NASA Technical Memorandum 106990

AIAA-96-1277

Layerwise Finite Elements for Smart

Piezoceramic Composite Plates in

Thermal Environments

Ho-Jun Lee

Lewis Research Center

Cleveland, Ohio

and

Dimitris A. Saravanos

Ohio Aerospace Institute

Cleveland, Ohio

Prepared for the

37th Structures, Structural Dynamics, and Materials Conference and Exhibit

cosponsored by AIAA, ASME, ASCE, AHS, and ASC

Salt Lake City, Utah, April 15-17, 1996

National Aeronautics and

Space Administration

LAYERWISE FINITE ELEMENTS FOR SMART PIEZOCERAMIC

COMPOSITE PLATES IN THERMAL ENVIRONMENTS

Ho-Jun Lee

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135

and

Dimitris A. Saravanos"

Ohio Aerospace Institute

Brook Park, Ohio 44142

ABSTRACT

Analytical formulations are presented which account for the coupled mechanical, electrical, and

thermal response of piezoelectric composite laminates and plate structures. A layerwise theory is

formulated with the inherent capability to explicitly model the active and sensory response of

piezoelectric composite plates having arbitrary laminate configurations in thermal environments.

Finite element equations are derived and implemented for a bilinear 4-noded plate element.

Application cases demonstrate the capability to manage thermally induced bending and twisting

defonnatiom in symmetric and antisymmetric composite plates with piezoelectric actuators, and show

the corresponding electrical response of distributed piezoelectric sensors. Finally, the resultant

stresses in the thermal piezoelectric composite laminates are investigated.

INTRODUCTION

The development of piezoelectric composite materials offers great potential for use in advanced

aerospace structural applications. By taking advantage of the direct and converse piezoelectric

effects, piezoelectric composite structures can combine the traditional performance advantages of

composite laminates along with the inherent capability of piezoelectric materials to adapt to their

current environment. Extensive development of analytical methods for modeling the isothermal

behavior of piezoelectric composite structures has been reported. The initial application of

piezoelectric materials as actuators involved the vibration control of beams _'2 and lead to the

development of simplified models for beams. Advances in theoretical models lead to both a

piezoelectric plate theory "_ and a piezoelectric thin shell theor3P, based on the assumptions of

classical plate and shell theory, respectively. Subsequent developments led to a variety of finite

element formulations for beam 6"7,plate _9, shelP °'13, and solid _qs elements.

* NASA Resident Research Associate at Lewis Research Center
Additionalapproacheshavebeenreportedonrefinedpiezoelectriclaminatetheorieswhichaccount

for thelimitationsof classicalaminateplatetheory (CLPT) 16"_. These approaches, called discrete

layer theories, provide a generalization of both the classical and higher order theories, and allow

separate displacement fields to be assumed in each layer of the laminate. A discrete layer beam theory

for analyzing piezoelectric beams was first reported by Robbins and geddy 6 using an induced strain

approach to approximate the piezoelectric effect. Heyliger and Saravanos 1.'2° subsequently presented

discrete layer beam and plate theories explicitly accounting for the coupled equations of

piezoelectricity, which have lead to generalized multi-field finite dement formulations.

All of the previously descn'bed works neglect the implication of thermal effects on both the active and

sensory response of piezoelectric structures, even though an area where piezoelectric materials may

provide dramatic advantages is in the development of smart thermal structures with the capability to

sense and actively compensate for thermally induced deformations. Thermal loads will typically affect

the response of smart piezoelectric laminates with four distinct physical mechanisms: (1) induction

of thermal strains due to coefficient of thermal expansion mismatch between the various composite

plies and piezoelectric layers, (2) pyroelectric effects on the electric displacement of the piezoelectric

material, (3) changes in the piezoelectric and dielectric properties of the piezoelectric materials, and

(4) changes in the elastic properties of the composite and piezoelectric materials. Only limited

research has been reported in this area. Two-dimensional thermopiezoelectric equations for plates

were formulated by l_fmdlin 21. A thermopiezoelectric laminate plate theory and a thin shell theory,

based on classical assumptions, were reported by Tauchert" and Tzou and Howard 23, respectively.

Subsequent developments in therrnopiezoelectric finite elements include a beam element by Rao and

S_, a shell dement using induced strain approximations for the electric and thermal effects by

Chandrashekhara and Kolli _, a solid dement using an induced thermal grain approach by Ha et al.27,

and a thin solid element by Tzou and Ye _'_s. Many of these previous approaches utilize simplified

laminate assumptions and neglect the coupling effects existing in thermopiezoelectric laminates. To

remedy these limitations, discrete layer approaches for smart thermopiezoelectric beam structures

were reported by Lee and Saravanos _32.

This paper presents the extension of the discrete layer mechanics _32 for the analysis of smart

thermopiezoelectric plate structures and addresses the problem of active thermal distortion

management with smart piezoelectric plates. The mechanics account for the coupled mechanical,

electrical, and thermal response of piezoelectric laminates at the material level through the thermopiezoelectric constitutive equations. The displacements, electric potential and temperature are

amaned to be layerwise (piecewise continuous) fields through the laminate thickness. This layerwise

generalization provides the capability to capture the locally induced piezoelectric effects, which leads

to increased accuracy in laminate stress predictions (especially for thick laminates, and/or laminates

with strong thermal and elastic inhomogeneities through-the-thickness). A corresponding finite

element formulation is presented using the layerwise laminate theory and a 4-noded plate element is

developed. Numerical studies demonstrate the capability to actively manage thermaUy induced

twisting and bending deformations in piezoelectric symmetric and antisymmetric composite plates

subjected to thermal gradients, as well as examining the corresponding voltage response of

piezoelectric sensors and the resulting intralaminar and interlaminar stresses in active thermo2
piezoelectricplates.

