by lateral interactions, and have an over-all thickness of a single tubulin. They function as frameworks to

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division, motility, intracellular transport, the movement of motorproteins, the formation of the moving cores of flagella and cilia,and the movement of chromosomes during cell division [2–4].The exploration of the elastic properties and subtle mechanicalbehavior of microtubules will hence be very valuable in biomedi-cine development and many bioindustry aspects.

Microtubules were first discovered and studied through exper-imental tests. For some decades, experimental instruments suchas optical tweezers, hydrodynamic flow, and atomic force micros-copy were used to measure their mechanical properties [5,6].However, these measurements may not accurately capture themechanical properties of individual microtubules due to the needfor very precise instruments. In the cytoskeleton, each individualfilament consisting of hundreds of tubulin monomers (a or b) hasmillions of atoms, which creates a large obstacle to the imple-mentation of molecular dynamics (MD) simulations of microtu-bules. Theoretical efforts regarding microtubules are much morelimited, which is the primary reason for working out the atomis-tic–continuum model presented in this study. A review of the lit-erature indicates that various efforts have been made in recentyears to elucidate the mechanical behavior of microtubules. Var-ious models of microtubules, including elastic beams [6–10],three-dimensional rods [5,11–13], and anisotropic elastic shells[14–16], have been developed, but none include interatomicinformation, which is critically important in micromechanics forsuch polyatomic bio-composites.

1810 K.M. Liew et al. / Composite Structures 93 (2011) 1809–1818

There have been some attempts to establish atomistic-basedcontinuum theories or quasicontinuum theories in recent decades.Tserpes and Papanikos developed finite element modeling for car-bon nanotubes by using a linkage between molecular and contin-uum mechanics [17–21]. Tadmor et al. [22,23], Miller et al. [24],and Shenoy et al. [25,26] proposed quasicontinuum models to con-nect atomistic simulations with continuum mechanics. Nakaneet al. [27], Ortiz and Phillips [22,28,29], and Arroyo and Belytschko[30] proposed methods that stem directly from semi-empirical en-ergy potential functions for lattice and crystal structures. Frieseckeand James [31] suggested an attempt to transform atomistic infor-mation into a continuum theory of nanostructures. Yakobson et al.[32] studied the carbon nanotubes with MD simulation and theclassical continuum shell theory simultaneously, and illustratedthat these buckling phenomena can be displayed through a contin-uum shell model. Chang and Gao [33] presented an analyticalmolecular mechanics model for single-walled carbon nanotubesto relate the elastic properties to the atomic structure. Hu et al.[34] and He et al. [35] linked the atomic structure with the non-local shell model and studied the buckling behavior and wavepropagation in carbon naotubes. Chandraseker and Mukherjee[36,37] developed an atomistic–continuum model for carbonnanotubes to estimate elastic moduli and take into account differ-ent deformation conditions. Liew and Sun [38–41] developed aninteratomic potential-related mesh-free method coupled with ahigher-order Cauchy–Born rule for the computation of carbonnanotubes. The established model has proved to be efficient andaccurate, with many advantages over conventional MD methods,and is thus adopted as the theoretical basis for this study. Jianget al. [42] have recently presented a one-dimensional atomistic-based model for biopolymers in longitudinal direction; however,the applicability of their model was quite limited. The drawbackshave attracted our attention to develop new theories to further ex-plore the problem on microtubules.

With the new aims in mind, this paper proposes (1) a concept offictitious-bond vector for microtubules to consider the polyatomicbio-composite structure, with which the interatomic energy is re-lated to the higher-order deformation gradient, therefore, a three-dimensional computing under arbitrary deformation states anddifferent loading conditions is possible, (2) a potential-based meth-od for computing the elasticity properties of microtubules, and (3)an inter-atomistic potential-based continuum constitutive relationand a material description for polyatomic composite material thatincludes microscale information and continuum properties andsuitable for computing in three-dimensional space. The technicalaspect of a quasicontinuum is achieved by incorporating thehigher-order Cauchy–Born rule into the constitutive model to en-sure the smooth computation of the mechanical behavior in latersteps, and some useful conclusions and results are obtained anddiscussed.

2. Microscale structure and continuum description

A comprehensive realization of the polyatomic bio-compositestructure of microtubules is vital in establishing the continuummodel. Along with atomistic–continuum modeling, the energy de-rived from the interatomic potential model matches the potentialin continuum mechanics, from which the orthotropic elastic mod-uli and nonlinear properties can be determined. A reliable equiva-lent atomistic model should reflect real geometry, containequivalent types and amounts of chemical elements, include con-figuration information, and be able to deal with arbitrary deforma-tions in interatomic energy. Microtubules are viewed as cyclicallygathered protofilaments that appear as longitudinal thin stripsformed by tubulins arrayed in a vertical line. Neighboring protofil-

