Architected, structural materials have been reported with promising enhancement of mechanical performance using the structural method (e.g., mechanical metamaterials MM) and the material method (e.g., composite materials). Here, we develop the extensible, plate-like mechanical metamaterials at the microscale using the functionally graded materials (FGM-MM) and the carbon nanotubes (CNT-MM) to obtain the advanced structural materials with good maneuverability over the postbuckling response. Theoretical models are developed to investigate the postbuckling response of the FGM-MM and CNT-MM subjected to the bilateral constraints, and numerical simulations are carried out to validate the theoretical results. The theoretical models are used to investigate the maneuverability of the postbuckling behaviors with respect to the material properties (i.e., the FGMs and CNTs) and geometric properties (i.e., the corrugated microstructures). The findings show that the corrugation in the MM and composition in the FGMs and CNTs assist in tuning the buckling mode transitions. The reported CNT-MM and FGM-MM provide a novel direction for programming the mechanical response of the artificial materials.

1. Thin–Walled Structures 159 (2021) 107264
Available online 29 November 2020
0263-8231/© 2020 Elsevier Ltd. All rights reserved.
Maneuverable postbuckling of extensible mechanical metamaterials using
functionally graded materials and carbon nanotubes
Talal Salem c,1
, Xiaoyun Xie a,1
, Pengcheng Jiao a,b,*
, Nizar Lajnef c
a
Institute of Port, Coastal and Offshore Engineering, Ocean College, Zhejiang University, Zhoushan, 316021, Zhejiang, China
b
Engineering Research Center of Oceanic Sensing Technology and Equipment, Zhejiang University, Ministry of Education, China
c
Department of Civil and Environmental Engineering, Michigan State University, East Lansing, MI, 48824, USA
A R T I C L E I N F O
Keywords:
Mechanical metamaterials (MM)
Maneuverable postbuckling
Microscale
Extensibility
Functionally graded materials (FGMs)
Carbon nanotubes (CNTs)
A B S T R A C T
Architected, structural materials have been reported with promising enhancement of mechanical performance
using the structural method (e.g., mechanical metamaterials MM) and the material method (e.g., composite
materials). Here, we develop the extensible, plate-like mechanical metamaterials at the microscale using the
functionally graded materials (FGM-MM) and the carbon nanotubes (CNT-MM) to obtain the advanced structural
materials with good maneuverability over the postbuckling response. Theoretical models are developed to
investigate the postbuckling response of the FGM-MM and CNT-MM subjected to the bilateral constraints, and
numerical simulations are carried out to validate the theoretical results. The theoretical models are used to
investigate the maneuverability of the postbuckling behaviors with respect to the material properties (i.e., the
FGMs and CNTs) and geometric properties (i.e., the corrugated microstructures). The findings show that the
corrugation in the MM and composition in the FGMs and CNTs assist in tuning the buckling mode transitions. The
reported CNT-MM and FGM-MM provide a novel direction for programming the mechanical response of the
artificial materials.
1. Introduction
Mechanical metamaterials (MM) provide the possibility of improving
the mechanical properties of bulk materials from the structural
perspective. Utilizing various engineered microstructures [1], MM have
been reported on their outstanding mechanical characteristics resulted
in the critical geometric nonlinearity. Studies have been conducted on
controlling and optimizing the mechanical response of MM [2], such as
the enhanced tension [3], programmable bending and buckling,
compression-induced twisting [4], ultra-light and ultra-stiff [5,6],
negative Poisson’s ratio [7], programmable unusual bulk modulus and
mass density, and complete recovery from large strains in compression
[8–11], etc. Exploring and programming the mechanical characteristics
in a desirable manner, MM have been used in various innovative ap­
plications, e.g., mechanical sensor [12], heat transferring [13,14], en­
ergy absorption [15,16], artificial muscles [17], drug delivery [19], and
soft robots [20]. However, MM have been facing the dilemma in
obtaining the better controllability in the mechanical response (i.e.,
tunability of performance) and less complexity in the microstructures (i.
e., feasibility of fabrication). To address such research dilemma, func­
tional composite materials can be used in MM, such as functionally
graded materials (FGMs) and carbon nanotubes (CNTs).
FGMs have been reported as the functional composite materials that
the material properties are varied in the length, width, or thickness di­
rections [21]. Studies have been conducted to explore the principles and
applications of FGMs [22,24]. FGMs have been applied in many fields,
including the reduction of the in-plane and lateral thickness stresses,
improvement of the residual stress distribution, increment of the frac­
ture toughness, reduction of the stress strength factor, and enhancement
of the thermoelectric energy conversion properties [24–26]. Studies
have been conducted to investigate the instability behavior (i.e., buck­
ling) of FGMs at the multiscale. The buckling behavior of the
Bernoulli-Euler nano-beams was studied under different boundary
conditions [47], and the size-dependent response of the functionally
graded nano-beams was investigated under the uniformly distributed
transverse load [48]. The nonlocal integral formulation was proposed
for the nano-beams in the non-isothermal environment by adopting the
proper Gibbs potential [49]. CNTs, on the contrary, have been
* Corresponding author. Institute of Port, Coastal and Offshore Engineering, Ocean College, Zhejiang University, Zhoushan, 316021, Zhejiang, China.
E-mail address: pjiao@zju.edu.cn (P. Jiao).
1
Authors contributed equally to work.
Contents lists available at ScienceDirect
Thin-Walled Structures
journal homepage: http://www.elsevier.com/locate/tws
https://doi.org/10.1016/j.tws.2020.107264
Received 28 August 2020; Received in revised form 2 November 2020; Accepted 3 November 2020

2. Thin-Walled Structures 159 (2021) 107264
2
developed by eventually mixing carbon nanotubes in the matrix [27],
which have been used in different applications [28–30]. Studies have
shown that the addition of a small percentage of nanotubes are likely to
considerably increase the mechanical, electrical, or thermal properties
of CNTs [32,33,36,50]. The integral elasticity model was proposed to
characterize the mechanical behavior of the single-walled CNTs under
axial force and kinematic boundary using the molecular dynamics
simulations [51]. Although the functionalities of FGMs and CNTs have
been investigated by tuning the material properties of the composites,
recent research interests have been dedicated to exploring their utilities
using the method of structural design (e.g., MM). As a consequence, MM
made by FGMs (i.e., FGM-MM) and CNTs (i.e., CNT-MM) are expected to
behave more effective maneuverability with less complicated
microstructures.
Here, we develop the microscale, extensible FGM-MM, and CNT-MM
by applying the structural design method (i.e., MM) to the functional
composite materials (i.e., FGMs and CNTs). The FGM-MM and CNT-MM
are designed in the plate-like configurations, which are placed between
the bilateral constraints to obtain the postbuckling response under the
axial compression. The material models are developed to characterize
the effective material properties of the reported engineered
metastructures, and the theoretical models are reported to investigate
the postbuckling response (i.e., buckling mode transitions and force-
displacement relations). Numerical simulations are carried out to vali­
date the theoretical results and satisfactory agreements are obtained.
The programmable mechanical behaviors of the FGM-MM and CNT-MM
are investigated in terms of the microstructures and material functions.
This work combines the structural method (i.e., corrugation) and ma­
terial method (i.e., FGMs and CNTs) to obtain the more maneuverable
postbuckling response, which is novel since studies have yet been car­
ried out on tuning the mechanical performance of MM using FGMs or
CNTs. The rest of the paper can be drawn as: Section 2 develops the
material models to characterize the effective materials of the microscale
FGM-MM and CNT-MM. Section 3 reports the theoretical models to
investigate the postbuckling response of the FGM-MM and CNT-MM.
Section 4 conducts the numerical simulations to validate the theoret­
ical results of the postbuckling behaviors. Section 5 carries out the
parametric studies to unveil the influences of the structure designs in the
MM and material functions in the FGMs and CNTs on the postbuckling
performance of the FGM-MM and CNT-MM. Section 6 summarizes the
main findings of this study. Appendices provide the detailed information
on the theoretical modelling and the numerical simulations.
Fig. 1. Principles of the maneuverable, extensible FGM-MM and CNT-MM designed using the structural method of the MM and the material method of the FGMs
and CNTs.
T. Salem et al.

