Original Article
Vibration analysis of a
multifunctional hybrid
composite honeycomb
sandwich plate
Journal of Sandwich Structures & Materials
0(0) 1–43
! The Author(s) 2018
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DOI: 10.1177/1099636218820764
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Paul Praveen A1,
Vasudevan Rajamohan2 ,
Ananda Babu Arumugam3 and
Arun Tom Mathew1
Abstract
In the present study, the free and forced vibration responses of the composite sandwich
plate with carbon nanotube reinforced honeycomb as the core material and laminated
composite plates as the top and bottom face sheets are investigated. The governing
equations of motion of hybrid composite honeycomb sandwich plates are derived using
higher order shear deformation theory and solved numerically using a four-noded
rectangular finite element with nine degrees of freedom at each node. Further, various
elastic properties of honeycomb core materials with and without reinforcement of
carbon nanotube and face materials are evaluated experimentally using the alternative
dynamic approach. The effectiveness of the finite element formulation is demonstrated
by performing the results evaluated experimentally on a prototype composite sandwich
plate with and without carbon nanotube reinforcement in core material. Various parametric studies are performed numerically to study the effects of carbon nanotube wt%
in core material, core thickness, ply orientations, and various boundary conditions on
the dynamic properties of composite honeycomb sandwich plate. Further, the transverse vibration responses of hybrid composite sandwich plates under harmonic force
excitation are analyzed at various wt% of carbon nanotubes and the results are
compared with those obtained without addition of carbon nanotubes to demonstrate
1
School of Mechanical Engineering, Vellore Institute of Technology (VIT), Vellore, India
Centre for Innovative Manufacturing Research (CIMR), VIT, Vellore, India
3
Department of Mechanical Engineering, Sharda University, Greater Noida, India
2
Corresponding author:
Vasudevan Rajamohan, CIMR, VIT, Vellore, India.
Email: vasudevan.r@vit.ac.in
2
Journal of Sandwich Structures & Materials 0(0)
the effectiveness of carbon nanotube reinforcement in enhancing the stiffness and
damping characteristics of the structures. The study provides the guidelines for the
designer on enhancing both the stiffness and damping properties of sandwich structures
through carbon nanotube reinforcement in core materials.
Keywords
Vibration analysis, sandwich plate, honeycomb core, hybrid composites, carbon
nanotube reinforcement, higher order shear deformation theory, finite element method
Introduction
The honeycomb sandwich composite structures are extensively being used in
military, aircraft, railroad cars, marine, automotive industries, and exclusively in
aerospace structures due to their superior structural properties such as high specific
stiffness, better bending stiffness, greater shear strength, and enriched damping
ratio which could not be achievable in the isotropic and composite plates. The
laminated composite sandwich honeycomb core becomes viable to the out-of-plane
shear and pressure loads and transfers the load between the two face sheets which
shows the better load-carrying capacity of the sandwich plate structures. The
vibration frequency spectrum of the sandwich composite plates affects the performance of the structure under the influences of number of layers, ply orientations,
fiber type, quality of resin, and core material and thickness.
Meunier and Shenoi [1] investigated analytically the dynamic responses of sandwich composite using the refinement of Reddy’s HSDT. It was shown that the
variation of viscoelastic properties of constitutive materials with respect to temperature and frequency significantly influences the dynamic response of sandwich
structure. Nayak et al. [2] studied the free vibration responses of the isotropic,
orthotropic, anisotropic, and sandwich composite plates with polyvinyl chloride
face sheets and HEREX C70 foam core using HSDT. It was observed that the
angle ply laminates significantly influence the natural frequency compared to those
of the structures with cross ply laminates. The increase in core thickness ratio
increases the natural frequency of the sandwich plate due to the influence of flexural rigidity. Abbadi et al. [3] investigated the static characteristics of honeycomb
sandwich panel by using experimental, analytical, and numerical methods. The
equivalent material properties of honeycomb core were identified by using the
homogenized approaches. The modeling of honeycomb sandwich structure was
carried out by adopting unified formulation (general kinematic model) subjected
to four-point bending test. The results show the models developed based on HSDT
Praveen et al.
3
are found to be very accurate than those obtained using CLT and FSDT theories.
Zhen et al. [4] demonstrated the accuracy of using C0 finite elements in HSDT to
identify the free vibration responses of composite sandwich plates with soft core
compared to those of three-dimensional elasticity solutions. Han et al. [5] investigated numerically the effectiveness of the sandwich core having the combination of
honeycomb with hybrid corrugation made of aluminum materials. It was observed
that the compressive strength and energy absorption could be improved considerably by the combined aluminum honeycomb corrugation. Vemuluri et al. [6]
presented numerically the vibration responses of sandwich plate with optimal
layouts of various partially treated magnetorheological elastomer (MRE) under
various magnetic fields. The results revealed that the modal loss factors and natural frequencies of the sandwich plate are greatly influenced by the MRE pockets
under magnetic field intensities.
Manoharan et al. [7, 8] investigated numerically and experimentally the free and
forced vibration responses of the laminated sandwich composite with fully and
partially treated magnetorheological (MR) fluid as the core layer and GFRP laminates as the face sheets. Vemuluri and Vasudevan [9] analyzed numerically and
experimentally the dynamic responses of the MR elastomer-based sandwichtapered composite plate. It was demonstrated that the stiffness and damping
could be influenced not only by the intensity of magnetic field but also with
changes in ply orientations of face layers, changes in aspect ratio, and taper
angle of the face layers of MRE sandwich plate. Arumugam et al. [10] studied
numerically the dynamic and instability responses of the rotating MR sandwich
composite plate under in-plane loading condition based on classical laminated
plate theory. The results revealed that the natural frequencies increase by increasing the speed of rotation and magnetic field while the instability regions of the
primary and secondary portion decrease with increase in magnetic field and rotation speed of the structures. Even though many researchers analyzed various core
materials such as honeycomb, MR fluid, and MR elastomer, the use of these
materials is insufficient due to their restricted viscoelastic and damping properties
of the fiber-fortified composite structures. After the revelation of carbon nanotubes (CNTs) by Iijima [11], their distinct physical properties have attracted a
massive attention from various researchers to fabricate the cross breed composite
materials with CNT fortification to accomplish the improved mechanical properties of the structure.
Zhou et al. [12] identified the substantial improvement in damping factors of the
half breed composite structures with the reinforcement of minimal quantity of
CNTs in the polymer composite. Various examinations have revealed that the
random dispersion of multiwalled CNTs (MWCNT) in the composite structure
enhances the mechanical, shear, flexural, damping, and thermal properties of the
polymer-based composites [13]. Gojny et al. [14] observed experimentally the
enhanced material properties of the composites with the reinforcement of singlewall CNTs (SWCNT) and double-wall CNTs (DWCNT). It was shown that the
strength, stiffness, and fracture toughness could be improved significantly by 10,
4
Journal of Sandwich Structures & Materials 0(0)
15, and 43%, respectively, by the addition of the 0.5 wt% of DWCNT than those
of the SWCNT. Warrier et al. [15] examined experimentally the effectiveness of the
reinforcement of fiber sizing with 0.5% CNT in the polymer-based composites.
The result shows that glass transition temperature and crack initiation fracture
toughness resistance increase with decrease in thermal expansion coefficient and
fracture toughness of the crack propagation. Khan et al. [16] examined experimentally a substantial improvement in the damping performance of carbon fiber reinforced composites with the addition of 0.5 and 1% MWCNTs in the
CFRP composites.
Godara et al. [17] inspected the shear strength responses of hybrid composite
through different techniques such as fiber sizing and CNT reinforced resin. It was
demonstrated that the addition of CNTs during fiber sizing increases the interfacial
shear strength by more than 90% compared to those of CNT reinforcement in
resin. Zhou et al. [18] investigated numerically and experimentally the shear
strength and fracture behavior of the CNT reinforced CFRP composite. It was
demonstrated that the strength and fracture toughness of the composite could be
improved significantly by the addition of 0.3 wt% of CNTs in polymer matrix than
those of CNTs in the fiber sizing of carbon fiber. Li et al. [19] presented experimentally the enrichment of damping property for CFRP composites. It was shown
that the internal sliding friction significantly improves the damping property of the
hybrid composites through multistick slip action of CNTs and multiscaled microstructures of the fibers.
Godara et al. [20] examined the mechanical responses of functionalized and
nonfunctionalized CNTs with 0.5 wt% dispersed in the CFRP composites.
It was shown that the functionalization of DWCNTs significantly decreases the
thermal expansion coefficient of hybrid composites by 32%. However, pristine
MWCNTs improve the fracture toughness of hybrid composites by 80%. Ma
et al. [21] observed the effectiveness of agglomeration, dispersion, and interfacial
agreement between the functionalized CNT and resin matrix. It was shown that the
flexural and thermomechanical properties are significantly increased by the functionalized CNTs than those of CNTs without functionalization. Rahaman and
Kar [22] inspected the characterization and enhanced dynamic properties of
CNT-coated E-glass fibers reinforced in the polymer-based laminated composites.
The result reveals that composite with CNT-coated fibers could increase the glass
transition temperature and storage modulus. Farrash et al. [23] inspected the
responses of free vibration of neat epoxy, glass/epoxy, and carbon/epoxy, without
and with 0.25 wt% reinforcement of CNT in the composite structures. The result
shows that the damping ratio of CNT/glass/epoxy composite decreases by 12.3%
while the natural frequency increases by 9.4%. Further, an increment of 31.5% in
damping ratio and 13.9% in natural frequency was observed when CNTs were
reinforced in carbon/epoxy composites. Jakkamputi and Vasudevan [24] identified
experimentally the free vibration responses of GFRP composites without and with
0.5 wt% CNT reinforcement in composite structure under clamped–clamped and
clamped–free boundary conditions exposed to higher thermal environment. The
Praveen et al.
5
result showed that the fundamental natural frequency significantly increases
by 18.46 and 17.43% over the composite without CNT addition under CC and
CF end conditions, respectively, at ambient temperature. Moreover, the fundamental natural frequency of CNT-GFRP decreases by 11.90 and 3.47%, whereas
damping factor increases by 35.41 and 22.64% under CC and CF conditions,
respectively, when the temperature raises from 30 to 60 C.
Wang and Shen [25] studied analytically the nonlinear flexural and vibration
responses of the CNT reinforced face sheets of the sandwich plates. The natural
frequencies of the sandwich plate increase considerably by the addition of CNTs in
face sheets and the thickness ratio of core/face sheet. Natarajan et al. [26] investigated the flexural and free vibration responses of the sandwich plates with the
reinforcement of CNT in the face sheets under mechanical and thermal loading
conditions using HSDT. The deflection of the CNT reinforced face sheets
decreases with increase in the CNT volume fraction. Mohammadimehr and
Mostafavifar [27] examined the free vibration responses of the functionally
graded (FG) CNT reinforcement into the face sheets of the sandwich plate subjected to magnetic field and temperature-dependent material properties based on
HSDT. The natural frequency and stiffness of the sandwich plate with FG-CNT in
face sheets increase significantly due to the influence of the parameters of the
material scale length, size effects, and the magnetic field intensity.