aments are separated by deep grooves due to the round shaped ofthe tubulins, and there are some holes with diameters of around10 Å on the external surface of the microtubules. A few heterodi-mer segments participate in lateral interactions that are responsi-ble for the unique mechanic behavior of microtubules. Greatdifferences between the interaction strength in the lateral and lon-gitudinal directions results in highly anisotropic elastic properties,as shown by recent experiments. Microtubule filaments are hollowcylinders made up of the protein tubulins: a tubulin and b tubulin.As the tubulin monomer appears to be a spatial spheroid and hasvery complicated atomic components with different kinds ofchemical elements and thousands of atoms, it is unrealistic to pre-cisely determine the interactions between every pair of atoms. Abetter approach that takes into account all of the atomistic interac-tions and at the same time more computationally efficient is badlyneeded. Due to its compact structure, we model the tubulin mono-mers and guanosine molecules as spatial spheroids with strictlydefined and homogenized volume densities for certain chemicalelements that correspond to the real chemical compositions ofmicrotubules. A representative unit cell is selected to evaluatethe system energy. Unlike carbon nanotubes, no covalent bondsexist between different macromolecules, the subunits of microtu-bules are gathered by weak interactions of innumerable pair atomsin the polyatomic composite. In order to accord mutual interac-tions to relative position of spheroids, a fictitious-bond that con-necting the central points between neighboring spheroids isassumed to stand for the interatomic potential between macro-molecules. The established model is schematically shown in Fig. 1.

The homogenization of all of the various types of atoms in tubu-lin and guanosine molecule leads to a new feasible approach tomodeling the interatomic potential in polyatomic bio-compositestructures: microtubules, see Ref. [42]. Each tubulin (a or b tubu-lin) has 400 residues consisting of 8,000 atoms, with each type ofatom distributed evenly across the whole body. Because a tubulinand b tubulin have almost the same components and also haveidentical topological structures, we ignore the differences betweenthem when calculating the interatomic potential. By treating theguanosine molecule as spheroids with smaller sizes than the tubu-lin monomers, protofilaments are then modeled as chains due tothe periodic structure of the tubulin-guanosine-tubulin heterodi-mers. The microtubule is then considered as several spheroidchains linked together by weak lateral interactions in a cylindricalfashion.

The volume of the modeled spheroids of the tubulin and guano-sine molecule can be obtained as

Vt ¼4pr3

t

3nm3; ð1Þ

and

VG ¼4pr3

G

3nm3; ð2Þ

where rt and rG are the radius and Vt and VG are the volume, respec-tively, of tubulin monomer and guanosine molecule.

The volume density q of each element in the spheroid is definedas the total atoms divided by the volume of the spheroid. The twosubscripts ‘t’ and ‘G’ following q, qt and qG, are adopted to repre-sent the volume density of each type of atom in the guanosine mol-ecule and tubulin monomer. For a specific atom type i, thereduction of the total atoms is obtained by

Nit ¼ qi

tV t; ð3Þ

and

NiG ¼ qi

GVG; ð4Þ

Fig. 1. Homogenized model for the polyatomic composite structure of microtu-bules. (a) A fraction of the cross-section of a microtubule that contain fiveprotofilaments. The a tubulin monomer and b tubulin monomer are modeled asspatial spheroids, lateral interactions are modeled by fictitious-bonds r1 � r6, whichare space vector connecting the central points of neighboring spheroids o1 � o5. (b)The wall structure of a microtubule with longitudinal fictitious-bond vectorsr1 � r4, and lateral fictitious-bond vectors r5 and r6. Unlike lateral fictitious-bondvectors, the longitudinal fictitious-bond vectors are built between tubulin monomerand guanosine molecule because of the function of guanosine molecules. Tubulinmonomers and guanosine molecules are all modeled as spatial spheroids, but aredifferent in size due to their differing geometries. The transformation is based onthe homogenization of all of the chemical elements in the polyatomic composite, bywhich the equivalent interatomic energy is ensured. The energy is derived throughan integration process of the adjacent spheroids and then concentrated to thefictitious-bond.

K.M. Liew et al. / Composite Structures 93 (2011) 1809–1818 1811

where Nit and Ni

G are the total number of atoms of type i in the tubu-lin monomer and guanosine molecule, respectively, and qi

t and qiG

are the volume densities of the chemical element i in tubulin andguanosine molecule, respectively.

Fig. 2. Schematic relationship for a cylindrical tube with a lattice structure; (a) and(b), the atomic models are replaced by a representative planar continuum sheetcontaining all of the interatomic information; (b) to (c), a cylindrical tube model isformed by rolling up the planar sheet created in the previous step.

3. Continuum model

To enable the application of a continuum process, an unde-formed microtubule is viewed as being formed by rolling up a con-tinuum planar sheet into a cylindrical shape, where the sheet isformed in a plane by fictitious-bonds that connecting the centerpoints of spheroids. This rolling process, however, is not a simplerigid transformation, and the micro-structure readjusts to achievethe minimum system energy during the rolling process. As the he-lix structure has less influence on the mechanical behavior ofmicrotubules [15], we ignore the helix angle parameter h.Fig. 2a–c shows that a continuum model for the microtubule canbe established by rolling up a sheet that consists of fictitious-bondvectors. The rolling is not a rigid mapping process, and is accompa-nied by the minimization of the interatomic energy to attain a sta-ble state in the microtubules.

