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Thickness ratio and d 33 effects on flexible piezoelectric unimorph energy conversion

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2016 Smart Mater. Struct. 25 035037

(http://iopscience.iop.org/0964-1726/25/3/035037)

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Thickness ratio and d33 effects on flexiblepiezoelectric unimorph energy conversion

Taewoo Ha1,2, John X J Zhang3 and Nanshu Lu1

1Center for Mechanics of Solids, Structures, and Materials, Department of Aerospace Engineering andEngineering Mechanics, Department of Biomedical Engineering, Texas Materials Institute, The Universityof Texas, Austin, TX 78712, USA2Department of Electrical and Computer Engineering, The University of Texas, Austin, TX 78712, USA3Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA

E-mail: nanshulu@utexas.edu

Received 13 October 2015, revised 11 January 2016Accepted for publication 18 January 2016Published 22 February 2016

AbstractPiezoelectric unimorphs are bilayer structures where a blanket piezoelectric film (with top andbottom electrodes) is uniformly laminated on an inactive but flexible substrate. Because oftheir simple construction and flexibility, unimorphs are widely used as a key element inflexible sensors and actuators. The response of a unimorph is governed by the materialproperties of the film and the substrate as well as their geometric parameters. For lowfrequency biological energy harvesting, structural optimization is critical due to thedimensional confinement imposed by curvilinear and deformable bio-tissues. Here we report acomprehensive theoretical framework to investigate the effects of the film-to-substratethickness ratio on voltage, charge, and energy outputs when the unimorph is subjected to eightdifferent boundary/loading conditions. A broad class of power generators can be designedusing such a framework under the assumption that the unimorph length is very large comparedto its thickness, where the only dimensionless variable is the film-to-substrate thickness ratio.We show that the analytical and finite element modeling results are in excellent agreement. Fornot so thin unimorphs, there is non-zero normal stress in the thickness direction (σ3) and d33can play a significant role in this case. Non-monotonic dependence of voltage and energygeneration on thickness ratio has been found in some cases and optimum thickness ratio forunimorph generator can be predicted. When the unimorph is actuated by voltage appliedacross the piezo-film thickness, non-monotonic maximum deflection versus thickness ratio isalso found. This work provides new physical insights on unimorphs and analytical solutionsthat can be used for the structural design and optimization of unimorphs under differentboundary/loading conditions.

Keywords: piezoelectric, unimorph, flexible, thickness ratio, d33, energy, generator

(Some figures may appear in colour only in the online journal)

1. Introduction

Piezoelectric materials convert electrical pulses from or tomechanical vibrations [1]. Such electromechanical energyconversion is ubiquitous in both inorganic materials and bio-logical systems. Piezoelectric materials in nature can be cera-mic or polymeric: ceramic piezoelectric materials includinglead zirconate titanate (PZT) [2], zinc oxide (ZnO) [3], bariumtitanate (BTO) [4] are known to have large intrinsic

piezoelectric coefficients but they are mechanically stiff andbrittle. Polymeric piezoelectric materials such as poly-vinylidene fluoride (PVDF) [5] and porous electromechanicalfilm (EMFi) [6] are intrinsically more compliant, but theirpiezoelectric coefficients are one order smaller than those of theceramic ones [7]. Both kinds of piezoelectric materials havefound wide applications in sensing, actuation, and energyharvesting [8] because of their unique combination ofmechanical form factors and energy transduction capabilities.

Smart Materials and Structures

Smart Mater. Struct. 25 (2016) 035037 (19pp) doi:10.1088/0964-1726/25/3/035037

0964-1726/16/035037+19$33.00 © 2016 IOP Publishing Ltd Printed in the UK1

Mechanical energy from nature, machines, and thehuman body provides an excellent alternative as clean, non-depletable power sources. Many ambient sources ofmechanical energy, including wind [9], water flow [10], andhuman motion [11], have been collected by piezoelectricdevices. With the emergence of wearable electronics and bio-integrated electronics [12], flexible and stretchable generatorsthat can harvest power from the natural processes of the bodysuch as the vibration of muscles, lungs, and heart haveattracted growing interest [13]. Examples include ZnOnanowire based nanogenerators [14], buckled PZT nanor-ibbon based stretchable biocompatible energy harvester [15],and BTO [16] and ZnO [17] thin film based bendable nano-generators, as well as PVDF based nanogenerators and bio-sensors [18–21]. Although the power generated is usually onthe order of microwatts to milliwatts, which may not besufficient to power most wearable or implantable devices, thegenerators can at least be used to recharge batteries.

Theoretical models for piezoelectric energy harvestersare widely available in the literature [22–25]. Two majorthreads of energy harvesting optimization theories are reso-nance frequency analysis and structural analysis. Asmechanical activities of the human body exhibit low fre-quency (1–30 Hz) [26], resonance frequency optimization isless relevant here. Instead, structural optimization becomes animportant problem due to the dimensional confinementimposed by the curvilinear and deformable human body andorgans [27]. Constituent equations for a cantileverunimorph under static conditions have been established bySmits and Choi [23]. Using those equations, the effects ofthickness ratio and elastic mismatch of a cantileverunimorph have been theoretically investigated by Wang et al[22], who found a non-monotonic relationship betweenthickness ratio and electric outputs. Subsequently, severalnumerical studies on the effect of thickness ratio of unim-orphs have been reported [28–30].

Existing unimorph models [22, 23, 31] are mostly limitedto cantilever configurations and the d33 contribution has beenneglected because σ3 is always assumed to be zero. However,unimorphs can also operate under the boundary conditions ofpure bending or simple–simple support and can be subjectedto a variety of types of load or displacement excitations.Moreover, σ3 is non-zero for thick unimorphs and may makesignificant contributions to the outputs. To find a remedy forthe aforementioned deficiencies, we derive closed-formsolutions for unimorphs in eight different boundary/loadingconditions. To validate our theory, finite element modeling(FEM) was performed using COMSOL Multiphysics. Non-monotonic electrical outputs versus thickness ratio curves arefound for several boundary value problems. To resolve thediscrepancy between theoretical and FEM results for somescenarios, the d33 effect had to be taken into consideration.Simple models to determine average σ3 along beam lengthhas been proposed and turn out to be very effective inaccounting for the d33 contribution. The d33 effect is found tobe more significant as piezo-layer thickness increases. Whensimple–simple and cantilever unimorphs are subjected tovoltage excitation, non-monotonic deflection versus thickness

ratio curves are obtained and the d33 effect is also revealed.Overall, this work provides new physical insights on flexibleunimorphs and analytical solutions that can be used for thestructural optimization of unimorphs under different bound-ary/loading conditions.