From the literature review, it can be concluded that various research works have
been focused on evaluation of the dynamic properties of sandwich structures made
up of face isotropic/orthotropic face sheets embedded without and with the mixture
of CNT and polymer foam filled in the hexagonal honeycomb cores and isotropic/
orthotropic materials. However, the dynamic analysis of multifunctional hybrid composite sandwich plates with CNT reinforcement in the corrugated honeycomb cores
has not yet been explored. The hybrid corrugated honeycomb cores’ sandwich composite plates would enhance the specific stiffness, bending stiffness, shear strength,
and damping ratio simultaneously without significant change in mass of the structure.
In this present study, the free and forced vibration responses of the composite
sandwich plate with CNT reinforced honeycomb as the core material and laminated composite plates as the top and bottom face sheets are investigated. The
governing equations of motion of hybrid composite honeycomb sandwich plates
are derived using higher order shear deformation theory and solved numerically.
Further, honeycomb core materials with and without reinforcement of CNT are
being manufactured and analyzed experimentally to identify the shear and loss
moduli of the hybrid composite material using the alternative dynamic approach.
The effectiveness of the finite element formulation is endorsed by performing the
experimental tests on the prototype of hybrid composite sandwich plates and comparing the numerical results in terms of natural frequencies and loss factors. Also,
the effects of CNT wt% in core material, core thickness, ply orientations, and
various boundary conditions on the dynamic properties of composite honeycomb
sandwich plate are investigated. Further, the transverse vibration responses of
hybrid composite sandwich plates under harmonic force excitation are analyzed
6
Journal of Sandwich Structures & Materials 0(0)
at various wt% of CNTs and the results are compared with those obtained without
addition of CNTs to demonstrate the effectiveness of CNT reinforcement in
enhancing the stiffness and damping characteristics of the structures.
Mathematical modeling of the multifunctional hybrid sandwich
composite plate
The sandwich composite plate comprising hybrid honeycomb as the core layer
made of CNT reinforced GFRP materials (CNT-GFRP) and uniform composite
laminated plates as the face sheets are considered for the development of the
mathematical modeling based on higher order shear deformation theory
(Figure 1). The assumptions considered for development of the governing equation
of motion are as follows: (i) each layer obeys the hooks law, (ii) all layers are
bonded together without any slip, and (iii) the core transmits the transverse shear
stress only. Since the honeycomb core material is stiff in shear but soft commonly,
the Young’s moduli of the core material are closely negligible than that of the
composite face layers. The normal stresses in the honeycomb layer are
also ignored.
The thickness of the face layers, ht and hb and honeycomb layer, hc are considered to be small than those of the length L and width B of the sandwich plate
structure. The damping at the top and bottom layer is assumed to be negligible and
the dissipation of energy at the core layer is assumed to exist because of the
Figure 1. (a) Illustration of number (N) of composite face sheet layers into an equivalent single
layer of composite plate and (b) hybrid honeycomb composite sandwich plate.
7
Praveen et al.
transverse shear between the top and middle layers and bottom and middle layers
only. Moreover, the transverse displacement at any given cross section is assumed
to be uniform along the sandwich plate.
Formulation of constitutive equation of top and bottom face sheet laminate
In general, the displacement field equation of the face sheet laminates can be
presented with higher order form as
uðx; y; zÞ ¼ uo ðx; yÞ þ zhx ðx; yÞ þ z2 kx ðx; yÞ þ z3 dx ðx; yÞ
vðx; y; zÞ ¼ vo ðx; yÞ þ zhy ðx; yÞ þ z2 ky ðx; yÞ þ z3 dy ðx; yÞ
(1)
wðx; y; zÞ ¼ w0
where u and v are the displacement field along x and y axes and w is the transverse
displacement.
The traction-free boundary condition of the top and bottom layers of the composite face layers is presented as
h
rXZ x;
¼0
2
h
¼0
rYZ x;
2
(2)
where rXZ and rYZ denote the stress developed along xz and yz planes.
By substituting the traction-free boundary condition as presented in equation
(2) in the displacement field equation (1), the number of independent variables is
reduced from nine to seven such that the displacement fields of the composite
sandwich plate can be represented as
ui ¼ u0i þ z hx
"
vi ¼ v0i þ z hy
4z2 @w0
þ
h
x
3h2 @x
#
4z2 @w0
ði ¼ t; bÞ
þ hy
3h2 @y
@w0
þ hx ¼ /x
@x
@w0
þ h y ¼ /y
@y
(3)
(4)
where hx and hy are the transverse functions with respect to the x and y axes, /x
and /y are the wrapping functions which include transverse normal rotations with
8
Journal of Sandwich Structures & Materials 0(0)
respect to the x and y axes. The wrapping function in equation (4) is substituted
into equation (3) to get the simplified displacement field equation (5)
ui ¼ u0i þ z hx
vi ¼ v0i þ z hy
4z2
/
ð
Þ
3h2 x
4z2
ði ¼ t; bÞ
/
3h2 y
(5)
The displacement field equations have been derived from equation (5) for the
top (ut , vt ) and bottom (ub , vb ) face sheets along x and y axes on the mid-plane of
the sandwich plate. Further ut , vt , ub , and vb denote the axial deformation along x
and y axes represented as a function of the distance, z, from the neutral axis of the
composite face sheets, respectively.
The transverse deformation (w) along z axis can be written as
ui ¼ u0i þ zhx
vi ¼ v0i þ zhy
w ¼ w0
4z3
/
3h2 x
4z3
/ ði ¼ t; bÞ
3h2 y
(6)
where uot , vot , uob , and vob are the top and bottom mid-plane deformations of the
composite face sheets along x and y directions. hx and hy are the rotation of normal
to mid-plane about x and y axes. /x and /y are the higher order terms to signify
the transverse deformation along x and y axes. w0 is the transverse deformation
field along the mid-plane of the sandwich plate.
Once the displacement field has been derived, the linear strain–displacement
relationships for the composite sandwich laminated plate with respect to global
axes x, y, and z can be expressed as follows:
The in-plane strain for the face sheets is given by
2x ¼ 20x þzj0x þ z3 j2x
2y ¼ 20y þzj0y þ z3 j2y
(7)
2xy ¼ 20xy þzj0xy þ z3 j2xy
The transverse shear strain of the face sheets is given by
2xz ¼ 2sxz þz2 jsxz
2yz ¼ 2syz þz2 jsyz
(8)
9
Praveen et al.
where
T
T
@v0
0 @v0 @u0
20 ¼ 20x ; 20y ; 20xy ¼ @u
;
;
þ
@x @y @y
@x
T
T
@hy
x @hy @hx
;
;
þ
j0 ¼ j0x ; j0y ; j0xy ¼ @h
@x
@x @y @y
T
@/
4 @/x
j2 ¼ j2x ; j2y ; j2xy ¼
; 3h42 @yy ;
3h2 @x
T
@w
2s ¼ 2sxz ; 2syz T ¼ hx þ @w
@x ; hy þ @y
T
ks ¼ ksxz ; ksyz T ¼ h24 /x ; h24 /y
4
3h2
@/x
@y
þ
@/y
@x
T
where 20 is the in-plane strain, k0 , k2 are the mid-plane curvatures, and 2s is the
transverse shear strain. Then, the strain displacement fields are simplified for perspective representation.
Then, the general constitutive relation for the stress–strain relationship of the
orthotropic composite lamina with respect to x, y, and z axes is given as
frg ¼ ½Q f2g
(9)
9 2
9
8
38
Q11 Q12 Q16 >
>
=
=
< rX >
< 2X >
7
6
¼ 4 Q12 Q22 Q26 5 2Y
rY
>
>
>
>
;
;
:
:
rXY
Q16 Q26 Q66
2XY
(
) "
#(
)
rXZ
Q55 Q54
2XZ
¼
rYZ
Q54 Q44
2YZ
(10)
where ½Qij is the coefficient of transformed material
1
½Qij ¼ ½T1 Qij k ½T1
1
½Qij ¼ ½T2 Qij k ½T2
T
T
ði; j ¼ 1; 2; 6Þ
ði; j ¼ 5; 4Þ
(11)
The transformation matrix T1 and T2 have been used to obtain the material properties on the fiber direction due to variation in the sequence of fiber arrangements
2
6
½T1 ¼ 6
4
½T2 ¼
"
sin2 h
cos2 h
sin2 h
coshsinh
cosh
sinh
sinh
cosh
cos2 h
coshsinh
#
2coshsinh
3
7
2coshsinh 7
5
cos2 h sin2 h
(12)
10
Journal of Sandwich Structures & Materials 0(0)
where T1 and T2 are the stress transformation matrices
2
Q11
Q12
0
3
6
7
½Qij k ¼ 6
0 7
4 Q12 Q22
5
Q66
0
0
"
#
Q55
0
Qij k ¼
0
Q44
ði; j ¼ 1; 2; 6Þ
(13)
ði; j ¼ 5; 4Þ
where
½Q11 ¼
½Q22 ¼
1
1
!
!
ð 12 E2 Þ
E1
; ½Q12 ¼
;
1 ð 12 21 Þ
ð 12 21 Þ
!
E2
; ½Q66 ¼ ðG12 Þ; ½Q55 ¼ ðG13 Þ; ½Q44 ¼ ðG23 Þ
ð 12 21 Þ
The stiffness coefficients [Qij] in equation (13) are related to engineering constants. where E1 and E2 are the Young’s modulus along the fiber and transverse
directions, respectively. v12 and v21 are the Poisson’s ratio; G12, G13, and G23 are the
shear moduli in material axes; and h is the ply sequence of the laminate.
The force, moment, and higher order terms are developed from the stress–strain
relationship, and the stress field of resultants for the composite sandwich plate is
presented as
8 9 2
38 9
Aij Bij Eij < 20 =
<N=
(14)
M ¼ 4 Bij Dij Fij 5 k0
: 2;
: ;
Eij Fij Hij
P
k
where N and M are the normal and bending moment force vectors
"
Q
R
#
¼
"
Asij
Dsij
Dsij
Fsij
#(
2s
ks
)
(15)
where P, Q, and R are the higher order terms.
The stiffness matrix of the composite face sheets connecting the stress and strain
is as follows
Z h=2
Qij 1; z; z2 ; z3 ; z4 ; z6 dz ði; j ¼ 1; 2; 6Þ
ðAij ; Bij ; Dij ; Eij ; Fij ; Hij Þ ¼
h=2
(16)
Z h=2
s
s
s
2 4
Qij 1; z ; z dz
Aij ; Dij ; ; Fij ¼
ði; j ¼ 5; 4Þ
h=2
11
Praveen et al.
Figure 2. Schematic illustration of undeformed and deformed configurations of the honeycomb
sandwich plate model.