The planar sheet formed by fictitious-bonds in the aforemen-tioned atomistic model is introduced into the continuum model.For each central point, there are four neighboring central pointsof adjacent spheroids. By considering the fictitious-bond vectorsfrom one center to the other four neighboring central points, thetotal interatomic potential can be determined by summing upthese four homogenized spheroid pair potentials. We ignore thepair potential of nonadjacent bodies because the energy storedamong nonadjacent bodies a long distance away is far less thanthat stored between adjacent bodies. Due to the periodicity ofthe structure, a representative unit cell is selected for theoreticalanalysis, as shown in Fig. 3. A microtubule can then be built seam-lessly by adding standard representative unit cells on both sides

Fig. 3. A representative unit cell for continuum analysis. The rectangular shadedarea is the continuum surface of the microtubules, the energy of which is derivedfrom spheroids based on homogenized atomistic models. Guanosine and tubulinmonomers are modeled by spheroids with different sizes. jRLj and jRCj are thedistances between the central points of adjacent spheroids, which are the fictitious-bond lengths in longitudinal and circumferential directions, and with the value ofjR�L j and jR�L j under equilibrium state determined by minimizing the total energy ofthe representative unit cell with a stable state.

1812 K.M. Liew et al. / Composite Structures 93 (2011) 1809–1818

and at the tips and tails. Unlike the atomic model, the planar sheetis defined in a continuous plane with an appropriate thickness. Thetubular structure of the continuum model for microtubules isformed by rolling up this planar thick sheet into a cylindrical tubeaccording to a mapping rule. The complete continuum model withhomogenized atomic information for the microtubule is shown inFig. 2c.

In the continuum model, energy is stored in the volumes ofspace occupied by the tubulin monomers and guanosine mole-cules. The volume of space for each component can be calculatedafter the appropriate wall thickness of the tube has been specified.An equivalent wall thickness equal to the diameter of a singletubulin monomer is employed. A review of the literature indicatesthat microtubules appear as spatial ring cycles with an inner radiusof 7.7 nm and an outer radius of 12.5 nm. The tubular continuummodel of a microtubule thus has an inner and outer radius of7.7 nm and 12.5 nm, respectively, in cross-section. The wall thick-ness is 4.8 nm and the cross-section area ACS can be calculated as

ACS ¼ pðR2 � r2Þ; ð5Þ

where R is the outer radius and r is the inner radius of the microtu-bule. In the continuum model, the volume of a representative unitcell Vce is the sum of the volume of space that a tubulin monomerVct and a guanosine molecule VcG occupy. Because the wall of themicrotubule is continuous, the volume of space that a single tubulinand guanosine molecule occupy can be calculated as

Vct ¼pðR2 � r2ÞLt

n; ð6Þ

VcG ¼pðR2 � r2ÞLG

n; ð7Þ

Vce ¼ Vct þ VcG; ð8Þ

(a) (b)Fig. 4. From (a) to (b), a planar sheet in two-dimensional vector space istransformed into a curved surface in three-dimensional space. The standardCauchy–Born rule maps the original vector R in (a) onto r00 in the tangential planepassing the emanating point in (b), and the higher-order Cauchy–Born rule maps Rto r0 , which is closer to the real displacement r.

where n is the total number of protofilaments in the microtubule,and Lt is the diameter of the tubulin monomer and LG the diameterof the guanosine molecule in the atomic model. The volumes ofspace in the continuum shell model are predefined, and contributeto the calculation of the strain energy densities in Section 4.

4. Continuum constitutive relationship

4.1. Constitutive response based on higher-order Cauchy–Born rule

Material properties of polyatomic bio-composite structures aredifficult to define in conventional continuum description. Based onthe fictitious-bond, here we developed a constitutive relationincorporated with the higher-order Cauchy–Born rule. The Cau-chy–Born rule is a fundamental kinematic assumption linking thedeformation of the lattice vectors of a crystal to that of a contin-uum deformation field; it provides a quasicontinuum techniquethat can smooth continuum approaches to nano and microscaleproblems. In general, the standard Cauchy–Born rule describesthe deformation of lattice vectors as

r � FðXÞ � R; ð9Þ

where R denotes the undeformed lattice vector, r represents thecorresponding deformed lattice vector, and F(X) is the deformationgradient. The application of the standard Cauchy–Born rule is ofteninsufficient and inaccurate because it requires a sufficiently homo-geneous deformation. For a deformation from a 2-D sheet to a 3-Dcurved surface, the standard Cauchy–Born rule can only map a pla-nar vector onto the tangential plane of the curved surface (seeFig. 4). The computational work of Arroyo and Belytschko [30] re-veals that the constitutive model based on the standard Cauchy–Born rule does not describe the bending effect, and the bucklingdeformation is not accurately displayed. Sunyk and Steinmann[43] pointed out that the Cauchy–Born rule requires sufficientlyhomogeneous deformations of the underlying crystal, and they sug-gested the application of the higher-order gradient in the contin-uum modeling of inhomogeneous deformations. In order tosmooth away these obstacles, various attempts have been involvedto modify the Cauchy–Born rule [30,44–46]. A higher-order Cau-chy–Born rule was later adopted and successfully used in mesh-freesimulation of single-walled carbon nanotubes [38–41]. In thisstudy, the higher-order Cauchy–Born rule is thus used to calculatethe deformation of the lattice structure formed by fictitious-bondvectors in the microtubules during the deformation and rollingprocess.

We apply X = (X1, X2) as the original reference configuration andx = (x1, x2, x3) as the current configuration. The deformation mapfrom the planar sheet to the curved surface is defined byx = x(X), then we have:

Fði; JÞ ¼ @x@X

; Gði; J;KÞ ¼ @2x@X2 ; Hði; J;K; LÞ ¼ @3x

@X3 ; ð10Þ

where i = 1, 2, 3; J, K, L = 1, 2. F(X), G(X), and H(X) are the first-, sec-ond-, and third-order deformation gradients, respectively.

