Original Article

Journal of Intelligent Material Systemsand Structures2018, Vol. 29(20) 3949–3959� The Author(s) 2018Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/1045389X18803441journals.sagepub.com/home/jim

Combined piezoelectric andflexoelectric effects in resonantdynamics of nanocantilevers

Adriane G Moura and Alper Erturk

AbstractWe establish and analyze an analytical framework by accounting for both the piezoelectric and flexoelectric effects inbimorph cantilevers. The focus is placed on the development of governing electroelastodynamic piezoelectric–flexoelectric equations for the problems of resonant energy harvesting, sensing, and actuation. The coupled governingequations are analyzed to obtain closed-form frequency response expressions via modal analysis. The combinedpiezoelectric–flexoelectric coupling coefficient expression is identified and its size dependence is explored. Specifically, atypical atomistic value of the flexoelectric constant for barium titanate is employed in the model simulations along withits piezoelectric constant from the existing literature. It is shown that the effective electromechanical coupling of a piezo-electric material, such as barium titanate, is significantly enhanced for thickness levels below 100 nm. The electromecha-nical coupling coefficient of a barium titanate bimorph cantilever increases from the bulk piezoelectric value of 0.065 tothe combined piezoelectric–flexoelectric value exceeding 0.3 toward nanometer thickness level. Electromechanical fre-quency response functions for resonant power generation and dynamic actuation also capture the size-dependentenhancement of the electromechanical coupling. The analytical framework given here can be used for parameter identifi-cation and design of nanoscale cantilevers that can be used as energy harvesters, sensors, and actuators.

KeywordsPiezoelectricity, flexoelectricity, energy harvesting, actuation, vibration

Introduction

Piezoelectric coupling is defined by a third-rank tensorand is limited to certain materials that are non-centro-symmetric, ranging from natural quartz to man-madematerials such as PZT (lead zirconate titanate) and BTO(barium titanate). Flexoelectricity, on the other hand, isthe generation of electric polarization by the applicationof a non-uniform mechanical strain field, that is, a straingradient (Maranganti and Sharma, 2009; Tagantsevet al., 2009; Yudin and Tagantsev, 2013). The phenom-enon of flexoelectricity is a higher-order effect and isexpected to be rather weak except for very small (submi-cron) dimensions, making the concept of interest mainlyfor microelectromechanical systems (MEMS) and espe-cially nanoelectromechanical systems (NEMS) applica-tions. Flexoelectricity is controlled by a fourth-ranktensor and is therefore allowed in materials of any sym-metry. Therefore, any piezoelectric material also exhibitsthe flexoelectric effect at very low thickness levels, in thepresence of non-homogeneous strain fields. As a gradi-ent effect, flexoelectricity is size dependent, while piezo-electric coupling has no size dependence. Widely used

piezoelectric cantilever models developed for devicesabove micron-level thickness have to be modified fornext-generation nanoscale devices since the effect offlexoelectric coupling will change the overall electroelas-tic dynamics at such small scales (Bhaskar et al., 2016).

Flexoelectric effect in solids has received suddenlygrowing attention especially after experiments by Maand Cross (2001a, 2001b, 2002, 2003, 2005, 2006) onelastic dielectrics (Cross, 2006) and for potential small-scale applications thanks to developments in MEMS/NEMS. In addition to experimental efforts by Ma andCross (Ma and Cross, 2001a, 2001b, 2002, 2003, 2005,2006) and others (Huang et al., 2011; Zubko et al.,2007), mainly for samples with high dielectric con-stants, atomistic simulations (Maranganti and Sharma,

The Georgia W. Woodruff School of Mechanical Engineering, Georgia

Institute of Technology, Atlanta, GA, USA

Corresponding author:

Alper Erturk, The Georgia W. Woodruff School of Mechanical

Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.

Email: alper.erturk@me.gatech.edu

2009) were presented to extract flexoelectric coeffi-cients, and, importantly, substantial difference (severalorders of magnitude) was reported between the simu-lated and identified flexoelectric coefficients (Cross,2006). A comprehensive article on the flexoelectriceffect in solids by Yudin and Tagantsev (2013) presentsa detailed discussion on the subject matter along with ahistorical account. It is no surprise that with its promiseof increased electromechanical coupling at small scales,flexoelectricity is of great interest for submicron levelenergy harvesting as well (Deng et al., 2014; Mouraand Erturk, 2017).