Formulation of constitutive equation of the honeycomb core layer
Honeycomb as the core layer made of without and with CNT reinforced GFRP is
considered in a laminated composite sandwich plate as the core material. The
displacement field has been derived based on the assumption that the core transmits only the transverse shear stress. The shear strain energy formulation of the
honeycomb core has been developed under the shear to shear condition based on
the top and bottom degrees of freedom (DOF) of the skin sheets. The diagrammatic representation of the sandwich plate before and after deformation under
transverse deformation is shown in Figure 2
@uc @w
þ
@z @x
@vc @w
ðcyz Þc ¼ @z þ @y
ðcxz Þc ¼
(17)
Shear deformation with respect to z direction with respect to top and bottom of
the skin is presented as
ht þ hb
ht þ hb
hx þ
/x
2hc
6hc
ht þ hb
ht þ hb
hy þ
/y
2hc
6hc
@uc uot u0b
¼
@z
hc
@vc vot v0b
¼
@z
hc
Shear along xz and yz direction is given by
ðcxz Þc ¼
ðcyz Þc ¼
uot
u0b
hc
vot
v0b
hc
ht þ hb
ht þ hb
@w0
hx þ
/x þ
2h
6h
@x
c
c
ht þ hb
ht þ hb
@w0
hy þ
/y þ
2hc
6hc
@y
12
Journal of Sandwich Structures & Materials 0(0)
Strain energy formulation for the laminated composite sandwich plate
Strain energy developed for top and bottom skin due to the axial deformation can
be presented as
Z
1
Ui ¼
(18)
frgT ð2Þ dv ði ¼ t; bÞ
2 v
Strain energy for top and bottom face sheet due to axial deformation. The strain energy
due to axial deformation on the top and bottom face sheet is derived by subsequent
substitution of equations (7) and (9) into equation (18) and can be presented as
08
9T
>
>
> @u0t
>
>
B>
>
>
>
@x
B>
>
>
>
B>
> @v0t
>
>
B>
>
>
>
B>
>
>
> @y
>
B>
>
>
B>
>
>
@u
@v
0t
0t
>
>
B>
>
þ
>
B>
>
>
@y
@x
>
B>
>
>
>
>
B>
>
@h
x
>
B>
>
>
>
B>
>
>
2
>
>
@x
>
>
Z LZ BB
Aij
<
=
B
@h
2
y
1
B
4 Bij
Ut ¼
B
B B> @y
>
2 0
>
>
2
Eij
B>
> @hx @hy >
>
>
B>
>
>
þ
>
B>
>
>
@x >
B>
> @y
>
>
B>
> @/x
>
>
B>
>
>
>
>
B>
>
@x
>
>
B>
>
>
>
@/
>
B>
y
>
>
>
B>
>
>
> @y
>
B>
>
>
B>
>
>
>
@>
@/
@/
y
>
>
x
>
>
þ
:
;
@y
@x
08
9T
>
>
> @u0b
>
>
B>
>
>
>
@x
B>
>
>
>
B>
> @v0b
>
>
B>
>
>
>
>
B>
>
> @y
>
B>
>
>
B>
>
>
@u
@v
0b
0b
>
>
B>
>
þ
>
B>
>
>
@y
@x
>
B>
>
>
>
>
B>
>
@h
x
>
B>
>
>
>
B>
>
>
2
>
>
@x
>
>
Z LZ BB
Aij
<
=
B
@h
2
y
1
B
4 Bij
Ub ¼
B
B B> @y
>
2 0
>
>
2
Eij
B>
> @hx @hy >
>
>
B>
>
>
þ
>
B>
>
>
@x >
B>
> @y
>
>
B>
> @/x
>
>
B>
>
>
>
>
B>
>
@x
>
>
>
B>
>
>
@/
>
B>
y
>
>
>
B>
>
>
> @y
>
B>
>
>
B>
>
>
>
@>
@/
@/
y
>
>
x
>
>
þ
:
;
@y
@x
Bij
Dij
Fij
Bij
Dij
Fij
8
91
@u0t
>
>
>
>
>
>
C
>
>
>
>
@x
C
>
>
>
>
C
> @v0t
>
>
>
C
>
>
>
>
C
>
>
> @y
>
C
>
>
>
C
>
>
@u0t @v0t >
>
>
C
>
>
þ
>
>
C
>
>
@y
@x >
>
C
>
>
>
>
C
>
>
@h
x
>
>
C
>
>
>
C
>
>
3 >
>
>
@x
>
>
Eij
< @hy
=C
C
Cdxdy
Fij 5
@y
>
>C
C
>
>
Hij
>
> @hx @hy >
>C
>
>
C
>
>
þ
>
C
>
>
@x >
>
>
> @y
>C
>
>
C
>
>
@/
x
>
>
>
>
>
>C
C
>
>
@x
>
>
>
C
>
>
>
@/
>
>
C
y
>
>
>
>
C
>
>
> @y
>C
>
>
>
>
C
>
>
>
A
@/y >
@/
>
>
>
>
: xþ
;
@y
@x
8
91
@u
>
>
0b
>
>
>
>
C
>
>
>
>
@x
C
>
>
>
>
C
> @v0b
>
>
>
C
>
>
>
>
C
>
>
> @y
>
C
>
>
>
C
>
>
@u0b @v0b >
>
>
C
>
>
þ
>
>
C
>
>
@y
@x >
>
C
>
>
>
>
C
>
>
@h
x
>
>
C
>
>
>
C
>
>
3 >
>
>
@x
>
>
Eij
< @hy
=C
C
Cdxdy
Fij 5
@y
>
>C
C
>
>
Hij
>
> @hx @hy >
>C
>
>
C
>
>
þ
>
C
>
>
@x >
>
>
> @y
>C
>
>
C
>
>
@/x
>
>
>
>
>
>C
C
>
>
@x
>
>
>
C
>
>
>
@/
>
>
C
y
>
>
>
>
C
>
>
> @y
>C
>
>
>
>
C
>
>
>
A
@/y >
@/
>
>
>
>
: xþ
;
@y
@x
(19)
(20)
13
Praveen et al.
where Aij is the extensional stiffness matrix; Bij is the coupling stiffness matrix; Dij
is the bending stiffness matrix; Eij , Fij , and Hij are the stiffness matrix which relate
the higher order terms.
Strain energy for top and bottom face sheet due to transverse deformation. Strain energy
due to transverse deformation on the top and bottom face sheet can be mentioned as
9T
8
@w0
>
>
þ
h
x
>
>
@x
>
>
>
"
>
>
Z LZ B>
@w
=
<
0
2
Asij
1
@y þ hy
U1 ¼
4
B >
Bsij
2 0
/ >
2 >
>
>
h2 x >
>
>
>
>
:
4
/ ;
h2
y
9
8
@w0
>
>
>
>
>
þ hx >
>
>
>
>
@x
>
>
>
>
>
>
>
> @w0
>
# >
>
þ hy >
=
<
Bsij
@y
dxdy
s
>
>
Dij
4
>
>
>
>
/
>
>
>
>
>
h2 x >
>
>
>
>
>
>
4
>
>
>
>
:
/
y ;
2
h
(21)
Asij, Bsij, Dsij are the higher order term stiffness matrix on shear terms.
Strain energy for core layer. Strain energy developed on core material exhibits only
transverse shear stress. It transmits the transverse shear deformation only. Strain
energy developed on core material of the sandwich composite plate can be
expressed as
Z n
o n
o
1
T
T
Uc ¼
ðcxz Þc ðGxz Þc ðcxz Þc þ ðcyz Þc ðGyz Þc ðcyz Þc dv
2 v
@uc @w
(22)
þ
ðcxz Þc ¼
@z @x
@vc @w
ðcyz Þc ¼ @z þ @y
Shear deformation along z direction with respect to top and bottom of the skin
is presented as
ht þ hb
ht þ hb
hx þ
/x
2hc
6hc
ht þ hb
ht þ hb
hy þ
/y
2hc
6hc
@uc uot u0b
¼
@z
hc
@vc vot v0b
¼
@z
hc
Shear along xz and yz direction is given by
ðcxz Þc ¼
uot
u0b
hc
ht þ hb
ht þ hb
@w0
hx þ
/x þ
2hc
6hc
@x
14
Journal of Sandwich Structures & Materials 0(0)
ðcyz Þc ¼
vot
v0b
hc
ht þ hb
ht þ hb
@w0
hy þ
/y þ
2hc
6hc
@y
The total strain energy (UT ) is derived by the summation of the top, bottom,
and core layer of the composite sandwich plate such that
U T ¼ U t þ U b þ 2 ðU 1 Þ þ U c
(23)
Kinetic energy for top and bottom face sheets. The general kinetic energy equation is
presented as
1
T
(24)
T ¼ fdg ½Ii fdg ði ¼ t; bÞ
2
The kinetic energy on the top and bottom face sheet is derived by subsequent
substitution of equation (6) and ½ It;b inertia matrices into equation (24) and can be
presented as
T¼
1
2
Z
L
Z
B
Z
H
2
@u0
@hx
q
þz
H
@t
@t
2
@/
z ðc1 Þ x
@t
0
0
9
8
>
@u0
@hx
@/ >
>
>
>
þz
z3 ðc1 Þ x >
>
>
>
@t >
@t
@t
>
>
>
>
=
<
@/
@hy
@v0
y
3
z ðc 1 Þ
þz
>
@t
@t
@t >
>
>
>
>
>
>
>
>
@w
0
>
>
>
>
;
:
@t
3
@hy
@v0
þz
@t
@t
@/y
z ðc1 Þ
@t
3
@w0
@t
The kinetic energy (Tt, Tb) for top and bottom face sheet laminate can be
presented as
08
9T
91
8
>
>
@u0t >
@u0t >
>
>
>
>
>
>
>
B>
C
>
>
>
>
>
>
>
B>
C
@t
@t
>
>
>
>
>
>
>
B>
C
>
>
> @v >
>
>
>
B>
C
@v
>
>
>
>
0t
0t
>
>
>
B>
C
>
>
>
>
>
>
>
B>
C
>
>
>
>
@t
@t
>
>
>
>
B>
C
>
>
>
>
>
>
B>
C
>
> @w0 >
>
@w0 >
>
>
B>
C
>
>
>
>
>
>
>
B>
C
>
>
>
>
@t >
>
>
C
>
>
> @t >
>
Z lZ B B
=
=
<
<
B @hx
2
1
@hx C
B
Cdxdy
(25)
½It
qB
Tt ¼
C
B
>
>
>
>
2 0
@t
@t
B
C
>
>
>
>
2
>
>
>
>
B>
>
>
>C
>
@h >
@h >
B>
C
>
>
>
>
> y>
> y>
B>
C
>
>
>
>
>
>
>
B>
C
@t
@t
>
>
>
>
>
>
B>
C
>
>
>
>
>
@/x >
@/x >
B>
C
>
>
>
>
>
>
>
>
B>
>
>
>
>
>C
> @t >
> @t >
B>
C
>
>
>
>
>
>
>
>
B>
C
>
>
>
>
>
>
>
@>
A
@/
@/
>
>
y>
y>
>
>
>
>
;
;
:
:
@t
@t
15
Praveen et al.