A vector emanating from a point X in the reference configura-tion is exactly mapped as

K.M. Liew et al. / Composite Structures 93 (2011) 1809–1818 1813

r ¼ xðXþ RÞ � xðXÞ: ð11Þ

Expanding x = x(X + R) in a Taylor series at X = (X1, X2), we have

xðXþ RÞ ¼ xðXÞ þ FðXÞ � ðRÞ þ GðXÞ : ðR � RÞ2!

þHðXÞ...ðR � R � RÞ

3!þ O Rk k4

� �: ð12Þ

Substituting Eq. (12) into Eq. (11), and retaining up to the third-or-der term gives the approximate expression for r

r � FðXÞ � ðRÞ þ GðXÞ : ðR � RÞ2!

þHðXÞ...ðR � R � RÞ

3!; ð13Þ

The accuracy of the approximation for deformed vectors is greatlyenhanced by the inclusion of the second and third deformation gra-dients, which results in a more reasonable approximation.

As has been stated, an undeformed microtubule can be viewedas being formed by rolling up a planar sheet into a cylindrical tube,during which the micro-structure readjusts to attain the minimumsystem energy. This transformation is decomposed into two stepsas shown in Fig. 5. Fig. 5a depicts an initial equilibrium graphitesheet with a length L1 and width L2 that is rolled up along axisX2. In Fig. 5b, the parameters k1 and k2, respectively, represent uni-form longitudinal and circumferential stretches that have a clearphysical meaning that will be further illustrated in the followingsections. k1L1 is equal to the perimeter of the tube. Given thecoordinate transformation x = x(X), the deformation map fromFig. 5a–c can be written as

x1 ¼ X1k1; ð14Þ

x2 ¼L2k2

2psin 2pX2

L2

� �; ð15Þ

x3 ¼L2k2

2p� L2k2

2pcos 2pX2

L2

� �: ð16Þ

In the undeformed lattice planar sheet, fictitious-bond vectors areformed by connecting the neighboring center points, from whichthe interatomic energy is determined. During the rolling process,the fictitious-bond vectors are treated using the higher-order Cau-chy–Born rule, which is slightly modified to ensure more accuratemapping. After obtaining the transformation map, the first-, sec-ond-, and third-order deformation gradients, F(X), G(X), and H(X),can be derived and included in the computational work to calculate

¦ Èo'

X 1

X 2

x1

x2

x3

(b)

(c)

X 1

X 2

(a)

L1

L2

L1

L2

1

2

Fig. 5. (a)–(c) A microtubule is formed by rolling an equilibrium planar sheet into acylindrical tube in two steps to form a microtubule with a cylindrical geometry andminimum system energy.

the deformation of fictitious-bond vectors under arbitrary deforma-tion gradients, then the system energy is obtained to derive the con-stitutive relations.

With the strain energy density unveiled, the first-order Piola–Kirchhoff stress tensor P and higher-order stress tensor Q and Tare given by:

P ¼ @W0

@F; Q ¼ @W0

@G; T ¼ @W0

@Hð17Þ

where W0 is the strain energy density determined by fictitious-bondvectors emanated from an evaluate point. Tangential slopes are gi-ven by:

MFF ¼@2W0

@F� @F; MFG ¼

@2W0

@F� @G; MFH ¼

@2W0

@F� @H;

MGF ¼@2W0

@G� @F; MGG ¼

@2W0

@G� @G; MGH ¼

@2W0

@G� @H;

MHF ¼@2W0

@H� @F; MHG ¼

@2W0

@H� @G; MHH ¼

@2W0

@H� @H; ð18Þ

F ¼ Fþ eF; G ¼ Gþ eG; H ¼ Hþ eH: ð19Þ

The modulus matrix is assembled as

M ¼MFF MFG MFH

MGF MGG MGH

MHF MHG MHH

264375: ð20Þ

We collect the elements F; G, and H together, and define them as

the gradient vector grad, i.e. grad ¼ F_

G_

H_

� �, where F

_

; G_

and

H_

are vectors made up of elements FiJ ; GiJK and HiJKL respectively,with i = 1, 2, 3 and J, K, L = 1, 2.

The first- and higher-order stress vector is defined as

r ¼ @W0

@grad; ð21aÞ

that gives

M ¼ @2W0

@grad2

" #¼ @r

@grad

� �� @2W0

@r2

" #� @r@grad

� �; ð21bÞ

where F; G and H are the first-, second-, and third-order deforma-tion gradients for (a defined) certain deformation states; F, G andH are deformation gradients when no loading is applied, and canbe analytically calculated using the mapping Eqs. (14)–(16); andeF; eG; eH are deformation gradients based on undeformed configu-ration after rolling.

