[american institute of aeronautics and astronautics 19th aiaa applied aerodynamics conference -...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A01-25162 42nd AIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics, and Materials Conference

and ExhibitSeattle, WA 16-19 April 2001 AIAA 2001-1403

HIGHER ORDER ZIG-ZAG PLATE THEORY FOR COUPLEDTHERMO-ELECTRIC-MECHANICAL SMART STRUCTURES

Maenghyo Cho, JinhoSchool of Mechanical and Aerospace Engineering, Seoul National University

San 56-1, Shinlim-dong, Kwanak-ku, Seoul 151-742, Korea

AbstractA higher order zig-zag plate theory is developed to

refine the predictions of the mechanical, thermal, andelectric behaviors fully coupled. Both thedisplacement and temperature fields through thethickness are constructed by superimposing linearzig-zag field to the smooth globally cubic varyingfield. Smooth parabolic distribution through thethickness is assumed in the transverse deflection inorder to consider transverse normal deformation.Linear zig-zag form is adopted in the electric field.The layer-dependent degrees of freedom ofdisplacement and temperature fields are expressed interms of reference primary degrees of freedom byapplying interface continuity conditions as well asbounding surface conditions of transverse shearstresses and transverse heat flux. Thus the proposedtheory is not only accurate but also efficient.Through the numerical examples of coupled anduncoupled analysis, the accuracy and efficiency ofthe present theory are demonstrated. The presenttheory is suitable in the predictions of fully coupledbehaviors of thick smart composite plate undermechanical, thermal, and electric loadings.

l.IntroductionRecently, development of adaptive structures are

paid great attentions due to a potential applicationsto vibration suppression, shape control, noiseattenuation and precision positioning. Complexityand full thermo-electric-mechanical coupling of the

* Assistant Professor, AIAA membert Graduate Research AssistantCopyright © 2001 by the American Institute of Aeronauticsand Astronautics Inc. All rights reserved

adaptive composite structures requires efficient andrefined models to predict mechanical behaviors underthermal environments.

In the early stage of the development of models,classical / first order shear deformation theory hasbeen employed to predict mechanical behavior ofembedded or surface bonded piezo-electric layers[l-3]. However, for the accurate prediction of static anddynamic behavior for general layup configurations ofadaptive laminated structures, classical and first ordershear theory is not adequate. Thus higher ordertheories with smeared displacements and layerwiseelectric potential fields[4, 5] and full layerwisetheories have been developed[6]. The smearedtheory is not enough to describe the deformationbehavior through the thickness because it cannotsatisfy static continuity conditions at the interfacesbetween layers. Layerwise theory can adequatelydescribe the deformation behavior through thethickness but it is not computationally efficientbecause it employs a large number of degrees offreedom which depend upon the number of layers [7].

Most of the plate theories developed until noware based on the plane stress assumption. Theyneglect transverse normal deformation effects. It hasbeen well known that the zig-zag pattern ofdisplacement through the thickness under the planestress assumption provide accurate prediction ofdeformations and stresses under mechanicalloadings[8-ll]. However, in thermo-mechanicalproblem, even in moderate thick plate configurations,the transverse normal deformation effect cannot beneglected since the thermal deformation is equallyimportant compared to those of the in-plane thermaldeformations [12]. In addition, for the complete

1American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


analysis of adaptive composite laminates underthermal environments, full coupling effects betweenthermal-mechanical-electricity should be consideredfor the reliable analysis. Chattopadhyay andcollaborator[ 13] used a finite element model basedon the smeared cubic higher order theory to analyzethe smart structures with the full coupling of thermo-mechanical-electricity. Thus it is still required todevelop accurate and efficient model which canpredict the static and dynamic behaviors of smartstructures under thermo-electric-mechanical coupledsituations.

In the present study, an efficient and accuratehigher order zig-zag theory for smart laminatedplates are developed. To predict reliable deformationbehaviors, transverse normal as well as transverseshear deformations are considered. For the efficientevaluation of the mechanical behaviors, transverseshear stress conditions are pre-imposed in thedisplacement field to reduce total active degrees offreedom.

