A helical Cauchy-Born rule for special Cosserat rodmodeling of nano and continuum rods

1Ajeet Kumar, 2Siddhant Kumar and 1Prakhar Gupta1Department of Applied Mechanics, IIT Delhi, Hauz Khas, New Delhi, India, 110016

2Department of Mechanical Engineering, IIT Delhi, Hauz Khas, New Delhi, India, 110016Correspondence: ajeetk@am.iitd.ac.in

Abstract. We present a novel scheme to derive nonlinearly elastic constitutive laws for specialCosserat rod modeling of nano and continuum rods. We first construct a 6-parameter (correspond-ing to the six strains in the theory of special Cosserat rods) family of helical rod configurationssubjected to uniform strain along their arc-length. The uniformity in strain then enables us todeduce the constitutive laws by just solving the warping of the helical rod’s cross-section (smallestrepeating cell for nanorods) but under certain constraints. The constraints are shown to be criticalin the absence of which, the 6-parameter family reduces to a well known 2-parameter family ofuniform helical equilibria. An explicit formula for the 6-parameter helical map is derived whichmaps atoms in the repeating cell of a nanorod to their images for the purpose of repeating cellenergy minimization. A scheme for the passage from nano to continuum scale is also presentedto derive the constitutive laws of a continuum rod via atomistic calculations of nanorods. Thebending, twisting, stretching and shearing stiffnesses of diamond nanorods and carbon nanotubesare computed to demonstrate our theory. We show that our scheme is more general and accuratethan existing schemes allowing us to deduce the shearing stiffness and several coupling stiffnessesof a nanorod for the first time.

Keywords: Cauchy-Born rule, special Cosserat rod, elastic constitutive modeling, helical sym-metry, molecular simulation.

Mathematics subject classification: 74B20, 74A25, 74Q15.

1 Introduction

Interest in rod theory has surged during the last few decades, due in large part to its applicabilityin biophysics, e.g., modeling of biological filaments such as DNA (Manning et al., 1996; Wang et al.,1997), collagen fibrils (Bozec et al., 2007), plant tendrils (Goriely and Tabor, 1998) etc. Recently,rod theory has also been employed for modeling cables (Goyal et al., 2005), curly hairs (Miller etal., 2014), carbon nanotubes and nanowires (Gould and Burton, 2006; Chandraseker et al., 2009;Kumar et al., 2011; Fang et al., 2013). The knowledge of accurate nonlinear constitutive laws isthe key to modeling such long and slender elastic bodies. The classical formulae for bending (EI),twisting (GJ) and extensional (EA) stiffnesses of a rod are well known, but they do not hold in thematerially nonlinear regime. Besides, these formulas do not hold for nanotubes and nanorods even inthe materially linear regime due to nanoscale effects. Here ‘E’ and ‘G’ denote the Young’s modulusand the shear modulus of elasticity respectively, whereas ‘A’, ‘I’ and ‘J’ have the usual meaningsof cross-sectional area, second area moment and polar moment of area, respectively. The classical

1

formula for the shearing stiffness (κGA) is a bit involved due to the shear correction factor (κ)which depends on a cross-section’s shape too (Timoshenko, 1940; Cowper, 1966; Hutchinson, 2001).There appears to be no work to derive this correction factor for nanorods. Chiral rods (Moroz andNelson, 1997; Upamanyu et al., 2008) have intrinsic twist in their stress-free reference configuration.They exhibit interesting coupling behavior, e.g., coupling between stretch and twist (Chandrasekerand Mukherjee, 2006; Upamanyu et al., 2008), shear and bending (Healey, 2002), radial stretch andtwist (Kumar and Mukherjee, 2011) etc. However, the formulas for these coupling stiffnesses havenot been derived yet. Simo and Vu-Quoc (1991) derived expressions for all the relevant stiffnesses oftheir rod model by substituting the constrained kinematics of their model in the three-dimensionallinear and isotropic constitutive laws. However, such a method does not yield correct expressionsof stiffnesses due to the assumed in-plane rigidity of a cross-section.

There have been several works on deducing the constitutive laws of bulk crystalline materialsfrom the inter-atomic energy of the relevant unit cell. These methods are primarily based on theCauchy-Born (CB) rule (Ericksen, 2008). The key here is the presence of translational periodicityin a uniformly deformed bulk crystal which reduces computation on the entire crystal to just theatoms of its unit cell. However, this periodicity is not retained when a nanorod is strained uni-formly along its arc-length, e.g., a uniformly twisted or uniformly bent configuration of a nanorodpossesses no translational periodicity (see Fig.1). This loss of translational periodicity also occursin case of “uniformly deformed” two-dimensional sheets which motivated Arroyo and Belytschko(2002) to propose the exponential Cauchy-Born rule. The same issue arises in case of thin filmswith “N” layers of atoms where the in-plane and bending strains need to obey certain compatibilityconditions in order to be integrable (Friesecke and James, 2000; Schmidt, 2008). We shall see thatthe configurations of a special Cosserat rod (Antman, 1995), when strained uniformly along its arc-length, form a 6-parameter family of rod configurations possessing helical symmetry. The number 6here corresponds to the strain measures in this theory (2 shears, 1 axial strain, 2 bending curvaturesand 1 twist) which reduces to 3 in case of Kirchoff rods (Love, 2000). Yang and Weinan (2006)proposed a generalized Cauchy-Born rule for rods. However, it has a major drawback: the methodat most allows a circular cross-section, for example, to turn into a planar ellipse. James (2006)proposed a theory of objective structures in which he uses objective boundary conditions to studydeformations of a nanorod based on a 2-parameter helical symmetry (e.g., coupled extension-twist,pure bending). Subsequently, Cai et al. (2008) proposed an alternate formulation based on periodicboundary conditions to simulate the 2-parameter helical deformations of James (2006). Hakobyanet al. (2012) proposed an objective Cauchy-Born rule to study planar deformations of nanorodssubjected to coupled bending and stretching. However, shearing is not allowed in their formulation.The case of non-planar deformations is even more involved requiring simultaneous bending, twisting,stretching and shearing. Palanthandalam-Madpusi and Goyal (2011) proposed an inverse techniquein which they presumed the functional form of the constitutive laws to be a truncated Fourier seriesand fitted the unknown parameters in the truncated series using equilibrium solutions of nanorods.In addition to the limitation of using truncated Fourier series, their approach also requires atomicequilibrium solutions of the full-length nanorod which is prohibitively expensive. Thus, no accurateand efficient method exists to deduce full nonlinear constitutive laws of nano and continuum rodsmodeled as a special Cosserat rod. In this paper, we present a helical Cauchy-Born rule (HCB rule)to achieve this objective.The paper is organized as follows. We begin with a brief description of the theory of special

Cosserat rods in Section 2. In Section 3, we derive an analytical formula for the deformation map

2

Figure 1: Loss of translational periodicity in a uniformly twisted nanorod.

of a continuum rod strained uniformly along its arc-length and show that such rods form a familyof 6-parameter helical configurations. We further show that to let the cross-sections warp in theseconfigurations and also preserve the uniform strain imposed in the helical rod, warping must beallowed but under certain constraints (see (20)). In Section 4, we then hypothesize that the discreteanalogues of the continuum formulas and the constraint equations derived in Section 3 hold fornanorods which gives us the desired discrete map (27) and the discrete constraint equations (28)for generating the 6-parameter family of a nanorod. We call these discrete analogues ((27) and(28)) the helical Cauchy-Born rule. The expressions for forces and moments that act in a nanorodas well as its various stiffnesses are then derived in terms of the relaxed positions of atoms in itsrepeating cell and the inter-atomic potential used. In Section 5, we compare our methodology withthe theory of Objective structures and also discuss an example of coupled bending-stretching defor-mation in a nanorod to illustrate the role of constraints. In Section 6, a scaling law is presented toderive constitutive laws of a continuum rod through a series of atomistic calculations on nanorodsof increasing radii. In Section 7, we then apply the HCB rule to compute various stiffnesses of adiamond nanowire. Interesting effects of surface relaxation on a nanorod’s constitutive responseare also observed in atomistic calculations based on the HCB rule. These effects are independentlyverified using linear elasticity theory augmented with surface stress effects. In Section 8, we fi-nally compare our methodology with existing techniques for deriving stiffnesses of both nano andcontinuum rods and establish that our method is more general and accurate. To verify that thestiffnesses obtained using our scheme are indeed accurate, we study the onset of Euler bucklingin a (10,10) single-walled carbon nanotube. We see that the critical buckling load obtained fromFEM simulation of the nanotube matches accurately with fully atomistic results when full nonlinearconstitutive laws are input to the FEM model. Section 9 concludes our paper and proposes severaldirections for further research.

