Electro-Mechanical Behavior of Smart Sandwich Plates With Porous Core and
Graphene-Reinforced Nanocomposite Layers
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Behavior of Smart Sandwich Plates With Porous Core and Graphene-
Reinforced Nanocomposite Layers." Proceedings of the ASME 2019
International Mechanical Engineering Congress and Exposition.
Volume 9: Mechanics of Solids, Structures, and Fluids. Salt Lake City,
Utah, USA. November 11–14, 2019. V009T11A016. ASME.
https://doi.org/10.1115/IMECE2019-10796
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1 © 2019 by ASME
International Mechanical Engineering Congress and Exposition IMECE2019
November 11-14, 2019, Salt Lake City, UT, USA
IMECE2019-10796
ELECTRO-MECHANICAL BEHAVIOR OF SMART SANDWICH PLATES WITH POROUS CORE AND GRAPHENE-REINFORCED NANOCOMPOSITE LAYERS
Kamran Behdinan1 Rasool Moradi-Dastjerdi2 Advanced Research Laboratory for Multifunctional Light Weight Structures (ARL-MLS),
Department of Mechanical & Industrial Engineering, University of Toronto Toronto, Ontario, Canada
1 Contact author, Email: behdinan@mie.utoronto.ca 2 Email: moradi@mie.utoronto.ca
ABSTRACT The use of piezoelectric sensor and/or actuator layers in
engineering structures provides smart sandwich structures with adaptive responses. Moreover, due to the brittle behavior of piezoceramic materials, inserting nanocomposite and porous layers between piezoelectric layers offers more flexible and lighter structures along with maintaining the advantages of nanocomposite materials. Therefore, in this paper, we have proposed smart sandwich plates consisting of a porous polymeric core and two graphene-reinforced composite (GRC) layers integrated with two piezoceramic layers. The distributions of porosities and randomly oriented graphene particles are assumed to be functionally graded (FG) along the thickness of core and nanocomposite layers, respectively. For the static behavior of the proposed sandwich plates, the coupled electro-mechanical governing equation has been extracted by minimizing potential energy equation with respect to displacement and electrical potential. The governing equation has been discretized by adopting a higher order shear deformation theory (HSDT) of plates and a developed mesh- free method. Using the developed solution framework, the effects of porosity and graphene characteristics, electro- mechanical loads, and layer thicknesses on the deflection behavior of the proposed FG piezoelectric porous nanocomposite sandwich plates (FG-PPNSPs) have been studied.
KEYWORDS Smart sandwich plate, Functionally graded porous core,
Graphene-reinforced nanocomposite, electro-mechanical static analysis, Mesh-free method
1. INTRODUCTION
Recently, the use of smart piezoceramic-actuated devices has been considerably increased due to the reliable, accurate, robust and controllable behavior of piezoceramic materials [1]. However, the main shortcoming of piezoceramics is their brittle behavior [1]. Embedding some passive and softer layers between piezoceramics could be helpful to make the resulted structures more flexible. Moreover, attaching piezoceramics on polymeric structures offers new smart structures. In case of sandwich structures, improving the stiffness of outer layers and/or reducing structural weight could introduce sandwich structures with higher strength-to-weight ratio [2–4]. Such sandwich structures could be integrated with piezoceramics to remarkably promote their application in advanced industries for energy harvesting, shape deformation monitoring, and vibration damping [1,5,6].
Phung-Van et al. [5] studied the electro-mechanical behavior of laminated composite plates made of graphite-epoxy integrated with two piezoceramic layers using cell-based smoothed finite element method (FEM) and first order shear deformation theory (FSDT). They also developed another framework consisting of HSDT and isogeometric FEM to present the electro-mechanical behavior of the same smart
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plates [6]. Talebitooti et al. [7] also considered the same smart plates and presented an optimal control analysis using HSDT and mesh-free method.