The temperature field is obtained bysuperimposing linear zig-zag field into the globalsmeared cubic field. The temperature degrees offreedom are suppressed by imposing top and bottomsurface heat flux conditions as well as interface heatflux continuity condition. The formulation includesfull coupling between thermo-mechanical-electricbehaviors. Even though the developed theory is atwo-dimensional plate version, full three-dimensional constitutive equations are used for theaccurate prediction of thermal deformations.

The developed theory does not have layer-dependent degrees of freedom of displacement fieldand temperature field but it has layer-dependentdegrees of freedom for electric potentials in order todescribe arbitrary distributions of electric through thethickness of smart structures.

2. FormulationThe free energy may be written as follows:

where the quantities Cijkl and eijk are the

elastic and the piezoelectric constant, respectively ,and aT is defined as CE /T0

x ̂ Composite Laminates

\

h{

Z i

\\ \

\\ -\\

. a \_''// */"

PiezoelectricActuator

Fig 1. Configuration of the adaptive laminatedcomposite plates

The constitutive equations for fully coupledthermo-mechanical-electric materials are given as,

°ij = a— = CijkiCki ~ eijkEk - k^e(2)

= - —— =kijeij+diEi

(3)

(4)where cr. and Dt are the components of the stress

tensor and electric displacement vector. S denotesentropy. 6 =T-TQis the temperature rise from theinitial temperature T0 . Ei is the components of theelectric field vector. btj is the dielectric permittivity

and ktj and di refer to the thermal-mechanical

and the thermal-piezoelectric coupling constants.Based on linear piezoelectricity, EI can be expressedfrom a scalar potential function 0 as follows:

£,. = -0, (i = l,2,3) (5)

Infinitesimal displacement and strain relationship isused and it is given as,

American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


2 (6)

The configuration of the smart laminated compositeplate is shown in Figl.

The governing equations are now derived usingthe variational principle, assuming no body force, asfollows:

= ~Jo° Jv (P Ui

8F2

D^.dVdt + J'° | qeS(j)dSdt = 08ne = 8®k +8®C+8F3 (8)

= -J'° f (KtjOjSO^ + S T0Se)dVdt

+ Jo'° lsqt80dSdt = 0

where a.(9)

In Eqn.(9), S is the entropy and S is the derivativeof S with respect to time. The quantity V is thelaminate volume, and the quantity t is the time spanfor the dynamic deformation.

For the efficient modeling without losing theaccuracy, the higher-order zig-zag is proposed. Azig-zag higher order in-plane displacement field isobtained by superimposing zig-zag linear field to theglobally cubic varying field. In order to include thetransverse normal effect which is significant inthermo-mechanical problems, the out-of-planedisplacement field is assumed as globally parabolicform through the thickness.

The starting displacement field can be written asfollows:

*=1 (10)U7>\xa » £) = w(xa) + rl(xa)z+ i"2\xa)Z

where H(z-zk) is a Heaviside unit step function.The general lamination layup configuration and

in-plane displacement field is shown in Fig 2. Byapplying top and bottom surface transverse shearfree conditions, the following two set of equations

are obtained.

7*3(11)

= 0

which reduce to the following equations.

(12)Transverse shear strains can be expressed as,

y«3=-

+ 1,«* + '2.«z (B)

Applying transverse shear stress continuityconditions at the interfaces between layers, thechange of slope S^ can be determined in terms of

the primary variables of the reference plane.5* = (14)

where S« is the change slope at each interfaces.The detailed expression of the coefficients a^9b^

are omitted here for the limited space. To avoid morecomplexity of the displacement field, the transversenormal stress continuity conditions through thethickness are not imposed in the present modeling.

Finally, substitution of Eqs.(12),(13) into Eq.(lO)yields equation (15)

_1_"2/z

(15)

The variables in the final displacement field aredefined only at the reference planes. The primaryvariables are w° ,w,<p a ,7 i , r 2 . Thus the number of

the primary variables does not depend upon thenumber of layers.

Similar to the displacement construction, thetemperature field through the thickness of the plate isobtained by superimposing linear zig-zag field ontothe global cubic smooth field. The starting temperature


Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


Fig 2. General lamination layup and in-planedisplacement Held configurations.

field can be written as follows:

zt) (16)

The temperature profile is shown in Fig3. In general,the plate may be subjected to thermal loads at bothtop and bottom surfaces. Thus the two thermalboundary conditions on both surfaces are expressedas

-!X'Z=* z~=h (17)Where qt and qb indicate the heat flux applied ontop and bottom surfaces, respectively. K1

33 denotes

the thermal conductivity in the thickness direction inz-th layer. Layer-dependent temperature variables,Osk is determined by the heat flux continuitycondition at each interface between layers.