Our paper is organized as follows. Section 2 describesmaterial properties and boundary/loading conditions.Section 3 offers both analytical and FEM results ofunimorph generators under eight different boundary/loadingconditions. Section 4 discusses the effect of elastic mismatchand unimorph actuators. Concluding remarks are drawn insection 5.

2. Problem description

Figure 1(a) is the schematic of a piezoelectric unimorphgenerator with the piezoelectric layer poled downward. Whena bending moment is applied to this unimorph as illustrated infigure 1(a), a dielectric polarization in the same direction asthe poling direction will be induced by the bending defor-mation of the piezoelectric layer. Suppose the piezoelectriclayer and the substrate have thicknesses of h1 and h2 andYoung’s moduli of Y1 and Y2, as labeled in figure 1(b), cal-culating the voltage, charge, and power output of such aunimorph first requires the determination of the neutral axisand the effective second moment of inertia of the bilayer. Thedistance from the neutral axis to the bottom surface of thesubstrate is represented by Δh1, as labeled in figure 1(b),where Δ has been given by [32]

1 2

2 1, 1

2

( )( )h h

h hD =

+ S + S+ S

Figure 1. (a) A schematic of a unimorph generator. When subjectedto bending moment, the polarization density varies linearly along thethickness direction. (b) Illustration of basic variables: the thicknessof the piezo-layer and the polymer substrate is h1 and h2,respectively, the Young’s modulus is Y1 and Y2, respectively, and thelength of the unimorph is labeled as L. The neutral axis of the bilayeris located Δh1 away from the bottom of the substrate, where Δ is adimensionless parameter given in equation (1).

2

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

where Y Y1 2S = is the film-to-substrate modulus ratio withY Y 1 2¯ ( )n= - being the plane strain modulus, Y being theYoung’s modulus, ν being the Poisson’s ratio, and h h1 2h =is the film-to-substrate thickness ratio.

The effective second moment of inertia of the bilayer isgiven by [32]

I h

h I

1 13

11

3, 2

23 2 2

3

23}

( ) ( )

( ) ( )

⎧⎨⎩⎡⎣⎢

⎤⎦⎥h h h h

h

h h

= S D - - D - +

+ D D - + =

where I represents the non-dimensional second moment ofinertia and hence the bending stiffness of the unimorph can bewritten as Y I.2 For given piezoelectric and substrate materials(i.e., Σ fixed), the only dimensionless variable in the problemis the thickness ratio η, whose effect is the focus of this study.

For eight different boundary/loading conditions, solu-tions have been obtained through both 2D plane strain linearpiezoelectric theory and FEM. Before describing the theory insection 3, the material parameters and model setup in thecommercial FEM software, COMSOL Multiphysics 4.4, aresummarized here. We choose Kapton as a representativeinactive flexible substrate material with Young’s modulus of2.5 GPa, Poisson’s ratio of 0.34, and mass density of1420 kg m−3, assuming it is isotropic [33]. For the piezo-electric layer, we choose poly(vinylidene fluoride-co-trifluoroethylene) (PVDF-TrFE) because of its popularity andmechanical robustness. Furthermore, PVDF-TrFE (75/25)has well-established experimental data of mass density(1879 kg m−3 [34]), elastic compliance, relative permittivity,and piezoelectric coefficients [35]:

-Elastic compliance tensor

s

3.32 1.44 0.89 0 0 01.44 3.24 0.86 0 0 00.89 0.86 3.00 0 0 00 0 0 94.0 0 00 0 0 0 96.3 00 0 0 0 0 14.4

10 Pa , 3

E

10 1 ( )

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

=

- -- -- -

´ - -

where Y1=1/s11 and v1=−s13/s11 are found to be 3 GPaand 0.268 GPa, respectively.

-Permittivity matrix

7.4 0 00 7.95 00 0 7.9

, 4T0 ( )

⎡⎣⎢⎢

⎤⎦⎥⎥e e= ´

where ε0=8.85×10−12 F m−1 is the vacuum permittivity.Hence the relative permittivity in the thickness directionis 7.9pe = .

-Piezoelectric coupling tensor

d0 0 0 0 36.3 00 0 0 40.6 0 0

10.7 10.1 33.5 0 0 0

10 C N . 512 1 ( )

⎡⎣⎢⎢

⎤⎦⎥⎥=

--

-´ ⋅- -

When the film is fixed to be PVDF-TrFE and the sub-strate to be Kapton, then the elastic mismatch Y Y1 2S = isfixed to be 1.148. To study the effect of the film-to-substratethickness ratio, the Kapton thickness is fixed to be 25 μmwhereas the thickness of PVDF-TrFE varies from 0.5 to20 μm with an incremental step of 0.5 μm. While themechanical boundary conditions vary from case to case, theelectrostatic boundary conditions remain the same for all eightcases of generators in section 3. The top surface of the piezo-layer is set as the ground and voltage is collected and aver-aged at the bottom of the piezo-layer. The absolute value ofcharge on top and bottom surfaces are collected and averagedto yield the total charge. All other boundaries are set to zerocharge. When meshing the model in COMSOL, we selected‘mapped mesh’ method with the maximum element size of0.5 μm and the minimum element size of 0.01 μm.

3. Formulation and results of generators

In this section, the analytical formulation of piezoelectricunimorph generator under eight different boundary/loadingconditions will be provided and the voltage, charge density,and energy density outputs will be predicted as functions ofthe film-to-substrate thickness ratio. When necessary, d33effects will be considered and the substrate thickness-to-length ratio is an added variable. Analytical results and FEMresults will be compared and discussed.