Notation: The set of unit vectors e1, e2, e3 represent a fixed, right-handed, orthonormal ba-sis for R3, X ≡ (X1,X2, s) denotes the position of a typical material point of a rod in its stress-freereference configuration whereas x denotes the same material point in deformed configuration. Here,(X1,X2) denote the cross-sectional coordinates and “s” denotes the undeformed arc-length coordi-nate (see Fig.2). Repeated Latin indices always sum from 1 to 3 whereas repeated Greek indices

3

DEFORMED

1

e

X1

X2

X3

___

d

dd3

(s)

e3

2 ΩUNDEFORMED (REFERENCE)

s

r(s)

2

1

(s)

(s)

e

Figure 2: A typical rod deforming from its straight state reference configuration.

sum from 1 to 2. Finally, the differentiation of a field variable with respect to the undeformedarc-length is denoted by a superscripted prime, i.e., d

ds≡ (·)′.

2 Brief description of a special Cosserat rod

Fig.2 shows a typical rod deforming from its straight state reference configuration. The kinematicvariable r (·) in the figure denotes the centreline of a rod. We also have two directors (d1(·),d2(·))which are images of two orthonormal lines (e1, e2) in the undeformed cross-section of a rod (typicallymaterial lines along the two principal axes). The third director d3(·) is assumed to be perpendicularto the other two. The directors rigidly rotate in this theory (Antman, 1995) and model rotation ofa rod’s cross-section, i.e., di(s) = R(s)ei where R(·) denotes a three-dimensional rotation tensor.The field quantities (r(·),R(·)) being the kinematic variables, their frame-indifferent derivatives(RT r′,RTR′) define strains in this theory. Let us define

r′ = Rv, v =[

ν1 ν2 ν3]T,

R′ = RK, k ≡ axial(K) =[

κ1 κ2 κ3]T. (1)

Here, the components of v are defined with respect to the global basis e1, e2, e3 which also equalthe components of r′ in the local frame of directors. The first two components of v represent shearwhile the third component represents axial stretch. Similarly, k is defined as the axial vector of theskew symmetric matrix K. The first two components of k represent the two components of localbending curvature while the third component represents twist. We then let n(·) and m(·) denotethe internal contact force and the internal moment respectively that act on a typical cross-sectionof a rod and write

n = nidi, m = midi. (2)

Here n1, n2 are the shear forces, n3 is the axial force, m1, m2 are the bending moments, and m3 is the

torque or twisting moment. Let us define the triples n =[

n1 n2 n3

]T, and m=

[

m1 m2 m3

]T.

Upon force and moment balance of an arbitrary segment of a rod, one can derive the following

4

differential equations for static equilibrium of a rod:

n′ + n = 0 and m′ + r′ × n+ m = 0. (3)

Here n and m are the distributed load and the distributed couple respectively that act on a rod. Fora hyperelastic rod, we assume the existence of a twice-differentiable, scalar-valued function Φ(v, k)which is strain energy per unit undeformed length in this theory. Using the energy minimizingprinciple and tools from calculus of variations, one can also derive the relevant Euler-Lagrangeequations of a rod which, when compared with (3), lead to

n =∂Φ

∂vand m =

∂Φ

∂k. (4)

The equations in (3) can then be written in the local frame of directors as

(

∂Φ

∂v

)′

+ k×∂Φ

∂v+ n = 0 and

(

∂Φ

∂k

)′

+ k×∂Φ

∂k+ v×

∂Φ

∂v+ m = 0. (5)

Based on the symmetry group relevant to circular rods, Healey (2002) classified them into trans-versely hemitropic and transversely isotropic rods. The relevant symmetry group for hemitropicrods is defined as follows:

Q ≡

Qθ : Φ(Qθv,Qθk) = Φ(v, k), ∀ 0 ≤ θ < 2π,Qθ ∈ SO(3),Qθe3 = e3

. (6)

This group can make a distinction between left and right handed helical patterns in a rod and isuseful in modeling intrinsically twisted rods, e.g., chiral nanotubes, DNA, collagen fibrils etc. Thesymmetry group for transversely isotropic rods, on the other hand, also contains improper rotationsin (6). Using the symmetry group of hemitropic rods (6) and certain “flip-symmetry” that modelshomogeneity in a rod along its arc-length, Healey (2002) derived a quadratic form of the strainenergy per unit undeformed length as shown below.

Φ(·) =1

2

[

Aκακα + Bκ23 + Cνανα +D(ν3 − 1)2 + 2E(ν3 − 1)κ3 + 2Fνακα

]

. (7)

Here, the coefficients A,B, C and D are the bending, twisting, shearing and extensional stiffnessesrespectively. The coefficient E denotes the coupling stiffness between axial strain and twist whileF denotes coupling between bending and shear. Both the coupling coefficients vanish in case oftransversely isotropic rods. Even for materially linear behavior, one needs to know the six coefficientsin (7). Deducing materially nonlinear behavior is even more difficult. One possibility would be todeduce Φ(·) from the three-dimensional elastic strain energy density W (·) by integrating the latterin the plane of the undeformed cross-section (Ω0), i.e.,

Φ(v, k) =

∫

Ω0

W (F(v, k)) dX1dX2. (8)

Here F() denotes the deformation gradient. Antman (1995) proposed several mathematical formsfor the three-dimensional deformation map of a rod in terms of its kinematic variables. One of themost widely used form is

x(X) = r(s) +XαR(s)eα. (9)

5

Using (9), the deformation gradient can be written as

F = [r′ +XαR′eα]⊗ e3 +Reα ⊗ eα = R [v⊗ e3 +XαKeα ⊗ e3 + eα ⊗ eα] , (10)

which could then be substituted in (8) to deduce Φ(·). However, the deformation map (9) restrictsa typical cross-section of a rod to be rigid. In reality, a typical cross-section of a continuum or nanorod does not remain rigid when strained. Forcing this rigidity leads to higher Φ(·) and incorrectstiffnesses. We make a note, however, that the deformation map (9) has been used widely byseveral researchers to illustrate the connection between the theory of special Cosserat rods and thethree-dimensional elasticity theory. The special theory of Cosserat rods does allow warping of across-section but warping is assumed to depend completely on the local value of the strain (v, k).Accordingly, no independent kinematic variable is needed to model warping. In the general theoryof Cosserat rods (Antman, 1995; Kumar and Mukherjee, 2011), the two cross-sectional directorscan also become non-orthogonal which allows a restricted form of in-plane warping (anisotropicexpansion and in-plane shearing) to be independent from other deformation modes but out-of-plane warping is still assumed to be dependent on all the other independent modes of deformation.Simo and Vu-Quoc (1991) introduced an additional kinematic variable in the theory of specialCosserat rods to let out-of-plane warping (due to twist only) be an independent deformation modebut in-plane warping is not allowed there. We show in the next section that accurate estimates ofstiffnesses for the special Cosserat rod theory can be obtained by allowing a typical cross-section towarp in the most general way (both in-plane and out-of-plane), but under certain constraints.

