Modeling and simulation of macro-fiber composite ...

Modeling and simulation of macro-fiber composite layered smart structures Shun-Qi Zhanga,c,∗, Ya-Xi Lib,c , R¨udiger Schmidtc a

School of Mechanical Engineering, Northwestern Polytechnical University, West Youyi Street 127, 710072 Xi’an, China b School of Aeronautics, Northwestern Polytechnical University, West Youyi Street 127, 710072 Xi’an, China c Institute of General Mechanics, RWTH Aachen University, Templergraben 64, D-52062 Aachen, Germany

Abstract Piezo fiber composite material, macro-fiber composite (MFC), is increasingly applied in engineering, due to its high flexibility and strong actuation forces. This paper develops a linear electromechanically coupled finite element (FE) model for composite laminated thin-walled smart structures bonded with orthotropic MFCs having arbitrary piezo fiber orientation. Two types of MFCs are considered, namely, MFC-d31 in which the d31 effect dominates the actuation forces, and MFC-d33 which mainly uses the d33 effect. The FE model is developed based on the ReissnerMindlin hypothesis using linear piezoelectric constitutive equations. The present results are compared with ANSYS and experimental results reported in the literature (Bowen et al., 2011). Afterwards, isotropic or composite structures with cross-ply laminates, integrated with MFC-d31 or -d33 patches having different fiber orientation, are simulated under a certain electric voltage on the MFC patches. Keywords: Macro-fiber composite, smart structures, piezoelectric, finite element 1. Introduction Structures integrated with smart materials, e.g. piezoelectric, magnetostrictive and shape memory alloy, are applied increasingly in many fields of technology for shape control, vibration control and health monitoring. Amongst them piezoelectric materials are the most widely used ones in industrial applications due to a number of beneficial properties that these materials possess. Piezoceramics, like lead zirconium titanate (PZT), have a high structural stiffness, which generates strong actuation forces. However, there are several practical limitations to implement this typically lead-based piezoceramic materials, for example the brittle nature of ceramics which makes them susceptible to fracture during handling and bonding procedures, and their extremely limited ability ∗

Corresponding author at: the School of Mechanical Engineering, Northwestern Polytechnical University, West Youyi Street 127, Postbox 554, 710072 Xi’an, China. Tel.: +86 029 88494271 Email address: [email protected], [email protected] (Shun-Qi Zhang)

Preprint submitted to Journal

February 10, 2015

to fit with curved surfaces [1]. Another very frequently used piezoelectric material is polyvinylidene fluoride (PVDF), which is much more flexible than piezoceramics, but with low actuation forces. The idea of a hybrid material consisting of piezoceramic fibrous phase embedded in epoxy matrix phase remedies many of the aforementioned limitations. The first type of this piezo fiber composite material is referred to as 1-3 composite, and is manufactured by Smart Material Corp. [2]. The second type are active fiber composite (AFC) actuators, which were originally developed by MIT and were the first composite actuators primarily used on structural actuation [1]. The third one, macro-fiber composite (MFC), was developed by NASA Langley Research Center in 1999 [1, 3, 4]. The flexible nature of MFC allows the material conforming to a curved surface easily. Additionally, an MFC patch even has larger actuation forces than a PZT patch, since the d33 effect dominates the actuation mode in MFCs. For more detailed information of active piezoelectric fiber composites, we refer to [5–7]. Because of these beneficial properties of piezoelectric fiber composites, considerable efforts have been made to integrate this material into metal structures for vibration control [8, 9] and health monitoring [10–12]. Due to the complexity of MFC structure, many researchers focused on the determination of material properties for homogenized MFC patches based on experimental or numerical investigations. The very early material properties of MFC, including the basic elasticity constants for orthotropic thin-walled structures in both the linear elastic region and the nonlinear constitutive behavior, were obtained experimentally by Williams et al. [13]. Later, Park and Kim [14], predicted the material properties of MFC using classical lamination theory and uniform fields model. Furthermore, Deraemaeker et al. [15–17] proposed a representative volume element (RVE) technique and mixing rules for the determination of equivalent material properties of both d31 - and d33 -type MFCs, while Biscani et al. [18] developed an asymptotic expansion homogenization (AEH) method for material parameters of d31 -type MFCs. Additionally, actuation properties for MFC under strong voltages were investigated by Williams et al. [19] using the theoretical piezoelectric constitutive model with higher-order electric field proposed in [20]. Analogously, Schr¨ock [21] investigated experimentally the hysteresis and creep effects in structural dynamics of MFC bonded plates. In order to get the structural response of MFC integrated smart structures, some papers available in the literature discussed simulation techniques with the assistance of commercial software, e.g. ANSYS [22, 23], ABAQUS [24, 25], and compared the results with those from experiments. Furthermore, Bowen et al. [23] predicted the cured shape and snap-through of asymmetric bistable laminates actuated by piezoelectric macro fiber composites attached to the laminate with the help of ANSYS. A similar prediction of piezoelectric-induced snap-through of a bistable carbon fiber reinforced plastic combined with an MFC was conducted by Giddings et al. [26]. Beyond usage of commercial softwares, Bilgen et al. [27] built a linear distributed parameter electro-mechanical model for frequency response analysis of MFC actuated clamped-free thin beams, and compared 2