3. Thin-Walled Structures 159 (2021) 107264
3
2. Problem formulation of the microscale FGM-MM and CNT-MM
2.1. FGM-MM and CNT-MM subjected to bilateral constraints
In this study, we theoretically formulate the FGM-MM and CNT-MM
by considering the structural characteristics of the MM and the material
characteristics of the FGMs and CNTs. Fig. 1 illustrates the maneuver­
able, extensible FGM-MM, and CNT-MM designed using the structural
method of the MM and the material method of the FGMs and CNTs. Due
to the slenderness of the plate-like structures (i.e., thickness ≪width),
the material properties of the FGMs are assumed to be distributed
following the material functions in the length direction x. In contrast,
the material properties of the CNT-MM are evenly distributed (i.e., the
material functions are not varied with x). Table 1 compares the material
functions between the FGM-MM and CNT-MM. In Table 1, fiand gi(i=1,
2) refer to the material functions, Vindicates the volume fraction in the
CNTs, and the subscripts FM and CM indicate the FGM-MM and CNT-
MM, respectively.
2.2. Effective material properties of the FGM-MM and CNT-MM
2.2.1. Structural characteristics of the MM
Fig. 2(a) demonstrates the postbuckling deformation for the exten­
sible FGM-MM and CNT-MM subjected to the bilateral constraints. Note
that the material properties of the FGMs and CNTs are considered
independently in this study. According to Ref. [37], the effective
stress-strain relations of the plate MM can be written as
EMM = ζMME
(
ρMM
ρhb
)2
(1)
where Eis the Young’s modulus of the material in the MM, ζMM, ρMMare,
respectively, the weight factor accounts for the influence of the cylin­
drical webbing patterns and the density of the MM, and ρhbis the density
of a hollow beam that has the equivalent length L, width ​ Wb, height ​ H,
thickness tMMand mass Mas the MM. The volume of the MM in the FGM-
MM and CNT-MM, and the volume of the hollow beam can be obtained
as [34]. Therefore, the effective Young’s modulus of the plate MM can be
rewritten as
EMM = ζMM
[
2(D + G)2
(Wb + H − 2tMM)
Wb
(
(D + G)2
+ πDH
)
]2
E (2)
where Dand Gare the cylindrical diameter and gap distance between two
cells in the cylindrical configuration, respectively, as shown in Appendix
B.
2.2.2. Material characteristics of the FGMs and CNTs
Assuming the postbuckling deformation is resulted in the normal
stress σxand couple stress mxy, the stress-strain relations of the FGM-MM
and CNT-MM can be written as [38],
σ =
{
σx
mxy
}
= Qε =
[
Q11
̂
Q44
]{
εx
χxy
}
(3)
The Poisson’s ratio is neglected for the slender structures in this
study [39]. Note that the material distributions of the FGM-MM are
varied in the height direction due to the corrugation, i.e., the material
properties of the corrugation walls are varied (see Fig. 1).
The components of the Qmatrix can be reduced to Ref. [38]:
{
Q11 = E11
̂
Q44 = G13l2 (4)
where E11, G13, and lare the Young’s modulus in the longitudinal di­
rection, shear modulus in the transverse direction, and the material
length scale factor, respectively.
First, the effective material properties of the FGM-MM are defined.
The coefficients in Eq. (4) can be written as
Q11 = E(x)G13 =
1
2
E(x) (5)
where E(x)is the material functions of the FGMs. In this study, we mainly
consider the two material functions for the FGMs as
⎧
⎪
⎨
⎪
⎩
E(x),1 = a(1 + bx)γ
E(x),2 = ae
bx
γ
(6)
where a, band γare the factors used to define the Young’s modulus.
Table 1
Comparison of the material functions between the FGM-MM and CNT-MM.
FGM-MM CNT-MM
Young’s modulus E EFM = f1(x) ECM = f2(V)
Shear modulus G GFM = g1(x) GCM = g2(V)
Fig. 2. (a) Demonstration of the postbuckling deformation for the extensible FGM-MM and CNT-MM subjected to the bilateral constraints. (b) Deformation analysis
of an arbitrary segment in the postbuckled FGM-MM and CNT-MM.
T. Salem et al.

4. Thin-Walled Structures 159 (2021) 107264
4
Taking Eqs. (2) and (6) into Eq. (5), the applied coefficients in Eq. (4)
can be written as
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Q11, FM = ζMM
[
2(D + G)2
(Wb + H − 2tMM)
Wb
(
(D + G)2
+ πDH
)
]2
E(x)
̂
Q44,FM =
1
2
ζMM
[
2(D + G)2
(Wb + H − 2tMM)
Wb
(
(D + G)2
+ πDH
)
]2
E(x)l2
(7)
where the subscript FM refers to the FGM-MM. Note that the Young’s
modulus function E(x)is given in Eq. (6).
Next, the properties of the effective materials for the CNT-MM are
determined. The Young’s modulus of the CNTs can be estimated by the
Mori-Tanaka scheme or the rule of mixtures. To consider the small scale
effect, the CNTs efficiency parameters (η1, η3)are introduced in the
extended version of this rule and evaluated by matching the effective
properties of the CNTs-reinforced composites determined from the
molecular dynamics simulations with those from the rule of mixture [9].
The Young’s modulus and shear modulus of the CNTs can be expressed
as [27,42].
⎧
⎪
⎨
⎪
⎩
E11 = η1rVEA
11 + (1 − rV)EB
G13 =
η3GA
13GB
(
rGB
+ GA
13
)
VB
(8)
where EA
11, GA
13, EB
, GB
are the CNTs’ Young’s modulus, shear modulus
and the matrix’s Young’s modulus and shear modulus, respectively. r =
VA
V describes the variation of the volume fraction, where VAand Vrefer to
the volumes of the CNTs and matrix, respectively. Substituting Eqs. (8)
and (2) into Eq. (4), the effective material properties of the CNT-MM can
be obtained by combining the Young’s modulus of the CNTs with the
MM as
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
Q11, CM = ζMM
[
2(D + G)2
(Wb + H − 2tMM)
Wb
(
(D + G)2
+ πDH
)
]2
(
η1rVBEA
11 + (1 − rVB)EB
)
̂
Q44,CM = ζMM
[
2(D + G)2
(Wb + H − 2tMM)
Wb
(
(D + G)2
+ πDH
)
]2
η3GA
12GB
(
rGB
+ GA
12
)
VB
l2
(9)
where the subscript CM refers to the CNT-MM.
3. Postbuckling analysis of the FGM-MM and CNT-MM subjected
to the bilateral constraints
3.1. General governing equations
According to the modified couple stress theory, the strain energy
density can be written as a function of the strain (i.e., conjugated with
stress) and curvature (i.e., conjugated with couple stress) as
U =
1
2
∫∫∫
(σε + mχ)dV (10)
where σ, ε, m and χ are the Cauchy stress tensor, conventional strain
tensor, deviatoric couple stress tensor, and symmetric curvature tensor,
respectively.
Based on the Euler-Bernoulli beam theory as shown in Fig. 2(b), the
displacement components of the FGM-MM and CNT-MM can be ob­
tained as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ux = − u(x, τ) − zφ(x, τ)
uy = 0
uz = w(x, t)
χxy = −
∂φ
∂x
(11)
where φ = ∂w
∂x, τ, and tare the rotation angle of the neutral axis, time
term, and thickness, respectively. u and ware the axial displacement and
transverse deflection, respectively. This study investigates the post­
buckling response of FGM-MM and CNT-MM subjected to the bilateral
constraints. Since the constraints net gap h is extremely small comparing
with the beam length L (e.g., L
h = 66.7in Table 2), the postbuckled FGM-
MM and CNT-MM can be assumed as small deformation. Therefore, the
small deformation assumptions are used.
The nonzero components of the strain curvature are obtained as
ε =
{
εx
χxy
}
=
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
−
∂u
∂x
− z
∂2
w
∂x2
−
∂2
w
∂x2
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(12)
According to the Hamilton’s principle, the total energy can be ex­
press as
∫
T1
T2
Ωdτ =
∫
T1
T2
(K − U + W)dτ = 0 (13)
where U, and Kare the virtual potential energy, virtual external work,
and virtual kinetic energy, respectively, which can be evaluated as
U =
∫L
′
0
∫b
0
⎡
⎣
∫t
0
σT
ε + mT
χdz
⎤
⎦dydx (14)
Table 2
Geometric and material properties, element size and type, and loading condition
of the FE models for the CNT-MM and FGM-MM.
Geometry
property
Overall Length L(mm) 2
Width Wb(mm) 0.4
Thickness tMM(μm) 10
Corrugation Height H(μm) 20
Constraints net gap
h(μm)
30
Diameter D(μm) 60
Rib width G(μm) 20
Material property CNTs [27] Young’s modulus
E11(TPa)
5.6466
Shear modulus
G13(TPa)
1.9445
Volume fraction VCNT 0.028
Efficiency parameters
η1
0.0058
Efficiency parameters
η3
0.642
FGMs Function E(X),1 EM(1 + 1.2X)0.9
Function E(X),2 EMe0.3X
Matrix Young’s modulus
EM(GPa)
2000
Length scale
factor
l tMM/10
Element size and
type
Size l(mm) FGM-MM and CNT-
MM
0.01
Bilateral constraints 0.05
Type S4R
T. Salem et al.