4.2. Evaluation of the fictitious-bond energy

Microtubules are formed by pair interaction forces among adja-cent sub-components, the fictitious-bond energy is the ensembleof interatomic energy of all pair atoms. The interatomic potentialof a pair of atoms can be expressed as

Etotal ¼Xbonds

Krðr � reqÞ2 þX

angles

Khðh� heqÞ

þX

dihedrals

ðVn=2Þ 1þ cosðnuþ cÞ½ �

þXi<j

AijbR12ij

� BijbR6ij

þqiqj

ebR" #

þX

hydrogenbonds

CijbR12ij

þ DijbR6ij

" #: ð22Þ

The first part of the total potential in Eq. (22) is the energy thatstems from the stretching of the bonds, the second part of the totalpotential is the energy that stems from the bending of the bonds,and the third part is the energy that stems from the torsion of

1814 K.M. Liew et al. / Composite Structures 93 (2011) 1809–1818

the bonds. These three parts together comprise the covalent bondenergy in the total potential. The last two parts of Eq. (22) describethe non-covalent bond energy, and are respectively the van derWaals energy plus the hydrogen bond and electrostatic energy.Non-bond energy is a kind of long-range interaction that is of sig-nificant importance in analyzing biological macromolecules. A re-view of the literature indicates that hydrogen bonds can beincluded in the van der Waals interaction terms [47], and thatthe electrostatic interactions can be ignored in a homogenizedmodel because of its electric neutrality [42]. In our model, thespheroids are thus connected by weak non-covalent bond energyprovided by the van der Waals interaction and hydrogen bonds.We use the Lennard–Jones potential for the van der Waals interac-tion between a pair of atoms, as follows:

Ev ¼AijbR12

ij

� BijbR6ij

or Ev ¼ 4eij

r12ijbR12ij

�r6

ijbR6ij

!; ð23Þ

where Ev is the interatomic energy that stems from the van derWaals interactions, bR is the interaction length equal to the spatialdistance between atom i and atom j, Aij and Bij are specific parame-ters that can be determined by eij and rij, and eij and rij are theparameters for certain pairs of atoms that are distributed in adja-cent spheroids. The values of eij and rij vary between pairs of atoms,see Ref. [47]. After determining the parameters eij and rij for differ-ent pairs of atoms, the interactions between the adjacent spheroidscan be calculated and included in a minimization process to find theequilibrium state of the microtubules.

In the computational work, a spherical coordinate system isadopted for the round spatial spheroids. After homogenization,large numbers of different types of atoms in the polyatomic bio-composite structure are replaced by specific volume densitieswithout loss of atomic information. For a specific pair of atoms oftype i in spheroid I and type j in spheroid II, the interaction poten-tial can be determined from the van der Waals interaction. Spher-oid I and spheroid II can be either a tubulin monomer or aguanosine molecule. For each monomer, a local spherical coordi-nate system is built up at the central point for the spatial distanceevaluation, as shown in Fig. 6.

We define wij as the interaction between single atom i and sin-gle atom j in adjacent spheroids. wij is a function of bR, such thatwijðbRÞ depends on the atom types i and j and the mutual distancebR. The total energy WijðbRÞ between atom i and atom j stored inthe pairs of spheroids is then calculated by an integration processfor the whole spheroid that can be expressed as

WijðbRÞ ¼ ZXI

ZXII

wijðbRÞqIiqIIj dV I dV II: ð24Þ

¦ Õ

r¦ È

¦ Õ

X

Y

Z

¦ È

r

(x , y , z )

(x , y , z )

atom i

atom j

o o

(Tubulin or Guanosine)Spheroid Spheroid

(Tubulin or Guanosine)

X

Y

Z

'

'

'

' '

'

'

i i i

j j j

Fig. 6. Interatomic energy of atomi and atomj between adjacent spheroids. i and jcan be any type of atom, and spheroid I and spheroid II are two adjacent spheroids.The schematic can be universally applied to neighboring tubulin–tubulin interac-tions, guanosine–guanosine interactions, and tubulin–guanosine interactions. o ando0 are the central points in the local spherical coordinate system.

Substituting Eq. (23) into Eq. (24) gives

WijðbRÞ ¼ ZXI

ZXII

AijbR12� BijbR6

� �qIiqIIj dV I dV II

¼Z

XI

ZXII

4eij

r12ijbR12ij

�r6

ijbR6ij

!qIiqIIj dV I dV II; ð25Þ

w�ij ¼ w�i þ w�j ; ð26Þ

eij ¼ ðeiejÞ1=2; ð27Þ

where w�i ; w�j ; ei; ej; eij; w�ij are the van der Waals parameters pre-sented by Cornell et al. [47]. We also have

Aij ¼ eijðw�ijÞ12 ¼ 4eijr12

ij ; ð28ÞBij ¼ 2eijðw�ijÞ

6 ¼ 4eijr6ij: ð29Þ

In the spherical coordinate system, the coordinate values of spher-oid I and spheroid II are expressed as

xi ¼ ri cos ui sin hi ð0 6 hi 6 p; 0 6 ui 6 2p; 0 6 ri 6 RiÞyi ¼ ri cos ui cos hi ð0 6 hi 6 p; 0 6 ui 6 2p; 0 6 ri 6 RiÞzi ¼ ri cos hi ð0 6 hi 6 p; 0 6 ri 6 RiÞ

8><>: ;

ð30Þ

xj ¼ rj cos uj sin hj ð0 6 hj 6 p; 0 6 uj 6 2p; 0 6 rj 6 RjÞyj ¼ rj cos uj cos hj ð0 6 hj 6 p; 0 6 uj 6 2p; 0 6 rj 6 RjÞzj ¼ rj cos hj ð0 6 hj 6 p; 0 6 rj 6 RjÞ

8><>: :

ð31Þ

The spatial relationship between a pair of atoms is schematicallyshown in Fig. 7. We define the distance vector of atom i in spheroidI and atom j in spheroid II as d. Then,

d ¼ fdx; dy; dzg; ð32Þ

where dx, dy, and dz are three coordinate components in orthogonalcoordinates that are determined by self-coordinate values and thespatial distance vector R = {Rx, Ry, Rz} between the central pointsof the two spheroids. During the rolling process, a vector R changesfrom an in-plane vector on the planar sheet into a spatial cord onthe tube surface, as described by Eqs. (14)–(16), and is dealt withby the higher-order Cauchy–Born rule. This results in a slight mod-ification of the bond length during the procedure.