Osk=Cke3+dk (18)where C*and d*are functions of heat conductionmaterial properties and thickness of the layers. Finaltemperature field is expressed in terms of twoprimary variables 00 and 03 in the reference plane

and prescribed heat fluxes at top and bottom surfaces.Finally, expressions for the temperature field can

be written as follows:

(19)3 +d")(z-zk)H(z-zk)

It must be noted that the higher order temperaturefield defines a non-uniform zig-zag temperaturedistribution in the plate. The function 60 (x,y) and6l (x,y) define the in-plane temperature variations. It

Piezo m ater ial

-*——— Zo

Q.J

Z Z N - I

Fig 3 Temperature field configurations throughthe thickness.

is important to note that although a lineartemperature field can address the in-planetemperature distribution, it cannot satisfy the surfacethermal boundary conditions nor the heat fluxcontinuity conditions at the interfaces between layers.Therefore, temperature variations through thethickness, which produce the most importantbending deformation, cannot be modeled accuratelyby the linear field nor smooth cubic temperature field.The present temperature field can describe accurateand simple distribution through the thickness.

The expressions for the potential function can bewritten as follows

K=l

The descriptions of the electric potential 0 isexpressed as layer-dependent form using linear zig-zag field through the thickness. Even though thelayer-dependent potential field is assumed throughthe thickness, the number of piezo-layers is relativelysmall composed to the total number of layers. Thusthis layer-dependent electric potential field does notincrease the number of degrees of freedomsignificantly. Variational functional based onEqs.(15), (19), and (20) can be constructed forgeneral materials with fully coupled constitutiverelations given in Eqs.(2-4). The equilibriumequations and boundary conditions can be derivedfrom the Hamilton principle. The fully coupledgoverning equations for the proposed deformation,temperature, and electric field are given here.


Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


Equilibrium equations

(21)

k / -*Hk H\ fc, k •[Qa +^3 J + X / C Ev x

— ~C[b -s — —— ^2AZ

The generalized stress resultants shown in Eq(21) aredefined as follows:[Nap Mafi R% ^] = J*^[l z z2 Z3}fe

_ hMl

al3 J - JQ aa/3 [1 (z - z.) ]H(z- Zi)dz

~

Af, =

.hJo 33 z

f a^zdzJo 33

=~~Rk=i

,~zn

v =

[" Fa° Fjl= \" Da[l z ][H (z - z0) - H (z -°

[TH

3, Numerical ExamplesConstitutive relations which are given in Eqs (2),

(3), and (4) account for the full coupling betweenmechanical, thermal and electric fields. Eqs (5) and(6) are used to replace the strains and the electricfields in eqs (2) and (3) by their expressions in termsof the displacements and electric potentials. Eqn(4)is not used for steady-state problems.

The decoupled theory is obtained by neglectingequations (8) and (9) and the temperature andelectric field terms given in eqn (7). The temperaturechange and the electric field are treated as external

Decoupled theory3.1 Thermo-mechanical-electrical Problems

Mechanical loads, thermal loads, and electric loadsare considered respectively. The extensive reviewcan be found in ref[14] for the mechanical behaviorsof various higher order laminated plate theories.

For the numerical evaluation of the performanceof the proposed model, rectangular plates undersimply supported boundary conditions areconsidered. In the mechanical loading case, doublesinusoidal transverse loading distributions areconsidered. In the thermal loading case, temperature


Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


is assumed sinusoidal in the reference plane andassumed linear through the thickness. In the electricloading, electric field is assumed as doublysinusoidal in the piezoelectric layer. The materialproperties of the numerical examples are given in thetablel.

In the mechanical loading problem, five-layeredcross-ply laminates and nine-layer cross-ply withpiezo material layer are considered. As shown in theFigs 3.(a)-(c), in the thick laminate configurations( hia = 114 , hi a =1/10 ), the deformations andstresses including transverse shear and normalstresses are predicted very accurately compared toPagano's exact elasticity solutions[15].

