Lumped model of bending electrostrictive transducers

Lumped model of bending electrostrictive transducers for energy harvesting Mickaël Lallart, Liuqing Wang, Claude Richard, Lionel Petit, and Daniel Guyomar Citation: Journal of Applied Physics 116, 124106 (2014); doi: 10.1063/1.4896185 View online: http://dx.doi.org/10.1063/1.4896185 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Fluoroethylenepropylene ferroelectret films with cross-tunnel structure for piezoelectric transducers and micro energy harvesters J. Appl. Phys. 116, 074109 (2014); 10.1063/1.4893367 A vibration energy harvester using magnet/piezoelectric composite transducer J. Appl. Phys. 115, 17E522 (2014); 10.1063/1.4867599 Mechanical characterization of an electrostrictive polymer for actuation and energy harvesting J. Appl. Phys. 111, 124115 (2012); 10.1063/1.4729532 Evaluation of energy harvesting performance of electrostrictive polymer and carbon-filled terpolymer composites J. Appl. Phys. 108, 034901 (2010); 10.1063/1.3456084 Modeling nonlinearity in electrostrictive sonar transducers J. Acoust. Soc. Am. 104, 1903 (1998); 10.1121/1.423761

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JOURNAL OF APPLIED PHYSICS 116, 124106 (2014)

Lumped model of bending electrostrictive transducers for energy harvesting €l Lallart,a) Liuqing Wang, Claude Richard, Lionel Petit, and Daniel Guyomar Mickae Universit e de Lyon, INSA-Lyon, LGEF EA 682, 8 rue de la Physique, Villeurbanne, F-69621, France

(Received 11 March 2014; accepted 10 September 2014; published online 25 September 2014) Electroactive polymers, and more particular dielectric electrostrictive polymers, have been of great interest over the last decade thanks to their flexibility, easy processing, conformability, and relatively low cost. Their application as actuators, sensors, or energy harvesters suits very well to systems that require high strain. In particular, bending devices are an important application field of such materials, especially when dealing with devices subjected to air or liquid flows. Nevertheless, the design of such devices and their associated electrical interface still requires starting from the local aspects of the electrostrictive effect. In order to provide a simple yet efficient design tool, this paper exposes a simple lumped model for electrostrictive dielectric devices working under flexural solicitation. Based on the analysis of the converted energy with respect to the provided energy, it is shown that electrostrictive systems can easily be reduced to a simple spring-mass-damper system with a quadratic dependence to the applied voltage on the mechanical side and to a current source controlled by the applied voltage with a capacitive internal impedance on the electrical side. Experimental measurements carried out to evaluate the mechanical to electrical conversion effect as well as the energy harvesting abilities in such systems also validate the proposed approach. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4896185] V

I. INTRODUCTION

Along with the development of smart materials for actuation, sensing, or energy harvesting, an increased attention has been placed on electroactive polymers (EAPs) over the last few years.1–8 Electrostrictive polymers are a particular branch of electroactive polymers that feature very low Young’s modulus, have an easy fabrication process, and are conformable and low cost. Electrostrictive polymers are divided into two categories: ionic9,10 and dielectric.11–18 While the former can exhibit very large deformation, its limits in terms of frequency range and high losses make the latter more adapted to applications involving energy harvesting at frequencies higher than 1 Hz. Energy harvesting is the process of converting energy from a source directly available to the system (for instance mechanical, thermal) to useful electrical energy that can be used by the connected electrical system. The spreading of energy harvesting products has been enabled by the dramatic decrease of electronic device consumption as well as the maintenance issue and environmental concerns raised by primary batteries.19,20 In case of mechanical energy harvesting, energy source can be found in various nature in terms of strain and stress levels and frequencies. While many works were devoted to piezoelectric,21 electromagnetic,22 or electrostatic23 devices for mechanical energy harvesting, addressing low frequency, high strain systems is better suited to electrostrictive polymers for the above-mentioned reasons. When using such materials, energy harvesting techniques may consider charge/discharge cycles based on electrostatic energy harvesting,24–26 or the use of pseudopiezoelectric operations.27–29 a)

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0021-8979/2014/116(12)/124106/8/$30.00

