Dynamics of nonlocal structures

Dynamics of nonlocal structures
S Adhikari
College of Engineering, Swansea University, Swansea UK
Email: S.Adhikari@swansea.ac.uk
National University of Defence Technology (NUDT), Changsha, China
April 16, 2014

Outline of this talk
1 Introduction
2 Finite element modelling of nonlocal dynamic systems
Axial vibration of nanorods
Bending vibration of nanobeams
Transverse vibration of nanoplates
3 Modal analysis of nonlocal dynamical systems
Conditions for classical normal modes
Nonlocal normal modes
Approximate nonlocal normal modes
4 Dynamics of damped nonlocal systems
5 Numerical illustrations
Axial vibration of a single-walled carbon nanotube
Transverse vibration of a single-layer graphene sheet
6 Conclusions

Nanoscale systems
Nanoscale systems have length-scale in the order of O(10−9
)m.
Nanoscale systems, such as those fabricated from simple and complex
nanorods, nanobeams [1] and nanoplates have attracted keen interest
among scientists and engineers.
Examples of one-dimensional nanoscale objects include (nanorod and
nanobeam) carbon nanotubes [2], zinc oxide (ZnO) nanowires and boron
nitride (BN) nanotubes, while two-dimensional nanoscale objects include
graphene sheets [3] and BN nanosheets [4].
These nanostructures are found to have exciting mechanical, chemical,
electrical, optical and electronic properties.
Nanostructures are being used in the ﬁeld of nanoelectronics,
nanodevices, nanosensors, nano-oscillators, nano-actuators,
nanobearings, and micromechanical resonators, transporter of drugs,
hydrogen storage, electrical batteries, solar cells, nanocomposites and
nanooptomechanical systems (NOMS).
Understanding the dynamics of nanostructures is crucial for the
development of future generation applications in these areas.

Nanoscale systems
(a) DNA
(b) Zinc Oxide ( ZnO)nanowire
(c) Boron Nitride nanotube (BNNT )
(d) Protein

Continuum mechanics at the nanoscale
Experiments at the nanoscale are generally difﬁcult at this point of time.
On the other hand, atomistic computation methods such as molecular
dynamic (MD) simulations [5] are computationally prohibitive for
nanostructures with large numbers of atoms.
Continuum mechanics can be an important tool for modelling,
understanding and predicting physical behaviour of nanostructures.
Although continuum models based on classical elasticity are able to
predict the general behaviour of nanostructures, they often lack the
accountability of effects arising from the small-scale.
To address this, size-dependent continuum based methods [6–9] are
gaining in popularity in the modelling of small sized structures as they
offer much faster solutions than molecular dynamic simulations for
various nano engineering problems.
Currently research efforts are undergoing to bring in the size-effects
within the formulation by modifying the traditional classical mechanics.

Nonlocal continuum mechanics
One popularly used size-dependant theory is the nonlocal elasticity
theory pioneered by Eringen [10], and has been applied to
nanotechnology.
Nonlocal continuum mechanics is being increasingly used for efﬁcient
analysis of nanostructures viz. nanorods [11, 12], nanobeams [13],
nanoplates [14, 15], nanorings [16], carbon nanotubes [17, 18],
graphenes [19, 20], nanoswitches [21] and microtubules [22]. Nonlocal
elasticity accounts for the small-scale effects at the atomistic level.
In the nonlocal elasticity theory the small-scale effects are captured by
assuming that the stress at a point as a function of the strains at all points
in the domain:
σij (x) =
V
φ(|x − x′
|, α)tij dV(x′
)
where φ(|x − x′
|, α) = (2πℓ2
α2
)K0(
√
x • x/ℓα)
Nonlocal theory considers long-range inter-atomic interactions and yields
results dependent on the size of a body.
Some of the drawbacks of the classical continuum theory could be
efﬁciently avoided and size-dependent phenomena can be explained by
the nonlocal elasticity theory.

FEM for nonlocal dynamic systems
The majority of the reported works on nonlocal finite element analysis
consider free vibration studies where the effect of non-locality on the
undamped eigensolutions has been studied.
Damped nonlocal systems and forced vibration response analysis have
received little attention.
On the other hand, significant body of literature is available [23–25] on
finite element analysis of local dynamical systems.
It is necessary to extend the ideas of local modal analysis to nonlocal
systems to gain qualitative as well as quantitative understanding.
This way, the dynamic behaviour of general nonlocal discretised systems
can be explained in the light of well known established theories of
discrete local systems.

Figure : Axial vibration of a zigzag (7, 0) single-walled carbon nanotube (SWCNT)
with clamped-free boundary condition.