08
9T
91
8
>
>
>
>
@u
@u
0b
0b
>
>
>
>
>
>
>
B>
C
>
>
>
>
>
>
>
B>
C
@t
@t
>
>
>
>
>
>
>
B>
C
>
>
>
>
>
>
>
B>
C
@v
@v
>
>
>
>
0b
0b
>
>
>
B>
C
>
>
>
>
> @t >
>
B>
C
>
>
>
>
@t
>
>
>
B>
C
>
>
>
>
>
>
>
B>
>
>
>
>
@w
@w
0 >
0 >C
>
B>
C
>
>
>
>
>
>
>
B>
C
>
>
>
>
@t
@t
>
>
>
>
B
>
>
>
>
Z Z B
=
=C
<
<
1 l 2 B
@hx
@hx C
Cdxdy
½ Ib
qB
Tb ¼
B
C
B
>
>
>
>
2 0
@t
@t
B
C
>
>
>
>
2
>
>
>
B>
C
>
>
>
>
>
@hy >
@hy >
B>
C
>
>
>
>
>
>
>
B>
C
>
>
>
>
>
>
>
B>
@t
@t
C
>
>
>
>
>
>
>
B>
C
>
>
>
>
>
>
>
@/
@/
B>
C
>
>
>
>
>
B>
>
>C
> x>
> x>
>
>
>
>
>
>
>
B>
C
@t
@t
>
>
>
>
>
>
>
B>
C
>
>
>
>
>
>
>
@>
A
@/
@/
>
>
y >
y >
>
>
>
>
;
;
:
:
@t
@t
(26)
where Tt and Tb are the kinetic energy of the top and bottom face sheet, q is the
density of each lamina, and ½It and ½Ib are the inertia matrix for the laminated face
sheets on top and bottom of the sandwich plate as presented in Appendix 1.
Kinetic energy for core layer. Kinetic energy developed on core material of the sandwich composite plate can be expressed as
Tc
ðcxz Þc
ðcyz Þc
n
@ ðcxz Þc
@t
n
@ ðcyz Þc
@t
o2
o2
"
2 #
2
Z
c
c
ð
Þ
1
ð
Þ
yz
xz
c
c
dv
q z2
þ
¼
@t
2 v c
@t
uot u0b
ht þ hb
ht þ hb
@w0
¼
hx þ
/x þ
hc
2hc
6hc
@x
vot v0b
ht þ hb
ht þ hb
@w0
hy þ
/y þ
¼
hc
2hc
6hc
@y
(
)
n
o
2 2
1
@u0t
1 @u0b
@wx
@/x
ht þhb
ht þhb
@ w
¼
þ
þ
þ 6hc
þ @x@t
@t
2hc
@t
hc
hc
@t
@t
(
)
n
o
2 2
1
@u0t
1 @u0b
@wy
@/y
ht þhb
ht þhb
@ w
þ
þ
þ @y@t
¼
þ 6hc
2hc
@t
@t
hc
hc
@t
@t
(27)
Total kinetic energy ðTÞ is the summation of kinetic energy on top, bottom, and
core layer of the sandwich plate which can be presented as
T ¼ Tt þ Tb þ Tc
(28)
16
Journal of Sandwich Structures & Materials 0(0)
where Tt , Tb , and Tc are the kinetic energy at the top, bottom, and core of the
sandwich composite plate.
Finite element formulation
The governing differential equations of motion in the finite element form have been
derived by developing a four-noded rectangular element at each corner with nine
DOF at each node being considered for modeling. The DOF comprises of ut , ub , vt ,
vb , w, hx , hy , /x , /y and the corresponding stiffness and mass matrices are developed
based on the formulation of strain and kinetic energies, where ut and ub are the top
and bottom in-plane displacement of composite layers along x direction, vt and vb
are the top and bottom in-plane displacement of composite layers along y direction,
w is the transverse displacement of sandwich plate, hx and hy are the rotational
displacements along x and y axes, and /x and /y are the transverse deformation
along x and y of the composite sandwich plate, respectively.
The in-plane and transverse displacement can be written in the form of nodal
displacement vectors and shape function as given by
9 2
8
>
uot >
Ni
>
>
>
>
> 6
>
>
>
>
>
v
ot >
6 0
>
>
>
6
>
>
>
>
6 0
>
uob >
>
>
6
>
>
>
>
>
>
6
>
>
v
0
>
>
ob
= 6
<
6
6
w
¼ 0
> 6
>
>
> 6
> hx >
>
>
6 0
>
>
>
>
6
>
>
>
>
6 0
hy >
>
>
>
6
>
>
>
>
6
>
> 4 0
> /x >
>
>
>
>
>
;
:/ >
0
y
such that
0
0
0
0
0
0
0
0
Ni
0
0
Ni
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Ni
0
0
Ni
0
0
0
0
0
0
0
0
0
0
0
0
Ni
0
0
0
0
0
0
0
0
0
0
0
0
0
Ni
0
0
Ni
0
0
0
0
0
0
0
0
0
Ni
fug ¼
4
X
½Ni ðui Þt
9
38
>
uot >
>
>
>
>
>
>
7>
>
>
v
ot >
7>
>
>
>
7>
>
>
7>
>
>
u
ob
>
>
7>
>
>
>
>
7>
>
>
v
>
7>
ob
=
<
7
7 w
7>
>
7>
>
> hx >
>
7>
>
>
>
7>
>
>
>
7>
h
>
>
y
>
7>
>
>
>
7>
>
>
> /x >
5>
>
>
>
>
;
:/ >
y
i¼1
4
X
½Ni ðui Þb
fug ¼
(29)
(30)
ði ¼ 1; 2; 3; 4Þ
i¼1
where Ni is the shape functions of the plate element as presented in Appendix 1.
ðui Þt ¼ u0t1 ; v0t1 ; w1 ; hx1 ; hy1 ; /x1 ; /y1 ; u0t4 ; v0t4 ; w4 ; hx4 ; hy4 ; /x4 ; /y4
ðui Þb ¼ u0b1 ; v0b1 ; w1 ; hx1 ; hy1 ; /x1 ; /y1 ; u0b4 ; v0b4 ; w4 ; hx4 ; hy4 ; /x4 ; /y4
T
T
17
Praveen et al.
where ðui Þt;b is the displacement vector of the element.
The relationship between the strain and the displacement field of an element is
mentioned as
fXgt;b ¼ ½Bi fui g ði ¼ 1; 2; 3; 4Þ
T
@/y
@/y
@hy
@hy
@/x
@/x
@v0t
@u0t
@v0t
@hx
@hx
0t
fXgt ¼ @u
@x
@y
@y þ @x
@x
@y þ @x
@y
@x
@y þ @x
@y
fXgb ¼
@u0b
@x
@v0b
@y
@u0b
@y
þ
@v0b
@x
@hx
@x
@hy
@y
@hx
@y
þ
@hy
@x
@/x
@x
@/y
@y
@/x
@y
þ
@/y
@x
T
(31)
where fXgt;b is the element strain vector of the top and bottom face sheets and ½Bi
is the strain displacement.
Derivation of element stiffness matrix
The element stiffness matrix for the sandwich plate has been derived from the
strain energy formulation. The general strain energy equation of the laminate
face sheet is represented as
1
T
ui ¼ fdg kei fdg
2
ði ¼ t; bÞ
(32)
By substituting equations (30) and (31) into equations (19) and (20), the element
stiffness matrix equation can be obtained from the strain energy equation for the
top, bottom, and core layers which are presented as
0
2
Aij
T 6
1
TB
f
g
ut ¼ d @
Bt ðx; yÞ 4 Bij
2
a
b
Eij
0
2
Aij
Z bZ a
T 6
1
TB
ub ¼ fdg @
Bb ðx; yÞ 4 Bij
2
a
b
Eij
Z
b
Z
a
Bij
Dij
1
3
Eij
C
7
Fij 5 Bt ðx; yÞ Adxdyfdg
Fij
Hij
Bij
Dij
1
3
Eij
C
7
Fij 5 Bb ðx; yÞ Adxdyfdg
Fij
(33)
Hij
Stiffness matrix of skin materials due to transverse shear strain can be mentioned as
1
u1 ¼ b
2
Z lZ
0
0
b
"
T Asij
fdg B1 ðx; yÞ
Bsij
T
#
!
Bsij
B1 ðx; yÞ fdgdxdy
Dsij
(34)
18
Journal of Sandwich Structures & Materials 0(0)
Stiffness matrix for the core layer is presented as
!
Z lZ b
T
1
T
fdg Bc ðx; yÞ ½Gxz c Bc ðx; yÞ fdgdxdy
uc ¼ b
2
0
0
(35)
Stiffness matrix for the top plate of the sandwich plate from equation (33) is
presented as
0
2
A
T 6 ij
B
ket ¼
@ Bt ðx; yÞ 4 Bij
b
a
Eij
02
0
0
0
B6
0
0
B6 0
B6 @Ni
@Ni
B6
0
B6 @x
@y
B6
B6
@Ni @Ni
B6 0
B6
@y @x
B6
6
Z bZ aB
0
0
B6 0
B6
e
k
¼
½ t
B6 0
0
0
b
a B6
B6
B6
B6 0
0
0
B6
B6
B6
B6 0
0
0
B6
B6
B6
@4
0
0
0
2
@Ni
0
6 0 0 @x
6
6
@Ni
60 0
0
6
@y
6
6
6 0 0 @Ni @Ni
6
@y @x
6
6
6
0
0
2
3 60 0
6
Aij Bij Eij
6
6
7 6
0
0
4 Bij Dij Fij 5 6 0 0
6
6
Eij Fij Hij
6
0
0
60 0
6
6
6
60 0
0
0
6
6
6
0
0
60 0
6
6
4
0 0
0
0
Z
b
Z
a
Bij
Dij
Fij
0
0
1
3
Eij
C
7
Fij 5 Bt ðx; yÞ Adxdy
Hij
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
@Ni
@x
0
0
0
0
@Ni
@y
0
@Ni
@y
@Ni
@x
0
0
0
@Ni
@x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
@Ni
@x
0
0
0
0
0
@Ni
@y
@Ni
@y
@Ni
@x
0
0
0
@Ni
@x
0
0
0
0
0
0
0
@Ni
@y
0
0
3
0
7
0 7
7
0
0 7
7
7
7
0
0 7
7
7
0
0 7
7
7
0
0 7
7
7
7
0
0 7
7
7
@Ni 7
7
0
7
@y 7
7
@Ni
5
0
@y
31
0
0
0 7C
7C
7C
C
0 7
7C
7C
7C
C
0 7
7C
7C
7C
7C
0 7C
7C
7C
C
0 7
7Cdxdy
7C
7C
7C
0 7C
7C
7C
7C
0 7C
7C
C
@Ni 7
7C
7C
@y 7C
7C
@Ni 5A
@x
(36)
Praveen et al.
