Supplement: Enhancing Mechanical Properties of
Multi-Wall Carbon Nanotubes via sp3 Inter-wall Bridging
1
Z. H. Xia1, P.R. Guduru2 and W.A. Curtin2
Department of Mechanical Engineering, The University of Akron, Akron, OH 44325
2
Division of Engineering, Brown University, Providence, RI 02912
Consider axi-symmetric shell buckling of a two walled carbon nanotube under uniform
axial compression, in which the walls resist relative shearing or sliding. The geometry of the
shell, the coordinate system used and the directions of the mid-plane displacements of the walls u,
v, w are shown in Fig. S1. It is assumed that the radius of the nanotube is much greater than the
spacing between the walls. Consider the kinematics of a buckled two-walled nanotube, shown in
Fig. S2. When two initially parallel nanotube walls undergo the same out-of plane deformation,
they slide with respect to each other. If  is the spacing between the two walls and dw/ dx is the
local slope, the relative sliding displacement s is given by
dw
dx
(S1)
The relative sliding induces a local shear force per unit area  on the walls, which can be
approximated to be proportional to s for small values of s.
s 
  ks  k
dw
dx
(S2)
where k is a spring constant. These quantities can also be described in the continuum parlance by
defining a “shear strain”  as
 
s


dw
dx
(S3)
and a “shear modulus” G as
G

 k

(S4)
Let the compressive force per unit length of the circumference on each wall just prior to buckling
be No and the strain be o. The corresponding pre-buckling homogeneous solution for the strains
and the displacements is given by
 x   o  
No
,
Eh
 y   x 
u   o x, v  0, w  
 No R
Eh
 No
Eh
 wo
1
(S5)
Immediately following buckling, the displacement field is assumed to be
u   o x  A cos(m x / L)
v0
w  wo  B sin(m  x / L)
(S6)
which represents an axisymmetric buckling mode with m half wavelengths along the tube length.
The constants A and B are small amplitudes of the buckling mode m. In shell buckling, it is well
known that within the assumptions of Timoshenko-Donnell theory [1], the critical buckling
strain for the general non-axisymmetric buckling is equal to that of axisymmetric case [2]. Hence,
it is reasonable to expect that the same might hold in the presence of shear coupling as well. The
mid-plane strains (x, y, xy) and curvatures (kx, ky, kxy) corresponding to the above perturbed
field are
m
sin(m  x / L)
L
w
w
B
 y  v , y    o  sin(m  x / L)
R
R R
 xy  u ,y v ,x  0
 x  u ,x   o  A
 x  w,xx   B
 y  w, yy  0
(S7)
m 2 2
sin(m  x / L)
L2
 xy  w,xy  0
In the above equations, a subscript following a comma represents partial differentiation.
Summing the elastic energy due to mid-plane stretching in each wall, the total membrane elastic
energy is given by
Um  2

Eh
 x   y 2  21      x y  1  xy2 dS
2 
2(1   ) S 
4 

(S8)
where S is the surface area of each wall. Using the strain expressions from Eq. S7, the membrane
elastic energy can be evaluated as
2(1   2 ) o m L

EhRL 
A2 m 2 2
B 2  m
2
2
Um  2
(1   ) o 


AB 
A sin( mx / L)dx
2 
2
2
2
(1   ) 
2L
2R
RL
L
0

(S9)
2
The total elastic energy in the two walls due to bending is
Ub  2


D
 x   y 2  2(1   ) x y   xy2  dS

2S
(S10)
D 5 m 4 RB 2

L3
where D 
Eh 3
.
12(1   2 )
The total stored energy due to relative sliding between the two walls or the shear-coupling
energy can be written as
2
1
1
 dw 
U s    s dS  G    dS
2
S 2
A  dx 
(S11)
Using the perturbed displacement w from Eq. S6, the expression in Eq. S11 can be evaluated as
1
m 2 3 RB 2
U s  G
2
L
(S12)
The total shortening of the MWCNT due to membrane strain and bending is given by
L
L
1
    x dx   w, 2x dx
20
0
(S13)
Using Eq. 14, the potential energy of the applied traction can be evaluated as
1 3
m L
2
2
V  2 (2 RN o )  4 R Eh  L   E h R o m B  4 R Eh  o
A sin( m x / L)dx
L
L 0
2
o
(S14)
The total potential energy is
(S15)
  U m  Ub  U s  V
From the minimum potential energy theorem,

 0,
A

0
B
(S16)
3
The first of these two equations gives
A
L
B
mR
(S17)
Using this in the second equation, an expression for the buckling strain o is obtained as
o 
1 L2
D m 2 2 1 G


2 Eh
R 2 m 2 2 Eh L2
(S18)
If there were no shear coupling (G = 0), this expression reduces to that of the classical shell
theory result. The effect of shear coupling is explicitly through the third term, which for the
current axisymmetric buckling, is independent of the mode number m. The critical buckling
strain is obtained by minimizing the above expression with respect to m as
o 
h
R 3(1   )
2