3 Configuration of a rod strained uniformly along its arc-

length

We now derive an expression for the configuration of a rod subjected to uniform strain (v, k) alongits arc-length in which the warping of the cross-section is also allowed. The cross-sectional strainenergy of such a configuration can then unambiguously be defined to be Φ(v, k). Assume the angle

between v and k to be φ (see Fig.3(a)). Furthermore, let k and k⊥denote the unit-normed tuples

parallel and perpendicular to k respectively, which lie in the plane formed by v and k. We thendecompose v as

v = |v|cos(φ)k + |v|sin(φ)k⊥. (11)

Upon substituting (11) in the definition of strain measures (1) and integrating, one can derive thefollowing analytical expressions for the configuration variables (r(·),R(·)):

r(s) = r0 + |v|R0

[

cos(φ)sk + sin(φ)

∫ s

0

elK dl k⊥]

,

R(s) = R0esK. (12)

Here, r0 and R0 denote the centerline position and the rotation respectively of a cross-sectionwhich was originally at s=0. Without affecting energy, one can rigidly translate and rotate thewhole deformed rod such that the “s=0” cross-section has its center at the origin and is oriented

6

(a) (b)

Figure 3: (a) A uniformly strained continuum rod (shown in pink-blue shades) whose centerline

forms a helix and lies on a cylinder (shown in light green) of radius rhelix =|v|sin(φ)

|k|. (b) A nanorod

with its repeating cells forming a helical configuration.

in the e1 − e2 plane. Further, substituting Rodrigues’ formula (47) for the rotation matrix elK in(12), we obtain

r(s) = |v|

[

cos(φ)sk + sin(φ)1

|k|

(

I− esK)

k× k⊥]

,

R(s) = esK. (13)

Let us define the following quantities:

τ = |v|cos(φ),

θ = |k|,

Rsθ = esK,

xf =|v|sin(φ)

|k|k× k

⊥. (14)

Substituting (14) into (13), the equation of the centerline then becomes

r(s) = sτ k +(

I−Rsθ

)

xf . (15)

Equation (15) represents a helix whose axis passes through xf (also called a fixed point) and is

directed along k (see Fig.3(a)). The radius of the helix (rhelix) equals|v|sin(φ)

|k|. Similarly, the pitch

of the helix equals 2πτ|k|

= 2π|v|cos(φ)|k|

. Thus, the parameters of this helix (fixed point, axis, radius and

pitch) get set by the prescribed uniform strain field in the deformed rod.

7

To allow a typical cross-section to also relax, let x0(X1, X2) denote the unknown relaxed shapeof the “s=0” cross-section. The deformation map for the whole rod can then be written as

x(X1,X2, s) = r(s) +Rsθx0(X1,X2)

= xf +Rsθ(x0(X1,X2)− xf) + sτ k (using (15)). (16)

From (16), we note that the deformed position of any cross-section (originally at axial location ‘s’)can be obtained in terms of the cross-section at s=0 by first rotating the latter about the fixedpoint (xf) of the helix by Rsθ and then translating the rotated cross-section by sτ k. Let us writex0(X1, X2) = Xαeα+u0(X1, X2) where u0(·) denotes the displacement of material points from theirundeformed reference position. For sake of brevity, we will now denote the relaxed position and thedisplacement of the cross-section at s=0 by just x() and u() respectively. In order to solve for theunknown relaxed position, the total strain energy of a rod’s cross-section must be minimized, i.e.,

minx()

∫

Ω0

W(

Frod(v, k,x();X1, X2, 0))

dX1dX2. (17)

Here Frod() denotes the deformation gradient of the map (16) and equals

Frod(v, k,x();X1, X2, s) = Rsθ

[(

K(x− xf ) + τ k

)

⊗ e3 +∂x

∂Xα

⊗ eα

]

= Rsθ

[

(v + k×Xαeα)⊗ e3 + eα ⊗ eα

+ (k× u)⊗ e3 +∂u

∂Xα

⊗ eα

]

.

(18)

Note that if x() minimizes (17) for r() and R() given by (13) then the map given by(

r(), R(), x())

→(

xf +Rlθ(r()− xf ) + tk, RlθR(), xf +Rlθ(x()− xf) + tk))

(19)

generates a two-parameter family of energetically equivalent configurations which also minimize(17) for any pair of (l, t).

It may be noted that (17) alone would only generate a two-parameter family of energetically-distinct configurations for the helical parameters (τ, θ) (Ericksen, 1977; Chouaieb and Maddocks,2004). To see this, note that (17) nowhere dictates the minimizing x() to be such that its center isrhelix distance away from the helical axis. However, rhelix must not change since it is fixed throughthe imposed uniform strain field (see (14)). Similarly, preserving the orientation of a relaxed cross-section is necessary so that the imposed curvature and twist values are not affected. In essence, theminimizing x() must not violate (13) at s=0. Thus, the center of the relaxed cross-section mustremain at the origin and its principal axes must be aligned along the global co-ordinate axes. Forthe theory proposed here, we take the center of a cross-section to mean its mass center. The otheroption could be the geometric center of a cross-section but the dynamical terms in the equations ofa rod are simpler about the mass center. Similarly, by principal axes of a cross-section, we meanthe principal axes of the moment of inertia tensor. The six algebraic equations enforcing theseconstraints are as follows:

∫

Ω0

ρ0(X1, X2) x(X1, X2) dX1dX2 = 0 (mass center at the origin),

∫

Ω0

ρ0(X1, X2) M dX1dX2 = 0 (mixed moments of inertia vanish), M =

x2()x3()x3()x1()x1()x2()

. (20)

8

Here, ρ0() denotes density of the body in its undeformed configuration. The first two components

of the unknown relaxed position vector of the cross-section(

x(X1, X2) =[

x1() x2() x3()]′ )

allowin-plane warping whereas the third component allows out-of-plane warping. The six constraintequations in (20) remove the 2-parameter energetically-equivalent solutions (see (19)) of the min-imization problem (17), making it well posed. In addition, they generate a 4-parameter family ofenergetically-distinct solutions for every pair of (τ, θ).

To solve the above constrained minimization problem, we define a constrained cross-sectionalenergy functional as follows:

G(

x(),λ, δ)

=

∫

Ω0

[

W (Frod) + ρ0(λ · x+ δ ·M)]

dX1dX2. (21)

Here λ and δ are the Lagrange multiplier vectors which enforce the constraints in (20). The firstvariation of (21) gives us the following Euler-Lagrange equations:

∇α · S− Se3 × k = ρ0(λ+ δx) in Ω0,

Sm = 0 in ∂Ω0 (traction-free boundary condition). (22)

Here S = ∂W∂F

denotes the 1st Piola-Kirchoff stress tensor, ‘m’ denotes the outward normal tothe boundary of a cross-section, (∇α·) denotes divergence in the plane of a cross-section and the

symbol δ =

0 δ3 δ2δ3 0 δ1δ2 δ1 0

is a matrix formed by the three components of δ. Equation (22) must

be supplemented with the constraint equations (20) in order to obtain the warped cross-sectionx() and the Lagrange multipliers λ and δ. The right hand side of (22)(a) can be termed theconstrained body force per unit undeformed volume, which is required to maintain the helical rodin static equilibrium. In the absence of the constraint equations (20), the constrained body force alsovanishes and (22) then only generates a two-parameter family of energetically distinct equilibriumconfigurations.

The strain energy per unit undeformed length can now be written as

Φ(v, k) =

∫

Ω0

W

(

Frod

(

v, k, x(X1, X2; v, k);X1, X2, 0)

)

dX1dX2. (23)

Here, the dependence of the relaxed cross-section x() on the imposed strain (v, k) has been madeexplicit.

