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A quadratic piezoelectric multi-layer shell element for FE analysis of smart laminated composite plates induced by MFC actuators To cite this article: Soheil Gohari et al 2018 Smart Mater. Struct. 27 095004
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Smart Materials and Structures Smart Mater. Struct. 27 (2018) 095004 (39pp)
https://doi.org/10.1088/1361-665X/aacc95
A quadratic piezoelectric multi-layer shell element for FE analysis of smart laminated composite plates induced by MFC actuators Soheil Gohari1,5 , Shokrollah Sharifi2 , Rouzbeh Abadi3 , Mohammadreza Izadifar4 , Colin Burvill1 and Zora Vrcelj2 1
Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia College of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia 3 Institute of Structural Mechanics, Bauhaus University-Weimar, Marienstr 15, D-99423 Weimar, Germany 4 Institute of functional interfaces, KIT Campus Nord, Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldschafen, Germany 2
E-mail:
[email protected],
[email protected] and
[email protected] Received 15 December 2016, revised 8 June 2018 Accepted for publication 14 June 2018 Published 31 July 2018 Abstract
Macro fiber composite (MFC) actuators developed by the NASA have been increasingly used in engineering structures due to their high actuation power, compatibility, and flexibility. In this study, an efficient two dimensional quadratic multi-layer shell element by using first order shear deformation theory (FOSDT) is developed to predict the linear strain–displacement static deformation of laminated composite plates induced by MFC actuators. FOSDT is adapted from the Reissner–Mindlin plate theory. An eight-node quadratic piezoelectric multi-layer shell element with five degrees of freedom is introduced to prevent locking effect and zero energy modes observed in nine-node degenerated shell element. Two types of MFC actuators are used: (1) MFC-d31 and (2) MFC-d33, which differ in their actuation forces. For result verification, the electro-mechanically coupled quadratic finite element (FE) model is compared with the ABAQUS results in various examples. Comparison of the results showed good agreement. The proposed quadratic FE formulation is simple and accurate, which eliminates the need for costly FE commercial software packages. It was observed that earlier studies have mostly emphasized on the effect of actuation power and MFC fiber orientations on mechanical shape deformation of smart composite plates. In this study, a more comprehensive, in-depth investigation is conducted into host structure performance such as boundary conditions, laminate stacking sequence configuration, and symmetry/asymmetry layups. Keywords: MFC actuators, quadratic piezoelectric multi-layer shell element, first order shear deformation theory, smart laminated composite plates, numerical simulation (Some figures may appear in colour only in the online journal) 1. Introduction
temperature resistance. Laminated plates and beams are typically adopted to achieve the desired stiffness and lightness for parts of load-bearing engineering structures [1–4]. Piezoelectric materials have recently drawn much attention due to their low power consumption, high material linearity, and quick response when induced by external forces. As a results, they are being used in the fields of mechanical and electrical engineering [5, 6]. Piezoelectric materials can be
Laminated and asymmetric composite structures are being used considerably in aerospace, automotive, civil, mechanical and structural engineering applications due to their high stiffness and strength to weight ratio, low density, and 5
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Figure 1. Schematic of the laminated composite plates sandwiched by a pair of MFC actuators: (a) cantilevered, (b) simply-supported.
integrated with laminated composite structures to provide smart-intelligent composite systems [7, 8]. Numerous engineering structures integrated with smart devices such as piezoelectric sensors and actuators have proved to be superior to their conventional counterparts. Static analysis of advanced composite structures under axial, transverse, twisting, and torsional loads has potential application in mechanical systems, helicopter rotor blades, and/or blades for turbomachinery [9]. Some other applications of piezoelectric materials in smart and adaptive engineering structures are acoustical noise reduction, structural shape control, damage identification, structural health monitoring, vibration suppression, deflection control in missile fins, and air foil shape changes [10–13]. One of the great advantages of piezoelectric materials is their ability to respond
to changing environment and control structural deformation, which has led to the new generation of aerospace structures like morphing airplanes [14]. There has been recently a dramatic achievement in development of the new generation of piezoelectric materials known as piezoelectric fiber composites (PFC), which have orthotropic material properties due to unidirectionally aligned piezoelectric fibers with circular cross-section. The fibers are impregnated into a resin matrix system to boost PFC properties when sharing mechanical load. PFCs were later upgraded to interdigitated electrode piezoelectric fiber composites which is also referred to as active fiber composites (AFCs) in order to increase the strain output, directional actuation, flexibility, and durability [15–17]. However, some difficulties were faced during the 2
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Figure 2. Schematic of a MFC actuator: (a) MFC-d33 components, (b) MFC-d31 components, (c) MFC-d33 polarization and local material coordinate systems, (d) MFC-d31 polarization and local material coordinate systems.
Figure 3. (a) Four-node quadratic element, (b) eight-node quadratic element, (c) Jacobian coordinates system.
similar study, mechanical properties of shear actuated fiber composites were analyzed using unit-cell approach by Raja and Ikeda [20]. Since piezoelectric materials are sensitive to thermal environment; therefore their effectiveness under thermal stresses was examined using equivalent layered and unit-cell approach [21]. MFC materials can be integrated with plate and shell structures due to their flexibilities and compatibilities. One of the great applications of MFCs is to control the effect of snapthrough due to unsymmetrical layup in a composite laminate. During manufacturing process, with no external loads, they can undergo multiple equilibrium shapes when cooled from the curing temperature to a lower operating temperature. Therefore, through applying appropriate voltage to MFC actuators, the undesired shapes can be controlled and modified [22]. The application of MFCs in shape, thermal and vibrational control
manufacture of AFCs such as misalignment of piezoelectric fibers and their limited contact with flat electrodes due to fiber cross sectional configuration. Therefore, the NASA developed MFC actuators whose piezoelectric fibers have rectangular cross sectional shape and are surrounded by polymer matrix. The other components of MFCs are protective and electrode layers bonded to piezoelectric fibers. MFC materials are composed of seven active layers, two electrodes, two kapton and two acrylic layers [18]. Developed by the NASA-Langley Research Center, MFC actuators and sensors present superior qualities among AFCs in performance, behavior, and manufacturability [19]. Understanding MFC properties is important to use them efficiently. There are several studies regarding MFCs modeling using numerical and analytical approaches. Park and Kim [13] modeled MFCs mechanical properties theoretically using classical laminated plate theory and uniform field model. In another 3
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Figure 4. Schematic of FE simulation procedures of the smart laminated composite plates induced by a pair of bonded MFC actuators:
(a) sketching, defining dimensions, and assigning material properties to the composite laminate (host structure) and MFC actuators, (b) applying the electro-mechanical boundary conditions, the piezoelectric actuators groundings, and the electrical surface charges, (c) applying meshing and defining element type of the host structure and MFC actuators.
models reported in the literature that investigate smart composite structures induced by MFC actuators [30, 31]. The literature survey reveals that the study on some critical key factors affecting the mechanical shape deformation of smart composite plates induced by MFC actuators is limited. Some of these factors include MFC and host structure orthotropic behavior, boundary condition, laminate stacking sequence, and symmetry/asymmetry layups. For instance, Lentzen et al [32] developed a geometrically nonlinear finite shell element considering piezoelectric layers. However, the piezoelectric actuators and host structures in their research had only isotropic properties. In addition, only cantilevered host structures were taken into account in their research. Marinković et al [33] developed an efficient nine-node shell element by extending the degenerate shell approach to modeling thinwalled piezoelectric active composite laminates. In spite of the fact that their model can successfully be used to overcome shear locking setback in extremely thin-layer smart structures, but there is still the possibility of zero-energy modes appearance. In addition, their model is not computationally efficient to analyze moderately-thick and thick laminates. Nine-node shell elements have five communicable spurious zero energy modes which plague its reliability. These modes disappear if the expensive third order quadrature is used. However, the element would then deteriorate substantially and exhibit both shear and membrane locking. Explicit hybrid-stabilization is one
Table 1. MFC piezoelectric actuators and host structure material
properties. Material properties
MFC-d31 [18]
MFC-d33 [18]
T300/976 CFRP [42]
E1 (GPa) E2 (GPa) E3 (GPa) v12 v13 v23 G12 (GPa) G13 (GPa) G23 (GPa) * = d31 (pm V-1) d 31 * = d32 (pm V-1) d 32 * = d11 (pm V-1) d 33 * = d12 (pm V-1) d 32
30.336 15.857 15.857 0.31 0.31 0.438 5.515 5.515 5.515 −170 −100 — —
29.4 15.2 15.2 0.312 0.31 0.31 5.79 6.06 6.06 — — 467 −210
150 9 9 0.3 0.3 0.3 7.1 7.1 2.5 0 0 0 0
Note: (*) is MFC actuator polarization coordinate system not local material coordinate system.
of plates and shells can be found in [23–25]. There are several studies regarding FE analysis of composite plates induced by conventional piezoelectric actuators [26–28]. Research has been completed into the numerical modeling of MFCs using commercial FE packages, with results compared to experimental data [24, 29]. There are also analytical and numerical 4
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Figure 5. Deflection of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d31 patches: (a) βa =0°, (b) βa =30°, (c) βa =45°, (d) βa =60°, (e) βa =90°.
be found in [18, 31], however the lack of research on host structure parameters and various boundary conditions is observed in those studies.
technique to be used in order to overcome those problems while increasing the case complexity [34]. The latest FE analysis of engineering structures induced by MFC actuators can
5
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Figure 6. Twisting of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d31 patches: (a) βa =0°, (b) βa =30°, (c) βa =45°, (d) βa =60°, (e) βa =90°.