In addition to the mismatch in the order of magni-tude of flexoelectric coupling between atomistic simula-tions (Maranganti and Sharma, 2009) and experimentalmeasurements (Cross, 2006), one of the issues in flexo-electric transduction and energy conversion has beenthe lack of a clear understanding and modeling of theconverse effect, as the subject has created confusionsince the converse effect is associated with a polariza-tion gradient (Cross, 2006). For instance, it was sug-gested (Chu et al., 2009; Cross, 2006) that mechanicalflexoelectric sensors could be made with no actuationproperty (which has no precedent or thermodynamicbasis). In a recent effort for finite samples, Tagantsevand Yurkov (2012) presented a consistent and sym-metric converse effect representation and its justifica-tion. Moura and Erturk (2017) implemented thisconverse coupling in a distributed-parameter electroe-lastic framework and showed the variation of the elec-tromechanical coupling in centrosymmetric cantileversfor a broad thickness range, along with a case study onstrontium titanate (STO) cantilevers. The present workaims to extend that effort to accommodate both piezo-electric and flexoelectric transduction mechanisms inpiezoelectric materials such as BTO.

In the following, an analytical framework is devel-oped and analyzed for combined transverse modepiezoelectric and flexoelectric effects in bimorph canti-levers for resonant energy harvesting and actuation. Inaddition to closed-form expressions for the electrome-chanically coupled voltage across the electrical loadand the shunted vibration response, the combinedpiezoelectric and flexoelectric coupling coefficientis extracted and studied. A case study is used for

analyzing the energy harvesting and actuation perfor-mance, coupling coefficient, and size effects for aBTO bimorph cantilever under bending vibrationusing atomistic flexoelectric constant and bulk piezo-electric constant from the existing literature in theproposed framework.

Direct and converse piezoelectric andflexoelectric effects

We consider the problem of a bimorph piezoelectriccantilever under bending vibrations as shown in Figure1 for linear behavior (i.e. linear-elastic material beha-vior and geometrically small oscillations). The samplegeometry justifies beam assumptions, such that thewidth (b) and thickness (h) of the rectangular cross-section are much shorter than the overhang length (L).We further assume that the beam dimensions are suchthat the continuum theory is applicable (the beamlength for the smallest case is orders of magnitudelarger than the lattice parameter of the respective mate-rial). ‘‘Static’’ flexoelectricity (Yudin and Tagantsev,2013) is applicable since the beam thickness (smallestdimension) is much smaller than the wavelength atvibration frequencies of interest. Assuming a linearconstitutive behavior, polarization for combined directpiezoelectric and flexoelectric effects in the transversemode can be written as

P3 = x33E3 + e311S11 +m1133

∂S11

∂x3

ð1Þ

where P3 is the polarization in thickness direction (3-direction is the thickness direction and 1-direction is theaxial direction in Figure 1), E3 is the electric field, S11 isthe axial strain, x33 is the dielectric susceptibility, e311 isthe piezoelectric constant, and m1133 is the flexoelectriccoefficient.

The mechanical stress accounting for the conversepiezoelectric and flexoelectric effects can be expressedas follows

T11 = c1111S11 + e311E3 + f1133

∂P3

∂x3

ð2Þ

or alternatively

Figure 1. Bimorph cantilevers undergoing bending vibrations (exhibiting combined piezoelectric and flexoelectric effects at verysmall thickness levels): (a) energy harvesting/sensing in response to mechanical excitation and (b) shape morphing or dynamicactuation under electrical excitation. The piezoelectric layers are oppositely poled in the thickness direction (series connection) andthe respective lateral faces have perfectly conductive and thin electrode layers.

3950 Journal of Intelligent Material Systems and Structures 29(20)

T11 = c1111S11 + e311E3 +m1133

∂E3

∂x3

ð3Þ

where T11 is the axial stress, c1111 is the elastic modulusof the piezoelectric material (under short-circuit condi-tion of the electrodes), and f1133 is the ‘‘flexocouplingcoefficient’’ (f1133 =x33

�1m1133). Note that the aboveform of flexoelectric coupling is suitable for basic ‘‘exo-genous’’ strain gradients, such as those due to mechani-cal bending, but would be limited for ‘‘endogenous’’ones, such as those due to domain boundaries andinterfaces, which are beyond the scope of this work(Yudin and Tagantsev, 2013).

In the following, we develop and explore a completeanalytical framework by accounting for both the piezo-electric and flexoelectric effects. The focus is placed onthe development of governing electroelastodynamicpiezoelectric–flexoelectric equations for the problems ofenergy harvesting, sensing, and actuation. The coupledgoverning equations are analyzed to obtain the fre-quency response functions such as the voltage outputacross the electrical load per base acceleration (in caseof mechanical excitation) or electromechanical admit-tance in dynamic actuation (in case of electrical excita-tion). Furthermore, the coupling coefficient for thebimorph configuration is identified and its size depen-dence is explored.

