On analytical and finite element modelling of piezoelectric extension and shear bimorphs

Computers and Structures 84 (2006) 1426–1437 www.elsevier.com/locate/compstruc

On analytical and finite element modelling of piezoelectric extension and shear bimorphs Christophe Poizat a, Ayech Benjeddou b

b,*

a Fraunhofer – Institute of Mechanics of Materials, Wo¨hlerstrasse 11, 79108 Freiburg, Germany Institut Supe´rieur de Me´canique de Paris, LISMMA-Structures, 3 rue Fernand Hainault, 93407 Saint Ouen Cedex, France

Received 24 January 2004; accepted 14 January 2005 Available online 8 May 2006

Abstract A number of well-established analytical and numerical modelling techniques of classical extension bimorphs are assessed and evaluated using general purpose finite element codes. This allows the clarification of several confusing common validation practices, using well-known experimental data and analytical formulas. New analytical formulas are presented and evaluated for the relatively new shear bimorph concept and a more general extension–shear one, resulting from the combination of the extension- and shear-piezoelectric modes via an off-axes polarisation. The corresponding new simple analytical formulas accurately reproduced the three-dimensional piezoelectric finite element results, confirming their usefulness for preliminary designs.  2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. Keywords: Analytical; Finite element; Piezoelectric actuators; Extension; Shear; Extension–shear

1. Introduction Piezoelectric bimorphs are the oldest form of basic piezoelectric adaptive structures. Their invention dates back to the early 1930s. A comprehensive and detailed historical review of the applications of piezoelectric bimorphs has been given by Smits et al. [1] in the beginning of the last decade. With the growing of the relatively new field of adaptive structures, piezoelectric bimorphs have been applied as sensors and actuators for high precision aerospace and medical applications as well as in shape, vibration and noise control. In particular, they are often used to validate experimentally [2], analytically [3] or numerically [4,5] new piezoelectric adaptive formulations. This has led to a very large research effort into closed-form one-dimensional (1D) [6,7], analytical two-dimensional (2D) [8,9] and three-dimensional (3D) exact [10,11] solutions for multilayer adaptive beams.

*

Corresponding author. Tel.: +33 1 49 45 29 79; fax: +33 1 49 45 29 29. E-mail address: [email protected] (A. Benjeddou).

The use of basic piezoelectric bimorph structures in validation tests for more elaborated beam and plate structures has often led to confusion. In many cases, this is due to the non-explicitly mentioned applied electric load (potential, difference of potentials or electric field), connectivity (series or parallel) and interface electrodes (present or not, grounded or not). Besides, by neglecting the shear effect, most of the above mentioned analytical [1] and exact [10,11] reference solutions are limited to thin bimorphs with relatively high length-to-thickness ratio. Most of these solutions [6,7,9] are based on equivalent single layer (ESL) formulations; thus, they do not really account for the multilayer aspect of the bimorph. To represent the piezoelectric effect, simplified solutions [1,3,4,6,7] use either inducedstrain or equivalent load approaches, both of which assume a constant electric field in the piezoelectric layer, thus neglecting the so-called induced electric potential [12,13]; hence, they take only partially the piezoelectric coupling into account [14]. Bimorph analyses and applications have so far exploited only the extension-mode [1–7,9–11], also called longitudinal-mode [15,16], where an electric field is applied

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C. Poizat, A. Benjeddou / Computers and Structures 84 (2006) 1426–1437

parallel to the initial poling direction of the piezoelectric material. Despite the growing research interest [12,13] in the so-called shear-mode, present when initial poling and electric field are perpendicular, only one paper [8] could be found. The present work aims to assess and evaluate some wellestablished analytical and numerical modelling techniques of classical extension bimorphs using general purpose finite element (FE) codes. By this, several common confusing validation practices [3,4] with some well-known experimental [2] and analytical formulas [1,11] will be clarified. Also the new (single and two-layer) shear and shear–extension [8] bimorph concepts will be studied closely. Several new analytical formulations are proposed and compared with finite element analyses for validation and discussion.

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Fig. 1. (a) Longitudinal (d33) and transverse (d31) piezoelectric effects and (b) shear (d15) piezoelectric effect.

This section aims first to recall the piezoelectric coupling modes and their application in benders technology, which is based mainly on the classical extension-mode. Then, emphasise is made on the shear-mode and its exploration to design new single and two-layer shear bimorphs and their combination with the extension ones.

For transversely isotropic piezoelectric materials, such as piezoceramics, only three piezoelectric strain constants are non-zero; these are the so-called d33, d31 and d15 constants. They define, respectively, the longitudinal, transverse and shear piezoelectric modes (Fig. 1). Typically, d31  d33/2. The longitudinal and transverse modes require parallel direction of poling and applied electric field. Perpendicular direction of the applied electric field to the poling direction results in the piezoelectric shear effect. The piezoelectric strain constant d15 for the shear-mode generally has the highest value among the strain piezoelectric constants.