their results with experiments. Similarly, Azzouz and Hall [28] developed a von K´arm´an nonlinear FE model based on the first-order shear deformation (FOSD) hypothesis for frequency response of a rotating MFC actuator. From the aforementioned publications, it can be seen clearly that most of them presented various applications and experimental investigations of MFC materials, as well as homogenization of MFC using analytical solutions or experimental results. Fewer papers developed a general FE model of thin-walled smart structures bonded with d31 - and d33 -type MFC layers or patches, after the first MFC manufactured in 1999. It is well known that since 1990’s the modeling and simulation of monolithic piezoelectric materials is booming. A large number of papers appeared in the literature developed linear models of thin-walled metal structures bonded with monolithic piezoelectric layers or patches using 2-dimensional (2D) finite elements based on various hypotheses, e.g. Kirchhoff-Love hypothesis [29, 30], Reissner-Mindlin hypothesis [31–34], third- or higherorder shear deformation hypotheses [35, 36], and zigzag hypothesis [37–39]. About one decade later, more and more researchers took geometrically nonlinear effects into account for FE modeling of piezoelectric layered smart structures undergoing large displacements, see e.g. in [40–47] among others. However, the above mentioned study on simulation of MFC bonded structures were carried out with commercial software, and they did not take the fiber angle variation into account, which may influence significantly the structural response. Moreover, very less work has been developed and presented for modeling and simulation of MFC bonded smart structures. In order to model and simulate fiber-based MFC piezo materials with various fiber angle arrangement, this paper is to develop a linear FE model for thin-walled structures integrated with d31 - and d33 -type MFC layers or patches using 2D finite elements based on the Reissner-Mindlin hypothesis. MFC is a piezo fiber material, consisting of monotonic piezoelectric material, epoxy matrix and electrodes with a specific arrangement, which can be considered as homogenized orthotropic materials with arbitrary piezo fiber angles like composite structures. The host structures are comprised of one isotropic metal layer or composite structures with cross-ply or angle-ply laminates. The model is first validated by one example from the literature, and then used for calculation of cantilevered plates with various fiber orientation for both host structures and MFCs. 2. MFC models Macro fiber composites mainly consist of piezoceramic fibers, epoxy matrix and electrodes, which have two different types of structures, yielding d31 or d33 modes. The first type, abbreviated as MFC-d31, has piezoelectric material polarized in the thickness direction normal to the fiber direction, thus the d31 effect is dominating the actuation forces. However, the second type of MFC, denoted as MFC-d33, is arranged in a specific manner such that the polarization of the piezoelectric material is along the piezo-fiber direction. Therefore, MFC-d33 can use the d33 effect 3

for generation of actuation forces, which is usually much larger (about 2 times larger) than the d31 effect. Additionally, actuation voltages for MFC-d31 patches can be applied in the range from −60 to 360 V (with the electrode separation of 0.18 mm), while those for MFC-d33 patches can vary between −500 and 1500 V (with center-to-center interdigitated electrode spacing of 0.5 mm) [2]. The schematic of these two kinds of MFCs are shown in Fig. 1a and Fig. 1b, respectively. Θ

PZT

2

Θ3

˜2 Θ ˘2 Θ P

Fiber

Electrodes ˜2 Θ

Epoxy

Θ3

PZT Epoxy

Θ2

Electrodes

˘2 Θ

hE

˜1 Θ

˜3 Θ Fibe r,

˘1 Θ

P

Θ

+

+

-

Fiber

Fiber hE

1

˘1 Θ P

+

Θ1

˜1 Θ

+

-

P

P

˜3 Θ

+

(a) MFC-d31 structure

(b) MFC-d33 structure

Figure 1: Schematic of different kinds of MFC models

This kind of arrangement for electrodes also improves structural flexibility. Because the electric field is applied at regular intervals, any damage or fracture of the piezoceramics or electrode merely reduces the performance of a small area surrounding the defect and will not significantly reduce the overall actuation effect [23]. In the present model, three coordinate systems are employed, as can be seen in Fig. 1, namely the curvilinear coordinate system represented by Θi (i = 1, 2, 3), the fiber coordinate system de˘ i , and the polarization coordinate system shown as Θ ˜ i . The curvilinear coordinate noted by Θ system is usually attached to thin-walled structures with the Θ3 -line defining the thickness direction, representing the geometry of the respective structures; the fiber coordinate system defines the ˘ 1 ; and the polarization coordinate fiber orientation, which gives the fiber angle between Θ1 and Θ ˜ 3 -line pointing along the direction of polarization of system is used for MFC material with the Θ piezoelectric material. From the previous study on homogenization of MFC materials, it is known that MFC can be treated as an orthotropic material, see e.g. [15–18] among others. In this paper we consider MFC layers or patches as electro-mechanically coupled composite structures with fiber angles. For the structural details of MFC material, we refer to [2, 5–7]. 3. Constitutive equations Using the assumptions of small strains and weak electric field for piezoelectric patches or layers, the constitutive equations in the fiber coordinate system can be expressed as [48] ε˘ij = s˘ijkl σ ˘kl + d˘mij E˘m , ˘ m = d˘mkl σ ˘n . D ˘kl + ǫ˘mn E 4

(1) (2)

Here, the Latin indices, i, j, k, l, m, n, assume the numbers 1, 2 or 3, while ij or kl denote 11, 22, 33, 12 or 21, 13 or 31, 23 or 32. In (1) and (2), ε˘ij and σ ˘kl , denote respectively the Green-Lagrange ˘ m and E˘n are the electric displacestrain and the second Piola-Kirchhof stress tensor components, D ment and electric field components. The coefficients s˘ijkl , d˘mkl and ǫ˘mn represent, respectively, the elastic compliance constants, the piezoelectric constants and the dielectric constants for constant stress field. Introducing the Voigt notations, as shown in Table 1, (1) and (2) can be written as Table 1: Voigt notations

ij or kl

11 22 33

p, q, r, s

1

2

23 or 32 13 or 31 12 or 21

3

4

5

6

ε˘p = s˘pq σ ˘q + d˘mp E˘m , ˘ m = d˘mq σ D ˘q + ǫ˘mn E˘n .