Euler–Bernoulli beam model forpiezoelectric–flexoelectric energyharvesting and actuation

Coupled mechanical equation and modal analysis

The partial differential equation governing the forcedvibration of a bimorph cantilevered piezoelectric thinbeam under base excitation (Figure 1(a)) is

�∂2M(x1, t)

∂x12

+ csI∂5wrel(x1, t)

∂x14∂t

+ ca

∂wrel(x1, t)

∂t

+m∂2wrel(x1, t)

∂t2= � m

d2wb(t)

dt2

ð4Þ

where wrel(x1, t) is the transverse displacement of thebeam (neutral axis) relative to its base and M(x1, t) isthe internal bending moment at position x1 and time t,ca is the viscous air damping coefficient (mass propor-tional damping), cs is the strain-rate damping coeffi-cient (stiffness proportional damping), I is thesecond moment of area of the rectangular cross-sec-tion, and m is the mass per unit length of the beam(m= rbh= 2rbhp, where b is the width of the beam, r

is the mass density of the material, hp is the thicknessof each piezoelectric layer), h= 2hp is the total beamthickness, and wb(t) is the transverse displacement ofthe base. The linear damping coefficients employed in

equation (4) satisfy the proportional damping condi-tion (Meirovitch, 2001) so that the correspondingundamped system’s mode shapes can be used in modalanalysis.

The internal bending moment in equation (4) is thefirst moment of the axial stress field over the cross-section of each layer

M(x1, t)= b

ð0�hp

T11x3dx3 +

ðhp

0

T11x3dx3

0B@

1CA ð5Þ

The axial strain component is due to bending onlyand at a certain level (x3) from the neutral axis is pro-portional to the curvature of the beam

S11(x1, x3, t)= � x3

∂2wrel(x1, t)

∂x12

ð6Þ

and it is clear from equation (6) that the strain gradient∂S11=∂x3 in this model is nothing but the curvature ofthe uniform Euler–Bernoulli beam (assuming the effectof the gradient ∂S11=∂x1 to be negligible).

Substituting equations (2) and (6) into the internalbending moment in equation (5) gives

Mðx1; tÞ ¼ b

ð0�hp

cE1111S11 � e311E3 + f1133

∂P3

∂x3

� �x3dx3

+

ðhp

0

cE1111S11 � e311E3 + f1133

∂P3

∂x3

� �x3dx3

�

ð7Þ

For a finite sample (in which the polarization variescontinuously from its bulk value to zero at the electrodeboundaries (Tagantsev and Yurkov, 2012)), the flexo-electric term can be evaluated using integration by partsto identify the role of this term in the bending momentequation

bf1133

ð0�hp

∂P3

∂x3

x3dx3 +

ðhp

0

∂P3

∂x3

x3dx3

0B@

1CA

=� bf1133

ð0�hp

P3dx3 +

ðhp

0

P3dx3

0B@

1CA

=� bf1133hp P3h i+ bf1133hp P3h i� �

ð8Þ

where P3h i is the average polarization induced by theelectric field in the beam. The spatial scale of the polar-ization variation at the interface is much smaller thanthe beam thickness; therefore, P3h i’P, where the

Moura and Erturk 3951

polarization in the bulk (Tagantsev and Yurkov, 2012)can be given by

P= x33E3 ð9Þ

and it is useful to note from equations (2) and (3) thatthe dielectric susceptibility x33 is

x33 =m1133

f1133

ð10Þ

The electric field component E3 should be expressedin terms of the respective voltage term for the series con-nection configuration shown in Figure 1(a). For the seriesconnection of two oppositely poled identical piezoelectriclayers, the voltage resultant is v(t). It is important to notethat for the series connection case, e311 has opposite signsfor the top and bottom layers (due to opposite poling);therefore, the instantaneous electric fields are in the samedirection (i.e. E3 = � v(t)=2hp in both layers) (Erturkand Inman, 2011).