Although a nine-node shell element has both advantages and disadvantages over an eight-node shell element, but one can simply retain the advantages while avoiding the disadvantages by using an eight-node shell element. Correctly representing quadratic displacements considering bilinear
shape function is one obvious advantage of employing a ninenode shell element. However, an eight node shell element can be easily constructed in an efficient way to present the same accuracy as a nine-node shell element does while avoiding the need to suppress spurious mechanisms [35]. Some additional 6
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Figure 7. Deflection of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d33 patches: (a) βa=0°, (b) βa=30°, (c) βa=45°, (d) βa=60°, (e) βa=90°.
advantages of using an eight-node shell element is to be able to model engineering structures with various thicknesses (thin, moderately thick, and thick). To prevent the shear
locking faced in extremely thin structures, one can simply adapt reduced integration technique while applying full integration for moderately-thick and thick structures [36]. 7
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Figure 8. Twisting of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d33 patches: (a) βa=0°, (b) βa=30°, (c) βa=45°, (d) βa=60°, (e) βa=90°.
static and dynamic structural analysis can be found in [37]. An eight-node quadratic shell element with five degrees of freedom is introduced for FE formulation. Two types of MFC actuators are used: (1) MFC-d31 and (2) MFC-d33, which differ in their actuation forces. MATLAB is to obtain the 2D FE results. Subsequently, for verification purposes, electro-
In the current research a simple quadratic piezoelectric multi-layer shell element using FOSDT is developed to predict the linear strain–displacement static deformation in smart laminated composite plates induced by MFC actuators. An indepth study on consideration of transverse shear deformation in laminated composite plate structures and its application in 8
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Table 2. Comparison of the results of the present study and Abaqus at various points (see figures 5–8).
wo (a, 0) (mm)
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure a
5(a) 5(b) 5(c) 5(d) 5(e) 7(a) 7(b) 7(c) 7(d) 7(e)
wo (a, b ) (mm)
Present study
Abaqus
Error (%)
−0.3056 −0.1422 −0.1461 −0.2031 −0.4354 −6.0980 −5.7110 −6.8800 −8.3200 −10.740
−0.3159 −0.1503 −0.1532 −0.2140 −0.4529 −6.2630 −5.9280 −7.0870 −8.5610 −11.040
3.37 5.69 4.79 5.36 4.01 2.70 3.79 3.00 2.89 2.79
a
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
Present study
Abaqus
Error (%)
−0.3056 −0.4638 −0.5032 −0.5308 −0.4354 −6.0980 −8.5070 −9.8460 −10.860 −10.740
−0.3159 −0.4881 −0.5283 −0.5607 −0.4529 −6.2630 −8.8080 −10.140 −11.190 −11.040
3.37 5.23 4.98 5.63 4.01 2.70 3.53 2.98 3.03 2.79
6(a) 6(b) 6(c) 6(d) 6(e) 8(a) 8(b) 8(c) 8(d) 8(e)
The value of relative error between two data can be computed using e =
X1 - X2 X1
´ 100.
• The lamina is not initially pre-stressed, thus, there are no residual stresses in presence of matrix and fibers; and • The matrix and fibers behave linearly within elastic domain.
mechanically coupled FE analysis using ABAQUS software package is used for numerical simulation of the smart laminated composite plates integrated with MFC actuators. Since the proposed quadratic FE formulation is simple and accurate, it eliminates the need for costly FE commercial software packages. It can also be adapted for composite structures without incorporated piezoelectric patches and layers. Furthermore, modeling smart structures requires training and is often time consuming. It becomes more complicated when considering assembly of smart structures with various piezoelectric patches in arbitrary positions. However, this issue can be easily overcome by using the proposed quadratic FE formulation.
For MFC actuators, the linear piezoelectricity theory is adapted with assumptions as follows [40]: • The strain-electric field varies linearly; • The piezoelectric coefficients are constant within the linear zone; thus, they cannot be electrically turned with a bias field; • The electric field is assumed to be constant across each lamina in all directions. u (x , y , z) = Ux (x , y) + zjx (x , y) = Ux (x , y) ⎡ ¶w ⎤ + z⎢ - gxz ⎥ , ⎣ ¶x ⎦
2. Composite laminates and MFC actuators modeling
(1 a )
v (x , y , z) = Uy (x , y) + zjy (x , y) = Uy (x , y) ⎤ ⎡ ¶w + z⎢ - gyz ⎥ , ⎦ ⎣ ¶y
Consider a laminated composite plate (host structure) composed of N orthotropic layers and with a total layup thickness of H. Each layer thickness is kept constant. The host structure is sandwiched by a pair of bonded MFC actuators and is either cantilevered (figure 1(a)) or simply supported at two edges (figure 1(b)). Considering material linearity for small displacements, plane stress assumption and the effect of transverse shear deformation, the general form of displacement fields can be derived based on FOSDT as stated in equations (1a)–(1c) [38, 39]. For composite laminates and MFC patches, some initial assumptions for the 2D FE formulation are made as follows [29, 40]:
w (x , y , z) = w0 (x , y) ,
(1 b ) (1 c )
where, Ux, Uy, and w0 are the mid-plane displacements along x, y, and z directions, respectively on the xy-plane [41]. jx and jy are rotations of transverse normal in the mid-plane along x and y directions, respectively. z is the vertical distance from the mid-plane to the kth layer, which is positioned between z=hk and z=hk+1 through laminate thickness [42, 43]. After obtaining the mid-plane displacements, the displacements of any arbitrary point, x, y, and z in three-dimensional (3D) space can be determined. The linear strain-displacement relation is stated in equations (2a)–(2d) [44]. It is assumed that all strain components change linearly through layup thickness in the entire laminate independent from changes in material properties.
• There is a perfect bonding between fibers and matrix, avoiding fibers dislocations and disarrangements through the matrix. • No slip occurs between the lamina interfaces; • Fibers distribution throughout the matrix is uniform; • There is a perfect bonding between fibers and matrix, avoiding fibers dislocations and disarrangements through the matrix; • The matrix is perfectly fabricated with no voids and impurity;
T ⎡ ¶u ¶n ¶u ¶n ⎤ + ⎥ [ exx eyy gxy ]T = ⎢ ¶x ⎦ ⎣ ¶x ¶y ¶y T f f ⎤T = ⎡⎣ e0xx e0yy t 0xy ⎤⎦ + z ⎡⎣ exx eyyf txy ⎦ ,
9
(2 a )
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Figure 9. Deflection of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d31 patches: (a) βa=0°, (b) βa=30°, (c) βa=45°, (d) βa=60°, (e) βa=90°.
10
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Figure 10. Twisting of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d31 patches: (a) βa = 0°, (b) βa = 30°, (c) βa = 45°, (d) βa = 60°, (e) βa = 90°.
⎡ ⎤T ⎡⎣ eoxx eoyy g oxy ⎤⎦T = ⎢ ¶Ux ¶Uy ¶Ux + ¶Uy ⎥ , ¶x ⎦ ⎣ ¶x ¶y ¶y
(2 b )
⎡ ⎤T ⎡ e f e f g f ⎤T = ⎢ ¶jx ¶jy ¶jx + ¶jy ⎥ , ⎣ xx yy xy ⎦ ¶x ⎦ ⎣ ¶x ¶y ¶y
(2 c )
⎤T ⎡ T = ⎡ g o g o ⎤T = j + ¶wo j + ¶wo g g yz xz ⎥ , (2 d ) ⎢ [ ] ⎣ yz xz ⎦ y x ⎣ ¶y ¶x ⎦
where, eoxx , eoyy , and g oxy are the laminate’s mid-plane strains, f f f are the flexural (bending) strains and wo exx , eyy , and g xy
11
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Figure 11. Deflection of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d33 patches: (a) βa = 0°, (b) βa = 30°, (c) βa = 45°, (d) βa = 60°, (e) βa = 90°.
12
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Figure 12. Twisting of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d33 patches: (a) βa = 0°, (b) βa = 30°, (c) βa = 45°, (d) βa = 60°, (e) βa = 90°.
is the transverse deflection of a composite laminate’s mid-plane. g oyz and g oxy are the transverse shear deformation in the yz and xz planes, respectively. Considering the plane stress assumption and neglecting the throughthickness stresses, the simplified 2D electro-mechanical
plate equations are derived from the 3D equations of theory of elasticity and three charged equilibrium equations of piezoelectric medium, as stated in equations (3a), (3b) for MFC-d31 and equations (3c), (3d) for MFC-d33, respectively [45]. 13
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Figure 13. The ABAQUS results of wo(x, y) in the laminated cantilevered GFRP composite plate induced by a pair of MFC-d31 patches: (a) βa = 0°, (b) βa = 30°, (c) βa = 45°, (d) βa = 60°, (e) βa = 90°.