2.1. Piezoelectric actuation modes

2.2. Piezoelectric bimorphs types

Coupled piezoelectric problems characterise an interrelationship between mechanical and electrical state variables. In piezoelectric materials, an electric potential gradient causes deformation (inverse effect), while a mechanical stress induces electrical charges (direct effect). The inverse effect is used for actuation while the direct effect is used for sensing. The coupling between mechanical strains SJ and electric fields Ek is materialised by the piezoelectric constitutive equations. In Eq. (1a,b), they are given in the stresses TI and electric displacements Dk form [17] that uses the elastic stiffness constants cIJ at constant (short-circuit) electric field, piezoelectric stress constants ekI and dielectric constants ekl at constant (clamped) strains

Bimorph actuators, often called benders, are mainly based on either the extension-mode, including both the longitudinal and transverse piezoelectric effects, or on the shear-mode. Some classifications of bimorphs consider not only the two-layers systems, that are mainly discussed here, but also monomorph, sandwich and multi-morph [18] configurations. The objective of this sub-section is to clarify the different configurations (series, parallel) of the classical extension mode-based bimorphs and to present new (single and two-layer) shear mode-based ones.

2. Piezoelectric bimorphs

T I ¼ cEIJ S J  ekI Ek ;

Dk ¼ ekJ S J þ eSkl El

ð1a; bÞ

where, the classical Einstein summation convention on repeated indices is applied, with I,J = 1, . . . , 6; k,l = 1, 2, 3. The piezoelectric constitutive equations can also be written in the alternative strains and electric displacement form [17], using the elastic compliance constants sIJ at constant (short-circuit) electric field, piezoelectric strain constants dkI and dielectric constants ekl at constant (free) stresses: S I ¼ sEIJ T J þ d kI Ek ;

Dk ¼ d kJ T J þ eTkl El

ð2a; bÞ

where the following relations between the material constants hold: 1

sEIJ ¼ ðcEIJ Þ ;

1

d kI ¼ ðcEIJ Þ ekI ;

eTkl ¼ eSkl þ ekJ d kJ ð2c; d; eÞ

2.2.1. Extension-mode benders Piezoelectric extension-mode bending actuators commonly use the (d31)-effect, even if the (d33)-effect can be used with interdigitated electrodes as reported in [15]. The simplest configuration consists of a slab made of two laminated piezoelectric layers (Fig. 2). If the top and bottom layers are appropriately poled and electroded, the application of an electric field across the two layers of the bender causes one layer to extend and the other one to contract. This generates moments that bend the bimorph. Depending on the electrode structure and poling direction, they are classified as series or parallel bimorphs [1]. The simplest series bimorph is made of two layers of the same thickness and material with opposite polarisation orientations (Fig. 2a). It is electroded on the top and bottom surfaces of the two-layers structure. Series bimorphs are also called antiparallel bimorphs. A parallel bimorph is made of two layers of the same thickness, material and polarisation

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Fig. 2. Two layers piezoelectric (a) ‘‘series’’ and (b) ‘‘parallel’’ bimorphs.

Fig. 3. Shear-mode ‘‘sandwich’’ (a) and ‘‘monomorph’’ (b) benders.

Fig. 4. Shear–extension bimorph [8].

orientations (Fig. 2b). An intermediate electrode allows to apply electric fields with opposite directions in the top and bottom layers. This also results in a bending moment. As pointed out in [9,11], the mathematical formulation for parallel bimorphs differs from that of the antiparallel ones because of the discontinuity of the normal component of the electric displacement on the interface electrode. Several drawbacks are known for these types of bimorphs. High bending and longitudinal stresses may lead to delamination or to damage of the active layer, that is generally made of a brittle piezoelectric ceramic. Furthermore, damage due to contact with foreign objects can not be excluded. 2.2.2. Shear-mode benders Piezoelectric benders using the shear-mode effect were mainly reported for the sandwich configuration [7,12,13] as represented in Fig. 3a. Here, the piezoelectric layer is embedded between two inactive sheets and its transverse shear deformations are predominant. One advantage of shear-mode benders is that, for the same tip deflection, the stress level inside the actuators is lower than that in the extension-mode ones [13]. In contrast to the exten-

sion-mode, one single shear layer or a shear ‘‘monomorph’’ (Fig. 3b) can provide a bending tip deflection as will be shown later. The two-layers shear bimorph concept is introduced in the following sub-section. 2.2.3. Extension–shear benders A combination of both extension- and shear-mode actuation mechanisms was recently proposed [8]. For this, a bimorph made of two piezoelectric layers of equal thickness with the axis of polarisation inclined of an angle a to the normal was considered (Fig. 4). When a = 0, the bimorph reduces to an extension bimorph, while for a = 90, the bimorph reduces to a shear bimorph. This hybrid configuration is very well suited for a performance comparison between extension and shear bimorphs. In Fig. 4, u and V (=u0/2, notation from [8]) denote the electric potential and electric potential difference, respectively. 3. Analytical modelling Several approaches are proposed in the literature to derive analytical solutions for bimorphs. They usually consider perfectly bonded layers and zero electrode thickness

C. Poizat, A. Benjeddou / Computers and Structures 84 (2006) 1426–1437

and stiffness. They mainly differ in their kinematic assumptions for the displacement field (ESL/layer-wise theories, first/higher-order approximations, shear effects, lateral effects) and the approximated form of the through-thethickness electric potential or field in each piezoelectric layer. In this section, a number of analytical formulas described in the literature for calculating the tip deflection of cantilever extension parallel bimorphs [1,11] are first reviewed. Then, new ones are also proposed here for the first time for the calculation of the tip deflection of shear and extension–shear cantilever bimorphs. Finally, both known and new formulas are assessed and evaluated in order to delimit their validity domains in the design of actuators by using feasible and common piezoelectric modelling techniques with commercial FE codes.

