dimensional reduction of an end-electroded piezoelectric composite rod

8/11/2019 Dimensional Reduction of an End-Electroded Piezoelectric Composite Rod

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Dimensional Reduction of a Piezoelectric

Composite Rod

Sitikantha Roy and Wenbin Yu Department of Mechanical and Aerospace Engineering

Utah State University, Logan, Utah 80322-4130, USA

Abstract

In the present paper, a new generalized Timoshenko model is constructed for a

composite rod with embedded or attached piezoelectric materials. This model isapplicable to composite rods without prescribed electric potential along the lat-eral surfaces. The Variational-Asymptotic Method (VAM) is applied as a mathe-matical tool to carry out the dimensional reduction process. The present reducedmodel captured the effects of dielectric as well as the polarization of the piezo-electric material, which justifies its coupled electromechanical nature. First, thethree-dimensional electromechanical enthalpy is asymptotically approximated byVAM using the slenderness of the rod as the small parameter and subsequently anequivalent one-dimensional electromechanical enthalpy is developed. Energy terms,which are asymptotically correct up to the second order are kept in the approxi-mated enthalpy expression. For engineering applications, the approximate enthalpy

is then transformed into a generalized Timoshenko model which has the traditionalsix mechanical degrees of freedom along with an extra one-dimensional electric de-gree of freedom.

Key words: Variational asymptotic method; Piezoelectric composite rod; Smartstructure

1 Introduction

Since their discovery by the Curie brothers [1], piezoelectric materials havebeen applied successfully in numerous scientific fields. Ultrasonic technol-ogy, MEMS/NEMS industry, micro acoustic generator, miniaturized electronic

Corresponding author.Email address: [email protected](Wenbin Yu).URL: www.mae.usu.edu/faculty/wenbin(Wenbin Yu).

Preprint submitted to European Journal of Mechanics -A/Solids


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transformer, and smart structures are among the well known application ar-eas of piezoelectric materials. What distinguishes a piezoelectric material isits remarkable property to create a conversion interface between two forms ofenergy, i.e., mechanical to electric or vice versa. This two-way coupling ca-pability, along with other properties, such as rapid response, high operating

bandwidth and low power consumption, make piezoelectric materials suitablefor use both as sensors and actuators [2].

Piezoelectric beam actuators and sensors are very common in scientific appli-cations. Rosen type piezoelectric transformer [3] is a perfect example wherepiezoelectric material is used to build a beam like structure. Inspite of ever in-creasing sophistication in modern three-dimensional (3D) computational tech-niques, sometime, it is neither computationally feasible nor desirable to makea full 3D analysis in an electromechanically coupled framework. As an alter-native, researchers try to exploit the slender nature of beam like structuresand simplify the analysis using one-dimensional (1D) reduced models. So, a

natural question always remain: can these reduced models properly capturethe electromechanical effect?

Most of the beam-modeling techniques in the existing literature use conven-tional displacement field based approaches, where the deformation patternsof a structure are assumed at the very beginning of the analysis [4, 5, 6, 7,8, 9, 10]. Sometimes, these models become oversimplified due to these a pri-ori assumptions, for example, existence of uniaxial stress state, plane strainstate etc. Also, it is very difficult, if not impossible, to assume correct defor-mational pattern to capture the physics of electromechanical coupling. These

assumption based models work reasonably well for uncoupled problems, buttheir applicability and authenticity in a coupled framework always remainquestionable. A comprehensive review on piezoelectric beam like transformermodeling can be obtained in [11].

Recently, the Variational Asymptotic Method (VAM) [12] has been appliedto model smart beams made of piezoelectric material. This method has bothmerits of variational methods (viz., systematic and easily implemented numer-ically) and asymptotic methods (viz., withoutad hockinematic assumptions).In the past,VAM has been successfully applied to model composite beams [13].Cesnik et al. used VAM to model smart beams made of piezoelectric fiber

composites. They also developed one-way coupled classical model for smartthin-walled beams [14], smart solid beams [15], and a coupled refined modelfor smart beams [16]. In the refined model, they applied a Ritz-type assumedmode approximation to accommodate the finite size effect of the cross-section.Very recently, the authors have developed a simpler theory for applying vari-ational asymptotic method in the modeling of piezoelectric composite beams[17]. The work was based on the previous work on the modeling of compositebeams [18] and piezoelectric beams [19].

