a coupled timoshenko model for smart structures

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Accepted Manuscript

A Coupled Timoshenko Model for Smart Slender Structures

Sitikantha Roy, Wenbin Yu

PII: S0020-7683(09)00044-4

DOI: 10.1016/j.ijsolstr.2009.01.029

Reference: SAS 6593

To appear in: International Journal of Solids and Structures

Received Date: 11 October 2007

Revised Date: 23 October 2008

Accepted Date: 20 January 2009

Please cite this article as: Roy, S., Yu, W., A Coupled Timoshenko Model for Smart Slender Structures, International

Journal of Solids and Structures(2009), doi: 10.1016/j.ijsolstr.2009.01.029

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A Coupled Timoshenko Model for Smart

Slender Structures

Sitikantha Roy and Wenbin Yu 1

Department of Mechanical and Aerospace Engineering

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

Abstract

In this paper, a generalized Timoshenko model has been developed for prismatic,

beam-like slender structures with embedded or surface mounted piezoelectric typesmart materials. Starting from a geometrically exact formulation of the original,three-dimensional electromechanical problem, we apply the variational asymptoticmethod to carry out a systematic dimensional reduction. In the process, the three-dimensional electromechanical enthalpy functional is approximated asymptoticallyusing the slenderness as the small parameter to find out an equivalent one-dimensionalelectromechanical enthalpy functional. For Timoshenko-like refinement over theEuler-Bernoulli beam model, terms up to the second order of the slenderness arekept in the enthalpy expression. As an unified analysis tool, the present model cananalyze embedded or surface mounted active layer with arbitrary cross-sectional ge-ometry as two cases of a general one, no special assumptions or modifications need

to be made for these two separate types of active inclusions.

Key words: Smart beams; Variational asymptotic method; VABS; CoupledAnalysis; Piezoelectricity.

1 Introduction

Last few decades have seen tremendous growth in the smart structure tech-nology and its implementations in various sectors in aerospace, mechanicaland civil engineering. Review papers like Chopra (2002), Chee et al. (1998),Loewy (1997), and Giurgiutiu (2000) discuss in great detail about the presentprogress and future prospect of this promising technology. Inspite of having

1 Corresponding author: +1-435-7978246 (tel.); +1-435-7972417 (fax)Email address: [email protected] (Wenbin Yu).

Preprint submitted to International Journal of Solids and Structures29 January 2009


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tremendous advancement in the smart structure modeling techniques alongwith the exploration of its diverse application areas, the analytical predictivecapabilities, specially in a multi-physics framework, for smart structures arestill very limited in comparison to those for conventional composite structures(Krommer and Irschik, 1999).

Many engineering structural components can be analyzed using beam models ifone dimension is much larger than the other two dimensions of the structure.For this very reason, smart slender structures are usually termed as smartbeams in the literature. Different researchers have proposed various smartbeam models to take advantage of this geometrical feature. These models tryto capture the behavior associated with the two small dimensions, eliminatedin the final one-dimensional (1D) beam analysis.

Roughly speaking, most of the studies in the literature can be classified asengineering models which are based on a priori kinematic assumptions and

asymptotic models which are derived by asymptotic expansions of the three-dimensional (3D) quantities in terms of the small parameters such as h/l, withh as the characteristic dimension of the cross section and l as the wavelengthof axial deformation. Engineering models begin with assuming some kind ofdistribution through the cross section for the 3D quantities, defined in theframework of 3D piezoelectricity, in terms of the 1D quantities defined onthe chosen beam axis. These models dominate the literature on the modelingof smart beams. Like, Sun and Zang (Sun and Zang, 1995; Zang and Sun,1996) developed a sandwich beam model based on this philosophy. Other no-table works in this line are that of Benjeddou et al. (1997) and Aldraihem and

Khdeir (2000, 2003). These models use assumptions mainly based on engineer-ing intuition and have clear physical meaning. The numerical implementationof these models can be developed straightforwardly from a variational state-ment. However, most of the a priori kinematic assumptions which are naturalextensions derived from the models of beam made of homogeneous, isotropicmaterial and cannot be easily extended or justified for heterogeneous struc-tures made with anisotropic materials. Moreover, there is no rational way forthe analysts to determine the loss of accuracy and what kind of refinement(that is, single-layer versus layerwise, first-order versus higher-order) should beundertaken to increase the accuracy while keeping a reasonable computationalcost.