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An approach combining the state space formulation with the asymptotic expansion technique has been proposed for series [10] and parallel [11] bimorphs. It allows to take into account the non-linear evolution of the electric potential through the layer thickness by truncating the expansions appropriately. In particular, closed-form solutions were derived for cantilever bimorphs with series [10] and parallel [11] configurations. However, although very interesting, the proposed formulas can not be used, probably due to some typing errors. In particular, it was noticed that the formula proposed for the parallel bimorph does not reduce to that of Smits et al. [1] when the electric field is supposed to be constant (b14 = 0, as recommended in [11]). In fact, in Eq. (43) of Lim et al. [11], reduced to the applied voltage case only, the second term has to be multiplied by b11d31, so that it becomes

3.1. Extension bimorphs Using a basic strength-of-materials energetic-type approach, Smits et al. [1] have derived the constituent equations of series and parallel extension bimorphs under both mechanical and electric loads. The analytic formulation has been detailed for an inward series configuration. The formulas corresponding to the parallel bimorphs are deduced by substituting the corresponding electric fieldpotential relation. For an applied electric field E only, the tip deflection d of a cantilever or clamped-free (CF) series or parallel bimorph, of length L and total thickness H (see Fig. 2), is given in [1] as 3d 31 L2 d¼ E 2H

d¼

3b11 b44 d 31 L2 V  2 4b11 b44  3b214 ðH =2Þ

ð4Þ

where  2 b11 ¼ ðcE11 e233 þ cE11 cE33 eS33  cE13 eS33  2cE13 e31 e33 þ cE33 e231 Þ=ðe233 þ cE33 eS33 Þ; b14 ¼ ðcE13 e33  e31 cE33 Þ=ðe233 þ cE33 eS33 Þ; b44 ¼ ðcE33 Þ=ðe233 þ cE33 eS33 Þ

It is noteworthy to recall that, in contrast to [1], in this approach, the electric field is not constant through a piezoelectric layer and the deflection is stiffness dependent. 3.2. Shear bimorphs

ð3Þ

where E ¼ HV for a series bimorph, and E ¼ hVb ¼ hVt ¼ 2V for H a parallel bimorph; V is the applied voltage. It appears then that, for a given voltage, the tip deflection of a parallel bimorph is twice that of the series one. It is worthy to notice also that, within this approach, the tip deflection appears independent of the material elastic stiffness. Eq. (3) is approximately valid for the same structure with simply supported (SS) boundary conditions (BC). A SS beam can namely be considered approximately as a beam clamped in the middle of the total length. Replacing the total length L with L/2 in Eq. (3) leads to the corresponding tip deflection. This was not commented in [4] but the results obtained with Eq. (3) are in good agreement with the analytical and numerical results given there (Fig. 3 in [4]). Of importance is the form of the electric field. Due to the coupling effect, the evolution of the electric potential through the thickness is not linear as in a purely dielectric material. As a consequence, the electric potential gradient, or electric field, is not constant in each layer, as supposed in [1]. In fact, Benjeddou et al. [12] showed that, for the extension-mode, the electric potential in a piezoelectric layer is the sum of two parts, where the first one depends linearly on the z coordinate, whereas the second one, representing the induced electric potential, depends quadratically on the z coordinate.

A closed-form solution for the bending rotation, axial and transverse displacements of a sandwich beam with shear piezoelectric core was recently derived for various BC under both mechanical and electric loads in [7]. The approach is based on the ESL first-order shear deformation theory (FSDT) and extends the formulation proposed in [6] for sandwich beams with extension piezoelectric patches. Unfortunately, probably due to some typing errors, some of the formulas do not fulfil the BC for which they were derived. Hence, it is the aim of this sub-section to derive new simple formulas for shear monomorph and two-layer bimorphs. It is well known [12,13] that the bending deformation of a cantilever sandwich beam with a piezoceramic shear core, due to the nature of shear actuation mechanism, is linear. FE simulations using the ABAQUS FE software [19] have confirmed this result for the monomorph of Fig. 5a as can be seen from Fig. 5b. This is exploited here after to present, for the first time, a formula for the tip deflection of shear bimorphs. Hence, within the small deformation assumption, the monomorph tip deflection is d ¼ Lc13

ð5aÞ

where the shear angle c13, as given in [12], is c13 ¼ d 15 E

ð5bÞ

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Fig. 5. Shear-mode monomorph (a) and its resulting (magnified) deformation (b).

Assuming E = V/H, where V = V+  V the applied voltage, Eqs. (5a) and (5b) give L ð5cÞ d ¼ d 15 V H It can be shown that, for a bimorph constituted of identical two-layers, this formula remains valid but with E = 2V/H. Notice that, as for the extension bimorph (see Eq. (3)), Eq. (5c) is also independent of the bimorph’s elastic stiffness. However, for sandwich shear bimorphs, it can be shown that the bimorph tip deflection depends on the different shear moduli of the faces and core materials. 3.3. Extension–shear bimorphs Since so far no analytical formula for the extension– shear bimorph exists, a formula is derived here using a superposition method. The basic idea consists in considering that the tip deflection is a superposition of the separate tip deflections resulting from the extension- and shearmodes, as illustrated in Fig. 6. The proposed equation consists then of the sum of Eqs. (3) and (5c), after projection of the electric field and use of the relation E = 2V/H: L2 L ð6Þ V cos a þ 2d 15 V sin a 2 H H where a is the polarisation angle. This formula will be assessed and evaluated later in this work by comparing the results with 2D (cylindrical) exact results of Vel and Batra [8] using a Stroh formalism, and with piezoelectric 2D and 3D FE obtained with ABAQUS [19]. Notice that, for Fig. 4, V = u0/2. d ¼ 3d 31