5. Thin-Walled Structures 159 (2021) 107264
5
W = P
∫L
′
0
[
sin φ
∂φ
∂x
u + sin φ
(
1 −
du
dx
)
φ
]
dx + P cos φu
⃒
⃒
⃒
⃒
x = L
′
x = 0
(15)
and
K =
∫L
′
0
∫b
0
⎡
⎣
∫t
0
ρ
(
∂u
∂τ
)(
∂u
∂τ
)
+ z2
(
∂2
w
∂x∂τ
)(
∂2
w
∂x∂τ
)
+
(
∂w
∂τ
)(
∂w
∂τ
)
dz
⎤
⎦dydx
(16)
Note that Pand L
′
are the axial force and beam length after the
postbuckling deformation, respectively.
The force, moments, unit mass, and moment of inertia of the FGM-
MM and CNT-MM can be written as
{
Nx, Mx, Mxy, m, I
}
=
∫b
0
∫t
0
{
σx, zσx, mxy, ρ,
z2
y
}
dzdy (17)
Substituting Eq. (17) into Eq. (12), we have
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Nx = Q11
∂u
∂x
− J11
∂2
w
∂x2
Mx = J11
∂u
∂x
− I11
∂2
w
∂x2
Mxy = − Q
=
44
∂2
w
∂x2
(18)
where Q11, J11, I11, and Q44can be expressed as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Q11 =
∫b
0
∫
−
t
2
t
2 Q11dzdy
J11 =
∫b
0
∫
−
t
2
t
2 zQ11dzdy
I11 =
∫b
0
∫
−
t
2
t
2 (z)2
Q11dzdy
Q44 =
∫b
0
∫
−
t
2
t
2 ̂
Q44dzdy
(19)
Taking Eqs. 14–17 into Eq. (13) and integrating by parts, we
obtainwhich leads to the coupled governing equations:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
P sin φ
∂φ
∂x
−
∂Nx
∂x
− m
∂2
u
∂t2
= 0
− P
(
1 −
du
dx
)
∂2
w
∂x2
+
∂2
Mx
∂x2
+
∂2
Mxy
∂x2
+ mI
(
∂2
w
∂x∂τ
)2
− m
∂2
w
∂τ2
= 0
(21)
and the boundary conditions:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
P cos φ + Nx + m
∂u
∂τ
= 0
P sin φ
(
1 −
du
dx
)
−
∂Mx
∂x
−
∂Mxy
∂x
+ m
∂w
∂τ
− mI
∂3
w
∂x∂τ2
Mx + Mxy + I
∂2
w
∂x∂τ
= 0
= 0
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⃒
⃒
⃒
⃒
x = L
′
x = 0
(22)
Substituting Eqs. (18) and (19) into Eqs. (21) and (22), the governing
equations and boundary conditions are obtained as
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
P sin φ
∂φ
∂x
− Q11
∂2
u
∂x2
+ J11
∂3
w
∂x3
− m
∂2
u
∂τ2
= 0
P
(
1 −
du
dx
)
∂2
w
∂x2
− J11
∂3
u
∂x3
+
(
I11 + Q44
)
∂4
w
∂x4
− mI
(
∂2
w
∂x∂τ
)2
+ m
∂2
w
∂τ2
= 0
(23)
and
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Pcosφ+Q11
∂u
∂x
− J11
∂2
w
∂x2
+m
∂u
∂τ
=0
Psinφ
(
1−
du
dx
)
− J11
∂2
u
∂x2
+
(
I11 +Q
=
44
)
∂3
w
∂x3
J11
∂u
∂x
−
(
I11 +Q
=
44
)
∂2
w
∂x2
+I
∂2
w
∂x∂τ
=0
+m
∂w
∂τ
− mI
∂3
w
∂x∂τ2
=0
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⃒
⃒
⃒
⃒
x=L
′
x=0
(24)
where Λ = du
dxis the extensible factor resulted in the cylindrical corru­
gation of the FGM-MM and CNT-MM.
It is worthwhile to point out that Eqs. (23) and (24) are the general
governing equations for the extensible FGM-MM and CNT-MM.
Neglecting the extensible terms (i.e., Λ = 0), Eqs. (23) and (24) are
reduced to:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
P sin φ
∂φ
∂x
− Q11
∂2
u
∂x2
+ J11
∂3
w
∂x3
− m
∂2
u
∂τ2
= 0
− J11
∂3
u
∂x3
+
(
I11 + Q
=
44
)
∂4
w
∂x4
+ P
∂2
w
∂x2
− mI
(
∂2
w
∂x∂τ
)2
+ m
∂2
w
∂τ2
= 0
(25)
and
∫
T1
T2
Ωdτ=
∫
T2
T1
∫L
′
0
[(
P sin φ
∂φ
∂x
−
∂Nx
∂x
− m
∂2
u
∂t2
)
u+
(
− P
(
1−
du
dx
)
∂2
w
∂x2
+
∂2
Mx
∂x2
+
∂2
Mxy
∂x2
+mI
(
∂2
w
∂x∂τ
)2
− m
∂2
w
∂τ2
)
w
]
dxdτ
+
∫
T2
T1
[(
P cos φ+Nx +m
∂u
∂τ
)
u+
(
P sinφ
(
1−
du
dx
)
−
∂Mx
∂x
−
∂Mxy
∂x
+m
∂w
∂τ
− mI
∂3
w
∂x∂τ2
)
w+
(
Mx +Mxy +I
∂2
w
∂x∂τ
)(
∂w
∂x
)]⃒
⃒
⃒
⃒
x = L
′
x = 0
dτ= 0 (20)
T. Salem et al.

6. Thin-Walled Structures 159 (2021) 107264
6
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
−
P cos φ + Q11
∂u
∂x
− J11
∂2
w
∂x2
+ m
∂u
∂τ
= 0
J11
∂2
u
∂x2
+
(
I11 + Q44
)
∂3
w
∂x3
+ P
∂w
∂x
+ m
∂w
∂τ
− mI
∂3
w
∂x∂τ2
= 0
J11
∂u
∂x
−
(
I11 + Q
=
44
)
∂2
w
∂x2
+ I
∂2
w
∂x∂τ
= 0
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⃒
⃒
⃒
⃒
⃒
x = L
′
x = 0
(26)
Further ignoring the time terms, the governing equations are the
same as the static micro-composite beams model proposed in Ref. [43].
In order to compare with the micro-isotropic beams in the existing
studies, the material coefficients of the FGMs and CNTs are reduced to
isotropic as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Q11 = Ebt
J11 = 0
I11 =
Ebt3
12
Q
=
44 = btGl2
(27)
Taking Eq. (27) into Eq. (25), the governing equation for the micro-
isotropic beams is
(
EI + GAl2
) ∂4
w
∂x4
+ P
∂2
w
∂x2
− mI
(
∂2
w
∂x∂τ
)2
+ m
∂2
w
∂τ2
= 0 (28)
Omitting the higher-order rotation term
(
∂2
w
∂x∂τ
)2
, Eq. (28) can be
simplified to
B
∂4
w
∂x4
+ P
∂2
w
∂x2
+ m
∂2
w
∂τ2
= 0 (29)
where B = EI + GAl2
is the bending stiffness. Eq. (29) is the same as the
governing equation of the micro-isotropic beams in Ref. [44]. Further
dropping the length scale factor land time-related term ∂2
w
∂τ2 , the gov­
erning equation is obtained for the classical Euler-Bernoulli beams.
3.2. Postbuckling analysis of the FGM-MM and CNT-MM
To solve the governing equations given in Eq. (23), the following
assumptions are taken into account in this section:
1) The FGM-MM and CNT-MM are assumed to be symmetric regarding
the neutral axis. Therefore, J11 = 0.
2) The axial force is gradually applied, which leads to the quasi-static
postbuckling response of the FGM-MM and CNT-MM. Thus, ∂2
w
∂x∂τ =
m ∂2
w
∂τ2 = 0.
According to those assumptions, the governing equations in Eq. (23)
can be reduced to
(
I11 + Q44
)
∂4
w
∂x4
+ (P − PΛ)
∂2
w
∂x2
= 0 (30)
and the boundary conditions are
w(x)
⃒
⃒
⃒
⃒
x = L
′
x = 0
=
dw(x)
dx
⃒
⃒
⃒
⃒
x = L
′
x = 0
= 0 (31)
where PΛ ∂2
w
∂x2 is the extensibility term caused by the cylindrical corruga­
tion. The extensible factor can be expressed as [39].
Fig. 3. Flowchart of numerically solving the postbuckling response of the FGM-MM and CNT-MM using the energy minimization.
T. Salem et al.