(3) (4)

Here, the elastic compliance constants s˘pq are related to the material properties in fiber coordinates as 1 ν˘12 ν˘21 1 , s˘12 = − = − , s˘22 = , Y˘1 Y˘1 Y˘2 Y˘2 1 1 1 , s˘55 = , s˘66 = , s˘44 = ˘ 23 ˘ 13 ˘ 12 κG κG G

s˘11 =

(5)

˘ ij are the Young’s moduli, the Poisson’s ratios and the shear moduli, κ is the where Y˘i , ν˘12 and G shear correction factor. For 2D plate/shell theory, introducing the usual assumption of σ ˘33 = 0, the constitutive equations can be arranged in matrix form with an inversed relation as ˘ σ ˘ =˘ c˘ ε−˘ eT E,

(6)

˘ =˘ ˘ D e˘ ε+χ ˘ E,

(7)

5

where     ε˘11   σ ˘   11                     ˘ ˘   ε ˘   σ ˘  D1   E1     22   22  ˘ ˘ ˘ ˘ ε = γ˘12 , D = D2 , E = E2 , σ ˘ = τ˘12 , ˘       ˘   ˘           D3 E3  γ˘23   τ˘23              γ˘13 τ˘13   c˘11 c˘12 0 0 0     c˘12 c˘22 0 χ ˘11 0 0 0 0     ˘ c= ˘ =  0 χ˘22 0  , 0 c˘66 0 0 0 , χ   0 0 χ˘33 0 0 c˘44 0  0 0

0

0

0

(8)

(9)

c˘55

with Y˘1 s˘12 ν˘12 Y˘2 s˘22 = , c˘12 = − = , s˘11 s˘22 − s˘12 s˘12 1 − ν˘12 ν˘21 s˘11 s˘22 − s˘12 s˘12 1 − ν˘12 ν˘21 s˘11 Y˘2 ˘ 23 , c˘55 = κG ˘ 13 , c˘66 = G ˘ 12 . c˘22 = = , c˘44 = κG s˘11 s˘22 − s˘12 s˘12 1 − ν˘12 ν˘21 c˘11 =

(10)

˘ Here σ ˘, ˘ ε denote the stress and strain vectors, ˘ c is the elasticity constant matrix. Furthermore, D, ˘ ˘ E, e and χ ˘ represent the electric displacement vector, the electric field vector, the piezoelectric constant matrix and the dielectric constant matrix, respectively, among which ˘ e depends strongly on the structure of MFC materials. 3.1. MFC-d31 As can be seen from Fig. 1, for MFC-d31 material, the polarization is along the thickness direction, while the fiber reinforcement is parallel to the mid-surface, thus the same coordinate axes can be used for both polarization and fiber orientation. Therefore the piezoelectric constant matrix can be expressed as 

0

0

0

0

 ˘ e= 0 0 0 e˘24 e˘31 e˘32 0 0

e˘15



 0 0

(11)

The electrodes exist only in the plane parallel to the reference surface, which implies that the electric field can be applied only in the thickness direction. Therefore, the constitutive equation for the direct effect reduces to h i ˘ 3 = e˘31 e˘32 0 0 0 ˘ D ε + χ˘33 E˘3 6

(12)

with d˘31 s˘22 − d˘32 s˘12 = d˘31 c˘11 + d˘32 c˘12 , s˘11 s˘22 − s˘12 s˘12 d˘31 s˘12 − d˘32 s˘11 = d˘31 c˘12 + d˘32 c˘22 , e˘32 = s˘12 s˘12 − s˘11 s˘22 χ˘33 = ǫ˘33 − d˘31 e˘31 − d˘32 e˘32 , e˘31 =

(13) (14) (15)

Assuming that the electric field through the thickness is constant yields ˘3 Φ E˘3 = − , hE

(16)

˘ 3 is the electric voltage applied along where hE denotes the distance between two electrodes, and Φ the thickness direction. In this case hE also refers to the thickness of the MFC-d31 layer, as shown in Fig. 1a. 3.2. MFC-d33 For MFC-d33 material, the polarization direction is parallel to the fiber reinforcement, and both of them are parallel to the mid-surface. The piezoelectric constant matrix will be organized as 

e˘11 e˘12

 ˘ e= 0 0

0 0

0

0

0



 e˘26 0 0  0 0 e˘35

(17)

Due to the specific arrangement of electrodes in MFC-d33 layers, the electric field can be applied only in the polarization direction. Similarly, the constitutive equation for the direct effect reduces to h i ˘ 1 = e˘11 e˘12 0 0 0 ˘ D ε + χ˘11 E˘1

(18)

with d˘11 s˘22 − d˘12 s˘12 = d˘11 c˘11 + d˘12 c˘12 , s˘11 s˘22 − s˘12 s˘12 d˘11 s˘12 − d˘12 s˘11 = d˘11 c˘12 + d˘12 c˘22 , e˘12 = s˘12 s˘12 − s˘11 s˘22 χ˘11 = ǫ˘11 − d˘11 e˘11 − d˘12 e˘12 , e˘11 =

(19) (20) (21)

With the interegigitated electrode in MFC-d33 patches, the electric field is very complex and distributed non-uniformly. The difference between the real electric field distribution and the ideal distribution were deeply investigated by Bowen et al. [49]. For simplicity, the paper follows the 7

work of Williams [50] that the electric field is assumed to be uniform and constant between the electrodes and distributed perfectly through the material, yields ˘1 Φ E˘1 = − . hE

(22)