The equation of motion of axial vibration for a damped nonlocal rod can
be expressed as
EA
∂2
U(x, t)
∂x2
+ c1 1 − (e0a)2
1
∂2
∂x2
∂3
U(x, t)
∂x2∂t
= c2 1 − (e0a)2
2
∂2
∂x2
∂U(x, t)
∂t
+ 1 − (e0a)2 ∂2
∂x2
m
∂2
U(x, t)
∂t2
+ F(x, t)
In the above equation EA is the axial rigidity, m is mass per unit length,
e0a is the nonlocal parameter [10], U(x, t) is the axial displacement,
F(x, t) is the applied force, x is the spatial variable and t is the time.
The constant c1 is the strain-rate-dependent viscous damping coefﬁcient
and c2 is the velocity-dependent viscous damping coefﬁcient.
The parameters (e0a)1 and (e0a)2 are nonlocal parameters related to the
two damping terms, which are ignored for simplicity.

Nonlocal element matrices
We consider an element of length ℓe with axial stiffness EA and mass per
unit length m.
1 2
le
Figure : A nonlocal element for the axially vibrating rod with two nodes. It has
two degrees of freedom and the displacement ﬁeld within the element is
expressed by linear shape functions.
This element has two degrees of freedom and there are two shape
functions N1(x) and N2(x). The shape function matrix for the axial
deformation [25] can be given by
N(x) = [N1(x), N2(x)]T
= [1 − x/ℓe, x/ℓe]T
(2)

Using this the stiffness matrix can be obtained using the conventional
variational formulation as
Ke = EA
ℓe
0
dN(x)
dx
dNT
(x)
dx
dx =
EA
ℓe
1 −1
−1 1
(3)
The mass matrix for the nonlocal element can be obtained as
Me = m
ℓe
0
N(x)NT
(x)dx + m(e0a)2
ℓe
0
dN(x)
dx
dNT
(x)
dx
dx
=
mℓe
6
2 1
1 2
+
e0a
ℓe
2
mℓe
1 −1
−1 1
(4)
For the special case when the rod is local, the mass matrix derived above
reduces to the classical mass matrix[25, 26] as e0a = 0 . Therefore for a
nonlocal rod, the element stiffness matrix is identical to that of a classical
local rod but the element mass has an additive term which is dependent
on the nonlocal parameter.

Figure : Bending vibration of an armchair (5, 5), (8, 8) double-walled carbon
nanotube (DWCNT) with pinned-pinned boundary condition.

For the bending vibration of a nonlocal damped beam, the equation of
motion can be expressed by
EI
∂4
V(x, t)
∂x4
+ m 1 − (e0a)2 ∂2
∂x2
∂2
V(x, t)
∂t2
+ c1
∂5
V(x, t)
∂x4∂t
+ c2
∂V(x, t)
∂t
= 1 − (e0a)2 ∂2
∂x2
{F(x, t)} (5)
In the above equation EI is the bending rigidity, m is mass per unit length,
e0a is the nonlocal parameter, V(x, t) is the transverse displacement and
F(x, t) is the applied force.

We consider an element of length ℓe with bending stiffness EI and mass
per unit length m.
1 2
le
Figure : A nonlocal element for the bending vibration of a beam. It has two
nodes and four degrees of freedom. The displacement ﬁeld within the element is
expressed by cubic shape functions.
This element has four degrees of freedom and there are four shape
functions.

The shape function matrix for the bending deformation [25] can be given
by
N(x) = [N1(x), N2(x), N3(x), N4(x)]
T
(6)
where
N1(x) = 1 − 3
x2
ℓ2
e
+ 2
x3
ℓ3
e
, N2(x) = x − 2
x2
ℓe
+
x3
ℓ2
e
,
N3(x) = 3
x2
ℓ2
e
− 2
x3
ℓ3
e
, N4(x) = −
x2
ℓe
+
x3
ℓ2
e
(7)
Using this, the stiffness matrix can be obtained using the conventional
variational formulation [26] as
Ke = EI
ℓe
0
d2
N(x)
dx2
d2
NT
(x)
dx2
dx =
EI
ℓ3
e




12 6ℓe −12 6ℓe
6ℓe 4ℓ2
e −6ℓe 2ℓ2
e
−12 −6ℓe 12 −6ℓ2
e
6ℓe 2ℓ2
e −6ℓe 4ℓ2
e



 (8)

Me = m
ℓe
0
N(x)NT
(x)dx + m(e0a)2
ℓe
0
dN(x)
dx
dNT
(x)
dx
dx
=
mℓe
420




156 22ℓe 54 −13ℓe
22ℓe 4ℓ2
e 13ℓe −3ℓ2
e
54 13ℓe 156 −22ℓe
−13ℓe −3ℓ2
e −22ℓe 4ℓ2
e




+
e0a
ℓe
2
mℓe
30




36 3ℓe −36 3ℓe
3ℓe 4ℓ2
e −3ℓe −ℓ2
e
−36 −3ℓe 36 −3ℓe
3ℓe −ℓ2
e −3ℓe 4ℓ2
e




(9)
For the special case when the beam is local, the mass matrix derived
above reduces to the classical mass matrix [25, 26] as e0a = 0.