Instead of relying on a priori kinematic assumptions, asymptotic methodscan reduce the original 3D problem into a sequence of 1D beam models bytaking advantage of the small parameter h/l (Altay and Dokmeci, 2003). Theconventional practice is to apply a formal asymptotic expansion directly to thesystem of governing differential equations of the 3D problem and successivelysolve the 1D field equations from the leading order to higher orders. Althoughthese models are mathematically elegant and rigorous, it is hard to identify to

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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

which behavior the equations deduced from a certain order correspond, andit is very difficult, if not impossible, to implement these theories numerically.This method becomes intractable for a complex problem such as smart beams.Although there are some conventional asymptotic models for smart platesdeveloped (Reddy and Cheng, 2001), such models for smart beams are rarelydeveloped.

Recently, the variational-asymptotic method (VAM) (Berdichevsky, 1979) hasbeen introduced to remedy the aforementioned shortcomings of asymptoticmethods. This method has both merits of variational methods (viz., systematicand easily implemented numerically) and asymptotic methods (viz., withouta priori kinematic assumptions). This method has been successfully appliedto model composite beams (Yu et al., 2002; Cesnik and Hodges, 1997). Cesniket al. used this method to model smart beams with active twist enabled bypiezoelectric fiber composites. They have developed classical models for smartthin-walled beams (Cesnik and Shin, 2001), smart solid beams (Cesnik andOrtega-Morales, 2001), and a refined model for smart beams (Palacios and

Cesnik, 2005). In the refined model they have used assumed mode techniqueand solved the cross-sectional analysis using a higher order state space so-lution. Very recently, based on the general framework of applying VAM tocomposite dimensionally reducible structures developed in Yu (2002), a gen-eralized, fully coupled classical model has been developed for smart beamscontaining piezoelectric materials (Roy et al., 2007). The present research iscarried out as a refinement incorporating transverse shear effects over theclassical model done in Roy et al. (2007).

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2 Three-dimensional Formulation

As sketched in Fig. 1, a beam can be represented by a reference line r measuredby x1, and a typical cross section s with h as its characteristic dimension and

described by the cross-sectional Cartesian coordinates x (Here and through-out the paper, Greek indices assume values 2 and 3 while Latin indices assume1, 2, and 3. Repeated indices are summed over their range except where explic-itly indicated.) At each point along r, an orthonormal triad bi is introducedsuch that bi is tangent to xi.

The position vector r of any material point in the undeformed beam structurecan be written as:

r(x1, x2, x3) = r(x1) + xb (1)

where r is the position vector of the points of the reference line, r = b1 and( ) means the partial derivative with respect to x1. When the beam deforms,

the triad bi rotates to coincide with a new triad Bi. B1 is not tangent to thedeformed beam reference line due to the transverse shear deformation. Forthe purpose of making the derivation more convenient, we introduce anotherintermediate triad Ti associated with the deformed beam (see Fig. 2), withT1 tangent to the deformed beam reference line, and T is determined by arotation about T1. The difference in the orientations of Ti and Bi is due tosmall rotations associated with transverse shear deformation. The relationshipbetween these two basis vectors can be expressed as:

B1

B2

B3

=

1 212 213

212 1 0

213 0 1

T1

T2

T3

(2)

where 212 and 213 are the small angles characterizing the transverse sheardeformation.

The material point having position vector r in the undeformed beam now canbe located by the vector function given as:

R(x1, x2, x3) = R(x1) + xT(x1) + wi(x1, x2, x3)Ti(x1) (3)

where R is the position vector to a point on the reference line of the deformedbeam and defined as the average of R(x1, x2, x3) over the reference cross sec-tion and wi are the components of warping expressed in Ti base system, bothin and out of the cross-sectional plane. In the present case we are trying toseek a solution which is valid in the interior domain of the slender structure,which means that, we will ignore the discrepancies in the boundary layer zonefollowing the Saint Venant principle. It is generally accepted that the trans-verse shear strains (212, 213) are one order higher than the strain measures

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1T

3T

3B

1B

132

C

Fig. 2. Coordinate systems used for transverse shear formulation

used in the classical model, which are; extension (11), twist (1), and twobendings (2 and 3) respectively.