3.1. Pure bending under constant moment (PB-M)

Figure 2(a) illustrates the boundary and loading conditions ofa unimorph subjected to pure bending under constant bendingmoment M0 (PB-M). Based on Euler–Bernoulli beam theory,the only non-zero stress component in this unimorph will bebending induced normal stress in x direction and its verticaldistribution is given by

M z h

h I, 61

0 1

23

( ) ( )s =- D S

where Δ is given in equation (1) and I in equation (2). Thesubscript 1 in σ denotes x direction and 3 will be used todenote z direction, where x and z directions are defined infigure 1(b).

Based on the linear piezoelectric theory [36], deformationinduced polarization density is proportional to stressthrough the piezoelectric coefficient, i.e. P di ij js=i j1, 2, 3, 1, 2, ,6 ,( )= = ¼ therefore z direction polar-ization density is given by

Pd M z h

h I. 73

31 0 1

23

( ) ( )=- D S

Since charges are only collected from the top and bottomsurface electrodes of the piezo-layer, only top (z=h1+h2)and bottom (z=h2) surface polarization density will beconsidered to calculate the total amount of charges.

3

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

As the total amount of charges and polarization densityare related throughQ a P Sd ,n¯ ¯ò= - ⋅ where an¯ stands for thesurface normal vector and S stands for the overall surface area(S=L in our plane strain model), the effective charge output,which is defined as Q Q Q 2MPB Top Bot(∣ ∣ ∣ ∣)= +- can beexpressed as

Qd LM

h I1

2. 8MPB

31 0

22

( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

hh=

S+ - D-

Therefore surface charge density can be readily given byQ S,r = i.e.

d M

h I1

2. 9MPB

31 0

22

( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥r

hh=

S+ - D-

Through the definition of capacitance C Q V ,= voltageis given by V Q C= where C L h ,p 1e= ¢ with pe ¢ being theeffective permittivity of the piezoelectric material [23]:

d

Y I1

6

1

11

2

1, 10p p

312

1 p

3( )⎜ ⎟

⎧⎨⎩⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

⎫⎬⎭e ee

hh

hh¢ =

S ++

+ - D

where the sign depends on the poling direction. Hence theoutput voltage is given by

Vd M

h I1

2. 11MPB

31 0

p 2( )⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥e

h hh=

¢S

+ - D-

Finally the generated energy density can be calculated byU V 2,r= i.e.

Ud M

h I_

1

21

2. 12MPB

312

02

p 23

2 2

( )⎜ ⎟ ⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥e

hh

h=¢

S+ - D

It is obvious from equations (9)–(12) that when film andsubstrate materials are fixed, i.e. when Σ is fixed, dimen-sionless ρ, V, and U will just depend on the thickness ratio, η,which will be plotted together with FEM results.

Our FEM model implements the pure bending conditionas illustrated by figure 2(a) with the substrate being Kapton,piezoelectric layer being PVDF-TrFE and with fixedh2=25 μm and M0=1×10−7 N m, which is estimatedfrom the change of heart curvature during heart beat [37, 38]times the bending stiffness of the unimorph. Bending momentwas applied on each side wall of the beam through theboundary condition named ‘rigid connector’. The FEM con-tour plots of stress and polarization density are given infigure 2(b) with h1=10 μm. Both stress and polarization

Figure 2. (a) A schematic for the pure bending with constant bending moment (PB-M) case which is discussed in section 3.1. (b) FEM resultsof σ1 and P3 when M0=1×10−7 N m. (c) The analytical (curves) and FEM (markers) results of normalized voltage and charge density asfunctions of the thickness ratio h h .1 2( )h = (d) The analytical and FEM results of normalized energy density as a function of the thicknessratio.

4

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

density vary linearly with z, as predicted by equations (7) and(8). Charge and voltage can also be directly output from FEMand U V 2r= is used to calculate the FEM output of elec-trical energy density.

Analytical and FEM results are compared in figures 2(c)and (d). Excellent overlap between equations (9) and (11) andFEM results demonstrates a strong validation of our linearpiezoelectric theory for unimorphs even without accountingfor the d33 effect in the formulation (d33 is included in FEM).The biggest mismatch between FEM results and equation (11)in figure 2(c) is less than 1.8%, which occurs at the largest η.The mismatch is due to the breakdown of Bernoulli’s beamtheory: when L and h2 are fixed and h1 is increased to increaseη, the overall unimorph thickness-to-length ratio increasesand σ3 can no longer be assumed zero. This effect is morepronounced in cantilever and simply supported unimorphsand will be specifically discussed in section 3.3. It is inter-esting to see that while the charge density monotonicallydecays as the piezo-film gets thicker, the voltage output isnon-monotonic, with the peak value reached at η=0.46.This is due to the tradeoff that under constant moment,increasing h1 will enlarge both distance from neutral axis,which enhances the voltage output, and second moment ofinertia, which diminishes the voltage output. Normalizedenergy density is plotted in figure 2(d). The optimizedthickness ratio for maximum energy output is η=0.18,which is different from the optimal η with maximum voltageoutput. Therefore, choosing the optimum thickness ratiodepends on whether maximum voltage output or energydensity is desired.

3.2. Pure bending under constant curvature (PB-κ)

Figure 3 shows the FEM and analytical solutions for theunimorph subjected to pure bending with constant curvatureκ0 (PB-κ). Following the same procedures outlined in 3.1 anddirectly substituting M Y h I0 0 2 2

3¯ ¯k= into equations (9) and(11), analytical expressions for the charge density and thevoltage output under PB-κ are

d Y h 12

13PB 31 1 0 2 ( )⎜ ⎟⎛⎝

⎞⎠r k

hh= + - Dk-

and

Vd Y h

12

, 14PB31 1 0 2

2

p( )⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

ke

hh

h=¢

+ - Dk-

respectively. Figure 3(b) shows the FEM results of σ1 and P3

where the piezo-layer is 10 μm thick and κ0 is fixed as 2 m−1.Figure 3(c) shows the normalized voltage and charge

density under the PB-κ condition. While the charge densitymonotonically decays with increasing η, which is similar tothe PB-M result, the voltage output increases with increasingη, which is very different from the PB-M result. This isbecause under constant curvature, the second moment ofinertia no longer shows up in the denominator of the voltageequation. Figure 3(d) plots the normalized electrical energydensity U V 2( )r= generated under the PB-κ condition,which monotonically increases with η.