3.1 Cauchy-Born rule for bulk crystals

The expression (23) to deduce Φ() requires knowledge of the three-dimensional elastic strain energydensity W (). However, this itself could be an unknown. The standard Cauchy-Born rule (Ericksen,2008) can be used to deduce W () using atomic level calculations. According to this, the latticevectors of a three-dimensional crystal, subjected to uniform deformation gradient field F, transformas

ai = FAi, i = 1, 2, 3. (24)

9

Here ai and Ai are the deformed and undeformed lattice vectors respectively. Due to periodicity inthe undeformed crystal lattice and the linearity of the map (24), the deformed lattice also retainsperiodicity. The positions of the atoms in the deformed crystal are then given by

x(n1,n2,n3),j = xj + nkak = xj + nkFAk. (25)

Here xjmj=1 denote deformed position vectors of the atoms within the unit cell and x(n1,n2,n3),j

denotes (n1, n2, n3)-image of the atom xj . The unknown positions xjmj=1 are obtained by mini-

mizing the unit cell energy with respect to xj while keeping the lattice vectors fixed and enforcingperiodicity as in (25). The three-dimensional elastic strain energy density of the crystal is thendefined as follows:

W (F) =1

V0

(

min(x1,··· ,xm)

E(x1, · · · ,xm;F)−Eref

)

=1

V0

(

E(x1(F), · · · , xm(F);F)−Eref

)

. (26)

Here V0 = (A1 × A2) · A3 denotes undeformed volume of the unit cell, E() denotes inter-atomicenergy of the deformed unit cell, Eref denotes the same for undeformed unit cell and xj(F) denotesposition of the jth atom obtained after minimizing the unit cell energy.

It does not seem appropriate to view one-dimensional nano-structures (e.g., long nanotubes andnanowires) as three-dimensional elastic bodies. Hence, deducing their Φ() through (23) would beincorrect. In the next section, we propose an independent methodology for such nanorods andnanotubes.

4 Nanorods uniformly strained along their arc-length

Having derived the formula (16) for uniformly strained continuum rods, we hypothesize that thediscrete version of the same formula holds for uniformly strained nanorods and nanotubes. Astraight crystalline nanorod, for example, will be periodic along its length. Let us assume itssmallest period to be L0. Then, the atoms within this periodic length will form the nanorod’srepeating cell. We call this repeating cell the fundamental domain (FD) from now on (as in James(2006)). A discrete variant of the map (16) can then be thought to act on the atoms in thefundamental domain (xj

mj=1) to generate their ith image as shown below (also see Fig.3(b)).

xi,j = RiθL0xj + iτL0k + (I−RiθL0

)xf . (27)

In order to obtain the unknown atomic positions xjmj=1 in the fundamental domain, we would

need to minimize inter-atomic energy of the fundamental domain (EFD) but under constraints (asdiscussed in Section (3)) which are

m∑

j=1

mjxj = 0 and M ≡m∑

j=1

mj

xj,2xj,3xj,3xj,1xj,1xj,2

= 0. (28)

Here ‘mj ’ is mass of the jth atom. We call the map (27) along with (28) the helical Cauchy-Born

rule. For constrained minimization, we then define the constrained fundamental domain energy asfollows:

Econs(v, k; xj ,λ, δ) = EFD (τ(v, k), θ(v, k); xj) + λ ·∑

mjxj + δ ·M. (29)

10

In order to obtain the strain energy per unit undeformed length, we then minimize (29) withrespect to the atomic positions and the Lagrange multipliers. This yields the following set of 3m+6nonlinear algebraic equations:

∂Econs∂xi

=∂EFD∂xi

+mi(λ+ δxi) = 0,

∂Econs∂λ

=∑

mjxj = 0,

∂Econs∂δ

= M = 0. (30)

The first term in (30)(a) denotes the force on atom ‘i’ due to all other atoms in a nanorod whilethe remaining terms denote “non-physical” forces due to constraints. Finally,

Φ(v, k) =1

L0

(

EFD (τ(v, k), θ(v, k); xj(v, k))−Eref

)

. (31)

Here Eref denotes the relaxed inter-atomic energy of the fundamental domain in its undeformedreference state. The procedure to compute EFD and its derivatives with respect to the atomicpositions and strains are described in Appendices A and B.

4.1 Expressions for forces, moments and the elasticity tensor

The presence of constraints in the formulation complicates derivation of the expressions for forces,moments and the elasticity tensor. In this section, we will obtain their simplified expressions bygetting rid of the constraint terms through a projection. The first derive of (31) with respect tostrain gives us the expressions for forces and moments , i.e.,

∂Φ

∂[v, k]=

1

L0

(

∂EFD∂[v, k]

∣

∣

∣

∣

xj

+

∂EFD∂xj

∣

∣

∣

∣

(v,k)

·

∂xj∂[v, k]

)

(using (31))

=1

L0

∂EFD∂[v, k]

∣

∣

∣

∣

xj

. (32)

We prove in Appendix C that the second term in (32)(a) vanishes. We then obtain the expressionfor the elasticity tensor by differentiating (32) with respect to strains in the following way.

∂2Φ

∂[v, k]2=

1

L0

(

∂2EFD∂[v, k]2

∣

∣

∣

∣

xj

+

∂2EFD∂[v, k]∂xi

·

∂xi∂[v, k]

)

=1

L0

(

∂2EFD∂[v, k]2

∣

∣

∣

∣

xj

−Q2T

∂2EFD∂[v, k]∂xi

·

(

Q2T

Kij + δijδ

Q2

)−1

Q2T

∂2EFD∂xj∂[v, k]

)

. (33)

Here, the matrix Kij =∂2EFD∂xi∂xj

whereas Q2 is a projection matrix (defined in Appendix C) which

is used to get rid of the constraint terms. The second term in (33) is the effect of relaxation ofatoms on the elasticity tensor.

11

5 Relation with the theory of Objective structures and the

role of constraints

Going by the definition of “Objective structures” in James (2006), the 6-parameter family of helicalnanorods can be called objective molecular structures (OMS). Indeed, the corresponding atoms ineach molecule (FD or its images) of a nanorod see precisely the same environment up to a rotationand translation (see (27) and Fig.3). This definition of “objective structures” is purely kinematic.A family of 2-parameter helical configurations emerges once the equilibrium of every atom in thestructure is enforced. On the other hand, in case of 6-parameter helical configurations (which donot intersect with the 2-parameter case), individual atoms do not remain in equilibrium. In fact,additional constraint forces (see (30)(a) and (22)) are needed to maintain such uniformly strainedconfigurations. Let us understand how such a state of strain can be attained locally in an actualrod which is in static equilibrium (without the constraint forces).

In uniform equilibrium configurations of a rod (without the distributed load and couple), the firstterm in both the equations of (5) vanishes due to uniformity and solving the rod equations then leadsto the 2-parameter helical equilibria (Chouaieb and Maddocks, 2004). However, if the strain (v, k)at an arc-length coordinate, say “s0”, of the rod do not correspond to any strain in the 2-parameterfamily of helical configurations, the rod then varies its strain field in the neighborhood of “s0” sothat the first term in (5) becomes non-zero to enforce local equilibrium. When a uniformly strainedrod is generated with the same strain (v, k) using our methodology, the equilibrium equations in (5)imply that the constraint body force generates a distributed load and couple which exactly matchthe first term in (5) (evaluated at “s0”) of the non-uniformly strained equilibrium rod. We thus seethat the constraint forces in our formulation actually mimic the non-uniformity of strain field in anactual equilibrium rod. We now present an example of uniform bending-stretching of a nanorod toillustrate the role of constraints.