The spatial distance between the pair of atoms is calculated by

jdoo0 j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

x þ d2y þ d2

z

q: ð33Þ

By considering the geometrical relationship, we have

dx ¼ xi � xj þ Rx

dy ¼ yi � yj þ Ry

dz ¼ zi � zj þ Rz

8><>: : ð34Þ

Fig. 7. With the established spherical coordinate system, the dashed line shows thecalculated spatial distance jdj between atom i and atom j in neighboring spheroids.

Table 1Van der Waals parameters.

Atom type w⁄ (Å) e (kcal/mol)

C 1.9080 0.1094H 1.3870 0.01570N 1.8240 0.1700O 1.6612 0.2100P 2.1000 0.2000S 2.0000 0.2500

K.M. Liew et al. / Composite Structures 93 (2011) 1809–1818 1815

The distance between atom i and atom j can then be expressed as

bRij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi � xj þ RxÞ2 þ ðyi � yj þ RyÞ2 þ ðzi � zj þ RzÞ2

q: ð35Þ

This gives

WijðbRijÞ¼Z

XI

ZXII

Aij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi�xjþRxÞ2þðyi�yjþRyÞ2þðzi�zjþRzÞ2

q� �12,"

�Bij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi�xjþRxÞ2þðyi�yjþRyÞ2þðzi�zjþRzÞ2

q� �6, #

qIiqIIj dV I dV II: ð36Þ

In Eq. (36), atom i and atom j can be any type of atom in either atubulin monomer or a guanosine molecule. In terms of chemicalcomposition, the tubulin monomer contains the elements C, H, O,N, and S, and the guanosine molecule contains the elements C, H,N, O and P. According to the spherical coordinate system, the totalenergy stored between the two spheroids is determined by sum-ming the paired energy of the different types of atoms. As the ele-ment volume in the spherical coordinate system spans the rangesr to r + dr, h to h + dh, and u to u + du, as given by dV = r2sinhdr dh -du, the interatomic energy between neighboring spheroids can bederived based on the specific distance represented by R.

4.3. Potential for microtubules

With aforementioned schemes available to calculate inter-atomic energy, the structural and elastic properties of microtu-bules can be determined by minimizing the energy of arepresentative unit cell. Applying the criteria that a structure canbe constructed by adding duplicated sub-components, a represen-tative unit cell is selected, as shown in Fig. 3, which is composed oftwo pairs of spheroids in the longitudinal and the circumferentialdirections. These four spheroids (two tubulin monomers and twoguanosine molecules) are named spheroid 1, spheroid 2, spheroid3, and spheroid 4. Correspondingly, we term the fictitious-bond en-ergy stored in the neighboring spheroids ve12, ve23, ve34, and ve41,respectively. The total energy vetotal stored in the representativevolume is calculated by

vetotal ¼ ðve12 þ ve23 þ ve34 þ ve41Þ=4: ð37Þ

After deriving the expression for the total energy vetotal, an equilib-rium state is found by minimizing the energy vetotal during the roll-ing process. The orthotropic elastic properties are obtained as atangential value along the orthogonal directions at this stage.

For ve12, the non-bond energy stored between specific types ofatomi in the tubulin and atom j in the guanosine is expressed as

ve1i2j ¼Z

X1

ZX2

wijðbR1i2jÞq1i dV1 q2j dV2; ð38Þ

where q1i is the volume density of atom type i in a tubulin mono-mer (the subscript i ranges from 1 to 5 to represent specific atomsof C, H, O, N, and S, respectively); q2j is the volume density of atomtype j in a guanosine molecule (the subscript j ranges from 1 to 5 torepresent specific atoms of C, H, O, N, and P, respectively); and bR1i2j

is the distance between the volume element of atom i in spheroid 1and the volume element of atom j in spheroid 2.

By summing all of the pairs of chemical elements, the fictitious-bond energy between spheroid 1 and spheroid 2 is calculated as

ve12 ¼X6

i¼1

X6

j¼1

ZX1

ZX2

wijðbR1i2jÞq1idV1q2jdV2: ð39Þ

whereR

X2

RX1

dV1 dV2 is the product ofRRRX1

r2 sin hdr dhdu andRRRX2

r02 sin h0 dr0 dh0 du0, and weijðbR1i2jÞ is the energy stored in per-unit

volume, which is a function determined by the fictitious-bond vec-tor R12 and can be expressed as

weijðbR1i2jÞ ¼ q1iq2j

eijðw�ijÞ12

bR121i2j

�2eijðw�ijÞ

6

bR61i2j

!: ð40Þ

Gaussian quadrature is adopted in the numerical process, whichrenders Eq. (39) as

ve12 ¼X6

i¼1

X6

j¼1

UðR12ÞweijbR1i2j

� �; ð41Þ

UðR12Þ ¼ nknlnmnnnpnqr2k r02l sin hm sin h0n ð42Þ

where nk, nl, nm, nn, np and nq are the gauss weights, and rk; r0l; hm; h0nare the gauss points. To simplify the equation, two irrelevant inde-pendent parts are extracted and calculated first, i.e.