3.3. Effects of d33 in cantilever and simply supportedunimorphs

While working on cantilever and simply supported unim-orphs, we realized that unlike unimorphs subjected to purebending, there is non-zero σ3 in cantilever and simply sup-ported unimorphs. Since σ3 generates additional P3 throughd33, it is important to account for its contribution when ana-lyzing cantilever and simply supported unimorphs.Figure 4(a) left panel depicts a cantilever unimorph subjectedto uniformly distributed load (the C–Q condition). A freebody diagram is drawn on the right panel when we make a cutalong Line AB. It is clear that the applied distributed load hasto be balanced by the reaction shear force applied by the walland the normal force on the AB plane. Let σ3 be the averagenormal stress on the AB plane, the force balance equation ishence established in the bottom of the free body diagram.Since the shear stress is parabolically distributed along the leftsurface as

z q L h I z h2 1 ,

15c 2

22

2( ) ( ) [( ) ( ) ]( )

t h h h= ⋅ + - D - - D

σ3 is a function of z.Figure 4(b) illustrates the case where a cantilever

unimorph is subjected to point load at the end (the C–Fcondition). Similar to the C–Q condition, the applied pointload is fully balanced by the reaction shear force applied bythe wall and the normal force on the AB plane. Again theaverage normal stress σ3 on the AB plane can be calculated bythe force balance equation in the bottom of the free bodydiagram where z( )t is simply given by equation (15) with q Lcreplaced by Fc.

Similarly, for simply supported unimorphs subjected touniformly distributed load (SS–Q condition) and point load atthe center (SS–F condition), figures 4(c) and (d) offer the freebody diagrams that help determine the average σ3 along theAB plane, both of which are constants, which is differentfrom the cantilever unimorphs.

3.4. Cantilever unimorph under uniformly distributed load(C–Q)

The expression for the moment distribution in C–Q conditionis

M xq L Lx x2

2, 16c

2 2

( )( )

( )=- +

where the qc is the distributed load per unit length, as labeledin figure 5(a). The charge density and voltage output can bederived following the same procedure outlined in section 3.1:

d q L

h I61

217C Q

31 c2

22

( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥r

hh=

S+ - D-

and

Vd q L

h I61

2, 18C Q

31 c2

p 2( )⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥e

h hh=

¢S

+ - D-

5

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

respectively, where the contribution of σ3 is not yetconsidered. Comparing equation (17) with (9) and (18) with(11), simply substituting M q L 60 c

2= can readily convert thePB-M solution to the C–Q solution. But in the C–Q solution,L is a new variable that does not exist in the PB-M solution.

Figure 5(b) shows the FEM contour plots of σ1 and P3when the piezo-layer thickness is 10μm under the C–Q con-dition and d33 is included. The uniformly distributed load ischosen such that the average moment in the C–Q conditionM q L 6c

2= equals to the moment applied in the PB-M con-dition. When we plot equations (17) and (18) as dash–dot lineagainst FEM results as shown in figure 5(c), a clear mismatchcan be observed, especially when h1/h2 is large. This is becausethe polarization density due to non-zero σ3 was not accountedfor. According to the equilibrium equation given in figure 4(a),average σ3 in the C–Q condition can be expressed as

qI

z

h

z

h

11 1

31

1

21

1

6. 19

3 c3

2

2

2

3

( )

( )

( )⎪

⎪

⎧⎨⎩⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎫⎬⎭

s h h

h h h

h

=- - + - D

- + - D - D

+ - D

Subsequently, the polarization density induced by σ3through the d33 coefficient is

P d qI

z

h

z

h

_ 11 1

31

1

21

1

6. 20

d3 33 c3

2

2

2

3

33 ( )

( )

( )⎪

⎪

⎧⎨⎩⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎫⎬⎭

h h

h h h

h

=- - + - D

- + - D - D

+ - D

Note d33 for PVDF-TrFE is a negative value as given inequation (5).

Finally the total charge density and voltage outputinduced by both σ1 and σ3 in the C–Q case are

d q L

h I

d

d

h

L

I

61

2

12 3 2 3

221

C Q31 c

2

22

33

31

22

2 ( ) ( )

⎜ ⎟⎜ ⎟

⎪

⎧⎨⎩⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

rh

h

h h h

¢ =S

+ - D -

´- + - D

S

-

Figure 3. (a) A schematic for the pure bending with constant bending curvature (PB-κ) case which is discussed in section 3.2. (b) FEM resultsof σ1 and P3 when κ0=2 m−1. (c), (d) The analytical and FEM results of normalized voltage, charge density, and energy density asfunctions of the thickness ratio.

6

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

and

Vd q L

h I

d

d

h

L

I

61

2

12 3 2 3

2,

22

C Q31 c

2

p 2

33

31

22

2 ( )

( )

⎜ ⎟⎜ ⎟

⎪

⎧⎨⎩⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

eh h

h

h h h

¢ =¢

S+ - D -

´- + - D

S

-

respectively. Note that in both equations the d33 termdiminishes as the length of the unimorph increases. Plottingequations (21) and (22) as solid curves in figure 5(c) andU V 2r= ¢ ¢ as a solid curve in figure 5(d), they have foundexcellent agreement with the FEM results, until again when h1is too large for the beam theory to stay valid as discussedbefore. The optimal η for the voltage and the energy density

are 0.58 and 0.2, respectively. The optimal η for the voltage isslightly shifted compared to PB-M case because of theadditional contribution from σ3.

Figures 5(e) and (f) show the normalized electric outputswhen the boundary and loading conditions stay the same butthe piezo-film and the substrate switch position. In this casepiezo-layer is still Layer 1 and substrate is still Layer 2, andΣ, η and Δ given by equation (1) stay the same. In this casethe piezo-layer is located below the neutral axis so the sign ofσ1 is changed. The σ3 equation in figure 4(a) is still valid butas the shear force term is larger than the piezo-layer-on-sub-strate case, the σ3 is smaller in the substrate-on-piezo-layercase. As a result, the σ3 contribution will be a smaller effectand it will counteract the σ1 contribution, as evidenced in

Figure 4. Schematics and free body diagrams for calculating average σ3 in cantilever and simply supported unimorphs. (a) Cantileverunimorph subjected to uniformly distributed load (C–Q case) where shear stress applied by the left wall has a parabolic distribution. (b)Cantilever unimorph subjected to point load (C–F case) where shear stress applied by the left wall also has a parabolic distribution. (c)Simply supported unimorph subjected to uniformly distributed load (SS–Q case) where q .3 ss = - (d) Simply supported unimorph subjectedto point load (SS–F case) where F L3 ss = - .