For the transverse bending vibration of a nonlocal damped thin plate, the
equation of motion can be expressed by
D∇4
V(x, y, t) + m 1 − (e0a)2
∇2 ∂2
V(x, y, t)
∂t2
+ c1∇4 ∂V(x, y, t)
∂t
+ c2
∂V(x, y, t)
∂t
= 1 − (e0a)2
∇2
{F(x, y, t)} (10)
In the above equation ∇2
= ∂2
∂x2 + ∂2
∂y2 is the differential operator,
D = Eh3
12(1−ν2)
is the bending rigidity, h is the thickness, ν is the Poisson’s
ratio, m is mass per unit area, e0a is the nonlocal parameter, V(x, y, t) is
the transverse displacement and F(x, y, t) is the applied force.

We consider an element of dimension 2c × 2b with bending stiffness D
and mass per unit area m.
x
y
(-c,-b)
(-c,b)
(c,-b)
(c,b)
12
3 4
Figure : A nonlocal element for the bending vibration of a plate. It has four nodes
and twelve degrees of freedom. The displacement ﬁeld within the element is
expressed by cubic shape functions in both directions.

The shape function matrix for the bending deformation is a 12 × 1 vector
[26] and can be expressed as
N(x, y) = C−1
e α(x, y) (11)
Here the vector of polynomials is given by
α(x, y) = 1 x y x2
xy y2
x3
x2
y xy2
y3
x3
y xy3 T
(12)
The 12 × 12 coefﬁcient matrix can be obtained in closed-form.

Using the shape functions in Eq. (11), the stiffness matrix can be
obtained using the conventional variational formulation [26] as
Ke =
Ae
BT
EBdAe (13)
In the preceding equation B is the strain-displacement matrix, and the
matrix E is given by
E = D


1 ν 0
ν 1 0
0 0 1−ν
2

 (14)
Evaluating the integral in Eq. (13), we can obtain the element stiffness
matrix in closed-form as
Ke =
Eh3
12(1 − ν2)
C−1T
keC−1
(15)
The 12 × 12 coefﬁcient matrix ke can be obtained in closed-form.

Me = ρh
Ae
N(x, y)NT
(x, y)
+(e0a)2 ∂N(x, y)
∂x
dNT
(x, y)
dx
+
∂N(x, y)
∂x
dNT
(x, y)
dx
dAe
= M0e +
e0a
c
2
Mxe +
e0a
b
2
Mye
(16)
The three matrices appearing in the above expression can be obtained in
closed-form.

Mxe =
ρhcb
630
×






















276 66b 42c −276 −66b 42c −102 39b 21c 102 −39b 21c
66b 24b2 0 −66b −24b2 0 −39b 18b2 0 39b −18b2 0
42c 0 112c2 −42c 0 −28c2 −21c 0 −14c2 21c 0 56c2
−276 −66b −42c 276 66b −42c 102 −39b −21c −102 39b −21c
−66b −24b2 0 66b 24b2 0 39b −18b2 0 −39b 18b2 0
42c 0 −28c2 −42c 0 112c2 −21c 0 56c2 21c 0 −14c2
−102 −39b −21c 102 39b −21c 276 −66b −42c −276 66b −42c
39b 18b2 0 −39b −18b2 0 −66b 24b2 0 66b −24b2 0
21c 0 −14c2 −21c 0 56c2 −42c 0 112c2 42c 0 −28c2
102 39b 21c −102 −39b 21c −276 66b 42c 276 −66b 42c
−39b −18b2 0 39b 18b2 0 66b −24b2 0 −66b 24b2 0
21c 0 56c2 −21c 0 −14c2 −42c 0 −28c2 42c 0 112c2






















(17)
Mye =
ρhcb
630
×






















276 42b 66c 102 21b −39c −102 21b 39c −276 42b −66c
42b 112b2 0 21b 56b2 0 −21b −14b2 0 −42b −28b2 0
66c 0 24c2 39c 0 −18c2 −39c 0 18c2 −66c 0 −24c2
102 21b 39c 276 42b −66c −276 42b 66c −102 21b −39c
21b 56b2 0 42b 112b2 0 −42b −28b2 0 −21b −14b2 0
−39c 0 −18c2 −66c 0 24c2 66c 0 −24c2 39c 0 18c2
−102 −21b −39c −276 −42b 66c 276 −42b −66c 102 −21b 39c
21b −14b2 0 42b −28b2 0 −42b 112b2 0 −21b 56b2 0
39c 0 18c2 66c 0 −24c2 −66c 0 24c2 −39c 0 −18c2
−276 −42b −66c −102 −21b 39c 102 −21b −39c 276 −42b 66c
42b −28b2 0 21b −14b2 0 −21b 56b2 0 −42b 112b2 0
−66c 0 −24c2 −39c 0 18c2 39c 0 −18c2 66c 0 24c2






















(18)

Nonlocal element matrices: Summary
Based on the discussions for all the three systems considered here, in
general the element mass matrix of a nonlocal dynamic system can be
expressed as
Me = M0e +Mµe (19)
Here M0e is the element stiffness matrix corresponding to the underlying
local system and Mµe is the additional term arising due to the nonlocal
effect.
The element stiffness matrix remains unchanged.