Moreover, the effects of adding nanofillers or embedding pores in polymeric matrices on the mechanical behavior of shells and plates have attracted the attention of researchers. Shen et al. [8] studied the effect of adding graphene on the nonlinear bending of FG-GRC plates resting on elastic foundations. Mechanical responses of sandwich plates reinforced with defected CNTs [9] and CNT agglomerations [3] were presented using mesh-free method. Wang and Zhang [10] examined the effect of porosity on the buckling and bending responses of FG-GRC foams using a refined plate theory. Zhao et al. [11] developed a semi-analytical approach to capture the effects of pores and their distribution on the vibrational analysis of functionally graded porous (FGP) curved panels and shells. Moreover, Setoodeh et al. [12] suggested a five-layer smart curved shell consisting of an FG porous core and two FG-CNT reinforced nanocomposite layers integrated with two piezoelectric layers. They presented the vibrational behavior of their suggested structures.
In this paper, smart sandwich plates consisting of an FGP core and two FG-GRC layers integrated with two piezoceramic layers have been proposed. This work is motivated by improving the flexibility of smart sandwich plates and the increase of the strength-to-weight ratio of passive sandwich plate. The static deflection of the proposed sandwich plates subjected to electro-mechanical loads is presented using a developed HSDT based mesh-free method. The coupled electro-mechanical governing equation has been extracted by minimizing potential energy equation. Effects of porosity and graphene characteristics, electro-mechanical loads, and layer thicknesses on the static deflection of the proposed FG-PPNSPs are presented.
2. MODELING OF PPNSP
As shown in Fig. 1, our proposed smart sandwich plates have one FG porous core, two nanocomposite layers reinforced with graphene particles and two piezoelectric layers. The geometrical dimensions of PPNSP are a, b, hc, hf, hp and h for length, width, core thickness, GRC layer thickness, piezoelectric layer thickness and total thickness, respectively. The proposed PPNSPs are assumed to be subjected to uniform dispersed mechanical load f0, surface electrical charge qs or electrical voltage input such that the outer faces of piezoelectric layers are connected to input voltage V0 and their inner faces are connected to ground. 2.1. Material properties of GRC and FGP layers
The considered GRC is assumed to be made of an isotropic polymeric matrix reinforced by randomly oriented graphene particles with length aG, width bG and thickness hG. The effective Young’s modulus E of such nanocomposites could be estimated using a modified Halpin-Tsai’s approach [13] by incorporating two efficiency parameters (η1 and η2). Shen et al.
( )
( )
G r
G r
f
− =
− =
+ (2)
where fr is graphene volume fraction, and G iiE and mE are
Young’s modulus of graphene particles and polymeric matrix, respectively. Shen et al. [8] reported η1 and η2 for some specific graphene volume fractions from 3% to 11% as shown in Table 1. However, we utilized a Spline interpolation through the reported data to define smooth variations of η1 and η2 with respect to fr. In addition, Poisson’s ratio υ of GRC is estimated using the rule of mixture method as below:
, 1G m r m m rf f f f= + = − (3)
where superscripts G and m are represented graphene and matrix, respectively.
The following profiles are considered for the distributions of graphene particles along the thickness of nanocomposite layers: Upper GRC layer:
( )min( ) 1 2 2 p
r r c f ff z f f z h h h−= + + − − (4)
Lower GRC layer:
( )min( ) 1 2 2 p
r r c f ff z f f z h h h−= + − + + (5)
where Δf is the difference between the maximum fr-max and minimum fr-min values of fr.
Furthermore, for the distribution of pores along the thickness of core, two types of functionally graded called FG-I and FG-II are considered as well as a uniformly dispersed (UD) type as a comparator. The elastic E and shear G moduli of porous core PEPM can be estimated as [11]:
UD: ( )01EPMP e P= − , 2
0 0 0
= − − − + (6)
FG-II: 01 cos 4 4EPM
c
= − + (8)
where P represents the maximum moduli (non-porous) of core and e0 is porosity coefficient.