4. Numerical modelling This section describes simple approaches to analyse piezoelectric bimorphs using the thermal analogy when piezoelectric finite elements are not available or using 3D and 2D (strain or stress plane behaviour) already available in some commercial general purpose FE codes, such as ANSYS or ABAQUS. The aim of this section is to clarify the inherent assumptions and the consequent limitations of the feasible (thermal analogy using shell FE) and common (2D plane strain and 3D piezoelectric FE) numerical techniques with commercial FE codes. 4.1. The thermal analogy Despite the nearly mature field of piezoelectric adaptive structures, commercial FE codes, such as ABAQUS and ANSYS, still do not offer the possibility to model piezoelectric structures with piezoelectric layered plate/shell elements. Rather, to analyse these adaptive structures, the analogy between piezoelectric induced strain and thermally induced strain can be used [14]. In this approach, the temperature change represents the electric field actuation and the piezoelectric strain constants replace the thermal expansion coefficients. Thus, standard thermo-mechanical elements, with anisotropic mechanical and thermal properties can be used. Denoting aI the thermal expansion coefficient in direction I, and h the change of temperature, the thermo-elastic constitutive equations can be written as S I ¼ sIJ T J þ aI h;

I; J ¼ 1; . . . ; 6

Fig. 6. Illustration of the superposition method for the extension–shear mode.

ð7Þ

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Table 1 Analysis and finite element types within ABAQUS Analysis type

Finite element

Description

Shell

Thermal analogy Piezoelectric analysis

S4R5 –

4 nodes shell element with 5 points reduced integration No standard piezoelectric shell element

2D plane strain

Thermal analogy

CPE4R CPE8R CPE4E CPE8E

2D 2D 2D 2D

4 8 4 8

C3D8R C3D20R C3D8E C3D20E

3D 3D 3D 3D

8 nodes element with reduced integration 20 nodes element with reduced integration 8 nodes piezoelectric element 20 nodes piezoelectric element

Piezoelectric analysis 3D

Thermal analogy Piezoelectric analysis

The thermal analogy can be retraced by looking at the converse piezoelectric constitutive Eq. (2a). It appears clearly that a thermo-mechanical formulation can be used, provided that aI takes the value of d3I and h takes the value of E3. Hence, this approach assumes implicitly non-zero normal component of the electric field only. Notice that the short-circuit elastic compliances sEIJ are to be used in Eq. (7) and that this approach is suitable for the extension-mode only because a4 = a5 = 0 for isotropic or orthotropic thermo-elastic materials. Otherwise, if available, FE with anisotropic thermo-elastic constitutive behaviour could be used. 4.2. Finite element modelling approaches The modelling possibilities of a piezoelectric bimorph using a general purpose FE code, such as ABAQUS [19], are summarised in Table 1. It shows that the user has the limited choice to apply either 2D/3D FE or the thermal analogy approach. Special attention has to be paid to the ABAQUS special matrix notations, which do not follow closely the standard engineering ones, when introducing the piezoelectric material properties. The same is true if the ANSYS FE code is used. 5. Evaluation and assessment In this section, with the help of the general purpose commercial FE code ABAQUS, the classical extension and new shear bimorph models are assessed and evaluated. First, some confusing validation practices are clarified regarding the experimental results of Lee and Moon [2] which are shown to be for a parallel and not a series bimorph as generally considered [3,4]. Then, the analytical formula, Eq. (3), proposed by Smits et al. [1] and the thermal analogy approach, Eq. (7), are evaluated in order to delimit their respective validity domains. Also, some results on the influence of the mechanical BC are presented for several piezoelectric extension bimorphs, that are totally or partially poled. The aim is to explore the drain channel phenomena due to the transverse (along the width) actuation, via d31 and d32 induced in-plane strains, which is inherent to the

nodes nodes nodes nodes

plane strain element with reduced integration plane strain element with reduced integration piezoelectric plane strain element piezoelectric plane strain element

extension coupling mode. Finally, efforts are concentrated on the evaluation of the shear and shear–extension analytical formulas, Eqs. (5c) and (6). 5.1. Extension bimorphs The first example to be discussed here is the piezoelectric polymer bimorph tested by Lee and Moon [2]. It consists of a cantilever beam made of two Kynar film layers with a thickness of h = 0.11 mm, a length of L = 80 mm and a width of B = 10 mm. Kynar is a piezoelectric polymer of PVDF type (PolyVinylDiene Fluoride). Its material properties, as described in [20], are given in the Appendix. In the experiment, the middle electrode is grounded, whereas the top and bottom ones are at the same potential (DC, with max. 500 V), so that this configuration is equivalent to a parallel bimorph (Fig. 7). A FE analysis is carried out with 4-nodes shell elements of ABAQUS/Standard (S4R5) and the thermal analogy. A convergence study shows that a mesh fineness of 10 elements over the length is sufficient. The results are presented in Fig. 7 in terms of the deflection versus the applied voltage for both series and parallel connections. Similar results were obtained with the 3D piezoelectric (C3D20E) and thermo-mechanical (C3D20R, with thermal analogy) elements. All FE results for the parallel connection are close to the experimental results of Lee and Moon [2], whereas those for the series connection differ strongly from them. This contradicts the common validation practice which considers these experimental results for a series connection. In fact, the experiment of Lee and Moon [2] is often cited in the literature with reference to a series bimorph [3,4]. When considering the series bimorph, confusion is often made between the potential and potential difference, so that an electric field twice as high as in the experiment of Lee and Moon [2] was considered. Confusion can also easily be made between series and parallel connections if the electrode structure and applied voltage are not explicitly mentioned. In order to discuss the validity of Eq. (3) from [1], a parametric study is carried out. For this, the following parameters are varied:

C. Poizat, A. Benjeddou / Computers and Structures 84 (2006) 1426–1437

Tip deflection [mm]

1432

5.00

Lee and Moon data

4.50 4.00

ABAQUS (parallel) ABAQUS (series)

3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0

100

200

300

400

500

Applied voltage [V] Fig. 7. FE analysis of the bimorph experimented by Lee and Moon [2].