Eq. (3) is four times redundant because of the way warping was introduced.To remove the redundancy, the following four integral constraints can be used:

wi = 0 x2w3 x3w2 = 0 (4)

where the notation means integration over the reference cross section. Theimplication of Eq. (4) is that warping does not contribute to the rigid-body dis-placement of the cross section. These constraints effectively define the meaningof 1D displacement variables including extension, bending and torsion. Usingthe concept of decomposition of rotation tensor (Danielson and Hodges, 1987)for small local rotation, we can express the Jaumann-Biot-Cauchy strains as:

ij =1

2(Fij + Fji) ij (5)

where ij is the Kronecker symbol, and Fij the mixed-basis component of thedeformation gradient tensor such that,

Fij = Ti Gkgk bj (6)

Here Gk are the covariant base vectors of the deformed configuration and gk

the contravariant base vectors for the undeformed configuration (Roy, 2007).The 3D strain field ij can be expressed in terms of the 1D generalized strainmeasures which are defined as:

11b1 = biTi R r

ibi = biTi K (7)

where K is the curvature vector of the deformed reference line, K = iTi.

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Until now, we have described the kinematics of the electromechanical system,which is the same as that of a conventional composite beam, the details ofwhich are fully described in Hodges (2006). The electrical counterpart in anelectromechanical system is characterized by the electric potential, (xi, t),which can be used to define the electric field as:

E = (8)

whose components in the bi system are:

Ei =

xi(9)

In a matrix form, we can lump the expressions for the strain and electric fieldas:

= h w + T + l w (10)

where = [ 11 212 213 22 223 33 E1 E2 E3 ]

T

, w = [ w1 w2 w3 ]

T

a column matrix of generalized warping functions, and T = [11 1 2 3]T

a column matrix of 1D strain measures of the classical model. From henceforward, the subscripts associated with the 1D strain array will denote thebase system it is defined on. The explicit forms of the operator matrices inEq. (10) 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 0 x3 x2

0 x3 0 0

0 x2 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

l =

1 0 0 0

0 1 0 0

0 0 1 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

(11)

The energetics of the smart beam can be described through the electromechan-ical enthalpy. For a linear piezoelectric material, twice the electromechanicalenthalpy per unit span can be expressed as:

2H =

T

CE eeT kS

(12)

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where CE is the 6 6 elastic material matrix at constant electric field, e isthe 6 3 piezoelectric coefficient matrix, and kS the 3 3 dielectric coeffi-cient matrix at constant strain. For regular composite material which is notpiezoelectric, the piezoelectric coefficients are zero.

So far, we have presented a 3D formulation for the electromechanically coupledproblem for smart beams in terms of unknown functions , wi, and . If weattempt to solve this problem directly, we will meet the same difficulty assolving any full 3D problem. Fortunately, VAM provides a useful techniqueto carry out the dimensional reduction to obtain asymptotically correct 1Dbeam models.

3 Dimensional Reduction

The dimensional reduction from the 3D continuum formulation to a 1D beamformulation can not be done exactly. We have to rely on some asymptoticanalysis in terms of the small parameter h/l inherent in the structure. Thispaper focuses on smart beams having electric potential known at least in asingle point of the cross section. The known potential could be zero such asthe grounded situation. This type of smart beams are those studied by mostof the existing literature. For example, a smart beam with electric potentialprescribed on electroded surfaces parallel to the beam reference line. For thistype of smart beams, we have no mechanism to introduce a 1D electric variableas we did in Roy and Yu (2008). Instead, we have to directly consider the 3D

electric potential in the dimensional reduction and solve it directly in thisprocess.

To deal with smart beams with arbitrary topology, we need to rely on thefinite element method for numerical solutions. To this end, we descretize thecross section by finite elements with generalized warpings (wi, ) as the fournodal variables.

w(x1, x2, x3) = S(x2, x3)V(x1, t) (13)

where S is the shape function matrix and V is the column matrix containingthe nodal variables.

Substituting Eqs. (10) and (13) back in Eq. (12), we obtain the electrome-chanical enthalpy expression asymptotically correct up to the second orderas:

2H =

O(2)

VTEV + 2VTDh + TD +

O(2 hl)

2VTDhlV + 2V

TDl +O(2 h

2

l2)

VTDllV

(14)

where denotes the order of material constants, denotes the order ofT.

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The integral constraints on the mechanical warping given in Eq. (4) can bewritten in a discretized form as:

VTH = 0 (15)

with H= STS and:

=

1 0 0 0

0 1 0 x3

0 0 1 x2

0 0 0 0

and = S (16)

Apart from the four integral constraints on the warping field, the prescribedelectric potential over the cross section comes as point constraints in the cross-

sectional problem, given as:

(x1, x2, x3) = i (17)

on the points of the cross section where the electric potential is prescribed.When expressed in discretized form, the prescribed potential over the cross-section forms a known array of electric potential denoted as Vk, an array ofdimension 4n, with n as the total number of nodes. Vk contains zero valuesat the places of unknown mechanical and electrical degrees of freedom andnon-zero values at the places of prescribed electrical potential.