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electrical loads
Table 1 graphene efficiency parameters at different fr [8] fr 0.03 0.05 0.07 0.09 0.11 η1 2.929 3.068 3.013 2.647 2.311 η2 2.855 2.962 2.966 2.609 2.260 2.2. HSDT for the determination of displacement field
in PPNSP Using plate theories, displacement field could be expanded
( ) ( )
x x x
y y y
u x y z u x y z x y z c w
v x y z v x y z x y z c w
w x y z w x y
= + + +
= + + +
=
(9)
where 2 1 4 3c h= − and, u0, v0, w0, x and y are displacements
and normal rotations of mid-plane. Considering Eq. (9), the in- plane bε and out-of-plane γ parts of strain vector ε are defined as:
T= bε ε γ , 3
1 1 3
T
xx yy xy z c z= = + +b 0ε ε κ κ ,
( )2 1 01 3
(10)
where:
0, ,
u v
u v = =
+ + ε κ
, 0, 0,
w w
w w
2.3. Constitutive low in PPNSP
The constitutive low of the proposed PPNSP can be associated with mechanical and electrical fields as follows [5,6]: = − = +
Tσ Qε e E D eε kE
(12)
,
=
= =
0 0 0
p s p s
k
(15)
Assuming the existence of the gradient of electric potential only along z direction, electric field is determined as [6]:
z z= −E (16) 3. MESH-FREE NUMERICAL ANALYSIS
The total energy function of the proposed PPNSP under electro-mechanical loads can be given as follows:
1 2
V
d f ψq (17)
where ψ , d, sq and df are potential, displacement, electrical surface charge and surface traction vectors, respectively.
Due to the use of mesh-free method, displacement field should be approximated in some predefined nodes. By means of
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MLS shape function [15], virtual nodal displacement field vector d is given by [9]:
0 0 0 1
x y i i i
u v w d =
= =d (18)
where N and i are the total number of nodes and MLS shape functions, respectively. Introduction of estimated displacement field (Eq. 18) in the equations of strain field (Eq. 11) and the linear variation of electric potential (Eq. 16) gives the following equations in terms of MLS shape functions and their derivations [3,9]:
( )3 2 0 1 1 3 1
ˆ ˆ, 1 3 sz c z c z= + + = +ε B B B d γ B d (19) ˆ= −E B ψ (20)
where:
, , ,
,
,
0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 ,
0 0 0 0 0 0
0 0 0 0 0 0 , 0 0 2
0 0 0 , 0 0 1 /
0 0 0
i xx i x
i yy i y
i x i s p
i y i h
(21)
By introducing Eqs. (12) and (18-20) into the total energy function of the proposed PPNSP (Eq. 17), the following coupled electro-mechanical system of equations can be achieved:
d
s
(22)
where the matrices of mechanical stiffness, coupling electro- mechanical stiffness and piezoelectric permittivity are represented by ddK , dψK and ψψK , respectively, and they are defined as below:
0 1 3 0 1 3 T T T T
A
A
dA
dA
B B Q B B (23)
0 1 3 T T T
b A
A
dA
dA
B B E B , T=ψd dψK K (24)
T
A
0 T
A
where 3
Sym c z−
=bQ Q , (27)
/2 2 1
1 3 3 9
4. NUMERICAL RESULTS AND DISCUSSIONS
In this section, the efficiency of the developed solution framework has been verified and static deflections of PPNSP under coupled electro-mechanical loads have been studied. In the proposed PPNSPs, piezoelectric and core layers are assumed to be made of PZT-G1195N and porous PMMA, respectively. In addition, the nanocomposite layers of PPNSP are assumed to be made of a mixture of randomly oriented graphene particles with aG=14.76, bG=14.77 and hG=0.188 nm embedded in a polymeric matrix of PMMA. The following material properties are considered for the utilized material [6,8]: Graphene: 11 1.8107GE = , 22 1.8078GE = TPa and υG=0.177, PMMA: Em=2.5 GPa and υm=0.34, PZT-G1195N: Ep=63 GPa, υp=0.3, e31=e32=22.86 C/m2 and k33=15×10-9 F/m.