Transverse shear and normal stresses areobtained by integrating 3-D local stress equilibriumequations through the thickness of laminates. Sameorder of accuracy of the present theory can also beobtained by the EHOPT (Efficient Higher OrderPlate Theory) with cubic zig-zag in-planedisplacement field[8,9] under the plane stressassumption.

The thermal loading cases are given in figures4.(a)-(d). The results of thick case (hi a = 1/4hi a = 1/10) with material properties in table 1 arecompared to the exact thermo-elasticitysolutions[12,17]. The transverse shear stress hascomplicated zig-zag pattern through the thicknessand it agrees very well to the exact elasticity solutionprovided by Noor et al[16]. It should be mentionedthat this accuracy cannot be guaranteed by applyingEHOPT with the plane stress assumption inmoderately thick range (h/a>l/2Q). The inclusionof transverse normal stress and strain effects in thepresent theory makes accurate the prediction oflaminates under thermal loading.

In the thermo-mechanical examples, theaccuracy and efficiency of the present theory havebeen demonstrated. The number of primary variablesof the present theory does not depend upon thenumber of layers. It has only two more primaryvariables compared to those of the previous EHOPTbecause it includes transverse normal deformation

effect through the thickness of laminates. Theelectric loading cases are given in Fig 5.(a)-(c).Composite laminates with surface bondedpiezoelectric actuators, subjected to externallyapplied electric field loads, are considered. As shownin Fig 5, the present theory accurately predictstransverse shear stresses as well as in-plane stresses.Fig 6 depicts deflection under mechanical, thermal,and electric loading, respectively. In mechanicalloading case, deflection through the thickness isalmost constant. In thermal and electric loading cases,the deflections change considerably through thethickness of plates. It is observed that transversenormal effect is significant in the thermal andelectric loading cases.

[90/0/90/0/0/90/0/90/Piezo]

z/h0.00 —

1.00E+0h/a=1/10

z/h

Figure 3 (a) Out-of plane displacement wunder mechanical loading

[0/90/0/90/0]

i0.00a

I0.40

I0.80

Figure 3(b) In-plane stress O^ h/a=l/4

under mechanical loading


Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


z/h0.00 —

[0/90/0/90/0]

z/h0.00 —

I0.00

I -0.80 -0.10

cr

[0/90/0]

Figure 3(c) Out-of-plane stress G ̂ h/a=l/4

under mechanical loading

Figure 4(c) Out-of-plane stress G^ h/a=l/4

under thermal loading

0.80 —

0.40 —

z/h0.00 —

-0.40 —

-0.80 —

-1

0.80 —

[90/0/90/0/0/90/0/90/Piezo]

0.40 —

- - present ' Z/h0.00 —

, - ' -0.40 —

—— • —— i —— - —— i —— • —— i —— • —— i -°-8° H00 -0.80 -0.60 -0.40 -0.20 -12C

[90/0/90/0/0/90/0/90]

-=r̂ H^——— exact ')- - present ^/

"̂̂' I ' I ' I ' I

.00 -80.00 -40.00 0.00 40. CW/Wmax h/a=1/10

Figure 4(a) Out-of plane displacement wunder thermal loading

[0/90/0]

z/h

Figure 4(d) Out-of-plane stress G ̂ h/a= 1/4


[90/0/90/0/0/90/0/90/P iezo]

z/h0.00 —

-2000.00 -1000.00 0.00 1000.00 2000.00

Figure 4(b) In-plane stress G^ h/a=l/4


— — I | I | I | i | I

-1.00 -0.98 -0.96 -0.94 -0.92 -0.90

Figure 5 (a) Out-of plane-displacement wunder electric loading


Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


z/h0.00 —

[90/0/90/0/0/90/0/90/Piezo]

present

exact

' I ' I ' I ' I ' I-0.80 -0.40 0.00 0.40 0.80 1.20

Figure 5(b) In-plane stress CF^ h/a=l/10

under electric loading

[90/0/90/0/0/90/0/90/Piezo]

z/h

exactpresent

I ' I0.00 0.40

crzx/a2,I

1.20

Figure 5(c) Out-of-plane stress (J^ h/a=l/10

under electric loading

[90/0/90/0/0/90/0/90/Piezo]

Z/h0.00 —

0.20 0.40 0.60 0.80——I

1.00

Figure 6 Out-of plane-displacement w h/a=l/10

0.36 —

0 0.32 —

O

0)•o

Nor

mal

ized

cen

tral

po

p0

£

00

1 1

1 1

1

[O/O/90/0/0/90/O/Piezo]

. *o o oo

o0

O coupled

O

_ l , I , I , I , I0.00 20.00 40.00 60.00 80.00 100.00

a/h

Fig 7. Deflection top reference surface of coupled

and uncoupled theories.