19
Stiffness matrix for the top plate of the sandwich plate from equation (33) can
be presented as
3
2
1
0
Z bZ a
Aij Bij Eij
T 6
C
7
B
keb ¼
@ Bb ðx; yÞ 4 Bij Dij Fij 5 Bb ðx; yÞ Adxdy
b
a
Eij Fij Hij
3
02
0
0
0
0
0
0
0
0
0
7
B6
0
0
0
0
0
0
0
0 7
B6 0
7
B6 @Ni
@N
i
B6
0
0
0
0
0
0
0 7
7
B6 @x
@y
7
B6
7
B6
@N
@N
i
i
B6 0
0
0
0
0
0
0 7
7
B6
@y @x
7
B6
6
B
Z b Z a B6 0
0
0
0
0
0
0
0
0 7
7
e
7
B6
@Ni
@Ni
kb ¼
B6 0
0
0
0
0
0
0 7
6
7
B
b
a
@x
@y
7
B6
7
B6
@Ni @Ni
B6 0
0
0
0
0
0
0 7
7
B6
@y @x
7
B6
B6
@Ni
@Ni 7
7
B6 0
0
0
0
0
0
0
7
B6
@x
@y 7
B6
7
B6
@Ni
5
@4
0
0
0
0
0
0
0
0
@y
2
31
@Ni
0
0
0
0
0
0
0
0
6
7C
@x
6
7C
6
7C
@N
i
60 0
C
0
0
0
0
0
0 7
6
7C
@y
6
7C
6
7C
6 0 0 @Ni @Ni 0
C
0
0
0
0 7
6
7C
@y @x
6
7C
6
7C
@Ni
6
7C
0
0
0 7C
0
0
0
3 60 0
2
6
7C
@x
Aij Bij Eij
6
7C
@Ni
7 60 0
6
C
0
0
0
0
0
0 7
4 Bij Dij Fij 5 6
7Cdxdy
@y
6
7C
6
7C
Eij Fij Hij
@Ni @Ni
6
7C
0
0 7C
0
0
0
60 0
6
7C
@y @x
6
7C
@Ni
6
7C
60 0
0
0
0
0
0
0 7C
6
7C
@x
6
C
@Ni 7
6
7C
0
0
0
0
0
0
60 0
7C
6
@y 7C
6
7C
4
@Ni @Ni 5A
0 0
0
0
0
0
0
@y @x
(37)
ket is represented as the stiffness matrix on top face sheet and keb is represented as
the stiffness matrix on bottom face sheet in equations (36) and (37).
20
Journal of Sandwich Structures & Materials 0(0)
Stiffness matrix derived for the top and bottom plates from equation (21) can be
presented as
½ke1 ¼
Z
bZ
b
"
02
B6 0 0
B6
a B6
B6 0 0
B6
a B6
B6
@4 0 0
As
Ds
0
0
0
@Ni
0
@x
@Ni
0
@y
0
0
0 0 0 0
2
0
6
6 0
6
6 0
6
6
# 6
6 0
6 @Ni
Ds
6
6 @x
s
F
6
6 Ni
6
6
6 0
6
6 0
4
0
0
Ni
0
0
0
0
Ni
0
0
0
0
c2 Ni
0
0
0
0
c2 Ni
31
0
0
0
0
0
0
0
0
0
0
@Ni
@y
0
0
0
0
0
0
0
Ni
0
0
c2 Ni
0
0
0
0
c2 Ni
7C
7C
7C
7C
7C
7C
7C
7C
7C
7Cdxdy
7C
7C
7C
7C
7C
7C
7C
7C
5A
3
7
7
7
7
7
7
7
5
(38)
where ke1 is the stiffness matrix on the top and bottom face sheet and c2 is a
constant as presented in Appendix 1.
Equation (38) can be simplified as
ke1
¼
Z
b
b
Z
a
a
B1 ðx; yÞ
" s
T A11
Ds11
#
Ds11
Fs11
B1 ðx; yÞ
!
dxdy
Stiffness matrix of the core material of the sandwich plate from equation (22) is
presented as
kec ¼
Z
b
b
Z
a
a
h
T
Bc ðx; yÞ ½Gc Bc ðx; yÞ
i
!
dxdy
The total stiffness matrix (kT ) is derived by the summation of the top, bottom,
and core layer of the composite sandwich plate such that
keT ¼ ket þ keb þ 2ke1 þ kec
(39)
21
Praveen et al.
Derivation of element mass matrix
The mass matrix for the sandwich plate has been derived from the kinetic energy
by substituting equation (31) into equations (25) and (26). The general kinetic
energy of the laminate face sheet due to inertia can be represented as
Z
1
T¼ b
2
b
b
Z
a
a
X
fdgT
Ni ðx; yÞ T ½Ii Ni ðx; yÞ fdg dxdy
(40)
Equation (40) can be simplified as
1
T
T ¼ fdg ½me fdg
2
(41)
The element mass matrix face sheet laminate that has been derived from equation (41) can be expressed as
mei ¼
Z lZ
B
2
B
2
0
X
Ni ðx; yÞ T ½Ii Ni ðx; yÞ dxdy
(42)
Element mass matrix for the top and bottom face sheet laminates of sandwich
plate is presented as
½met ¼
½meb ¼
Z
Z
b
Z
a
a
b
bZ a
a
b
Ni ðx; yÞ T ½It Ni ðx; yÞ dxdy
Ni ðx; yÞ T ½Ib Ni ðx; yÞ dxdy
The element mass matrix for the core material from equation (27) can be
expressed as
½mec ¼ b
hc 3
12
Z
b
b
Z
a
a
Ni ðx; yÞ T ½Ic Ni ðx; yÞ dxdy
The total mass matrix produced by the summation of the top, bottom, and core
layer of the composite sandwich plate can be represented as
meT ¼ met þ meb þ mec
(43)
Therefore, the Lagrange’s equations considered for developing the governing
differential equations of motion of the sandwich structure can be presented in finite
22
Journal of Sandwich Structures & Materials 0(0)
element form such that
meT fd€g þ keT fdg ¼ ffe g
(44)
½M fd€g þ ½K fdg ¼ fFg
(45)
where ½meT and ½keT are the element mass and stiffness matrices of the sandwich
plate; {d} is the displacement vector; and ffe g is the element force vector at each
node of the rectangular element, respectively. By assembling the element mass and
stiffness matrices and the force vector together for all of the elements yields the
governing differential equations of motion of the honeycomb sandwich composite
plate without and with CNT reinforcement in the form of finite element methods,
which can be presented as
The differential equations of motion (equation (44)) can be rewritten as for the
free vibrational analysis of the honeycomb sandwich composite plate as
½M fd€g þ ½K fdg ¼ f0g
(46)
where ½M, ½K, and fFg are the mass and stiffness matrices, and the force vector,
respectively.
The generalized solution assumed from equation (46) can be presented as
fug ¼ fug ðeÞk
(47)
The reduced generalized characteristics value from equation (46) can be
expressed as
½K f/g ¼ ðkÞ2 ½M f/g
(48)
where fug is denoted as the arbitrary constant and ½K is the global stiffness matrix
expressed in combination of real ½Kr and imaginary ½Ki parts of the stiffness
matrix such that
½ K ¼ ½ K r þ ðj Þ ½ K i
(49)
The normalized eigen vector f/r g has been evaluated using the real part of the
stiffness matrix to resolve the characteristic value problem. Equation (48) can be
represented as
2
ðki Þ 1 þ j ðgi Þ
¼
f/r gT ½Kf/r g
f/r gT ½Mf/r g
!
þ
!
f/r gT ½Ki f/r g
f/r gT ½Kr f/r g
(50)
23
Praveen et al.
The real and imaginary part of equation (50) can be expressed to identify the
natural frequency ðki Þ and loss factor ðgi Þ at each mode based on modal strain
energy [28] such that
!
(51)
!
(52)
ðki Þ ¼
f/r gTi ½Kf/r gi
f/r gTi ½Mf/r gi
ðgi Þ ¼
f/r gTi ½Ki f/r gi
f/r gTi ½Kr f/r gi
2
where f/r gi is the normalized eigen vector corresponding to each mode “i” and
½Kr and ½Ki are real and imaginary stiffness matrices.
Experimental investigation
Materials and methods
The MWCNTs functionalized with carboxylic acid supplied by United Nanotech
Innovations Private LimitedVR , with more than 95% chemical purity level having an
average dimensions of 10 mm length and 17 nm outer diameter were considered for
the present study to identify the material and mechanical characterization, and
structural responses of honeycomb composite sandwich plates. Initially, the CNTs
were taken for the identification of disintegration of nanotubes, dispersion, and
distribution of CNT reinforcement in the honeycomb composite sandwich plate
using SEM. The laminated composite test specimens were fabricated by adding the
hardener (HY951) with 10:1 ratio in the epoxy resin without CNT reinforcement to
evaluate the elastic properties of the top and bottom skin. The corrugated honeycomb composite laminated core specimens were fabricated by adding the hardener
(HY951) with 10:1 ratio in the sonicated solution with and without CNT to identify the shear modulus along corrugated (Gxz) and joined (Gyz) directions.
Following this, honeycomb composite sandwich plates were fabricated without
and with CNT reinforcement to identify the dynamic responses under various
boundary conditions.
Fabrication of hybrid honeycomb sandwich composites plates
The COOH-functionalized CNTs are strongly entangled with each other that may
develop random shapes of clusters. At this stage, the nanoparticles are not found in
the disintegrated pattern from each other as displayed in Figure 3 (a) and (b). It
can limit the homogenous scattering of CNTs into the epoxy matrix resin. The
aggregate of CNT has to be separated into individual tubes for improving the
efficacy of the hybrid composites by dispersing the CNTs randomly in epoxy. In
24
Journal of Sandwich Structures & Materials 0(0)
Figure 3. SEM images of (a) agglomerated CNTs with 2 mm scaling, (b) agglomerated CNTs
with 200 nm scaling, (c) disintegrated CNTs with 1 mm scaling, and (d) disintegrated CNTs with
epoxy with 2 mm scaling.
order to achieve this practically, the most efficient liquid processors for disintegrating the CNTs and dispersing it in the epoxy uniformly are the ultrasonicator
and shear mixer for the enrichment of the CNT/epoxy before fabrication.
In this study, ultrasonic liquid processor is applied to the entangled CNTs
mixed in an organic solvent to separate into individual nanotubes that lead to
homogeneous dispersion in the matrix resin. CNT nanoparticles were measured
in a beaker using the high precision balance to get the required quantity of precalculated CNT weight percentage in the composites. Initially, the known quantities of CNTs were blended into the dissolvable organic solvent using the shear
mixture. The 750 W power ultrasonicator consisting of 12.5 mm diameter of titanium horn capable of providing 20 kHz ultrasonic frequency was used to separate
the clustered CNTs in the solvent with pulsed on (5 s)–off (5 s) time interval for 1
h sonication. The glass beaker comprising the blend was put in the ice bath to keep
up the sonication process under the operational temperature. The precalculated
quantity of epoxy resin LY556 was heat treated at 75 C to condense the polymer
Praveen et al.
25
viscosity and was dispersed into the mixture of CNT/solvent to get the required
solution of CNT/epoxy. Further, the sonication was repeated to disperse the
CNTs/solvent homogenously into the viscous less epoxy resin. The mixture of
CNT/solvent/epoxy was kept in the vacuum-assisted oven to evaporate the solvent
absolutely at 75 C for 48 h. Afterwards, the mixture of CNT/epoxy was stirred for
30 min using shear mixture to enrich the CNTs homogeneously in epoxy. The
CNT/epoxy was mixed to the hardener HY951 by 10:1 ratio with respect to the
weight of CNT mixture. Figure 3(c) and (d) shows the dispersion of separated
CNTs in the epoxy polymer resin. From Figure 3, it was identified that the entanglement of clustered CNTs was significantly disintegrated and dispersed in the
epoxy resin using the ultrasonic liquid processor.
E-glass fiber reinforced laminated composite sandwich plate without and with
reinforcement of CNT was fabricated using the hand layup technique. The unidirectional glass fabrics 92145 were set over the flat smooth surface, and blend of
MWCNT/epoxy LY556 was applied on the fiber lamina by utilizing the hand
roller. The porous peel ply and breather cloth were set over the stacked sequence
of plies only after the roller-based hand layup process. The total layup was
enclosed with vacuum bag, which was concealed by using the sealant tapes.