The polarization in equation (9) can then be substi-tuted into equation (8) along with the appropriate elec-tric field equations for the series connection case. Thisgives the following contribution from the flexoelectriceffect

� bf1133hp P3h i+ bf1133hp P3h i� �

= bm1133v(t) ð11Þ

The flexoelectric and piezoelectric coupling termsresulting from equation (7) are only a function oftime, and therefore, it must be multiplied by½H(x1)� H(x1 � L)� (where H(x1) is the Heavisidefunction) to ensure its survival when the bendingmoment is substituted into equation (4). The internalbending moment is then

M(x1, t)= � YI∂2wrel(x1, t)

∂x12

+qv(t)½H(x1)� H(x1 � L)�

ð12Þ

where the coefficient of the backward coupling term (q)for the series connection case is

q= 12e311bhp +m1133b ð13Þ

and the bending stiffness term YI of the compositecross-section (in short circuit) is

YI =2b

3c1111h3

p ð14Þ

The coupled beam equation can then be obtainedfrom equation (4) as

YI∂4wrel(x1, t)

∂x14

+ csI∂5wrel(x1, t)

∂x14∂t

+ ca

∂wrel(x1, t)

∂t

+m∂2wrel(x1, t)

∂t2� qv(t)

dd(x1)

dx1

� dd(x1 � L)

dx1

� �

= � md2wb(t)

dt2

ð15Þ

where d(x1) is the Dirac delta function that satisfies thefollowing equation for a smooth test function g(x1)

�

d(n)d(x1 � p)

dx(n)1

g(x1)dx1 =(� 1)ndg(n)(p)

dx(n)1

ð16Þ

The vibration response relative to the moving basecan be expressed as

wrel(x1, t)=X‘

r= 1

fr(x1)hr(t) ð17Þ

Here, hr(t) is the modal mechanical coordinate andfr(x1) is the mass-normalized eigenfunction (obtainedfrom the short-circuit problem) for the rth vibrationmode for the series connection of the piezoelectriclayers given by

fr(x1)=

ffiffiffiffiffiffiffi1

mL

r

coslr

Lx1 � cosh

lr

Lx1 +sr sin

lr

Lx1 � sinh

lr

Lx1

� �� �ð18Þ

where sr is

sr =sinlr � sinh lr

coslr + coshlr

ð19Þ

and the eigenvalues (lr.0, r = 1, 2, . . . ) are the rootsof the characteristic equation (for the short-circuit andclamped-free boundary conditions)

1+ cos l cosh l= 0 ð20Þ

The mass-normalized eigenfunctions in equation (17)satisfy the following orthogonality conditions

ðL0

mfr(x1)fs(x1)dx1 = drs,

ðL0

YIfr(x1)d4fs(x1)

dx41

dx1 = drsv2r

ð21Þ

where drs is the Kronecker delta and vr is theundamped natural frequency of the rth vibration modeat short circuit (Rl ! 0)

3952 Journal of Intelligent Material Systems and Structures 29(20)

vr = l2r

ffiffiffiffiffiffiffiffiYI

mL4

rð22Þ

The mechanical equation in modal coordinates canbe obtained after substituting equation (17) into equa-tion (15) (then multiplying the latter by the mode shape,integrating over the beam length, and applying ortho-gonality conditions) as

d2hr(t)

dt2+ 2zrvr

dhr(t)

dt+vr

2hr(t)� urv(t)= fr(t) ð23Þ

where the modal piezoelectric–flexoelectric electrome-chanical coupling term is

ur =qdf(x1)

dx1

x1 = L

= 12e311bhp +m1133b

� �df(x1)

dx1

x1 = L

ð24Þ

and the modal mechanical forcing function can beexpressed as

fr(t)= � md2wb(t)

dt2

ðL0

fr(x1)dx1 ð25Þ

Coupled electrical circuit equation and modal analysis

To derive the governing electrical circuit equations ofthe bimorph series configuration, we first examine a sin-gle layer under bending vibrations. The only source ofmechanical strain is assumed to be the axial strain dueto bending, yielding the following electric displacementD3

D3 = e33E3 + e311S11 +m1133

∂S11

∂x3

ð26Þ

where e33 is the dielectric permittivity of the material,e33 = e0 + x33 =(1+ �x33)e0 (note that for high-Kmaterials, which are of interest in flexoelectricity,�x33 � 1, and e33’x33).

The piezoelectrically and flexoelectrically coupledelectrical circuit equation can be obtained from

d

dt

ðA

D � ndA

0@

1A=

v(t)

Rl

ð27Þ

where D is the vector of electric displacement compo-nents, n is the unit outward normal of the electrodes,and the integration is performed over the electrode areaA. The only contribution to the inner product of theintegrand is from D3. Using equation (26) in equation(27), the following circuit equation is obtained

e33bL

hp

dv(t)

dt+

v(t)

Rl

+(12e311hpb+m1133b)

ðL0

∂3wrel(x1, t)

∂x12∂t

dx1 = 0

ð28Þ

which can be extended to the resultant of two layers inseries connection (as in the case of a purely piezoelectricbimorph (Erturk and Inman, 2011)

Cdv(t)

dt+

v(t)

Rl

+X‘

r= 1

ur

dhr(t)

dt= 0 ð29Þ

where the modal electromechanical coupling is the sameas equation (24), and the equivalent capacitance of twolayers combined in series is

C =e33bL

2hp

ð30Þ

Equations (23) and (29) are the governing electrome-chanical piezoelectric–flexoelectric bimorph cantileverequations in modal coordinates.