1 G
2 Eh
(S19)
This result can be readily generalized to an n-walled nanotube as
o 
h
R 3(1   )
2

n  1 G
n Eh
(S20)
Thus, an increase in shear modulus due to sp3 inter-wall coupling increases the resistance to
inter-wall sliding and thus increases the stability of the structure. Note that in the above
expression, R is the average radius of the MWNT, E = 5.5 TPa, h = 0.066 nm,  = 0.34nm and 
= 0.19.
Transversely Isotropic Shell (TIS) Theory
The SCS theory considers the discrete structure of a MWCNT, i.e., the presence of each
shell and its shear coupling with its neighbors is considered explicitly. According to Eq. S20, as
G →, the correction due to shear coupling increases unboundedly. On the other hand, as G →
and n becomes large, the MWCNT becomes an elastic continuum shell, whose behavior ought to
be well described by the classical shell theory. Denote the Young’s modulus of graphite in the
plane of the graphene sheets by Eg (assumed to be approximately isotropic in the plane of the
sheets). If the inter-wall shear modulus G = Eg/2(1+), then the wall of the MWCNT becomes
an isotropic continuum; else, it can be viewed as a transversely isotropic continuum.
According to the classical shell theory, the buckling strain of an elastic shell of thickness
H is given by the first term on the RHS of Eq. S20, in which h is replaced with H. This equation
is derived based on the so called Love-Kirchoff assumption, according to which the normals to
4
the mid-plane remain normal and straight after deformation. In other words, the shear
deformations are neglected completely, which is equivalent to assuming that G →. However,
Eq. S20 does not recover the isotropic continuum result in the limit G → . This pathology in
the SCS theory arises because it ignores the influence of G in modifying the stress state in each
wall. Thus, the SCS theory is valid only for “small” values of G/Eg (<< 1). As the inter-wall
shear coupling increases due to increased density of sp3 bonds, the SCS theory is expected to
breakdown. It would then be necessary to view the wall of a MWCNT as a transversely isotropic
continuum in order to recover the classical shell theory result when G → .
Note that the derivation of the classical shell theory result does not distinguish between
isotropy and transverse isotropy. For isotropic materials, the error due to Love-Kirchoff
assumption is usually small if the plate/shell thickness is small compared to its in-plane
dimensions. However, as G/Eg decreases, the contribution to the overall deformation by the
transverse shear strain increases and it becomes necessary to use shear deformation theories of
plates and shells. Thus, in order to capture the effect of increasing sp3 coupling within a
continuum framework, it is necessary to employ a shear deformation theory of transversely
isotropic shells.
Such theories have been developed in the analysis of composite materials, where severe
transverse isotropy is quite common. Buckling of transversely isotropic shells with shear
deformation has been considered by Ambartsumyan [3]. For a given buckling mode, if n is the
number of half wavelengths along the circumference, m is the number of half wavelengths along
m R
the length L, R is the mean radius,  
, H is the wall thickness, Eg is the in-plane Young’s
L
modulus,  is the in-plane Poisson’s ratio and G is the transverse shear modulus, then the
corresponding buckling strain is given by [3]
c 
D 

2
2
n
Eg H 3

2 2

D
 2  n2

E g HR 2
2

2
1

1    H   2  n 2
 10  R 

2

(S21)



and  is the anisotropy parameter, defined as  
Eg
. The
G 1   2 
12(1   )
critical buckling strain is the lowest value of  c among all integer values of m and n. Here, the
value of G is the same as that in the SCS theory. In Eq. S21, as G  ,   0 and minimizing
where
2
 c with respect to  2  n 2  / n 2 recovers the classical shell theory. Thus, the transversely
2
isotropic shell (TIS) theory can be used as a continuum description of a shear coupled MWCNT,
which is capable of recovering the classical result at large values of the shear modulus.
In the limit G0, which corresponds to the absence of any shear interaction between the
shells, Eq. S21 predicts that the critical buckling strain approaches zero. However, according to
Ru’s analysis, the buckling strain of a MWCNT with no shear coupling is given by the first term
on RHS of Eq. S20. Being a continuum theory, TIS theory ignores the finite buckling resistance
5
of every individual wall and can not recover the buckling strain in the limit G0. Thus, the SCS
theory works better in the limit G0 as discussed in the previous section and the TIS theory is
more accurate in the limit G. The two theories agree reasonably well in the intermediate
range as discussed in the manuscript. It can be anticipated that, as the density of sp3 bonds
increases, the buckling strain of a MWCNT will show a transition from the SCS regime to the
TIS regime, as illustrated in Fig. 4 of the manuscript.
References
1. L.H. Donnell, Beams, Plates, and Shells (McGraw-Hill, New York, 1976).
2. S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability (McGraw-Hill, New York,
1961).
3. S.A. Ambartsumian. Fragments of the theory of anisotropic shells. World Scientific,
Singapore, 1991.
6
2R
h
L
x, u
z, w
y, v
Figure S1. Coordinate system for the shell buckling problem.
w
x
slip  
dw
dx

slope =
dw
dx
Figure S2. Geometrically necessary relative sliding between the walls of a multi-walled
carbon nanotube.
7