• Length-to-thickness ratio L/H, from 5 to 364 (L/H = 364 in [2]), • Material: Kynar and PZT-5A (see Appendix for their material properties), • Modelling approach: analytical, Eq. (3), and FE models, Table 1. It’s worth noting here that the deflection obtained with a ratio L/H = 364 (as taken in [2]) corresponds to a very high electric field (4545 V/mm), so that the material behaviour of typical piezoelectric ceramics is no longer linear. The corresponding very high deflection has to be considered as a theoretical value, helping in the discussion of the different approaches. Mesh convergence was carried out with particular attention to the maximum-to-minimum element length ratio to avoid numerical locking effects. The results

are given in Table 2 for Kynar and Table 3 for PZT-5A. The following conclusions can be drawn: • The discrepancy between the analytical results, Eq. (3), and the FE results is material dependent. For instance, for L/H = 50, the discrepancy is in the order of 0.5% with Kynar and 5% with PZT-5A when using C3D20E piezoelectric finite elements. • The discrepancy between analytical results, Eq. (3), and FE results is smaller when the finite element analysis is based on the thermal analogy than when using piezoelectric elements, in particular the plane strain ones. • The discrepancy between the analytical results and the numerical results obtained with piezoelectric plain strain elements is higher with the material PZT-5A than with Kynar.

Table 2 Parallel bimorph tip deflection in [mm] versus L/H, obtained with Eq. (3) from Smits et al. [1], and FE results obtained with several element types (see Table 1) L/H

364 100 50 10 5

Analytic [1]

ABAQUS S4R5

da

dFE

Err. [%]

ABAQUS CPE8R dFE

Err. [%]

ABAQUS CPE8E dFE

Err. [%]

ABAQUS C3D20R dFE

Err. [%]

ABAQUS C3D20E dFE

Err. [%]

4.560 0.345 0.086 0.003 0.001

4.560 0.345 0.086 0.003 0.001

0.0 0.0 0.0 0.0 0.0

4.560 0.345 0.086 0.003 0.001

0.0 0.0 0.0 0.1 0.3

4.824 0.365 0.091 0.004 0.001

5.5 5.5 5.5 5.6 5.8

4.559 0.347 0.087 0.003 0.001

0.0 0.5 0.4 0.3 0.5

4.589 0.346 0.086 0.003 0.001

0.6 0.3 0.3 0.2 0.4

Err. = 1da/dFE. Material: Kynar.

Table 3 Parallel bimorph tip deflection in [mm] versus L/H, obtained with Eq. (3) from Smits et al. [1], and FE results obtained with several element types (see Table 1) L/H

364 100 50 10 5

Analytic [1]

ABAQUS S4R5

da

dFE

Err. [%]

dFE

Err. [%]

dFE

Err. [%]

dFE

Err. [%]

dFE

Err. [%]

33.92 2.565 0.641 0.026 0.006

34.03 2.574 0.643 0.026 0.006

0.3 0.3 0.3 0.3 0.3

33.90 2.574 0.639 0.025 0.006

0.0 0.3 0.3 1.9 3.5

39.20 2.968 0.742 0.030 0.007

13.5 13.6 13.5 13.2 11.7

34.97 2.635 0.655 0.026 0.006

3.0 2.7 2.1 0.2 2.2

32.32 2.446 0.611 0.024 0.006

4.9 4.9 4.9 6.5 7.3

Err. = 1  da/dFE. Material: PZT-5A.

ABAQUS CPE8R

ABAQUS CPE8E

ABAQUS C3D20R

ABAQUS C3D20E

z/H

C. Poizat, A. Benjeddou / Computers and Structures 84 (2006) 1426–1437 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.00

400.07 300.11 416.03 200.11 324.01 224.21 100.08 116.32 116.32 100.08

Kynar dielectric

324.01 300.11

416.03 400.07

100

PZT5A

224.21 200.11

0

500

200

300

400

500 500

Electric potential [V]

Fig. 8. Electric potential distribution through the normalised thickness z/H for Kynar, PZT-5A and a purely dielectric material (L/H = 10).

