NASA/TM--2001-210892

Finite Element Analysis of Active and Sensory

Thermopiezoelectric Composite Materials

Ho-Jun Lee

Glenn Research Center, Cleveland, Ohio

National Aeronautics and

Space Administration

Glenn Research Center

May 2001

https://ntrs.nasa.gov/search.jsp?R=20010069273 2018-05-14T14:47:17+00:00Z
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CONTENTS

LIST OF ILLUSTRATIONS ..................................................... v

LIST OF TABLES ........................................................... vii

CHAPTER

1.1

1.2

1.3

1. INTRODUCTION ................................................. 1

Overview of Smart Structures .......................................... 1

Piezoelectric Materials ................................................ 1

Objectives of This Research ........................................... 2

CHAPTER

2.1

2.2

2. ANALYTICAL MODELING OF PIEZOELECTRIC MATERIALS ......... 3

Introduction ........................................................ 3

Induced Strain Models ................................................ 3

2.2.1 Actuator Models ............................................. 4

2.2.2 Actuator and Sensor Models ................................... 6

2.3 Coupled Electromechanical Models ..................................... 7

2.3.1 Analytical Models ........................................... 7

2.3.2 Finite Element Models ........................................ 7

2.4 Coupled Thermoelectromechanical Models ............................... 9

2.4.1 Analytical Models ........................................... 9

2.4.2 Finite Element Models ....................................... 10

2.5 Limitations of Existing Analytical Models ............................... 11

CHAPTER 3.

3.1

3.2

3.3

THEORETICAL DEVELOPMENT ................................. 13

Current Formulation Assumptions ...................................... 13

Governing Material Equations ......................................... 13

3.2.1 Strain and Electric Field in Cartesian Coordinates ................. 14

3.2.2 Strain and Electric Field in Curvilinear Coordinates ................ 14

Thermopiezoelectric Constitutive Equations .............................. 17

3.3.1 Piezoelectric Beam .......................................... 18

3.3.2 Piezoelectric Plate and Shell .................................. 18

CHAPTER 4. LAMINATE MECHANICS ........................................ 21

4.1 Layerwise Laminate Theory ........................................... 21

4.1.1 Beam Laminate Theory ...................................... 21

4.1.2 Plate Laminate Theory ....................................... 22

4.1.3 Shell Laminate Theory ....................................... 23

4.2 Generalized Laminate Matrices ........................................ 24

4.2.1 Beam Laminate Matrices ..................................... 24

4.2.2 Plate Laminate Matrices ...................................... 27

4.2.3 Shell Laminate Matrices ..................................... 29

NASA/TM--2001-210892 iii
CHAPTER5. FINITE ELEMENTFORMULATION ............................... 33

5.1 BeamElement ..................................................... 34

5.2 PlateElement ...................................................... 36

5.3 ShellElement...................................................... 38

CHAPTER6. CASESTUDIESAND DISCUSSIONOF RESULTS .................... 47

6.1 Comparisonswith RoomTemperatureExperimentalResults................. 47

6.1.1 PiezoelectricBimorphBeam ................................. 47

6.1.2 CompositePlatewith PiezoelectricPatches ...................... 47

6.2 Comparisonswith AnalyticalResultsfor TemperatureEffects................ 54

6.2.1 PiezoelectricBimorphBeam .................................. 54

6.2.2 CompositePlatewith PiezoelectricPatches ...................... 54

6.3 Comparisonswith aCommercialFiniteElementAnalysisProgram............ 60

6.3.1 SensoryMode ............................................. 60

6.3.2 ActiveMode ............................................... 60

6.4 Additional NumericalStudies ......................................... 66

6.4.1

6.4.2

6.4.3

6.4.4

6.4.5

PyroelectricEffects ......................................... 66

TemperatureDependentMaterialProperties...................... 70

SelectiveShapeControlof BendingandTwisting ................. 78

ThermalStresses........................................... 83

Influenceof CurvatureonActive andSensoryResponse............ 89

CHAPTER7. CONCLUSIONSAND SUGGESTIONSFORFUTUREWORK .......... 101

7.1 SuggestedPlanof FutureExperimentalStudies .......................... 101

REFERENCES............................................................. 111

NASA/TM--2001-210892 iv
ILLUSTRATIONS

°

2.

3.

4.

,

6.

7.

8.

,

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

Curvilinear piezoelectric laminate and coordinate system ........................ 15

Piezoelectric bimorph beam (Lee and Moon, 1989) ............................ 48

Comparisons of deflections of a PVDF bimorph beam .......................... 49

Cantilevered composite plate with discrete piezoelectric patches

(Crawley and Lazarus, 1991) .............................................. 50

Comparisons of deflections of a [0/_+45]+ graphite/epoxy plate .................... 52

Comparisons of deflections of a [30/02] s graphite/epoxy plate .................... 53

Piezoelectric bimorph beam (Sunar and Rao, 1997) ............................ 55

Comparisons of thermal effects on the deflections of a

PVDF bimorph beam .................................................... 56

Simply supported composite plate with discrete piezoelectric patches

(Ha et al., 1992) ........................................................ 57

Comparisons of applied active voltage effects on the centerline

deflection of a [0/_+45]+ graphite/epoxy plate .................................. 59

Clamped composite plate with a piezoelectric layer ............................ 61

Comparison of centerline deflections of a clamped [08/p] plate ................... 63

Comparison of sensory electric potentials of a clamped [0s/p] plate ................ 64

Comparison of applied active voltage effects on the centerline

deflection of a clamped [08/p] plate ......................................... 65

Pyroelectric effects on the centerline deflections of a clamped [08/p] plate .......... 68

Pyroelectric effects on the sensory electric potentials of a

clamped [08/p] plate ..................................................... 69

Temperature dependence of AS4-3501-6 properties ............................ 71

Temperature dependence of PZT-5A properties d3t and _33 ....................... 72

Temperature dependence of PZT-5A properties at l, and P3 ...................... 73

Influence of temperature dependence on deflections at the center of a

[0/_+45]+ carbon/epoxy plate for different applied thermal gradients ................ 75

Influence of temperature dependence on sensory voltages at the center of a [0/_+45]+

carbon/epoxy plate for different applied thermal gradients ....................... 76

Influence of temperature dependence on applied voltages to minimize

the center deflection of a [0/_+45]+ carbon/epoxy plate for different

applied thermal gradients ................................................. 77

Initial bending and twisting deformation of a [453/-453] plate ..................... 79

Selective shape control of the bending deformation of a [453/-453] plate ............ 80

Selective shape control of the twisting deformation of a [453/-453] plate ............ 81

Shape control of bending and twisting deformations of a [453/-453] plate ........... 82

Cantilevered [0s/p] beam ................................................. 84

Comparison of normal stresses at the middle of a [08/p] beam .................... 86

Comparison of normal stresses at the free end of a [08/p] beam ................... 87

Comparison of shear stresses at the free end of a [08/p] beam ..................... 88

Geometry of a [09/p] circular cylinder with an attached piezoelectric layer .......... 90

Different types of applied thermal loads ..................................... 91

NASA/TM--2001-210892 v
33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

Influence of pyroelectric effects on the displacements of a [09/p] circular

cylinder under a uniform thermal load ....................................... 92

Influence of pyroelectric effects on the sensory electric potential of a

[09/p] circular cylinder under a uniform thermal load ........................... 93

Influence of pyroelectric effects on the displacements of a [09/p] circular

cylinder under a cosine thermal load ........................................ 94

Influence of pyroelectric effects on the sensory electric potential of a

[09/p] circular cylinder under a cosine thermal load ............................ 95

Influence of pyroelectric effects on the displacements of a [09/p] circular

cylinder under a double cosine thermal load .................................. 96

Influence of pyroelectric effects on the sensory electric potential of a

[09/p] circular cylinder under a double cosine thermal load ...................... 97

Displacements of a [0_/p] circular cylinder under a uniform thermal load ........... 98

Displacements of a [09/p] circular cylinder under a cosine thermal load ............. 99

Displacements of a [09/p] circular cylinder under a double

cosine thermal load .................................................... 100

Dynamic experimental setup ............................................. 103

High temperature test configuration of a cantilevered beam with

one piezoelectric patch .................................................. 104

High temperature test configuration of a cantilevered beam with

two piezoelectric patches ................................................ 106

Test configuration of a flat plate with attached piezoelectric patches .............. 108

Test configuration of a curved plate with attached

piezoelectric patches ................................................... 109

NASA/TM--2001-210892 vi
TABLES

°

,-)

_°

3.