Here hE denotes the distance between two electrodes, which is not equal to the thickness of the ˘ 1 is the electric voltage applied along the Θ ˘ 1 -axis. MFC-d33 layer, as can be seen in Fig. 1b, and Φ 3.3. Parameter configuration The hybrid MFC material, which consists of isotropic piezoceramic fiber and epoxy matrix, can be homogenized to an orthotropic material. The fiber reinforced direction usually has a larger Young’s modulus than the other two, and the parameters in the directions normal to the fiber reinforcement are equal. Therefore, the equivalent MFC material has 7 elastic material parameters and 3 electrical material parameters, as shown in Table 2. Table 2: Description of material parameters for MFC

MFC in fiber coordinates Y˘1

Y˘2

ν˘12

ν˘23

˘ 12 G

˘ 13 G

˘ 23 G

d˘31

d˘32

ǫ˘33

d˘11

d˘12

˘ǫ11

MFC-d31

Y˜1

Y˜2

ν˜12

ν˜23

˜ 12 G

˜ 13 G

˜ 23 G

d˜31

d˜32

ǫ˜33







MFC-d33

Y˜3

Y˜2

ν˜32

ν˜21

˜ 32 G

˜ 31 G

˜ 21 G







d˜33

d˜32

˜ǫ11

3.4. Multi-layer piezo composites In order to model cross-ply or angle-ply laminated composite structures with bonded multilayers of MFC materials, as shown in Fig. 2, the constitutive equations must be transformed from Θ2 ˘2 Θ

Θ3

˘1 Θ Fib

er

˘1 Θ

θ

˘1 Θ

˘1 Θ

Θ1

Figure 2: Multi-layer composites with MFCs

8

˘ i to the curvilinear coordinate system Θi . Finally, the conthe fiber-oriented coordinate system Θ stitutive equations can be obtained as σ = cε − eT E,

(23)

D = eε + χE,

(24)

with c = TT ˘ cT,

e=˘ eT,

χ=χ ˘

(25)

where T is a transformation matrix, given as [45, 51] 

cos 2 θ sin 2 θ

sin 2 θ cos 2 θ

sin θ cos θ − sin θ cos θ

0 0

0 0



      2 2  T= −2 sin θ cos θ 2 sin θ cos θ cos θ − sin θ 0 0     0 0 0 cos θ − sin θ   0

0

0

sin θ

(26)

cos θ.

Here θ represents the fiber angle, as shown in Fig. 2. Since the electric field is always along the polarization direction when applying an electric voltage on electrodes, there is no necessity to ˘ transform the electric field to fit the polarization direction, which yields E = E. Considering smart structures with multi-MFC layers, the vectors D, E and the matrices e, χ can be arranged as follows:  (1)   (1)   (1)  (1)   (1)     D E e e 0 0 0 χ 0 · · · 0 s s ss s1 s2            (2)     Ds(2)    Es(2)    es1 e(2)  0 χ(2) 0 0 0 ··· 0  ss s2  D= ..  , E =  ..  , e =  .. .. .. ..  .. ..  ..  .. .  , χ =  ..  . . . . . . . . . . .                (N )    (N )   (N ) (N ) (N ) Ds Es 0 0 · · · χss es1 es2 0 0 0

(27)

Here s = 3 for MFC-d31 materials, while s = 1 for MFC-d33 materials, and N represents the number of MFC layers. Since the electric field is applied always along the polarization direction, the electric field in the structural coordinates can be related to the electric voltage without accounting

9

for the angles of piezo fibers, which results in    (1)   (1)  1 0 · · · 0 − (1)   E Φs  h s      E          1  Es(2)      − (2) · · · 0  Φ(2)  0 s hE   E= ..  =  . .. ..   ..  = Bφ Φ, ..  .. . .     . . .          (N )    (N )   1 Es Φs 0 0 · · · − (N)

(28)

hE

where Bφ denote the electric field matrix, and Φ is the electric voltage vector applied on MFC patches. 4. Reissner-Mindlin plate theory Since thin-walled plates with bonded MFCs are mainly considered in later calculations, a 2D finite element with the Reissner-Mindlin hypothesis, known as FOSD hypothesis, is adopted. According to this FOSD hypothesis, the displacements through the thickness are assumed to be linearly distributed and the shell director in the deformed configuration not necessarily normal to the mid-surface, yielding additional transverse shear strains compared to Kirchhoff-Love hypothesis. The displacements in the shell space then can be expressed by five components at the mid-surface as 0

1

vα = vα + Θ3 v α ,

(29)

0

v3 = v3 . 0

(30)

0

1

Here v α and v 3 denote the translational displacements at the mid-surface, while v α are the midsurface rotations, with the Greek letters varying from 1 to 2. These five generalized displacements, in the linear case, can also be understood as five degrees of freedom, as depicted in Fig. 3. Alternatively, the displacements in the shell space can be written in matrix form as 1

v2 1

v1 0

Θ2

a2 n

v3

0

v2

a1

Θ3

0

Θ1

Figure 3: Degrees of freedom

10

v1

   1 0 0 Θ3  v1    u = v2 = 0 1 0 0     0 0 1 0 v3

0     v 1    0     v 0   2    0 3 Θ  v3 = Z · v,  1    0    v   1    1   v2

(31)

where v is the generalized displacement vector. With the consideration of small strains and displacements, geometrically linear strain-displacement relations for plate structures are employed, given as   ∂vβ ∂vα 0 0 1 3 1 + = v + v + Θ v + v α,β β,α α,β β,α , ∂Θβ ∂Θα ∂vα ∂v3 1 0 = + = v α + v 3,α , 3 α ∂Θ ∂Θ

εαβ =

(32)