Global system matrices
Using the finite element formulation, the stiffness matrix of the local and
nonlocal system turns out to be identical to each other.
The mass matrix of the nonlocal system is however different from its
equivalent local counterpart.
Assembling the element matrices and applying the boundary conditions,
following the usual procedure of the finite element method one obtains
the global mass matrix as
M = M0+Mµ (20)
In the above equation M0 is the usual global mass matrix arising in the
conventional local system and Mµ is matrix arising due to nonlocal nature
of the systems:
Mµ =
e0a
L
2
Mµ (21)
Here Mµ is a nonnegative definite matrix. The matrix Mµ is therefore, a
scale-dependent matrix and its influence reduces if the length of the
system L is large compared to the parameter e0a.

Nonlocal modal analysis
Majority of the current finite element software and other computational
tools do not explicitly consider the nonlocal part of the mass matrix. For
the design and analysis of future generation of nano electromechanical
systems it is vitally important to consider the nonlocal influence.
We are interested in understanding the impact of the difference in the
mass matrix on the dynamic characteristics of the system. In particular
the following questions of fundamental interest have been addressed:
Under what condition a nonlocal system possess classical local normal
modes?
How the vibration modes and frequencies of a nonlocal system can be
understood in the light of the results from classical local systems?
By addressing these questions, it would be possible to extend
conventional ‘local’ elasticity based finite element software to analyse
nonlocal systems arising in the modelling of complex nanoscale built-up
structures.

The equation of motion of a discretised nonlocal damped system with n
degrees of freedom can be expressed as
[M0 + Mµ] ü(t) + C ˙u(t) + Ku(t) = f(t) (22)
Here u(t) ∈ Rn
is the displacement vector, f(t) ∈ Rn
is the forcing vector,
K, C ∈ Rn×n
are respectively the global stiffness and the viscous damping
matrix.
In general M0 and Mµ are positive definite symmetric matrices, C and K
are non-negative definite symmetric matrices. The equation of motion of
corresponding local system is given by
M0ü0(t) + C ˙u0(t) + Ku0(t) = f(t) (23)
where u0(t) ∈ Rn
is the local displacement vector.
The natural frequencies (ωj ∈ R) and the mode shapes (xj ∈ Rn
) of the
corresponding undamped local system can be obtained by solving the
matrix eigenvalue problem [23] as
Kxj = ω2
j M0xj , ∀ j = 1, 2, . . . , n (24)

Dynamics of the local system
The undamped local eigenvectors satisfy an orthogonality relationship
over the local mass and stiffness matrices, that is
xT
k M0xj = δkj (25)
and xT
k Kxj = ω2
j δkj , ∀ k, j = 1, 2, . . . , n (26)
where δkj is the Kroneker delta function. We construct the local modal
matrix
X = [x1, x2, . . . , xn] ∈ Rn
(27)
The local modal matrix can be used to diagonalize the local system (23)
provided the damping matrix C is simultaneously diagonalizable with M0
and K.
This condition, known as the proportional damping, originally introduced
by Lord Rayleigh [27] in 1877, is still in wide use today.
The mathematical condition for proportional damping can be obtained
from the commutitative behaviour of the system matrices [28]. This can
be expressed as
CM−1
0 K = KM−1
0 C (28)
or equivalently C = M0f(M−1
0 K) as shown in [29].

Considering undamped nonlocal system and premultiplying the equation
by M−1
0 we have
In + M−1
0 Mµ ¨u(t) + M−1
0 K u(t) = M−1
0 f(t) (29)
This system can be diagonalized by a similarity transformation which also
diagonalise M−1
0 K provided the matrices M−1
0 Mµ and M−1
0 K
commute. This implies that the condition for existence of classical local
normal modes is
M−1
0 K M−1
0 Mµ = M−1
0 Mµ M−1
0 K (30)
or KM−1
0 Mµ = MµM−1
0 K (31)
If the above condition is satisﬁed, then a nonlocal undamped system can
be diagonalised by the classical local normal modes. However, it is also
possible to have nonlocal normal modes which can diagonalize the
nonlocal undamped system as discussed next.

Nonlocal normal modes can be obtained by the undamped nonlocal
eigenvalue problem
Kuj = λ2
j [M0 + Mµ] uj , ∀ j = 1, 2, . . . , n (32)
Here λj and uj are the nonlocal natural frequencies and nonlocal normal
modes of the system. We can deﬁne a nonlocal modal matrix
U = [u1, u2, . . . , un] ∈ Rn
(33)
which will unconditionally diagonalize the nonlocal undamped system. It
should be remembered that in general nonlocal normal modes and
frequencies will be different from their local counterparts.

Nonlocal normal modes: Damped systems
Under certain restrictive condition it may be possible to diagonalise the
damped nonlocal system using classical normal modes.
Premultiplying the equation of motion (22) by M−1
0 , the required condition
is that M−1
0 Mµ , M−1
0 C and M−1
0 K must commute pairwise. This
implies that in addition to the two conditions given by Eqs. (28) and (31),
we also need a third condition
CM−1
0 Mµ = MµM−1
0 C (34)
If we consider the diagonalization of the nonlocal system by the nonlocal
modal matrix in (33), then the concept of proportional damping can be
applied similar to that of the local system. One can obtain the required
condition similar to Caughey’s condition [28] as in Eq. (28) by replacing
the mass matrix with M0 + Mµ. If this condition is satisﬁed, then the
equation of motion can be diagonalised by the nonlocal normal modes
and in general not by the classical normal modes.