4.

5.

6.

7.

8.

9.

10.

Material properties of PVDF piezoelectric polymer

(Lee and Moon, 1989) ................................................... 49

Material properties of piezoceramic (G-1195) and

Graphite/Epoxy (AS4/3501) (Crawley and Lazarus, 1991 ) ...................... 51

Material properties of PVDF piezoelectric polymer

(Sunar and Rao, 1997) ................................................... 56

Material properties of piezoceramic (G- 1195) and

Graphite/Epoxy (AS4/3501) (Ha et al., 1992) ................................ 58

Material properties of piezoceramic and Carbon/Epoxy ......................... 62

Material properties of piezoceramic and Carbon/Epoxy ......................... 67

Material properties of piezoceramic (PZT-5A) and

carbon/epoxy (AS4/3501-6) composite ...................................... 74

Material properties of piezoelectric and Graphite/Epoxy ........................ 85

Series of tests for a cantilevered beam with one piezoelectric patch ............... 105

Series of tests for a cantilevered beam with two piezoelectric patches ............. 107

NASA/TM--2001-210892 vii

CHAPTER 1

INTRODUCTION

Smart composite structures offer the capability to combine the low density, superior

mechanical and thermal properties of composite materials along with the inherent capabilities of

smart materials to sense and adapt to their environments. Thus, the use of smart structures (also

referred to as intelligent or adaptive structures) offers the potential to significantly improve the

performance of aerospace structural components. However, before these materials can be

implemented into actual structures, the coupled mechanical, electrical, and thermal behavior of smart

materials must be fully characterized. This has led to extensive research since the 1980's to assess

both the sensory and active responses of smart materials.

1.1 Overview of Smart Structures

Smart structures are distinguished from conventional structures by the presence of integrated

actuator and sensor elements. In a typical smart structure application, the sensors are used to monitor

the mechanical response of the structure through changes in the displacements, strains, or

accelerations. Once an adverse or undesirable structural response is detected in the sensors, a

controller generates the required input to the actuators. The actuators respond to this input and

produce a corresponding change in the mechanical response of the structure to a more benign or

acceptable state. The capability of smart structures to sense and adapt to their environment leads to

a wide range of potential applications including: vibration suppression of aircraft structures; noise

contr31 of helicopter rotors; health monitoring of bridges; shape control of large space trusses;

aeroelastic control of aircraft lifting components; and seismic control of buildings. Crawley (1993)

and Loewy (1997) provide detailed overviews of the current state of smart structures research for

aerospace applications.

A variety of different materials can be utilized as either sensor or actuator elements in smart

structure applications. Depending on the specific material used, the sensor and actuator elements are

controlled through either electric, magnetic, thermal, or light energy. Some of the common actuator

and sensor materials include: piezoelectric materials, shape memory alloys, fiber optics,

electrostrictive materials, magnetostrictive materials, and electro-rheological fluids. Of the different

materials available for use in smart structures, only piezoelectric materials have the unique capability

to be used effectively as both actuator and sensor elements. Other advantages of piezoelectric

materials which help account for their widespread popularity include: simple integration into the

structure; a readily obtainable commercial supply of piezopolymers and piezoceramics; and

familiarity in using these materials gained from previous applications in transducers.

1.2 Piezoelectric Materials

Historically, piezoelectric materials have been utilized mainly as active structural elements

in transducers for application in strain gages, accelerometers, and sonar. Recently, the focus of

research has shifted away from the transducer applications toward the development of smart structure

applications which combine the active and sensory behavior of piezoelectric materials.

NASA/TM--2001-210892 1
Thebasiccharacteristicsof piezoelectricmaterialswhichallow fortheiruseassensorsand

actuatorsarethedirectpiezoelectriceffect,conversepiezoelectriceffect,andthepyroelectriceffect.

In thedirectpiezoelectriceffect,theapplicationof a mechanicalloadon thepiezoelectricmaterial

inducesanelectricalresponse.Throughmeasurementof thiselectricalresponse,themechanicalstate

of deformationinthestructurecanbedeterminedandmonitored,leadingto thesensoryapplication.

In contrast,the conversepiezoelectriceffect transformsanelectrical input in the piezoelectric

materialintoacorrespondingmechanicalstrain.Thisleadstotheactiveapplicationsof piezoelectric

materials,in whichthestateof deformationof thestructurecanbecontrolledor alteredby applying

theappropriateelectricalinput.Thethirdcharacteristicbehavioristhepyroelectriceffect,in which

thepiezoelectricmaterialrespondsto changesin temperatureby producinganelectricalresponse,

which will influenceboththedirect andconversepiezoelectriceffectsin changingtemperature

environments.

1.3 Objectives of This Research

The renewed interest in the application of piezoelectric materials in smart structures has lead

to extensive efforts to fully characterize the sensory and active behavior of these materials. Due to

the complex coupled mechanical, electrical, and thermal behavior of piezoelectric materials, new

analytical formulations and mechanics are required to accurately analyze the response of smart

structures which contain piezoelectric elements. Thus, the purpose of this research is to develop

comprehensive mechanics for accurately predicting the sensory and active behavior of composite

laminates which contain piezoelectric elements. Special emphasis is placed on capturing the coupled

response at the material level through the thermopiezoelectric constitutive equations and to introduce

the displacements, electric potentials, and temperature as state variables in the analysis. This unified

representation of the coupled response leads to the inherent capability to model both the sensory and

active responses of piezoelectric composite laminates.

The mechanics incorporate a layerwise laminate theory for more accurate analysis of

displacements, strains, stresses, electric fields, and thermal fields, especially for thick laminates and

laminates which contain strong inhomogeneties through-the-thickness. Thermal effects which arise

from coefficient of thermal expansion mismatch, pyroelectric effects of the piezoelectric materials,

and temperature dependent material properties are explicitly accounted for in the formulation.

Corresponding finite element formulations are developed for piezoelectric beam, plate, and shell

elements to provide a more generalized capability for the static and dynamic analysis of piezoelectric

composite structures under arbitrary thermoelectromechanical loads and boundary conditions. The

accuracy of the current formulation is verified by comparison with published experimental data and

results from other analytical models. Additional numerical studies are also conducted to demonstrate

additional capabilities of the formulation to represent the sensory and active behaviors. A future plan

of experimental studies is also provided to characterize the high temperature dynamic response of

piezoelectric composite materials.

NASA/TM--2001-210892 2
CHAPTER2

ANALYTICAL MODELING OFPIEZOELECTRICMATERIALS

2.1 Introduction

The initial use of piezoelectric materials dates back to 1880, when the Curie brothers first

discovered the direct piezoelectric effect. Until recently, the use of piezoelectric materials has been

limited to a variety of transducer applications. The widespread use of piezoelectric materials as

distributed actuators began only in the 1980's, when advances in design and manufacturing

technologies made these applications feasible. The experimental work of Bailey and Hubbard (1985)

is usually cited as the first application of piezoelectric materials as actuators for vibration control.

Using a piezoelectric polymer film as the active element on a cantilevered beam, they were able to

demonstrate active damping of the first vibrational mode. This new actuator application has led to

renewed interest in the development of piezoelectric materials for advanced aerospace structures.

As research into characterizing the active and sensory behavior of piezoelectric materials

progressed, a variety of different analytical models were developed. These models can be classified

into three broad categories as induced strain models, coupled electromechanical models, and coupled

thermoelectromechanical models. The induced strain models use approximate theories to incorporate

the piezoelectric effects and are generally limited to predicting only the active response of

piezoelectric materials since the electric potential is neglected as a state variable in the formulation.