7. Thin-Walled Structures 159 (2021) 107264
7
Fig. 4. (a) Meshed CNT-MM and FGM-MM with the cylindrical corrugation under the axial force and the bilateral constraints. (b) Postbuckling shape configurations
of the CNT-MM, FGM1-MM and FGM2-MM in the first buckling mode (Φ1), flatten first buckling mode (Φ1 flatten), third buckling mode (Φ3), flatten third buckling
mode (Φ3 flatten), and fifth buckling mode (FGM1-MM denote the MM made by the first type of FGMs, and FGM2-MM are the MM made by the second FGMs).
d = KT (74)
Fig. 5. Comparison of the force-displacement relations and postbuckling shape configurations at the buckling mode transitions between the theoretical and nu­
merical results for the (a) CNT-MM and (b) FGM1-MM and FGM2-MM.
T. Salem et al.

8. Thin-Walled Structures 159 (2021) 107264
8
Λ =
P
Q11
cos φ (32)
where φis the rotation angle of the FGM-MM and CNT-MM resulted in
the postbuckling deformation. For the small deformations, φ ≈ 0, and
cos φ ≈ 1.
3.2.1. Postbuckling analysis of the FGM-MM
The material factors of the FGM-MM are given as
{
Q11, J11, I11, Q44
}
FM
= bt
{
EFM(x), 0,
t2
EFGM− MM(x)
12
,
EFGM− MM(x)l2
2
}
(33)
Therefore, the governing equation for the postbuckling behavior of
the FGM-MM can be rewritten as
[
EFM(x)I +
1
2
EFM(x)Al2
]
∂4
w
∂x4
+ (P − PΛ)
∂2
w
∂x2
= 0 (34)
Introducing the non-dimensional variables X = x
L
′ and W(X) = w(XL
′
)
h
,
Eq. (34) is normalized as
∂4
W
∂X4
+ N(X)
∂2
W
∂X2
= 0 (35)
where the correlated normalized axial load N(X)is obtained as
N(X) =
2L
′ 2
(P − PΛ)
(
2I + Al2
)
EFM
(36)
Taking Eq. (7) into Eq. (36), N(X)can be rewritten as
N(X) =
(P − PΛ)b2
(
(D + G)2
+ πDH
)2
L
′ 2
2ζMM
(
2I + Al2
)
(D + G)4
(Wb + H − 2tMM)2
α(1 + βX)γ
(37)
According to Ref. [45], the second-order general solution of Eq. (35)
can be written as
d2
W(X)
dX2
= k1S1(X) + k2S2(X) (38)
where the integral functions are expressed as
⎧
⎪
⎪
⎨
⎪
⎪
⎩
S1(X) =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βX
√
J 1
2− γ
(
n
2
2 − γ
(1 + βX)
1
4− 2γ
)
S2(X) =
̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βX
√
J 1
γ− 2
(
n
2
2 − γ
(1 + βX)
1
4− 2γ
) (39)
in which Jindicates the first Bessel functions, and nis:
n =
b
[
(D + G)2
+ πDH
]
L
′
2β(D + G)2
(Wb + H − 2tMM)
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
2(P − PΛ)
(
2I + Al2
)
ζMMα
√
(40)
Integrating Eq. (39), we obtain
and
W(X) = k1
∫X
0
∫t
0
̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + αξ
√
J 1
2− γ
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξdt + k2
∫X
0
×
∫t
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
γ− 2
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξdt + k3X + k4 (42)
where ki(i = 1,…,4)are the integral constants. Taking Eqs. (41) and (42)
into Eq. (31), we have
Eq. (43) represents the eigenvalue problem for ni, which has the
characteristic equation as
dW(X)
dX
= k1
∫X
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
2− γ
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξ + k2
∫X
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
γ− 2
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξ + k3 (41)
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
k3 = 0
k4 = 0
k1
∫1
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
2− γ
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξ + k2
∫1
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
γ− 2
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξ + k3 = 0
k1
∫1
0
∫t
0
̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + αξ
√
J 1
2− γ
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξdt + k2
∫1
0
∫t
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
γ− 2
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξdt + k3 + k4 = 0
(43)
T. Salem et al.

9. Thin-Walled Structures 159 (2021) 107264
9
A set of values of nican be obtained by numerically solving Eq. (44).
Assuming α = 2300, β = 1.2, γ = 0.3and using the first material
function in Eq. (6), the general form of the buckling shape function is
obtained as
where Ajare the coefficients to determine the contributions of the
buckling modes to the postbuckling shape functions of the FGM-MM.
njand kjare determined as nj = 1.78π, 2.55π, 3.56π, 4.38π…,and kj =
1.80, − 14.10, 2.03, 0.79…for j = 1, 2, 3, 4…., respectively.
In the same manner, assuming α = 2300,β = 1, and γ = 2and using
the second material function in Eq. (6), the postbuckling shape functions
of the FGM-MM can be obtained as
W(X) =
∑
∞
j=1
Aj
∫X
0
∫t
0
J0
(
2nje− 0.5ξ
)
− kjY0
(
2nje− 0.5ξ
)
dξdt (46)
where Yindicates the second Bessel functions, and njand kjare obtained
as nj = 2.58π, 3.56π, 5.10π, 6.21π…,and kj = − 0.81, − 0.06, − 0.36,
0.86…for j = 1, 2, 3, 4…, respectively.
3.2.2. Postbuckling analysis of the CNT-MM
In this section, the postbuckling behavior of the CNT-MM is
analyzed. The governing equation can be expressed as
∂4
W
∂X4
+ N2∂2
W
∂X2
= 0 (47)
where the correlated normalized axial load and coefficients of the CNT-
MM are
N2
=
P(1 − Λ)L
′ 2
I11CM + Q44CM
(48)
and
{
Q11, J11, I11, Q44
}
CM
= bt
{
Q11, CM, 0,
t2
Q11, CM
12
, ̂
Q44,CM l2
}
(49)
The general solution of Eq. (47) is:
d2
W(X)
dX2
= k1 sin NX + k2 cos NX (50)
where k1and k2are the constants. Integrating Eq. (50), we have
dW(X)
dX
= − k1
cos NX
N
+ k2
sin NX
N
+ k3 (51)
and
W(X) = − k1
sin NX
N2
− k2
cos NX
N2
+ k3X + k4 (52)
where ki(i = 1, 2, 3, 4) indicate the integral constants. Using the
boundary conditions in Eq. (31), the relations of the constants can be
determined as
∫1
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
2− γ
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξ ⋅
∫1
0
∫t
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
γ− 2
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξdt −
∫1
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
γ− 2
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξ ⋅
∫1
0
×
∫t
0
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + βξ
√
J 1
2− γ
(
n
2
2 − γ
(1 + βξ)
1
4− 2γ
)
dξdt = 0 (44)
W(X) =
∑
∞
j=1
Aj
∫X
0
∫t
0
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1 + 1.2ξ
√ [
J0.588
(
nj1.176(1 + 1.2ξ)0.294)
+ kjJ− 0.588
(
nj1.176(1 + 1.2ξ)0.294)]
dξdt (45)
Table 3
Geometric and material properties used in the comparison between the pre­
sented and existing studies [46].
Geometric
Properties (mm)
Length L 250
Width Wb 30
Thickness tMM 2.3
Constraints gap H 4
Material properties Young’s modulus E (MPa) 2300
Fig. 6. Comparison of the force-displacement relations between the reduced
CNT-MM and FGM-MM models in this study, and the theoretical results in the
existing study [46].
T. Salem et al.