5.1 Uniform bending-stretching of a nanorod

In this case, we have v =[

0 0 ν3]T

and k =[

κ1 0 0]T. Thus, the helix degenerates into a

circle whose parameters are τ = 0, θ = κ1, xf = − ν3κ1e2 and rhelix = ν3

κ1(see Fig.4). Under 2-

parameter family of helical equilibria, only pure bending of a nanorod can be realized. Thus, onlyθ is prescribed but there is no way to impose ν3 too. Even here, the helical nanorod degeneratesinto a circle but the radius of the circle almost equals 1

κ1. Furthermore, to maintain equilibrium

of every individual atom, the nanorod cannot sustain any internal force in it. Let us see how theconstraints in our formulation do the trick of fixing the radius to be ν3

κ1and also generate internal

force in the nanorod.In case the FD has reflection symmetry about the (e1 − e2) and (e2 − e3) plane, the FD will

have no tendency to rotate when the nanorod is bent about e1 axis (see Fig.4). Hence, the threeconstraints employed to preserve the orientation of the FD do not get “active”. Two of the remainingconstraints that fix the mass center of the FD to be at the origin do the job of selecting just onesolution from the 2-parameter energetically equivalent solutions depicted by (19). Hence, they donot affect the energy of the FD but remove non-uniqueness from the minimization problem. Theconstraint which fixes the e2 co-ordinate of the mass center is the only non-zero Lagrange multiplierin this case which also fixes the radius of the circle to be ν3

κ1. Upon constrained relaxation, the

12

Figure 4: A portion of a nanorod uniformly bent (κ1) and stretched (ν3). The fundamental domainof relaxed axial length L0 is shown in green.

equation (30)a implies that the net force on the jth atom in the FD, due to all other atoms in thenanorod, is mjλ2e2. Accordingly, the net force on the FD is Mλ2e2 where “M” is the total mass ofthe FD (shown as the blue vector R in Fig.4). As shown in Fig.4, this net force can be generatedonly through the resultant of the force F that acts on the FD from the left and right portions ofthe nanorod. This force F is actually the internal axial force generated in a nanorod due to itsstretching.

Hakobyan et al. (2012) also proposed a scheme to simulate non-uniform bending in a nanoroddue to terminal load. They generate “local objective structures” corresponding to uniform bending-stretching deformations by fixing both radius (rhelix) and θ during relaxation but without introduc-ing any constraint explicitly. However, the formula used by them (see eq.(9) in Hakobyan et al.(2012)) to generate a locally uniform bent-stretched configuration couples every atom in the FD toevery other atom of the FD (even if they lie outside the radius of influence of the relevant inter-atomic potential). This generates a “non-physical” force on every jth atom of the FD (see equation(A13) in their paper) which equals the sum of the forces on the FD from the left and right portionsof the nanorod multiplied by mj/M . This is nothing but our constraint force mjλ2e2. Thus, ourformulation reduces to theirs in the special case of uniform bending-stretching deformation of ananorod.

We clearly see the role of constraints in fixing strains in the 6-parameter uniform helical config-urations. Although not all constraints get active for a particular strain, the formulation is generaland automatically activates the necessary constraints.

13

6 Passage from nano to continuum scale

Having discussed the methodology at both continuum and nano scales, we now discuss the passagefrom nano to continuum scale. Friesecke and James (2000) proposed a similar scheme in the contextof thin films and gave a brief idea of its extension to nanotubes. The idea here is to deduce theconstitutive laws of a continuum rod (having cross-section Ω0) through a series of computations onnanorods of increasing cross-sectional dimension (denoted by λΩ0) but subjected to uniform strain(vλ, kλ). Taking a cue from the scaling law of elasticity, i.e., if x(X) is a elasticity solution for abody D then xλ(X) = λx( 1

λX) is also a elasticity solution for a scaled body (λD), it is easy to show

that the solutions obtained from solving (17) under the constraints (20) also possess this scalingproperty. If we set

vλ = v, kλ =1

λk,

xλ(X1, X2, 0) = λx(X1/λ,X2/λ, 0), (34)

then one can show (using (23)) that Φ(

v, k)

= 1λ2Φλ(

vλ, kλ)

. Accordingly, forces and moments scaleas follows:

∂Φ

∂v=

1

λ2∂Φλ∂vλ

,∂Φ

∂k=

1

λ3∂Φλ∂kλ

. (35)

Similarly, the stiffnesses scale according to

∂2Φ

∂v2=

1

λ2∂2Φλ∂v2λ

,∂2Φ

∂v∂k=

1

λ3∂2Φλ∂vλ∂kλ

,∂2Φ

∂k2=

1

λ4∂2Φλ

∂k2λ. (36)

It may be noted that the solutions of nanorods at a given λ do not exactly follow the scaling lawdue to various nanoscale effects. Miller and Shenoy (2000), in fact, proposed an analytical modelto include the effect of surface stress on bending and stretching stiffnesses of square nanorods.However, with increase in λ, the effect of surface stress and other nanoscale effects diminish andthe scaled solutions then converge to the desired continuum solution.

7 Deducing constitutive laws of a diamond wire

We now use the theory proposed in preceding sections to deduce the constitutive laws of a continuumdiamond wire through a series of simulations on nanowires of increasing radii. The nanowire iscarved out from a bulk diamond crystal with the wire’s axis aligned along the [1 0 0] direction. TheTersoff potential (Tersoff, 1988) is used to model interaction between carbon atoms. The relaxedlattice constant for diamond is obtained by applying periodic boundary conditions to a unit cellof the diamond crystal and letting the length of the unit cell relax until it becomes stress free.Similarly, the relaxed material constants of the crystal are extracted from the matrix of secondderivative of the unit cell energy with respect to the deformation gradient (Tadmor et al., 1999).These constants are tabulated in Table 1 which match closely with the available literature data(Yoshikawa et al., 1993; Klein and Cardinale, 1993). We note that the shear modulus does notfollow the isotropy relation (G = E

2(1+ν)) exhibiting cubic symmetry of the crystal. Fig.5 shows

a typical fundamental domain of the nanowire that we use in our calculations. The fundamentaldomain spans one unit cell of the crystal along the nanowire’s axis and comprises of four layers ofatoms (see Fig.5(b)).

14

Lattice constant 0.3561 nmYoung’s modulus (E) 1052 GPaShear modulus (G) 641 GPaPoisson’s ratio (ν) 0.09

Table 1: Lattice constant and material parameters for diamond.

−2 −1 0 1 2−2

−1

0

1

2

nm

nm

[0 0 1]

[0 1 0]

(a)

−2 −1 0 1 2−2

−10

12

0

0.1

0.2

0.3

nmnm

nm[1 0 0]

axial direction

(b)

Figure 5: A typical fundamental domain of a circular diamond nanowire of 1.6 nm radius (509atoms): (a) axial view. (b) side view.

7.1 Effect of surface relaxation

In order to understand the passage from nano to continuum scale, it is important to study theeffect of surface relaxation on constitutive laws. We focus on a simple case of axial straining of acircular nanowire which we first present from a purely continuum perspective. We then comparethe continuum results with those from atomistic calculations. Assume the wire to be made up ofisotropic material governed by linear elasticity. Due to circular symmetry, no shear strain woulddevelop in it. In fact, any material point of the wire would displace only in the radial direction.Hence,

ǫRR =∂u

∂R, ǫθθ =

u

R. (37)

Here u(R) denotes the radial displacement. The strain ǫzz is prescribed to the nanowire. Uponsubstituting (37) in the equilibrium equations of elasticity, we get u(R) = cR for some constant ‘c’.Let ‘r’ denote the relaxed radius of the nanowire and ‘R’ its unrelaxed radius in the bulk crystal.We then obtain ǫRR = ǫθθ = r−R

R. To account for the effect of surface relaxation, the energy per

15

0 1 2 3 40.35

0.36

0.37

0.38

radius of nanowire (nm)

rela

xed

leng

th o

f sim

ulat

ion

cell

(L 0)

relaxed length (HCB, PBC)3−d lattice constant = 0.3561 nm0.3561(1+2νη/(RE))

(a)

0 1 2 3 4−1,000

−500

0

500

1,000

radius of nanowire (nm)

Axi

al fo

rce

(eV

/nm

)

ε= −0.003 (HCB, PBC)ε= 0.000 (HCB, PBC)ε= 0.003 (HCB, PBC)ε= 0.006 (HCB, PBC)

πR2Eε − 2πRνη

surface stress = 54.9 N/m

(b)