Following this, the laminate was kept in an oven with the temperature
control rate of 1 C min 1 up to 75 C and then by dwelling for 2 h. Further,
the cured composite face sheets were taken out from the oven and kept at room
temperature for 24 h before the cutting of laminated composite into required
geometry. This procedure was followed for the fabrication of the hexagonal honeycomb core through the joining of corrugated sheets made without and with
CNT reinforcement by using the High carbon–high chromium steel honey comb
die (Figure 4(a)).
Mechanical characterization of the composite face layers
The dynamic elastic properties of top and bottom face sheets material were identified by performing the experiments on various samples of composite materials.
The top and bottom face layers made of glass fiber/epoxy composite with 0.26 mm
ply thickness for the cross section 150 mm 50 mm 5 mm having [0 ]20 and
[90 ]20 ply orientations were manufactured using vacuum-aided hand layup process
as discussed in the previous section. The volume fraction of glass fiber was found
as 0.28 for the top and bottom face layers of the sandwich composite plate. The
ASTM E1876 was followed to perform the simply supported in-plane and out-ofplane flexure test. The uniaxial accelerometer is placed at the 0.25 times of length
(0.25L) diagonal to the impact hammer excitation node point of rectangular
plate to measure the Young’s modulus, shear modulus, and Poisson’s ratio in
real time using the system connected with data acquisition and analysis software
(Dewesoft 7.1.1VR ). The evaluated elastic properties of the composite face layers are
mentioned in Table 1.
26
Journal of Sandwich Structures & Materials 0(0)
Figure 4. (a) High carbon–high chromium steel honey comb die for the fabrication of (b)
composite corrugated sheet and (c) hexagonal honeycomb core without and with CNT
reinforcement.
Table 1. The various mechanical properties of face sheets of the honeycomb sandwich plates.
Composite fiber
E1
E2
v12
G12
G23
qcf
22.920 GPa
5.699 GPa
0.269
2.246 GPa
2.061 GPa
1592 kg m 3
Resin
E
v12
G12
qr
3.450 GPa
0.3
1.330 GPa
1200 kg m
3
Mechanical characterization of the honeycomb core material
The hexagonal honeycomb core was fabricated through the joining of corrugated
sheets with 0.26 mm ply thickness (Figure 4(b) and (c)). The cross section 80 mm
80 mm 5 mm having [0 /90 ] ply orientation without and with 0.5 wt% of CNT
reinforcement was manufactured using vacuum-aided hand layup process and high
chromium–high carbon honeycomb die is shown in Figure 4(a). The volume fraction of hybrid honeycomb core made up of CNT/glass/epoxy was found to be
0.283. The shear moduli Gxz and Gyz of the unidirectional glass fiber reinforced
polymer composite without and with CNT reinforcement were measured in terms
of storage and loss moduli from the dynamic test as shown in Figure 5.
The vibration test was carried out with the combination of impact hammer and
accelerometer placed in the excitation direction on the steel block to capture the
dynamic responses of the honeycomb core from the data acquisition and analysis
software (Dewesoft 7.1.1VR ). The fundamental natural frequencies of the composite
honeycomb core with and without CNT reinforcement, along the transverse direction with fixed–fixed end conditions, were identified from the dominant resonant
response peaks shown in the frequency response function by experimental test
software. Following this, the damping factor at each resonant peak was calculated
by using the half-power bandwidth method and then the loss factor was also
obtained from the logarithmic decrement relations. The shear moduli of the hexagonal honeycomb core along corrugated (Gxz) and joined (Gyz) directions could
27
Praveen et al.
Figure 5. Schematic representation of honeycomb core test setup. CPU: central processing
unit; DAS: data acquisition system.
be identified from the measured resonant responses’ frequencies of the honeycomb
composite sandwich plate.
Finally, the alternative dynamic approach [29] was followed to identify the
storage and loss moduli [30] by measuring the out-of-plane Young’s modulus
and shear modulus for the honeycomb core layer from the resonance response
frequencies, mass moment of inertia, and geometrical parameters such as
lc=0.08 m, wc=0.08 m, hc= 0.01 m, m = 1.36 kg, hm= 0.02 m, ac=0.006 m,
sc=0.008 m, tc=0.0006 m, dtc=0.0016 m, Ez= 4.80 GPa, x ¼ 2pxn , of the system
is shown in Figure 6. These measured and calculated values are substituted in
equations (53) and (54) to predict the shear modulus along Gxz and Gyz directions
for the composite honeycomb core with and without CNT reinforcement
1
Gyz vG þ hm h a2c
2
¼0
m€
vG þ
hc
2
Ez wG a c
€G þ
mw
¼0
hc
Imx €h þ
Gxz ¼
Gyz ¼
1 Ez h a4c 1
þ
12 hc
2
Gyz
(53)
1
vG þ hm h hm a2c
2
¼0
hc
12Imx x2hx hc þ a4c mx2hx hc
3x2hx h2m hc m þ 12Imx x2hx hc a2c Ez ac 2
12Imx x2hy hc þ a4c mx2hy hc
3x2hy h2m hc m þ 12Imx x2hy hc a2c Ez ac 2
(54)
where Ez is the out-of-plane Young’s modulus of the core, xhx and xhy are the
resonant frequencies from experimental, m is the mass of steel block, hm is the
height of steel block, Imx is the mass moment of inertia, ac is the side of the cell, and
28
Journal of Sandwich Structures & Materials 0(0)
Figure 6. Schematic representation of honeycomb core geometry.
Table 2. Complex shear modulus of honeycomb core layer.
CNT (wt%)
Corrugation direction, Gxz (MPa)
Joining direction, Gyz (MPa)
0
0.5
1
1.5
182.29
253.76
303.18
565.45
166.16
216.05
231.42
512.28
(1þi
(1þi
(1þi
(1þi
0.1565)
0.1849)
0.1989)
0.2481)
(1þi
(1þi
(1þi
(1þi
0.1524)
0.1747)
0.1868)
0.2358)
CNT: carbon nanotube.
vG and wG are the displacement center of gravity of the mass. The complex modulus of the honeycomb core in terms of storage and loss moduli evaluated from the
measured directional moduli is presented in Table 2.
The storage modulus (s) and loss modulus (l) of the honeycomb core at various
wt% of CNT could be presented as an empirical relation such that
ðGxz Þs
ðGyz Þs
ðGxz Þl
ðGyz Þl
2
¼ 2E þ 12ðCNT wt%Þ
2E þ 10ðCNT wt%Þ þ 2E þ 08
2
1E þ 09 ðCNT wt%Þ þ 2E þ 08
¼ 2E þ 12ðCNT wt%Þ
2
ð
Þ
¼ 308 CNT wt% þ 1:756ðCNT wt%Þ þ 0:1595
2
¼ 367ðCNT wt%Þ þ 0:341ðCNT wt%Þ þ 0:1553
(55)
where “CNT wt%” is the weight percentage of the CNT content reinforced in the
honeycomb core and E is the exponential term.
It can be observed from Table 2 that that the addition of CNT in unidirectional
glass fiber/epoxy increases the complex shear modulus Gyz and Gxz than those of
without CNT reinforcement which consequently increases the storage modulus
Praveen et al.
29
Figure 7. SEM images of the (a) CNTs randomly distributed in epoxy and (b) CNTs randomly
distributed in GFRP-HCC.
and loss modulus of the core material. This can be related to the fact that the
random dispersion and distribution of nanoparticles in fiber reinforced polymer
composites, as shown in Figure 7, increases the interlamina shear force between the
matrix and CNT. Further, it can be seen that the complex shear modulus along the
corrugated direction is higher than that of joining direction of the honey comb core
materials irrespective of CNT content. This can be due to the continuous corrugated double cell wall direction which leads to have better stiffness and shear force
distribution of the materials. Hence, it can be concluded that with the addition of
CNT in the fiber reinforced core material both the storage modulus and loss
modulus of the core materials of the sandwich structure could be increased
simultaneously.
Validation
The efficacy of developed FEM formulation in identifying the dynamic characteristics of the hybrid honeycomb composite sandwich plate is ensured by comparing
the results in terms of natural frequencies and loss factors of composite sandwich
plate without and with CNT reinforcement in honeycomb core layer obtained
through experimental tests. The top and bottom face layers of the sandwich
plate having the dimension of 300 mm 200 mm 2 mm were fabricated with
5 mm core thickness of the hybrid and nonhybrid honeycomb composite sandwich
plates using the hand layup technique. The fabricated sandwich plates without and
with CNT reinforcement in core layer are shown in Figure 8.
The top and bottom composite face sheets having [0 /90 /0 ]s ply orientation of
unidirectional glass fiber with 0.26 mm ply thickness and volume fraction of 0.28
were fabricated. The core layers were fabricated using glass fibers with [0 /90 ] ply
orientation with identical thickness and volume fraction without and with 0.5 wt%
of CNT reinforcement as mentioned in the previous section to yield honeycomb
pattern. The experimental investigation was performed with a steel fixture used to
30
Journal of Sandwich Structures & Materials 0(0)
Figure 8. Fabrication of honeycomb sandwich composite plates (a) before assembly (b)
after assembly.
Figure 9. Experimental investigation of honeycomb sandwich plates. (a) Snapshot of experimental setup under CFFF end conditions and (b) diagrammatic illustration of the test setup under
FFFF end conditions. CPU: central processing unit; DAS: data acquisition system.
clamp the composite sandwich plates under various boundary conditions such as
clamped–free (CFFF), clamped–clamped (CFCF), and free–free (FFFF) end conditions as shown in Figure 9.
The impulse roving hammer-5800SL was used to energize the sandwich structure and the accelerometer was mounted on the flat top surface along the z direction on the sandwich plates to obtain the responses with respect to the applied
excitation energy. The vibration acceleration signals from transducer were converted by the four-channel data acquisition system (Model: ATA-DAQ042451)
into the functional frequency response signals which were processed through the
31
Praveen et al.
Table 3. Comparison of numerical and experimental results on natural frequencies of composite sandwich plates without and with CNT reinforcement in core materials under various
boundary conditions.
Natural Frequencies (Hz)
FEM
Experimental
Percentage deviation
Boundary
conditions
Mode
(m,n)
Without
CNT
With
CNT
Without
CNT
With
CNT
Without
CNT
With
CNT
CFFF
(1,1)
(1,2)
(2,1)
(2,2)
(1,1)
(1,2)
(2,1)
(2,2)
(2,2)
(1,1)
(3,2)
(2,3)
63.36
133.04
381.23
494.87
272.86
318.89
718.9
780.24
203.96
398.41
562.42
758.08
70.942
148.61
429.45
556.35
308.71
359.93
819.17
875.38
227.1
446.77
629.9
848.4
58.44
143.84
355.94
536.27
253.09
341.01
709.45
803.11
214.87
392.80
560.37
745.46
71.57
154.68
405.64
557.52
288.44
386.76
759.43
896.91
227.63
412.62
596.25
802.35
7.77
8.12
6.63
8.37
7.25
6.94
1.31
2.93
5.61
1.41
0.36
1.66
0.89
4.08
5.54
0.21
6.57
7.45
7.29
2.46
0.23
7.64
5.34
5.43
CFCF
FFFF
CNT: carbon nanotube; CFCF: clamped–clamped; CFFF: clamped–free; FEM: finite element method; FFFF:
free–free.