The coupled electromechanical models provide a more consistent representation of both the sensory

and active responses of piezoelectric materials by incorporating both the displacements and electric

potential as state variables in the formulation. Typically, these models are implemented as finite

element codes to provide a more general analysis tool and a wide variety of different beam, plate,

shell, and solid elements have been developed. A natural extension of the coupled electromechanical

models is to also incorporate thermal effects. These coupled thermoelectromechanical models

include temperature as an additional state variable to account for thermal effects in addition to the

piezoelectric effects. A more limited number of finite element codes have been developed with this

capability.

2.2 Induced Strain Models

The induced strain models can be separated into two categories: (1) actuator models and (2)

actuator and sensor models. The actuator models are concerned only with analyzing the active

behavior of piezoelectric materials. They typically approximate the strain generated in the

piezoelectric material by an applied electric voltage using statically equivalent forces and moments.

The combined actuator and sensor models were developed to include predictions of the sensory

response of piezoelectric materials. Although these models introduce the piezoelectric constitutive

equations into their formulation, the electric potential is usually not included as a state variable, the

conservation of electric flux is not considered in the equations of motion, and the sensory voltages

are back calculated using the charge equation.

NASA/TM--2001-210892 3
2.2.1ActuatorModels

CrawleyanddeLuis (1987)developedaninducedstrainactuatormodelfor beams.They

formulatedstaticanddynamicanalyticalmodelsbasedonthegoverningequationsfor beamswith

attachedandembeddedpiezoelectricactuatorsto modelextensionandbending.Experimentswere

performedon both isotropic and compositecantileveredbeamswith attachedand embedded

piezoelectricactuatorstovalidatetheirmodels.Thestudyfoundthatsegmentedactuatorsarealways

moreeffective thancontinuousactuatorssincethe outputof eachactuatorcanbe individually

controlled. Theyalsoshowedthatembeddedactuatorsin compositesdegradetheultimatetensile

strength,but havenoeffectontheelasticmodulus. BazandPoh(1988)investigatedmethodsto

optimizethe locationof piezoelectricactuatorson beamsto minimize thevibration amplitudes.

Numericalstudiesdemonstratedthepotentialtocontrol vibrationsin largeflexible structuresusing

asmallnumberof bondedpiezoelectricactuators.Im andAtluri (1989)presentedamorecomplete

beammodelwhich accountedfor transverseandaxial deformationsin addition to extensionand

bending.Governingequationswereformulatedfor abeamwith bondedpiezoelectricactuatorsfor

applicationsin dynamicmotioncontrolof largescaleflexible spacestructures.

TzouandGadre(1989)formulatedaninducedstrainpiezoelectricshelltheory. A dynamic

modelwasderivedfrom Love'sshelltheoryfor applicationto multi-layeredthin shellswith active

distributedactuators. A casestudywas validatedwith experimentalresults for the vibration

suppressionof a cantileveredbeamwith a piezopolymeractuatorfilm. CrawleyandAnderson

(1990)developedaBernoulli-Eulermodelto moreaccuratelymodelactuationinducedextension

andbendingin one-dimensionalbeamsthanthemodelof CrawleyanddeLuis (1987). Themodel

neglectedsheareffectsandwasshownto bebestsuitedfortheanalysisof thinbeamsandactuators.

Clarket al.(1991)alsoextendedthemodelof CrawleyanddeLuis (1987)to studytheresponseof

multiplepiezoelectricactuatorsonbeamexcitations.BasedonEulerbeamtheory,theirmodelwas

validatedwithexperimentalresultsforthevibrationresponseof asimplysupportedisotropicbeam.

The modelwas found to bebestsuited for performing initial studiesto determinetheoptimal

locationof actuatorsfor excitingspecificvibrationmodes.

CrawleyandLazarus(1991)developedinducedstrainactuationmodelsfor plates.Equations

ofstrainactuationwerederivedforbothisotropicandanisotropicplates.Exactsolutionswerefound

for simplegeometriesandboundaryconditions,while approximatesolutionswereusedto solve

morecomplexproblems.Themodelswereverifiedwith experimentalresults.Staticanalysisof a

cantileveredcompositeplate with attachedpiezoceramicactuatorswereconductedto showthe

potentialfor shapecontrol of structures.Dimitriadis et al. (1991)extendedtheone-dimensional

inducedstrainbeammodelof CrawleyanddeLuis (1987)to two-dimensionalplateswith bonded

piezoelectricactuators. Dynamic analyseswere performed on simply supportedplates to

demonstratetheuseof actuatorsto exciteselectivemodesandtheinfluenceof actuatorgeometry

onthe modalresponse.RobbinsandReddy(1991)developeda piezoelectriclayerwiselaminate

theorywhichwasimplementedintoabeamelement.Numericalcomparisonswereconductedusing

fourdifferentdisplacementtheories(twoequivalentsinglelayertheoriesandtwolayerwiselaminate

theories)to demonstratetheincreasedaccuracyin displacementandstresspredictionsobtainedfrom

usingthelayerwisetheories.

Wangand Rogers(1991)formulatedan inducedstrainpiezoelectricplate theory. The

formulationwasbasedonclassicalaminatedplatetheoryandusedHeavisidefunctionstorepresent

the distributedpiezoelectricactuators. The model was verified with the analytical work of

Dimitriadis et al. (1991)andadditionalcasestudieswereperformedto demonstratethecapability

of actuatorstoinducebendingandextensioninlaminatedplates.Clarketal.(1993)experimentally

NASA/TM--2001-210892 4
validated the induced strain plate model developed by Dimitriadis et al. (1991 ). Results from the free

vibration analysis demonstrated capabilities to excite selective vibration modes of a simply supported

plate using bonded actuators, as well as the significance of actuator location and excitation frequency

on the structural response. Pai et al. (1993) accounted for geometric nonlinearities (large rotations

and displacements) in piezoelectric composite plates. The developed partial differential equations

characterize the dynamic response of composite plates through elastic and geometric couplings

between the extension, bending, and twisting motions.

Mitchell and Reddy (1995b) formulated a power series solution for axisymmetric composite

cylinders with either attached or embedded piezoelectric laminae. The solution was verified with

finite element analysis. Numerical studies were performed to damp vibrations in truss-type

structures using both an embedded cylindrical truss actuator element and an attached actuator patch.

Linet al. (1996) presented an induced actuation plate finite element based on first order shear

deformation theory. Numerical studies were verified with analytical solutions and demonstrated

capabilities to control the deflection of composite plates using piezoelectric actuators. Park and

Chopra (1996) formulated one-dimensional models to predict the extension, bending, and torsion

behavior of beams under piezoelectric actuation. Comparisons with experimental data showed good

correlation only for applications in which actuators have low orientation angles (less than 45 ° ) with

respect to the beam neutral axis.

Sonti and Jones (1996) developed differential equations of motion for a composite cylindrical

shell with surface bonded piezoelectric elements. Approximate analytical solutions were obtained

for the equivalent forces exerted by the actuator and shell. Chandrashekhara and Varadarajan (1997)

implemented a finite element model for laminated composite beams with integrated piezoelectric

actuators derived from a higher order shear deformation theory. Numerical studies investigated the

effect of stacking sequence and boundary conditions on the actuator voltages and demonstrated the

capability to achieve adaptive shape control of beam structures. Charette et al. (1997) formulated

analytical models based on a variational approach to study plates with piezoelectric actuators. The

model was specialized for a simply supported plate and used to determine the effects of piezoelectric

actuator on the dynamic behavior of the plate. Results of the study were verified with experiments

and showed only slight changes in the mode shapes of the plate.

Chattopadhyay and Seeley (1997) implemented a third-order laminate theory into a finite

element formulation to investigate piezoelectric actuators. Numerical results were verified with

published experimental data. Additional comparisons were conducted to demonstrate the limitations

of classical laminate theory in analyzing through-the-thickness stress and strain variations. Librescu

et al. (1997) presented a model for composite beams with piezoelectric actuators that included

structural tailoring and boundary-moment control. Numerical results demonstrated the potential to

improve the dynamic responses of thin-walled cantilevered structures. Bhattacharya et al. (1998)

developed a finite element formulation based on first order shear deformation theory for beams and

plates. An eight noded isoparametric piezoelectric element was developed to perform free vibration

analysis of laminated composite beams and plates. Numerical studies assessed the impact of stacking

sequence, boundary conditions, and applied electric potentials on the free vibration response.