Figure 6: Effect of surface relaxation: (a) convergence of relaxed length of the fundamental domainto diamond’s lattice constant. (b) contour plots for axial force vs radius of the nanowire at constantaxial strain.

unit unrelaxed length (Φ) of the wire can be written as

Φ = πR2

(

1

2σ :: ǫ

)

+ 2πrη

=πE

(1 + ν)(1− 2ν)

(

(r − R)2 + 2νǫzzR(r − R) + .5R2(1− ν)ǫ2zz)

+ 2πrη. (38)

Here η denotes the surface stress of diamond’s lateral surface. Upon minimizing Φ in (38) withrespect to the unknown relaxed radius r and further using Hooke’s law, we obtain the followingexpression for axial force in the wire:

F = πR2Eǫzz − 2πRνη. (39)

Using formula (39), we can also find the axial strain (ǫzz) at which the axial force vanishes, i.e.,ǫzz =

2νηER

. Accordingly, the relaxed length of the fundamental domain for a nanowire of unrelaxedradius (R) would be

L0(R) = 0.3561

(

1 +2νη

ER

)

. (40)

We now perform minimization of the fundamental domain energy (based on the HCB rule) fordifferent radii of the nanowire and at different axial strains. The “zero axial strain” state is takento be the one for which the axial length of the fundamental domain equals the diamond’s latticeconstant. The energy of the fundamental domain is then minimized using (30) and the axial force is

16

computed using (32). To obtain the relaxed length of the fundamental domain for a given radius, thefundamental domain is strained such that the axial force in it vanishes. Since, the nanowire is onlybeing axially strained (at zero twist), we can also use conventional molecular statics using periodicboundary conditions (PBC) to compute the axial force as well as the relaxed length. The resultsfrom both the HCB and PBC computations match exactly since the HCB formulation reduces tothe PBC technique for the special case of axial straining. These results are plotted in Figs.6(a,b).

According to (39), the axial force is quadratic in the unrelaxed radius of the wire but becomeslinear when the imposed axial strain (ǫzz) is zero. The results from atomistic calculations in Fig.6(b)clearly support these observations. The coefficients obtained from data fitting of the axial forceatomistic data with a quadratic polynomial are tabulated in Table 2. On comparing with (39), thesurface stress “η” of the diamond crystal can be obtained as

η = −B

2πν= 54.9 N/m. (41)

With the value of the surface stress (η) taken from (41) and the Young’s modulus (E) and thePoisson’s ratio (ν) taken to be that of the bulk diamond crystal (see Table 1), we use formulae(40) and (39) to plot the relaxed fundamental domain length and the axial force in Fig.6. We notethat the continuum formulas reproduce the atomistic data obtained from the HCB rule accurately.This illustrates that continuum mechanics, when augmented with surface effects, holds even at thenanoscale.

F = A R2 +B R + Cǫzz A (πEǫzz) B (-2πνη) C (0)

0.000 -0.69 -197.24 19.74

Table 2: Coefficients of a quadratic polynomial obtained after fitting the axial force data.

7.2 Convergence of stiffnesses to their continuum limits

We now show that as the radius of the nanowire increases, its various stiffnesses converge to acontinuum limit. Fig.7(a) plots the ratio of stiffnesses of the nanowire in its relaxed configuration(obtained using formula (33)) against the nanowire’s unrelaxed radius. Based on classical formulasfor stiffnesses of a circular rod, the ratio of these stiffnesses can be obtained as follows:

twisting stiffness

bending stiffness≡

B

A=

GJ

EI=

2G

E,

shearing stiffness

extensional stiffness≡

C

D=

κGA

EA= κ

G

E,

shear correction factor(κ) =C

D×

2A

B. (42)

We indeed note from Fig.7(a) that the ratio of twisting to bending stiffness converges to the con-tinuum limit of 2G

Ewhile the ratio of shearing to extensional stiffness allows us to deduce the shear

17

0 1 2 3 40

0.4

0.8

1.2

radius of the nanowire (nm)

ratio

of m

odul

i

0.5* twisting stiffness/bending stiffness (HCB)shearing stiffness/extensional stiffness (HCB)G/E = 0.61 (diamond lattice)

shear correction factor = 0.88

(a)

0 0.1 0.2 0.30

400

800

1200

1600

Scaled twist (κ3=λ x κ

3,λ)

Sca

led

torq

ue (

T=

Tλ/λ

3 )

0.5342 nm (HCB)1.2464 nm (HCB)1.9586 nm (HCB)2.6708 nm (HCB)3.3830 nm (HCB)

(b)

Figure 7: (a) ratio of the nanowire’s stiffnesses in its stress-free reference configuration. (b) nonlineartorque vs twist response of the nanowire for different radii.

correction factor atomistically. We obtained the shear correction factor to be 0.88 which is veryclose to 0.87 based on its continuum formula (Cowper (1966), see equation (43) below).

κ =6(1 + ν)

7 + 6ν(for circular beams). (43)

Similarly, Fig.7(b) plots the scaled twisting response of a diamond wire (at zero axial strain) fordifferent radii of the nanowire. As per the discussion in Section 6, the twist value is scaled tok3 = λk3,λ while the torque is scaled to T = Tλ/λ

3 and then plotted in Fig.7(b). We clearly notethat as the radius of the nanowire increases, the scaled response converges to a nonlinear continuumresponse.

8 Comparison of HCB formulation with existing techniques

We now compare our formulation with existing techniques for deriving stiffnesses of nano andcontinuum rods.

8.1 Continuum rods

In Table 3, we show stiffnesses of a circular isotropic continuum rod derived using various tech-niques. We see that, except for twisting stiffness, all other stiffness values obtained using the rigidkinematics assumption (eq.(9)) are inaccurate. The formulation of Simo and Vu-Quoc (1991) isalso based on the assumption of in-plane cross-sectional rigidity. But, they later make an ad hocassumption for normal traction on a typical cross-section to be “E(ν3− 1)” which, although correctfor small deformations, is inconsistent with their rigidity assumption. Due to this, they are able to

18

A B C D coupling nonlinearity

Rigid kinematics (eq.(9)) EI (1−ν)(1+ν)(1−2ν)

GJ GA EA (1−ν)(1+ν)(1−2ν)

− −

Simo & Vu-Quoc (1991) EI GJ GA EA − ×Mora and Muller (2003) EI GJ × × × ×

Classical solution EI GJ 67GA EA −− −−

HCB rule EI GJ 67GA EA X X

Table 3: Stiffnesses of an isotropic circular rod obtained using various techniques: entries in redare inaccurate. The meaning of other symbols are as follows. “×”: cannot be derived, “−”: givesinaccurate result, “−−”: no technique exists, “X”: accurate estimate can be deduced.

obtain correct values for stretching and bending stiffnesses but shearing stiffness is still inaccurate.Mora and Muller (2003) rigorously derived a general expression to obtain the linear constitutivelaws of arbitrarily shaped cross-sections using Γ−convergence. But their method, being applicableonly for Kirchoff rods, cannot deduce shearing and stretching stiffnesses. We make a note that dif-ferent expressions of the shear correction factor (derived using different kinematic approximations)for a Timoshenko beam have been proposed in the literature (Timoshenko, 1940; Cowper, 1966;Hutchinson, 2001) but they all match at “0” poisson’s ratio (6

7for isotropic circular rods). Since

our formulation does not impose any ad hoc kinematic approximation on warping, we do not obtainany analytical expression for the correction factor but we found the obtained numerical value tomatch with 6

7. We also emphasize that the HCB formulation is unique with regards to obtaining

the coupling stiffnesses as well as material non-linearity (the rigid kinematic model can also be usedto deduce them but the values obtained would be highly inaccurate). The analytical expression forthe extension-twist coupling stiffness (E) based on our formulation turns out to be

E =(

E1 − 2G12(1 + ν21))

J θ (for thin tubes having low intrinsic twist θ). (44)

The formula (44) holds only for thin circular tubes made up of orthotropic material. Here “E1” isthe Young’s modulus along the principal material direction which twists about the tube axis withtwist “θ” whereas “G12” and “ν21” are the shear modulus and the poisson’s ratio in the lateralsurface of the tube. The formula for a thin tube but with large intrinsic twist is relatively longer(Singh et al., 2016). As per the formula, the coupling stiffness is proportional to intrinsic twist onlyfor low intrinsic twist but becomes identically zero for isotropic rods even at large intrinsic twist.