GOVERNING MATERIAL EQUATIONS

This section outlines the governing equations for thermopiezoelectric materials.

response is represented by the stress-equilibrium equation,

O#d + ft = P _

The mechanical

(1)

where oi, _ p, u_ are the stress, body force per unit volume, density, and displacement, respectively;

with i, j = 1, 2, 3. Maxwelrs equation for the conservation of electric displacements Di is

Di,i = 0 (2)

The constitutive equations for a thermopiezoelectric material 33 employing standard contracted notation are

s --=._(_o_ _" +a&cr) " E + ,,_(r)o (3)

D. = d/,(r) o + _i(T) E_ + p2x(r) 0 (4)

or in semi-inverted form

g c.o ( T) So - e.(r) E - 1") (s)

D = e=.,(T) " S + e_..,E _',r_ k + p;"'(r) e (6)

where S, represents the strain tensor; E m the electric field vector; 0 = AT = T - To is the temperature

difference fi,om the current temperature T and the thermally stress _ee reference temperature To; and

=c.=,:(r) e.

_.(T) = c.p_ (T) _(T)

= e,. t (T) - d_,.,(T) e._Cr)

(7)

p_._(r) = p.0""t,J - d.:..(T) " _.._"(T)

where C,p and s,p are the elastic stiffness and compliance tensors; d.m and e,,m are the different forms
ofthe piezoelectric tensor due to inversion; e,_ is the electric permittivity tensor; a, and _,,, are the

different forms of the coefficient of thermal expansion due to inversion; p, is the pyroelectric

constant, superscripts E, o, S, and T, represent constant voltage, constant stress, constant strain, and

constant temperature conditions, respectively, a, [3 = 1, ..,6; and k, m = 1, 2, 3.

The small deformation strain-displacement relations are

1

s¢ - i (''_ ÷ ''_) (s)

and the electric field vector is related to the electric potential _ by

E, = - ,_ (9)

Through use of the divergence theorem and neglecting body forces, Eqs. (1) and (2) can be expressed

in an equivalent variational form as

f (p _,,,,,+ o,,,s,- 1),aE,)_v=f ,, ,,,, dr+ f q 6, dr (1o)

v r rp

where t_are the surface tractions applied on the bounding surface P; q is the electrical charge applied

on the surface Pp of the piezoelectric material; and V represents the whole volume including both

composite and piezoelectric materials.

DISCRETE LAYER LAMINATE THEORY

A discrete layer laminate theory for thermopiezoelectric composite plates is formulated by

introducing the following piecewise continuous approximations for the state variables

N

u(x,y,z,t) = EUJ(x,y,,) _r/(z) (11)

/-1

N

_(_,y,z,t) = _ vJ(x, xt) ¥(z) (12)

j-I

w(x,y,z,t) = w*(x,y,t) (13)

4
N

_(x,y,z,O = _ t_(x,y,t) eli(z) (14)

.f-1

N

O(z,t) = _ oJ(t)¥(z)

j'l

(15)

where N is the number of interpolation functions qlj. Currently, qJJis represented using linear Lagrangian interpolation functions, although in general any order of interpolation functions can be used. U j,

V j, W °, _, and 0 i are the generalized laminate state variables, in which a constant through-thethickness displacement is assumed for w. This assumption of constant through-the-thickness

displacement leads to a simplified set of constitutive equations, derived from Eqs. (3)-(6), shown in

Appendix A.

By substituting Eqs. (5)-(9) and (11)-(15) into Eq. (10) and integrating through-the-thickness, the

following variational form is obtained

5
J¢ N

_ f ¢p._'-., -6., ÷e_-6v',aA +f e,:,.a_

L--I m-I A A

H H

E E f {a,7..',.2 ÷A: (.,".v,_* _,",..>' *

J_l m-I A

,4,';'(,,, """2 ÷","""1÷";'"J÷ ,,,",,,,>'* +A,7,,2",,2•

_kin. m. k m k m k m k

A2. tu_, ova, + v_, 6u_,+ v. 6%,+ v_, 6v.) +

km m k m k m k m k

A_ (n_, 6n_, + ,,_, 6v. + v. 6v. + v. 6n_,) }dA +

N

E f {n_(v" 6w +w_,6v')+Bg(,,'6w _, +w. 6,,')+

•,,-1 A

B_s(u'6w o, + v "6w + w.6v" + w 6,,') ) dA +

f (_.,,,6,, +c._(_6_,+.,6.,)+c,-qw6.,),a

A

N N

EEf {D4_" v"Sv _ + D4t's (n"6v k + v'6u to) + Dss._""6. '

1_1 m-I A

m-] A

--f ,, ,.,at +f ¢ .,dr

r r,

÷

+ q_,_64,"} aa

(16)

in which the dependence of the z-coordinate has been separated into the generalized density matrix

[P], the stiffness matrices [A], [B], [C], [D], the piezoelectric matrix [E]. the dielectric permittivity

matrix [G], and the thermal force matrices [f,_] and [q,th]- These generalized laminate matrices are

listed in Appendix B.

FIN ITE ELEMENT FORMULATION

The finite element formulation for a composite piezoelectric plate is obtained by incorporating

additional local in-plane approximations to the state variables

M N

u(x,y,z,O = _ _ U _ (t) R '(x,y) _l_(z) (17)

t-I j-I

6
M At

vCx,xz, O = ¥(z) (is)

t-1 J-I

M

= w°'(t) r '(x,y) (19)

t=l

M N

t-I J'l

(20)

where M is the number of in-plane shape functions R. Currently, R is represented using bilinear

Lagrangian interpolation functions.

By combining Eqs. (16)-(20), the following finite element matrix formulation is obtained for the case

of a constant through-the-thickness displacement plate

[X,,l[x,_][xt,][;c,J

[K,,][K=] [K_ [/c_J

[K,,I[Zs2][X,sl o

l_'4d [K42] o [KJ

'IM1,I 0 0 0 '{0tl

0 IX&_] 0 0 t_[.

o 0 [M,r] o {ff'}l

o o o o .{_)J

(21)

where the elements of the submatrices above are calculated in accordance with the generalized

discrete layer laminate matrices defined previously in Appendix B. The finite element submatrices

for the mass, stiffness, external force, and thermal force are included in Appendix C.