Oguamanam et al. (1998) formulated a piezoelectric composite beam finite element using

von Karman nonlinear strain-displacement relations. Numerical studies demonstrated the influence

of stress stiffening effects on the natural frequency of slender beams. Tong et al. (1998) developed

a two dimensional thin plate finite element to investigate shape control applications. Numerical

NASA/TM--2001-210892 5
studies were performed to determine the optimum applied voltage, actuator layout, and actuator

number for shape control of composite plates with distributed piezoelectric actuators. Chattopadhyay

et al. (1999) implemented a third-order laminate theory into a finite element formulation to

investigate the dynamic response of delaminated smart composite plates. Numerical studies were

conducted on piezoelectric composite plates with single and multiple delaminations to demonstrate

changes in the dynamic response. Hong and Chopra (1999) developed an induced strain plate finite

element for composite plates based on classical laminate theory. The model was verified with

experimental results and demonstrated the capability to achieve shape control using piezoelectric

actuators.

2.2.2 Actuator and Sensor Models

Lee and Moon (1989) studied the control and sensing of bending and torsional deformations

produced by an applied electric field using piezopolymer bimorphs. The experimental results

validated the analytical model and demonstrated the use of piezopolymers as actuators. Lee (1990)

developed a model that incorporated the piezoelectric constitutive relations. The model was based

on classical laminated plate theory and was able to predict both the active and sensory behavior of

piezoelectric materials. Lee et al. (1991) investigated the use of sensor actuator pairs for active

damping control. Experimental studies were conducted on the active damping control of the first

mode of a cantilevered plate using a sensor and actuator pair to validate the piezoelectric plate

theory.

Chandrashekhara and Agarwal (1993) developed a laminated piezoelectric plate element

based on a first-order shear deformation theory applicable to both thin and moderately thick

laminates. Numerical results were verified with previously published results for a cantilevered plate

with attached piezoceramic actuators subjected to a static electric field. Hwang and Park (1993)

developed a piezoelectric plate element based on classical laminate theory and Hamilton's principle.

The piezoelectric constitutive relations were used to formulate a four noded, two-dimensional

quadrilateral plate element for both sensory and active applications. Case studies were performed

to investigate the static response of a piezoelectric bimorph beam and the vibration control of a

cantilevered plate with attached piezoelectric sensors and actuators. Koconis et al. (1994a, 1994b)

developed separate analytical models to investigate the sensory and active behavior of piezoelectric

composite beams, plates, and shells. One model is used to predict the change in shape when a

specified electric voltage is applied to the actuator, while the second model is used to determine the

electric voltages necessary to achieve a desired shape. Both models were formulated using a twodimensional, linear, shallow shell theory which includes transverse shear effects and validated with

other numerical, analytical, and experimental results.

Sung et al. (1996) derived sensor and actuator equations for a cylindrical piezoelectric

composite shell. Based on classical laminate theory, these equations formed the basis of a sensor

and actuator design methodology to control flexural and torsional vibrations in cylindrical shells.

This methodology was used to design an experimental rig to demonstrate capabilities of the modal

sensor and actuator to monitor and control the different vibration modes. Lam et al. (1997)

developed a finite element model based on classical laminated plate theory for the active vibration

control of composite plates with distributed piezoelectric actuators and sensors. Numerical studies

were conducted on a cantilevered composite plate to demonstrate capabilities for static and dynamic

analysis. Plettner and Abramovich (1997) implemented a consistent methodology based on KirchoffNASA/TM--2001-210892 6
Love thin shell theory to model the static and dynamic responseof anisotropic laminated

piezoelectricshells.Theformulationreplacedtheinducedpiezoelectricstrainwith anequivalent

mechanicalload. The model was verified with experimentaland finite elementresults for a

rectangularisotropicplate.

Penget al.(1998)implementedabeamfinite elementusingathirdorderlaminatetheoryfor

activevibrationcontrol of piezoelectriccompositebeams.Numerical studieswereconductedto

assesshapecontrolapplicationsandto investigatetheeffectof sensorandactuatorlocationsonthe

responseof thebeam.Liu etal.(1999)formulatedaplatefiniteelementbasedonclassicalaminated

plate theory for modeling the static and dynamic responseof compositeplates containing

piezoelectricactuatorsandsensors.Numericalstudieswereconductedto verify the modelwith

resultsfrom previouslydevelopedmodelsand to studythe influenceof stackingsequenceand

sensor/actuatorpositionontheresponseof compositeplates.

2.3 Coupled Electromechanical Models

Attempts to develop a more comprehensive representation of piezoelectric material behavior

led to the development of more consistent models that captured the coupled response between the

mechanical and electrical behavior. In these coupled models, the charge equation is incorporated into

the equations of motion and the electric potential is introduced as an additional degree of freedom

in the analysis. These models are generally also implemented as finite element programs to provide

a more flexible and general purpose analytical tool.

2.3.1 Analytical Models

Mitchell and Reddy (1995a) formulated a refined hybrid theory for laminated piezoelectric

composite plates. The displacement fields are modeled using third order shear deformation theory

while electric potentials are represented using a layerwise laminate theory. An analytical solution

was developed for simply supported boundary conditions and numerical results demonstrated the

limitations of the induced strain methods in modeling thick laminates. Heyliger and Saravanos

(1995) developed exact solutions to predict the vibration characteristics of simply supported

laminated piezoelectric plates. Numerical studies were conducted to determine the influence of

different laminations and aspect ratios on the natural frequencies and mode shapes. Batra and Liang

(1997) developed three-dimensional elasticity solutions for the simply supported rectangular

laminated plate with embedded piezoelectric layers. Numerical studies examined the steady state

vibration of both thin and thick plates containing one actuator layer and one sensor layer.

2.3.2 Finite Element Models

The groundwork for much of the current research into finite element formulations was

initiated by Allik and Hughes (1970) with the development of a three-dimensional tetrahedron

element. Derived at a time when piezoelectric materials were used mainly as crystals in transducer

applications, the finite element formulation incorporated the piezoelectric constitutive relations and

NASA/TM--2001-210892 7
demonstrated the potential advantages of utilizing the finite element method. Naillon et al. (1983)

formulated a finite element model to study the resonance phenomena of single piezoelectric

structures typically used in ultrasonic transducers. Numerical studies were performed on the

resonance characteristics of two-dimensional parallelepiped bars to assess potential applications in

the design of ultrasonic probes. Lerch (1990) formulated two and three dimensional finite elements

for performing vibrational analysis of piezoelectric sensors and actuators. The models were verified

with experimental data. The natural frequencies and mode shapes of various piezoelectric structures

were determined and used to optimize applications as transducers.

Tzou and Tseng (1990) developed a thin piezoelectric solid element. Derived from

Hamilton's principle and the piezoelectric constitutive relations, the element is specifically

formulated for thin plate and shell structures with distributed piezoelectric sensors and actuators.

Numerical studies were performed on the vibration response of a cantilevered plate with both an

active and sensory layer of polymeric piezoelectric material. Lammering (1991) developed a

piezoelectric shell element. Based on the Reissner-Mindlin shell theory and incorporating the

piezoelectric constitutive equations, a shell element was formulated for thin shell structures with

attached piezoelectric layers. Case studies were conducted on a cantilevered beam with an attached

piezoelectric polymer layer. Ha et al. (1992) formulated a three-dimensional brick element. Using

a variational principle and the piezoelectric constitutive relations, they developed an eight-noded

solid element. Results from static and dynamic case studies were verified with experimental results

for composite plates with attached piezoceramic actuator and sensor patches.