Approximate nonlocal normal modes
Majority of the existing ﬁnite element software calculate the classical
normal modes.
However, it was shown that only under certain restrictive condition, the
classical normal modes can be used to diagonalise the system.
In general one need to use nonlocal normal modes to diagonalise the
equation of motion (22), which is necessary for efﬁcient dynamic analysis
and physical understanding of the system.
We aim to express nonlocal normal modes in terms of classical normal
modes.
Since the classical normal modes are well understood, this approach will
allow us to develop physical understanding of the nonlocal normal modes.

Projection in the space of undamped classical modes
For distinct undamped eigenvalues (ω2
l ), local eigenvectors
xl , ∀ l = 1, . . . , n, form a complete set of vectors. For this reason each
nonlocal normal mode uj can be expanded as a linear combination of xl :
uj =
n
l=1
α
(j)
l xl (35)
Without any loss of generality, we can assume that α
(j)
j = 1
(normalization) which leaves us to determine α
(j)
l , ∀l = j.
Substituting the expansion of uj into the eigenvalue equation (32), one
obtains
−λ2
j (M0 + Mµ) + K
n
l=1
α
(j)
l xl = 0 (36)
For the case when α
(j)
l are approximate, the error involving the projection
in Eq. (35) can be expressed as
εj =
n
l=1
−λ2
j (M0 + Mµ) + K α
(j)
l xl (37)

Nonlocal natural frequencies
We use a Galerkin approach to minimise this error by viewing the
expansion as a projection in the basis functions xl ∈ Rn
, ∀l = 1, 2, . . . n.
Therefore, making the error orthogonal to the basis functions one has
εj ⊥ xl or xT
k εj = 0 ∀ k = 1, 2, . . . , n (38)
Using the orthogonality property of the undamped local modes
n
l=1
−λ2
j δkl + M′
µkl
+ ω2
k δkl α
(j)
l = 0 (39)
where M′
µkl
= xT
k Mµxl are the elements of the nonlocal part of the modal
mass matrix.
Assuming the off-diagonal terms of the nonlocal part of the modal mass
matrix are small and α
(j)
l ≪ 1, ∀l = j, approximate nonlocal natural
frequencies can be obtained as
λj ≈
ωj
1 + M′
µjj
(40)

Nonlocal mode shapes
When k = j, from Eq. (39) we have
−λ2
j 1 + M′
µkk
+ ω2
k α
(j)
k − λ2
j
n
l=k
M′
µkl
α
(j)
l = 0 (41)
Recalling that α
(j)
j = 1, this equation can be expressed as
−λ2
j 1 + M′
µkk
+ ω2
k α
(j)
k = λ2
j

M′
µkj
+
n
l=k=j
M′
µkl
α
(j)
l

 (42)
Solving for α
(j)
k , the nonlocal normal modes can be expressed in terms of
the classical normal modes as
uj ≈ xj +
n
k=j
λ2
j
λ2
k − λ2
j
M′
µkj
1 + M′
µkk
xk (43)

Equations (40) and (43) completely deﬁnes the nonlocal natural frequencies
and mode shapes in terms of the local natural frequencies and mode shapes.
The following insights about the nonlocal normal modes can be deduced
Each nonlocal mode can be viewed as a sum of two principal
components. One of them is parallel to the corresponding local mode and
the other is orthogonal to it as all xk are orthogonal to xj for j = k.
Due to the term λ2
k − λ2
j in the denominator, for a given nonlocal mode,
only few adjacent local modes contributes to the orthogonal component.
For systems with well separated natural frequencies, the contribution of
the orthogonal component becomes smaller compared to the parallel
component.

Frequency response of nonlocal systems
Taking the Fourier transformation of the equation of motion (22) we have
D(iω)¯u(iω) = ¯f(iω) (44)
where the nonlocal dynamic stiffness matrix is given by
D(iω) = −ω2
[M0 + Mµ] + iωC + K (45)
In Eq. (44) ¯u(iω) and ¯f(iω) are respectively the Fourier transformations of
the response and the forcing vectors.
Using the local modal matrix (27), the dynamic stiffness matrix can be
transformed to the modal coordinate as
D′
(iω) = XT
D(iω)X = −ω2
I + M′
µ + iωC′
+ Ω2
(46)
where I is a n-dimensional identity matrix, Ω2
is a diagonal matrix
containing the squared local natural frequencies and (•)′
denotes that the
quantity is in the modal coordinates.