Closed-form steady-state response in energyharvesting

For harmonic base excitation with wb(t)=W0ejvt, themodal forcing function given by equation (25) can beexpressed as fr(t)=Fre

jvt, where the amplitude Fr is

Fr =v2mW0

ðL0

fr(x1)dx1 ð31Þ

Then the steady-state modal mechanical response ofthe beam and the steady-state voltage response acrossthe resistive load are also harmonic at the same fre-quency as hr(t)=Hre

jvt and v(t)=Vejvt, respectively,where the amplitudes Hr and V are complexed values.Therefore, equations (23) and (29) yield

(v2r � v2 + j2zrvrv)Hr � urV =Fr ð32Þ

1

Rl

+ jvC

� �V + jv

X‘

r= 1

urHr = 0 ð33Þ

where zr is the modal mechanical damping ratio.The steady-state voltage response is obtained as

v(t)=

P‘r = 1

�jvurFr

v2r�v2 + j2zrvrv

1Rl+ jvC +

P‘r = 1

jvur2

v2r�v2 + j2zrvrv

ejvt ð34Þ

Moura and Erturk 3953

Once the voltage across the electrical load isobtained, the current and power output can be calcu-lated easily. For the case of a real-valued electrical load(i.e. a resistive load), the current delivered to the load isi(t)= v(t)=Rl and the instantaneous power output isP(t)= v2(t)=Rl.

The steady-state modal mechanical response of thebeam (that accounts for the converse piezoelectric–flexoelectric effect) can be obtained as

wrel(x1, t)=X‘

r= 1

Fr � ur

P‘r= 1

jvurFr

v2r�v2 + j2zrvrv

1Rl+ jvC +

P‘r= 1

jvur2

v2r�v2 + j2zrvrv

0BB@

1CCA fr(x1)e

jvt

v2r � v2 + j2zrvrv

2664

3775 35Þ

Closed-form steady-state response in actuation

The governing equations in energy harvesting can bemodified to represent the actuation problem, such thatthere is no base excitation (Fr = 0), and the excitationis due to harmonic voltage input (Figure 1(b))

(v2r � v2 + j2zrvrv)Hr = urV ð36Þ

� I + jvCV + jvX‘

r= 1

urHr = 0 ð37Þ

where the V=Rl term is replaced with the current input�i(t)= � Iejvt. The steady-state mechanical responseand actuation current are obtained by solving equations(36) and (37) as

wrel(x1, t)=X‘

r= 1

urfr(x1)

v2r � v2 + j2zrvrv

Vejvt ð38Þ

i(t)= jv C +X‘

r = 1

u2r

v2r � v2 + j2zrvrv

!Vejvt ð39Þ

Piezoelectric–flexoelectricelectromechanical coupling coefficientsand size effects

The electromechanical coupling coefficient k is a directmeasure of energy conversion as commonly used inpiezoelectricity (Lesieutre and Davis, 1997). A dynamicdefinition of the modal electromechanical couplingcoefficient can be obtained based on the differencebetween the open-circuit and short-circuit natural fre-quencies (Lesieutre and Davis, 1997)

k2 =voc

r

� �2 � vscr

� �2

vocr

� �2ð40Þ

where k is the combined piezoelectric-flexoelectric cou-pling coefficient for the rth vibration mode (the focus in

the simulations of this work will be placed on the fun-damental mode, r = 1).

To express the coupling coefficient using equation(40), recall the undamped short-circuit natural fre-quency of the rth vibration mode

vscr =vr = l2

r

ffiffiffiffiffiffiffiffiYI

mL4

rð41Þ

Then, for modal vibrations under open-circuit condi-tions, equation (29) can be reduced to

v(t)=�urhr(t)