The coupled finite element formulation can also be expressed in a compact form with the electric

potential partitioned into active and sensory components such that

where the superscripts s and a indicate the partitioned electric potential vectors in either sensory or

active configurations, respectively. This form has the advantage of positioning the unknown variables

(displacements and sensory electric potentials) in the left-hand terms, while the known quantities

(mechanical loads, thermal loads, electric charges, and active voltages) are included in the right-hand

terms.

The partitioned finite element formulation of Eq. (22) can be easily uncoupled into the following

independent equations for the structural displacements

7
[K.] • -

(23)

and the sensory electric potentials

; + - {Q'}- (24)

APPLICATIONS

Simply Supported [0/4-45]: Graphite/Epoxy Plate with Discrete Piezoceramic Patches

This case study is a problem of active thermal distortion management, examined previously by Ha

et al.2_, of a 37.2 cm x 22.8 cm x 0.75 mm [0/+45], graphite/epoxy plate with fitteen 6.0 cm x 6.0

cm x 0.13 mm piez, oceramic patches uniformly attached to each top and bottom surface (see

Figure 1). A thermal gradient (50°C on the top surface and -50°C on the bottom surface) is

applied to the plate. Figure 1 illustrates the geometry and finite element mesh of the plate. The

plate is simply supported along the two edges parallel to the y-axis and is flee on the two edges

parallel to the x-axis. The material properties used for both the composite and piezoceramic

materials are listed in Table I. The objective was to apply increasing active voltages on both the

top and bottom piezoceramic patches to minimize the out-of-plane deflection induced by the

thermal gradient. Additional studies investigate the sensory response of the piezoceramic patches

operating in a combined sensor and actuator mode, as weU as the development of local stresses in

the plate with all piezoceramic patches operating as actuators.

Active Thermal Distortion Control. The resulting bending deformation with 0 Volts applied on

both surfaces of the piezoceramic patches is shown in Figure 2(a). Through application of

increasing active voltages in the piez,oceramic patches, the thermally induced deformation of the

plate can be gradually eliminated. Figures 2(o) and 2(c) demonstrate the reduction in the out-ofplane deflection achieved by applying active voltages of 34 V and 70 V, respectively. The active

voltage is applied on the outer surface of each of the piezoceramic patches, while the inner surface

(i.e. the surface in contact with the plate) is grounded. The same voltages are applied to both the

upper and lower piezoceramic patches. A comparison of centerline deflections (along y/b=0.5)

predicted by Ha et al.2, and by the current formulation, along with the corresponding active

voltages, is shown in Figure 3. In order to obtain the same reduction in centerline deflection as

Ha et al., slightly higher active voltages were applied using the current method (34 and 70 Volts,

as compared to 31 and 61 Volts for Ha et al.). These differences between the two analyses are

most likely due to the different formulations and elements used. Ha et al. uses a three-dimensional brick element in their analysis, while the current method utilizes a layerwise plate theory and

$
dement which assumes a constant through-the-thickness displacement w. This use of a plate theory

neglects certain material properties that Ha et al's. analysis considers (specifically C13, C_, Cs3, d_,

ds3, and as3). Another difference is the explicit incorporation of thermal terms in the constitutive

equations of the current formulation, which are neglected in Ha et al's. analysis.

Sensory Voltages. Besides the fully active configurations previously studied (active potentials applied

to all piezoceramic patches), other electric configurations yielding combinations of piezoceramic

sensors and actuators are sometimes more desirable. The current formulation has the capability to

explicitly predict sensory voltages. These sensory capabilities are demonstrated on an active/sensory

configuration of the [0/+45], graphite/epoxy plate. The upper layer of piezoceramic patches are used

as sensors, while the bottom layer of patches are maintained as actuators. In the sensory application,

the inner surface of each piezoceramic patch is grounded while the outer surface is free to develop

a sensory electric potential. Figures 4(a)-4(c) show the sensory voltages that develop on the upper

piezoceramic patches upon combined application of a thermal gradient with active voltages of 0, 34,

and 70 Volts on the lower piezoelectric patches, respectively. Generally, there is a reduction in the

sensory voltage values which corresponds to the reduction in out-of-plane deflection with increasing

active voltages. However, additional contributions remain from local stresses and pyroelectric

effects. In actual applications, these measured sensory voltages will be used in a control loop to

provide the inferred feedback essential for monitoring and actively controlling the thermal distortions.

Local Stresses. The development of high local stress fields in a thermopiezoelectric laminate from

the mismatch in induced thermal and piezoelectric grains is also an area of primary concern, since it

will affect the integrity of these structures. As stated previously, one strong advantage of the

layerwise approach is the accurate calculation ofintralaminar in-plane stresses and interlaminar shear

stresses in piezoelectric composite laminates (I-Ieyliger and Saravanoslt'_. Consequently, this section

presents the stress fields for the [0/±45], graphite/epoxy plate problem with both piezoceramic layers

used as actuators. In these studies, a continuous piezoceramic layer is used instead of many

distributed piezoceramic patches to eliminate the need for a highly refined mesh. The stresses are

nondimensionalized using an equivalent laminate moduli defined as

h

0

Figures 5(a)-5(c) show the variation of normal stresses (o=, o,) and in-plane shear stress (o_)

through-the-thickness at x/a=0.5 and y/b=0.4 with the different applied voltages, while Figures 5(d)-

5(e) show the variation of interlaminar shear stresses (o_, o_ through-the-thickness near the free

edge at x/a=0.9 and y/b=0.4. The normal stresses in Figures 5(a) and 5(b) increase as higher active

voltages are applied reflecting the increase in piezoelectric strains, which indicates that one potential

tradeofffor minimizing thermal distortions would be increased normal stresses. Thus, the stress state

in thermopiezoelectric laminates (and particularly in the actuators) should be among the primary

design considerations. The normal stresses can also be separated into individual components as

shown in Figures 6(a) and 6(b) for the 70 V case (the shear stresses contain only an elastic component

in the current study). The three components correspond to the three terms found on the right hand

side of Eq. (5) and represent the individual stresses which are induced from the elastic, piezoelectric,

9
and thermal effects. This shows how the individual stress components are affecting the overall stress

state of the plate.