7

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

figures 5(e) and (f). The optimal η for maximum voltage andenergy density outputs are 0.44 and 0.18, respectively.

3.5. Cantilever unimorph subjected to point load at the end(C–F)

Point load is an easily achievable loading condition inexperiment. As depicted in figure 6(a), a point load (Fc) isapplied at the end of the cantilever unimorph. The momentdistribution in the cantilever can be expressed by

M x F L x . 23c( ) ( ) ( )= -

After averaging equation (23) over L and substituting theresult FcL/2 into M0 in equation (7), the charge density and

the voltage output without accounting for σ3 contribution canbe readily obtained as

d F L

h I21

224C F

31 c

22

( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥r

hh=

S+ - D-

and

Vd F L

h I21

2. 25C F

31 c

p 2( )⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥e

h hh=

¢S

+ - D-

But as illustrated in figure 4(b), the C–F case has nonzeroσ3, Based on the equation given in figure 4(b), σ3 and the

Figure 5. (a) A schematic for the cantilever unimorph subjected to uniformly distributed load (C-Q case), which is discussed in section 3.4.(b) FEM results of σ1 and P3 when qc=15 N m−1. (c), (d) The analytical and FEM results of normalized voltage, charge density, and energydensity as functions of the thickness ratio. Dash–dot curves do not account for d33 effects, whereas solid curves do. (e), (f) The analytical andFEM results of normalized voltage, charge density, and energy density as functions of the thickness ratio when the piezo-layer and substrateare flipped. Dash–dot curves do not account for d33 effects, whereas solid curves do.

8

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

resulted polarization in the C–F condition are given by

F

L I

z

h

z

h

11 1

31

1

21

1

626

3c 3

2

2

2

3

( )

( )

( )⎪

⎪

⎧⎨⎩⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎫⎬⎭

s h h

h h h

h

=- - + - D

- + - D - D

+ - D

and

P dF

L I

z

h

z

h

_ 11 1

31

1

21

1

6, 27

d3 33c 3

2

2

2

3

33 ( )

( )

( )⎪

⎪

⎧⎨⎩⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

⎫⎬⎭

h h

h h h

h

=- - + - D

- + - D - D

+ - D

Figure 6. (a) A schematic for the cantilever unimorph subjected to point load (C–F case), which is discussed in section 3.5. (b) FEM results ofσ1 and P3 when Fc=1 mN. (c), (d) The analytical and FEM results of normalized voltage, charge density, and energy density as functions ofthe thickness ratio. Dash–dot curves do not account for d33 effects, whereas solid curves do. (e), (f) The analytical and FEM results ofnormalized voltage, charge density, and energy density as functions of the thickness ratio when the piezo-layer and substrate positions areflipped. Dash–dot curves do not account for d33 effects, whereas solid curves do.

9

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

respectively. The total charge density and voltage outputinduced by both σ1 and σ3 in the C–F case are therefore

d F L

h I

d

d

h

L

I

21

2

12 3 2 3

628

C F31 c

22

33

31

22

2 ( ) ( )

⎜ ⎟⎜ ⎟

⎪

⎧⎨⎩⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

rh

h

h h h

¢ =S

+ - D -

´- + - D

S

-

and

Vd F L

h I

d

d

h

L

I

21

2

12 3 2 3

6.

29

C F31 c

p 2

33

31

22

2 ( )

( )

⎜ ⎟⎜ ⎟

⎪

⎧⎨⎩⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

eh h

h

h h h

¢ =¢

S+ - D -

´- + - D

S

-

Figure 6(b) shows the FEM results of σ1 and P3 when thepiezo-film is 10 μm thick and the averaged moment equals tothe M0 used in the PB-M case. To reduce the singularity errorof FEM, the point load is located L/200 away from the endedge of the cantilever because if the point load is appliedexactly on the end edge of the unimorph, the COMSOLMultiphysics only applies half of the load to the unimorph.This technique of leaving a small margin towards the end isalso applied to other FEM simulations which have loadingsapplied up to the edge, including distributed loads and applieddisplacement.

The comparison between FEM and analytical solution ismade in figures 6(c) and (d). The good agreement also vali-dates our rationale that there is σ3 contribution in C–F case.Comparing figures 5(c) and (d) with 6(c) and (d), it is con-cluded that the d33 effect on the electric outputs of C–Q isgreater than that of C–F. This is because σ3 of C–Q is threetimes greater than that of C–F as we dictatedF L q L M2 6 .c c

20= = The optimal η for the voltage and the

energy density were 0.5 and 0.18, respectively.Similar to figures 5, 6(e) and (f) show the normalized

electrical outputs when film and substrate are flipped. The σ3contribution will again counteract the σ1 contribution but theeffect of σ3 is much less significant than the C–Q case. Theoptimal η for the voltage and the energy density are 0.46 and0.18, respectively, which are the same as the PB-M case.

3.6. Cantilever unimorph subjected to fixed end displacement(C–W)

Figure 7 offers the FEM and analytical results for the C–Wunimorph as depicted in figure 7(a). Plugging the cantileverload–displacement relation: F Y Iw L3 22 c

3¯ ( )= intoequation (23), the moment in the cantilever can be expressedas

M xY Iw

LL x

3

2. 302 c

3( ) ( ) ( )= -

Following the same procedure outlined before we canapply equation (30) to calculate the charge density and the

voltage outputs just induced by σ1 as

d Y h 12

31C W 31 1 1 2 ( )⎜ ⎟⎛⎝

⎞⎠r k

hh= + - D-

and

Vd Y h

12

, 32C W31 1 1 2

2

p( )⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

ke

hh

h=¢

+ - D-

respectively. Similar to the C–F case, the modified chargedensity and voltage of C–W condition with the d33 effectincluded are given by

d Y hd

d

h

L

I

12

12 3 2 3

633

C W 31 1 1 233

31

22

2 ( ) ( )

⎜ ⎟⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

r kh

h

h h h

¢ = + - D -

´- + - D

S

-

and

Vd Y h

d

d

h

L

I

12

12 3 2 3

6, 34

C W31 1 1 2

2

p

33

31

22 2 ( ) ( )⎜ ⎟

⎜ ⎟⎧⎨⎩

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

ke

hh

h

h h h

¢ =¢

+ - D

-- + - D

S

-

respectively.Figure 7(b) shows the FEM result of σ1 and P3 under the

C–W condition when the piezo-layer is 10 μm thick, theunimorph is 200 μm long, and wc=53.3 nm such that theaverage curvature, w L3 2 ,1 c

2k = is the same as theκ0=2 m−1 used for the PB-κ case.