C,Rl ! ‘ ð42Þ

Substituting equation (42) into the modal mechani-cal equation of motion in equation (23), the undampednatural frequency of the rth vibration mode underopen-circuit conditions becomes

vocr

� �2= vrð Þ2 1+

ur2

v2r C

� �ð43Þ

yielding

k2 =

ur2

v2r C

1+ ur2

v2r C

=ur

2

v2r C + ur

2=

1

1+v2

r C

ur2

ð44Þ

Equation (44) can be simplified by substituting theexpression for the equivalent capacitance and electro-mechanical coupling

k2 =1

1+4cE

1111e33

3a2r e311 + 4m1133=hð Þ2

ð45Þ

where ar = � sin lr � sinh lr +sr( cos lr � cosh lr)and recall that h= 2hp. Equation (45) clearly capturesthe thickness dependence of the flexoelectric effect andshows that with decreased thickness (h), the couplingcoefficient (k) increases. This equation also shows theeffect of material properties on the coupling coefficientand gives insight into the sign of the flexoelectric con-stant (m1133), which has been reported with differentsigns in the literature as pointed out by Zubko et al.(2013). According to equation (45), the flexoelectricand piezoelectric constants should have the same signto prevent non-monotonic dependence of the couplingcoefficient on the thickness (and its vanishing at a cer-tain thickness value)—that is a negative e311 should beaccompanied with a negative m1133. It is worth

3954 Journal of Intelligent Material Systems and Structures 29(20)

mentioning that the coupling coefficient depends on thevibration mode, electrode coverage, and so on.Typically, the first bending mode is of interest (r = 1)for which full electrode coverage yields no chargecancelation.

Case studies and results

Electromechanical coupling and size effects

We consider BTO in our simulations using the atomis-tic value of m1133 = � 5:463 3 10�9 C=m (Marangantiand Sharma, 2009). The electromechanical couplingcoefficient due to combined piezoelectric and flexoelec-tric energy conversion is plotted for a range of cantile-ver thicknesses in Figure 2. The focus is placed on thefundamental bending vibration mode (r = 1), and thebeam thickness in the simulations ranges from 1 to1 nm. As stated previously based on equation (45), thecoupling coefficient increases with decreased thicknessand is illustrated graphically in Figure 2. The isolatedpiezoelectric and flexoelectric coupling coefficients arealso shown in this figure and it is seen that only forthickness levels below 100 nm does the flexoelectriceffect become appreciable, and it strongly enhances theoverall electromechanical coupling. For micron thick-ness levels and above, the overall electromechanicalcoupling is merely due to bulk piezoelectricity; how-ever, the electromechanical coupling is dramaticallyenhanced due to flexoelectricity for thickness levelsapproaching the nanoscale.

Resonant energy harvesting: electromechanicalfrequency response and size effects

The electromechanical frequency response behavior ofa bimorph cantilevered piezoelectric and flexoelectric

energy harvester under base excitation is simulated witha focus on the first bending mode (r = 1) for a range ofelectrical load resistive values. Three different geo-metric scales are explored ranging from millimeter scaleto nanometer scale. The bimorph is made of BTOand has perfectly conductive surface electrodes onthe faces that are perpendicular to the transverse baseexcitation (Figure 1). The atomistic value (Marangantiand Sharma, 2009) of m1133 =� 5:463 3 10�9C=m isused in the following simulations along with thenecessary material properties (Bechmann, 1956):e311 =� 4:4C=m2, cE

1111 = 166GPa, es33 = 12:56nF=m,

and es33 = 12:56nF=m. A mechanical quality factor (Q)

of ;50 is assumed, yielding an approximate modalmechanical damping ratio of 1% of the critical damp-ing for resonant vibrations. Three cases with totalthicknesses (h) of 1 mm, 1 mm, and 1 nm (thickness ofone layer of the bimorph is hp = h=2) are analyzedwhile keeping a constant aspect ratio of L/b/h fixed at100/5/1. The mechanical excitation is harmonic baseacceleration, d2wb(t)=dt2 = � v2W0ejvt. Therefore, theresults are presented as frequency response magnitudemaps normalized by the base acceleration quantified interms of gravitational acceleration (g = 9:81m=s2). Tocapture optimal load in power generation and respec-tive trends with changing load, a range of electricalresistive load values spanning from short- to open-circuit conditions (100 O to 1 GO) are simulated foreach case.