At this point we write the generalized warping field as a combination of knownpart (Vk) and an unknown part (Vu). The unknown part of the warping con-tains zero terms at those nodes which correspond to the prescribed electricpotentials. The unknown part is further expanded as an asymptotic series interms ofh/l. We can write the total warping function as:

V = Vk + Vu

= Vk + V0 + V1 + V2 + O(hh3

l3) (18)

where V0 O(h), V1 O(hlV0), and V2 O(

hlV1).

3.1 Zeroth-order Solution

Details of the zeroth-order solution have been given in Roy et al. (2007) wherea classical model for the smart beam was constructed. In the present case, we

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briefly describe the zeroth-order solution procedure to maintain continuitywith the first-order solution presented later. For the first approximation of theelectromechanical enthalpy given in Eq. (14), we keep terms up to O(2).Substituting Eq. (18) into Eq. (10) and neglecting all the terms higher thanO(2), we get:

2H0 = (Vk + V0)TE(Vk + V0) + 2(Vk + V0)

TDh + TD (19)

It can easily be shown that integral constrains in Eq. (15) for the zeroth-orderapproximation turn out to be:

VT0 H = 0 (20)

As described in Roy et al. (2007), we minimize the electromechanical enthalpygiven in Eq. (19) along with the constraints given in Eq. (20). The final linearsystem involving V0 turns out to be:

EV0 = (HT I)DhT EVk (21)

The Eq. (21) is solved in two stages as follows:

EV(1)0 = (H

T I)DhT V(1)0 = V0T

EV(2)0 = EVk V

(2)0 = V (22)

The final solution for V0 can be written as:

V0 = V0T + V (23)

3.2 First-order Solution

For the first-order approximation, we keep terms up to O(2 h2

l2) in the expres-

sion of the electromechanical enthalpy. The expression ofV from the Eq. (18)along with Eq. (23) is substituted in Eq. (14). Neglecting all the terms higherthan O(2 h

2

l2), we get an expression:

2H1 = TT(V

T0 Dh + D)T +

TTD

Th(Vk + V) + V

T1 EV1

+ 2(Vk +V0T + V)

T

Dhl(V0

T + V

) + 2VT

k DhlV

1 + 2(V0T + V)

T

DhlV

1

+ 2VT1 Dhl(V0

T + V

) + 2(V0

T + V

)TDlT (24)

+ (V0

T + V

)TDll(V0

T + V

) + 2VT1 DlT

In the present analysis, it is assumed that the cross-sectional distribution ofthe externally given electric potential remains same along the beam reference.In other words, Vk = 0. Often, only some portions of the smart beam areelectroded and have prescribed electric potential. For such cases, the smart

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beam should be divided into several beams with different cross-sectional mod-els because the cross sections are essentially different. It can also be provedthat V2 and any higher order perturbation beyond V2, do not contribute tothe electromechanical enthalpy asymptotically correct to O(2 h

2

l2). Integrat-

ing Eq. (24) by parts and neglecting the constant terms, the leading terms

with respect to the unknown V1 from Eq. (24) can easily be obtained as:

2H = VT1 EV1 + 2VT1 DhlV

0 2VT1 D

ThlV

0 2VT1 Dl

T (25)

Similar to the zeroth-order warping, the first-order warping should also satisfythe following integral constraints:

VT1 H = 0 (26)

Minimizing the electromechanical enthalpy functional in Eq. (25) subject tothis constraint, we derive the following Euler-Lagrange equation for the first-order warping V1:

EV1 = (HT I)

(DhlV0 D

ThlV0 Dl)

T + (Dhl DThl)V

(27)

Now, the linear system in Eq. (27) can be solved for V1 similarly as what hasbeen done for V0 and given as:

V1 = V1S

T + V

1 (28)

3.3 Some Clarifications

As Vk = 0 in the present study, so from the second part of Eq. (22) we canderive:

EV = 0 (29)

This equation tells us V comes from the kernel of E matrix, i.e., V

.The actual representation of V is unknown, hence we can not solve V

1 inan explicit form. This may suggest we need to incorporate an electric degreeof freedom in the refined 1D model, which is beyond the scope of the presentwork and will be investigated in future research. But as we have the notionof the order ofV, it is possible for us to find an approximate solution for V1instead. If we write V as an asymptotic series as:

V =O(h)

V(1) +

O(hhl)

V(2) +

O(hh2

l2)

V(3) + O(h

h3

l3) (30)

where V(i) are the terms at each order level, then we have

V =O( h

l)

V(1) +

O( h2

l2)

V(2) +

O( h3

l3)

V(3) + O(

h4

l4) (31)

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Now as we know

V1 V O(h) and V

1 V

O(h

l) (32)

Looking into the Eqs. (30), (31) and (32), we can formulate an approximate

subordinate equation from Eq. (27) by integrating both sides of the followingequation:

EV 1 = (HT I)(Dhl D

Thl)V

(33)

to get an equation like,

EV1 = (HT I)(Dhl D

Thl)V (34)

It is clear that Eq. (34) is an approximate one because we have neglectedsome constant terms in the electromechanical enthalpy. Assuming that the

contribution of the neglected terms is small, we can solve for V1 from Eq. (34).Substituting Eq. (28) back in Eq. (24) and neglecting boundary layer relatedterms resulted from integration by parts for eliminating derivatives of V andV1, we get an expression of the electromechanical enthalpy asymptoticallycorrect up to the second order as:

2H1 = TTAT + 2

TTB

T + TT C

T + 2TTD

T + 2TTf + 2

TT f + 2

TT f (35)

Here we have dropped the quadratic terms with respect to the electric poten-tial in Eq. (35) because it will not affect the 1D beam model. It is also notedthat Eq. (35) will be asymptotically correct only if the neglected integration

constant is small in Eq. (34). Later we will show that dropping of this integra-tion constant will not affect the construction of a generalized Timoshenknomodel, which is the main purpose of this study. The expressions ofA, B, Cand D are given as:

A = V0Dh + D

B = V0DhlV0 + DTlV0

C= VT1SDThlV0 + V

T1SDl + V

T1SDhlV0 + V0DllV0

D = VT0 DhlV1S + DTlV1S (36)

and the actuation forces are given as,

f =1

2DTh(Vk + V)

f = VT0 D

Thl(Vk + V) V

T0 DhlV D

TlV (37)

f = VT1SD

Thl(Vk + V)

1

2VT1S(Dhl + D

Thl)V V

T0 DllV

+1

2(VT0 Dhl V

T0 D

Thl + D

Tl)V1

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3.4 Transformation to a Generalized Timoshenko Model

Because of the special choice of triad Ti, the 1D force-strain measures associ-ated with transverse shear deformation are zero, leaving only an extensional

force strain and curvature strains. In other words,

11 = 11|21=0 i = i|21=0 (38)

A kinematical identity can be derived between these two sets of 1D strainmeasures in Ti and Bi basis as:

T = B + Q

B (39)

where

Q =

0 0

0 0

0 1

1 0

(40)

where B = [11 1 2 3]T are the classical strain measures and B =

[212 213]T. All these strain measures are associated with a Timoshenko

model measured in the Bi basis.

Substituting Eq. (39) in Eq. (35), we can express the electromechanical en-thalpy in terms of the 1D strain measures of the Timoshenko model relatedto Bi base as:

2H1 = TBAB + 2

TBAQ

B + 2TBB

B + TB C

B + 2TBD

B + 2TBf

+ 2TB Q

Tf + 2TB f + 2

TB f (41)

Such a model is not convenient for engineering applications because it involvesderivatives of 1D strain measures. Our purpose here is to eliminate thesetroublesome derivatives to construct a generalized Timoshenko model which

will look like:

2HT = TBXB + 2

TBFB +

TBGB 2

TBF

a1 2

TBF

a2 (42)

where Fa1 = [fa1 m

a1 m

a2 m

a3]T and Fa2 = [f

a2 f

a3 ]

T. To facilitate thistransformation, we will fit the first five terms of the electromechanical enthalpyas given in Eq. (41) into a quadratic form in terms of B and B, i.e., the firstthree terms in Eq. (42). Following Yu et al. (2002), we can obtain the following

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expressions for the derivatives of strain measures in Eq. (41):

B = N1QFTB N

1QGB

B = G1FTN1QFTB + G

1FTN1QGB (43)

B = 0

It is timely noted here that the actuation force f will not affect the gener-alized Timoshenko model because B vanishes, which further means that theconstant neglected in obtaining V1 in Eq. (34) has no effect in the model wewant to construct because V1 only appears in f .