The normalized voltage, charge and energy density of thepiezo-layer-on-substrate case are plotted in figures 7(c) and(d) and that of the substrate-on-piezo-layer case are offered infigures 7(e) and (f). Although the d33 effect is included assolid curves in both cases, the σ3 contribution is small withinthe current range of h1/h2 and the results are very similar tothe PB-κ results. But when h1/h2 gets very large (e.g. largerthan 4), the voltage and energy density curves in figures 7(e)and (f) will become non-monotonic.

It is interesting to notice that the C–F results (figure 6)are very similar to the PB-M results (figure 2) and the C–Wresults (figure 7) are very similar to the PB-κ results(figure 3), we can conclude that the pure bending conditionand cantilever subjected to end load condition are almostequivalent except that the d33 effect in cantilever unimorphshas to be accounted for when h2/L is large. More discussionson h2/L effect will be given in sections 3.9 and 4.1.

3.7. Simply supported unimorph with uniformly distributed load(SS–Q)

Simple support is another popular boundary condition forbeams. When subjected to uniformly distributed load qs asillustrated in figure 8(a), the moment function can be writtenas

M xq x

L x2

. 35s( ) ( ) ( )= -

10

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

Hence the charge density and the voltage output inducedby σ1 are obtained using equations (7) and (35) as

d q L

h I121

236SS Q

31 s2

22

( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥r

hh=

S+ - D-

and

Vd q L

h I121

2, 37SS Q

31 s2

p 2( )⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥e

h hh=

¢S

+ - D-

respectively. Comparing equations (36) and (37) with (17)and (18), SS–Q is equivalent to C–Q if q q 2.s c= Thereforesimilar to the C–Q case, SS–Q should also account for theeffect of σ3 as calculated in figure 4(c) and the modified

charge density and voltage become

d q L

h I

d

d

h

L

I

121

2

1238

SS Q31 c

2

22

33

31

22

( )⎜ ⎟

⎜ ⎟⎧⎨⎩

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

rh

h¢ =S

+ - D

+S

-

and

Vd q L

h I

d

d

h

L

I

121

2

12. 39

SS Q31 c

2

p 2

33

31

22

( )⎜ ⎟

⎜ ⎟⎧⎨⎩

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

eh h

h¢ =¢

S+ - D

+S

-

Figure 7. (a) A schematic for the cantilever unimorph subjected to end displacement (C–W case), which is discussed in section 3.6. (b) FEMresults of σ1 and P3 when wc=53.3 nm. (c), (d) The analytical and FEM results of normalized voltage, charge density, and energy density asfunctions of the thickness ratio. Dash–dot curves do not account for d33 effects, whereas solid curves do. (e), (f) The analytical and FEMresults of normalized voltage, charge density, and energy density as functions of the thickness ratio when the piezo-layer and substratepositions are flipped. Dash–dot curves do not account for d33 effects, whereas solid curves do.

11

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

In equations (38) and (39), the d33 term always vanishesas h2/L decreases, which is similar to the C–Q case. How-ever, unlike the C–Q case, the d33 term minifies both chargedensity and voltage output since σ1 in the piezo-layer isnegative in the SS–Q case. This effect is clearly observable infigures 8(c) and (d), where the solid curves that account forthe d33 effects are much lower than the dash–dot curves anddemonstrate much better agreement with the FEM results. Inthis case, the optimal η for the voltage and the energy outputsare 0.32 and 0.14, respectively.

Similar to the C–Q case that the d33 effect presents in thisboundary/loading condition, we also provide the solutionswhen the piezo-layer and the substrate are flipped, as shownin figures 8(e) and (f). Now the d33 effect enhances the outputas expected and its contribution is as significant as the piezo-layer-on-substrate case. For the given range of h1/h2, the

voltage curve is monotonic and hence there is no optimal ηfor the voltage output. If the beam length increases, therewould be an optimal η which will be similar to the C–Q case.The optimal η for the energy density is 0.22.

3.8. Simply supported unimorph subjected to point load at thecenter (SS–F)

Figure 9 shows the results for the SS–F case. As the pointload Fs is applied at the middle of the beam, the analyticalexpression of the moment is given by

M x

F xx

L

F L x Lx L

2for 0

2

2for

2

. 40

s

s( ) ( ) ( )

⎧⎨⎪⎪

⎩⎪⎪

=

-

Figure 8. (a) A schematic for the simply supported unimorph subjected to uniformly distributed load (SS–Q case), which is discussed insection 3.7. (b) FEM results of σ1 and P3 when qs=30 N m−1. (c), (d) The analytical and FEM results of normalized voltage, charge density,and energy density as functions of the thickness ratio. Dash–dot curves do not account for d33 effects, whereas solid curves do. (e), (f) Theanalytical and FEM results of normalized voltage, charge density, and energy density as functions of the thickness ratio when the piezo-layerand substrate positions are flipped. Dash–dot curves do not account for d33 effects, whereas solid curves do.