8.2 Nanorods

We now compare our approach with existing schemes for deriving stiffnesses of nanorods. Wetake a specific example of a (10,10) single-walled carbon nanotube (SWCNT). The Tersoff-Brennerpotential (Brenner, 1990) is used to model the interaction between carbon atoms. We clearly seefrom Table 4 that our scheme is versatile since accurate estimates of all stiffnesses can be obtained.The other schemes either cannot derive one or more stiffnesses or they give inaccurate result. Forexample, if we impose the general Cosserat rod kinematics in which the cross-sectional directorscan change their magnitude and also the angle between them, a typical circular cross-section of aCNT will at most deform into a planar ellipse (Kumar and Mukherjee, 2011). Yang and Weinan

19

A(

eV·nm)

B(

eV·nm)

C(

eVnm

)

D(

eVnm

)

E(

eV)

F(

eV)

Periodic boundary condition × × × 6363 × ×General Cosserat rod kine-matics

2261 1068 1978 6363 0 0

2-parameter family (James(2006))

1544 1068 × 6363 0 ×

Hakobyan et al. (2012) 1544 × × 6363 × ×HCB rule 1544 1068 1086 6363 0 0

Table 4: Stiffnesses of a (10,10) SWCNT in its relaxed state obtained using various techniques:entries in red are inaccurate. The symbol “×” implies the particular stiffness cannot be derived.

(2006) also imposed this kinematics in their model. We see from the Table that this restrictedform of in-plane warping leads to higher bending and shearing stiffnesses. In fact, if we imposeeven more restrictive special Cosserat rod kinematics (rigid cross-section), the stiffnesses obtainedwould be even higher (see Fig.8(a)). Chandraseker et al. (2009) also estimated the stiffnesses ofa (9,6) SWCNT by imposing periodic boundary conditions (PBC) on its unit cell. To derive thebending stiffness, e.g., they imposed clamped-clamped bending deformation on a periodic unit cellof a (9,6) SWCNT and assumed that the unit cell itself bends like a rod. The same deformationwas also imposed on the images of the unit cell to enforce periodicity. Being just 2 nm in length,the unit cell actually deforms like a shell. They accordingly derive inaccurate bending stiffness(about 1/60 of EI). Hakobyan et al. (2012) proposed a novel scheme to simultaneously bend andstretch a nanorod. However, since shearing is not allowed in their model, their method cannotdeduce several of the stiffnesses. With regards to coupling stiffnesses, the 2-parameter approachof James (2006) can be used to deduce accurate estimate of the extension-twist coupling whilebending-stretching coupling can be deduced using the scheme of Hakobyan et al. (2012). However,several other coupling stiffnesses (e.g., bending-shear, bending-twisting etc.) cannot be derivedusing any of the existing schemes. The HCB rule can accurately deduce all of them (see Fig.8(b)for extension-twist (E) and bending-shear coupling (F)).To establish that the stiffness values obtained using our approach are indeed accurate (in particularthe shearing stiffness for which no reference exists), we present in Fig.9 the critical Euler bucklingload of a (10,10) SWCNT based on various continuum rod models - the constitutive laws for allthe models are derived using the HCB rule. The boundary condition is of clamped-clamped type.Accordingly, the buckling load is normalized by 4π2A

L2 where “L” is the length of the SWCNT and Ais its bending stiffness in the relaxed state. As evident from Fig.9, the results from fully atomisticcalculations match most closely with the finite element special Cosserat rod solutions (Kumar andHealey, 2010) when full nonlinear constitutive laws are input to the finite element model. Thereis a slight mismatch in their values at smaller lengths where the SWCNT also has the traits of atwo-dimensional shell (Buehler et al., 2004). There are two other curves generated using nonlinearconstitutive laws: one in which the shear correction factor is forced to be unity and the other inwhich the rod is unshearable (shear correction factor being infinity; the nonlinear constitutive lawsrequired for this model can also be deduced using the scheme of Hakobyan et al. (2012)). Both thecurves are very different from fully atomistic results while the one in which unmodified shearingstiffness is input to the FEM model (special Cosserat rod, nonlinear) matches closely with fully

20

0 0.01 0.02 0.03

1

2

3

compressive strain

bend

ing

stiff

ness

(no

rmal

ized

)

without relaxationwith relaxation

(a)

−0.4 −0.2 0 0.2 0.4−300

−100

0

100

300

twist (κ3)

coup

ling

stiff

ness

(eV

)

bending−shearingextension−twist

(b)

Figure 8: (a)Effect of relaxation of atoms in the fundamental domain on bending stiffness.(b)Variation in coupling stiffnesses of a (10,10) SWCNT as it is twisted.

20 30 40 50 60 70

0.9

1

1.1

Length of SWCNT (nm)

Buc

klin

g lo

ad (

norm

aliz

ed)

fully atomisticspecial Cosserat, nonlinearspecial Cosserat, linearspecial Cosserat, nonlinear, κ = 1unshearable, nonlinearKirchoff limit

Figure 9: Buckling load of a (10,10) SWCNT vs the length of the SWCNT for different rod models.

21

atomistic results. This suggests that the shearing stiffness obtained using the HCB rule is accurate.We can also note better accuracy of the results obtained from the special Cosserat rod solutionbased on nonlinear constitutive laws over that of linear constitutive laws.

9 Conclusions and discussion

We presented a novel helical Cauchy-Born rule (HCB rule) to deduce linear as well as nonlinearelastic constitutive laws for nano and continuum rods. Just as translational periodicity is employedto generate images of a unit cell in the standard CB rule, we employ helical symmetry to generateimages (see (27)). The key concept in our method is the introduction of constraints (see (20)and (28)) using which we generate full 6-parameter uniform helical configurations relevant for thetheory of special Cosserat rods. In the absence of these constraints, the HCB rule reduces to the 2-parameter approach of James (2006). We also showed that for the case of coupled bending-stretchingdeformations, our method reduces to that of Hakobyan et al. (2012). Our method allows us to deduceshearing stiffness and several coupling stiffnesses of a nanorod for the first time. We also presenteda scheme to derive the stiffnesses of a continuum rod through atomistic calculations of nanowires ofincreasing radii. An interesting effect of surface relaxation on axial force of a nanorod (observed inHCB computations) was explained accurately using linear elasticity theory augmented with surfaceeffects. We showed finally that our method is more general and accurate than existing techniqueswith regards to deducing the stiffnesses of both nano and continuum rods. Having established theversatility of our scheme, the method can now be employed to investigate interesting deformationsin nanorods and nanotubes, e.g., buckling phenomena due to large compression and twist, wherematerial linearity is not sufficient and the knowledge of accurate nonlinear material laws is crucial(Gupta and Kumar, 2016).

An immediate next step is to deduce analytical formulas for various coupling stiffnesses of anintrinsically twisted continuum rod in its stress-free reference configuration. We would need tosolve the linearized version of the minimization problem (17) in presence of linearized form of theconstraint equations (20).

Since the method presented allows us to deduce the constitutive laws of a rod at zero temper-ature, an obvious extension of the proposed rule is to include the effect of finite temperature andderive nonlinear thermoelastic constitutive laws for a rod. This is more important from the pointof view of biomolecules (e.g., collagen, DNA etc.) where thermal fluctuations play significant rolein their elastic response.

A limitation of the method is that it allows us to only study elastic deformations in rod-likenanostructures or continua. Often defects or fracture occurs in rod-like nanostructures due to severestretching, twisting or bending. In such a scenario, our model can be readily employed within thelarge portion of the domain away from these defects and bridged to a fully atomistic model in theregion of defect. However, the details of coupling the atomistic and continuum domains in the rodtheory setting need to be investigated in detail.