Cantilevered [45_/-451] Graphite/Epoxy Plate with Discrete Piezoceramic Patches

A numerical study was also conducted on a [453/-453] antisymmetric graphite/epoxy laminate with

attached piczzw, eranfic patches. The plate has the same geometry and material properties used in the

previous section and is clamped along the x=0 edge, while the other three edges remain free. A

thermal gradient (15°C on the top surface and - 15°C on the bottom surface) is applied to the plate

with 0 Volts applied on both surfaces of the piezoceramic patches. The objective of this study is to

minimize, both individually and in combination, the thermally induced bending and twisting

deformations through application of active voltages.

Active Thermal Distortion Control. Application of a 30°C thermal gradient, with all piezoceramic

patches grounded, results in the combined bending and twisting deformation shown in Figure 7(a).

Through application of active voltages of varying polarities in the piezoceramic patches, the bending

and twisting deformations can be inhibited either individually or in combination. As demonstrated

in the previous problem, the bending deformation can be managed through application of voltages

which have the opposite polarity. By applying 40 V to both the top and bottom layers of piezoceramic patches, the bending deformation can be eliminated as shown in Figure 7(b). The twisting

deformation remains unaffected and is now more apparent due to the removal of thermal bending.

In a similar manner, the twisting deformation of the plate can be eliminated, without affecting the

bending behavior as shown in Figure 7(c). This requires the application of voltages with the same

polarities, specifically 145 V is applied to the top layer of piezoceramic patches, while -145 V is

applied to the bottom layer ofpiezoceramic patches. Finally, in order to inhibit both the bending and

twisting deformations simultaneously, the previous voltages are superposed. Thus, by applying 185

V on the upper and -105 V on the lower piezoceramic patches, all deformations in the plate can be

compensated, as shown in Figure 7(d). These results demonstrate the potential of piezoelectric

structures to selectively manage desired thermal distortions.

Sensory Voltages. The sensory capabilities of the current formulation are demonstrated on a

combined active/sensory configuration of the [453/-453] graphite/epoxy plate. The upper layer of

piezoceramic patches are used as sensors, while the bottom layer of patches are maintained as

actuators. Figure 8 shows the voltages that develop on the upper surfae, e sensors upon combined

application of a thermal gradient and 0 Volts on the lower piezoelectric actuators. These sensory

voltages will provide the electric feedback required to monitor the state of deformation in the plate.

Local Stresses. This section presents the stress fields for the active [453/-45s] graphite/epoxy plate

problem. As before, a continuous piezoceramic layer is used instead of many distributed piezoceramic patches to eliminate the need for a highly refined mesh in this study. Figures 9(a)-9(c) show

the variation of normal stresses (o_, oyy) and in-plane shear stress (O_y) through-the-thickness at

x/a=0.5 and y/b=0.4 with the various applied voltages, while Figures 9(d)-9(e) show the variation of

10
intedaminar shear stresses (o,_ or, ) through-the-thickness near the flee edge at x/a==0.9 and y/b---0.4.

The normal stresses in Figures 9(a) and 9(b) indicate that similar high stress states arise from

minimizing the twisting of the plate individually (145 V on top and -145 V on bottom layer) or

simultaneously with bending 085 V on top and -105 V on bottom layer). In contrast, minimizing the

bending behavior independently (40 V on both top and bottom layers) has little effect on the normal

stresses. It is recalled that the twisting was compensated by taking advantage of the extensiontwisting coupling in this laminate. The in-plane shear stress shown in Figure 9(c) indicates that

regardless of the applied active voltage, high shear stresses will occur in the plate. The normal

stresses can also be separated into the individual components as shown in Figures lO(a) and lO(b)

for the case of combined active elimination of thermal twisting and bending (185 V applied on the

upper piezoceramic patches and -105 V applied on the lower piezoceramic patches). The three

components correspond to the three terms found on the right hand side of Eq. (5) and show the effect

of the individual stresses on the overall stress state of the plate.

SUMMARY

Layerwise laminate and structural mechanics were described to model the coupled mechanical,

electrical, and thermal behavior of smart piezoelectric composite laminates and plate structures.

A finite element formulation and a bilinear 4-noded plate element were developed and encoded

into prototype sottware. Thus, a necessary analytical and computational capability was developed

to analyze the global and local response of smart piezoceramic plate structures operating in

thermal environments.

The accuracy of the formulation was compared with previously published analytical results.

Numerical studies were performed on symmetric [0/+45], and antisymmetric [453/-453] composite

plates subject to thermal gradient with discrete piezoceramic patches attached on both surfaces,

which demonstrated the capability to actively manage thermally induced bending and twisting

deformations. The corresponding electric voltages developed at distributed piezoelectric sensors

were also calculated. Finally, the severe stress fields through-the-thickness of the composite

plates were predicted and the contributions of the thermal and piezoelectric components

quantified. The numerical studies have indicated the significance of thermal effects on the

performance of piezoelectric structures in extreme temperature environments and demonstrated

the capabilities of the mechanics to accurately model such behavior.

11
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Dynamics, and Materials Conference, New Orleans, LA, 1628-1636, 1995.

27 Ha, S.K., Keilers, C., and Chang, F.-K., "Finite Element Analysis of Composite Structures

Containing Distributed Piezoceramic Sensors and Actuators," AJAA J., Vol. 30, No. 3, 772-780,

1992.