εα3

(33)

where εαβ are the strain components parallel to the mid-surface, εα3 are the transverse shear strains, and ,α represents the partial derivative with respect to Θα . They can be arranged in matrix form as   ∂ 3 ∂ 0 0 Θ 0  0     1 1 ∂Θ ∂Θ  v1    ε11     ∂ ∂         3 0      0 0 0 Θ       2 2 v  ∂Θ ∂Θ    2   ε22    ∂ ∂ ∂   ∂ 0 3 3 (34)  v 2ε12 =  2 0 Θ Θ 2 1   3  ∂Θ ∂Θ1   ∂Θ ∂Θ     1  v   2ε23    ∂      1   0 0 1         0 2 1   ∂Θ 2ε13   v2 ∂ 0 0 1 0 ∂Θ1 =⇒ ε = Lv, (35) where ε is the strain vector, and L represents the derivative operation matrix. 5. Finite element formulations The displacement vector v for an arbitrary point at the mid-surface can be approximated by nodal displacements via shape functions. In the present model, an eight-node Serendipity shell element is considered. For simplicity, the curvilinear coordinates are transferred to the natural ones (ξ, η) using the Jacobian matrix J, as shown in Fig. 4. The shape functions and the Jacobian matrix are standard, which can be found in many books, e.g. [52–54]. Finally, the strain vector can be related to nodal displacements as ε = Lv = LNq = Bu q, 11

(36)

4

7

Θ

3

8 6 1

5

η

(-1,1) 7 4

2

3 (1,1)

J J−1

8

6 ξ

Θ1 1 (-1,-1)

2

(1,-1) 2

5

Figure 4: Eight-node shell element

where Bu is the strain field matrix, N denotes the shape function matrix, and q represents the nodal displacement vector. The principle of virtual work is given as δWint − δWext = 0,

(37)

with
δεT σ − δET D dV, ZV Z Z T T T = δu f b dV + δu f s dΩ + δu f c − δΦT ̺ dΩ − δΦT Qc .

δWint = δWext

Z

V

Ω

(38) (39)

Ω

Here δ represents the variational operator, δWint and δWext denote respectively the variation of the internal work and the external work. Furthermore, f b , f s and f c are the body, surface and concentrated force vectors, respectively, ̺ and Qc represent the surface and concentrated charge vectors. Substituting (38)-(39) into (37) and considering the linear strain-displacement relations, one obtains a linear electro-mechanically coupled static FE model, including static equilibrium equations and sensor equations, which are respectively given as Kuu q + Kuφ Φa = Fue ,

(40)

Kφu q + Kφφ Φs = Gφe .

(41)

Here Kuu , Kuφ , Kφu , Kφφ , represent the stiffness matrix, the piezoelectric coupled matrix, the coupled capacity matrix and the piezoelectric capacity matrix, respectively. Furthermore, Fue , Gφe , q, Φa and Φs are respectively the external force vector, the external charge vector, the nodal displacement vector, the actuation voltage vector and the sensor voltage vector. The above system

12

matrices and vectors are calculated by BT u cBu dV, V Z T T = Kφu = − BT u e Bφ dV, V Z =− BT φ ǫBφ dV, V Z Z T T T T T NT = Nv Zu f b dV + v Zu f s dΩ + Nv Zu f c , Ω V Z = − ̺ dΩ − Qc .

Kuu = Kuφ Kφφ Fue Gφe

Z

(42)

Ω

Since (41) are sensor equations, they will not be used when only shape control problems are considered. 6. Numerical examples In this section, the present FE model is validated by a cantilevered plate comprised of one isotropic host structure and two MFC-d33 patches, which was first investigated by Bowen et al. [23] using ANSYS and through experiments. Later the plate is expanded to composite structures with angle-ply laminates, considering various fiber reinforcement angles in MFCs for both d31 and d33 modes. In the following simulations, voltages are only applied to one MFC patch placed on the top surface of the plate, and the patch on the bottom surface is under closed-circuit boundary conditions. 6.1. Validation test The first example is a cantilevered plate with MFCs, proposed by Bowen et al. [23], as shown in Fig. 5. The host structure is made of aluminum, which is a typical isotropic material with the Θ2

111 000 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111

d

Θ3

111 000 000 111 000 111 000 111 000 111

Host structure

MFC

WMFC

Back line

B A

WBeam

Θ1

Central line

Front line

LMFC

C

LBeam hMFC

hBeam Θ1

Figure 5: Schematic figure of the MFC bonded smart plate

13

Young’s modulus Y = 70 GPa and the Possion’s ratio ν = 0.32. The length and width of the plate are respectively 300 mm and 75 mm, with the thickness of 1.97 mm. Two MFC-d33 patches (M8557P1, Smart Material Corp. [2]), with the dimensions of 85 × 57 × 0.3 mm3 , are bonded to the top and bottom surfaces of the aluminum plate, with a distance d = 15 mm away from the clamped edge, see Fig. 5. The material properties of MFC-d33 are employed according to the study by Williams et al. [50] and Bowen et al. [23], shown in Table 3, which are slightly different from those provided by Smart Material Corp. [2]. Table 3: Material properties

MFC-d33

MFC-d31

T300/976

Y˜3 = 29.4 GPa

Y˜1 = 30.336 GPa

Y˜1 = 150 GPa

Y˜2 = Y˜1 = 15.2 GPa

Y˜2 = Y˜3 = 15.857 GPa

Y˜2 = Y˜3 = 9 GPa

˜ 32 = G ˜ 31 = 6.06 GPa G

˜ 12 = G ˜ 13 = 5.515 GPa G

˜ 12 = G ˜ 13 = 7.1 GPa G

˜ 21 = 5.79 GPa G

˜ 23 = 5.515 GPa G

˜ 23 = 2.5 GPa G

ν˜32 = ν˜31 = 0.312

ν˜12 = ν˜13 = 0.31

ν˜12 = ν˜13 = 0.3

ν˜21 = 0.312

ν˜23 = 0.438

ν˜23 = 0.3

d˜33 = 467 × 10−12 m/V

d˜31 = −170 × 10−12 m/V

d˜32 = −210 × 10−12 m/V d˜32 = −100 × 10−12 m/V hE = 0.5 mm

hE = 0.18 mm

The MFC-d33 patch on the top surface is actuated by a constant voltage of 400 V (electric field 400/0.5 V/mm). The vertical deflections of the points on the central line are calculated by the present model, and compared with the ANSYS simulation and experimental results reported in Bowen et al. [23], as shown in Fig. 6. As can be seen, perfect agreement is achieved between the 1