• When using the thermal analogy, a FE coarse mesh with only 10 elements over the length provides, for both materials, right results with only max. 0.3% discrepancy with the analytical model of Smits et al. [1], for all length-to-thickness ratios. These results can be discussed as follows: The increasing discrepancy between the analytical results and the numerical results obtained with piezoelectric plane strain elements with an increasing piezoelectric coupling is mainly due to the artificial piezoelectric coupling effect induced by the plane strain assumption. This point is discussed later in more details in Section 5.2, and in the recent contribution of Wang [21]. The increasing discrepancy between Kynar and PZT-5A regarding the FE results with C3D20E elements and analytical results (Eq. (3) from [1]) can be explained by the neglect of the quadratic induced potential (Section 3.1) in the analysis of Smits et al. [1]. This key point is illustrated in Fig. 8, obtained by using C3D20E finite elements: this figure indicates that the through-the-thickness distribution of the electric potential for Kynar is only slightly different from that of a purely dielectric material (the piezoelectric constants of Kynar or PZT-5A are set to zero). On the contrary, the electric potential may differ from the linear case to more than 15% for PZT-5A (Fig. 8). The discrepancy between the analytical results from Eq. (3) and FE results is higher when using piezoelectric (C3D20E) instead of thermal (C3D20R) elements. In the thermal analogy, the imposed temperature is constant

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through the layer thickness. This is representative of a constant through-the-thickness electric field, as in the analysis of Smits et al. [1]. The overall discrepancy remains small, below 5% for PZT-5A and L/H > 10, but increases to 7.3% for L/H = 5 (Table 3). Another reason for the discrepancy between analytical [1], Eq. (3), and 3D FE results (Table 3) lies in the transverse (width) bimorph effect. As underlined in Lim et al. [11], a bending effect is also present in the transverse direction y. Due to the free edges and the contraction and elongation in the transverse direction y, the shape of the bimorph far from the clamped edge warped in both longitudinal (x) and transverse (y) directions. This leads to a ‘‘drain channel’’ effect which depends on the bimorph width B, on length-to-thickness ratio, material properties, and mechanical BC. Thus, the latter effect is now studied in details. The influence of the mechanical BC on an extension parallel bimorph is analysed for three types of BC: • One edge is clamped, the other one is free (clamped-free, CF) – Fig. 2b, • Both edges are simply supported (SS), • Both edges are mechanically clamped in rotation (‘‘clamped–clamped’’, CC) and in the direction z. The extension bimorph considered in the FE analysis is a two layers parallel bimorph of length L = 80 mm, of width B = 10 mm, of thickness H = 0.22 mm, made of piezoelectric ceramic PZT-5A. Clearly, from Table 4, the mechanical BC significantly influence the resulting maximum deflection. The given maximum deflection is obtained for the CF BC. CC BC leads to a small deflection. A significant ‘‘drain channel’’ effect is observed for CC BC (Fig. 9a). It is smaller for CF BC, and absent in the shear-mode, since there E2 = 0.

Table 4 Maximum tip deflection for three different boundary conditions Boundary condition

Maximum deflection [mm]

Clamped–clamped (Fig. 9a) Simply supported Clamped-free

0.058 (x = L/2) 1.587 (x = L/2) 6.465 (x = L)

Fig. 9. Deformation of a CC parallel bimorph (a) fully (drain channel effect) and (b) partially (Lp/L = 25%) poled (Fig. 10). For symmetry reasons, just one half of the structure is modelled and represented here.

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C. Poizat, A. Benjeddou / Computers and Structures 84 (2006) 1426–1437

Fig. 10. Parallel bimorph with partial poling for (a) CF and (b) CC and SS BC.

It is suggested that this ‘‘drain channel’’ effect can be reduced by only partially poling the piezoelectric bimorph (see Fig. 9b). This is confirmed by the analysis made for several Lp/L ratios, where Lp is the length of the poled ceramic (Fig. 10), for CC boundary conditions. For the non-poled part, the piezoelectric coefficients are, for simplicity, set to zero in the simulations. The results are shown in Fig. 11 as the maximum deflection in terms of Lp/L ratio (with 10% increments) for the three considered mechanical BC. Details of the mid-plane deformation profiles for CC and CF BC are given in Fig. 12 for the poling ratios of Lp/L = 50% and Lp/L = 100%. The maximum deflection is obtained for Lp/L = 100% with CF and SS boundary conditions (Fig. 11). The ratio

7 Deflection [mm]

6 5 4

CC SS CF

of the maximum deflection between the CF and SS BC is near 4. Interestingly, the maximum deflection in the CC case is for Lp/L = 50% ‘‘only’’ 10 times smaller than for the corresponding CF BC, with the advantage to support a higher mechanical load (Figs. 11 and 12). This last point is under investigation since it is relevant in actuator applications like the peristaltic micropump presented by Sester and Poizat [15] and studied later by Poizat et al. [16]. 5.2. Shear and extension–shear bimorphs The cantilever shear–extension bimorph shown in Fig. 4 is now considered to analyse both shear and extension– shear bimorphs. Both layers consist of a piezoceramic PZT-5A (material properties are given in the Appendix). The following dimensions are retained: H = 2 mm, hb = ht = 1 mm, B = 10 mm. The length-to-thickness ratio L/H takes the values L/H = 5, 10 and 50. The deflection, longitudinal stress and electric field are normalised (e.g. u0 = 100 V) as

3

d¼

2

cE11 d e31 u0

T1 ¼

LT 1 e31 u0

E3 ¼ 

E3 H u0

ð8Þ

1 0 0%

25%

50%

75%

100%

L p /L Fig. 11. Extension bimorph maximum deflection versus Lp/L for CC, SS and CF BC.

7

Deflection [mm]

6

CF, 100%

5 4 3

CF, 50% CC, 100%

2

CC, 50%

1 0 -1 0

10

20

30

40 x [mm]

50

60

70

80

Fig. 12. Extension bimorph deflection (y = B/2, z = H/2) with Lp/L = 50% and Lp/L = 100% for CC and CF boundary conditions.