The principles of the electromechanical conversion effect in dielectric electrostrictive polymers originate from the Maxwell stress induced by charges on the electrodes of the material and by the strain-dependent orientability of the dipoles.30–35 Nevertheless, while local constitutive equations of electrostriction have been derived from formal analysis (using phenomenological approach30,36–38 or physical interpretation34,35), general models linking macroscopic quantities usually rely on numerical integration.39,40 No simple high-level model (such as electromechanically coupled spring-mass-damper for piezoelectric element) linking macroscopic quantities (force, displacement, voltage, and current) has been proposed in order to be able to efficiently design the structure and the associated interface (e.g., electrical circuit), although some studies empirically deduced the possible law using analogy with piezoelectricity when the polymer is subjected to a bias voltage.41 Hence, this work aims at giving a lumped model for electrostrictive beam devices working under flexural solicitations. While previous works established the dependency of the flexural displacement with the electric field,42 no real energy conversion analysis has been performed so far, preventing linking macroscopic and local parameters for the considered structure type. Hence, the objective of this paper is to give a simple yet effective formulation of the macroscopic electromechanical equations (motion equation and current generation equation) for 1D structures based on the total converted energy (both from mechanical to electrical and from electrical to mechanical), allowing the derivation of a set of mechanical and electrical macroscopic equations that permit both quickly evaluating the performance of a given device or optimizing the global system, for instance, for energy harvesting purposes. Hence, while in Ref. 42 the aim was to investigate and optimize the strain/displacement

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for a given applied electric field, therefore focusing on actuator only (the displacement was arising from the application of an electric field—no external force was considered), without any energy conversion analysis and concerns, and without a simplified lumped model, the present paper is more concerned with the energy and conversion analysis, especially from the mechanical to electrical aspect (externally applied force—although the proposed model can also consider the electrical to mechanical conversion), and with a final objective to set up a high-level, simplified lumped model that can be easily used for the design of the transducer environment, like electrical interfaces. The paper is organized as follows. Section II exposes the derivation of this simple lumped model, based on the electrical and mechanical energy densities for obtaining the simplified global response of the system. The particular case of a cantilever beam is discussed in Sec. III to give an illustrative example of the model. Then, the proposed approach is experimentally assessed in Sec. IV. Finally, a brief conclusion is presented in Sec. V, summarizing the main findings of this work.

are the compliance, electrostrictive coefficient, and absolute permittivity of the material. Superscripts denote the constant parameter. Then, assuming Euler-Bernoulli behavior (S2 ¼ 0, T3 ¼ 0, i.e., plane sections remain plane), it is possible to find the expression of the stress along axis 1 as a function of the strain along the same direction and the electric field as

II. SYSTEM MODELING

where Yb and b, respectively, refer to the Young modulus and Poisson’s ratio of the substrate. Finally, from the stress-strain expressions of the polymer and substrate layers (Eqs. (2) and (3) respectively), and considering the neutral axis hn

This section aims at exposing the general theoretical development for the lumped model of electrostrictive-based beam devices. Starting from the mechanical behavior of a structure with a bonded electrostrictive element on its surface and subjected to flexural vibration, an energy analysis is then performed in the electrical domain to get the electrical expression and then to the mechanical aspect for obtaining the motion equation. The global investigated system is presented in Figure 1, where hp and hb are the respective thickness of the polymer and the substrate beam, Lp and Lb their length. Finally, the width of the device will be denoted w in the following.

T1 ¼ cEp S1 1 pE cEp M31 E3 2 with cEp ¼

YE ; (2) 1 pE 2

with YE and pE the Young modulus and Poisson’s ratio of the short-circuited polymer, respectively. However, in order to have a non-null response in the case of bending, the neutral axis should not be in the middle of the polymer. Hence, a substrate is added to the structure. For the latter and according to Hooke’s law, the relationship between the stress along axis 1 T1 and strain S1 along the same axis is given by T1 ¼ cb S1

hn ¼

with

cb ¼

Yb ; 1 b 2

hb 2 cb þ 2hb hp cEp þ hp 2 cEp ; 2 hb cb þ hp cEp

(3)

(4)

it is possible to get the expression of the strain as a function of the flexural displacement u3(x1) S1 ¼ ðx3 hn Þ

@ 2 u 3 ðx 1 Þ : @x1 2

(5)

A. Mechanical behavior

The mechanical behavior of the structure featuring flexural bending beam can be obtained starting from the constitutive equations of electrostrictive polymers assuming transverse isotropic material and polarization along axis 3 given as, considering Voigt’s notation 8 E E E 2 > < S1 ¼ s11 T1 þ s12 T2 þ s13 T3 þ M31 E3 (1) S2 ¼ sE12 T1 þ sE11 T2 þ sE13 T3 þ M31 E3 2 > : T D3 ¼ 33 E3 þ 2M31 T1 E3 þ 2M31 T2 E3 þ 2M33 T3 E3 ; with S, T, E, and D, respectively, referring to the strain, stress, electric field, and electric displacement. s, M, and

FIG. 1. Schematic of the considered structure.