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In most of the reported studies on piezoelectric beams, it is assumed thatthe lateral surfaces are fully or partially electroded with prescribed potentials.The present work attempts to study the complementary cases where there isno prescribed electric potential on the lateral surfaces. Thus, the focus of thisstudy is to use VAM to rigorously reduce the dimension of such piezoelec-

tric composite rods to develop a reduced 1D beam model. The constructedmodel and the accompanying numerical code, can have significant applica-tion potential in the MEMS/NEMS industry, acoustic field, smart structures,where beam like actuators and sensors made of piezoelectric composites arefrequently used [11].

Three-dimensional Formulation

The Hamiltons principle governing the 3D behavior of a piezoelectric rod isstated as t2

t1

(K H) +W

dt= 0 (1)

wheret1 and t2 are arbitrary fixed times,KandHare the kinetic and electricenthalpy, respectively, andWis the virtual work of applied loads and electriccharges (if exist), The bar is used to indicate that the virtual work needs notto be the exact variations of functionals. The electric enthalpy of piezoelectricmaterial is:

H= 12 V

( T : CE : 2Ee : ET E) dV (2)

where CE is the elastic tensor at constant electric field, is the strain tensor,e is the piezoelectric tensor, Eis the electric field vector, is the dielectrictensor at constant strain field, andVis the space occupied by the structure. Itis noted that although the focused application is piezoelectric rods, the presentformulation is equally applicable to smart rods made of other smart materialscharacterized by a constitutive model with the same mathematical structureas Eq. (2).

In Fig. 1, a beam is represented by a reference line r measured by coordinatex1. A typical cross section s withh as its characteristic dimension is described

by cross-sectional Cartesian coordinates x (Here and throughout the paper,Greek indices ,... assume values 2 and 3 while Latin indices assume 1, 2,and 3. Repeated indices are summed over their ranges except where explicitlyindicated). At each point along r, an orthonormal triad bi is introduced suchthat bi is tangent to the coordinate curve xi. The position vector r of anabitrary point in the undeformed structure is given by

r(x1, x2, x3) =r(x1) +xb (3)

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B1

B2

B3

Deformed State

Undeformed State

r

R

R

s

r

u

x1

b1

b2

b3

R

r

Fig. 1. Schematic of beam deformation

wherer is the position vector of a point on the reference line and r =b1. Here( ) denotes the partial derivative with respect to x1. When the beam deforms,the triad bi rotates to coincide with a new triad Bi. Here B1 is normal tothe unwarped cross-section, but not tangent to the beam deformed referenceline due to transverse shear deformation. For the convenience of derivation,we introduce another triadTiassociated with the deformed beam (see Fig. 2),with T1 tangent to the deformed beam reference line and T is determinedby a rotation about T1. The difference in the orientations ofTi andBi is dueto small rotations associated with transverse shear deformations, as shownin Fig. 2. In Fig. 2, 213 is a small angle in 1-3 plane caused by the sheardeformation while another rotation due to 212 is not sketched for clarity.

1T

3T

3B

1B

132

Fig. 2. Coordinate systems used for transverse shear formulation

The material points having position vector r in the undeformed beam can belocated after deformation by the vector function given by

R(x1, x2, x3) =R(x1) +xT(x1) +wi(x1, x2, x3)Ti(x1) (4)

whereR is the position vector to a point on the reference line of the deformedbeam and defined as the average ofR(x1, x2, x3) over the reference cross sec-

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tion. In Eq. 4, w1 denotes the out of plane warping while w2 and w3 denotethe inplane components of the warping. Eq. (4) is four times redundant be-cause of the way warping was introduced. For simplicity of analysis we arechoosing here a centroidal cross-sectional coordinate system, which can easilybe relaxed if necessary. To remove the redundancy of the warping field, the

following four constraints can be used,

wi= 0 w3,2w2,3= 0 (5)

where the notation means integration over the reference cross section.Using the concept of decomposition of rotation tensor [20] for small localrotation, we can express the Jaumann-Biot-Cauchy strains as

ij =1

2(Fij+ Fji) ij (6)

whereij is the Kronecker symbol and Fij the mixed-basis component of the

deformation gradient tensor such that

Fij =Ti Gkgk bj (7)

Here Gk are the covariant base vectors of the deformed configuration andgk the contravariant base vectors for the undeformed configuration. The 3Dstrain field ij can be expressed in terms of the 1D generalized strain measureswhich are defined as

11b1= biTi R ribi= biTi

K

k (8)

where K is the curvature vector of the deformed reference line, k is the cur-vature vector of the undeformed reference line, 11 is the extensional strain,1 is the elastic twist, and denotes the elastic bending curvature in the xdirection.