28 Tzou, H.S. and Ye, 1L, "Piezothermoelasticity and Precision Control of Piezoelvctfic Systems:

Theory and Finite Element Analysis," J'. Vib. Acoust., Vol. 116, 489-495, 1994.

29 Tzou, H.S. and Ye, R., 'Tyroeleotric and Thermal Strain Effects of Piezoelectric (PVDF and PZT)

Devices," Proceedings of Adaptive Structures and Composite Materials: Analysis and Application,

ASME, AD-Vol. 45, 125-132, 1994.

30 Lee, H.-J. and Saravanos, D.A_, "On The Response of Smart Piezoelectric Composite Structures

in Thermal Environments," Proceedings of the 36th Structures, Structural Dynamics, and Materials

Conference, New Orleans, LA, pp. 2876-2885, 1995.

31 Lee, H.-J. and Saravanos, D.A., "A Coupled Layerwise Analysis oft_he Thermopiezoelectric

Response of Smart Composite Beams," NASA TechnicalMemoranclum ]06889, 1995.

32 Lee, H.-J. and Saravanos, D.A., "Coupled Layerwise Analysis of Thermopiezoelectric Composite

Beams," A/A/I ,]., in press.

33 Nye, J.F., Physical Properties of Crystals, The Clarendon Press, Oxford, 1964.

14
TABLE I: Material propertiesof piezoceramic(PZT G1195N)and graphite/epoxy

(T300/976)composite2_

Piezoceramic Graphite/Epoxy

ElasticModuli (GPa):

Ett 63.0 150.0

F__ 63.0 9.0

233 63.0 9.0

Poisson's Ratio:

Ul2 0.3 0.3

u23 0.3 0.3

u3t 0.3 0.3

Shear Moduli (GPa):

G12 24.2 7.10

G23 24.2 2.50

G31 24.2 7.10

Density (kg/m3):

p 7600. 1600.

Piezoelectric Charge Constant (pm/V) :

d14 0. 0.

dis 0. 0.

d u 0. 0.

d25 0. 0.

d3t 254. 0.

d32 254. 0.

d36 0. 0.

Electric Permittivity (nffm):

e n 15.3 0.

e22 15.3 0.

e33 15.0 0.

Thermal Expansion Coefficient (_tm/m °C):

a n 0.9

ct22 0.9

1.1

25.2

Reference Temperature, T_, (*C): 20. 20.

15
Appendix A:

Thermopiezoelectric Constitutive Equations for a Constant Through-the-Thickness

Displacement Plate

The simplified constitutive equations for a constant through-the-thickness displacement plate are

_n t!2 0 0 sl6

_!2 %2 0 0 su

0 0 s44 S4s 0

0 0 S4s Sss 0

r16 s_ 0 0 a_

11

21

, 4(+

sl

61

g,

s,

s4=

%

%

o o 4(

o o d32

d)4 44 0

d,_ d_ 0

o o d_

"¢sl =2

._ .r.,_,. ! ) '0

J

,I

(A-l)

= 0 0 d24 d25

Id32 0 0

e

Cl I

c,_+l o _ o i%,_+

,/_ jcsl [o o e. I%1

lC sJ

P,I

P2( 0

Pal

(A-2)

or in semi-inverted form

'11

721

34_" =

Js]

C n cl2 0 0 c16

c n cm 0 0 czs

0 0 c44 c4s 0

0 0 045 c55 0

c16 c_ 0 0 c_

rt.f_

L,1

!

"2

i rl

j°s

0 0 ¢31

0 0 e32

e14 _24 0

elS e2s 0

0 0 e_

IE, I

;E.} -

I%1

0 0

0

0

(A-3)

r

D

'D

LD

[l ) 0 el4 el5

= [_0 0 e24 e25 3x es2 0 0

,SL

I _'22 '_ E2,_ . 4 P2( 0

0 0 %3 E'*I P,I

(A-4)

16
Appendix B:

Generalized Laminate Matrices for a Constant Through-the-Thickness Displacement Plate

IfL is the number of composite plies and piezoelectric layers, the laminate density matrix is

L

p,2"= E f p ¢<=)*'_=)a=

l-I ,

(B-l)

for ij = 11 and 22, while

L

p,,=E fp_

1ol :

(B-2)

The laminate stiffness matrices are

L

f c_j tk(z) t'(z) dz

z

for ij = 11, 12, 16, 22, 26, 44, and 66;

i. ao/k(z) dz n:ffiEfc_

t-t : @z

(B-3)

(B-4)

L

_=Efc_a=

1-1 z

(B-S)

for ij = 44, 45, and 55.

_. L f axp_(z) at'(z)

D# = _ j C¢ dz

t-, , az az

The laminate piezoelectric matrices are

r.

E?= E f "_**_=)a*'c=)d=

t-I , az

011-6)

(B-7)

forij = 31, 32, and 36.

e,y" = _" f % a*'(z),'(z) az

1=1 : _Z

The laminate dielectric permittivity matrix is

(B-S)

17
forij= 11 and 22, while

L

1=1 #

L

1-1 •

0tk(z) 0*=(z) dz

03-9)

(B-IO)

The laminate thermal force vector is

L

ft =

l=l z

for i -- 1 and 2. The laminate thermal electric displacement vector is

L

k

q,._= E f P,*jej**(z) dz

1-1 s

(B-11)

(B-12)

for i =1 and 2, while

L

t f 0it(z)

t-1 , az

t_-13)

18
Appendix C:

Finite Element Submatrices for a Constant Through-the-Thickness Displacement Plate

The mass submatrices are

(ol)

The stiffness submatrices are

/" ORp OR* OR" OR q OR" ORq

[Kll ]Pq { [All] + [A_] ( + )

J

ox ox o= oy oy a=

aR p ORq

+ [Dss] gr R q + [A_] } dA

ay ay

(c-2)

f OR p ORq OR p ORq aR p ORq

[K12 ]/',= { [AI2] + [AI_] + [A2s] J

A ax ay ax ax _y ay

_R p ORq

+ [D4s]R rRq + [A_] } dA

ay ax

(o3)