Substituting Eq. (43) back into Eq. (41) and by inspection we obtain thefollowing matrix equations for X, F and G:

X= A + 2AQG1FTN1QFT 2BN1QFT + FQTN1CN1QFT

F = AQG1

FT

N1

QGBN1

QG + FQT

N1

CN1

QG (44)G = GTQTN1CN1QG

where N = X FG1FT. The equations can be solved in a similar manneraccording to the procedure in Yu et al. (2002) for X, F and G. The finalexpressions for active forces Fa1 and F

a2 are given as:

Fa1 = FQTN1

f FG

1QTf f

Fa2 = GQTN1

f FG

1QTf

(45)

The 1D constitutive relations for a generalized Timoshenko model can bewritten as:

F1

F2

F3

M1

M2

M3

=

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

11

12

13

1

2

3

fa1

fa2

fa3

ma1

ma2

ma3

(46)

It is noted that for a beam analysis of the smart structure, only mechanicalvariables exist and the effects due to piezoelectric coupling and prescribedelectric potential exhibit in the actuation forces (fai ,m

ai ). Furthermore, due

to electromechanical coupling, the stiffness values (sij for i = 1, . . . ,6 andj = 1, . . . ,6) are different from beams made of non-piezoelectric materialswith the same elastic and dielectric properties.

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Table 1Cross-sectional geometry and material properties of the three-layer Beam

Properties Aluminium PZT5H

EE11 = EE22(GPa) 70.3 60.00

EE

33(GPa) 70.3 48.16E12 0.345 0.2906

E23 = E13 0.345 0.5099

GE12(GPa) = GE13 26.13 23.0

GE23(GPa) 26.13 23.3

e33(C

m2 ) 0 23.3

e31 = e32(C

m2 ) 0 6.5

e24 = e15(C

m2 ) 0 17.0

kS11 = kS22(

CVm ) 10.18 10

11 1.503 108

kS33(C

Vm) 10.18 1011 1.3 108

Thickness(mm) 16 1 (top and bottom)

Width(mm) 10 10

4 Model Verification

The present theory has been implemented into VABS, a computer program

capable of general-purpose cross-sectional modeling of beams having arbitrarycross-sectional geometry and made of general anisotropic material. One uniquefeature of the present model is that it decouples the original 3D electrome-chanical analysis into a two-dimensional (2D) cross-sectional analysis and a1D beam analysis. If only the global behavior is of interest, one can carry outthe cross-sectional analysis first to obtain the constitutive model as shownin Eq. (46) and then use this model as inputs for necessary beam analyses,static or dynamic. If one is also interested in detailed distribution of the 3Dvariables, we need to recover the 3D variables using VABS based on the globalbehavior. To validate the theory and the numerical implementation in VABS,we studied the following examples.

4.1 Example I

The first example is taken from Zang and Sun (1996), where two piezoelectriclayers are mounted on an aluminium core to form a three-layer construction.Piezoelectric material is polarized along the thickness direction. The beam is

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clamped at one end and the length of the beam is 0.1 m. The dimensions ofthe cross-section and the material properties are given in Table 1, where E, and G denote Youngs modulus, Poissons ratio and shear modulus, respec-tively. The superscript E indicates that the mechanical properties have beenmeasured at constant electric field. The terms e and k denote the electrome-

chanical coupling coefficients and the dielectric properties of the material,respectively. The superscript S indicates that the dielectric properties havebeen measured at constant strain. These 3D material properties contribute tothe cross-sectional stiffness matrix formulation in Eq. (46) via 3D electrome-chanical enthalpy functional given in Eq. (12). The interfaces between thepiezoelectric layer and aluminum core are grounded and the electric potentialof the top and bottom surfaces are prescribed to be -10 V. A 3D finite elementmodel of the beam is constructed in ANSYS using piezoelectric elements. ForVABS 2D cross-sectional discretization, we divide the width by 20 8-nodedquadrilateral elements and along the thickness each PZT5H layer is dividedinto two elements and the aluminium core is divided into 16 elements. The