A complete description of a piezoelectric composite rod requires not only themechanical field but also the electric field, which can be characterized by theelectric potential,(xi). The electric field is defined as

E=

=

xigi (9)

The energetics of the structure can be described using the concept of electricenthalpy. For linear piezoelectric material, twice the electric enthalpy per unitspan can be expressed as

2H=

T

CE eTeS

= TC (10)

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where =

g

with g as the determinant of the 3D metric tensor ofthe undeformed configuration. We define a generalized 3D strain vector as, = [11 212 213 22 223 33 E1 E2 E3]T. In Eq. (10), CE is the6 6 elastic material matrix measured at constant electric field, e is the 3 6piezoelectric coupling coefficient matrix, andS is the 3

3 dielectric coefficient

matrix measured at constant strain. For regular composite material which isnot piezoelectric, the piezoelectric coefficients are zero.

To calculate the kinetic energy, we need to know the absolute velocity of ageneric point in the structure by taking a time derivative of Eq. (4), such that

v= V +(+ w) + w (11)where ( ) is the partial derivative with respect to time, V is the absolutevelocity of a point in the deformed reference line, is the inertial angular

velocity ofBi bases, and the notation( ) forms an antisymmetric matrix froma vector according to( )ij =ijk( )k using the permutation symbol ijk.In Eq. (11), the symbols v,V, , w denote column matrices containing thecomponents of corresponding vectors in Bi bases, and =0 x2 x3T. Thekinetic energy of a beam can be obtained by

K= 12

V

vTvdV=K1D+K (12)

where is the mass density and

K1D=12

L

0(VTV + 2T

V + Ti)dx1 (13)

K =12

V

(w+ w)T(w+ w) + 2(V +)T(w+ w) dV (14)

with L as the length of the beam, and , , and i defined as mass per unitlength, the first and second distributed mass moments of inertia respectively,which can be trivially obtained through simple integrals over the cross section[21].

If no electric charges applied on the surfaces or inside the body, the virtualwork of the active beam is completely done by applied loads and can becalculated as

W= L0

F Rg

+

Q Rds

dx1+

Q R|x1=Lx1=0 (15)

where denotes the lateral surface of the beam, F = FiBi is the appliedbody force, Q = QiBi is the applied surface tractions. R is the Lagrangianvariation of the displacement field, such that

R= qiBi+ xB+ wiBi+ wjBj (16)

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where the virtual displacement and rotation are defined as

qi = R Bi Bi = jBj Bi (17)

where qi and contain the components of the virtual displacement and

rotation in theBi system, respectively. Since the warping functions are small,one may safely ignore products of the warping and virtual rotation in Randobtain the virtual work due to applied loads as

W=W1D+ W (18)

where

W1D= L0

(fiqi+ mii)dx1+Qi qi|x1=Lx1=0 + eijxQj i|x1=Lx1=0 (19)

W

= L

0 Fi

gw i

+ Qiwids dx1+Qiwi |

x1=Lx1=0 (20)

with the generalized forces fi and moments mi defined as

fi=Fig+

Qids mi= eij

xFjg+

xQjds

(21)

Then the Hamiltons principle in Eq. (1) becomes

t2t1

(K1D+K H) +W1D+ W

dt= 0 (22)

So far, we have presented a 3D formulation for the electromechanically cou-pled problem of smart beams in terms of 1D displacements (represented byR r) and rotations (represented by biBi) and 3D warping functions (wi and). Hereafter, we use warping functions to include both the 3D mechanicalwarping functions wi and the electric potential except where explicitly in-dicated. If we attempt to solve this problem directly, we will meet the samedifficulty as solving any full 3D problem. The main complexity comes fromthe unknown 3D warping functions wi and . The common practice in theliterature is to assume wi, a priori, in terms of 1D displacements and rota-tions, and in terms of applied electric potential to straightforwardly reducethe original 3D continuum model into a 1D beam model. However, for beams

made with generally composite materials, the imposition of such ad hocas-sumptions may introduce significant errors. The accuracy of the reduced beammodel becomes worse when there exist multiple fields in the structure. For-tunately, VAM provides a powerful technique to obtain wi and through anasymptotical analysis of the variational statement in Eq. (22) in terms of smallparameters inherent in the structure to construct asymptotically correct 1Dbeam models. The rest of the paper will show how this could be accomplishedfor smart slender structures.