The voltage output (per base acceleration) frequencyresponse map for the 1-mm-thick BTO bimorph(100 mm 3 5 mm 3 1 mm) is shown in Figure 3(a).With increased electrical load resistance, the voltageincreases monotonically at all frequencies, as a typicaltrend in energy harvesting (Erturk and Inman, 2011). Itis shown that the resonance frequency for the 1-mm-thick STO cantilever is unaffected by the change inresistive load, that is, the frequency of peak magnitudedoes not change as the electrical load resistance value isswept from short- to open-circuit conditions. This indi-cates very low electromechanical coupling such that thefeedback in the mechanical domain due to induced lowvoltage is negligible. The combined piezoelectric andflexoelectric coupling coefficient (for the 1 mm thick-ness level and BTO material properties) is obtainedfrom equation (45) or Figure 2 as k =0:0652 (which isroughly the bulk piezoelectric value) confirming negli-gible contribution from flexoelectricity. The beamthickness is then decreased to 1 mm while keeping thesame aspect ratio (i.e. the dimensions are now 100 mm3 5 mm 3 1 mm). The voltage output frequencyresponse map for this case is shown in Figure 3(b). Aswith the 1 mm thickness case, the 1-mm-thick BTObimorph shows no noticeable shift in the fundamentalresonance frequency with changing load resistance.The combined piezoelectric and flexoelectric couplingcoefficient for this case is k =0:0655, which, again,

Figure 2. Transverse mode coupling coefficient (k) versusbimorph thickness (h) of a BTO cantilever for combinedpiezoelectric and flexoelectric, piezoelectric only, andflexoelectric only effects (for the first bending mode).

Moura and Erturk 3955

indicates negligible flexoelectric contribution. Thebeam thickness is further decreased to 1 nm (beamdimensions of 100 nm 3 5 nm 3 1 nm) and theanalysis is repeated. The nanometer-thick bimorphexhibits a shift in resonance from short- to open-circuit conditions, as shown in Figure 3(c). Thisshows significant electromechanical coupling as con-firmed by the coupling coefficient of k = 0:365 andFigure 2.

The electric current flowing to the resistive load issimply obtained from the voltage output using Ohm’slaw. The current output (per base acceleration) fre-quency response maps are also generated for theBTO bimorph for each geometric scale, as shown inFigure 4. The electrical current output decreases withincreased electrical load resistance, which is the oppo-site trend as compared to voltage output. At all fre-quencies, the maximum current is achieved undershort-circuit conditions of the surface electrodes. As

with the voltage output frequency response maps, similartrends are observed for each case study in terms of thecoupling coefficient. The thickness levels of 1 mm and1 mm show no noticeable shift in resonance frequency(Figure 4(a) and (b)), indicating low electromechanicalcoupling. The 1 nm thickness case shows significant fre-quency shift (Figure 4(c)), revealing strong electromecha-nical coupling as discussed previously for the voltageoutput, as a result of flexoelectric contribution.

As a product of the two quantities which have oppo-site trends with changing load resistance, the electricalpower exhibits more interesting trends, such as the pres-ence of an optimal electrical load resulting in the maxi-mum power output at a given frequency. The electricalpower output is calculated for each of the three geo-metric scales with the fixed aspect ratio. The resultinggraphs are shown in Figure 5. The optimal load formaximum power output can be determined for eachcase from the power output frequency response maps.

Figure 4. Current output frequency response versus loadresistance maps (in magnitude form and per base acceleration)for cantilevered BTO harvesters with a fixed aspect ratio of 100/5/1 (L/b/h) for three different geometric scales with thefollowing thickness (h) values: (a) 1 mm, (b) 1 mm, and (c) 1 nm.

Figure 3. Voltage output frequency response versus loadresistance maps (in magnitude form and per base acceleration)for cantilevered BTO harvesters with a fixed aspect ratio of 100/5/1 (L/b/h) for three different geometric scales with thefollowing thickness (h) values: (a) 1 mm, (b) 1 mm, and (c) 1 nm.

3956 Journal of Intelligent Material Systems and Structures 29(20)

Both the 1 mm and 1-mm-thick harvesters result in apeak power output around 100 kO (Figure 5(a) and(b)). As with the previous frequency response maps, the1 mm and 1 mm power output frequency responsemaps show the resonance frequency to be insensitive tothe resistive load due to low electromechanical cou-pling. Consequently, a single optimal load is observedin the power map for the fundamental vibration modefor each case in Figure 5(a) and (b). However, the 1-nm-thick harvester exhibits two peak values for two dis-tinct optimal electrical loads, 100 kO and 1 MO, respec-tively, at the short-circuit and open-circuit resonancefrequencies, yielding the same power output (Figure5(c)). This is an indication of a relatively stronglycoupled harvester configuration, as a result of the elec-tromechanical coupling enhancement due to the flexo-electric effect.