Heyliger et. al. (1994) implemented a layerwise laminate theory into a finite element

formulation for plates. Two separate layerwise models were developed which incorporated the

coupled equations of piezoelectricity to account for both the active and sensory behavior of

laminated plates with piezoelectric layers. Numerical results for a simply supported composite plate

with attached polymer piezoelectric layers were verified with exact solutions. Ray et al. (1994)

developed a two-dimensional quadrilateral element using a higher order laminated plate theory. An

eight-noded quadratic isoparametric quadrilateral element was formulated. Results from the static

analysis of a simply supported cross-ply laminated plate bonded with a piezoelectric polymer were

verified using previously reported exact solutions. Shieh (1994) developed a multiaxially active and

sensory laminated piezoelectric beam element. Based on adjusted elementary beam assumptions to

account for warping effects, the element can simultaneously model axial extension, biaxial bending,

and torsional twisting of the beam. Numerical studies were performed on a space antenna frame to

demonstrate capabilities for three-dimensional multiaxial vibration control.

Saravanos and Heyliger (1995) developed a layerwise finite element formulation for beams.

Two separate theories were used to perform static and free vibration analysis of composite beams.

Numerical results were verified with previously published analytical studies and demonstrated the

increased accuracy of stress-strain predictions with the layerwise theory. Shen (1995) developed

a finite element formulation for beams containing piezoelectric actuators and sensors. Based on

Timoshenko beam theory, the methodology captured the coupling between the longitudinal and

bending motions. The theory was validated with previously published analytical results for a

piezoelectric polymer bimorph beam and experimental results for a cantilevered beam with attached

piezoelectric sensors and actuators. Suleman and Venkayya (1995) formulated a plate element for

analyzing composite plates with layered piezoelectric actuators and sensors. Based on classical

laminate theory, a four-noded bilinear Mindlin plate element was developed. Previously published

NASA/TM--2001-210892 8
experimental and analytical results for a cantilevered piezoelectric bimorph beam and a cantilevered

composite plate with distributed piezoelectric actuators were used to validate the formulation.

Donthireddy and Chandrashekhara (1996) also formulated a layerwise theory for beams.

Results from the static response of a cantilevered composite beam with attached piezoelectric

actuators was validated with previous analytical results. Additional parametric studies were

conducted to study the influence of boundary conditions and ply orientation on the shape control of

beams. Heyliger et al. (1996) implemented a layerwise laminate theory into a finite element

formulation for shells. Results were verified with exact solutions for the static and free vibration

response of a simply supported plate. Additional studies were performed on the active and sensory

response of a cylindrical shell. Kim et al. (1996) developed a transition element to connect threedimensional solid elements to flat shell elements. The reported approach used solid elements to

provide detailed models of the piezoelectric material, while flat shell elements were used to provide

a more flexible model for the substrate structure, and transition elements were used to connect the

two regions. The model was verified experimentally for a cantilevered plate.

Samanta et al. (1996) extended the eight-noded quadrilateral element developed by Ray et

al.(1994) for dynamic analysis. Based on a higher order shear deformable displacement theory, the

model was developed for active vibration control of laminated plates with integrated piezoelectric

layers. Numerical results were performed on a simply supported cross-ply plate with attached

piezopolymer layers to demonstrate the potential to achieve significant reductions in vibration

amplitude. Kim et al. (1997) provided additional details of the theoretical development of the

transition element reported by Kim et al. (1996). Numerical studies were conducted to demonstrate

convergence characteristics and to show the increased computational efficiency of this approach.

Saravanos (1997) presented a shell element for curvilinear piezoelectric laminates which combined

a first order shear deformation theory for the displacements along with a layerwise theory for the

electric potential. The quadratic element was intended for static and dynamic analysis of thin to

moderately thick shell structures. Numerical studies quantified the effects of curvature on the active

and sensory response of piezoelectric shells.

2.4 Coupled Thermoelectromechanical Models

All of the previously described models neglect the implication of thermal effects on both the

active and sensory response of piezoelectric structures. Although Mindlin (1974) derived the twodimensional thermopiezoelectric equations for plates over twenty years ago, only limited research

has been performed into this area. The development of models for thermopiezoelectric materials can

be separated into two categories: (1) analytical models and (2) finite element models. The analytic

models extend existing piezoelectric laminate theories to account for thermal effects and obtain

solutions for specific problems. The finite element models provide a more general purpose tool to

efficiently analyze complex problems.

2.4.1 Analytical Models

Tauchert (1992) developed a thermopiezoelectric laminate plate theory. His theory extended

classical laminate theory to account for thin laminated plates with thermopiezoelectric layers.

Specific solutions were developed for both free and simply supported composite plates with attached

piezoelectric polymer layers. Numerical results demonstrated the capability to reduce thermal

deformations through application of active electric voltages. Tzou and Howard (1994) formulated

NASA/TM--2001-210892 9
athermopiezoelectricthinshelltheoryfor applicationsto activestructures.Thegenericshelltheory

wasderivedusing Kirchoff-Love shell theoryandHamilton'sprinciple. Using a simplification

procedurebasedontheLameparametersandradii of curvatures,specificsolutionswereobtained

for a cylindrical piezoelectricring, a piezoelectricring, anda piezoelectricbeam.TangandXu

(1995) developeddynamic solutions for a simply supportedanisotropic piezothermoelastic

compositeplate.Thecouplingbetweentheelasticfield andthe electricand thermalfields were

neglectedto simplify the analysis. Numerical resultsdemonstrateda significant reductionin

deflectionof aplatebytheadditionof apiezoelectriclayerwith a harmonicelectricfield.

TzouandBao(1995)extendedthethermopiezoelectricthin shelltheorydevelopedbyTzou

and Howard(1994)for applicationsto anisotropicshell laminateswith distributedsensorsand

actuators. The governingequationswere simplified and applied to a thin piezothermoelastic

laminatedshellmadeof apiezoelectricpolymer.Applicationsdemonstratedthecouplingbetween

theelastic,electric,andthermalfields andtheimportanceof all threefieldsontheoverallbehavior

of theshell.StamandCarmen(1996)presentedaxisymmetricthermoelectromechanical solutions

for concentricpiezoelectriccylinders.Theanalyticalapproachwasusedto modelthequasistatic

responseof a linear piezoelectricmotor. Resultsof the studydemonstratedthe capability to

extrapolatethenonlineardependenceof thepiezoelectriccoefficientswith electricfields to lower

temperaturesusingthe constitutiveequations.Friswell et al. (1997)developeda linearmodelto

investigateactivedampingof thermallyinducedvibrations.Numericalstudieswereconductedon

a simply supportedaluminumbeamwith a piezoelectricsensor/actuatorpair to demonstratethe

influenceof pyroelectriceffectson vibrationcontrol.

2.4.2FiniteElementModels

RaoandSunar(1993)developedafiniteelementformulationwithapplicationsforintegrated

sensingandcontrol of thermopiezoelectricmaterials. Numerical studieswere performedon a

piezoelectricbimorphbeamandanisotropicbeamwith attachedpiezoelectricpolymerlayers.The

resultsdemonstratedthesignificanceof thermaleffectson theperformanceof distributedcontrol

systems.Jonnalagaddaet al. (1994)implementeda nine-nodedLagrangianplateelementusinga

first-order sheardeformationtheory. Numericalstudieswereperformedon a simply supported

compositeplatewith anattachedpiezoelectricactivelayeranddemonstratedtheimportanceof a

higherorderlaminatetheoryto accuratepredictsheardeformationsin thick laminates.TzouandYe

(1994)extendedthepreviouslydevelopedsolidelementof TzouandTseng(1990)to accountfor

thermal effects. The resulting three-dimensionalthin hexahedronelement representedthe

displacements,electric potential,and temperatureas statevariables. Numerical studieswere

performedonacantileveredisotropicbeamwith attachedpiezoelectriclayersto demonstratethe

influenceof thermaleffectsonsensingandcontrol.

Chandrashekharand Tenneti (1995)developeda nine-nodedfinite elementfor active

thermal control of compositeplates with piezoelectricactuatorsand sensors.The approach

incorporatedan inducedstrainapproachto approximatethe piezoelectricand thermal strains.