--f { [B4s] R m ORq + [Bss] .R" _aR q } d.,,l

.a Oy

(o4)

[r,,],, = f { [E,,] aR---Z" R, + [E,,] aR---Z"R, }

dA

, ax ay

(c-s)

/" ORr OR q ORr ORq

[K2z]r_ -- j { [A2zl + [A2sl ( +

ay _y or ay

aR J' ORq

+ [D,t,t] RrR q + [A_] 0x 0x } dA

ORP ORq

Oy Ox (o6)

19
[r_]"= f ([B.]R'--+[B4s]R'--} c_Rq aR_ (c-7)

/. 0R___p aR___p

(c-s)

[x3,],, -- + [c,,] (

+ [c,--_]aR" a_, } ,_

ax ax

aR p aR q

+- )

(c-9)

[x:,,],, -- Jr{ [On] OR p ORq + [O221 ORP OR q + [O33]R PR q } dA (c4o)

The external forces are

[F,]" = f t, R fix, y)

,4

(C-11)

/- (Q],--j ¢ R,(x,y)dA (c42)

The thermal forces are

.,,I

(C-13)

f 0R V(x,y) (F_r ffij s_ d_

,, as

(C-14)

f _P [Q'Y= (q_" _x

a,

_P

+ q_'_ ay

+ q_R _' } dA (C-15)

20
,I

u

m

_4

< 37.2 ©m >

Figure 1" Model of Graphite/Epoxy Plate with Attached Piezoceramic Patches

(a) Top View

0.13 mm Piezoeeramie

\ Graphite/Epoxy

Figure 1: Model of Graphite/Epoxy Plate with Attached Piezoceramic Patches

(b) Front View

21
I

1.00

0.60

0.20

-0.20

O.O0

0.20

0.40

0.60

0.80

Figure 2: Active Compensation of Thermal Deformation of [0/±45], Plate Subject to

100 °C Gradient (a): 0 Volts Applied on Upper and Lower Piezoceramic Patches

22
1.8e

1.4o

1.OO

0.60

1.21

-0.20

,,3t..]a 0.8O 1.011

7_

/_

i_ O.OO

_O.2O

\ _.- o.4o

 o.60

/-- o so

t.OO

Figure 2: Compensation of Thermal Deformation of [0/±45]. Plate Subject to 100°C

Gradient (b): 34 Volts Applied on Upper and Lower Piezoceramic Patches

23
1,6e -

1.20

0.80

0.40

,.oo "Ix,,"_ _"-W-._/ / _ 7-o.oo

1_ _ _o.2o

L/''./..../"'d-....L_/ / / /J./__ 7-o.,o

0.40 _ 0.SO

It/. 1.00 1.0tl

Figure 2: Compensation of Thermal Deformation of [0/_=45]sPlate Subject to 100 °C

Gradient (c): 70 Volts Applied on Upper and Lower Piezoceramic Patches

24
M

ae

1.80

1.48

l.t0

0.60

0.20

-0.20 I I I I

0.O0 0.20 0.40 0.60 0.80 1.00

Ha etal.:OV

Current: I V

Ha et al.: 31 V

Current: 34 V

Ha et al.: 61 V

Current: 70 V

x/a

Figure 3: Centerline (y/b=0.5) Deflections of [0/+45]s Plate

25
6O

Figure 4: Sensory Voltages on Upper Piezoceramic Patches of a [0/+45],Plate

Subject to 100°C Gradient (a): 0 Volts Applied on Lower Piezoceramic Patches

26
75-

60

30

e 15

I_ / /

I

!

/

7

/

/i

Figure 4: Sensory Voltages on Upper Piezoceramic Patches of a [0/+45]_Plate

Subject to 100°C Gradient (b): 34 Volts Applied on Lower Piezoceramic Patches

27
75"

6O

_45

r.

o 15

O

• " I I

J

i

I

i

lr-,It_

Figure 4: Sensory Voltages on Upper Piezoceramic Patches of a [0/±45],Plate

Subject to 100°C Gradient (c): 70 Volts Applied on Lower Piezoceramic Patches

28
V

v-4

M

N

m

N

t_

].20

1.76

11.32

-e.12

meow

0 56 -

-- • ..o0.. °

-1.00

-0.50

o

.__ ° .i../m ,m.I

- I

+,...._ +'-*,.+

I

I.---"" I ........ +

°++,+.+++

+

I I I I

-O.30 -O.lO O.)O O.30 O.50

J

i i Jl

,.,,,,,

Iv

34 V

70 V

z/h

Figure 5: Through-The-Thickness Stresses in [0/±45]s Plate Subject to 100°C

Gradient with Different Applied Voltages on Upper and Lower Piezoceramic

Patches (a): o_

29
O,60

•,-+ 0.20

h

r,r,1

_ -o.2o

-0.60

J

-,+,++,

***++..

Ira' _m' _ 'll'll i'ulm 'I_ +If+"

-1.go I I I I

4.50 -0.30 -0.10 O.lO 0.30 0.50

-- IV

.... 34 V

....... ?IV

z/h

Figure 5: Through-The-Thickness Stresses in [0/±45], Plate Subject to 100 °C

Gradient with Different Applied Voltages on Upper and Lower Piezoceramic

Patches (b): oyy

30
1.00

O.60

0.20

r_

h

N -0.20

-0.60

I I

.1.00 i I I

-0.50 -0.30 -O.lO 0.10 0.30 0.50

i J ii

.....a.