⎡ 0 0 ⎤ ⎡ s11 ⎤k ⎢ C1111 C1122 C1133 0 ⎥ C C C 0 0 0 ⎥ ⎢ s22 ⎥ ⎢ 1122 2222 2233 ⎢ t12 ⎥ = ⎢ 0 0 0 C1212 0 0 ⎥ ⎢ t23 ⎥ ⎢ 0 0 0 0 C 0 ⎥ 2323 ⎢⎣ t13 ⎥⎦ ⎢ ⎥ 0 0 0 0 C3131⎦ ⎣ 0 ⎡ e11 ⎤k ⎡ 0 0 e31 ⎤ ⎢ ⎥ ⎡ F ⎤k ⎢ e22 ⎥ ⎢ 0 0 e32 ⎥ ⎢ 1 ⎥ ⎢ ⎥ g ´ 12 - ⎢ 0 0 0 ⎥ ⎢ F2 ⎥ , ⎢g ⎥ ⎢ 0 e24 0 ⎥ ⎣ F3 ⎦ ⎢ 23 ⎥ ⎢⎣ e ⎣ g13 ⎦ 0 ⎥⎦ 15 0
⎡ e11 ⎤k k k ⎡ ⎤ r 0 0 0 0 e15 ⎢ e22 ⎥ ⎡ 1⎤ ⎢ r 2 ⎥ = ⎢ 0 0 0 e24 0 ⎥ ⎢ g12 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ r ⎥⎦ 3 ⎣ e31 e32 0 0 0 ⎦ ⎢ g23 ⎥ ⎣ g13 ⎦ ⎡z11 0 0 ⎤k ⎡ F ⎤k 1 ⎢ ⎥ + ⎢ 0 z22 0 ⎥ ⎢⎢ F2 ⎥⎥ , ⎢⎣ 0 0 z 33⎥⎦ ⎣ F3 ⎦
(3 a )
14
(3 b )
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Figure 14. The ABAQUS results of wo(x, y) in the laminated cantilevered GFRP composite plate induced by a pair of MFC-d33 patches: (a) βa = 0°, (b) βa = 30°, (c) βa = 45°, (d) βa = 60°, (e) βa = 90°.
⎡ 0 0 ⎤ ⎡ s11 ⎤k ⎢ C1111 C1122 C1133 0 0 0 0 ⎥⎥ C C C ⎢ s22 ⎥ ⎢ 1122 2222 2233 ⎢ t12 ⎥ = ⎢ 0 0 0 C1212 0 0 ⎥ ⎢ t23 ⎥ ⎢ 0 0 0 0 0 ⎥ C 2323 ⎢⎣ t13 ⎥⎦ ⎢⎣ 0 0 0 0 0 C3131⎥⎦ ⎡ e11 ⎤k ⎡⎢ e11 0 0 ⎤⎥ k ⎢ e22 ⎥ e12 0 0 ⎥ ⎡ F1 ⎤ ⎢ ⎢ ⎥ ´ ⎢ g12 ⎥ - ⎢ 0 e26 0 ⎥ ⎢ F2 ⎥ , ⎢ g23 ⎥ ⎢ 0 0 0 ⎥ ⎣ F3 ⎦ ⎢⎣ g ⎥⎦ ⎢⎣ 0 0 e ⎥⎦ 13 35
⎡ e11 ⎤k k ⎡ r1 ⎤k ⎡ e11 e12 0 0 0 ⎤ ⎢ e22 ⎥ ⎢ r 2 ⎥ = ⎢ 0 0 e26 0 0 ⎥ ⎢ g12 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ r ⎥⎦ 3 ⎣ 0 0 0 0 e35⎦ ⎢ g23 ⎥ ⎣ g13 ⎦ ⎡z11 0 0 ⎤k ⎡ F ⎤k 1 ⎢ ⎥ + ⎢ 0 z22 0 ⎥ ⎢⎢ F2 ⎥⎥ , ⎣⎢ 0 0 z 33⎥⎦ ⎣ F3 ⎦
(3 c )
(3 d )
where, σij, εij, Cijkl, eij, and Φi are the stresses, the strains, the elastic stiffness elements of the matrix [Q], the 15
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piezoelectric coefficients, and the components of the electric fields, respectively in the orthotropic material orientation. pi and ζij are the electric displacement and the piezoelectric dielectric constants, respectively. Global stress–strains and electrical displacements as shown in equations (4a)–(4c) in the kth ply can be calculated by transforming 2D stress–strains and electrical displacements in the material direction through transformation matrix [T] in equation (4d) [46]. ⎡ sxx ⎤k ⎡ s11 ⎤k ⎢ syy ⎥ ⎢ s22 ⎥ ⎢ sxy ⎥ = [T ]-1 ⎢ s12 ⎥ = [s ] , ⎢s ⎥ ⎢ s23 ⎥ ⎢ yz ⎥ ⎢⎣ s13 ⎥⎦ ⎣ sxz ⎦
Q12 = C1122
⎡ exx ⎤k ⎡ e11 ⎤k ⎢ eyy ⎥ ⎢ e22 ⎥ ⎢ gxy ⎥ = [T ]-1 ⎢ g12 ⎥ = [e] , ⎢g ⎥ ⎢ g23 ⎥ ⎢ yz ⎥ ⎢⎣ g ⎥⎦ 13 ⎣ gxz ⎦ (4 a )
k ⎡z xx 0 0 ⎤k ⎡z 0 0⎤ ⎢ ⎥ ⎢ 11 ⎥ ⎢ 0 zyy 0 ⎥ = ⎢ 0 z22 0 ⎥ = [z ] , ⎢⎣ 0 0 z ⎥⎦ ⎣⎢ 0 0 z 33⎥⎦ zz
⎡ C2 S2 SC ⎢ 2 2 - SC C ⎢ S [T ] = ⎢- 2SC 2SC C 2 - S 2 ⎢ 0 0 0 ⎢⎣ 0 0 0
0 0 ⎤ ⎥ 0 0 ⎥ 0 0 ⎥, C -S ⎥ S C ⎦⎥
2 C1133 E1 = , C3333 1 - n12 n 21
(5 c )
Q66 = C1212 = G 12,
(5 d )
* - Q12 d32 *, e31 = Q11d31 - Q12 d32 = Q11d31
(5 e )
* - Q22 d32 *, e32 = Q12 d31 - Q22 d32 = Q12 d31
(5 f )
* - Q12 d32 *, e11 = Q11d11 - Q12 d12 = Q11d33
(5 g )
* - Q22 d32 *, e12 = Q12 d11 - Q22 d12 = Q12 d33
(5 h )
2.1. Electrical properties of MFC actuators (d31-effect)
The polarization direction in MFC-d31 type is through thickness (z direction) and perpendicular to the piezoelectric fibers, which are parallel to the mid-plane (figures 2(b), (d)). Thus, the piezoelectric matrix in global coordinate system can be rearranged to equation (6a). The electrical field in MFC-d31 is obtained using equation (6b).
(4 b )
⎡ 0 0 0 0 e15⎤ [e] = ⎢⎢ 0 0 0 e24 0 ⎥⎥ [T ] , ⎣ e31 e32 0 0 0 ⎦ (4 c )
F3 = -
V3 , ta
(6 a )
(6 b )
where, V3 and ta are electrical voltage applied through thickness and a MFC-d31 thickness, respectively. 2.2. Electrical properties of MFC actuators (d33-effect)
(4 d )
In this type of MFC actuator, both the polarization direction and the piezoelectric fibers are parallel to the mid-plane (figures 2(a), (c)). Thus, the piezoelectric matrix in the global coordinate system can be simplified to equation (7a). The electrical filed through thickness for MFC-d33 is obtained using equation (7b).
where, c is cos(β) and s is sin (β). β is the winding angle between either fibers and x axis in a fiber-reinforced composite ply (βp) or between piezoelectric fibers and x axis in a MFC actuator (βa) (figures 1(a), (b)). Qij and eij are the reduced elastic stiffness and the piezoelectric modules as stated in equations (5e)–(5h), respectively. It must be noted that local material and the MFC actuators polarization coordinate systems are different. In this study, local material and actuators polarization coordinate systems are defined as (123) and (123)*, respectively. Thus, the MFC coefficients in polarization coordinate system must first be converted into their equivalents in the local material coordinate system, and then substituted into the 3D equations of theory of elasticity and three charged equilibrium equations of piezoelectric medium (figures 2(a)–(d)). Q11 = C1111 -
(5 b )
where, E1, E2, v12, and G12 are in-plane local elasticity modules of an orthotropic layer in local material coordinate system. dij is the piezoelectric dielectric constant.