Dewesoft 7.1.1 experimental test software to identify the natural frequencies from
the dominant peaks of the functional response graphical data.
Thereby, the half-power bandwidth method has been adapted for finding the
damping ratio corresponding to each model peak response. Then, the loss factor
was calculated by using the logarithmic decrement relations. The results are presented in Tables 3 and 4. Very good agreement between the results derived from
FEM and experimental tests was observed. It can also be observed that the natural
frequency and loss factors of CNT reinforced honeycomb sandwich plates are
much higher than those of without CNT reinforcement.
Further, it can be seen that that the natural frequencies and loss factor of the
CNT reinforced sandwich composite plate at first mode increase by 5.61, 12.26,
and 18.35% under FFFF, CFCF, and CFFF end conditions and 14.25, 22.81, and
31.15% under CFCF, CFFF, and FFFF end conditions, respectively, as compared
to those of GFRP-honeycomb sandwich plates.
Parametric study
The dynamic characteristics of hybrid honeycomb laminated composite plates are
highly influenced by the volume fraction of CNT, thickness of the core, ply orientations, and various boundary conditions. The developed finite element formulation is used to carry out the parametric studies to examine the effects of all those
32
Journal of Sandwich Structures & Materials 0(0)
Table 4. Comparison of loss factors from FEM with the experimental loss factors for the
composite sandwich plates under various boundary conditions.
Loss factor (g)
FEM
Experimental
Percentage deviation
Boundary
conditions
Mode
(m,n)
Without
CNT
With
CNT
Without
CNT
With
CNT
Without
CNT
With
CNT
CFFF
(1,1)
(1,2)
(2,1)
(2,2)
(1,1)
(1,2)
(2,1)
(2,2)
(2,2)
(1,1)
(3,2)
(2,3)
0.0141
0.0138
0.0271
0.0261
0.0325
0.0306
0.0479
0.0459
0.0180
0.0194
0.0185
0.0209
0.0184
0.0158
0.0310
0.0299
0.0380
0.0349
0.0534
0.0539
0.0258
0.0222
0.0236
0.0305
0.0132
0.0127
0.0255
0.0239
0.0301
0.0288
0.0441
0.0429
0.0168
0.0179
0.0171
0.0195
0.0171
0.0150
0.0296
0.0281
0.0351
0.0324
0.0491
0.0503
0.0244
0.0207
0.0218
0.0289
6.38
7.97
5.90
7.66
7.38
5.88
7.93
6.54
6.67
7.73
7.57
6.70
7.07
5.06
4.52
6.02
7.63
7.16
8.05
6.68
5.43
6.76
7.63
5.25
CFCF
FFFF
CNT: carbon nanotube; CFCF: clamped–clamped; CFFF: clamped–free; FEM: finite element method; FFFF:
free–free.
properties on the natural frequencies, mode shapes, and forced vibration responses
of honeycomb composite sandwich plate without and with CNT reinforcement.
The developed finite element simulations are performed by considering the glass
epoxy laminated hybrid and nonhybrid honeycomb composite sandwich plate of
length (L) 300 mm length, width (W) 200 mm, and each ply thickness 0.26 mm of a
lamina at various contents of CNT in core materials. The volume fraction of all the
honeycomb composite sandwich plates has been considered as identical. The various mechanical properties of the honeycomb sandwich plate considered for the
simulation are as shown in Table 1. The various boundary conditions are focused
along the plate edges including clamped–clamped (CCCC), CFCF, CFFF, and
FFFF and indicated from the left end of the sandwich plate.
Influence of CNT content on natural frequencies and loss factors
The effect of variation in the volume fraction of CNT on natural frequencies and
loss factors of the CNT reinforced honeycomb composite sandwich plate under
CFFF end conditions are studied by varying the CNTs’ volume fractions from 0 to
1.5% and the obtained results are presented in Table 5. The natural frequencies
and the corresponding loss factors for the first four bending modes of composite
sandwich plate without and with CNT reinforcement in core material. It can be
observed that the fundamental resonant frequency of the plate with CNT is 8.30%
higher than those of the plate without reinforcement of CNT. This can be due to
33
Praveen et al.
Table 5. Effects of variation in volume fraction of CNTon natural frequencies and loss factors of
honeycomb composite sandwich plate without and with CNT reinforcement.
Natural frequencies (Hz)
Loss factor (g)
Mode
(m,n)
0%
0.5%
1%
1.5%
0%
0.5%
1%
1.5%
(1,1)
(1,2)
(2,1)
(2,2)
65.36
136.91
392.43
508.67
66.501
138.94
404.48
522.94
67.235
139.95
411.24
530.17
70.855
144.12
439.42
559.36
0.0146
0.0145
0.0285
0.0275
0.0213
0.0159
0.0335
0.0299
0.0326
0.0174
0.0454
0.0329
0.0537
0.0240
0.0618
0.0453
CNT: carbon nanotube.
the presence of CNT in core material which increases the storage modulus and
consequently increases the natural frequencies of the sandwich structure. This can
further be endorsed to the strong bonding of the nanotubes with the polymer
chains which limits the motion of the honeycomb composite sandwich plate.
This increases the bending stiffness of the sandwich plate, thereby enhancing the
natural frequencies of the CNT reinforced sandwich plate. It can be seen that the
natural frequencies of the sandwich plate increase at all the modes considered with
the increase in volume fraction of CNT.
Moreover, an increase in the loss factors can be observed with the glass/epoxy/
CNT sandwich plate. The CNT reinforced honeycomb sandwich plate perceived
an increase in their fundamental mode loss factor by 67.5% than those of the
sandwich plate with CNT reinforcement in core material. This is due to the existence of stick–slip mechanism between the epoxy resin, glass fiber, and the nanoparticles’ reinforcement. Further, the loss modulus of the honeycomb material
increases with increase in CNT content which consequently increases the dissipation energy and the loss factor of the sandwich plate.
Effect of core thickness on variation of natural frequencies and loss factors of
honeycomb sandwich plate
The effects of variation in the core thickness on natural frequencies and loss factors
of a sandwich plate without and with CNT reinforcement in core material are
studied under CFFF end conditions and the results are tabulated in Table 6.
The simulation is carried out with consideration of hybrid honeycomb core layer
with 1.5 wt% of CNT reinforcement of the sandwich plate. It can be seen that the
natural frequencies increase at all the modes considered while the loss factor
decreases with the increase in thickness of core layer.
The increase in stiffness of the sandwich structure is more dominant than an
increase in mass of the sandwich structure, which results in an increase in natural
frequencies with increase in the thickness of the core layer. However, the increase
in thickness of the honeycomb core layer could decrease the shear strain energy
distribution, which results in a decrease in loss factor of the honeycomb core layer.
34
Journal of Sandwich Structures & Materials 0(0)
Table 6. Effects of variation in core thickness on natural frequencies and loss factors of honeycomb composite sandwich plate without and with CNT reinforcement.
Mode
(m,n)
(1,1)
(1,2)
(2,1)
(2,2)
Natural frequencies (Hz)
Loss factor (g)
2.5 mm
5 mm
7.5 mm
10 mm
2.5 mm
5 mm
7.5 mm
10 mm
48.435
96.722
303.42
382.65
70.855
144.12
439.42
559.36
90.73
186.34
557.36
713.53
108.79
224.63
662.04
850.08
0.0394
0.0231
0.0453
0.0265
0.0454
0.0339
0.0575
0.0384
0.0537
0.0426
0.0618
0.0472
0.0633
0.0544
0.0671
0.0589
CNT: carbon nanotube.
Table 7. Effects of variation in boundary conditions on natural frequencies
and loss factors of honeycomb composite sandwich plate without and with
CNT reinforcement.
Boundary conditions
Mode (m,n)
Natural
frequencies (Hz)
Loss factor (g)
CCCC
(1,1)
(1,2)
(2,1)
(2,2)
(1,1)
(1,2)
(2,1)
(2,2)
(1,1)
(1,2)
(2,1)
(2,2)
(2,2)
(1,1)
(3,2)
(2,3)
791.49
1185.7
1885.7
1940.6
323.80
370.68
874.26
887.80
70.855
144.12
439.42
559.36
215.80
449.86
623.65
838.98
0.0579
0.0745
0.0798
0.0945
0.0644
0.0598
0.0792
0.0801
0.0537
0.0240
0.0618
0.0553
0.0563
0.0367
0.0342
0.0311
CFCF
CFFF
FFFF
CCCC: clamped–clamped; CNT: carbon nanotube; CFCF: clamped–clamped; CFFF:
clamped–free; FFFF: free–free.
Effect of various boundary conditions on natural frequencies and loss factors of
honeycomb sandwich plate
The influence of boundary conditions on the variation of the natural frequencies and
loss factors of the honeycomb sandwich composite plates are examined by considering
the top and bottom skin sheets with symmetric ply orientations [0 /90 /0 ]s and [0 /
90 ] and the core layer reinforced with 1.5 wt% of CNT. The various boundary
conditions adopted for the simulation are CCCC, CFCF, CFFF, and FFFF and
Praveen et al.
35
Figure 10. Variation of normalized deflection of honeycomb sandwich composite plate under
various boundary conditions. CCCC: clamped–clamped; CFCF: clamped–clamped; CFFF:
clamped–free; FFFF: free–free.
the results are shown in Table 7. Further, the contour plots on the variation of normalized deflection with normalized length and width along x, y directions are shown
in Figure 10 at various modes corresponding to Table 7. The natural frequencies and
loss factors are identified based on the conforming mode shapes obtained along the
longitudinal (m) and transverse (n) directions of the honeycomb sandwich plates.
From Table 7, it can be seen that the honeycomb sandwich composite plate
under CCCC and CFFF end conditions exhibits the highest and lowest natural
frequencies, respectively, among all the boundary conditions considered.
This can be due to the fact that the honeycomb composite plates with CCCC and
CFFF end conditions have the highest and lowest stiffness, respectively. It can also
be observed that the natural frequencies of honeycomb sandwich plate under CFCF
boundary conditions are higher than those of FFFF and CFFF boundary condition.
36
Journal of Sandwich Structures & Materials 0(0)
Table 8. Effects of variation in ply orientations on natural frequencies and loss factors of honeycomb composite sandwich plate without and with CNT reinforcement.
Mode
(m,n)
(1,1)
(1,2)
(2,1)
(2,2)
Natural frequencies (Hz)
Loss factor (g)
[0 /90 /45 ]s
[0 /90 ]2s
[45 ]2s
[0 /90 /45 ]s
[0 /90 ]2s
[45 ]2s
53.62
93.82
339.57
436.17
54.33
130.21
346.34
477.36
57.915
178.61
357.76
559.65
0.0698
0.0285
0.0841
0.0663
0.0762
0.0301
0.0864
0.0519
0.0820
0.0437
0.0955
0.0488
CNT: carbon nanotube.
This can be due to the fact that camping at both ends of the composite sandwich
plate exhibits higher stiffness than those obtained by free at four edges of the sandwich plate. It can be seen that the honeycomb composite sandwich plate under
CFCF end condition exhibits the highest loss factors, respectively, among all the
boundary conditions considered. This can be due to the existence of stick–slip mechanism between the epoxy resin, glass fiber, and the nanoparticles reinforcement.