We separate the diagonal and off-diagonal terms as
D′
(iω) = −ω2
I + M
′
µ + iωC
′
+ Ω2
diagonal
+ −ω2
∆M′
µ + iω∆C′
off-diagonal
(47)
= D
′
(iω) + ∆D′
(iω) (48)
The dynamic response of the system can be obtained as
¯u(iω) = H(iω)¯f(iω) = XD
′−1
(iω)XT ¯f(iω) (49)
where the matrix H(iω) is known as the transfer function matrix.
From the expression of the modal dynamic stiffness matrix we have
D
′−1
(iω) = D
′
(iω) I + D
′−1
(iω)∆D′
(iω)
−1
(50)
≈ D
′−1
(iω) − D
′−1
(iω)∆D′
(iω)D
′−1
(iω) (51)

Substituting the approximate expression of D
′−1
(iω) from Eq. (51) into the
expression of the transfer function matrix in Eq. (49) we have
H(iω) = XD
′−1
(iω)XT
≈ H
′
(iω) − ∆H′
(iω) (52)
where
H
′
(iω) = XD
′
(iω)XT
=
n
k=1
xk xT
k
−ω2 1 + M′
µkk
+ 2iωωk ζk + ω2
k
(53)
and ∆H′
(iω) = XD
′−1
(iω)∆D′
(iω)D
′−1
(iω)XT
(54)
Equation (52) therefore completely deﬁnes the transfer function of the
damped nonlocal system in terms of the classical normal modes. This
can be useful in practice as all the quantities arise in this expression can
be obtained from a conventional ﬁnite element software. One only needs
the nonlocal part of the mass matrix as derived in 2.

Nonlocal transfer function
Some notable features of the expression of the transfer function matrix are
For lightly damped systems, the transfer function will have peaks around
the nonlocal natural frequencies derived previously.
The error in the transfer function depends on two components. They
include the off-diagonal part of the of the modal nonlocal mass matrix
∆M′
µ and the off-diagonal part of the of the modal damping matrix ∆C′
.
While the error in in the damping term is present for non proportionally
damped local systems, the error due to the nonlocal modal mass matrix
in unique to the nonlocal system.
For a proportionally damped system ∆C′
= O. For this case error in the
transfer function only depends on ∆M′
µ.
In general, error in the transfer function is expected to be higher for
higher frequencies as both ∆C′
and ∆M′
µ are weighted by frequency ω.
The expressions of the nonlocal natural frequencies (40), nonlocal normal
modes (43) and the nonlocal transfer function matrix (52) allow us to
understand the dynamic characteristic of a nonlocal system in a qualitative
and quantitative manner in the light of equivalent local systems.

Figure : Axial vibration of a zigzag (7, 0) single-walled carbon nanotube (SWCNT)
with clamped-free boundary condition.

A single-walled carbon nanotube (SWCNT) is considered.
A zigzag (7, 0) SWCNT with Young’s modulus E = 6.85 TPa, L = 25nm,
density ρ = 9.517 × 103
kg/m3
and thickness t = 0.08nm is used
For a carbon nanotube with chirality (ni , mi ), the diameter can be given by
di =
r
π
n2
i + m2
i + ni mi (55)
where r = 0.246nm. The diameter of the SWCNT shown in 7 is 0.55nm.
A constant modal damping factor of 1% for all the modes is assumed.
We consider clamped-free boundary condition for the SWCNT.
Undamped nonlocal natural frequencies can be obtained as
λj =
EA
m
σj
1 + σ2
j (e0a)2
, where σj =
(2j − 1)π
2L
, j = 1, 2, · · · (56)
EA is the axial rigidity and m is the mass per unit length of the SWCNT.
For the ﬁnite element analysis the SWCNT is divided into 200 elements.
The dimension of each of the system matrices become 200 × 200, that is
n = 200.

Nonlocal natural frequencies of SWCNT
2 4 6 8 10 12 14 16 18 20
0
5
10
15
20
25
30
35
40
Normalisednaturalfreqency:λj
/ω1
Frequency number: j
e0
a=2.0nm
e
0
a=1.5nm
e
0
a=1.0nm
e0
a=0.5nm
local
analytical
direct finite element
approximate
First 20 undamped natural frequencies for the axial vibration of SWCNT.

Nonlocal mode shapes of SWCNT
0 5 10 15 20 25
-1.5
-1
-0.5
0
0.5
1
1.5
Modeshape
Length (nm)
(a) Mode 2
0 5 10 15 20 25
-1.5
-1
-0.5
0
0.5
1
1.5
Modeshape
Length (nm)
(b) Mode 5
0 5 10 15 20 25
-1.5
-1
-0.5
0
0.5
1
1.5
Modeshape
Length (nm)
(c) Mode 6
0 5 10 15 20 25
-1.5
-1
-0.5
0
0.5
1
1.5
Modeshape
Length (nm)
e0
a=0.5
e0
a=2.0
approximate
(d) Mode 9
Figure : Four selected mode shapes for the axial vibration of SWCNT. Exact ﬁnite
element results are compared with the approximate analysis based on local
eigensolutions. In each subplot four different values of e0a, namely 0.5, 1.0, 1.5 and
2.0nm have been used.