Numericalstudiesdemonstratedthecapabilityto suppressthermallyinduceddeformationsthrough

theapplicationof electricalvoltagesin piezoelectricpatches.ShenandWeng(1995)implemented

a three-dimensionalbrick finite elementto investigatecompositeplateswith piezoelectriclayers.

Numericalstudiesdemonstratedthesignificantcouplingbetweenthestrainandelectricfieldsand

thecapabilityto achievethermalshapecontrolof asimplysupportedpiezoelectriccompositeplate.

NASA/TM--2001-210892 10
Sunar and Rao (1997) formulated finite element equations for the design of thermopiezoelectric

sensors and actuators. Numerical studies were conducted on cantilevered beams with piezoelectric

actuators to show the significance of temperature effects on distributed control.

2.5 Limitations of Existing Analytical Models

The previously described analytical models contain three limitations that restrict the general

application of these models. The first limitation is found in the induced strain models and arises from

the use of approximate forces to represent the piezoelectric strains. This approximate representation

fails to capture the coupled mechanical and electrical response and limits these models for use in

predicting only the actuator behavior of piezoelectric materials. The induced strain limitations are

overcome in the coupled electromechanical and thermoelectromechanical models through the use

of a more consistent representation of the coupling which occurs between the electrical and

mechanical responses. However, both the coupled electromechanical and thermoelectromechanical

models are restricted by a second limitation arising from the predominant use of single layer laminate

theories to predict the laminate response. In general, the single layer theories only provide good

predictions for thin laminates and are inaccurate for both thick laminates and laminates which

contain strong inhomogeneities through-the-thickness. The third limitation is found in both the

induced strain and coupled electromechanical models, and arises due to neglecting thermal effects,

even though these effects can be significant in changing temperature environments. Even in the

coupled thermoelectromechanical models which account for thermal effects, many of the analytical

formulations fail to account for all the mechanisms which produce thermal effects in piezoelectric

materials: induction of thermal strains due to thermal expansion mismatch, pyroelectric effects, and

temperature dependent material properties. Currently, no analytical model addresses all three of these

limitations.

Thus, in this research, each of these three limitations are addressed to provide an enhanced

analysis capability for modeling arbitrary structures which contain piezoelectric materials. A

consistent representation of the coupled mechanical, electrical, and thermal response of piezoelectric

materials is implemented to overcome the induced strain limitations and provides an inherent

capability to model both the sensory and active behaviors of piezoelectric materials. A layerwise

laminate theory is implemented to address the single layer laminate theory limitations and leads to

more accurate analysis of thick laminates and laminates with contain strong inhomogeneities

through-the-thickness. Thermal effects arising from thermal expansion mismatch, pyroelectric

effects, and temperature dependent material properties are explicitly accounted for in the

formulation. The development of this enhanced formulation has been reported for a linear beam

element (Lee and Saravanos, 1996), a bi-linear plate element (Lee and Saravanos, 1997, 1998) and

a quadratic curvilinear shell element (Lee and Saravanos, 2000).

NASA/TM--2001-210892 11

CHAPTER 3

THEORETICAL DEVELOPMENT

This section outlines the foundations and steps for developing the analytical formulation for

piezoelectric composite materials. This development is based on the theory of linear piezoelectricity

which have been previously reported in detail by Cady (1964), Tiersten (1969), Nye (1972), and

Parton and Kudryavstev (1988).

3. l Current Formulation Assumptions

The current formulation is based on the linear theory of piezoelectricity in which the

equations of linear elasticity are coupled to the charge equations of electrostatics through the

piezoelectric constants. Due to the assumption of a linear theory, both mechanical and electric body

forces and couples are neglected in the derivation. The current model also assumes infinitesimal

deformations and strains, and, thus, no distinction is made between the initial and final positions.

The current formulation also neglects the two way thermopiezoelectric coupling and focuses only

on the one way heat conduction analysis. This assumption essentially provides that temperature

changes due to changes in strains and electric fields are small compared to the magnitude of the

thermal load. Thus, temperature changes only produce mechanical and electrical forcing on the

piezoelectric material.

3.2 Governing Material Equations

The mechanical response of the piezoelectric material is represented by the equation of

motion,

°ij,j + fi = P iii ( 1 )

while the electrical response is described by the electrostatic equation for the conservation of electric

flux,

Di, i = 0 (2)

where oij are the stresses, fi are the body forces per unit volume, P is the density, ui are the

displacements, and D i are electrical fluxes. In addition, i, j ranges from 1 to 3, superscript dots

represent time derivatives, and subscript commas represent differentiation. Through use of the

divergence theorem and neglecting body forces, Equations (1) and (2) can be expressed in an

equivalent variational form as

f u-, + as, - Di i) dV= f t i idE+ f qf)_ dF (3)

V F t [p

where Sij represent the strains, Ei are the electric fields, _ are the surface tractions applied on the

NASA/TM--2001-210892 13
surface Ft, q is the electrical charge applied on the surface Fv of the piezoelectric material, q) is the

electric potential, and V represents the whole volume including both composite and piezoelectric

materials.

3.1.1. Strains and Electric Field in Cartesian Coordinates

Two additional relationships for the strain and electric field are required in the equivalent

variational form (Equation 3). In a Cartesian coordinate system, the small deformation straindisplacement relations are

1

Sij - 2 ( ui'i + u'i ) (4)

while the electric field vector is related to the electric potential by

Ei = - q),i (5)

3.1.2. Strains and Electric Field in Curvilinear Coordinates

The curvilinear coordinate system employed in the formulation is shown in Figure 1. Each

ply of the laminate remains parallel to a reference curvilinear surface A o and an orthogonal

curvilinear coordinate system O_n; is defined, such that the axes _ and 1"1lie on the curvilinear

reference surface A o, while the axis _ remains straight and perpendicular to the layers of the

laminate. A global Cartesian coordinate system Ox_._ is used to define A o, hence, a point r = (x,y,z)

on the curvilinear laminate is,

r(_,rl,_) = ro(_,rl)+_ (6)

where, ro = (Xo,Yo,Zo) are the Cartesian coordinates of the reference surface A o and _ indicates the

unit vector perpendicular to the reference surface.

The strain and electric field in this curvilinear coordinate system take the following form

(Fung, 1965)

S_j - I [(u;j+tS.;) _ uk(F_,j+i+)]

2_ (7)

E. - (P.i

, _ (8)

NASA/TM--2001-210892 14
O

/x

Z

I:11

Figure 1" Curvilinear piezoelectric laminate and coordinate systems.

where gii are the metric tensors and Fij are the Christofel symbols. The following form of the

Euclidian metric tensors are employed in the current formulation (Soedel, 1993),

gll = (1 + _ )2 o-"

gll (9)

02

gz'- = (1 + -- )_" g22 (10)

g33 = 1 (1 1)

where Ri are the local radii of curvature and gij° are the components of the metric tensor on the

surface A o (_=0) defined as

g1°1 = _/Xo,_2 +Yo42 + Zo,_2

(12)

o ] 2 2 2

g22 = _/Xo,n +Yo,n +Zo,n

(13)

NASA/TM--2001-210892 15
Substituting Equations (9)-( 11 )into Equation (7) produces the following relationships (Soedel, 1993)

o o

_ gll ,, Sl I 1 (tt,_ + gll'qv +--t_)

o " R 1

glL(1 + ---_-q) g2°

Rl

(14)

o O

S,,,, : 1 (g_..... u+v +--w) g22 .

-- g,'_(l + ¢ o 'n R_

.. _2 ) gll

(15)

S = W.