10. Thin-Walled Structures 159 (2021) 107264
10
Fig. 8. Influence of the cylindrical corrugation on the postbuckling response of the CNT-MM, FGM1-MM and FGM2-MM. Force-displacement relations of the (a)
CNT-MM at V = 12%(b) FGM1-MM at b = 0.6, and (c) FGM2-MM at b = 0.6that affected by the diameter-to-height ratio D
H. Postbuckling shape configurations of the (d)
CNT-MM, (e) FGM1-MM and (f) FGM2-MM in Φ1 for D
H = 0.2, Φ3 for D
H = 1, and Φ5 for D
H = 5. (D
H are obtained from the cases of 20 μm-to-100 μm, 20 μm-to-20 μmand 100
μm-to-20 μm). Postbuckling response of the FGM1-MM and FGM2-MM, respectively, and the material function factor is fixed as b = 0.6.
Fig. 7. Influence of the material functions on the postbuckling response of the CNT-MM, FGM1-MM and FGM2-MM. Force-displacement relations of the (a) CNT-MM
affected by the volume fraction V, (b) FGM1-MM affected by the index b, and (c) FGM2-MM affected by the index b. Postbuckling shape configurations of the (d) CNT-MM
in Φ1 for V = 2.8%, Φ3 for V = 12%, and Φ5 for V = 28%, and the (e) FGM1-MM and (f) FGM2-MM in Φ1 for b = 0.3, Φ3 for b = 0.6, and Φ5 for b = 0.9(L = 2 mm,
Wb = 0.4 mm, tMM = 10 μm, h0 = 30 μm, and EM = 130 GPa).
T. Salem et al.

11. Thin-Walled Structures 159 (2021) 107264
11
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
−
k1
N
+ k3 = 0
− k1
cos N
N
+ k2
sin NX
N
+ k3 = 0
−
k2
N2
+ k4 = 0
− k1
sin N
N2
− k2
cos N
N2
+ k3 + k4 = 0
(53)
Expressing k1, k2, and k3in terms of k4, the eigenvalues and buckling
shape functions ψjof the CNT-MM can be numerically obtained. There­
after, the general form of the postbuckling shape function is obtained as
ψ(X) = 1 − cos NX +
N sin N
1 − cos N
(
sin NX
N
− X
)
(54)
Eq. (54) can be expressed in terms of the symmetric and asymmetric
terms as [46].
ψj(X) = 1 − cos NjX
Nj = (j + 1)π
}
j = 1, 3, 5, … (55)
and
ψj(X) = 1 − 2X − cos NjX +
2 sin NjX
Nj
Nj = 2.86π, 4.92π, 6.94π, 8.95π, …
⎫
⎪
⎬
⎪
⎭
j = 2, 4, 6, … (56)
Using the superposition theory, the postbuckling shape functions of
the CNT-MM can be obtained as
W(X) =
∑
∞
j=1,3,5…
Aj
(
1 − cos NjX
)
+
∑
∞
j=2,4,6…
Aj
(
1 − 2X − cos NjX +
2 sin NjX
Nj
)
(57)
where Ajare the coefficients to determine the contributions of the
buckling modes to the postbuckling shape functions.
3.2.3. Energy method to solve the postbuckling response of the FGM-MM
and CNT-MM
In this section, the postbuckling behaviors of the FGM-MM and CNT-
MM are resolved by determining the weight coefficients Ajusing the
energy method. In particular, Ajare obtained by minimizing the total
potential energy of the postbuckled FGM-MM and CNT-MM. The total
potential energy of the FGM-MM and CNT-MM consists of the bending
strain energy Ub, compressive strain energy Uc, and external work Up,
which yields
U = Ub + Uc =
1
2
∫L
′
0
(
I11 + Q44
)(
∂2
w
∂x2
)2
dx +
1
2
PΔc (58)
and the external work caused by the axial force Pis:
Up = −
1
2
PΔ (59)
where the longitudinal displacement Δ, axial shortening Δc, and varia­
tion of length Δbcaused by the buckling-induced midplane rotation are:
Fig. 9. Comparison of the material and geometric influences on the force-displacement relations for the (a) CNT-MM with D
H = 0.2and V = 2.8%, D
H = 1and V = 12%,
and D
H = 5and V = 28%, and (b) FGM1-MM and (c) FGM2-MM with D
H = 0.2and b = 0.3, D
H = 1and b = 0.6, and D
H = 5and b = 0.9.Distributions of A1, A3, A5and A7for the
(d) CNT-MM with D
H = 0.2and b = 0.3, and (e) FGM1-MM and (f) FGM2-MM with D
H = 0.2and b = 0.3. distributions of the first four buckling mode coefficients (A1, A3,
A5, and A7) for the CNT-MM with D
H = 0.2and b = 0.3. Fig. 9(e) and (f) show the distributions of A1, A3, A5, and A7for the FGM1-MM and FGM2-MM, respectively,
with D
H = 0.2and b = 0.3. It can be seen that certain mode coefficient dominants the postbuckling response in the stable buckling modes (i.e., Φ1, Φ3, and Φ5), and
significant coefficient fluctuations are obtained in buckling snap-throughs (i.e., Φ1-Φ3, Φ3-Φ5, and Φ5-Φ7).
T. Salem et al.

12. Thin-Walled Structures 159 (2021) 107264
12
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Δ = Δc + Δb
Δc = L − L
′
Δb =
1
2
∫L
′
0
(
∂w
∂x
)2
dx
(60)
As a consequence, the total potential energy Πcan be written as
Π = Ub + Uc + Up (61)
The normalized length after the postbuckling deformation can be
written as
L
′
= L − Δc = L − Δ +
h2
2L′
∫1
0
(
∂W
∂X
)2
dX (62)
Considering an arbitrary segment in the FGM-MM and CNT-MM, the
shortening of the element might be written as
dΔc =
Pdx
Q11
(63)
Integrating Eq. (63) and substituting it into Eq. (60), the axial force
can be written as
P =
1
EmL
⎡
⎣Δ −
h2
2L′
∫1
0
(
∂W
∂X
)2
dX
⎤
⎦ (64)
where Emis written as
Em =
∫1
0
1
Q11
dX (65)
Substituting Eq. (64) into Eqs. (58) and (59), the normalized po­
tential energies and axial force are expressed, in terms of the normalized
axial displacement d, as
Vb =
1
2Im
⎡
⎣
∑
∞
j=1
Aj
2
∫1
0
(
I11 + Q44
)(
∂2
ψj(X)
∂X2
)2
dX
⎤
⎦ (66)
Vc =
kh2
2EmIm
⎡
⎣d −
1
2
∑
∞
j=1
Aj
2
∫1
0
(
∂ψj(X)
∂X
)2
dX
⎤
⎦
2
(67)
VP = −
kh2
2EmIm
⎡
⎣d2
−
d
2
∑
∞
j=1
Aj
2
∫1
0
(
∂ψj(X)
∂X
)2
dX
⎤
⎦ (68)
and
P̃ =
kh2
EmIm
⎡
⎣d −
1
2
∑
∞
j=1
Aj
2
∫1
0
(
∂ψj(X)
∂X
)2
dX
⎤
⎦ (69)
Fig. 10. Variation of the maximum axial force in the buckling mode transitions Φ1-Φ3, Φ3-Φ5, and Φ5-Φ7 for the (a) CNT-MM, (b) FGM1-MM and (c) FGM2-MM.
T. Salem et al.