Nonlocal frequency response of SWCNT
0 1 2 3 4 5 6 7 8
10
−3
10
−2
10
−1
10
0
10
1
10
2
Normalisedresponseamplitude:Hnn
(ω)/δst
Normalised frequency (ω/ω
1
)
(a) e0a = 0.5nm
0 1 2 3 4 5 6 7 8
10
−3
10
−2
10
−1
10
0
10
1
10
2
Normalisedresponseamplitude:H
nn
(ω)/δ
st
1
)
(b) e0a = 1.0nm
0 1 2 3 4 5 6 7 8
10
−3
10
−2
10
−1
10
0
10
1
10
2
nn
(ω)/δ
st
1
)
(c) e0a = 1.5nm
0 1 2 3 4 5 6 7 8
10
−3
10
−2
10
−1
10
0
10
1
10
2
nn
(ω)/δ
st
1
)
local
exact − nonlocal
approximate − nonlocal
(d) e0a = 2.0nm
Figure : Amplitude of the normalised frequency response of the SWCNT at the tip for
different values of e0a. Exact ﬁnite element results are compared with the approximate
analysis based on local eigensolutions.

A rectangular single-layer graphene sheet (SLGS) is considered to
examine the transverse vibration characteristics of nanoplates.
The graphene sheet is of dimension L=20nm, W=15nm and Young’s
modulus E = 1.0 TPa, density ρ = 2.25 × 103
kg/m3
, Poisson’s ratio
ν = 0.3 and thickness h = 0.34nm is considered
We consider simply supported boundary condition along the four edges
for the SLGS. Undamped nonlocal natural frequencies are
λij =
D
m
β2
ij
1 + β2
ij (e0a)2
where βij = (iπ/L)
2
+ (jπ/W)
2
, i, j = 1, 2, · ·
(57)
D is the bending rigidity and m is the mass per unit area of the SLGS.
For the ﬁnite element analysis the DWCNT is divided into 20 × 15
elements. The dimension of each of the system matrices become
868 × 868, that is n = 868.

Nonlocal natural frequencies of SLGS
2 4 6 8 10 12 14
1
2
3
4
5
6
7
8
9
10
11
12
Normalisednaturalfreqency:λ
j
/ω
1
Frequency number: j
e0
a=2.0nm
e0
a=1.5nm
e0
a=1.0nm
e0
a=0.5nm
local
analytical
approximate
First 15 undamped natural frequencies for the transverse vibration of SLGS.

Nonlocal mode shapes of SLGS
0
5
10
15
20
0
5
10
15
−0.02
0
0.02
X direction (length)
Y direction (width)
(a) Mode 2
0
5
10
15
20
0
5
10
15
−0.02
0
0.02
Y direction (width)
(b) Mode 4
0
5
10
15
20
0
5
10
15
−0.02
0
0.02
Y direction (width)
(c) Mode 5
0
5
10
15
20
0
5
10
15
−0.02
0
0.02
Y direction (width)
(d) Mode 6
Figure : Four selected mode shapes for the transverse vibration of SLGS for
e0a = 2nm. Exact ﬁnite element results (solid line)are compared with the approximate
analysis based on local eigensolutions (dashed line).

Nonlocal frequency response of SLGS
0 1 2 3 4 5 6 7 8 9 10
10
−3
10
−2
10
−1
10
0
10
1
10
2
Normalisedamplitude:Hij
(ω)/δst
1
)
(a) e0a = 0.5nm
0 1 2 3 4 5 6 7 8 9 10
10
−3
10
−2
10
−1
10
0
10
1
10
2
Normalisedamplitude:H
ij
(ω)/δ
st
1
)
(b) e0a = 1.0nm
0 1 2 3 4 5 6 7 8 9 10
10
−3
10
−2
10
−1
10
0
10
1
10
2
ij
(ω)/δ
st
1
)
(c) e0a = 1.5nm
0 1 2 3 4 5 6 7 8 9 10
10
−3
10
−2
10
−1
10
0
10
1
10
2
ij
(ω)/δ
st
1
)
local
exact − nonlocal
approximate − nonlocal
(d) e0a = 2.0nm
Figure : Amplitude of the normalised frequency response Hij (ω) for i = 475,j = 342
of the SLGS for different values of e0a. Exact ﬁnite element results are compared with
the approximate analysis based on local eigensolutions.

Conclusions
Nonlocal elasticity is a promising theory for the modelling of nanoscale
dynamical systems such as carbon nantotubes and graphene sheets.
The mass matrix can be decomposed into two parts, namely the classical
local mass matrix M0 and a nonlocal part denoted by Mµ. The nonlocal
part of the mass matrix is scale-dependent and vanishes for systems with
large length-scale.
An undamped nonlocal system will have classical normal modes
provided the nonlocal part of the mass matrix satisfy the condition
KM−1
0 Mµ = MµM−1
0 K where K is the stiffness matrix.
A viscously damped nonlocal system with damping matrix C will have
classical normal modes provided CM−1
0 K = KM−1
0 C and
CM−1
0 Mµ = MµM−1
0 C in addition to the previous condition.