Vertical displacements (mm)

Present ANSYS (Bowen et al. 2011)

0.8

Experiments (Bowen et al. 2011)

0.6

0.4

0.2

0

0

50 100 150 200 250 Distance from the clamped edge (mm)

300

Figure 6: Central line deflection of the MFC-d33 bonded plate for validation test

14

Table 4: Numerical values for the present result in Fig. 6

Θ1 (mm) Deflection (mm) 0

0

15

6.1445 × 10−4

55

0.0372

100

0.1593

150

0.3453

200

0.5344

250

0.7242

300

0.9141

present results and those of Bowen et al. [23]. Therefore, the present FE model is accurate enough to be applied in the simulation of MFC actuated structures. 6.2. Isotropic plate bonded with MFC-d31 patches The next example for the simulation of MFC bonded smart structures is a clamped plate, similar to the one in Fig. 5, comprised of a host plate made of isotropic aluminum but now with two MFCd31 patches (M8528-P3, Smart Material Corp. [2]) bonded on its top and bottom surfaces. The material properties of aluminum are the same as those in the previous validation test, and those for the MFC-d31 patch are given in Table 3. The dimensions for the plate and MFC-d31 patches are respectively 300 × 75 × 2 mm3 and 85 × 28 × 0.3 mm3, with MFC-d31 bonded at d = 15 from the clamped edge. A voltage of 200 V (electric field 200/0.18 V/mm) is applied onto the top MFC-d31 patch. Vertical deflections of the central line are obtained by the present FE model and presented in Fig. 7. The figure shows that the deflection curve consists of three parts, namely, a straight line (from 0 to 15 mm in Θ1 -axis direction), a curved line (from 15 mm to 100 mm, where the MFC-d31 patches are bonded), and another straight line (from 100 mm to 300 mm). It can be seen that with a similar structure the displacements are much smaller than those in the previous validation test. The reason is that on one hand the MFC-d31 patches have smaller width, only 28 mm compared to 57 mm, and on the other hand the d33 coefficient in MFC-d33 patches is 2.75 times larger than d31 in MFC-d31 material. 6.3. Isotropic plate with MFC-d33 patches having arbitrary fiber orientation The next example considers the same aluminum plate (300 × 75 × 2 mm3) as the previous one, but bonded with MFC-d33 patches having various material angles. The dimensions for MFC-d33 15

0.3

Vertical displacement (mm)

0.25 0.2 0.15 0.1 0.05 0 −0.05

0

50 100 150 200 250 Distance from clamped edge (mm)

300

Figure 7: Central line deflection of the aluminum plate bonded with MFC-d31 patches Table 5: Numerical values for the curve in Fig. 7

Θ1 (mm) Deflection (mm) 0

0

15

−5.8704 × 10−4

55

0.0089

100

0.0462

150

0.1020

200

0.1571

250

0.2122

300

0.2672

patches are 85 × 57 × 0.3 mm3, which are provided by Smart Material Corp. [2] as a standard size. The material properties of MFC-d33 patches are listed in Table 3. The upper MFC-d33 patch is subjected to a voltage of 400 V, and the lower patch is short circuited. The vertical deflections at point A (see Fig. 5) and the twist of the plate (wB − wC ) are presented in Fig. 8 with respect to the material angles of the MFC patches. The present numerical values of vertical deflection and twist are given in Table 6. The figure shows that the vertical deflections at point A decrease as the material angle increases until to about 60◦ , and then turn into the opposite direction. An increasing trend can be seen for the twist when the material angle of MFC-d33 increases from 0◦ to 45◦ , followed by a decline as the material angle of MFC-d33 continuously increases until to 90◦ . The change of the sign of the displacements results from the opposite sign of the d33 and d31 coefficients. Theoretically, there is no twist when the MFC-d33 angle is 0◦ or 90◦ . The small deviations are caused by the error margin of the numerical method. 16

1 Deflection Twist

0.8

Deflection (mm)

0.6 0.4 0.2 0 −0.2 −0.4

0

20

40 60 Fiber angle (degree)

80

Figure 8: Vertical deflections and twist of the aluminum plate with MFC-d33 patches Table 6: Numerical values for the vertical deflections and twist (mm)

Piezo fiber angle Deflection (wA )

Twist (wB -wC )

0◦

0.8897

7.3074 × 10−11

15◦

0.8114

0.0935

30◦

0.5968

0.1624

45◦

0.3019

0.1886

60◦

3.3226 × 10−3

0.1644

75◦

−0.2182

0.0955

90◦

−0.3001

2.1209 × 10−11

In the next step, the 3D deformed shapes of the isotropic plate with MFC-d33 patches having fiber angles of 0◦ , 30◦ , 45◦ , 60◦ , 75◦ and 90◦ are plotted in Fig. 9. Furthermore, the corresponding vertical deflections of the front line, central line and back line (marked in Fig. 5) are shown in Fig. 10, which clearly illustrates the twist of the plate. The results show that when the fiber angle of MFC is 0◦ or 90◦ , the vertical deflections of the front and back lines are theoretically identical, since there is no twist occurring in the structure. It can be seen from Fig. 10a that there is only a slight difference between the displacements at the central and front (or back) lines at the area bonded with MFC for the case of MFC-d33 with the fiber angle of 0◦ . This difference will be enlarged when the fiber angle of MFC-d33 patches change to 90◦ , as can be seen from Fig. 10f. This is due to the fact that the d31 effect is much much smaller than d33 effect. However, when the fiber angles of the MFC are not equal to 0◦ or 90◦ , the vertical deflections of those lines become much different. The in-plane longitudinal stresses ε11 and the transverse shear stresses ε13 are presented in Fig. 11. The figures show that the longitudinal stresses are very strong in the piezo bonded area 17