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2 Dimensional Reduction

The dimensional reduction from the original 3D formulation to a 1D formula-tion can only be done approximately. The best way to do it is to take advantage

of the small parameters in the formulation to construct the 1D formulation sothat an asymptotically correct approximation to the original 3D formulationcan be achieved.

2.1 Asymptotical Order Estimation

For a structure to be modeled as a beam, it should be slender, which meansh/l1 and h/R1, whereh is the characteristic size of the cross section, lis the characteristic wavelength of axial deformation andRis the characteristic

radius of initial curvatures and twist of the beam. For simplicity, we assumeRand l are of similar order, which meanshh/lh/R1.

The strain is also assumed to be small if we are only interested in a geomet-rically nonlinear but physically linear 1D theory, i.e., = max{ij} 1.Here, denotes the characteristic magnitude of the 3D strain. From the 1Dequations of motion [21], we can estimate the following orders of the appliedforces and moments

F1O(h/h) Q1O(h) FO(h2/h) QO(h2) (23)

with denoting the order of the elastic constants. In the present case we arerestricting ourselves to those low-frequency vibration problems for which thefollowing equation holds

h

cs h

l (24)

where is the characteristic scale of change of the function wi in time andcs is the minimal velocity of shear waves inside the piezoelectric material. Weestimate the following orders as follows

csO(

) wO(h

)

ui1DO(l) ui1DVi1DliO(l

) (25)

Using the orders estimated in Eqs. (24) and (25), we can estimate the orderofK in Eq. (14). The variableK can be shown to be higher than O(2 h2

l2)

and it can be neglected in O(2 h2

l2) level correction to the governing 3D vari-

ational statement. Thus, the 3D unknown functions inK do not affect the

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cross-sectional minimization process up to the second order correction to thegoverning variational statement. In a similar manner, with proper restrictionof the order of the applied loads, we can safely neglect W in the governing3D variational statement in Eq. (22). The leading terms of the variationalstatement in Eq. (22) are given as

t2t1

(K1D H) +W1D

dt= 0 (26)

In Eq. (26), only the enthalpic termH contains the unknown 3D functions,wi and . This means that wi and can be solved in terms of unknown 1Dfunctions from a much simpler variational statement defined over the cross-section, given as,

H= 0 (27)

Le [19] has shown that the arrangement of electrodes for which the end surfacesof a piezoelectric beam are fully or partially covered and when electric potentialvaries along the axial direction, an extra electric degree of freedom is needed inthe 1D model. In fact, as long as the electric potential is not prescribed at anypoint along the lateral surfaces, the electric degree of freedom will appear inthe 1D model. The existence of an extra 1D electric degree of freedom, definedas a derivative of the electric potential can also be justified by the followingasymptotic argument. It is easy to assess the following asymptotic orders

11hiO() 1O(hl) wi/hO()

cijklO() eijkO(e) ijO() (28)cijkleijk2ij

where is a factor which is used to tackle the disparity of order among thegeneralized electro-mechanical stiffness coefficients in Eq. (10), and it helps toavoid numerical instability while solving the linear system for the unknownwarping field. Now considering the electric field existent in the cross-sectionalplane we can derive,

eijEO(eh

)O(e h

)O() (29)

From Eq (29) we can estimate,

O(h) (30)

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Similar to Eq. (30), considering the existence of axial electric field we canestimate the order of electric potential as

e1ijE1O(el)O(e

l)O()

O(h

l1 h) (31)

Eqs. (30) and (31) can simultaneously be true if and only if the electric fieldis of the form,

(x1, x2, x3) = (x1) +(x1, x2, x3) (32)

where (x1)

O((hl

)1h) and O(h).