total number of 2D elements in the cross-section is 20 20. In the ANSYSmodel, we divide the cross-sections by the same mesh and we divide the lengthinto 200 elements. Thus, the ANSYS model uses a total of 2002020 SOLID5, 8-noded coupled brick elements. Fig. 3 compares the transverse centroidaldisplacements between VABS and ANSYS. To demonstrate the predictive ca-pability of VABS for detailed distributions of 3D variables, we also recoveredthe 3D field using VABS based on the global beam behavior at the mid span(x1 = 0.05 m). Fig. 4 plots the distribution of the axial displacement throughthe thickness. Both ANSYS and VABS predict the same linear behavior, whichimplies that the cross section remains as a plane after deformation. Fig. 5 plots

the voltage distribution along the thickness of the structure. It is clear that theelectric potential is zero inside the aluminum core and assume an almost lineardistribution within the piezoelectric layers. The non-zero stress componentsare plotted in Figs. 6-9. Excellent agreement between VABS and ANSYS hasalso been observed for these quantities.

4.2 Example II

The second example is the sandwich beam given in Zang and Sun (1996),

where the top and the bottom layers are made of aluminium, each 8 mmthick. A PZT5H layer of thickness 2 mm is sandwiched between them. Thepiezoelectric core is given a voltage drop of 20 V. In this case, piezoelectricmaterial is polarized along the axial direction and electric field is perpendic-ular to the polarization. The geometry and material properties are listed inTable 2. The cross section is meshed with 10 (along the width) 18 (alongthe thickness), 8-noded, quadrilateral elements. We compare the generalizedTimoshenko model, Eq. (46), with UM/VABS (Palacios and Cesnik, 2005), a

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0 0.02 0.04 0.06 0.08 0.10.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

7

Axial Location (m)

TransverseDispla

cement(m)

VABS

ANSYS

Fig. 3. Transverse deflection along the span between VABS and ANSYS

4 3 2 1 0 1 2 3 4

x 108

0.01

0.008

0.006

0.004

0.002

0

0.002

0.004

0.006

0.008

0.01

u1(x

1,x

2,x

3) (m)

Thickness(m)

VABS

ANSYS

Fig. 4. Axial displacement distribution along the thickness

computer code for cross-sectional analysis of smart beams developed at Uni-versity of Michigan. Since the current version of UM/VABS can not produce afully coupled Timoshenko model, the results obtained using the temperature

analogy of UM/VABS, denoted as UM/VABST, are used instead. The tem-perature analogy essentially is an uncoupled approach with an assumed linear

distribution of electric potential in the piezoelectric layer. The non-zero stiff-ness and actuation forces are listed in Table 3. The results are compared with

UM/VABST. The results with temperature analogy is almost identical. Thisis expected because the thickness of the PZT5H layer is relatively small sothe stiffness properties are mostly contributed by the passive aluminium andthe actual voltage variation within the PZT core is almost linear as verifiedby both ANSYS and VABS in Fig. 10, in other words the constant electricfield assumption in the temperature analogy of UM/VABS actually reflects

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10 8 6 4 2 00.01

0.008

0.006

0.004

0.002

0

0.002

0.004

0.006

0.008

0.01

Voltage (V)

Thickness

(m)

VABS

ANSYS

Fig. 5. Voltage distribution along the thickness

2 1.5 1 0.5 0 0.5 1 1.5 2

x 105

0.01

0.008

0.006

0.004

0.002

0

0.002

0.004

0.006

0.008

0.01

11

(N/m2)

Thickness

(m)

VABS

ANSYS

Fig. 6. Axial stress (11) distribution along the thickness m

the reality.

Fig. 11 compares the transverse deflections computed by other beam theories

and VABS, where FOBT stands for first-order beam theory and HOBT standsfor higher-order beam theory obtained from Aldraihem and Khdeir (2000),and Analytical refers to the analysis of Sun and Zang (1995). Also, as a benchmark, we calculated the centroidal deflections from the direct 3D multiphysicssimulation of ANSYS. It is observed that the analytical results in Sun and Zang(1995) have an excellent agreement with ANSYS, VABS slightly over predictsthe results, and HOBT under predicts the results and FOBT significantlyunder predict the results.