10 Acknowledgments

We thank the anonymous reviewers for their several useful suggestions. P.G. acknowledges supportfrom the DST-Inspire fellowship.

22

A Computing inter-atomic energy of the fundamental do-

main (EFD)and its derivatives

The energy of the fundamental domain depends on positions of the atoms within the FD as well astheir images which lie within the cut-off radius. The locations of the image atoms are calculatedusing formula (27). Once we know these atomic locations, we can use a standard procedure tocompute the inter-atomic energy of the fundamental domain. In order to minimize EFD (see (30))and also to deduce the constitutive laws (see (32) and (33)), we need to calculate the derivatives ofEFD with respect to atomic positions (xj) as well as strains which essentially implies computing thederivatives of atomic positions and their images (both xj and xi,j) with respect to xj and strains.The map (27) does not turn out to be useful for this purpose. In fact, the derivative of the map(27) with respect to the strain “k” (evaluated at zero strain) possesses a non-removable singularity.We, therefore, use the following formula equivalent to (27) for this purpose:

xi,j =

(∫ iL0

0

Rsθ ds

)

v +

(

RiL0θ

)

xj . (45)

From (45), we note that computing the derivatives of xi,j with respect to xj and v are straightfor-ward. However, computing the derivatives with respect to k necessitates taking derivatives of therotation matrix (as well as its integral) with respect to k. Analytical formulas for these derivativesare now presented for the readers’ convenience.

B Derivative of atomic positions with respect to strains

From (45), the derivatives of xi,j with respect to strains follow as:

∂xi,j∂v

=

∫ iL0

0

Rsθ ds,

∂2xi,j∂v2

= 0,

∂2xi,j∂v∂k

=

∫ iL0

0

∂Rsθ

∂kds,

∂xi,j∂k

=

(∫ iL0

0

∂Rsθ

∂kds

)

v +∂RiL0θ

∂kxj ,

∂2xi,j

∂k2=

(∫ iL0

0

∂2Rsθ

∂k2ds

)

v +∂2RiL0θ

∂k2xj . (46)

Now, we show the derivatives of the rotation matrix as well as its integral with respect to k. Letψ = |k|, u = k

ψdenote the axis of rotation and θ = sψ be the angle of rotation. Also, let [u]×

and [ei]× denote the skew symmetric matrices whose axial vectors are u and ei respectively. Notethat the definition of θ here is different from the earlier definition but simplify our notations below.Using Rodrigues’ formula for rotation, we get

R(s) = exp(sK) = cos(θ)I+ sin(θ) [u]× + (1− cos(θ))u⊗ u. (47)

23

Now, u being a constant, Rodrigues’ formula (47) can be integrated analytically which is shownbelow in (48). The formulas for the derivatives of (47) and (48) are also shown below. Since theseformulas contain a removable singularity at θ = 0, a Taylor expanded version (with singularityremoved) is used in such situations.

∫

R = s

(

sin(θ)

θI+

1− cos(θ)

θ[u]× +

θ − sin(θ)

θu⊗ u

)

.

= s

(

(1−θ2

6)I+

θ

2(1−

θ2

12) [u]× +

θ2

6(1−

θ2

20)u⊗ u

)

(if θ ≈ 0).

= sI (if θ = 0). (48)

∂R

∂ki= s

[

− sin(θ)uiI+cos(θ)θ − sin(θ)

θui [u]× +

sin(θ)

θ[ei]× +

sin(θ)θ − 2(1− cos(θ))

θuiu⊗ u +

1− cos(θ)

θ(ei ⊗ u + u⊗ ei)

]

.

= s

[

− sin(θ)uiI−θ2

3(1−

θ2

10)ui [u]× + (1−

θ2

6+

θ4

120) [ei]× −

θ3

12uiu⊗ u

+θ

2(1−

θ2

12) (ei ⊗ u + u⊗ ei)

]

(if θ ≈ 0).

= s [ei]× (if θ = 0). (49)

∫

∂R

∂ki= s2

(

cos(θ)θ − sin(θ)

θ2uiI+

θsin(θ)− 2(1− cos(θ))

θ2ui [u]× +

1− cos(θ)

θ2[ei]×

+3sin(θ)− 2θ − cos(θ)θ

θ2uiu⊗ u +

θ − sin(θ)

θ2(ei ⊗ u + u⊗ ei)

)

.

= s2(

−θ

3(1−

θ2

10)uiI−

θ2

12ui [u]× +

1

2(1−

θ2

12) [ei]× −

1

60θ3uiu⊗ u

+θ

6(1−

θ2

20)(ei ⊗ u + u⊗ ei)

)

(if θ ≈ 0).

=s2

2[ei]× (if θ = 0). (50)

Similarly, the second derivative of the rotation matrix and its integral can be obtained.

C Derivative of [xj , λ, δ] with respect to strains

In this section, we first derive an expression for the derivative of [xj , λ, δ] with respect to strainsand finally show that the second term in eq.(32)(a) vanishes, i.e.,

∂EFD∂xj

∣

∣

∣

∣

(v,k)

·

∂xj∂[v, k]

= 0. (51)

24

We first observe that (30) holds at all strains. Differentiating the three equations in (30) withrespect to strains, we then obtain

∂2Econs∂[xj ,λ, δ]∂[v, k]

+∂2Econs

∂[xj ,λ, δ]2∂[xj , λ, δ]

∂[v, k]= 0. (52)

Let us define a stiffness matrix K3m×3m and the linearized constraint matrix C3m×6 such that itsblock-components are as shown below:

K3×3ij =

∂2EFD∂xi∂xj

, C3×6i =

[

∂2Econs∂xi∂λ

∂2Econs∂xi∂δ

]

=[

I3×3 xi3×3]

. (53)

Here I is an identity matrix whereas xi =

0 xi,3 xi,2xi,3 0 xi,1xi,2 xi,1 0

is a matrix formed by three components

of the position vector of ith atom. We can then write (52) in the following matrix form:

[

Kij + δijδ3m×3m

C3m×6

CT 06×6

]

∂xi

∂[v,k]

3m×6

∂[λ, δ]∂[v,k]

6×6

= −

∂2EFD

∂xi∂[v,k]

3m×6

06×6

. (54)

The equation (54) can be inverted to obtain the derivatives of [xj , λ, δ] with respect to strains.

Now, upon block multiplication in (54), we see that CT

∂xi∂[v, k]

= 0. This further implies that

∂xi∂[v, k]

is perpendicular to columns of C. Using Q-R factorization of C, we then obtain

C =[

Q13m×6 Q23m×(3m−6)] [

R16×6 06×(3m−6)]T

= Q1R1 ⇒

∂xi∂[v, k]

= Q2 z3m−6. (55)

Upon pre-multiplying the first set of block equations in (54) by Q2T and substituting (55), we thenobtain

(

Q2T

Kij + δijδ

Q2

)

z = −Q2T

∂2EFD∂xi∂[v, k]

⇒

∂xi∂[v, k]

= −Q2

(

Q2T

Kij + δijδ

Q2

)−1

Q2T

∂2EFD∂xi∂[v, k]

. (56)

Now using (30) and (53), we get:

∂Econs∂xi

=

∂EFD∂xi

+C

[

λ

δ

]

⇒ Q2T

∂Econs∂xi

= Q2T

∂EFD∂xi

. (57)

Finally,

∂EFD∂xi

·

∂xi∂[v, k]

=

∂EFD∂xi

·Q2

(

Q2T

Kij + δijδ

Q2

)−1

Q2T

∂2EFD∂xi∂[v, k]

= Q2T

∂Econs∂xi

·

(

Q2T

Kij + δijδ

Q2

)−1

Q2T

∂2EFD∂xi∂[v, k]

= 0 (using (57) and (30)a). (58)

25

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