13. Thin-Walled Structures 159 (2021) 107264
13
where the variables and normalization factors are introduced as
d =
ΔL
′
h2
⋅k =
L
′
L
⋅Q =
t
h
⋅P̃ =
PL
′2
Im
⋅Vb =
UbL
′3
Imh2
⋅Vc =
UcL
′3
Imh2
⋅VP =
UPL
′3
Imh2
⋅Im
=
∫1
0
I11dX
(70)
Substituting the buckling shape functions in Eqs. (45), (46) and (57)
into Eqs. 66–68, respectively, the normalized total potential energy for
the FGM-MM is
and the normalized total potential energy for the CNT-MM is
Minimizing the total potential energies with respect to the deflection
W(X)between the bilateral constraints, the weight coefficients Ajof the
postbuckling shape functions can be determined for the FGM-MM and
CNT-MM as
{
Min
[
Π
(
Aj
)]
0 ≤ W(X) ≤ 1
(73)
Given the complexity of the total potential energies, the numerical
approach is used to solve the energy minimization in Eq. (73). In
particular, to improve the calculation efficiency while ensuring the ac­
curacy, the first 20 modes are used to express the general buckling shape
function, the finite total potential energy expression is then obtained. By
using the built-in functions in Mathematica, the minimum potential
energies with different axial displacement are obtained. Fig. 3 illustrates
the flowchart of numerically solving the postbuckling response of the
FGM-MM and CNT-MM using the energy minimization.
4. Numerical modelling and validation of the FGM-MM and CNT-
MM
In this section, numerical simulations are conducted to validate the
presented theoretical model. The FGM-MM and CNT-MM are developed
to obtain the postbuckling response in Abaqus R2016x. The linear
perturbation buckle algorithm is used in the buckling analysis and the
dynamic implicit with Nlgeom is applied in the postbuckling analysis.
Buckling imperfection is accounted for the finite element (FE) model by
modifying the input files [35]. User-defined subroutine USDFLD is
developed to characterize the material properties of the FGM-MM and
CNT-MM (see Appendix B).
4.1. Numerical setup and modelling
Fig. 4(a) displays the meshed FE models of the FGM-MM and CNT-
MM subjected to the bilateral confinements. One ends of the FGM-MM
and CNT-MM (i.e., x = 0) are fixed and the other ends can slide in the
loading direction. To investigate the effect of the bilateral constraints on
the postbuckling response, the contact interaction is considered in the
FE models. In particular, the interaction properties are defined as “hard”
contact that allows separation after frictionless contact for the normal
and tangential behaviors. Linearly increasing loading (i.e., axial
displacement d) is gradually applied to the end edges of the FGM-MM
and CNT-MM (x = L), which is defined aswhere the amplitude and
time period are K = 60 ​ μmand T = 10 s, respectively.
Fig. 4(b) presents the postbuckling shape configurations between the
CNT-MM, FGM1-MM, and FGM2-MM. In particular, the postbuckled
beams are demonstrated in the first buckling mode (Φ1), flatten first
buckling mode (Φ1 flatten), third buckling mode (Φ3), flatten third
buckling mode (Φ3 flatten), and fifth buckling mode. It can be seen that
the postbuckling shapes are significantly affected by the material func­
tions of the FGMs and CNTs, especially for the ones in the higher
buckling modes. Note that FGM1-MM denote the plate-like MM made by
the first type of FGMs and FGM2-MM are the MM by the second FGMs.
Following Eq. (6), the first and second types of the FGMs are given as
{
FGM1 : EM(1 + 1.2X)b
FGM2 : EMebX (75)
where bis varied from 0.3 to 0.9. The geometric and material properties
of the FE models are summarized in Table 2 [27].
4.2. Comparison of the theoretical and numerical results
Fig. 5 compares the postbuckling response of the CNT-MM and FGM-
MM between the theoretical and numerical results (see Supplemental
Video). The geometric and materials in Table 2 are used to obtain the
results. Fig. 5(a) presents the comparison of the CNT-MM for the force-
displacement relations, and the postbuckling shape configurations are
provided in the buckling mode transitions (i.e., Φ1- Φ3 and Φ3- Φ5).
ΠFM =
(
1
2Im
+
Al2
4Im
)
⎡
⎣
∑
∞
j=1
Aj
2
∫1
0
EFM(X)
(
∂2
ψj(X)
∂X2
)2
dX
⎤
⎦ +
6k
Q2
⎡
⎣d −
1
2
∑
∞
j=1
Aj
2
∫1
0
(
∂ψj(X)
∂X
)2
dX
⎤
⎦
2
−
6k
Q2
⎡
⎣d2
−
d
2
∑
∞
j=1
Aj
2
∫1
0
(
∂ψj(X)
∂X
)2
dX
⎤
⎦ (71)
ΠCM =
1
2
∫1
0
⎛
⎜
⎝1 +
Q44CM
I11CM
⎞
⎟
⎠
(
∑
∞
j=1,3,5…
AjN2
j cos NjX +
∑
∞
j=2,4,6…
Aj
(
N2
j cos NjX − 2Nj sin NjX
)
)2
dX +
Q11CMh2
k
2I11CM
⎡
⎣d −
1
2
∫1
0
(
∑
∞
j=1,3,5…
AjNj sin NjX
+
∑
∞
j=2,4,6…
Aj
(
2 + Nj sin NjX + 2 cos NjX
)
)2
dX
⎤
⎦
2
−
Q11CMh2
k
2I11CM
⎡
⎣d2
−
d
2
∫1
0
(
∑
∞
j=1,3,5…
AjNj sin NjX +
∑
∞
j=2,4,6…
Aj
(
2 + Nj sin NjX + 2 cos NjX
)
)2
dX
⎤
⎦ (72)
T. Salem et al.