B. Electrical aspect

Now the mechanical expression of the strain (which depends on the stress and electric field) obtained, and using the electrical constitutive equation in Eq. (1), it is possible to derive the electric displacement as a function of electric field E3 and strain S1 in the active layer as 2 1 D3 ¼ T33 2 E þ 1 pE cEp M31 2 E3 2 E3 s 11 þ 2M31 1 pE cEp S1 E3 T33 E3 þ 2M31 1 pE cEp S1 E3 : (6) From this equation and from the strain-flexural displacement expression in the polymer given by Eq. (5), the variation of the provided electrical energy to the polymer (only the latter layer receives the electrical energy, so that the thickness ranges from hb hn to hb þ hp hn in the following expression) is given by, assuming that the thickness hp is small enough to approximate the relationship between the electric field E3 and voltage V as E3 ¼ V/hp


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dWprov;elec ¼

ð w ð Lb ð hb þhp hn 0

J. Appl. Phys. 116, 124106 (2014)

0

hb hn

wLp 33 VdV hp

The provided electrical energy being given by the voltage and charge Q provided by the source, the lumped expression of the electrical relationship is then given as

E3 dD3 dx3 dx1 dx2 M31 1 pE hb cb cEp w hp

!

hp þ hb @u3

2 V d hp cEp þ hb cb @x1 x1 ¼Lp ! M31 1 pE hb cb cEp w hp þ hb hp hp cEp þ hb cb !

@u3

VdV: @x1 x1 ¼Lp !

Q ¼ C0 V aðxÞVu3 ðxÞ:

C. Mechanical aspect

(7)

From the previous expression, it is possible to decompose the energy variation into a purely dielectric energy and a coupling energy (part of the provided electrical energy is converted into mechanical energy) dWprov;elec ¼ dWelec þ dWcoupling;elec ;

(8)

Although the scope of this paper focuses on the mechanical to electrical conversion, this subsection proposes to investigate the mechanical aspect of bending electrostrictive systems. It is now considered that the energy is provided by a mechanical source. Starting from the stress expressions in both the polymer and substrate (Eqs. (2) and (3)), it is possible to derive the provided mechanical energy variation as dWprov;mech ¼

ð w ð Lb ð hb þhp hn

T1 dS1 dx3 dx1 dx2 ! ð Lb 2 @ u3 ðx1 Þ @ 2 u3 ðx1 Þ ¼ cb Ib d dx1 @x1 2 @x1 2 0 ! ð Lp 2 @ u3 ðx1 Þ @ 2 u3 ðx1 Þ þA d dx1 @x1 2 @x1 2 0 ! 1 hp þ hb E E þ M31 1 p hb cb cp w hb cb þ hp cEp 2 !

1 2 @u3

V d ; (13) h @x

0

which can be identified as dWelec ¼ C0 VdV dWcoupling;elec ¼ aðxÞVdðu3 ðxÞVÞ ¼ aðxÞV 2 du3 ðxÞ þ aðxÞVu3 ðxÞdV ;

(9)

with C0 the equivalent clamped capacitance and a(x1) the coupling coefficient (called “force factor” in the following in a similar manner than piezoelectric systems43), which actually depends on the considered position x along the x1 axis of the flexural displacement u3(x) C0 ¼

au3 ðxÞ ¼

wLp 33 ; hp

M31 1 pE hb cb cEp w hp

hp þ hb hp cEp þ hb cb

(10) !

(12)

hn

0

p

1 x1 ¼Lp

where Ib denotes the second moment of area of the substrate

!