OV

34V

7O V

z/h

Figure 5: Through-The-Thickness Stresses in [0/+45], Plate Subject to 100°C

Gradient with Different Applied Voltages on Upper and Lower Piezoceramic

Patches (c): o_y

31
1.O0

'I"1

el

M

o.6o I- .A'A-A--A--A A_ It>---'--_ /- ,

o.,,_..--.-,_- /..._\ I

-0.20

-0.60 -

-l.O0

-O.$O

I I I I

-0.30 -O.lO O.iO O.30 0.50

_ii_ _lV

,-m,. 7,v

z/h

Figure 5: Through-The-Thickness Stresses in [0/_:45], Plate Subject to 10O°C

Gradient with Different Applied Voltages on Upper and Lower Piezoceramic

Patches (d): o_

32
r_

4.00

2.80

'" IT'"_ I-.-

"+'_ "'_d_+,,,+...,,.._. e_. |.. "

-2.00 I l l l l l

-1,51 -0.30 -1.11 1.11 1,31 0.50

O V

_V

70 V

z/h

Figure 5: Through-The-Thickness Stresses in [0/+45], Plate Subject to 100*C

Gradient with Different Applied Voltages on Upper and Lower Piezoceramic

Patches (e): Oy_

33
1.20

e.76

,'_ 1,,.+2

14

M

M

" -0.12

-0.$6

,+..

I I I I

ll_lUnm

,,+i,l.

-1.00

-0.50 4.30 -0.10 0.1 O 0.30 0.50

Torsi

mtJtlc

+l'l_rmal

FiI_eLIctI_

z/h

Figure 6: Elastic, Thermal, and Piezoelectric Stress Components in [0/±45], Plate

Subject to 100°C Gradient with 70 Volts Applied on Upper and Lower Surface

Piezoceramic Patches (a): Oxx

34
1.00

0.60

e4_

0.20

.jm,,,,

-o,2o

-0.60

°lil'*'' o.,,.,, !

I

....... ....... : I

-- "..t,*...

J

__ ,._..

I

-1.00 I I I I

-O.SO -0.3O -0.10 0.10 O.30 O.SO

i

i I i I

,,,,,,i

III

Total

mastic

Thermal

_oelectric

z/h

Figure 6: Elastic, Thermal, and Piezoelectric Stress Components in [0/+45]s Plate

Subject to 100°C Gradient with 70 Volts Applied on Upper and Lower Surface

Piezoceramic Patches (b): Oyy

35
1,Qe -

0.67 -

Figure 7: Active Compensation of Thermal Bending and Twisting Deformation of

[453/-453] Plate Subject to 30°C Gradient (a): Initial Thermal Deformation: 0 Volts

Applied on Upper and Lower Piezoceramic Actuators

36
1.II -

0.67

0.33

,__ O.O0

-0.33

-o.,7 I __ __ °'°°°°

_"* _ _ Z-""' -'_*

.... o.4o _ o.so "_"

.r/b 0.so 1.o0 1.o0

Figure 7: Active Compensation of Thermal Bending and Twisting Deformation of

[453/-453] Plate Subject to 30°C Gradient Co): Active Compensation of Thermal

Bending: 40 Volts Applied on Upper and Lower Piezoceramic Actuators

37
1.00 --

0.67 -

0.33 _

....

-0.33

-o._7_ /._/____I-7-o.oo

I _ _ __o.2o

_ _ 7- o_o

Y/Z) o.so ].o0 x.o0

Figure 7: Active Compensation of Thermal Bending and Twisting Deformation of

[453/--453] Plate Subject to 30°C Gradient (c): Active Compensation of Thermal

Twisting: 145 Volts Applied on Upper and -145 Volts on Lower Piezoceramic

Actuators

38
1.00 -

0.67

O.33

Figure 7: Active Compensation of Thermal Bending and Twisting Deformation of

[453/-453] Plate Subject to 30°C Gradient (d): Complete Compensation of Thermal

Deformation: 185 Volts Applied on Upper and -105 Volts on Lower Piezoceramic

Actuators

39
7S _

60

i

_7

/

!

Figure 8: Sensory Voltages on Upper Piezoceramic Patches of a [453/--453] Plate

with 0 Volts Applied on Lower Piezoceramic Patches

4O
m

m

N

2.11

1.20

-1.20

;....... .......-+.-,+.:._._.._.__._

" "1 I

! !

i I

-- I I

I I I I

-2.00

-0.5O -0.30 -O.tO 0.10 0.30 O.SO

"--'--- IV

.... 40 Y

....... 14S.t-145 Y

-- - 18S/-115 V

zlh

Figure 9: Through-the-Thickness Stress in [453/-453] Plate Subject to 30°C Gradient

with Different Applied Voltages on Upper and Lower Piezoceramic Patches (a): O_x

41
e4_

h

h

1..gO

0.90 -

0.30 -

-0.30

-o.9o 2J

-1.$0

-0.50

F 4urn' _ _ B

i_ O

I I I I

-0.30 -0.10 O.IO O.30

mm

.n

0.50

OV

_V

14st-14s

,tSl-les

z/h

Figure 9: Through-the-Thickness Stress in [453/--453] Plate Subject to 30°C Gradient

with Different Applied Voltages on Upper and Lower Piezoceramic Patches 03): Oyy

42
r_

N

1,50 O,gO l

-0.90 I

-1.50 i I I I

-0.SO -0.30 -O.lO 0.10 0.30 0.50

z/h

Figure 9: Through-the-Thickness Stress in [453/-453] Plate Subject to 30°C Gradient

with Different Applied Voltages on Upper and Lower Piezoceramic Patches (c): Oxy

43
v,,q

14

3.00 .2t _%_,, 1

1.80 - ,_,..._.