33 ,_

(16)

812

g22._ 1

_ 1 (un - ----_v) +

g¢_'( 1 +--_-_) gll gl_(1 +---_-_)

-- R2 Rl

(V _. - --

gl l.q.u)

o

g22

(17)

O

1 gll

S13 = //- + (W:-_/g)

"_" gn(l+ ¢ '_ Rl

(18)

O

1 g22 ,

$23 : v4 + (w q---v)

g,°(1 +--_-_) R2

"" R2

(19)

where u, v, w are the displacements of the curvilinear system. Similarly, introducing Equations (9)-

(11) into Equation (8) produces the following relationships

1

E l : _ q),¢

gl]( 1 + _'_1)

(20)

NASA/TM--2001-210892 16
1

E_ -

- o + _ q)'q

g22( 1 -_, )

(21)

E 3 = - ¢P.¢ (22)

3.3 Thermopiezoelectric Constitutive Equations

The constitutive equations for a linear thermopiezoelectric material employing standard

contracted notation (Nye, 1964) are

Sa = s_E.T(T) cyl_ + da,.(T) r E m + a_'V(T) 0 (23)

Dm T ¢_.T 6,T( = dm_(T) _,_ + _.,k(T) E k +Pm T) 0 (24)

or in semi-inverted form,

ET T

_a = C_ (T) Sf_ - e_(T) E m - _._'r(T) 0 (25)

T S.T S,T

D m -- em_(T) S_ + _._(T) E k ÷ Pm (T) 0 (26)

with the following relationships in the various piezoelectric and thermal properties arising from the

inversion,

e_,,(T) T = cloT(T) d_(T) E m (27)

_.aE,T(T) = Ca_ET (T) a_,T(T) (28)

S,T o,T T T

Emk(T) = Ernk (T) - dm_(T) e_(T) (29)

S T ¢_.T T

Pm" (T) =Pm (T) - dm_(T) _.Ea'T(T) (30)

where C._ and s._ are the elastic stiffness and compliance tensors; d_m and e_ are the different forms

of the piezoelectric tensor due to inversion; emk is the electric permittivity tensor; % and _. are the

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

NASA/TM--2001-210892 17
constant; 0 = AT = T - T,, is the temperature difference from the current temperature T and the

thermally stress free reference temperature To; and; superscripts E, c_, S, and T, represent constant

voltage, constant stress, constant strain, and constant temperature conditions, respectively; Gt,13= 1,

.... 6; and k, m = 1, 2, 3.

3.3.1 Piezoelectric Beam

The specific form of the thermopiezoelectric constitutive equations for a piezoelectric beam

are simplified since only axial variations of the displacements are assumed,

(31)

D3 -- d31 c_1 + _;33 E3 + P3 0 (32)

or in semi-inverted form

} ]{ 1{} t

CY1 Cll 0 $11 0 e31 E 1 k 1

_5 0 C55 $5 ] ! el5 0 E 3 0

0 (33)

03 = e31 Sl + E33 E3 + P3 0 (34)

3.3.2 Piezoelectric Plate and Shell

For the piezoelectric plate and shell, the thermopiezoelectric constitutive equations for a

monoclinic orthotropic piezoelectric material (Mason, 1950) are used,

$1

S 2

' S 4

$5

S 6

_11 S12

_12 $22

_= 0 0

0 0

0 0

0 0

0 0

s44 0 0

0 s55 0

0 0 s66

0 ol]

0 a_l

_34_' +

0 0 d31

0 0 d32

t14 0 0

dr5 0 0

0 0 0

r i} E_ +

E

_2

0_0 °l

0

(35)

NASA/TM--2001-210892 18
r

_)l[ [ 0 0 0 dis

_V._-- 1_91 10d31 d320 d240 00

(_4_" +

%1

iio:l/ it ,J l, C22 E, +" _2_0

0 f'33 E 11_1

(36)

or in semi-inverted form

152

'l (_4 _ =

(35

(_6

Cll Cl2 0 0 0

C12 C22 0 0 0

0 0 C44 0 0

0 0 0 C_5 0

0 0 0 0 C66

S I

S.

S:}

S,

,i

0 0 e31

0 0 e32

0 e24 0

el5 0 0

0 0 0

kl

E2 -' 0

E 3 0

0

(37)

0,1

[_ 0 0 et5

0 e24 0

e31 e32 0 0

S,I

,s4_ +

S_l

r!1

E22

0 El

E, + 'P2{Pl}0

0 E33 E P3

(38)

NASA/TM--2001-210892 19

CHAPTER 4

LAMINATE MECHANICS

In order to provide more general and accurate analysis of piezoelectric laminates, different

theories are implemented for the beam, plate, and shell laminates. A layerwise laminate theory is

used as the basis for the different formulations since this theory represents a generalization of other

single and higher order laminate theories (Reddy, 1993). The layerwise theory allows separate fields

to be assumed for each state variable which provides for more accurate analysis of thick laminates

and laminates containing strong inhomogeneities. In the current formulation various simplifying

assumptions and approximations are used for the beam, plate, and shell laminates.

4.1 Layerwise Laminate Theory_

The general layerwise laminate theory is formulated by introducing piecewise continuous

approximations through-the-thickness for each state variable,

N

U(X,y,z,t) -- __, blJ(x,y,t) "%t"J"..) (39)

j 1

N

v(x,y,z,t) = __, vJ(x,y,t) V{(Z)

j:l

(40)

N

w(x,y,z,t) =__. wJ(x,y,t) _Jw(Z)

j:l

(41)

N

tp(x,y,z,t) = __. g/(x,y,t) gi_(z)

j=l

(42)

N

O(x,y,z,t) = _ OY(x,y,t) _(Z)

j=l

(43)

where N is the number of interpolation functions VJ and u j, v j, v/, q9, and 0j are the generalized

laminate state variables. In general, any order of interpolation functions can be used and different

interpolation functions can be selected for each state variable depending on the level of detail desired

in the analysis.

4.1.1. Beam Laminate Theory

In deriving the beam laminate theory, only axial and through-the-thickness variations of the

state variables are assumed. In addition, the displacement w is assumed to be constant through-thethickness. This leads to the following simplified approximations from the general layerwise theory,

NASA/TM--2001-210892 21
u(x,-,t)

N

= _ .J(x,t) _/(z)

) i

(44)

w(x,z,t) = w°(x,t) (45)

N

q_(x,z,t) -- _ 4(x,t) v"(z)

,i _ 1

(46)

N

O(z,t) -- _ O/(t) _;(z)

j:l

(47)

_(z) = [1-{ {1 (48)

where N is the number of interpolation functions WJ, _=z/h, and h represents the thickness of each

discrete layer. In this formulation, the same linear Lagrangian interpolation functions are used for

each state variable.

4.1.2 Plate Laminate Theory

The plate laminate theory is formulated with the displacement w assumed to be constant

through-the-thickness, which leads to the following simplified approximations from the general

layerwise theory,

N

u(x,y,z,t) = Y_ uJ(x,y,t) _J(z) (49)

j:l

N

v(x,y,z,t) = _ vJ(x,y,t) _J(Z)

j:l

(50)

w(x,y,z,t) = w °(x,y,t) (51)

N

(p(x,y,z,t) = _ q_(x,y,t) _J(z)

j=l

(52)

NASA/TM--2001-210892 22
0(z,t)

N

j 1

(53)

_(z) = [1-_ _] (54)

where N is the number of interpolation functions _J and _=z/h. For the current plate laminate theory,

the same linear Lagrangian interpolation functions are used for each state variable.

4.1.3 Shell Laminate Theory

The shell laminate theory combines linear displacement fields through-the-thickness of the

laminate for the displacements u and v (along the _ and 1"1axes respectively) with layerwise electric

potential and temperature fields through the laminate, consisting of N discrete continuous segments.

Consequently, the present shell theory is specialized for modeling the deformation of thin and

moderately thick piezoelectric shells, while maintaining the capability to capture the through-thethickness electric and thermal inhomogeneities. The various field variables are approximated by the

following form,

u({,q,_,t)= u°(¢,rl,t) + _[3¢(_,n,t) (55)

v({,n,¢,t)-- v°({,rl,t) + ¢[3n({,n,t) (56)

w({,q,¢,t) = w°(¢,q,t) (57)

N

+({,n,;,t)-- wJ(;) (58)

j=l

N

O({,rl,_,t )= y_ _({.rl,t ) qg(_) (59)

j=l

_(_) = [1-4 _] (60)

where N is the number of interpolation functions @, _=_]h, and [3_and 13n are the flexural rotation

NASA/TM--2001-210892 23
angles. The same linear Lagrangian interpolation functions hu(_,) are used for the electric potential

and temperature state variables.