2.2 Electromechanical Cross-sectional Analysis

We are free to define as the average of the 3D electric potential over thecross section, i.e.,

(x1) =(x1, x2, x3) / 1 (33)which implies

(x1, x2, x3)= 0 (34)Using Eq. (9) and defining E1D = andT = [11 1 2 3 E1D]T as 1Dstrain vector related to Ti base system [21] andw = [w1 w2 w3 ]

T, we canexpress as

= hw+ T O()

+ Rw O( h

R)

+ lw

O(hl)

(35)

The explicit forms of the operator matrices are given as

h=

0 0 0 0

x2

0 0 0

x3

0 0 0

0 x2

0 0

0 x3

x2

0

0 0 x3

0

0 0 0 0

0 0 0 x2

0 0 0 x3

= 1

g

1 0 x3x2 00x3 0 0 00 x2 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

(36)

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R= 1

g

k k3 k2 0k3 k

k1 0k2 k1 k 0

0 0 0 00 0 0 0

0 0 0 0

0 0 0 k

0 0 0 0

0 0 0 0

l= 1

g

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 00 0 0 0

0 0 0 0

0 0 010 0 0 0

0 0 0 0

(37)

wherek = k1(x3x2

x2 x3 ), and k1 and k are the initial twist and initialcurvatures respectively. In order to deal with arbitrary cross-sectional geome-try and general anisotropic materials, we need to rely on a numerical approach,such as the finite element method to seek a solution for the unknown warpingfunctions. Thus, the warping field can be discretized as

w(x1, x2, x3) =S(x2, x3) V(x1) (38)

with S(x2, x3) representing the element shape function and V as a columnmatrix of the nodal values of the warping functions over the cross section.Substituting Eq. (35) along with Eq. (38) back into Eq. (10) and neglecting theterms higher than (h/l)2 and (h/R)2,(h2/Rl) we obtain the discretized form

of the electric enthalpy containing terms upto the second order of slendernessas

2H=VTE V + 2VT (Dh T+ DhRV + DhlV) +TTD T+

VTDRRV + VTDllV

+ 2VTDRT+ 2VTDlT+ 2V

TDRlV (39)

where the operator matrices are calculated using the following formulas

E=

[hS]TC [hS]

DhR=

[h S]

T C [R S]

Dh= [hS]

TC [] Dhl=

[h S]T C [l S]D= []TC [] DRR= [R S]T C [R S] (40)

Dll =

[l S]T C [l S]

DR=

[R S]

T C []

Dl =

[l S]T C []

DRl=

[R S]

T C [l S]

Following a similar procedure as outlined in [13], we can seek stationary pointsof the enthalpy for Eq. (39) with respect to the unknown warping V, at each

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order level, along with the discretized constraints. The final expression of thewarping solution can be written as [22]

V =V0T

O()+ V1RT

O( hR )+ V1S

T

O(hl)(41)

Substituting Eq. (41) into Eq. (39), we can get an asymptotically correctexpression of the 1D enthalpy containing terms up to the second order of theslenderness, i.e.,O(2 h

2

l2), O(2 h

2

R2) and O(2 h

2

Rl).

2H=TTAT+ 2TTB

T+ TTC

T+ 2TTD

T (42)

whereA,B,Cand D matrices are given as

A= VT0 Dh+ D+ VT0 (DhR+ D

ThR+ DRR)V0+ 2V

T0 DR+ D

TRV1R

B= VT0 DhlV0+ DTlV0+ V

T0 DhlV1R+ D

TlV1R+ V

T0 DRlV0

+1

2(DTRV1S+ V

T1RDS)

C=VT1SDS+ VT0 DllV0

D= (DTl+ VT0 Dhl)V1S (43)

with DR and DSas expressions in terms of the matrices defined in Eq. (40)

and the Kernel matrix ofE.