12

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

Following the same procedure introduced in section 3.5,the charge density and voltage outputs just induced by σ1are

d F L

h I81

241SS F

31 s

22

( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥r

hh=

S+ - D-

and

Vd F L

h I81

2. 42SS F

31 s

p 2( )⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥e

h hh=

¢S

+ - D-

The effective moment term in the SS–F case is FsL/8while that of the C–F case is FcL/2. If the effective moments

are required to be equal, then F F4 .s c= Figure 9(b) shows theFEM results of σ1 and P3, when the piezo-layer is 10 μmthick and F F4 ,s c= where Fc is the end point load applied inthe C–F case. The magnitude of σ1 is similar to that in the C–F case, except the sign is opposite. To account for the effectof σ3 as illustrated in figure 4(d), we need to modifyequations (41) and (42) to be

d q L

h I

d

d

h

L

I

81

2

843

SS F31 c

2

22

33

31

22

( )⎜ ⎟

⎜ ⎟⎧⎨⎩

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

rh

h¢ =S

+ - D

+S

-

Figure 9. (a) A schematic for the simply supported unimorph subjected to point load (SS–F case), which is discussed in section 3.8. (b) FEMresults of σ1 and P3 when Fs=4 mN. (c), (d) The analytical and FEM results of normalized voltage, charge density, and energy density asfunctions of the thickness ratio. Dash–dot curves do not account for d33 effects, whereas solid curves do. (e), (f) The analytical and FEMresults of normalized voltage, charge density, and energy density as functions of the thickness ratio when the piezo-layer and substratepositions are flipped. Dash–dot curves do not account for d33 effects, whereas solid curves do.

13

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

and

Vd q L

h I

d

d

h

L

I

81

2

8. 44

SS F31 c

2

p 2

33

31

22

( )⎜ ⎟

⎜ ⎟⎧⎨⎩

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

eh h

h¢ =¢

S+ - D

+S

-

In equations (43) and (44), the d33 term always vanishesas h2/L decreases, which is similar to the SS–Q case. How-ever, the polarization density induced by σ3 in the SS–F caseis less than that of SS–Q. To achieve the same effectivemoment, we need q F L3 2 ,s s ( )= which means SS–Q casehas 1.5 times larger effective σ3 than the SS–F case. The sameconclusion can also be made by comparing the coefficient ofthe d33 term between equations (38) and (43): 12 in SS–Qcase versus 8 in the SS–F case. Therefore, the electric outputvariation caused by the d33 effect in SS–Q is greater than thatin SS–F, which can be observed from the comparisonbetween figures 8(c) and 9(c). The optimal η for maximumvoltage and the energy density outputs are 0.36 and 0.16,respectively in figures 9(c) and (d). When piezo layer andsubstrate are flipped, figures 9(e) and (f) suggest the optimal ηare 0.84 for maximum voltage output and 0.2 for maximumenergy density output, respectively.

3.9. Simply supported unimorph subjected to fixed centerdisplacement (SS–W)

Figure 10(a) illustrates the SS–W condition where a simplysupported unimorph is subjected to a fixed center displace-ment, ws. The load–displacement relation of a simply sup-ported beam subjected to center load or displacement isF Y Iw L48 ,2 s

3¯= therefore the moment function of the SS–Wcase can be obtained by substituting this new F inequation (40):

M x

Y Iw

Lx x

L

Y Iw

LL x

Lx L

24for 0

224

for2

. 45

2 s3

2 s3

( )

¯

¯( )

( )

⎧⎨⎪⎪

⎩⎪⎪

=

-

Following the same procedure in section 3.8 (the SS–Fcase), the analytical solutions for the charge density and thevoltage output for the SS–W case without considering the d33effect are

d Y h 12

46SS W 31 1 2 2 ( )⎜ ⎟⎛⎝

⎞⎠r k

hh= + - D-

and

Vd Y h

12

, 47SS W31 1 2 2

2

p( )⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

ke

hh

h=¢

+ - D-

respectively. When the d33 effect is considered as in the SS–Fcase, equations (46) and (47) have to be modified to

d Y hd

d

h

L

I1

2

8

48

SS W 31 1 2 233

31

22

( )

⎜ ⎟⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥r k

hh¢ = + - D +

S-

and

Vd Y h

d

d

h

L

I

12

8, 49

SS W31 1 2 2

2

p

33

31

22

( )⎜ ⎟

⎜ ⎟⎧⎨⎩

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

ke

hh

h¢ =¢

+ - D

+S

-

respectively. Figure 10(b) shows the FEM results of σ1 and P3

distribution under the SS–W condition when ws=13.3 nm,which is chosen to ensure that the average curvature

w L62 s2k = is the same as the κ0=2 m−1 used for the

PB-κ case. Without considering the d33 effect, the normalizedelectric outputs are plotted as dash–dot curves in figures 10(c)and (d), which are similar to the solid curves in the PB-κcases but very different from the FEM results. When the d33effect is considered, the solid curves find much betteragreement with the FEM outputs. When piezo-layer is ontop of the substrate, the d33 effect minifies the outputs as d33is negative. Consequently, the optimal η for maximumvoltage output is 0.84 and 0.62 for maximum energy density.Since both voltage and charge density drop below 0 as η

increases, we extend the range of η to 2 and find that as η

increases, the energy density drops to a minimum afterreaching a maximum, and then takes off again. This behavioris distinctly different from PB-κ case where voltage, chargedensity, and energy density are always monotonicallyincreasing with η. Still, as L increases, the d33 effect willdiminish according to equations (48) and (49).

Because there is the d33 effect in the SS–W case, swap-ping piezo-layer and substrate would yield different outputs,as shown in figures 10(e) and (f). All three outputs aremonotonic in this case because the σ1 and σ3 contributionshave the same sign. The d33 effect is clearly visible in thiscase, but now enhances the outputs.

4. Discussions

4.1. Effects of Y1/Y2 and h2/L

Throughout section 3, the elastic mismatch, Y Y ,1 2S = hasbeen fixed to be 1.148 for unimorphs composed of PVDF-TrFE bonded to Kapton. Here we would like to investigatethe effects of Σ. As Wang et al [22] described, theoptimal thickness ratio η of unimorph cantilevers also varieswith Σ.

Figure 11 plots the normalized voltage, charge density,and energy density as functions of η in log–log scale undervarious Σ for the PB-M condition in which case d33 plays norole. As can be seen in figure 11(a) (plotted fromequation (11)), (b) (plotted from equation (9)) and (c) (plotted

14

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

from equation (12)), the optimal η shifts to the left as Σ

increases in all electric output cases and are in general dif-ferent for different outputs. When η is small (i.e. η=1), thelarger Σ always yields the higher output but when η is large,larger outputs are achieved for smaller Σ.