The normalised deflections versus the poling angle, obtained using Eq. (6) and 2D/3D FE are shown in Fig. 13 together with the results of Vel and Batra [8]. Quantitatively, all methods lead to the same results for the shear-mode configuration (a = 90). For smaller angles, as expected, the results of Vel and Batra [8] (see their Fig. 7) are in good agreement with those obtained with 2D plane strain piezoelectric FE (CPE8E element of ABAQUS) for L/H = 5 as well as for L/H = 10. The numerical results obtained with a 3D model (C3D20E elements of ABAQUS) are in better agreement with the proposed Eq. (6). This relatively good agreement between the 3D FE results and the proposed Eq. (6) is explained as follows: with increasing extension-mode actuation, the coupling effect in the transverse direction y leads to the ‘‘drain channel’’ effect and discrepancy with the 1D model of Smits et al. [1], but this discrepancy is small in comparison to the coupling effect imposed by the plane strain assumption. This is coherent with the fact that all approaches converge to the same results when the angle a tends to 90. In the case of the shear-mode actuation mechanism, the transverse direction y does not play any role.

C. Poizat, A. Benjeddou / Computers and Structures 84 (2006) 1426–1437

1435

500

140

450 Normalized deflection δ

Normalized deflection δ

120 100 80 60

ABAQUS 3D

Equation 6

40

ABAQUS 2D

20

0

10

20

30

40

50

350 300 250 200

ABAQUS 3D

150

Equation 6

100

ABAQUS 2D

50

Vel and Batra

0

400

Vel and Batra

0 60

70

80

90

0

10

20

30

40

50

60

70

80

90

Angle α [degree]

Angle α [degree]

(a) L/H=5

(b) L/H=10 1000

Normalized deflection δ

900 800 700 600 500 400 300

ABAQUS 3D

200

Equation 6

100 0 0

10 20 30 40 50 60 70 80 90 Angle α [degree]

(c) L/H=50

Fig. 13. Normalised deflection versus poling angle a (Fig. 4) for shear–extension bimorph.

This comparison between several approaches underlines the importance of an appropriate model choice. The optimal angle for a maximum tip deflection as well as the predicted tip deflection is namely significantly different, depending on the modelling method. Here, both analytical solutions ([8] and proposed Eq. (6)) fail in predicting the optimal angle for a maximum tip deflection obtained with ABAQUS 3D analyses.

Fig. 13, as expected, shows that the tip deflection is smaller for the shear-mode actuators. The main advantage of the shear or shear–extension actuation mechanisms consists in the lower stress level [13]. As can be seen from Fig. 14, the maximum axial stress is two times smaller for a = 45, without significant reduction of the tip deflection. In Figs. 15 and 16, the through-the-thickness electric potential and normal non-dimensional electric field

1.00 1.00

0.80

0.80

extension mode (0˚)

extension mode (0˚)

0.60

0.60

0.40

shear-extension mode (45˚)

shear mode (90˚)

z/H

z/H

shear mode (90˚)

0.40

shear-extension

mode (45˚) 0.20

0.20

0.00 -60

-40

-20

0

20

40

60

Normalized stress T1 [MPa], x=L/2

Fig. 14. Through-the-thickness longitudinal stress T1 at x = L/2 for three angles a.

0.00 -50

0 -25 25 Electric potential [V]

50

Fig. 15. Through-the-thickness electric potential at x = L/2 for three angles a.

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C. Poizat, A. Benjeddou / Computers and Structures 84 (2006) 1426–1437 Table 5 PZT-5A elastic (GPa), piezoelectric (C m2) and dielectric (108 F m1) constants

1.00 0.80 extension mode (0˚)

z/H

0.60

shear mode (90˚) 0.40

shear-extension mode (45˚)

0.20

C E11

C E22

C E33

C E12

C E13

C E23

C E44

C E55

C E66

99.201

99.201 86.856 54.016 50.778 50.778 21.100 21.100 22.593

e31

e32

e33

e24

e15

eS11

eS22

eS33

7.209

7.209

15.118

12.322

12.322

1.53

1.53

1.5

Appendix. Piezoelectric materials properties

0.00 0.8

0.9 1.0 1.1 1.2 1.3 Normalized electric field gradient E3 [V/mm]

Fig. 16. Normalised electric field E3 at x = L/2 for three angles a.

distributions are shown at x = L/2. In agreement with the assumptions made in [12], the electric potential is linear for the shear actuation mechanism and quadratic for the extension actuation one. The shear–extension bimorph with a = 45 lies between these two configurations. 6. Conclusion Extension, shear and extension–shear two-layers bimorphs were studied with analytical and numerical approaches. In this frame, some simple well known (extension) and new (shear and extension–shear) analytical formulas, as well as the thermal analogy approach, with their common assumption of a through-the-thickness constant electric field, were evaluated against 2D plane strain and 3D piezoelectric FE available in some commercial codes. The FE method is pertinent to validate these approaches and to study more complex structures than series or parallel cantilever bimorphs. It can help, for example, in designing optimal CC bimorphs and their associated 3D ‘‘drain channel’’ effect. In particular, it was shown that a CC bimorph with a poling ratio Lp/L of 50% leads to a reasonable maximum deflection with, probably, the advantage of a higher blocking pressure, a parameter of particular interest in the design of peristaltic micropump [16], in comparison to the corresponding cantilever configuration. This latter aspect is under work. Besides, the FE analyses have shown that, the analytical and thermal analogy solutions for extension bimorphs can be used with a reasonable agreement with 3D piezoelectric FE for large to medium bimorph length-to-thickness ratios. It was also found that, while shear bimorphs lead to smaller maximum deflection, they also create much smaller and homogeneous stresses in the bimorphs interface. The new extension–shear bimorph shows an intermediate behaviour between extension and shear ones taking advantage of each of them. The corresponding new analytical formula has reproduced with good accuracy the 3D piezoelectric FE results confirming its usefulness for preliminary designs.