Cae12 ¼

X6

i¼1

X6

j¼1

q1iq2jeijðw�ijÞ12; ð43Þ

Cbe12 ¼

X6

i¼1

X6

j¼1

2q1iq2jeijðw�ijÞ6; ð44Þ

where eij and w�ij are the parameters for specific pair of atoms be-tween the neighboring spheroid 1 and spheroid 2. This gives

ve12 ¼Xn

k¼1

Xn

l¼1

Xn

m¼1

Xn

n¼1

Xn

p¼1

Xn

q¼1

UðR12ÞCa

e12bR1212

� Cbe12bR612

!ð45Þ

where bR12 is the spatial distance of the volume elements betweengauss points of neighboring spheroids, which is a function of theindependent variable gauss points rk; r0l; hm; h0n; up u0q and thespatial distance vector R12 between the central points of spheroid1 and spheroid 2.

The van der Waals potential parameters for the pairs of proteinsare derived from the work of Cornell et al. [47]. The parameter val-ues w� and e for certain chemical elements in microtubules are pre-sented in Table 1, which are responsible for the longitudinalinteratomic energy of the atom pairs of the guanosine moleculeand tubulin monomer, and the lateral interatomic energy betweenneighboring protofilaments.

The values of ve23, ve34, and ve41 are then calculated in the sameway as ve12, the expressions for which are given as

ve23 ¼X6

i¼1

X6

j¼1

ZX2

ZX3

wijbR2i3j

� �q2i dV2 q3j dV3; ð46Þ

ve34 ¼X6

i¼1

X6

j¼1

ZX3

ZX4

wijbR3i4j

� �q3i dV3 q4j dV4; ð47Þ

ve41 ¼X6

i¼1

X6

j¼1

ZX4

ZX1

wijbR4i1j

� �q4i dV4 q1j dV1: ð48Þ

It can be seen that vetotal is a function of R12, R23, R34, and R41. Giventhe geometrical characteristic that R12 and R23 are equal to R34 andR41 for an undeformed microtubule, only two unknown fictitious-

Table 2Volume density of the tubulin monomer and GDP molecule in a microtubule.

q (nm3) C H N O S P

Tubulin 37.41 72.78 9.860 18.53 0.7771 –GDP 114.8 172.2 57.40 126.3 – 22.96

1816 K.M. Liew et al. / Composite Structures 93 (2011) 1809–1818

bond vectors remain to be defined by minimizing the systemenergy.

Because the spheroids are round and formed with regularshape, as shown in Fig. 3, the system energy in the unit can beviewed as depending on the fictitious-bond RL and RC parallel tocoordinate axis in orthogonal directions, with jR12j = jR34j = jRLjand jR23j = jR41j = jRCj. In the continuum model, the microtubuleis treated as a hollow cylinder of one tubulin molecule in thickness,and the volume strain energy density refers to the average energyper-unit volume of the entire cylinder. For a representative unitcell, this can be expressed as

w ¼ vetotal

Vce; ð49Þ

where Vce is the volume of space occupied by a representative unitcell. This can be further expressed as

w ¼ 1jRLj

4n � vetotal

p D2outer � d2

inner

� � ; ð50Þ

where n is the total number of protofilaments in a microtubule,Douter and dinner are, respectively, the outer and inner diameter ofthe microtubule, jRLj is the distance between the central points oftubulin and the guanosine in longitudinal direction, and Vce is re-garded as a constant that takes the value of jR�Lj under the equilib-rium state of a minimum system energy.

Given an unknown state, the continuum strain is uniformly dis-tributed and can be defined as

kl ¼jRLj � jR�LjjR�Lj

; kh ¼jRC j � jR�C jjR�C j

ð51Þ

where jRCj is the distance between the central points of neighboringspheroids in circumferential direction, and jR�C j is the equilibriumfictitious-bond length after optimization. To achieve equilibriumstatus, the system must have a minimum energy, and thus we have

@w@kl

kl¼k�l

¼ @w@kh

kh¼k�h

¼ 0; ð52Þ

k�l ¼jR�Lj � R� rjR�Lj

; k�h ¼jRC j � 2RjR�C j

; ð53Þ

where k�l and k�h are parameters determined by minimizing strainenergy density.

Newton’s method is applied to this nonlinear problem with twovariables. The first- and second-order derivatives of w with respectto independent parameters are required at each iteration step. Inorder to find the equilibrium state, we introduce the unknownvector

k ¼ ðkl; khÞT : ð54Þ

By setting an initial value k(0) and an accuracy parameter d, the un-known vector can be determined by iteratively solving the incre-mental equation

kðkþ1Þ ¼ kðkÞ �M0ðkðkÞÞ�1MðkðkÞÞ; ð55Þ

and satisfying

kðkþ1Þ � kðkÞ 6 d; ð56Þ

where M and M0 are the first- and second-order derivate matrix,respectively, of w with respect to k.