⎡ exx ⎤k ⎢ eyy ⎥ ⎡ rxx ⎤k ⎡ 0 0 0 0 e15⎤k ⎥ [T ] ⎢ gxy ⎥ ⎢ ryy ⎥ = ⎢ 0 0 0 e 0 24 ⎢ ⎥ ⎢g ⎥ ⎢r ⎥ ⎣ xy ⎦ ⎣ e31 e32 0 0 0 ⎦ ⎢ yz ⎥ ⎣ gxz ⎦ ⎡z xx 0 0 ⎤k ⎡ F ⎤k 1 ⎢ ⎥ + ⎢ 0 zyy 0 ⎥ ⎢⎢ F2 ⎥⎥ = [r ] , ⎢⎣ 0 0 z ⎥⎦ ⎣ F3 ⎦ zz
2 C2233 E2 = , 1 - n12 n 21 C3333 C C n12 E2 - 1133 2233 = , C3333 1 - n12 n 21
Q22 = C2222 -
⎡ e11 e12 0 0 0 ⎤ [e] = ⎢⎢ 0 0 e26 0 0 ⎥⎥ [T ] , ⎣ 0 0 0 0 e35⎦ F1 = -
V1 , Dx
(7 a )
(7 b )
where, V1 and Δx are electrical voltage applied along the piezoelectric fibers and the distance between the piezoelectric electrodes in a MFC-d33, respectively. 2.3. Quadratic FE formulation considering first order shear deformation theory
In this study, a pair of MFC actuator patches is bonded to the laminated composite plate (host structure). Various
(5 a )
16
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Figure 15. The ABAQUS results of wo(x, y) in the laminated simply-supported GFRP composite plate induced by a pair of MFC-d31 patches: (a) βa = 0°, (b) βa = 30°, (c) βa = 45°, (d) βa = 60°, (e) βa = 90°.
boundary conditions (Cantilever and simply-support) are prescribed to the smart piezo composite plates. Each MFC actuator has embedded piezoelectric fibers with different orientation angle with respect to x axis (βa = {0°, 30°, 45°, 60°, 90°}). Applied electrical voltage (Va) induces the host structure and creates electro-mechanical strains. The effect of MFC-d31 and MFC-d33 for each case needs to be examined. Considering N number of MFC actuators, the electric displacements, the electrical potentials, the piezoelectric coefficients, and the piezoelectric dielectric constants, the 2D FE matrixes can be derived as stated in equation (8), respectively.
⎡ r1 ⎤ ⎡ F1 ⎤ ⎡ e1 e1 ⎢ s⎥ ⎢ s⎥ ⎢ s1 s2 2 2 ⎢ rs ⎥ ⎢F ⎥ ⎢ e 2 e1 [r ] = ⎢ ⎥, [F] = ⎢ s ⎥, [e] = ⎢ s1 s2 ⎢ ⎥ ⎢ N ⎥ ⎢ N N ⎢⎣ r sN ⎥⎦ ⎣ Fs ⎦ ⎣ es1 es2 ⎡ z1 z1 0 0 0 ⎤ ⎢ s1 s2 ⎥ ⎢z 2s1 z1s2 0 0 0 ⎥ [z ] = ⎢ ⎥, ⎢ ⎥ ⎢⎣z sN1 z sN2 0 0 0 ⎥⎦
17
0 0 0⎤ ⎥ 0 0 0⎥ ⎥, ⎥ 0 0 0⎦
(8 )
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Figure 16. The ABAQUS results of wo(x, y) in the laminated simply-supported GFRP composite plate induced by a pair of MFC-d33 patches: (a) βa = 0°, (b) βa = 30°, (c) βa = 45°, (d) βa = 60°, (e) βa = 90°.
18
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Figure 17. Deflection of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d31 patches: (a) [Piezo/0/30]s
versus [Piezo/0/30/−30/0/Piezo], (b) [Piezo/0/45]s versus [Piezo/0/45/−45/0/Piezo], (c) [Piezo/0/90]s versus [Piezo/0/90/0/90/Piezo].
Table 3. Comparison of the results of the present study and Abaqus at various points (see figures 9–12).
wo
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
9(a) 9(b) 9(c) 9(d) 9(e) 11(a) 11(b) 11(c) 11(d) 11(e)
(
a , 2
)
wo
0 (mm)
Present study
Abaqus
Error (%)
0.0192 0.0210 0.0337 0.0441 0.0551 0.5418 0.3245 1.0120 1.2750 1.5380
0.0204 0.0223 0.0353 0.0459 0.0567 0.5786 0.3463 1.0590 1.3220 1.5840
5.88 5.94 4.52 3.83 2.80 6.36 6.29 4.43 3.55 2.90
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
19
10(a) 10(b) 10(c) 10(d) 10(e) 12(a) 12(b) 12(c) 12(d) 12(e)
(, a 2
b H , 2 2
) (mm)
Present study
Abaqus
Error (%)
0.0836 0.0825 0.0856 0.0927 0.1044 1.1530 0.7631 1.3990 1.6100 1.8820
0.0885 0.0864 0.0897 0.0971 0.1087 1.2340 0.8023 1.4640 1.6760 1.9480
5.51 4.50 4.50 4.51 3.95 6.56 4.88 4.43 3.93 3.38
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Figure 18. Deflection of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d33 patches: (a) [Piezo/0/30]s versus [Piezo/0/30/−30/0/Piezo], (b) [Piezo/0/45]s versus [Piezo/0/45/−45/0/Piezo], (c) [Piezo/0/90]s versus [Piezo/0/90/0/90/Piezo]. Table 4. Comparison of the results of the present study and Abaqus at various points (see figures 17, 18).
wo (a, 0) (mm)
[0 30 30 0]
Figure 17(a) Figure 18(a)
[0 30 −30 0]
Present study
Abaqus
Error (%)
Present study
Abaqus
Error (%)
0.4289 10.1700
0.4490 10.5800
4.47 3.87
0.3699 9.7940
0.3451 9.4230
6.70 3.78
[0 45 45 0]
Figure 17(b) Figure 18(b)
[0 45 −45 0]
Present study
Abaqus
Error (%)
Present study
Abaqus
Error (%)
0.4615 10.650
0.4825 11.060
4.35 3.70
0.3858 10.070
0.3937 10.390
2.00 3.07
[0 90 90 0]
Figure 17(c) Figure 18(c)
[0 90 0 90]
Present study
Abaqus
Error (%)
Present study
Abaqus
Error (%)
0.4354 10.740
0.4693 11.330
7.22 5.20
0.7753 10.570
0.8018 10.060
3.30 4.82
20
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Figure 19. Deflection of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d31 patches: (a) [Piezo/0/30]s versus [Piezo/0/30/−30/0/Piezo], (b) [Piezo/0/45]s versus [Piezo/0/45/−45/0/Piezo], (c) [Piezo/0/90]s versus [Piezo/0/90/0/90/Piezo].
⎡ -1 ⎢ N ⎡ F1 ⎤ ⎢ Dxa s ⎢ ⎥ ⎢ 0 ⎢ F2 ⎥ [F] = ⎢ s ⎥ = ⎢⎢ ⎢ N ⎥ ⎢ ⎣ Fs ⎦ ⎢ ⎢ 0 ⎣
⎤ 0 ⎥ ⎥ ⎡ V1 ⎤ ⎥⎢ 1 ⎥ -1 0 ⎥ ⎢ V12 ⎥ = [B ][V ] , DxaN m 3j ⎥⎢ ⎥ ⎥⎢ N ⎥ - 1 ⎥ ⎣V1 ⎦ 0 ⎥ DxaN ⎦ (9 b ) where, m=3 and j=1 are for MFC-d31 and m=1 and j=3 are for MFC-d33. N=2 for a pair of MFC patches bonded to the host structure. In the next step, a 2D FE formulation is developed using FOSDT, which is based on the Reissner–Mindlin plate theory. Consideration of the transverse shear strains in composite laminates is important since the deformed configuration of composite structures is not necessarily normal to the mid-plane. Thus, The
where, the term S=3 is for MFC-d31 and S=1 is for MFC-d33. N stands for the number of MFC actuators. N=2 since only a pair of bonded MFC patches is considered. Using equation (6b) and equation (7b), the electric field matrix for multiple MFC-d31 and MFC-d33 can be derived as stated in equations (9a), (9b), respectively. ⎡ -1 0 ⎢ N ⎡ F1 ⎤ ⎢ ta ⎢ s⎥ ⎢ -1 0 ⎢ F2 ⎥ [F] = ⎢ s ⎥ = ⎢⎢ taN ⎢ N⎥ ⎢ ⎣ Fs ⎦ ⎢ 0 ⎢ 0 ⎣
⎤ 0 ⎥ ⎥ ⎡ V1 ⎤ ⎥⎢ 3 ⎥ 0 ⎥ ⎢ V32 ⎥ ⎥ ⎢ ⎥ = [B3j ][Vm] , ⎥⎢ N ⎥ - 1 ⎥ ⎣V3 ⎦ N ⎥ ta ⎦
(9 a )
21
0
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Figure 20. Deflection of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d33 patches: (a) [Piezo/0/30]s versus [Piezo/0/30/−30/0/Piezo], (b) [Piezo/0/45]s versus [Piezo/0/45/−45/0/Piezo], (c) [Piezo/0/90]s versus [Piezo/0/90/0/90/Piezo].