Conclusions
Natural frequency of a general nonlocal system can be expressed as
λj ≈
ωj
1+M′
µjj
, ∀j = 1, 2, · · · where ωj are the corresponding local
frequencies and M′
µjj
are the elements of nonlocal part of the mass matrix
in the modal coordinate.
Every nonlocal normal mode can be expressed as a sum of two principal
components as uj ≈ xj + (
n
k=j
λ2
j
(λ2
k
−λ2
j )
M′
µkj
1+M′
µkk
xk ), ∀j = 1, 2, · · · . One of
them is parallel to the corresponding local mode xj and the other is
orthogonal to it.

Further reading
[1] E. Wong, P. Sheehan, C. Lieber, Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes, Science (1997) 277–1971.
[2] S. Iijima, T. Ichihashi, Single-shell carbon nanotubes of 1-nm diameter, Nature (1993) 363–603.
[3] J. Warner, F. Schaffel, M. Rummeli, B. Buchner, Examining the edges of multi-layer graphene sheets, Chemistry of Materials (2009) 21–2418.
[4] D. Pacile, J. Meyer, C. Girit, A. Zettl, The two-dimensional phase of boron nitride: Few-atomic-layer sheets and suspended membranes, Applied
Physics Letters (2008) 92.
[5] A. Brodka, J. Koloczek, A. Burian, Application of molecular dynamics simulations for structural studies of carbon nanotubes, Journal of Nanoscience
and Nanotechnology (2007) 7–1505.
[6] B. Akgoz, O. Civalek, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams,
International Journal of Engineering Science 49 (11) (2011) 1268–1280.
[7] B. Akgoz, O. Civalek, Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory, Materials &
Design 42 (164).
[8] E. Jomehzadeh, H. Noori, A. Saidi, The size-dependent vibration analysis of micro-plates based on a modified couple stress theory, Physica
E-Low-Dimensional Systems & Nanostructures 43 (877).
[9] M. H. Kahrobaiyan, M. Asghari, M. Rahaeifard, M. Ahmadian, Investigation of the size-dependent dynamic characteristics of atomic force microscope
microcantilevers based on the modified couple stress theory, International Journal of Engineering Science 48 (12) (2010) 1985–1994.
[10] A. C. Eringen, On differential-equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54 (9)
(1983) 4703–4710.
[11] M. Aydogdu, Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E 41 (5) (2009) 861–864.
[12] M. Aydogdu, Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mechanics Research
Communications 43 (34).
[13] T. Murmu, S. Adhikari, Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems, European Journal of Mechanics -
A/Solids 34 (1) (2012) 52–62.
[14] T. Aksencer, M. Aydogdu, Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory, Physica
E-Low-Dimensional Systems & Nanostructures 43 (954).
[15] H. Babaei, A. Shahidi, Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the galerkin method,
Archive of Applied Mechanics 81 (1051).
[16] C. M. Wang, W. H. Duan, Free vibration of nanorings/arches based on nonlocal elasticity, Journal of Applied Physics 104 (1).
[17] R. Artan, A. Tepe, Nonlocal effects in curved single-walled carbon nanotubes, Mechanics of Advanced Materials and Structures 18 (347).
[18] M. Aydogdu, S. Filiz, Modeling carbon nanotube-based mass sensors using axial vibration and nonlocal elasticity, Physica E-Low-Dimensional
Systems & Nanostructures 43 (1229).
[19] R. Ansari, B. Arash, H. Rouhi, Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal
elasticity, Composite Structures 93 (2419).
[20] T. Murmu, S. C. Pradhan, Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory,
Journal of Applied Physics 105 (1).
[21] J. Yang, X. Jia, S. Kitipornchai, Pull-in instability of nano-switches using nonlocal elasticity theory, Journal of Physics D-Applied Physics 41 (1).
[22] H. Heireche, A. Tounsi, H. Benhassaini, A. Benzair, M. Bendahmane, M. Missouri, S. Mokadem, Nonlocal elasticity effect on vibration characteristics
of protein microtubules, Physica E-Low-Dimensional Systems & Nanostructures 42 (2375).
[23] L. Meirovitch, Principles and Techniques of Vibrations, Prentice-Hall International, Inc., New Jersey, 1997.
[24] M. Géradin, D. Rixen, Mechanical Vibrations, 2nd Edition, John Wiely & Sons, New York, NY, 1997, translation of: Théorie des Vibrations.
[25] M. Petyt, Introduction to Finite Element Vibration Analysis, Cambridge University Press, Cambridge, UK, 1998.
[26] D. Dawe, Matrix and Finite Element Displacement Analysis of Structures, Oxford University Press, Oxford, UK, 1984.
[27] L. Rayleigh, Theory of Sound (two volumes), 1945th Edition, Dover Publications, New York, 1877.

[28] T. K. Caughey, M. E. J. O’Kelly, Classical normal modes in damped linear dynamic systems, Transactions of ASME, Journal of Applied Mechanics 32
(1965) 583–588.
[29] S. Adhikari, Damping modelling using generalized proportional damping, Journal of Sound and Vibration 293 (1-2) (2006) 156–170.

Dynamics of nonlocal structures

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Dynamics of nonlocal structures