Coupled theory3.2 Heat Flux Problems

The thermo-piezoelectric-mechanical couplingeffects are investigated for cross-ply graphite/epoxylaminates. The plate is subject to a thermal load. Aheat flux of lOOOW/w2 is applied. The dimensionsof plate are a=b=l.The composite plate includes top surface with

bonded piezo actuator. The results indicate that thecoupling effects play an important role in smartcomposite applications under thermal load. Thedeflection predicted by the uncoupled theory isalways larger, compared to the coupled theory by theamount of 10-20%.

In Fig 7, the transverse deflections of coupled anduncoupled models are given. The material propertiesfor this case is given in table 2. In the thick range,the nondimensional deflections are decreased but thediscrepancy from the thin limit case is marginal. Theuncoupled model overestimates the deflections about13% compared to those of the coupled one.

4. ConclusionA higher order zig-zag theory is developed to

enhance the prediction of fully-coupled mechanical,thermal, and electric responses of smart compositeplates.

By imposing transverse shear stress free condition


Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


of top and bottom surfaces and interface continuityconditions between layers, layer-dependentdisplacement variables can be eliminated. In thesimilar way, by imposing top and bottom surfaceheat flux boundary conditions and interface heat fluxcontinuity conditions between layers, temperatureunknowns reduced to the temperature degrees offreedom of reference surface. Thus the final form ofdisplacement and temperature fields have onlyreference primary variables. Thus only layer-dependent degrees of freedom comes from theelectric potential degrees of freedom. However, theformulation still keeps the efficiency since thenumber of the piezoelectric layers are not so large inthe practical applications.

Through the numerical examples of uncoupledresponses, it is observed that the transverse normaldeformation effect is not negligible in the situationsthat electric and thermal loads are applied. Thepresent theory demonstrated its performance inpredicting deformations and interlaminar stressesbecause it includes the effect of transverse normaldeformation. Coupled and uncoupled analyses underthermal loads indicate that uncoupled analysis mayoverestimate the deflections compared to those of thecoupled analysis. The deviation of the results ofuncoupled analysis from those of coupled case is upto 20 %. Thus fully coupled analysis should beperformed in the case of thermal environments.

The present theory should be implemented in thefinite element framework for the practical thicksmart composite plate problem with generalgeometry, boundary, and loading conditions.

Reference[1] Crawley, E. F. "Use of Piezoelectric Actuators as Elements of

Intelligent Structures," AIAA J, Vol. 25, No. 10, 1987, pp.1373-

1385.

[2] Ha, S.K. and Keilers, C. and Chang, F "Finite element

analysis of composite structures containing distributed

piezoceramic sensors and actuators, AIAA J. Vol 30, No 3, 1992,

pp772-780.

[3] Lee, C. K. "Theory of Laminated Piezoelectric Plates for the

Design of Distributed Sensors/Actuators. Part I: Governing

Equations and Reciprocal Relationships," Journal of the

Acoustical Society of America, Vol. 87. No.3, 1990, pp. 1144-

1158

[4] Mitchell, J. A. and Reddy, J. N., "A Refined Hybrid Plate

Theory for Composite Laminates with Piezoelectric Laminate."

Int. J Solids Struct, Vol.32, No.16, 1995, pp. 2345-2367.

[5] Victor M. Franco Correia, and Maria A. Aguiar Gomes, and

Afzal Suleman, and Cristovao M. Mota Soares, and Carlos A.

Mota Soares."Modelling and design of adaptive composite

structures." Comput. Methods Appl. Mech. Engrg. Vol 185,

2000, pp 325-346.

[6] Saravanos, D.A. and Heyliger, PR. and Hopkins, D.A.

"Layerwise mechanics and finite element for the dynamic

analysis of piezoelectric composite plates." Int. J Solids Struct,

Vol 34, No 3, 1997, pp 359-378.