effect of additional transverse shear strains must be considered for more accurate perdition of strain–displacement values. To formulate the 2D FE equations, an eight-node quadratic piezoelectric multi-layer shell element with five degrees of freedom is introduced. A four-node element requires a much finer mesh than an eight-node element in order to give convergent displacements and stresses in models involving out-of-plane bending. Thus, eight-node shell element is computationally more accurate and efficient. The five generalized displacements in the shell space can be defined as the derivative operation matrix [Z] and the generalized displacement vector [U] as stated in equation (10). ⎡ Ux ⎤ ⎢U ⎥ ⎡ u ⎤ ⎡1 0 0 z 0⎤ ⎢ y ⎥ ⎢ n ⎥ = ⎢ 0 1 0 0 z ⎥ ⎢ w0 ⎥ = [Z u][U ]. ⎥ ⎣ w ⎦ ⎢⎣ 0 0 1 0 0 ⎦ ⎢ fx ⎥ ⎢ ⎥ ⎣ fy ⎦
Considering the small linear strain–displacement deformation and using equations (2a)–(2d), the strain components can be rearranged in form of a matrix in equation (11), which represents the product of the derivative operation matrix [D] and the generalized displacement vector [U]. ⎡¶ ⎤ ¶ 0 0 z 0 ⎥ ⎢ ¶x ⎢ ¶x ⎥ ¶ ¶ ⎥ ⎢ 0 0 z ⎥ ⎡ Ux ⎤ ⎡ exx ⎤ ⎢ 0 ¶y ¶y ⎢ ⎥ ⎢ eyy ⎥ ⎢ ⎥ U ¶ ¶ ⎥⎢ y ⎥ ⎢g ⎥ ⎢ ¶ ¶ xy = 0 z z w0 = [D][U ]. ⎢ ⎥ ⎢ ¶y ¶x ¶y ¶x ⎥ ⎢ j ⎥ ⎢ ⎢ gyz ⎥ ⎢ ⎥ x⎥ ¶ ⎢ ⎥ ⎢⎣ gxz ⎥⎦ ⎢ 0 0 0 1 ⎥ ⎣ jy ⎦ ¶y ⎢ ⎥ ⎢ ⎥ ¶ ⎢ 0 0 1 0 ⎥ ⎣ ⎦ ¶x
(10)
22
(11)
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Figure 21. The ABAQUS results of wo(x, y) in the laminated cantilevered GFRP composite plate induced by a pair of MFC-d31 patches: (a) [Piezo/0/30]s, (b) [Piezo/0/30/−30/0/Piezo], (c) [Piezo/0/45]s, (d) [Piezo/0/45/−45/0/Piezo], (e) [Piezo/0/90]s, (f) [Piezo/0/90/0/90/Piezo].
To approximate the displacements and rotations of transverse normal for any point at the mid-plane, nodal displacements with the use of Lagrange shape functions are introduced. For the 2D eight-node quadratic shell elements modeled in the present work, the shape functions Ni are defined in local coordinate system known as Jacobian coordinates (equations (12a)–(12c)). The schematic of various quadratic shell elements and their representation in the Jacobian coordinate system is illustrated in figures 3(a)–(c), respectively.
å Ni Uix , Uy = å Ni Uiy, jx = å Ni jix , jy = å Ni j iy , x = å Ni xi , y = å Ni yi ,
Ux =
functions. The 2D eight-node isoparametric quadratic shell elements are given by equations (13a)–(13c). 1
1
Ni = 4 (1 + xxi )(1 + hhi ) - 4 (1 - x 2)(1 + hhi ) 1
- 4 (1 - h 2)(1 + xxi ) , i = 1, 3, 5, 7, 1
Ni = 2 (1 - x 2)(1 + hhi ) , i = 2, 6,
(12a)
1
Ni = 2 (1 - h 2)(1 + xxi ) , i = 4, 8.
(12b)
(13a)
(13b) (13c)
Differentiation of the displacements operator expressed through the shape function in equations (12a)–(12c) leads to the displacement differentiation matrix [B] in equation (14).
(12c)
where, Ni for i=1:8 are used for the coordinates and the displacements Lagrange interpolations shape 23
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Figure 22. The ABAQUS results of wo(x,y) in the laminated cantilevered GFRP composite plate induced by a pair of MFC-d33 patches: (a) [Piezo/0/30]s, (b) [Piezo/0/30/−30/0/Piezo], (c) [Piezo/0/45]s, (d) [Piezo/0/45/−45/0/Piezo], (e) [Piezo/0/90]s, (f) [Piezo/0/90/ 0/90/Piezo].
[Bu] = [D][N ] ⎡ ¶N1 0 ⎢ ⎢ ¶x ¶N1 ⎢ ⎢ 0 ¶y ⎢ ⎢ ¶N1 ¶N1 =⎢ ¶y ¶x ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ 0 0 ⎢⎣
0
z
0 0 ¶N1 ¶y ¶N1 ¶x
¶N1 ¶x 0
z
¶N1 ¶y
0
¶N2 ¶x
0
¶N1 ¶N2 0 ¶y ¶y ¶N ¶N2 ¶N2 z 1 ¶x ¶y ¶x
z
0
1
0
0
1
0
0
0
0
z
0 0 ¶N2 ¶y ¶N2 ¶x
¶N2 ¶x 0
z
¶N2 ¶y
0
¶N8 ¶x
0
¶N2 ¶N8 0 ¶y ¶y ¶N ¶N8 ¶N8 z 2 ¶x ¶y ¶x
z
0
1
0
0
1
0
0
0
24
0
z
0 0 ¶N8 ¶y ¶N8 ¶y
¶N8 ¶x 0
z
¶N8 ¶y 0 1
⎤ 0 ⎥ ⎥ ¶N8 ⎥ z ¶y ⎥ ⎥ ¶N z 8 ⎥. ¶x ⎥ ⎥ 1 ⎥ ⎥ ⎥ 0 ⎥ ⎥⎦
(14)
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Figure 23. The ABAQUS results of wo(x, y) in the laminated simply-supported GFRP composite plate induced by a pair of MFC-d31 patches: (a) [Piezo/0/30]s, (b) [Piezo/0/30/−30/0/Piezo], (c) [Piezo/0/45]s, (d) [Piezo/0/45/−45/0/Piezo], (e) [Piezo/0/90]s, (f) [Piezo/0/90/ 0/90/Piezo].
Computation of the Jacobian matrix [J] entries at any quadrilateral location requires differentiation from the shape functions in equations (12a)–(12c) with respect to the quadrilateral coordinates. Therefore, the matrix [J] can be formed as stated in equation (15). ⎡J [J ] = ⎢ 11 ⎣ J21
⎡ ¶x ⎢ ⎤ ¶ (x , y ) J12 ⎢ ¶x = = ⎢ ¶x J22 ⎥⎦ ¶ (x , h ) ⎢ ⎣ ¶h
¶y ⎤ ⎥ ¶x ⎥ ¶y ⎥ ⎥ ¶h ⎦
⎡ ¶N1e ¶N2e ¶N8e ⎤ ⎡ x1 y1 ⎤ ¼ ⎢ ⎥ ¶x ¶x ¶x ⎥ ⎢ x2 y2 ⎥ =⎢ ⎢ ⎥. ⎢ ¶N1e ¶N2e ¶N8e ⎥ ⎢ ⎥ ¼ ⎢ ⎥ x y ¶h ¶h ⎦ ⎣ 8 8 ⎦ ⎣ ¶h
The shape function derivatives in global coordinate as the elements of the displacement differential matrix [Bu] is obtained by inversing the Jacobian matrix [J] multiplied to the shape function derivatives in the Jacobian coordinates. The results are shown in equation (16). ⎡ ¶Nie ⎤ ⎡ ¶Nie ⎤ ⎥ ⎢ ⎥ ⎢ ⎢ ¶x ⎥ = [J ]-1 ⎢ ¶x ⎥. e e ⎢ ¶Ni ⎥ ⎢ ¶Ni ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ ¶y ⎦ ⎣ ¶h ⎦
(16)
Finally, the strain–displacement matrix [Bu] required for computation of the element stiffness matrix and the 2D FE nodal displacement vector [U] lead to computation of the strains. Therefore, considering the small displacements and material linearity, the strain–displacement nodal parameter matrix can be obtained by using equation (17).
(15)
25
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Figure 24. The ABAQUS results of wo(x, y) in the laminated simply-supported GFRP composite plate induced by a pair of MFC-d33 patches: (a) [Piezo/0/30]s, (b) [Piezo/0/30/−30/0/Piezo], (c) [Piezo/0/45]s, (d) [Piezo/0/45/−45/0/Piezo], (e) [Piezo/0/90]s, (f) [Piezo/0/90/ 0/90/Piezo].