1 Θ3 (mm)

Θ3 (mm)

1

0.5

0

0.5

0 50

Θ2 (mm)

50 0

0

100 Θ1 (mm)

200

300 Θ2 (mm)

(a) MFC-d33 with fiber angle of 0◦

0

0

100 Θ1 (mm)

200

300

(b) MFC-d33 with fiber angle of 30◦

1 Θ3 (mm)

Θ3 (mm)

0.6 0.5

0.4 0.2 0 −0.2

0 50 Θ2 (mm)

50 0

0

100 Θ1 (mm)

200

300 Θ2 (mm)

(c) MFC-d33 with fiber angle of 45◦

0

100 Θ1 (mm)

200

300

(d) MFC-d33 with fiber angle of 60◦ 0.5 Θ3 (mm)

0.5 Θ3 (mm)

0

0

−0.5

0

−0.5 50

Θ2 (mm)

50 0

0

100 Θ1 (mm)

200

300 Θ2 (mm)

(e) MFC-d33 with fiber angle of 75◦

0

0

100 Θ1 (mm)

200

300

(f) MFC-d33 with fiber angle of 90◦

Figure 9: Surface shapes of the aluminum plate with MFC-d33 patches having different fiber angles

compared to other area, and they will reduce with increasing the piezo fiber angles to 90◦ . Moreover, the transverse shear stresses are extremely large for the piezo fiber angle of 30◦ , 45◦ and 60◦ , with positive and negative stagger arrangement, at the bonded area.

18

1

0.8 Front line Central line Back line

0.6 Vertical deflection (mm)

Vertical deflection (mm)

0.8

0.6

0.4

0.2

0

0

50

100 150 200 250 Distance from clamped edge (mm)

0.2

−0.2

300

0

50

100 150 200 250 Distance from clamped edge (mm)

300

(b) MFC-d33 with fiber angle of 30◦

0.4

0.2 Front line Central line Back line

Front line Central line Back line

0.15 Vertical deflection (mm)

0.3 Vertical deflection (mm)

0.4

0

(a) MFC-d33 with fiber angle of 0◦

0.2

0.1

0

−0.1

Front line Central line Back line

0.1 0.05 0 −0.05

0

50

100 150 200 250 Distance from clamped edge (mm)

−0.1

300

(c) MFC-d33 with fiber angle of 45◦

0

50 100 150 200 250 Distance from clamped edge (mm)

300

(d) MFC-d33 with fiber angle of 60◦ 0.1

Vertical deflection (mm)

0.1 0.05 0 −0.05 −0.1 −0.15 −0.2

Front line Central line Back line

0.05 Vertical deflection (mm)

Front line Central line Back line

−0.25

0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.3

0

50 100 150 200 250 Distance from clamped edge (mm)

300

0

(e) MFC-d33 with fiber angle of 75◦

50 100 150 200 250 Distance from clamped edge (mm)

300

(f) MFC-d33 with fiber angle of 90◦

Figure 10: Line shapes of the aluminum plate with MFC-d33 patches having different fiber angles

19

0

−1 0

100

200

0

300 ε13 (MPa)

200

0

300

(a) MFC-d33 with fiber angle of 0◦

0

200

0

0

300

−0.5 100

200

0

300

(c) MFC-d33 with fiber angle of 45◦

0

200

0

0

300

−1 100

200

0 0

100

200

300

−1

ε13 (MPa) 0 0

100

200

300

ε11 (MPa)

−1

0

300

(e) MFC-d33 with fiber angle of 75◦

1 0 −1

0

100

200

300 ε13 (MPa)

50

0 0

300

ε11 (MPa)

1

ε13 (MPa)

50

200

50

−1 100

100

(d) MFC-d33 with fiber angle of 60◦ 0

0

−1 0

1

ε11 (MPa)

50

0

50

0 0

ε13 (MPa)

1

0.5

ε13 (MPa)

50

300

1

−0.5 100

200

50

0 0

100

(b) MFC-d33 with fiber angle of 30◦ 0.5

ε11 (MPa)

50

−1 0

50

−1 100

0

1

1 0

0

1

ε11 (MPa)

50

0

50 0

1

ε11 (MPa)

50

1 0 −1

0

100

200

300

(f) MFC-d33 with fiber angle of 90◦

Figure 11: Stress (ε11 ,ε13 ) distribution of the aluminum plate with MFC-d33 patches having different fiber angles

20

6.4. Composite plate with MFC-d33 patches having arbitrary fiber orientation In the last example, the host structure is considered to be a composite plate with symmetrical cross-ply laminates. The host plate is made of T300/976 orthotropic material with the stacking sequence of [90/0]s . The dimensions in this example are exactly the same as those in the previous example in Section 6.3 (300×75×2 mm3 for the host structure, and 85×28×0.3 mm3 for the MFC patches). The thickness for each substrate layer of the host composite structure is 0.5 mm, with the total thickness of 2 mm. The two MFC-d33 patches are bonded on the top and bottom surfaces with the same dimensions and material properties as those in Section 6.3. The upper MFC-d33 patch is subjected to a voltage of 400 V. Analogously, the 3D surface shapes of the composite plate with arbitrary fiber orientation MFC-d33 patches are presented in Fig. 12. The corresponding line shapes of the front, central and back lines are presented in Fig. 13.