14. Thin-Walled Structures 159 (2021) 107264
14
When buckling mode transitions happen, the force-displacement curves
sharply dropped in the axial force due to the displacement-control
loading conditions. The theoretical model accurately captures the
snap-through of the postbuckled CNT-MM in the force-displacement
relations and deformed shape configurations. Fig. 5(b) compares the
postbuckling behaviors of FGM-MM between the theoretical and nu­
merical results, and good agreements are obtained for the force-
displacement response and postbuckling shape configurations. Fig. 5
demonstrates the accuracy of the developed theoretical model in pre­
dicting the postbuckling performance of the CNT-MM and FGM-MM.
The theoretical and numerical results show clear differences of the
postbuckling response resulted in the material properties of the CNTs
and FGMs, which indicate the possibility of using the material properties
(e.g., material functions of the CNTs and FGMs) to harness the post­
buckling response.
4.3. Comparison of the reduced theoretical and numerical models with the
existing study
In this section, the CNT-MM and FGM-MM models are reduced to the
uncorrugated, isotropic beams (i.e., without the corrugation in the MM
and the composition in the FGMs and CNTs) under the bi-walls boundary
conditions. The reduced theoretical and numerical results are compared
with the postbuckling response reported in the existing study [46].
Table 3 shows the geometric and material factors used in the compari­
son. Fig. 6 compares the postbuckling response for the uncorrugated,
isotropic beams under the bilateral constraints between the theoretical
and numerical models in this study and the theoretical results in the
literature [46]. In particular, the developed CNT-MM and FGM-MM
models are simplified to compare the force-displacement relations
with the existing study. The findings indicate that the reported theo­
retical models accurately predict the stiffness (i.e., slope), and more
importantly, the postbuckling snap-throughs are obtained happening
under the same axial force and displacement. The satisfactory agree­
ments demonstrate the accuracy of the reported theoretical models.
5. Maneuverability of the postbuckling response for the FGM-
MM and CNT-MM
In this section, the presented theoretical models are used to inves­
tigate the influence of the material functions in the FGMs and CNTs and
the corrugation in the MM on the postbuckling response (i.e., force-
displacement relations and postbuckling shape configurations) of the
CNT-MM and FGM-MM. Note that the material influence is studied in
terms of the volume fraction Vfor the CNTs (see Eq. (8)) and index bfor
the FGMs (see Eq. (6)), and the geometric influence is investigated with
respect to the diameter Dand height Hof the corrugation in the MM (see
Fig. 1).
5.1. Material influence of the FGMs and CNTs
The influence of the material functions is studied on the postbuckling
response of the CNT-MM and FGM-MM. The beam length L, width Wb,
thickness t, constraints net gap h0, and matrix Young’s modulus EMare
fixed as 2 mm, 0.4 mm, 10 μm, 30 μm, and 130 GPa, respectively. Fig. 7
(a) shows the effect of the volume fraction Von the force-displacement
relations of the CNT-MM. Since different volumes have different effi­
cient parameters [27], the effective bending stiffness of the CNT-MM is
critically changed, especially for the CNTs with extremely high
stiffness-to-weight ratios. Fig. 7(b) and (c) display the effect of the index
bon the force-displacement curves of the FGM1-MM and FGM2-MM,
respectively, using the FGMs functions defined in Eq. (75). Although
the index b has shown the influence on the effective bending stiffness,
less significance is observed comparing with V. The variations of the
buckling mode transitions follow the same patterns as the
force-displacement relations, which indicates the programmability of
the postbuckling behavior by changing the material properties. Fig. 7(d)
shows the postbuckling shape configurations for the CNT-MM in the first
buckling model (Φ1) with V = 2.8%, the third buckling model (Φ3) for
V = 12%, and the fifth buckling mode (Φ5) for V = 28%. The presented
buckling snap-through events are highlighted in Fig. 7(a). Fig. 7(e) and
(f) present the postbuckling shape configurations for the FGM1-MM and
FGM2-MM, respectively, in the first buckling model (Φ1) for b = 0.3,
the third buckling model (Φ3) for b = 0.6, and the fifth buckling model
(Φ5) for b = 0.9.
5.2. Geometric influence of the cylindrical corrugation
Fig. 8 depicts the effect of the cylindrical corrugation on the post­
buckling response of the CNT-MM and FGM-MM. Fig. 8(a) indicates the
influence of the diameter-to-height ratio (i.e., D
H = 0.2 − 5) of the
corrugation on the force-displacement relations of the CNT-MM. Note
that the volume fraction is fixed as V = 12%. Fig. 8(b) and (c) show the
effects of D
Hon the
The D
Hratios are: 20 μm-to-100 μm, 20 μm-to-20 μm, and 100 μm-to-20
μm. In the same manner, Fig. 8(d)-8(f) present the deformed shape
configurations of the postbuckling events highlighted in Fig. 8(a)-8(c). It
can be seen that changing the materials from the FGMs to CNTs signif­
icantly affects the postbuckling characteristics, while adjusting the
FGMs functions is likely to play a less important role.
5.3. Material and geometric influences on the FGM-MM and CNT-MM
Next, the material and geometric influences are investigated and
compared on the postbuckling performance of the CNT-MM and FGM-
MM. Fig. 9(a) compares the force-displacement relations for the CNT-
MM with 1) diameter-to-height ratio is D
H = 0.2and volume fraction is
V = 2.8%, 2) D
H = 1and V = 12%, and 3) D
H = 5and V = 28%. Fig. 9(b)
and (c) compares the force-displacement relations for the FGM1-MM
and FGM2-MM, respectively, with 1) D
H = 0.2and b = 0.3, 2) D
H = 1and
b = 0.6, and 3) D
H = 5and b = 0.9.According to the comparison, the
volume fraction of the CNTs provide a more significant influence on the
CNT-MM, since the effective bending stiffness is not critically decreased
when D
His reduced from 1 to 0.2 and Vis increased from 2.8% to 12%.
However, the FGM-MM are found to be more sensitive to D
Hrather than b.
Different geometric and material functions lead to different postbuck­
ling response for the CNT-MM and FGM-MM. Fig. 9(d) presents the
Fig. 10 shows the variation of the maximum axial force during the
buckling mode transitions (i.e., FΦ3for the Φ1-Φ3 snap-through) for the
(a) CNT-MM, (b) FGM1-MM and (c) FGM2-MM. Different corrugation
factors are used to investigate the influences of the material and geo­
metric functions. In particular, the corrugation factor is defined as
[
2(D+G)2
(W+H− 2tMM)
W((D+G)2
+πDH)
]2
based on Eq. (2). It can be seen that higher corru­
gation factors (i.e., more significant corrugation) and larger volume
fraction and index b(i.e., more obvious functional materials) tend to
provide higher controllability on the buckling mode transitions. In
addition, the CNTs are likely to be more maneuverable over the FGMs on
the postbuckling response of the bilaterally constrained CNT-MM and
FGM-MM.
6. Conclusions
In this study, we reported the microscale, extensible mechanical
metamaterials made by functionally graded materials (FGM-MM) and
carbon nanotubes (CNT-MM). The corrugated metastructures were ob­
tained using the material method (i.e., the FGMs and CNTs) and struc­
tural method (i.e., the corrugations). The postbuckling response under
the bilateral constraints was theoretically and numerically investigated.
Systematic studies were conducted to unveil the influences of the
T. Salem et al.