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2 1.5 1 0.5 0 0.5 1 1.5 2

x 105

0.01

0.008

0.006

0.004

0.002

0

0.002

0.004

0.006

0.008

0.01

22

(N/m2)

Thickness

(m)

VABS

ANSYS

Fig. 7. Transverse normal stress (22) distribution along the thickness

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5

x 104

0.01

0.008

0.006

0.004

0.002

0

0.002

0.004

0.006

0.008

0.01

33

(N/m2)

Thickness

(m)

VABS

ANSYS

Fig. 8. Transverse normal stress (33) distribution along the thickness

4.3 Example III

To show the effect of electromechanical coupling to the 1D generalized Timo-shenko model, we study another example which has similar geometric dimen-sions as the previous two examples, but completely made of PZT5H material

and polarized along the axial direction. The top surface is given 180 V andthe bottom surface is grounded. The non-zero stiffness constants are listed

in Table 4. The comparison with an uncoupled approach using UM/VABST

clearly sows the increased significance of the electromechanical coupling in thecross-sectional stiffness constants when the whole beam is made of piezoelectricmaterial. It suggests that with increasing percentage of the piezoelectric mate-rial with respect to the base non-piezoelectric material, the electromechanicalcoupling becomes increasingly important, which implies it is necessary to use

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2500 2000 1500 1000 500 0 500 1000 15000.01

0.008

0.006

0.004

0.002

0

0.002

0.004

0.006

0.008

0.01

23

(N/m2)

Thickness

(m)

VABS

ANSYS

Fig. 9. Transverse shear stress (23) distribution along the thickness

Table 2Geometry and material properties of the sandwich beam

Properties Aluminium PZT5H

EE11 = EE22(GPa) 70.3 60.01

EE33(GPa) 70.3 48.16

E12 = E13 0.345 0.4092

E23 0.345 0.2906

GE12(GPa) = GE13 26.13 23.0

GE23(GPa) 26.13 23.3

e11(C

m2 ) 0 23.3

e12 = e13(C

m2 ) 0 6.5

e26 = e35(C

m2 ) 0 17.0

kS11(C

Vm ) 10.18 1011 1.3 108

kS22 = kS33(

C

Vm) 10.18 1011 1.503 108

Thickness(mm) 8 (top and bottom) 2

Width(mm) 10 10

a fully coupled approach such as the one developed in this study.

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Table 3Cross-sectional constants of the sandwich beam

VABS UM/VABST Diff (%)

s11(N) 0.1245114 108 0.1245114 108 0.0000

s22(N) 0.3647259 107

0.3646642 107

0.0064s33(N) 0.3809688 10

7 0.3809688 107 0.0000

s44(Nm2) 0.1004871 103 0.1004871 103 0.0000

s55(Nm2) 0.3415910 103 0.3415911 103 0.0000

s66(Nm2) 0.1037487 103 0.1037487 103 0.0000

fa3 (N) 0.4658982 101 0.4669564 101 0.2271

10 5 0 5 100.01

0.008

0.006

0.004

0.002

0

0.002

0.004

0.006

0.008

0.01

Thickness(m)

Voltage (V)

Fig. 10. Voltage distribution in the sandwich beam (example II)

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

7

Axial Location (m)

TransverseDeflection(m)

VABS

Analytical

FOBT

HOBT

ANSYS

Fig. 11. Comparison of transverse deflection with axial location

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Table 4Cross-sectional constants of a beam completely made of PZT5H

VABS UM/VABST Diff (%)

s11(N) 0.10802340 108 0.10802340 108 0.0000

s22(N) 0.32989070 107

0.3185522 107

3.4370s23(N) 0.21008010 10

2 0 100

s33(N) 0.34786730 107 0.34455350 107 0.9526

s44(Nm2) 0.90017817 102 0.9001782 102 0.0000

s55(Nm2) 0.29166318 103 0.2916632 103 0.0000

s66(Nm2) 0.90019500 102 0.9001950 102 0.0000

fa3 (N) 0.10043127 102 0.2546698 102 -153.576

5 Conclusion

A generalized Timoshenko model is constructed using the variational asymp-totic method through a rigorous dimensional reduction of the original 3D, fullycoupled electromechanical analysis. The developed model is implemented nu-merically using the finite element method into VABS, a computer code nowcapable of a general-purposes cross-sectional analysis of smart beams. The fairpredictive capability of the present model is demonstrated through comparison

with the results available in the literature and with the direct 3D multiphysicssimulation of ANSYS. The significant affects of electromechanical coupling tothe 1D beam constitutive model are also disclosed through comparison withan uncoupled approach.

Acknowledgement

This work was supported in part by the Army Research Office under grant49652-EG-II with Drs. Gary Anderson and Bruce LaMattina as the technicalmonitors, and by the Georgia Tech Vertical Lift Research Center of Excellencewith Dr. Michael J. Rutkowski as the technical monitor. The authors also wantto thank Drs. Cesnik and Palacios at University of Michigan for technicaldiscussions and their generosity of letting us use UM/VABS.

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