Θ3 (mm)

Θ3 (mm)

1.5 1 0.5

1 0.5 0

0

50

50 Θ2 (mm)

0

0

100 Θ1 (mm)

200

300

Θ2 (mm)

1 0.5

100 1 Θ (mm)

200

300

1 0.5 0

0

50

50 2

Θ (mm)

0

0

100 Θ1 (mm)

200

300 2

Θ (mm)

(c) MFC-d33 with fiber angle of 45◦

0

0

100 Θ1 (mm)

200

300

(d) MFC-d33 with fiber angle of 60◦ 1 Θ3 (mm)

1 Θ3 (mm)

0

(b) MFC-d33 with fiber angle of 30◦

Θ3 (mm)

Θ3 (mm)

(a) MFC-d33 with fiber angle of 0◦

0

0.5 0

0.5 0

50 2

Θ (mm)

50 0

0

100 Θ1 (mm)

200

300 2

Θ (mm)

(e) MFC-d33 with fiber angle of 75◦

0

0

100 Θ1 (mm)

200

300

(f) MFC-d33 with fiber angle of 90◦

Figure 12: Surface shapes of the composite plate with MFC-d33 patches having different fiber angles

The figures show that no twist deformations occur when the material angle of MFC-d33 is 0◦ or 90◦ , because of the symmetrical stacking sequence of the composite. However, the plate occurs 21

1.6

Front line Central line Back line

1.2 1 0.8 0.6 0.4

1 0.8 0.6 0.4 0.2 0

0.2 0

Front line Central line Back line

1.2 Vertical deflection (mm)

Vertical deflection (mm)

1.4

0

50 100 150 200 250 Distance from clamped edge (mm)

−0.2

300

(a) MFC-d33 with fiber angle of 0◦

0

50 100 150 200 250 Distance from clamped edge (mm)

300

(b) MFC-d33 with fiber angle of 30◦ 0.6

Front line Central line Back line

0.6 0.4 0.2 0 −0.2

Front line Central line Back line

0.5 Vertical deflection (mm)

Vertical deflection (mm)

0.8

0.4 0.3 0.2 0.1 0 −0.1

0

50 100 150 200 250 Distance from clamped edge (mm)

−0.2

300

(c) MFC-d33 with fiber angle of 45◦

0

50 100 150 200 250 Distance from clamped edge (mm)

300

(d) MFC-d33 with fiber angle of 60◦ 0.05 Front line Central line Back line

0

0.1

Vertical deflection (mm)

Vertical deflection (mm)

0.15

0.05 0 −0.05 −0.1 −0.15 Front line Central line Back line

−0.2 −0.25 0

50 100 150 200 250 Distance from clamped edge (mm)

−0.05 −0.1 −0.15 −0.2 −0.25

300

0

(e) MFC-d33 with fiber angle of 75◦

50 100 150 200 250 Distance from clamped edge (mm)

300

(f) MFC-d33 with fiber angle of 90◦

Figure 13: Line shapes of the composite plate with MFC-d33 patches having different fiber angles

obviously twist deflections illustrated both in the 3D surface and the line shapes when the material angle of MFC-d33 changes to 30◦ and 45◦ . The reason is that the piezo fiber orientation is not parallel to the in-plane coordinate axes.

22

7. Conclusion In this paper, a linear electro-mechanically coupled FE model for composite laminated thinwalled smart structures bonded with MFC-d31 or -d33 patches, which have arbitrary fiber reinforcement angles, has been developed. The present FE model is based on the Reissner-Mindlin hypothesis using linear piezoelectric constitutive equations. The FE model first has been validated by a cantilevered plate with MFC patches, and then was used for simulation of isotropic and crossply laminated structures bonded with MFCs having different fiber orientation. The results show that MFC-d33 patches produce larger actuation forces since the d33 coefficient is much larger than the d31 coefficient. Additionally, using MFC-d33 patches with different fiber reinforcement angles the structural deformation of isotropic and cross-ply laminated plates have been investigated. Acknowledgements The authors would gratefully acknowledge the financial support from CHINA SCHOLARSHIP COUNCIL for the first author (with the No.2010629003), and from the Graduate School of Northwestern Polytechnical University, China, for the second author. Furthermore, the work is partially supported by ”111 project” with the Grand No. B13044. References [1] R B Williams, B W Grimsley, D J Inman, and W K Wilkie. Manufacturing and mechanicsbased characterization of macro fiber composite actuators. In ASME 2002 International Mechanical Engineering Congress and Exposition, pages 79–89, 2002. [2] Smart Material Corp.: www.smart-material.com. [3] NASA: www.nasa-usa.de. [4] W K Wilkie, R G Bryant, J W High, R L Fox, R F Hellbaum, A Jalink, B D Little, and P H Mirick. Low-cost piezocomposite actuator for structural control applications. In SPIE’s 7th Annual International Symposium on Smart Structures and Materials, volume 3991, 2000. [5] R. B. Williams, W. K. Wilkie, and D. J. Inman. An overview of composite actuators with piezoceramic fibers. In Proceedings of IMAC-XX: Conference & Exposition on Structural Dynamics, the Westin Los Angeles Airport, Los Angeles, California, USA, 4-7 February 2002. [6] H. A. Sodano, J. Lloyd, and D. J. Inman. An experimental comparison between several active composite actuators for power generation. Smart Materials and Structures, 15:12111216, 2006. 23

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