It can also be due to the poor load-carrying capacity of the randomly dispersed
CNTs throughout the cross section of the plate which may also be relative to the
clamping toward the width of the sandwich plate.
Effect of ply orientations of face sheets on natural frequencies and loss factors
of honeycomb sandwich plate
The simulation is carried out to study the influence of ply orientations of face sheets
on the natural frequencies and loss factors of hybrid honeycomb composite plate. The
symmetric laminate with various ply orientations of the face layers is considered under
CFFF boundary conditions at 1.5 wt% of CNT reinforced in the honeycomb sandwich plate. The various ply orientations including [0 /90 ]2s, [45 ]2s, and [0 /90 /
45 ]s at the top and bottom face layers of the hybrid honeycomb composite sandwich plate are considered and the results are presented in Table 8. It can be noted that
the natural frequencies at all modes increase in the order of [0 /90 /45 ]s, [0 /90 ]2s,
and [45 ]2s under CFFF end conditions. This can be due to the fact that the [45 ]2s
and [0 /90 /45 ]s ply orientations have highest and lowest stiffness which yields the
highest and lowest natural frequencies of sandwich composite plate. Further, it can be
seen that the presence of 45 ply orientation induces shear force which leads to
increase in the stiffness of the honeycomb composite sandwich plate.
Correlation of mode shapes of honeycomb sandwich plates with and without
CNT reinforcements
The influences of CNT content in core material of honeycomb sandwich plates on
the variation of the transverse vibration mode shape are investigated by
Praveen et al.
37
Figure 11. Comparison of free vibration mode shapes and wavelength of (a) Mode (1,1), (b)
Mode (2,1), and (c) Mode (3,1) obtained along the midline of longitudinal plane of the honeycomb
sandwich composite plate with various wt% of CNT under CFFF boundary conditions. CNT:
carbon nanotube.
considering various wt% (0, 0.5, 1, and 1.5%) of CNT reinforcements. The mode
shapes are plotted between the obtained normalized transverse deflections and the
normalized length obtained along the midline of the sandwich plate.
The comparison of lowest three vibration mode shapes of a symmetric laminated composite sandwich plate under CFFF boundary condition is shown in
Figure 11. A significant reduction in transverse peak amplitude of laminated sandwich plate is observed with increase in addition of CNT wt% which shows the
effectiveness of CNT in reduction in vibration amplitude of the honeycomb composite sandwich plate.
Transverse vibration responses of sandwich plates
The transverse vibration responses of a composite honeycomb sandwich plate with
various wt% of CNTs (0, 0.5, 1, and 1.5%) under CFFF end condition are analyzed by considering a sinusoidal excitation of magnitude 1 N applied at a distance
of 100 mm 100 mm from the lower left corner end of the plate. The structural
velocity spectrum in decibels is calculated over the frequency range of 1–700 Hz
and the results are shown in Figure 12. The results reveal that the root mean
velocity spectrum of honeycomb composite sandwich plate without the addition
of CNT reinforcement is higher than those of hybrid honeycomb composite sandwich plate under CFFF end conditions. This can be due to the enhancement of
stiffness of hybrid honeycomb composite sandwich plate and subsequently the
load-carrying capacity of the CNT reinforced honeycomb sandwich structure
could be increased. Hence, it can be observed that the stiffness and flexibility of
the hybrid honeycomb composite sandwich plate is significantly altered by the
addition of various wt% of CNT reinforcement without any significant change
in the mass of structure.
38
Journal of Sandwich Structures & Materials 0(0)
Figure 12. Comparison of transverse vibration responses of different wt% of CNT reinforced in
the honeycomb sandwich composite plate under CFFF boundary condition. CNT:
carbon nanotube.
Conclusions
In this study, the free and forced vibration responses of the composite sandwich
plate with CNT reinforced honeycomb as the core material and laminated composite plates as the top and bottom face sheets are investigated. The governing
equations of motion of hybrid composite honeycomb sandwich plates are derived
using higher order shear deformation theory and solved numerically.
The following are the important conclusions derived from the present study:
• It is seen that that the fundamental natural frequency and loss factor of the
sandwich composite plate could be increased by 5.61, 12.26, and 18.35% under
FFFF, CFFF, and CFCF end conditions and 14.25, 22.81, and 31.15 under
CFCF, CFFF, and FFFF end conditions with 0.5 wt% CNT reinforcement in
core material.
• It was also shown that the natural frequencies and loss factors of the honeycomb composite sandwich plates at all modes considered increase with increase
in the wt% of CNT.
• The natural frequencies increase with increase in the core thickness of the sandwich plate while the loss factor decreases at all the modes considered.
39
Praveen et al.
• The CCCC end condition yields the highest natural frequency while CFFF end
conditions yield the lowest natural frequency among all the end conditions considered.
• The comparison of lowest three vibration mode shapes of a laminated sandwich
plate is observed with increase in addition of CNT which shows the effectiveness
of CNT in reduction in vibration amplitude of the honeycomb composite sandwich plate.
The reinforcement of CNTs in core materials not only enhances the stiffness of
the hybrid sandwich composite by limiting the motion of the polymer chain but
also enhances the damping capability due to the interfacing sliding of matrix resin
and CNT particles. The forced vibration analysis was performed to compare the
transverse vibration response of various wt% of honeycomb sandwich plate without and with CNT reinforcement. It can be concluded that the peak transverse
displacement of honeycomb composite sandwich plates could be decreased
approximately by 132, 124, 120, and 114 dB, respectively, with 0, 0.5, 1, and
1.5% of CNT reinforcement.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication
of this article.
ORCID iD
Vasudevan Rajamohan
http://orcid.org/0000-0002-1849-3814
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Appendix 1
Inertia matrix for the top and bottom face laminate can be presented as
2
I011
6
0
6
6
0
6
6
I111
½It;b ¼ 6
6
0
6
6 ðI c Þ
6
311
1
4
0
0
0
0
I011
0
0
I111
0
I011
0
0
0
ðI311 c1 Þ
I111
0
0
I211
0
ðI411 c1 Þ
0
ðI311 c1 Þ
0
0
ðI411 c1 Þ
0
ðI611 ðc1 Þ2 Þ
0
I111
0
0
I211
0
ðI411 c1 Þ
0
0
0
ðI311 c1 Þ 7
7
7
0
7
7
0
7
ðI411 c1 Þ 7
7
7
0
7
5
2
I611 ðc1 Þ
R h=2
where I011 ; I111 ; I211 ; I311 ; I411 ; I611 ; ¼ h=2 q 1; z; z2 ; z3 ; z4 ; z6 dz; c1 ¼ 3h42 .
Inertia matrix for the core layer can be presented as
2
c3 2
6
6 0
6
6 c3 c4
6
6 0
6
½I c ¼ 6 0
6
6 c3 c5
6
6 0
6
4 c3 c6
0
where c3 ¼
0
c3 2
0
c3 c4
0
0
c3 c5
0
c3 c6
1
hc c 4 ¼
c3 c4
0
c4 2
0
0
c4 c5
0
c4 c6
0
1
hc
; c5 ¼
0
c3 c4
0
c4 2
0
0
c4 c5
0
c4 c6
d hc
hc
c3
0
0
0
q tc
0
0
0
0
; c6 ¼
c3 c5
0
c4 c5
0
0
c5 2
0
c5 c6
0
d hc
3ðhc Þ
.
0
c3 c5
0
c5 c6
0
0
c5 2
0
c5 c6
3
c3 c6
0
c4 c6
0
0
c5 c6
0
c6 2
0
3
0
7
c3 c6 7
7
0 7
7
c4 c6 7
7
0 7
7
0 7
7
c5 c6 7
7
0 5
2
c6
42
Journal of Sandwich Structures & Materials 0(0)
The shape functions of the plate element can be presented as
1
ð1 nÞð1 gÞ
N1 ¼
4
1
ð1 þ nÞð1 gÞ
N2 ¼
4
1
ð1 þ nÞð1 þ gÞ
N3 ¼
4
1
ð1 nÞð1 þ gÞ
N4 ¼
4
Strain displacement
presented as
2
@Ni
6
6 @x
6
6 0
6
6
6 @Ni
6
6 @y
6
6
6 0
6
6
6
Bt;b ðx; yÞ ¼ 6
6 0
6
6
6 0
6
6
6
6 0
6
6
6
6 0
6
6
4
0
matrix due to axial deformation for top face sheet can be
0
3
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0
0 0
0
@Ni
@x
0
0
0
0 0
0
0
0
0 0
0
@Ni
@y
@Ni
@y
@Ni
@x
0
0 0
0
0
0
@Ni
@x
0
0 0
0
0
0
0
0
0 0
0
0
0
@Ni
@y
@Ni
@y
@Ni
@x
0
0
0 7
7
7
0 7
7
7
7
0 7
7
7
7
0 7
7
7
7
0 7
7
7
7
0 7
7
7
7
0 7
7
@Ni 7
7
7
@y 7
@Ni 7
5
@x
ði ¼ 1; 2; 3; 4Þ
Strain displacement matrix due to transverse deformation for top and face
sheets can be presented as
2
@Ni
6 @x
6
6
@Ni
B1 ðx; yÞ ¼ 6
6 @y
6
4 0
0
where c7 ¼
4
ð hc Þ 2
.
Ni
0
0
0
0
Ni
0
0
0
0
0
0
c7 Ni
0
0
c7 Ni
3
7
7
7
7 ði ¼ 1; 2; 3; 4Þ
7
7
5
43
Praveen et al.
Strain displacement matrix due to transverse deformation for the core layer can
be presented as
2
@Ni
d hc
ðd hc ÞNi
Ni
0
@x
3
@Ni
ðd hc ÞNi
Ni
0
Ni e
0
0
@x
Gyz
Gxz
0
0
Gxz
0
;
; Gyz ¼
; G ¼
Gc ¼
0 Gyz
0 Gyz xz
0 Gxz
ht þ hb
1
c2 ¼
; c3 ¼
2
hc
hc
hc
e¼
a
6 Ni
Bc ðx; yÞ ¼ c3 6
4
0
0
0
Ni
3
0
e
d
hc
3
Ni
7
7
5
where ði ¼ 1; 2; 3; 4Þ, hc is the thickness of the core layer, and c3 and e
are constants.
Mass matrix for top and bottom skin can be presented as
½met
½meb
¼
¼
Z
Z
b
b
b
b
Z
Z
a
a
a
a
Ni ðx; yÞ T ½It Ni ðx; yÞ dxdy
Ni ðx; yÞ T ½Ib Ni ðx; yÞ dxdy
where
2
Ni
6 0
6
6 0
6
6 0
6
N¼6
6 0
6 0
6
6 0
6
4 0
0
0
Ni
0
0
0
0
0
0
0
0
0
Ni
0
0
0
0
0
0
0
0
0
Ni
0
0
0
0
0
0
0
0
0
Ni
0
0
0
0
0
0
0
0
0
Ni
0
0
0
0
0
0
0
0
0
Ni
0
0
0
0
0
0
0
0
0
Ni
0
3
0
0 7
7
0 7
7
0 7
7
0 7
7 ði ¼ 1; 2; 3; 4Þ
0 7
7
0 7
7
0 5
Ni