⎡ ¶N1 ¶N ¶N2 ¶N 0 0 z 1 0 0 0 z 2 0 ⎢ ¶x ¶x ¶x ⎢ ¶x ⎢ ¶N1 ¶N ¶N2 ¶N 0 0 0 0 0 z 1 z 2 ⎢ 0 ¶y ¶y ¶y ¶y ⎢ ⎢ ¶N1 ¶N1 ¶N1 ¶N1 ¶N2 ¶N2 ¶N2 ¶N2 0 z 0 z z z [e] = [Bu][U ] = ⎢ ¶y ¶x ¶y ¶x ¶y ¶x ⎢ ¶y ¶x ⎢ ¶N1 ¶N2 0 0 0 1 0 0 0 1 ⎢ ¶y ¶y ⎢ ⎢ ¶N1 ¶N2 0 1 0 0 0 1 0 ⎢ 0 ¶x ¶x ⎣
26
¶N8 ¶x
0
¶N8 ¶y
0 ¶N8 ¶y ¶N8 ¶x
0
0
0
0
0
z
¶N8 ¶x
0
0
0
¶N z 8 ¶y
¶N8 ¶y ¶N8 ¶y
0 1
⎡ U1x ⎤ ⎢ y⎥ ⎢ U1 ⎥ ⎢ w01 ⎥ ⎢ x⎥ ⎤ ⎢ j1 ⎥ 0 ⎥⎢ y ⎥ ⎥ ⎢ j1 ⎥ ¶N8 ⎥ ⎢ U2x ⎥ z ⎥ ¶y ⎥ ⎢ U2y ⎥ ⎥ ⎢ ¶N8 ⎥ ⎢ w02 ⎥ z ⎥⎢ j x ⎥ ¶x ⎥ 2 ⎥ ⎢ ⎥ jy 1 ⎥⎢ 2 ⎥ ⎥ ⎢ ⎥⎢ ⎥ ⎥ ⎢ U8x ⎥ 0 ⎥ ⎦ ⎢ U8y ⎥ ⎥ ⎢ ⎢ w08 ⎥ ⎢ jx ⎥ ⎢ 8y ⎥ ⎢⎣ j 8 ⎥⎦
(17)
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Figure 25. Deflection of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d31 patches-upper and lower patches voltages: (a) V = {+100, −40} (V), (b) V = {+150, −80} (V), (c) V = {+240, −120} (V), respectively.
Table 5. Comparison of the results of the present study and Abaqus at various points (see figures 19, 20).
wo
(
a , 2
)
0 (mm)
[0 30 30 0]
Figure 19(a) Figure 20(a)
[0 30 −30 0]
Present study
Abaqus
Error (%)
Present study
Abaqus
Error (%)
−0.0334 −1.2800
−0.0340 −1.3190
1.64 3.04
−0.0369 −1.3700
−0.0382 −1.4160
3.49 3.35
[0 45 45 0]
Figure 19(b) Figure 20(b)
[0 45 −45 0]
Present study
Abaqus
Error (%)
Present study
Abaqus
Error (%)
−0.0377 −1.3760
−0.0385 −1.4280
2.19 3.77
−0.0404 −1.4280
−0.0415 −1.4800
2.74 3.64
[0 90 90 0]
Figure 19(c) Figure 20(c)
[0 90 0 90]
Present study
Abaqus
Error (%)
Present study
Abaqus
Error (%)
−0.0551 −1.5380
−0.0579 −1.6080
5.11 4.55
−0.1433 −1.8510
−0.1525 −1.9430
6.42 4.97
27
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Figure 26. Deflection of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d33 patches-upper and lower patches voltages: (a) V = {+1000, −250} (V), V = {+1300, −350} (V), and V = {+1500, −500} (V), respectively.
where, the subscripts e stands for the element on which the total internal/external work is done and δ is the variation operator. fb, fs, and fc represent the body, surface, and concentrated charge vector, respectively. ψs and ψc are the surface and concentrated charge vectors in an element, respectively. Substituting equations (6a), (7a), (9a), (9b), (10), (14) into (18b) and taking into account the 2D linear strain–displacement deformation, the static equilibrium equations representing the 2D electromechanically coupled FE model are obtained (equation (19)). Computation of the system matrix in equation (19) leads to obtaining the generalized displacements vector [U]. Therefore, through the derivative operation matrix [Zu], the general form of displacement fields is computed.
To drive an assembly algorithm and formulate the 2D FE equations, the principle of virtual work is employed to obtain the 2D linear electro-mechanically coupled static FE equations. The FE modeling can be extended for sensor equations. However, since the aim of this research is to investigate the inverse effect of piezoelectric materials on shape deformation of smart laminated composite structures, the sensor equations are neglected. The principle of virtual work in equations (18a), (18b) stipulates that the total internal work done on a mechanical system should be equal to the total external work. dWint = dWext ,
(18a)
òV (d [ee]T [s e] - d [Fe]T [r e]) dV = òV (d [U e]T fb ) dV + ò (d [U e]T fs ) d W-ò (d [Vme]T ys) d W W W
òV [Bue]T [T ]T [C ][T ][Bue] dV - Vm òV [Bue]T ´ [e]T [B3ej ] dV = ò [N e]T [Zue]T fb dV V e T e T + ò [N ] [Zu ] fs dV + [N e]T [Zue]T fc . W
[U e ]
+ d [U e]T fc - d [Vme]T yc, (18b)
28
(19)
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Figure 27. Deflection of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d31 patches-upper and lower patches voltages: (a) V = {+100, −40} (V), (b) V = {+150, −80} (V), (c) V = {+240, −120} (V), respectively.
Table 6. Comparison of the results of the present study and Abaqus at various points (see figures 25–28).
wo
wo (a, 0) (mm)
Figure Figure Figure Figure Figure Figure
25(a) 25(b) 25(c) 26(a) 26(b) 26(c)
Present study
Abaqus
Error (%)
−0.1116 −0.1989 −0.3056 −3.5210 −4.7410 −6.0980
−0.1171 −0.2027 −0.3126 −3.3630 −4.8690 −6.2070
4.92 1.91 2.29 4.69 2.69 1.78
Figure Figure Figure Figure Figure Figure
27(a) 27(b) 27(c) 28(a) 28(b) 28(c)
(
a , 2
)
0 (mm)
Present study
Abaqus
Error (%)
0.0053 0.0128 0.0185 0.2695 0.3758 0.5318
0.0060 0.0137 0.0193 0.3005 0.3862 0.5523
11.62 6.34 4.43 10.31 2.69 3.71
composite plate) and a pair of two MFC patches bonded to the host structure (figures 1(a), (b)). Assigning material properties to MFC patches and host structure in local material ordination (figure 4(a)), prescribing boundary conditions and applying electrical loads to MFC patches after part assembly (figure 4(b)), and meshing MFC patches and composite laminate (figure 4(c)) are done step by step during
2.4. Numerical simulation
For the present study verification, the electro-mechanical coupled FE simulation of smart composite plates induced by a pair of MFC-d31 and FMC-d33 actuator patches is implemented by using ABAQUS. A smart laminated piezo composite structure consists of the host structure (laminated fiber-reinforced
29
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Figure 28. Deflection of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d33 patches-upper and lower patches voltages: (a) V = {+1000, −250} (V), V = {+1300, −350} (V), and V = {+1500, −500} (V), respectively.
order to verify the present study results at any nodal point or various points, a path is defined in the ABAQUS contour plots. The deformation values from any specific path is picked and then compared with the results of the present study. In the present study, the effect of various parameters including MFC actuator type, applied electrical voltage, laminates stacking sequence, twisting-bending coupling due to unsymmetrical stacking sequence configuration, and prescribed boundary conditions are considered. Material properties of MFC actuator patches and CFRP composite laminate in the following examples are summarized in table 1 [18, 42]. The plate thickness and dimensions are tp =1 (mm), a=0.24 (m), and b=0.06 (m). MFC patches thickness and dimensions are ta =0.2 (mm) and La =wa =0.06 (m) in both MFC-d31 and MFC-d33. The distance between the electrodes in MFC-d33 is Δx = 0.2 (mm). MFCs placement and composite laminate properties, and composite laminate’s number of layers (fourthlayered fiber-reinforced CFRP composite plate) are kept constant. Two types of boundary conditions are prescribed to the host structure, which are same for all samples (figures 1(a), (b)).
FE simulation. In ABAQUS software, U1, U2, and U3 represent the displacements and UR1, UR2, and UR3 as the rotational angles along x, y, and z directions, respectively. Since MFC actuator type varies during the analysis, the piezoelectric coefficients d31 and d32 for MFC-d31 type and d11 and d12 for MFC-d33 type are considered. The piezoelectric and the host structure local material orientations are defined, thereafter. An 8-node quadrilateral in-plane general-purpose continuum shell, reduced integration with hourglass control, finite membrane strains (SC8R) is defined as the element type for the host structure while a 20-node quadratic piezoelectric brick, reduced integration (C3D20RE) is used for MFC patches (figure 4(c)).
3. Results and case-study application examples In this section, various examples are intended to analyze reliability and accuracy of the 2D quadratic piezoelectric multi-layer shell element for effective FE analysis of smart laminated composite plates induced by MFC actuators. In 30
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Figure 29. The ABAQUS results of wo(x, y) in the laminated cantilevered GFRP composite plate induced by a pair of MFC-d31 patches-upper and lower patches voltages: (a) V = {+100, −40} (V), (b) V = {+150, −80} (V), (c) V = {+240, −120} (V), respectively.