o.6o • ._ " "-"," -.u ._.- \_ ._.-'" "_,Jn /

I'_ .A.,,- -.... -.nn \.-% _"1

I-_ L- o" ".:., .o/... % A,." I

-1.80 I

-3.00

-0.50

I 1 J I

-0.30 -0.10 O.]O 0.30 0.50

OV

4QV

145[..]A 5

_si-le5

z/h

Figure 9: Through-the-Thickness Stress in [453/-453] Plate Subject to 30°C Gradiem

with Different Applied Voltages on Upper and Lower Piezoceramic Patches (d): o,_

44
2.00

-2.00 I I I I

-O.SO -0.30 4.10 O.lO O.3O 0.50

z/h

Figure 9: Through-the-Thickness Stress in [453/'-453] Plate Subject to 30 °C Gradient

with Different Applied Voltages on Upper and Lower Piezoceramic Patches (e): Oy_

45
•-,q 0.60

M

M

= -O.6O

-1.80

m

_ eum luem mere • ,e ,,m

........ ._.--. .... ---0-,,_

-3.00 I I I I

-0.50 -0.30 -0.10 0.1O 0.30 O.50

ilii

,,oJo.,

Tots]

mt, oLic

Tl_mti

l_o_sctrtc

z/h

Figure 10: Elastic, Thermal, and Piezoelectric Stress Components in [453/-453] Plate

Subject to 30°C Gradient with 185 Volts Applied on Upper and -105 Volts Applied

on Lower Piezoceramic Actuators (a): o,_

46
2.00

1.2O

,t,-+ 1.41

Ii,,,+

_ -o.4o

-1.20 J

-- Total

I'" ... .,.+ _uo li,,l it iii_,i,,,

I

....... Thermad

i

i. IqQ2oeloetrie

-2.00 ' I I I I

-O.$O 4.30 -0.10 O.IO O.3O O.$O

z/h

Figure 10: Elastic, Thermal, and Piezoelectric Stress Components in [453/-453] Plate

Subject to 30°C Gradient with 185 Volts Applied on Upper and -105 Volts Applied

on Lower Piezoceramic Actuators Co): Oyy

47
REPORT DOCUMENTATION PAGE I Fonn OMBAoWom# No.o;,z_l_

Put)Ik:mpoflJn9bunion kx tllb colleOIkmof _ Is esUmatKI to avoraoo 1 hour per roq=ome, InctJdlngItm lknt _ _ _, _ _ _ m,

me _ the data medea, and comploanOand mvlming ttm collecaonol In_-ma_on. Send commmm mganllno dye buxkm eeUmam or my omef asfect d this

_¢_,¢_d trdmmaZk_ _ =,ug0==l_ for mdoclng_ I=,umkm.to waadn0t_ I..Ioadqoanw=Swvl_. DVocsm'm for Infm'n'aWt _ and _ 1215 Jelfmon

Dav_Hlghway. SullD 1204. Attlng10n.VA 22202-430_and_theOffk:oolMana0ementand Budget, Pap,onmrk_Pro/pct(0"/04-018_.W, mhlnglm. DC 2060_

1. AGENCY USE ONLY (Leave b/ank) 2. REPORT DATE

July 1996

4. TITLE AND SUBTITLE

Layerwise F'mim Elements for Smart Piezoceramic Composite

Plates in Thermal Enviromem

e.AUTHOR(S)

Ho-Jun Lee and DimitrisA. Saravanos

7. PERFORMINGORGANIZATIONNAMe(S)ANDADDRESS(ES)

National Aeronautics and Space Adminislration

Lewis Research Center

Cleveland, Ohio 44135-3191

9. SPONSORI_NErORING AGENCYNAME(S)ANDADDRESS(ES)

NationalAeronautics and Space Administration

Washington, D.C. 20546-0001

& REPORT TYPE AND DATES COVERED

Technical Memorandum

5. FtR4DING NUMBERS

WU-505--63--5B

6. PERFORMING ORGANIZATION

REPORT NUMBER

E-l(Y290

10. SPO_NG

AGENCY REPORT NUMBER

NASA TM- 106990

AIAA-96-1277

11. SUPPLEMENTARY NOTES

Prepped for the 37th Stmctmes, S_ Dynamics, and Materials Conference and Exh'bit cosponsored by AIAA, ASME, ASCE,

AHS, mad ASC, Salt Lake City, Utah, Aprn 15-17, 1996. Ho-Jun Lee, NASA Lewis Research Center and DimlY'is A. Sarwanos,Ohio

Aerospace Institute, 22800 Cedar Point Road, Cleveland, Ohio 44142 and Resident Research Associate at Lewis Research Center.

Responsible perso_ Ho-Jun Lee, organization code 5210, (216) 433--3316.

1211. msvmmn_Avmumaarv SrATB_srr

Unclassified - Unlimited

Subject Category 24

This publication is available li'mn the NASA Center for AeroSpace Infonv.aficn, (301) 621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maxknum 200 tmrds)

Analytical formulations are presented which account for the coupled mechanical, electrical, and thermal response of

piezoelectriccomposite laminatesand plate muctures. A layerwisetheoryisformulatedwith theinl_rent capability to

explicitly model the active and sensory response of piezo¢leclric composite plates having arbime-y laminate configurations

in thermal environments. Finite element equations are derived and implemented for a bi]inear 4-noded plate element.

Application cases demonswate the capability to manage thennaUy induced bending and twisting deformations in symmetric and antisymmctriccompositeplateswithpiezoelectricactua_s, and show thecorrespondingelectricalresponseof

dislxibuted piezoelectric sensors. Finally, the resultant stresses in the thermal piezoelectric composite laminates are

investigated.

14. SUBJECT TERMS

Actuators; Composite materials; Finite element method; Laminates;

Piezoelectric ceramics; Piezoelectricity; Sensors; Smart snucmres

17. SECURITY CLASSiFICATiON

OF REPORT

Unclassified

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18. SECURITY CLASSIFICATION

OF THIS PAGE

Unclassified

19. SECURITY CLASSIFICATION

OF ABSTRACT

Unclassified

15. NUMBER OF PAGES

49

16. PRICE CODE

A03

20. UMITATION OF ABSTRACT

Standard Form 298 (Rev. 2-89)

Prescrbed by ANSI Sld. Z39-18

296-102

O _

_ CO. _

• _

o

z _0

_J

cD

t3