[7] Reddy, J.N. and Robbins Jr, D.H. "Theories and computational

models for composite laminates.", Appl. Mech. Rev Vol 47, No

6, 1994, pp 147-169.

[8] Cho, M., and Parmerter, R.R., "An efficient higher-order plate

theory for laminated composites," Composite Structures, Vol.

20, 1992,pp.ll3-123

[9] Cho, M., and Parmerter, R.R., " Efficient higher-order

composite plate theory for general lamination comfigurations,"

AIAA J, Vol. 31, 1993, pp.1299-1306.

[10] Toledano, A. and Murakami, H. "A Composite Plate Theory

for Arbitrary Laminate Configurations.", ASME J. of Appl

Mech., Vol 54,1987, ppl81-189.

[11] Di Sciuva, M. "An Improved Shear-Deformation Theory for

Moderately Thick Multilayered Anisotropic Shells and Plates.",

J. ApplMech., Vol 54, 1987, pp 589-596.

[12] Ali, J.S.M. and Bhaskar, K. and Varadan, T.K. "A new theory

for accurate thermal/mechanical flexural analysis of symmetric

laminated plates." Composite Structures, Vol 45, 1999, pp 227-

232

[13] Chattopadhyay, A. and Li, Jingmei and Zhou, Xu and Gu,

Haozhong. "Dynamic Response of smart composites using a

coupled thermo-piezoelectric-mechanical model", A/AA-99-

1481.

[14] Noor, AK and Burton, WS. Assessment of shear deformation

theories for multilayered composite plates. Appl Mech Rev Vol

42, 1989; pp 1-13.

[15] Pagano, NJ. "Exact solutions for rectangular bi-directional


Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03


composites and sandwich plates." J Compos Mater ,Vol 4,

1970; pp 330-343.

[16] Noor, A. K. and Kim, Y. H. and Peters, J. M. "Transverse

shear stresses and their sensitivity coefficients in multilayered

composite panels." AIAA Journal, Vol 32,1994, pp. 1259-1269.

[17] Savoia, M. and Reddy, JN. "Three-dimensional thermal

analysis of laminated composite plates." Int J Solids Struct Vol

32, 1995; pp 503-608.

Table 1 Material properties usedin decoupled numerical examples

Material property[0/90/0/90/0] [0/90/0]

EL=25xl06

ET=lxl06

GLT=0.5xl06

Grr=0.2xl06

VLT=0.25

=lXlO'8

Table 2. Material properties usedin numerical example of coupled case

Material property[Piezo/O/90/0/0/90/O/Piezo]

Material propertyGraphite-epoxy layer

____E1=63xl09

E3=E2=63xl09

G12=24.6xl09

G13= G23=24.6xl09

v 12=0.28Vi3=V 23=0.28

a11=oc22=0.9xlO~6

a33=0.9xlO'6

d3i=150x!0'12

d32=150x!0-12

d33=-336.8xlO'12

d24==-513xlO-12

di5=-513xlO-12

Kn=2.lK22=2.l

EL=144.23xl09

ET=9.65xl09

GLT=3.45xl09

Grr=4.14xl09

vLT=0.3

aLL=25.2xlO'

Kn =4.48K22=3.2lK33=3.2l

Material property[90/0/90/0/0/90/0/90/Piezo]

PZT-5Alayer

Material propertyGraphite-epoxy layer

E!=61xl09

E3=E2=53.2xl09

G12=22.6xl09

G13=G23=21.1xl09

v 12=0.35v13=v 23=0.38au=a22=1.5xlO"6

a33=2.0x!0'6

d31=-171xlO-12

d32=-171xlO-12

d33=374xlO'12

d24==584x!0-12

EL=181xl09

Er=10.3xl09

GLT=7.17xl09

Grr=2.87xl09

VLT=0.28Vrr=0.33aTT=22.5xlO-6

aLL=0.02xlO'6

10American Institute of Aeronautics and Astronautics

Dow

nloa

ded

by U

nive

rsita

ets-

und

Lan

desb

iblio

thek

Dus

seld

orf

on D

ecem

ber

23, 2

013

| http

://ar

c.ai

aa.o

rg |

DO

I: 1

0.25

14/6

.200

1-14

03

[american institute of aeronautics and astronautics 19th aiaa applied aerodynamics conference -...

Documents