@u3

: @x1 x1 ¼Lp (11)

Ib ¼

hb 3 w 12

(14)

and A is defined as

! w cb 2 hb 4 þ 4cb cEp hb 3 hp þ 6cb cEp hb 2 hp 2 þ 4cb cEp hb hp 3 þ cEp 2 hp 4 : A¼ cb hb þ cEp hp 12

Again, this expression can be decomposed into available mechanical energy variation and part of the energy variation converted into electricity dWprov;mech ¼ dWmech þ dWcoupling;mech ;

(16)

with K(x) and a(x) are defined as the short circuit stiffness and force factor, respectively #2 #2 ð Lb " 2 ð Lp " 2 @ u x @ u x ð Þ ð Þ 3 1 3 1 dx1 þA dx1 ; K ðxÞu3 2 ðxÞ ¼ cb Ib @x1 2 @x1 2 0 0

which can be further identified as, with u3(x) the flexural displacement at the considered point x dWmech ¼ K ð xÞu3 ðxÞdu3 ð xÞ 1 dWcoupling;mech ¼ að xÞV 2 du3 ð xÞ; 2

au3 ð xÞ ¼ (17)

(15)

M31 1 pE hb cb cEp w hp

hp þ hb hp cEp þ hb cb

!

(18) !

@u3

: @x1 x1 ¼Lp

(19)


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It can be noted that the stiffness K depends not only on the displacement at the considered position x but also on the flexural shape of the structure. Additionally, this parameter can also be found from the short-circuit response of the device. It is also possible to verify that the expression of a is identical to the one obtained in Sec. II B. Once the static mechanical behavior is obtained, and by considering that the dynamic deformed shape is similar, it is also possible to derive the dynamic mass M as ð Lb 1 mðx1 Þu3 2 ðx1 Þdx1 ; (20) M ð xÞ ¼ 2 u3 ð xÞ 0 with m(x1) the lineic mass of the beam, defined from the material dimensions and the respective densities qb and qp of the substrate and polymer ( mðx1 Þ ¼ wðhb qb þ hp qp Þ for x1 2 ½0; Lp (21) for x1 2Lp ; Lb : mðx1 Þ ¼ whb qb D. Global lumped model

For the electrical aspect, the differentiation of the expression of the charge (Eq. (12)) allows expressing the outgoing current I from the polymer as I¼

dQ ¼ að xÞV_ u3 ðxÞ þ að xÞV u_ 3 ð xÞ C0 V_ ; dt

(22)

while for the mechanical part, the global provided energy, taking into account the kinetic energy (Eqs. (17) and (20)) is given as a function of the externally applied force F ð ð ð :: Fdu3 ðxÞ ¼ Mu3 ð xÞdu3 ð xÞ þ Ku3 ð xÞdu3 ðxÞ ð 1 aðxÞV 2 du3 ð xÞ; (23) þ 2 which allows getting the equivalent lumped model for an electrostrictive-based electromechanical structure operating in bending mode and considering the displacement at the position x 8 > < MðxÞu:: ðxÞ þ Ku ð xÞ ¼ F 1 að xÞV 2 3 3 2 (24) > : I ¼ að xÞV u_ 3 ðxÞ þ að xÞu3 ðxÞV_ C0 V_ : Furthermore, dynamic losses can also be introduced through a structural damping coefficient C, yielding the global lumped model 8 > < Mð xÞu:: ðxÞ þ Cu_ ð xÞ þ K ð xÞu ðxÞ ¼ F 1 að xÞV 2 3 3 3 2 (25) > : I ¼ að xÞV u_ 3 ðxÞ þ að xÞu3 ðxÞV_ C0 V_ ; where the expression of the model parameters is summarized as follows:

8 ð Lb > 1 > > mðx1 Þu3 2 ðx1 Þdx1 ; > M ð xÞ ¼ 2 > > u3 ð xÞ 0 > > > 8 > #2 #2 9 > > < = ð Lb " 2 ð Lp " 2 > > @ u3 ðx1 Þ @ u 3 ðx 1 Þ 1 > > c I dx þ A dx K x ¼ > ð Þ b b 1 1 ;; > > @x1 2 @x1 2 u3 2 ðxÞ : < 0 0 2 3 > ! !