3 Transformation to a Generalized Timoshenko Model

The enthalpy of the form in Eq. (42) is not convenient for engineering applica-tions because it involves derivatives of the 1D generalized strains. To get rid ofthese derivatives, we can transform this asymptotically correct energy expres-

sion to a generalized Timoshenko model following the equilibrium-equationapproach as employed by Yu et al.[13] for the regular composite beams. Con-catenating the 1D electric degree of freedom, E1D, with the 1D generalizedmechanical strain measures associated with the two base systems (Ti andBi), we can write the identity relating the generalized 1D strain measuresassociated with these two bases as

T =B+ Q

B+ P B (44)

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where, B = [11 1 2 3 E1D]T, B= [212 213]

T,Q andPare given as

Q=

0 0

0 0

0 11 0

0 0

P =

0 0

k2 k3

k1 00 k10 0

(45)

Substituting Eq. (44) in Eq. (42) and dropping the higher order terms, onecan express the enthalpy in terms of the generalized Timoshenko beam strainmeasures as

2H1= TBAB+ 2

TBAQ

B+ 2TBAP B+ 2

TBB

B+ TBC

B+ 2TBD

B (46)

The generalized Timoshenko enthalpy can be written as2HT =

TBXB+ 2

TBF B+

TBGB (47)

If both the Eqs (46) and (47) are asymptotically correct up to the same order,then they would have been equivalent. Such equivalency can be used to deter-mine the unknown stiffness matrices (X,F,G) in the generalized Timoshenkomodel in Eq. (47), by eliminating the derivatives in Eq. (46). Following [23],one can easily derive the 1D electromechanical equilibrium equation as

F

2

F3

+ D1 F2F3

+ D2

F1

M1

M2

M3

F

= 0 (48)

F1

M1

M2

M3

F

+ D3

F1

M1

M2

M3

F

+ D4 F2F3 = 0 (49)

where

D1 =

0 k1k1 0

; D2= k3 0 0 0 0

k2 0 0 0 0

; (50)

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D3 =

0 0 0 0 0

0 0 k3 k2 00 k3 0 k1 0

0 k2 k1 0 00 0 0 0 0

(51)

andD4= Q DT2. Here Fi andMi have the usual meaning of cross-sectionalresultant forces and moments and F is the 1D electric induction which isconjugate to the electric degree of freedom. These quantities are defined as

F1=HT11

Mi =HT

i

F=1

2

HT1

F =HTE1D

(52)

Using the definitions in Eq. (52), one can express B, B and

B in terms of

B and B from Eqs. (48) and (49) as

B=N1(A3B+ A4B) (53)

B=G1[(FTN1A3+D1G+D2F)B+(FTN1A4+D1FT+D2X)B] (54)where

A3 = (F G1D1D4)G + (F G1D2 D3)F

A4 = A3G1FT + (F G1D2 D3)N

N=X F G1

FT

(55)

Differentiating both sides of Eq. (53), one can express B in terms ofB andB using Eq. (54).

B =N1(F G1D2 D3)A3 A3G1(D1G+ D2F)

B

+ N1(F G1D2 D3)A4 A3G1(D1FT + D2X)

B (56)

Substituting Eqs. (53), (54) and (56) back into Eq. (46), one will get a Timoshenko-like energy expression. The next step is to set this form equal to Eq. (47), after

which one can obtain by inspection the following matrix equations as

X=A2AQG1(FTN1A4+ D1FT + D2X) + 2BN1A4+ AT4 N1CN1A4+ 2DN1

(F G1D2 D3)A4 A3G1(D1FT + D2X)

(57)

F =AQG1(FTN1A3+ D1G+ D2F) + BN1A3+ AT4 N1CN1A3+ AP+ DN1

(F G1D2 D3)A3 A3G1(D1G+ D2F)

(58)

G= AT3 N1CN1A3 (59)

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We need to solve this set of complicated nonlinear matrix equations for theunknown matricesX, F and G. These equations can be simplified as

A=X F G1FT =N (60)P =QG1(FTN1A3+ D1G+ D2F)

(F G1D2

D3)

TN1CN1A3 (61)

A1BN1A3A1DN1 (F G1D2 D3)A3 A3G1(D1G+ D2F)G=AT3 N

1CN1A3 (62)

which can be solved using a perturbation technique as shown in [13]. Afterthe unknown matrices are solved, the final generalized Timoshenko stiffnessmodel for 1D electromechanical beam analysis will look like

F1

F2F3

M1

M2

M3

F

=

s11 s12 s13 s14 s15 s16 e11

s12 s22 s23 s24 s25 s26 e12s13 s23 s33 s34 s35 s36 e13

s14 s24 s34 s44 s45 s46 e14

s15 s25 s35 s45 s55 s56 e15

s16 s26 s36 s46 s56 s66 e16

e11 e12 e13 e14 e15 e16 77

11

1213

1

2

3

E1D

(63)