15. Thin-Walled Structures 159 (2021) 107264
15
material functions and corrugation geometries on the postbuckling
performance of the FGM-MM and CNT-MM. The findings indicate that
more significant corrugation and more critical functional materials
provided higher controllability on the buckling mode transitions of the
FGM-MM and CNT-MM. The CNTs were found with higher maneuver­
ability over the FGMs on the postbuckling response. The presented FGM-
MM and CNT-MM can be fabricated using the 4D printing techniques (i.
e., 3D printing of the functionally graded materials), and the experi­
mental results can be used to validate the developed theoretical and
numerical models. The reported FGM-MM and CNT-MM open a striking
path to program the mechanical response of corrugated metastructures
for advanced applications.
Author statement
Data are available upon reasonable request from the corresponding
author.
Declaration of competing interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgements
This study is supported in part by the Fundamental Research Funds
for the Central Universities, China (2020-KYY-529112-0002). P.J. ac­
knowledges the financial support of the startup funding at the Zhejiang
University, China.
Appendix A. Effective Young’s Modulus of the FGM-MM and CNT-MM
⎧
⎨
⎩
VMM = LWtMM
(
1 +
πDH
(D + G)2
)
Vhb = 2LtMM(W + H − 2tMM)
. (A1)
Taking Eq. (A1) into Eq. (1) leads to Eq. (2). In addition, the weight factor ζMMof the cylindrical corrugation for the FGM-MM and CNT-MM is
numerically calibrated using Abaqus R2016x. The mean value is 0.43 for the parameters in Table 2.
Appendix B. Supplementary data
Supplementary data to this article can be found online at https://doi.org/10.1016/j.tws.2020.107264.
References
[1] X. Yu, J. Zhou, H. Liang, Z. Jiang, L. Wu, Mechanical metamaterials associated with
stiffness, rigidity and compressibility: a brief review, Prog. Mater. Sci. 94 (2018)
114–173.
[2] D.D. Vescovo, I. Giorgio, Dynamic problems for metamaterials: review of existing
models and ideas for future research, Int. J. Eng. Sci. 80 (2014) 153–172.
[3] A. Rafsanjani, A. Akbarzadeh, D. Pasini, Snapping mechanical metamaterials under
tension, Adv. Mater. 27 (2015) 5931–5935.
[4] T. Frenzel, M. Kadic, M. Wegener, Three-dimensional mechanical metamaterials
with a twist, Sci 358 (2017) 1072–1074.
[5] S.C. Han, J.W. Lee, K. Kang, A new type of low density material: Shellular, Adv.
Mater. 27 (2015) 5506–5511.
[6] X. Zheng, H. Lee, T.H. Weisgraber, M. Shusteff, J. DeOtte, E.B. Duoss, J.D. Kuntz,
M.M. Biener, Q. Ge, J.A. Jackson, S.O. Kucheyev, N.X. Fang, C.M. Spadaccini,
Ultralight, ultrastiff, mechanical metamaterials, Sci 344 (2014) 1373–1377.
[7] K. Bertoldi, M. Reis, S. Willshaw, T. Mullin, Negative Poisson’s ratio behavior
induced by an elastic instability, Adv. Mater. 22 (2010) 361–366.
[8] L.R. Meza, S. Das, J.R. Greer, Strong, lightweight, and recoverable three-
dimensional ceramic nanolattices, Sci 345 (2014) 1322–1326.
[9] J. Bauer, J.R. Meza, T.A. Schaedler, R. Schwaiger, X. Zheng, L. Valdevit,
Nanolattices: an emerging class of mechanical metamaterials, Adv. Mater. 29
(2017) 1701850.
[10] M.J. Mirzaali, R. Hedayati, P. Vena, L. Vergani, M. Strano, A.A. Zadpoor, Rational
design of soft mechanical metamaterials: independent tailoring of elastic properties
with randomness, Appl. Phys. Lett. 111 (2017), 051903.
[11] S. Janbaz, M. McGuinness, A.A. Zadpoor, Multimaterial control of instability in soft
mechanical metamaterials, Phys. Rev. Appl. 9 (2018), 064013.
[12] P. Jiao, H. Hasni, N. Lajnef, A.H. Alavi, Mechanical metamaterial piezoelectric
nanogenerator (MM-PENG): design principle, modelling and performance, Mater.
Des. 187 (2020) 108214.
[13] S. Basu, M. Francoeur, Penetration depth in near-field radiative heat transfer
between metamaterials, Appl. Phys. 99 (2011) 59.
[14] K. Joulain, J. Drevillon, P. Philippe Ben-Abdallah, Noncontact heat transfer
between two metamaterials, Phys. Rev. B. 81 (2010) 165119.
[15] Q. Chen, X. Zhang, B. Zhu, Design of buckling-induced mechanical metamaterials
for energy absorption using topology optimization, Struct. Multidiscip. Optim. 58
(2018) 1395–1410.
[16] M. Mohsenizadeh, F. Gasbarri, M. Michael Munther, A. Beheshti, K. Davami,
Additively-manufactured lightweight metamaterials for energy absorption, Mater.
Des. 139 (2018) 521–530.
[17] S. Li, D.M. Vogt, D. Rus, R.J. Wood, Fluid-driven origami-inspired artificial
muscles, Proc. Nat. Assoc. Sci. 114 (2017) 13132–13137.
[19] Y. Jin, Engineering plasmonic gold nanostructures and metamaterials for
biosensing and nanomedicine, Adv. Mater. 24 (2012) 5153–5165.
[20] A. Rafsanjani, K. Bertoldi, A.R. Studart, Programming soft robots with flexible
mechanical metamaterials, Sci. Rob. 4 (2019), eaav7874.
[21] E. Taati, Analytical solutions for the size dependent buckling and postbuckling
behavior of functionally graded micro-plates, Int. J. Eng. Sci. 100 (2016) 45–60.
[22] B. Kieback, A. Neubrand, H. Riedel, Processing techniques for functionally graded
materials, Mater. Sci. Eng. 139 (2003) 81–106.
[24] V. Birman, L.W. Byrd, Modeling and analysis of functionally graded materials and
structures, Appl. Mech. Rev. 60 (5) (2007) 195–216.
[25] E. Müller, C. Drašar, J. Schilz, W.A. Kaysser, Functionally graded materials for
sensor and energy applications, Mater. Sci. Eng. 362 (2002) 17–39.
[26] Y W.A. Miyamoto, W.A. Kaysser, A. Rabin, Kawasaki, R.G. Ford, Functionally
Graded Materials: Design, Processing and Applications, Springer, 2013, https://
doi.org/10.1007/978-1-4615-5301-4.
[27] H.S. Shen, Postbuckling of nanotube-reinforced composite cylindrical shells in
thermal environments. Part I: axially-loaded shells, Comp. Struct. 93 (2011)
2096–2108.
[28] G.M. Spinks, S.R. Shin, G.G. Wallace, P.G. Whitten, S.I. Kim, S.J. Kim, Mechanical
properties of chitosan/CNT microfibers obtained with improved dispersion, Sensor.
Actuator. B Chem. 115 (2005) 678–684.
[29] Q. Cheng, J. Bao, J.G. Park, Z. Liang, C. Zhang, B. Wang, High mechanical
performance composite conductor: multi-walled carbon nanotube sheet/
bismaleimide nanocomposites, Adv. Funct. Mater. 19 (2009) 219.
[30] A.M.K. Esawi, K. Morsi, A. Sayed, M. Taher, S. Lanka, Effect of carbon nanotube
(CNT) content on the mechanical properties of CNT-reinforced aluminium
composites, Compos. Sci. Technol. 70 (2010) 2237–2241.
[32] Y.S. Song, J.R. Youn, Modeling of effective elastic properties for polymer based
carbon nanotube composites, Polymer 47 (2006) 1741–1748.
[33] H. Zhu, H. Cao, X. Liu, M. Wang, X. Meng, Q. Zhou, L. Xu, Nacre-like composite
films with a conductive interconnected network consisting of graphene oxide,
polyvinyl alcohol and single-walled carbon nanotubes, Mater. Des. 175 (2019)
107783.
[34] P. Jiao, A.H. Alavi, Buckling analysis of graphene-reinforced mechanical
metamaterial beams with periodic webbing patterns, Int. J. Eng. Sci. 131 (2018)
1–18.
[35] P. Jiao, A.H. Alavi, W. Borchani, N. Lajnef, Micro-composite films constrained by
irregularly bilateral walls: a sizedependent post-buckling analysis, Compos. Struct.
195 (2018) 219–231.
[36] J. Wuite, S. Adali, Deflection and stress behaviour of nanocomposite reinforced
beams using a multiscale analysis, Compos. Struct. 71 (2005) 388–396.
T. Salem et al.

16. Thin-Walled Structures 159 (2021) 107264
16
[37] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, second ed.,
Cambridge University Press, 1999. ISBN-10:0521499119.
[38] M. Mohammad-Abadi, A.R. Daneshmehr, Modified couple stress theory applied to
dynamic analysis of composite laminated beams by considering different beam
theories, Int. J. Eng. Sci. 87 (2015) 83–102.
[39] H. Zhang, X. Xie, Y. Xie, C.M. Wang, P. Jiao, Extensible beam-like metastructures at
the microscale: theoretical and modified Hencky bar-chain modeling, Int. J. Mech.
Sci. 18 (2020) 105636.
[42] H.S. Shen, Nonlinear bending of functionally graded carbon nanotube-reinforced
composite plates in thermal environments, Compos. Struct. 91 (2009) 9–19.
[43] M.M. Abadi, A. Daneshmehr, An investigation of modified couple stress theory in
buckling analysis of micro composite laminated Euler–Bernoulli and Timoshenko
beams, J. Int. Eng. Sci. 75 (2014) 40–53.
[44] W. Xia, L. Wang, L. Yin, Nonlinear non-classical microscale beams: static bending,
postbuckling and free vibration, Int. J. Eng. Sci. 48 (2010) 2044–2053.
[45] Q.S. Li, Buckling of multi-step non-uniform beams with elastically restrained
boundary conditions, J. Constr. Steel Res. 57 (2001) 753–777.
[46] W. Borchani, N. Lajnef, R. Burgueno, Energy method solution for the postbuckling
response of an axially loaded bilaterally constrained beam, Mech. Res. Commun.
70 (2015) 114–119.
[47] R. Barretta, F. Fabbrocino, R. Luciano, F. Marotti de Sciarra, G. Ruta, Buckling
loads of nano-beams in stress-driven nonlocal elasticity, Mech. Adv. Mater. Struct.
27 (11) (2020) 869–875.
[48] R. Apuzzo, S.A. Barretta, R. Faghidian, & F. Luciano, Marotti de Sciarra, Nonlocal
strain gradient exact solutions for functionally graded inflected nano-beams,
Compos. B Eng. 164 (2019) 667–674.
[49] M. Barretta, F. Canađija, Marotti de Sciarra, Nonlocal integral thermoelasticity: a
thermodynamic framework for functionally graded beams, Compos. Struct. 225
(2019) 111104.
[50] R. Acierno, R. Barretta, F. Luciano, Marotti de Sciarra, P. Russo, Experimental
evaluations and modeling of the tensile behavior of polypropylene/single-walled
carbon nanotubes fibers, Compos. Struct. 174 (2017) 12–18.
[51] R. R.M. Barretta, F. Canađija, Marotti de Sciarra, Modified nonlocal strain gradient
elasticity for nano-rods and application to carbon nanotubes Appl. Sci. 9 (3) (2019)
514.
T. Salem et al.