Subsequently, the effect of MFC-d33 on shape deformation of the host structure is taken into account. The 2D deflection and twisting results of the present study and their comparison with ABAQUS for MFC-d33 case are shown in figures 7(a)–(e) and 8(a)–(e), respectively. The same deflection and twisting patterns occurred when using MFC-d33 as observed in MFC-d31 case. However, due to higher dielectric constants and electrical voltage limitation in MFC-d33 actuators, the intensity of bending-twisting deformation caused by MFC-d31 and MFC-d33 varies considerably. Comparison of the results presented in figures 5–8 at various points and their relative errors are tabulated in table 2. In the second attempt, the simply-supported plates are considered. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d31 case are shown in figures 9(a)–(e). Like the cantilevered case, figures 9(a)–(e) results at wo (x, 0) demonstrate that any changes in βa can significantly affect bending deformations in the host structure. However, unlike the cantilever plates, the bending patterns in the simply-supported plates show positive and negative displacements when βa ≠{0°, 90°}. In the simply-support case, mechanical deformation of the upper MFC patch at W(a/2, y, H/2) is also obtained using the present study and then compared with the ABAQUS results as seen in figures 10(a)–(e). The 2D deflection and twisting results of the present study
3.1. Effect of MFC actuator fiber angle
The cross-ply laminated cantilevered and simply-supported plates with stacking sequence [0/90]s are initially considered. To examine the sole effect of MFC actuator fibers angle, the same amount of electrical voltage is applied to all actuators. The host structure stacking sequence is also kept constant. The MFC actuator fiber angles are βa ={0°, 30°, 45°, 60°, 90°}. The electrode angle causes the host structure to be deflected and twisted when electrical voltage is applied to MFC actuators. The electrical voltage applied to MFC-d31 is upper patch: +240 (V) and lower patch: −120 (V) and to MFC-d33 is upper patch: +1500 (V) and lower patch: −500 (V). The magnitude and pattern of applied electrical voltage to MFC patches are the same in both cantilevered and simplysupported cases. In the first attempt, the cantilever plates are considered. The 2D deflection and twisting results of the present study and their comparison with ABAQUS for MFC-d31 case are shown in figures 5(a)–(e) and 6(a)–(e), respectively. Figures 5(a)–(e) results at wo(x, 0) demonstrate that any changes in βa can significantly affect bending deformations in the host structure. Furthermore, the results from both approaches at wo(a, y) demonstrates the effectiveness of βa≠{0°, 90°} on creating twist in the host structure.
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Figure 30. The ABAQUS results of wo(x, y) in the laminated cantilevered GFRP composite plate induced by a pair of MFC-d33 patchesupper and lower patches voltages: (a) V = {+1000, −250} (V), V = {+1300, −350} (V), and V = {+1500, −500} (V), respectively.
and their comparison with ABAQUS for MFC-d33 case are shown in figures 11(a)–(e) and 12(a)–(e), respectively. The same deflection and twisting patterns occurred when using MFC-d33 as observed in MFC-d31 case. Like the cantilevered case, the higher deformation rate is caused by in MFC-d33 actuators due to having the higher maximum voltage limit as well as dielectric constants. Like both the present study and ABAQUS results in the cantilevered case, any fiber angle other than 0° and 90° results in twisting deformation while 0° and 90° result in pure cylindrical bending in the simply-supported case regardless of MFC actuator type. It is also obvious that any changes in piezoelectric fiber angle result in different shape deformation regardless of boundary condition. Comparison of the results presented in figures 9–12 at various points and their relative errors are tabulated in table 3. The 3D FE simulations of the laminated cantilevered GFRP composite plate induced by MFC-d31 and MFC-d33 actuators with various piezoelectric fiber angles are illustrated in figures 13(a)–(e) and 14(a)–(e), respectively. The 3D FE simulations of the laminated simply-supported GFRP composite plate induced by MFC-d31 and MFC-d33 actuators with various piezoelectric fiber angles are illustrated in figures 15(a)–(e) and 16(a)–(e), respectively.
3.2. Effect of the host structure stacking sequence configurations
In this example, the effect of various symmetrical and unsymmetrical stacking sequence configurations on shape deformation of the smart laminated composite plates induced by MFC actuators is investigated. The relationship between stacking sequence and composite laminate stiffness can result in shape deformation changing considerably. Thus, it is important to choose suitable layup to control the structural shape deformation of composite laminates to our advantage. The samples with symmetrical stacking sequence are: [Piezo/0/30]s, [Piezo/0/45]s, [Piezo/0/90]s, and unsymmetrical: [Piezo/0/30/−30/0/Piezo], [Piezo/0/45/−45/0/ Piezo], [Piezo/0/90/0/90/Piezo]. Each layer thickness and applied electrical voltage is kept constant. Same boundary condition is prescribed to the host structure as in example.1. To investigate the sole effect of stacking sequence variation, all MFC actuators fiber angle orientation is kept constant at 0°. The electrical voltage applied to MFC-d31 is upper patch: −120 (V) and lower patch: +240 (V) and to MFC-d33 is upper patch: −500 (V) and lower patch: +1500 (V). The magnitude and pattern of applied electrical voltage to MFC patches are the same in both cantilevered and simply-supported cases. The deflection results at wo(x, 0) for both
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Figure 31. The ABAQUS results of wo(x, y) in the laminated simply-supported GFRP composite plate induced by a pair of MFC-d31 patchesupper and lower patches voltages: (a) V = {+100, −40} (V), (b) V = {+150, −80} (V), (c) V = {+240, −120} (V), respectively.
cantilevered and simply supported cases are considered. For each case, both the MFC-d31 and MFC-d33 results obtained from the current study and ABAQUS are discussed. In the first attempt, the cantilever plates are considered. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d31 case are shown in figures 17(a)–(c), which demonstrate the effectiveness of stacking sequence in various shape deformations. For instance, among symmetrical laminates, the maximum and minimum shape deformations were observed in [Piezo/0/90]s and [Piezo/0/30]s, respectively, while no significant changes in shape deformation were observed among unsymmetrical laminates. However, the comparison between symmetrical and unsymmetrical laminates indicated higher shape deformation in all cases regardless of the host structure winding angle changes. Furthermore, shape deformation variation between [Piezo/0/90]s and [Piezo/0/90/0/90/Piezo] was observed to be the most noticeable one in comparison with other cases. Subsequently, the effect of MFC-d33 on shape deformation of the host structure with various stacking sequence configuration is taken into account. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d33 case are shown in figures 18(a)–(c). The same deflection patterns occurred when using MFC-d33 as it was observed in MFC-d31 case. However, due to the higher dielectric constants and electrical voltage limitation in MFC-d33 actuators, the intensity of bending deformation caused by MFC-d31 and MFC-d33 varies. In addition, unlike the MFC-d33 actuator results, the shape deformation results of the host structure induced by MFC-d33 actuators are not significantly different, regardless of
changes in the laminates winding angle and layup symmetry/ asymmetry status. Comparison of the results presented in figures 17, 18 at various points and their relative errors are tabulated in table 4. In the next attempt, the simply-supported plates are considered. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d31 case are shown in figures 19(a)–(c). It can be observed that the lamination layup can significantly affect shape deformation of the simply supported plates induced by MFC-d31 actuators. According to figures 19(a), (b), which are the results for the angle-ply laminates (βp at the Kth layer ≠ {0°, 90°}), the magnitude of deformation seems to be the same when comparing the symmetrical and unsymmetrical layups. However, that is not the case when the laminates are not angle-ply (figure 19(c)). The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d33 case are shown in figures 20(a)–(c). Like MFC-d31, the host structure can be significantly induced by MFC-d33 actuators. However, like the cantilevered case, higher magnitude of deformation is created by MFC-d33 due to their higher dielectric constants and electrical voltage limitation. Comparison of the results presented in figures 19, 20 at various points and their relative errors are tabulated in table 5. The 3D FE simulations of the laminated cantilevered GFRP composite plates induced by MFC-d31 and MFC-d33 actuators with various stacking sequence configurations are illustrated in figures 21(a)–(e) and 22(a)–(e), respectively. The 3D FE simulations of the laminated simply-supported GFRP composite plates induced by MFC-d31 and MFC-d33 33
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Figure 32. The ABAQUS results of wo(x, y) in the laminated simply-supported GFRP composite plate induced by a pair of MFC-d33 patchesupper and lower patches voltages: (a) V = {+1000, −250} (V), V = {+1300, −350} (V), and V = {+1500, −500} (V), respectively.