Kynar [20]: The piezoelectric polymer KYNAR has the following properties: Elastic properties Piezoelectric properties Dielectric properties

E = 6.85 GPa, m = 0.29 (isotropic behaviour) d31 = 22.9 pC/N and d32 = 4.6 pC/N eT = 3 · 109 F m1

PZT-5A [8]: The piezoceramic PZT-5A has the elastic (GPa), piezoelectric (C m2) and dielectric (108 F m1) material properties reported in Table 5. For clarity, the elastic (GPa), piezoelectric (C m2) and relative dielectric (xe0, e0 = 8.854e12 F m1) matrices, after some manipulations (see e.g. [12]) for a PZT-5A piezoelectric ceramic polarised along the global x-axis, are: 3 2 86:86 50:78 50:78 0 0 0 6 50:78 99:20 54:01 0 0 0 7 7 6 7 6 7 6 50:78 54:01 99:20 0 0 0 E 7 6 ½c xpoled ¼ 6 0 0 22:6 0 0 7 7 6 0 7 6 4 0 0 0 0 21:1 0 5 0 15:12

0 0

6 7:21 6 6 6 7:21 ¼6 6 0 6 6 4 0

0

2

½expoled

2

0 1694

 S 6 er x–poled ¼ 4 0 0

0 0 0 12:32 0 1728 0

0 3 0 0 7 7 7 0 7 7 0 7 7 7 12:32 5 0 0

0

0

21:1

3

7 0 5 1728

References [1] Smits JG, Dalke SI, Cooney TK. The constituent equations of piezoelectric bimorphs. Sens Actuat A 1991;28:41–61. [2] Lee CK, Moon FC. Laminated piezopolymer plates for torsion and bending sensors and actuators. J Acoust Soc Am 1989;85:2432–9. [3] Koconis DB, Kolla´r LP, Springer GS. Shape control of composite plates and shells with embedded actuators. I. voltages specified. J Compos Mater 1994;28:415–58.

C. Poizat, A. Benjeddou / Computers and Structures 84 (2006) 1426–1437 [4] Donthireddy P, Chandrashekhara K. Modeling and shape control of composite beams with embedded piezoelectric actuators. Compos Struct 1996;35:237–44. [5] Carrera E. An improved Reissner–Mindlin-type model for the electro-mechanical analysis of multilayered plates including piezolayers. J Intell Mater Sys Struct 1997;8:232–48. [6] Abramovich H. Deflection control of laminated composite beams with piezoceramic layers – closed form solutions. Compos Struct 1998;43:217–31. [7] Abramovitch H. Piezoelectric actuation for smart sandwich structures – closed form solutions. J Sandwich Struct Mater 2003;5:377–96. [8] Vel SS, Batra RC. Analysis of piezoelectric bimorphs and plates with segmented actuators. J Thin-Walled Struct 2001;39:23–44. [9] Fernandes A, Pouget J. Analytical and numerical approaches to piezoelectric bimorph. Int J Solids Struct 2003;40:4331–52. [10] He LH, Lim CW, Soh AK. Three-dimensional analysis of an antiparallel piezoelectric bimorph. Acta Mech 2000;145:189–204. [11] Lim CW, He LH, Soh AK. 3D electromechanical responses of a parallel piezoelectric bimorph. Int J Solids Struct 2001;38:2833–49. [12] Benjeddou A, Trindade MA, Ohayon R. A unified beam finite element model for extension and shear piezoelectric actuation mechanisms. J Intell Mater Sys Struct 1997;8:1012–25.

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[13] Benjeddou A, Trindade MA, Ohayon R. Piezoelectric actuation mechanisms for intelligent sandwich structures. Smart Mater Struct 2000;9:328–35. [14] Benjeddou A. Advances in piezoelectric finite element modelling of adaptive structural elements: a survey. Comput Struct 2000;76: 347–63. [15] Sester M, Poizat C. Simulation techniques for piezoelectric composite materials and their applications to smart structures. In: Wereley NM, editor. Smart structures and integrated systems, vol. 3985. SPIE; 2000. p. 750–8. [16] Poizat C, Benevolenski O, Thielicke B. From bending actuators to a peristaltic micropump: modelling and design analysis. In: Lipitakis EA, editor. Proceedings of HERCMA, Athens, September 24–27, 2003, vol. 1. Athens: LEA Publishers; 2003. p. 141–8. [17] Ikeda T. Fundamentals of piezoelectricity. Oxford: Oxford University Press; 1996. [18] Ha SK, Kim YH. Analysis of a piezoelectric multimorph in extensional and flexural motions. J Sound Vib 2002;253:1001–14. [19] ABAQUS Standard and Theory Manuals, Version 6.4, HKS, 2003. [20] ATOCHEM, Kynar Piezo film technical manual 1987. p. 9–20. [21] Wang SH. A finite element model for the static and dynamic analysis of a piezoelectric bimorph. Int J Solids Struct 2004;41:4075–96.