E E > > M 1 c c w h b b p > p 7 1 6 31 hp þ hb @u3

> > 4 5; aðxÞ ¼ >

> E > hp hp cp þ hb cb @x1 x1 ¼Lp u3 ð xÞ > > > > > > > wLp 33 > > > : C0 ¼ h : p

Hence, this globally coupled equation set shows the importance of the electromechanical factor a(x) for the energy conversion process, as the latter fixes the amount of charge generated for a given mechanical excitation of the global structure and/or the electromechanical force induced by the application of a voltage on the electroactive material. More precisely, this parameter is tightly related to the coupling coefficient k whose expression is given as, using definition from Refs. 44 and 45 and assuming a(x)V 2u Ku2 coupled energy að xÞV k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi : elastic energy dielectric energy C0 K

(27)

(26)

III. CASE STUDY

As an application example of the previous development, this section aims at expliciting the particular case of a cantilever beam clamped at x1 ¼ 0 (so that the polymer is bonded close to the fixed side) driven by a force F at its free end (x1 ¼ Lb). In this case, the flexural displacement can be derived through the expression of the moment Mðx1 Þ exerted by the force F given by M ðx1 Þ ¼ ðLb x1 ÞF:

(28)

As this moment is related to the longitudinal stress T1 following


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M ðx1 Þ ¼ w

ð hb þhp hn T1 x3 dx3 ;

(29)

hn

with the neutral axis hn given by

8 2 h c þ 2h h cE þ h 2 cE > > > b b b p p p p in the active zone ð0 < x1 Lp Þ < 2 hb cb þ hp cEp (30) hn ¼ > > hb > : in the passive zone ðLp < x1 Lb Þ; 2 it is possible to derive the expression of the flexural displacement u3(x1), using previous expressions in combination with Eq. (5) and considering boundary conditions (null displacement and slope at x1 ¼ 0) as well as the continuity of the displacement and slope at the interface between the active and passive zones as 8 > 1 B 2 Lb 1 > > E3 F x2 þ Fx3 in the active zone ð0 x Lp Þ < 2 A 2 6A (31) u3 ð xÞ ¼ > 2 > 2 > ð x 3Lb Þx F þ a1 x þ a2 in the passive zone ðLp < x Lb Þ; : cb hb 3 w with

8 ! > cb 2 hb 4 þ 4cb cEp hb 3 hp þ 6cb cEp hb 2 hp 2 þ 4cb cEp hb hp 3 þ cEp 2 hp 4 > w > > A¼ ; > > cb hb þ cEp hp 12 > > > > ! > > > w h þ h b p > > 1 pE cEp cb hp hb M31 ; > > BLp 2 12Lb Lp 6Lp 2 Lb Lp Lp 2 > > E3 þ þ F; a1 ¼ > > > A cb hb 3 w cb hb 3 w A 2A > > > > > BLp 2 2 6Lb Lp 2 4Lp 3 Lb Lp 2 Lp 3 > > E3 þ þ þ F: : a2 ¼ 2A cb hb 3 w cb hb 3 w 2A 3A Hence, from the displacement u3(x), it is possible to get the expression of the force factor a(x) such as 82 3 2 > ! 3 E E > M 1 c c w h > 31 b b p p > 2BLp E3 2 þ 2Lb Lp þ Lp 2 F 7 hp þ hb >6 > 4 56 7; > > 4 5 hp hp cEp þ hb cb > 1 3 > 2 2E 2 þ > F x L x Bx 3 b > > 3 > > < aðxÞ ¼ 2 in the active zone ð0 x Lp Þ 3 > > ( ) ! > E E > M31 1 p hb cb cp w 2 2 3 > 2BL E þ 2L L þ L h w F c > 6 7 h þ h p 3 b p p b b p b >4 > 5

: > > hp hp cEp þ hb cb 2A ð6Lb x2 2x3 ÞF þ ð a1 x þ a2 Þcb hb 3 w > > > > > : in the passive zone ðLp < x Lb Þ

(32)

(33)

However, the contribution of the electric field is negligible regarding the one from the force with typical and realistic parameter values. Therefore, the force factor can be approximated as follows: 8 !0 1 > E > hb cb cEp w M 1 31 > p h þh 2Lb Lp þLp 2 > p b > > A > > hp hp cEp þhb cb @ 1 3 2 > > x L x b > > 3 > > > > > > in theactive zone ð0xLp Þ > > > 2 3 > > > ! > < M31 1pE hb cb cEp w 6 7 hp þhb aðxÞ 4 5 E > hp hp cp þhb cb > > > > > > 9 8 > > > > > > 2Lb Lp þLp 2 cb hb 3 w < > > = > >

> 2 3 > > > ; :4Ax3 12ALb x2 þ ð 24A2cb hb 3 wÞLb Lp þ ðcb hb 3 w12AÞLp 2 xþ ðcb hb 3 w12AÞLb Lp 2 þ 8A cb hb w Lp 3 > > > 3 > > > > > : in thepassivezone ðL