5. Elastic properties of microtubules

The orthotropic elastic moduli of microtubules in the longitudi-nal and circumferential directions are El and Eh. Lateral interactionsplay an important role in the stability, rigidity, and intensity of

microtubules, but highly orthotropic properties indicate that Eh isseveral orders of magnitude weaker than El due to the energy inthe longitudinal direction being relatively stronger. Because jR�Ljand jR�C j are equilibrium lengths in the circumferential and longitu-dinal directions, they are unknown before the minimization of therepresentative unit cell strain energy density w. For non-equilib-rium energy state, fictitious-bond length jRLj and jRCj are the dis-tances between central points of spheroids in orthogonaldirections, which are remained to be optimized. The strain andstress are subsequently expressed as

el ¼ kl � 1; eh ¼ kh � 1; er ¼ eh ð57Þ

rl ¼@w@el¼ @w@kl

; rh ¼@w@eh¼ @w@kh

; ð58Þ

p ¼ 4trh=ðDouter þ dinnerÞ ð59Þ

where el, eh and er are the longitudinal, circumferential and radialstrain, respectively, rl and rh are the longitudinal and circumferen-tial stress, respectively, p is the hydrostatic pressure, and t is thewall thickness.

Nonlinear strain and stress are defined for orthotropic moduli[48], and are closely related to the system energy. The nonlinearlongitudinal, circumferential and radial moduli can be obtained as

El ¼@2w

@k2l

� @2w@kl@kh

!2,@2w

@k2h

; ð60Þ

Eh ¼@2w

@k2h

� @2w@kh@kl

!2,@2w

@k2l

; ð61Þ

Er ¼ 2Eh Douter � dinnerð Þ= Douter þ dinnerð Þ: ð62Þ

The longitudinal modulus and circumferential modulus are the ini-tial tangential moduli El and Eh in the minimum energy system,which are calculated at the initial equilibrium state withjRLj ¼ jR�Lj and jRC j ¼ jR�C j.

Computation is carried out for a microtubule by a self-devel-oped computing program. The spheroid model of the tubulinmonomer has a diameter of 4.8 nm and that of the guanosine mol-ecule has a diameter of 0.55 nm. The spheroid volume of the tubu-lin monomer and guanosine molecule are Vt = 57.90 nm3 andVG = 0.08711 nm3. The volume density is calculated and shown inTable 2. Because geometrical dimensions are considered in thecontinuum model, the microtubule is modeled as a homogeneouscontinuum hollow tube with a thickness of 4.8 nm. The volumeof space occupied by a single tubulin monomer and a guanosinemolecule in the cylindrical model is 113.6 nm3 and 12.9 nm3,respectively. The equilibrium state is determined by minimizingthe interaction energy in a representative unit cell. After this pro-cess, the equilibrium distance between lateral protofilaments inthe circumferential direction and the equilibrium distance be-tween the guanosine and the tubulin in the longitudinal directionare obtained. The results show that the equilibrium fictitious-bondlengths in the two directions are 2.903 nm and 5.006 nm respec-tively, and the longitudinal modulus El, circumferential modulusEh, and radial modulus Er evaluated by the model are 1.85 GPa,3.03 MPa and 1.44 MPa, respectively. It has been reported thatthe circumferential modulus of microtubules is two orders of mag-nitude lower than the longitudinal modulus [15], and that the lon-gitudinal and lateral moduli are in the range of 1–4 MPa [15] and

Table 3Orthotropic elastic moduli.

Elastic modulus Longitudinal (GPa) Circumferential (nm MPa)

Reference values 1–2.55 1–4Present results 1.85 3.03

K.M. Liew et al. / Composite Structures 93 (2011) 1809–1818 1817

1–2.55 GPa [42], respectively. These reference values are con-trasted with the current results in Table 3. After checking thenumerical results, it can be seen that the elastic moduli obtainedfrom the continuum constitutive model in this study match the re-cently obtained experimental and theoretical values well.

6. Conclusions

This study proposes a continuum framework for polyatomicbio-composite structure, and a constitutive model is developedthat strategically couple atomistic simulations with continuummechanics. The philosophy and methods of this approach areextensively explored, and generate new insights in determiningsystem energy, reflecting real geometry and predicting orthotropicelastic properties of microtubules, which are computed and dis-cussed based only on the interatomic potential and a quasicontin-uum technique. Good results are obtained from the computersimulations.

The atomistic interactions in both the longitudinal and circum-ferential directions of microtubules are considered by assemblinginteratomic energy of innumerable pair atoms to a fictitious-bond,and the real configuration of microtubules is taken into account topredict their elastic properties by incorporating the higher-orderCauchy–Born rule in the rolling process. An attainable atomistic–continuum method for computing the nonlinear orthotropic elasticproperties of microtubules is proposed that is based on the inher-ent interatomic energy. By introducing a fictitious-bond, the pro-posed computational model related the system energy withfictitious-bond vectors and current deformation gradients inthree-dimensional configurations. A feasible continuum constitu-tive model is then established suitable for considering arbitrarydeformation states and different loading conditions of this poly-atomic bio-composite structure, thus it has profound significancein polyatomic composite material study and breaks new ground inresearch into the various mechanical properties of microtubulessubjected to specific displacement and loading conditions. The pro-posed model can enable computational simulation for microtu-bules in future study.

Acknowledgements

The work described in this paper was fully supported by a CityUniversity of Hong Kong Strategic Research Grant (Project No.7008112) and the China National Natural Science Foundation (Pro-ject No. 10902129).

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