4.1. Validation and convergence
The accuracy of utilized solution framework has been verified by comparing the deflections of four-layer graphite/ epoxy laminated plates integrated with two PZTG1195N layers with results reported in [5,6]. The thicknesses of each piezo and laminate layers were assumed to be 0.1 and 0.25 mm, respectively. The configuration of the proposed plates is denoted by [p/θ1/θ2]s or [p/θ1/θ2]as where p and θ represent piezo layer and the angle of fiber orientation. The subscripts ‘as’ and ‘s’ show asymmetric and symmetric laminate configurations, respectively. For the considered plates subjected to f0=100 N/m2 and different input voltages, Table 2 compares the central deflections obtained in the present work with those reported using FSDT [5] and HSDT [6] based-FEMs. Very good agreements, especially with Ref. [6] which employed HSDT, have been observed.
The convergence of the developed solution framework is also examined for the proposed PPNSPs. Figure 2 illustrates the central deflections of PPNSP as a function of node numbers in each direction. Here, PPNSP is assumed to be subjected to mechanical load f0=2 KN/m2; however, input voltage is set to V0=0 which means the surfaces of piezo layers are connected to ground or the structure does not have piezoelectric effect. In addition, the PPNSP is assumed to have a perfect core e0=0 and UD distribution of graphene particles with fr=5%. The use of 17 nodes in each direction (17×17) provides results with almost the same accuracy with those obtained by using more nodes, therefore the convergence of the developed solution framework is verified.
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Note that in order to report high accurate results, 21×21 node arrangement has been selected in the following simulations.
Table 2 Central deflection of simply supported laminated
plates integrated with two piezoelectric layers under different input voltages (×1e-4 m)
V0 (V)
[p/- 30/30]as
[p/- 45/45]as
[p/- 45/45]s
0 [5] -0.7442 -0.6688 -0.6323 -0.6326 [6] -0.7452 -0.6617 -0.6239 -0.6375 Present -0.7435 -0.6602 -0.6224 -0.6359 5 [5] -0.3259 -0.2957 -0.2801 -0.2863 [6] -0.3283 -0.2968 -0.2817 -0.2842 Present -0.3274 -0.2960 -0.2809 -0.2835 10 [5] 0.0924 0.0774 0.0601 0.0721 [6] 0.0886 0.0682 0.0606 0.0691 Present 0.0886 0.0682 0.0605 0.0690
The convergence of the developed solution framework is
Figure 2. Central deflection of the proposed PPNSP as a
function of the number of nodes in each direction
4.2. Electro-mechanical deflections of the proposed FG-PPNSPs
In the following examples, except otherwise mentioned, square simply supported PPNSPs with length a=b=0.4 m, piezoelectric layer thickness hp=0.5 mm, nanocomposite layer thickness hf=3 mm, core layer thickness hc=12 mm have been considered.
The effect of input voltage on deflection shape is shown in Fig. 3 for PPNSP with perfect core e0=0 and graphene volume fraction of fr=5% subjected to uniform mechanical load f0=2 KN/m2. Figure 3 shows that pure mechanical load on PPNSP without piezoelectric effect (V0=0) causes a downward deformation. By enhancing input voltage, deflections of PPNSP are reduced. The reason is that input voltage causes a reverse deflection on the plate. In addition, it is observed that the induced electrical deflection can recover all the mechanical deflections when V0=40-50 V. The application of higher voltages like V0=80 V are able to generate completely reverse deformations.