respectively (figures 25(b), (c)). MFC-d31 actuators are then replaced with MFC-d33 types, whose results can be found in figures 26(a)–(c). Like the MFC-d31 results, a same trend in variation of the plate tip deflection was observed. By increasing the upper and lower MFC patches electrical voltage to V={+1000, −250} (V), V={+1300, −350} (V), and V={+1500, −500} (V), respectively, the plate tip deflections increased to −3.521 (mm), −4.741 (mm), −6.098 (mm), respectively. The results were confirmed through both the proposed 2D piezoelectric multi-layer shell element and ABAQUS. Good agreement was observed between the results obtained from both approaches. Furthermore, it was noticed that the plate tip deflections are significantly higher when choosing MFC-d33 actuators, which make them ideal for implementation of shape deformation. Comparison of the results presented in figures 25, 26 at various points and their relative errors are tabulated in table 6. In the next attempt, the simply-supported plates are considered. Like the cantilevered case, the host structure shape deformation differs as the intensity of applied electrical voltage varies. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d31 case are shown in figures 27(a)–(c). Unlike the cantilevered case, the maximum deflection takes place in the middle of the plate in the simply-supported case. By increasing the upper and the lower MFC patches electrical voltage to V={+100, −40} (V), V={+150, −80} (V), and V={+240, −120} (V), respectively, the plate mid-deflections increased to 0.005 (mm), 0.012 (mm), 0.018 (mm), respectively. MFC-d31 actuators are then replaced with MFC-d33 types, whose results
actuators with various stacking sequence configurations are illustrated in figures 23(a)–(e) and 24(a)–(e), respectively. 3.3. Effect of MFC actuator electrical voltage intensity
Composite laminates shape can be significantly induced by changing electrical voltage. Therefore, in this example, the effect of applied electrical voltage intensity on shape deformation of the laminated composite plates induced by MFC actuators is investigated. For that purpose, a series of arbitrarily selected electrical voltage ranging from lowest to maximum voltage limit are applied to MFC actuators. Like the previous examples, the deflection results at wo(x, 0) for both cantilevered and simply supported cases are considered. For each case, both the MFC-d31 and MFC-d33 results obtained from the current study and ABAQUS are discussed. The host structure stacking sequence configuration is [0, 90]s which is kept constant. In the first attempt, the cantilever plates are considered. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d31 case are shown in figures 25(a)–(c). As predicted before, by increasing the intensity of applied electrical voltage, higher mechanical displacements are observed. For instance, when applying V={+100, −40} (V) to the upper and lower patches, respectively, the plate tip deflection of −0.111 (mm) occurred (figure 25(a)). However, by increasing the upper and lower MFC patches electrical voltage to first V={+150, −80} (V) and then to V={+240, −120} (V), respectively, the plate tip deflections increased to −0.198 (mm) and −0.305 (mm), 34
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Figure 33. In-plane deformation of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d31 patches: (a) Ux(x, 0), (b) Ux(a, y), (c) Uy(x, 0), (d) Uy(a, y).
can be found in figures 28(a)–(c). Increasing electrical voltage seemed to have the same effect on shape deformation variation as observed in the results obtained in MFC-d31 case. However, in both cantilevered and simply-supported cases, MFC-d33 seemed to have higher actuation power for shape control problems than MFC-d31. For example, by increasing the upper and lower MFC-d33 patches electrical voltage to V={+1000, −250} (V), V={+1300, −350} (V), and V={+1500, −500} (V), respectively, the plate mid-deflections increased approximately to 0.269 (mm), −0.375 (mm), −0.531 (mm), respectively. Good agreement between the results of the present study and ABAQUS was observed. Comparison of the results presented in figures 27, 28 at various points and their relative errors are tabulated in table 6. The 3D FE simulations of the laminated cantilevered GFRP composite plates induced by MFC-d31 and MFC-d33 actuators undergoing various applied electrical voltages are illustrated in figures 29(a)–(e) and 30(a)–(e), respectively. The 3D FE simulations of the laminated simply-supported GFRP composite plates induced by MFC-d31 and MFC-d33
actuators undergoing various applied electrical voltages are illustrated in figures 31(a)–(e) and 32(a)–(e), respectively.
3.4. Further study on in-plane deformation caused by MFC-d31 and MFC-d33
In the final example, the mid-plane displacements Ux and Uy of the smart laminated GFRP composite plates induced by a pair of MFC actuators are taken into account. To observe the effect of MFC actuators electrode rotational angle (βa) on the in-plane shape deformation results, MFC actuators with βa={0°, 45°} are selected. The results are investigated by using the 2D piezoelectric multi-layer shell element and ABAQUS to verify the proposed method. Like the previous examples, both the cantilevered and simply-supported plates are considered during the analysis. The electrical voltage and the host structure stacking sequence configuration are kept constant. The electrical voltage applied to MFC-d31 is upper patch: −120 (V) and lower patch: +240 (V) and to MFC-d33 35
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Figure 34. In-plane deformation of the laminated cantilevered GFRP composite plate induced by a pair of MFC-d33 patches: (a) Ux(x, 0), (b) Ux(a, y), (c) Uy(x, 0), (d) Uy(a, y).
is upper patch: −500 (V) and lower patch: +1500 (V). The host structure stacking sequence configuration is [0, 90]s. In the first attempt, the cantilever plates are considered. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d31 case are shown in figures 33(a)–(d). The results demonstrate the effectiveness of MFC electrode angle in creating different shape deformation in the host structure. It was also noticed that Ux and Uy vary bilinearly along the plate’s length (figures 33(a), (c)) while they present rather linear behavior along the plate’s width (figures 33(b), (d)). These changes are irrespective of MFC electrode angle. Like the MFC-d31 results, the same trend in variation of in-plane deformation was observed when choosing MFC-d33 but with different intensity due to the major difference in MFC actuators material crystallization, dielectric coefficient matrixes, and applied electrical voltage limits. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d33 case are shown in figures 34(a)–(d).
In the next attempt, the simply-supported plates are considered. The 2D deflection results of the present study and their comparison with ABAQUS for MFC-d31 and MFC-d33 cases are shown in figures 35(a)–(d) and 36(a)–(d), respectively. Like the cantilevered case, both MFC-d31 and MFC-d33 proved to influence in-plane deformation in the host structure. However, due to the different type of boundary condition, less deformation was observed in the simply-supported case. Furthermore, the magnitude of in-plane deformation was observed to be much less than the vertical displacements in both cantilevered and simply-supported case, which are irrespective of the MFC actuator type.
4. Concluding remarks In this research, a quadratic piezoelectric multi-layer shell element using FOSDT was developed to predict the linear strain–displacement deformation in the laminated composite 36
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Figure 35. In-plane deformation of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d31 patches: (a) Ux(x, 0), (b) u(a/2, y, H/2), (c) Uy(x, 0), (d) v(a/2, y, H/2).
(1) Comparison of the results between the 2D FE formulation developed by the authors and commercial FE software ABAQUS showed good agreement, which demonstrates the reliability and cost-effectiveness of the model proposed in this paper for FE analysis of smart laminated piezo composite plates under various boundary conditions. (2) MFC actuators can induce mechanical deformation in laminated composite structures through the application of electrical voltage. The intensity of shape deformation varies considerably by changing piezoelectric fibers angle, stacking sequence configuration in the host structure, layup asymmetry, applied electrical voltage intensity, and MFC actuator type. (3) Strain-displacement deformation induced by MFCs is affected by MFC actuator type due to changes in the elements of piezoelectric dielectric coefficient matrix (dij). Typically, MFC-d33 produces higher actuation power than MFC-d31 due to having much larger
plates induced by MFC actuators. FOSDT was adapted from the Reissner–Mindlin plate theory. An eight-node quadratic piezoelectric multi-layer shell element with five degrees of freedom was proposed to overcome locking effect and zero energy modes observed in nine-node degenerated shell element. Two types of MFC actuators were used: (1) MFC-d31 and (2) MFC-d33. For verifying the results of the present study, the electro-mechanical coupled FE simulation of the smart laminated composite plates induced by MFC-d31 and MFC-d33 was implemented using ABAQUS. Good agreement between the results was observed. Furthermore, the effect of actuation power, MFC electrode orientations, prescribed boundary conditions, laminate stacking sequence configuration, and symmetry/asymmetry layups was comprehensively investigated. A step by step guideline for FE simulation of smart piezo composite structures was also proposed. Variety of mechanical and electrical factors affecting the electro-mechanically coupled FE modeling was accounted for design recommendations. The research contributions reported in this paper are summarized as follows: 37
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Figure 36. In-plane deformation of the laminated simply-supported GFRP composite plate induced by a pair of MFC-d33 patches: (a) Ux(x, 0), (b) u(a/2, y, H/2), (c) Uy(x, 0), (d) v(a/2, y, H/2).
Colin Burvill https://orcid.org/0000-0002-6294-4467 Zora Vrcelj https://orcid.org/0000-0002-1403-7416
dielectric constants. Thus, they can be more effective in controlling larger shape deformations.
References
Acknowledgments
[1] Sharifishourabi G, Ayob A, Gohari S, Yahya M Y B, Sharifi S and Vrcelj Z 2015 Flexural behavior of functionally graded slender beams with complex crosssection J. Mech. Mater. Struct. 10 1–16 [2] Rajaneesh A, Satrio W, Chai G B and Sridhar I 2016 Longterm life prediction of woven CFRP laminates under three point flexural fatigue Composites B 91 539–47 [3] Sharifi S, Gohari S, Sharifiteshnizi M and Vrcelj Z 2016 Numerical and experimental study on mechanical strength of internally pressurized laminated woven composite shells incorporated with surface-bounded sensors Composites B 94 224–37 [4] Gohari S, Sharifi S, Vrcelj Z and Yahya M Y 2015 First-ply failure prediction of an unsymmetrical laminated ellipsoidal woven GFRP composite shell with incorporated surfacebounded sensors and internally pressurized Composites B 77 502–18
The first author would like to thank A/Professor Colin Burvill in the Department of Mechanical Engineering at the University of Melbourne for his help and support during this research project.
ORCID iDs Soheil Gohari https://orcid.org/0000-0002-2165-448X Shokrollah Sharifi https://orcid.org/0000-0002-3423-9672 Rouzbeh Abadi https://orcid.org/0000-0003-1649-9996 Mohammadreza Izadifar https://orcid.org/0000-00028153-4834 38
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