4 Model Validation

4.1 Linear Solution of a 1D Electromechanical Beam Problem

After the electromechanical stiffness model has been obtained, the problemreduces down to a 1D problem defined on the reference line of the beam. Weneed to solve this 1D problem to get the unknown 1D mechanical as well as

the electric field. To illustrate the use of the developed 1D electromechanicalbeam model, we consider a simple case of a prismatic cantilever piezoelectricbeam. The cantilever is subjected to zero external loads at the tip and hasprescribed voltages at its root and tip. The displacements, rotations, and po-tential described byui(x1),i(x1), and (x1), respectively. Eqs. (48) and (49)can be simplified for a static prismatic case as

F = 0 M + e1F = 0 F = 0 (64)

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where

F = [F1 F2 F3]T M= [M1 M2 M3]

T e1 =

0 0 0

0 010 1 0

(65)

The boundary conditions for the present problem are given by

@x1 = 0 : ui(0) = 0 i(0) = 0 (0) =02

@x1 = l : F(l) = 0 M(l) = 0 (l) =0

2 (66)

For the linear case, the generalized strains can be written as [21]

= u + e1 = E1D = (67)

with= [11212 213]T, = [123]T,u = [u1 u2u3]T, and = [12 3]T.

Solving Eqs. (64), (66), and (67), the voltage distribution turns out to be

=0

l x1 0

2 (68)

We can express the displacements u(x1) and rotations of the cross section(x1) through any generic point on the beam reference line as

u(x1)(x1) =

x1R x2

1

2 e1ST x1S x

2

1

2 e1T

STx1 T x1

f

a

ma

0l (69)wherefa = [e11 e12 e13]T, ma = [e14 e15 e16]T, and R, S, T are 33 matricessuch that

R SST T

=

s11 s12 s13 s14 s15 s16

s12 s22 s23 s24 s25 s26

s13 s23 s33 s34 s35 s36

s14 s24 s34 s44 s45 s46

s15 s25 s35 s45 s55 s56

s16 s26 s36 s46 s56 s66

1

(70)

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E

x1

x3

x2

Electrode

Electrode

P

Fig. 3. Sketch of a rectangular piezoelectric beam polarized along the axis

4.2 Numerical Validation

The above dimensional reduction process has been implemented into VABS,a finite-element based, general-purpose cross-sectional analysis code [13]. Fornumerical validation, we applied VABS to analyze two cases depending onwhether the polarization direction in piezoelectric material is normal or par-allel to the beam reference line.

4.2.1 Axial polarization

For this case we have taken an example given in [19] for a prismatic beamof rectangular cross-section made of piezoelectric material polarized along theaxial direction and having evenfold rotation axis tangential to the centerline;see Fig. 3. Closed-form expressions of cross-sectional stiffness of a classicalbeam model were derived in [19]. The material properties taken from [24] aregiven in Table 1. Where the elastic properties (cijkl) are given in GPa, thepiezoelectric coefficients (eijk) are given in C/m

2, and the dielectric properties(ij) are given in 10

12

C/V-m. We use VABS to analyze a rectangle cross

section with width 0.01 m (along x2 direction) and depth 0.02 m (alongx3 di-rection) and the cross sectional properties according to the constitutive modelin Eq. (63) are listed in Table 2 along with the available analytical solutions,where only non-zero stiffness terms are listed for clarity. As shown in Table 2,transverse shear stiffnesses (s22 and s33) and torsional stiffness (s44) are notavailable in the analytical solution [19], while for all those terms having ana-lytical solutions, the numerical prediction of VABS achieves a perfect matchwith the analytical solution.