Figure 3. Centerline deflection of the proposed PPNSP
The effects of graphene volume fraction and its dispersion
are shown in Fig. 4 for PPNSPs with a perfect core and FG- GRC layers when fr-min=3%. The increase of fr-max from 3% to 7% sharply decreases the values of central deflections. However, higher values of fr-max show a slight improvement on the reduction of their deflections. The reason could be the formation of graphene agglomeration in matrix which restricts the performance of reinforcement part. Furthermore, the decrease of volume fraction index p shows a higher enhancement in reducing the deflection of the proposed PPNSP. According to Eqs. (4-5), the decrease of p means higher volume fractions of graphene.
Figure 4. Central deflection of the proposed PPNSP as a
function of fr-max
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Figure 5 shows the central deflection of the proposed PPNSPs as a function of porosity parameter for uniform and the two aforementioned FG dispersions of pores. The increase of only less than 1.3 % in deflection is observed by the increase of porosity parameter from zero to 0.6. This observation means that by cutting the weight of core to almost its half, the proposed smart sandwich plates with FGP core show almost the same structural stiffness and deflections in comparison with those have perfect cores. In addition, PPNSPs with FG porous cores, especially FG-I, have lower deflections than those have UD porous core.
Figure 5. Central deflection of the proposed PPNSP as a
function of porosity parameter The effect of layer thicknesses on the deflection of PPNSPs
subjected to electro-mechanical loads have been illustrated in Figs. 6. The effect of piezoelectric layer thickness on the deflection of PPNSPs is shown Fig 6a. The considered PPNSPs are assumed to be subjected to f0=2 KN/m2 and two different electrical conditions: (i) V0=0 which means the surfaces of piezo layers are connected to ground and the structure has no piezoelectric effect, (ii) qs=0 which means that the structure has piezoelectric effect; however, there is no external surface electrical charge. Under both electrical conditions, a slight increase in piezo layer thickness dramatically reduces the deflection of PPNSP. Compared with the case of V0=0, lower deflections are observed in PPNSPs with piezoelectric effect, qs=0, because of inducing reverse (upward) deflections. Figure 6b shows that the increase of core thickness from hc=5 mm to hc=40 mm considerably reduces the deflection of PPNSPs such that a sharp reduction is observed by the increase of thickness up to hc=15 mm. Furthermore, increasing the thickness of graphene-reinforced layers from hf=1 mm to hf=10 mm leads to a significant reduction in PPNSP deflection as shown in Fig. 6c. The comparison between Figs. 6b and 6c shows the effect of the thickness of nanocomposite layers on deflection is much
stronger than that of the thickness of core because of their distances from the mid-plane (z=0).
(a)
(b)
(c)
Figure 6. Central deflection of the proposed PPNSP as a function of the thickness of (a) piezoelectric (b) core (c)
nanocomposite layers
5. CONCLUSIONS Sandwich plates consisting of one FG porous polymeric
core and two FG graphene-reinforced nanocomposite layers integrated with two piezoceramic layers are proposed in this paper to determine smart, lightweight, stiff and more flexible plates. In a framework of mesh-free method and HSDT, the static deflections of the proposed FG-PPNSPs subjected to coupled electro-mechanical loads were studied. The obtained results indicated that:
• The increase of fr-max from 3% to 7% sharply decreases the deflections of PPNSP; however, higher values of fr-max slightly reduce their deflections.
• Although the use of porous core reduces around half of the weight of core, the proposed smart plates with perfect and porous cores show almost the same deflections.
• PPNSPs with FG porous cores have lower deflections than those have UD porous core.
• A slight increase in piezo layer thickness dramatically reduces the deflection of PPNSP.
• PPNSPs with piezoelectric effect can have lower deflections than those without it.
• The increase of the thickness of nanocomposite or core layers reduces the deflections of PPNSP.
ACKNOWLEDGMENTS The work described in this paper was supported by Natural
Sciences and Engineering Research Council of Canada (NSERC under grant RGPIN-217525). The authors are grateful for their support.
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