Finally, it is of interest to understand the structuralresponse of the BTO bimorph while generating electric-ity from strain (piezoelectric effect) and strain gradient

(flexoelectric effect) fluctuations in response to mechan-ical base excitation. The motion of the cantilever isevaluated at the tip (x1 = L) using equation (35).Figure 6 shows the tip displacement maps for all threegeometric scales of the bimorph using the same loadresistances and normalized excitation frequency range.For the 1 mm and 1 mm-thick bimorphs, the vibrationresponses of the cantilevers are insensitive to change inelectrical load resistance, again, showing negligible elec-tromechanical coupling at these thickness levels (Figure6(a) and (b)). Therefore, as a result of weak electrome-chanical coupling, Joule heating in the resistive loaddoes not create any significant dissipation in the vibra-tion response of the BTO cantilever. However, for thebimorph with 1 nm thickness, the electromechanicalcoupling is relatively strong, as seen from previous elec-trical output graphs (Figures 3(c), 4(c), and 5c), andtherefore, mechanical to electrical energy conversion israther significant. Consequently, the structuralresponse of the bimorph is sensitive to changing load

Figure 5. Power output frequency response versus loadresistance maps (in magnitude form and per base acceleration)for cantilevered BTO harvesters with a fixed aspect ratio of 100/5/1 (L/b/h) for three different geometric scales with thefollowing thickness (h) values: (a) 1 mm, (b) 1 mm, and (c) 1 nm.

Figure 6. Tip displacement frequency response versus loadresistance maps (in magnitude form and per base acceleration)for cantilevered BTO harvesters with a fixed aspect ratio of 100/5/1 (L/b/h) for three different geometric scales with thefollowing thickness (h) values: (a) 1 mm, (b) 1 mm, and (c) 1 nm.

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resistance near the resonant frequency (Figure 6(c)).Certain load resistance values result in significant shuntdamping, confirming thermodynamic consistency ofthe fully coupled model.

Resonant actuation: electromechanical frequencyresponse and size effects

The same modeling framework is used to understandthe electromechanical response of the bimorphs in thecase of electrical excitation (voltage input) for the sameset of system parameters. Of interest is the tip displace-ment frequency response (structural response for unitactuation voltage input) from equation (38) and theadmittance frequency response (amount of current

drawn for unit actuation voltage input) calculated fromequation (39). The tip displacement and admittance fre-quency responses are shown in Figures 7 and 8, respec-tively. Particularly, in the admittance graphs shown inFigure 8, the relative frequency difference between theresonance and antiresonance frequencies is a measureof electromechanical coupling. These frequencyresponse functions show, once again, that the overallcoupling is enhanced due to flexoelectricity only at thenonometer scale.

Conclusion

An electromechanical framework is developed and ana-lyzed for combined transverse mode flexoelectric andpiezoelectric energy harvesting as well as resonantactuation for the bending vibration of a piezoelectric

Figure 7. Tip displacement frequency response functions (inmagnitude) for resonant actuation of BTO cantilevers with afixed aspect ratio of 100/5/1 (L/b/h) for three differentgeometric scales with the following thickness (h) values: (a)1 mm, (b) 1 mm, and (c) 1 nm.

Figure 8. Admittance frequency response functions (inmagnitude) for resonant actuation of BTO cantilevers with afixed aspect ratio of 100/5/1 (L/b/h) for three differentgeometric scales with the following thickness (h) values: (a)1 mm, (b) 1 mm, and (c) 1 nm.

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cantilever by accounting for two-way electromechanicalcoupling. The modeling framework is based on theEuler–Bernoulli beam theory and properly accounts forthermodynamically consistent, symmetric, direct, andconverse coupling terms, and it captures the size effecton the combined flexoelectric–piezoelectric couplingcoefficient. Based on a modal analysis procedure,closed-form solutions of the electromechanical fre-quency response functions are presented along withvarious case studies for a broad range of geometricparameters. Thickness dependence of the electromecha-nical coupling (which is a measure of energy conver-sion) is analytically extracted and its size dependence isobserved also in simulations of the electromechanicalfrequency response functions. The flexoelectric–piezoelectric coupling increases from the bulk piezo-electric value of k = 0.0652 at the millimeter scale tok = 0.365 at the nanometer scale due to flexoelectriccontribution. Overall, since the coupling coefficient isthickness dependent, the energy conversion dramati-cally increases in submicron thickness levels due to theflexoelectric effect. The proposed model can be usedfor parameter identification as well as performancequantification and optimization in combined flexoelec-tric and piezoelectric energy harvesting. The model wasalso implemented for dynamic actuation, which couldbe of interest for next-generation NEMS conceptsinvolving actuation for nanocantilevers with submicronthickness levels.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This work was supported by the National ScienceFoundation Grant CMMI-1463339, which is gratefullyacknowledged.

ORCID iD

Alper Erturk https://orcid.org/0000-0003-0110-5376

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