The effect of h2/L is also investigated because it is acomponent that always appears in the d33 term. When h2/L=1, the d33 effect is negligible but when h2/L is finite,the d33 effect sometimes can become obvious or evenoverwhelms σ1 contribution. Figures 11(d), (e), and (f) plotnormalized voltage, charge density, and energy density asfunctions of η under various h2/L for the SS–Q condition.For voltage and energy density, the optimal η shiftsslightly to the right as h2/L decreases and stabilizes when

h2/L drops below 1/20, which is consistent with theobservation in [39]. Similar plots can be constructed forother boundary/loading conditions which have to accountfor the d33 effect.

4.2. Unimorph actuators

Thickness ratio variation also affects the displacement of aunimorph subjected to applied electric field [40] as shownin figure 12. By applying the electric field E V h0 1=across the thickness direction of the piezoelectric layer, auniform longitudinal strain is induced in the piezoelectriclayer. Consequently the magnitude of the longitudinalstress within the piezoelectric layer is d Y V h ,1 31 1 0 1s =

Figure 10. (a) A schematic for the simply supported unimorph subjected to central displacement (SS–W case), which is discussed insection 3.9. (b) FEM results of σ1 and P3 when ws=13.3 nm. (c), (d) The analytical and FEM results of normalized voltage, charge density,and energy density as functions of the thickness ratio. Dash–dot curves do not account for d33 effects, whereas solid curves do. (e), (f) Theanalytical and FEM results of normalized voltage, charge density, and energy density as functions of the thickness ratio when the piezo-layerand substrate positions are flipped. Dash–dot curves do not account for d33 effects, whereas solid curves do.

15

Smart Mater. Struct. 25 (2016) 035037 T Ha et al

while the stress at the substrate layer remains zero.Therefore, the resultant moment of the unimorph can bewritten as

M d Y Vz

hzd , 50

h

h h

E 31 1 012

1 2

( )⎛⎝⎜

⎞⎠⎟ò= ⋅ - D

+

which is calculated to be

M d Y V h 12

. 51E 31 1 0 2 ( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

hh= + - D

Hence, the maximum displacement of the simply sup-ported unimorph subjected to constant voltage (SS–V) as

shown in figure 12(a) is

wd V L

h I

d

d

h

L

I

_8

12

8. 52

s max31 0

2

22

33

31

22

( )⎜ ⎟

⎜ ⎟⎧⎨⎩

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

hh=

S+ - D

+S

The second term in equation (52) accounts for thethickness direction deformation from σ3 induced by theconstant electric potential.

Figure 12(b) plots the normalized maximum actuationdisplacement obtained from FEM and equation (52) withrespect to the thickness ratio when Σ is fixed to be 1.148. Thed33 effect slightly diminishes the displacement. The optimalthickness ratio for maximum displacement is 1.89.

Figure 11. (a)–(c) The effects of stiffness ratio on the normalized voltage, charge density, and energy density under the PB-M condition. (d)–(f) The effects of substrate thickness-to-length ratio on the normalized voltage, charge density, and energy density under the SS–Q condition.

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Smart Mater. Struct. 25 (2016) 035037 T Ha et al

Similarly, the maximum displacement of the cantileverunimorph subjected to constant electric potential (C–V) asshown in figure 12(c) is

wd V L

h I

d

d

h

L

I

_2

12

2. 53

c max31 0

2

22

33

31

22

( )⎜ ⎟

⎜ ⎟⎧⎨⎩

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎫⎬⎭

hh=

S+ - D

-S

Figure 12(d) plots the normalized maximum displace-ment with respect to the thickness ratio. The optimal thicknessratio of the maximum displacement of the C–V is 1.89, whichis the same as the SS–V case. The σ3 effect is also consideredin this case and it enlarges the displacement but is less sig-nificant compared to the SS–V case. Due to symmetry, theactuation of unimorphs under pure bending is identical to theC–V case.

It is worth mentioning that the moment induced by theapplied electric potential does not equal any moment inducedby the mechanical loadings as discussed in section 3 since thestress distributions are very different. Therefore, the

generation mode and actuation mode of the sameunimorph are not reciprocal to each other.

5. Conclusion

We investigate the electromechanical behaviors of flexibleunimorph power generators and actuators. Analytical andnumerical models are built to unveil the effects of piezo-layer-to-substrate thickness ratio and piezoelectric material con-stants on energy conversion under eight different boundary/loading conditions. Our theory reveals that when theunimorph is subjected to displacement-controlled loadingconditions, the charge, voltage, and energy outputs aremonotonic functions of the thickness ratio, whereas when theunimorph is subjected to load-controlled conditions, optimalthickness ratios for maximum voltage and energy outputsexist. Our linear piezoelectric theory has been fully validatedby FEM. We have also found that except pure bending con-ditions, all cantilever and simply supported unimorphs shouldcare about the d33 (i.e. σ3) contribution when theunimorph length is not much larger than the thickness. Asimplified average stress model is proven effective in

Figure 12. (a) A schematic of a simply supported unimorph actuator subjected to constant electric potential (SS–V case). (b) The analyticaland FEM results of normalized maximum displacement of the unimorph as a function of the thickness ratio. Dash–dot curves do not accountfor d33 effects, whereas solid curves do. (c) A schematic of a cantilever unimorph actuator subjected to constant electric potential (C–V case).(d) The analytical and FEM results of normalized maximum displacement of the unimorph as a function of the thickness ratio. Dash–dotcurves do not account for d33 effects, whereas solid curves do.

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accounting for the d33 effect. d33 effect may also change theoutputs of displacement controlled problems from monotonicto non-monotonic. The effects of elastic mismatch andthickness-to-length ratio have been discussed and analyticalsolutions for unimorph based actuators are also offered. Thiswork provides a comprehensive and accurate solution for thedesign and optimization of unimorph based power generatorsand actuators.

Acknowledgments

This work is supported by the NSF CMMI award under GrantNo. 1351875 and the AFOSR award under Grant No.FA9550-15-1-0112 (PI: Lu). Partial financial support is alsoprovided by National Science Foundation grants ECCS1509369, ECCS 1309686 (PI: Zhang). TH is thankful to theCharles M Simmons Endowed Presidential Fellowship inEngineering from the University of Texas at Austin.

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