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Table 1Material properties for the case of axial polarization

cE1111 = 86.74 cE2222= 129.77 c

E3333 = 102.83 c

E1122 =8.25 cE1133 = 27.15

cE1123=3.66 cE2233=7.42 cE2223 = 5.7 cE3323 = 9.92 cE2323 = 9.92

cE

1313 = 68.81 cE

1312 = 2.53 cE

1212= 29.01 e111 = 0.171 e122=0.152e133 =0.0187 e123= 0.067 e213 = 0.108 e212 =0.095 e313=0.0761e312= 0.067 S11 = 39.21

S22= 39.824

S23= 0.86

S33= 40.42

Table 2Cross-sectional constants of a rectangular section with axial polarization

Le [19] VABS

s11(N) 1.566107 1.566107

s22(N) 0.463107

s33(N) 1.172107s44(Nm

2) 0.244103

s55(Nm2) 0.522103 0.522103

s66(Nm2) 0.130103 0.130103

e11(C) 0.361104 0.361104

77(C-m/V) 0.7841014 0.7841014

E

x1

x3

x2

Electrode

Electrode

P

Fig. 4. Sketch of a rectangular piezoelectric beam polarized normal to the axis

4.3 Cross-sectional polarization

It is also possible that the piezoelectric material is polarized in the plane ofthe cross section along a certain direction; see Fig. 4. The present model is alsoable to provide a generalized Timoshenko beam model for this application ofpiezoelectric beams. To test this capability of the present model and VABS, weconsider another rectangular cross section with the same geometric dimensionsas we considered in the previous case but made with a different piezoelectric

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Table 3Material properties of PZT4

Properties PZT4

E11 = E22(GPa) 81.3

E33(GPa) 64.512 0.329

13 = 23 0.432

G12(GPa) 30.6

G13 = G23(GPa) 25.6

e31= e32(C/m2) 5.2

e33(C/m2) 15.08

e24= e15(C/m2) 12.7

11= 22(C/V-m) 6.761109

33(C/V-m) 5.874109

material, PZT4, and the material is polarized along the thickness direction(alongx3). The material properties are listed in Table 3. The non-zero cross-sectional stiffness properties are listed in Table 4, where one can observe thatthere is a significant shear actuation. To verify whether the present modelcaptures the essential behavior of the structure, we consider a 0.2 m longcantilever beam made of this cross section. The root surface is applied with-100 V and the tip surface is applied with 100 V. A 3D finite element model

was created for this structure suitable for ANSYS multiphysics simulation.The tip deflection was calculated and compared with that obtained from thepresent theory using Eq. (69). As one can observe that an excellent match wasfound between the present 1D model and the 3D multiphysics simulation.

5 Conclusion

A generalized Timoshenko model has been constructed using the variational

asymptotic method for piezoelectric composite beams without prescribed elec-tric potential along the lateral surfaces. The novel contribution of this paper isthe development of a generalized Timoshenko model along with the extra 1Delectric degree of freedom for piezoelectric composite beams and its applicationin electromechanically coupled problems. As expected, for axial polarizationwe get an axial actuation phenomenon and for the cross-sectional polariza-tion case we get a shear actuation phenomenon. The developed model hasbeen augmented into VABS, a general-purpose cross-sectional analysis code.

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Table 4Cross-sectional constants of a rectangular section with cross-sectional polarization

Stiffness VABS

s11(N) 1.821107

s22(N) 4.559106

s33(N) 4.291106

s44(Nm2) 1.216102

s55(Nm2) 6.069102

s66(Nm2) 1.400102

e13(C) 0.213102

77(C-m/V) 0.1561011

Table 5Tip deflection of the cantilever beam with cross-sectional polarization

Displacement ANSYS VABS % Error

u3(m) 0.99219107 0.99278107 0.06

A simple closed-form solution is provided for the linear statics of cantileverpiezoelectric beams. A couple of numerical examples are used to demonstratethe accuracy of the present model in comparison to analytical closed formresults and the 3D multiphysics simulations. The excellent agreement found

between the present beam model and the ANSYS 3D multiphysics simulation,demonstrates that one can use this model to greatly simplify the analysiswithout significant loss of accuracy.

Acknowledgement

This work was supported in part by the Army Research Office under grant49652-EG-II and by the Army Vertical Lift Research Center of Excellence atGeorgia Institute of Technology and its affiliate program through subcontractat Utah State University. The views and conclusions contained herein are thoseof the authors and should not be interpreted as necessarily representing theofficial policies or endorsement, either expressed or implied, of the fundingagencies. Authors gratefully acknowledge Dr. Hans help in getting the cross-sectional polarization results from ANSYS 3D multiphysics simulation.

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