4.2 Generalized Laminate Matrices

The next step in the development of the formulation is the introduction of the layerwise

approximations into the variational form of the equation of motion and the electrostatic equation

(Equation 3). By performing an integration through-the-thickness, the dependence of the throughthe-thickness coordinate can be separated from the in-plane coordinates into generalized laminate

matrices for the density, stiffness, piezoelectric, dielectric permittivity, thermal expansion, and pyroelectric properties.

4.2.1 Beam Laminate Matrices

The equivalent variational form is derived by substituting Equations (4), (5), (20), (21), and

(31 )-(34) into Equation (3), and integrating through-the-thickness,

tl tl

_f,_ Pll iim 6u , dx + f 1033_i; 8w dx +

k _ 1 m : 1

X X

_ f(D,,,,a. _ Oa.* ,_ ,

k l m:l "_.i -dx + D55 um 6u )dx +

II

EfB, ow ,Oaw f

ss ( -- auk+ u -- ) dx +

k=l 0X C_X A55

X X

dw d6w

Ox Ox

dx +

k 1 m:, _X + 0"-;

A"

-- 6q¢" ) dx -

ll II II k Ogu k

Z E fo*'_m5_*d_33 Z f(A,

k=l m 1 k 1 OX

X X

P3 k 6q) k ) dx

=ft i 6uidF + fq 6q_dF

v vp

(61)

NASA/TM--2001-210892 24
in which the dependence of the z-coordinate has been separated into the generalized density matrix

[p], the stiffness matrices [A], [B], [D], the piezoelectric matrix [E], the dielectric permittivity matrix

[G], the thermal expansion matrix [A_], and the pyroelectric matrix [P3].

The laminate density matrices are

kFtl

Pll = Y_

I 1

f: b p _k(z) _'(z) dz (62)

L

P33 = Z f b p dz (63)

1_,1 Z

the laminate stiffness matrices are

A55 = _ f b C55 dr, (64)

l_-1 z

L

_. OW_(z)

Bss = _, f b Cs5 _ dz (65) l: t : " Oz

Ol_ m = E f. b Cll_l/k(z) _/m(_,) d_ (66)

l=1 -

L

I") km f_

"-'55 = _ b C55 O_1*(z) OuF(z) dz (67)

The piezoelectric laminate matrix is

L

--E f. b e3, V*(Z) ovm(z) dz (68) l._ . Oz

The dielectric permittivity laminate matrix is

L cqv_'(z) OV"(z) dz (69)

G3_'m = Z f.b_33

=_ _ 8z Oz

NASA/TM--2001-210892 25
The laminate thermal force vector is

L

A_,: y_ f bX 1

/ I

(70)

The laminate thermal electric displacement vector is

L

pk f

3 : _ bP3 @ _J(z) OVk(z) dz

I: 1 O_,

X

where L is the number of plies and b is the width of the beam.

(71)

NASA/TM--2001-210892 26
4.2.2.PlateLaminateMatrices

Theequivalentvariationalform is derivedby substitutingEquations(4), (5),(24)-(25),and

(36)-(40)into Equation(3), andintegratingthrough-the-thickness

N

N N

f km k km_m_vk)dA + f 033_4.,_wd A

__, (011 ii "8H + 022

k : 1 m _ 1

A A

N N

ZZf,_,mmill U,x _ll.v + .,,,,,,,,e, "*12 _ Ig.v Ol',y + 1',3,,,,e, Oil i, ) +

k:l m=l

A

m k me k me k me k --kin me k

A_6"(u,. BUy + U.y oua.+ u,. or.,.+ v,. otG.) + "_22 V,y OVy +

A k6 m e k m e k m k m e k (u v ov_ + v v our+ v_.Svv+ v v ovv) +

Akin. ,,18 k ., e k me k ., e X

66 (lgy bly + Uy OVa + V,. OV,. + Va. OU.y) }dA +

N

f

( B"44(_"aw,, + w,,.av") +B_(u"aw .,. + w..,.au "1) +

m_l " "

A

B45m( u m 8_,y + v m 8W.,. + Wa 8V m + W..,8U " ) } dA +

f {C44Wv614' + C45(wvfwv + w v614',:) + C5514'rrwr } dA

A

N

f {DkmVm_vk + r..km_, m k m_lik r_km m_llk +

44 /-345 tH _,' + l' ) + /-355 H

k:l m=l

A

/_31 r.,km U_.me Oq) k + Eking32I, V,ym fi_ k + E_6( U sSq_ m k + v_Sq_) m k +

Ekm me k r, km me k

31 q) OU.r + E'32 (P OV,y + E:6 ( q)m Buyk + q_m 8v_)k

...,,kin me, k .-., km me k

022 (p,y Oq),y - Gff_ _"8_* }

N

Z f {- x;'a.: - a;a_,r'. + P?Sq)_ + P26q)., m. + P_6q_ m } dA

m=l

A

F Fp

(72)

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], the thermal expansion matrix [A], and the pyroelectric matrix [P].

NASA/TM--2001-210892 27
The laminate density matrix is

km

9ij = __,

I 1

f 9 _k(z) _'"(z) dz (73)

for ij = 11 and 22, while

L

p3_=E/=1fpd= (74)

the laminate stiffness matrices are

Akin

ij = Z

l 1

f cij _k(z) _"(z) d" (75)

forij = 11, 12, 16, 22, 26,44, and 66;

L

Bij = Y_ C o dz

t:l Oz

(76)

L

c,-,:E f c,,d:

l:l "

(77)

L

Dijkm=Z

l= _

f Ci_0¢(:)

Oz

0_(:)

3z

&

.7.

(78)

for ij = 44, 45, and 55. The laminate piezoelectric matrices are

L

km f (_l].lm ( z ) &

Eij : Z eij tltk(z) 3Z 1:1

(79)

Eij km = Z f eij _lm(z)

I= 1 OZ

_m(z) d- (80)

NASAFFM--2001-210892 28
for ij - 31, 32, and 36. The laminate dielectric permittivity matrix is

km

G,; = F_, f _,jVk(:) V"'(:) d:

I=1

2,

(81)

for ij = 11 and 22, while

L O_k(z) o_m(=) d: km

Z

(82)

The laminate thermal force vector is

L

k

A, : Z f )_i 14/j Oj It/k(_,)NZ,

l=l

(83)

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

k

P, =_ f p, _,_%.,v_(:)d:

l=1

(84)

for i = 1 and 2, while

L &g_(z)

e; : Z f p3w_o_ d: l:l c3z

(85)

where L is the number of composite plies and piezoelectric layers.

4.2.3 Shell Laminate Matrices

The equivalent variational form is derived by substituting Equations (11 ), (12), (24)-(25), and

(40)-(45) into Equation (3), and integrating through-the-thickness

NASA/TM--2001-210892 29
{ (p'a 17,,

.4

(p_a"

N

_-" f { (A,1SI ° + AI2S,[ + Bllkl

j 1 ,4

" A c o (AI2S 1 + 220, + Bl2k I +

(A44S4" + ES4--__ )as4 ° + (A66S6 °

o

g_L

+ pef3_) 8u o + (9,4 i; o

+

+ Oef3,l)&," + pa_i;°Sw°+

+ pD_.) 5[3 }d_dr I +

o,2k +Ej,¢ A 0J)as?+

B22k 2 + E_2qfl A_OJ)aS2 ° +

+ O66k 6)as: + (B66S:' + O66k 6)ak 6} +

(B,,S, ° + B,2S2 ° + D,,k, + D,zk 2 + E_,ff_ - fk_Oi)ak, }d_dq +

(B12S I

" + B22S 2" + D12k t + O22k 2 + E32q) ^J /" - £_oJ)& _ . +

s ° ,p'; ,.

-- + At- +

{(E 5o - G[_ ,,, o )5q_'4

j:l t=l .4 glt (gll)- gll

(ES4 S: ,* q_,*n jk(F * - G22_ + P )aq) +

o O

( of g22 gll - g22

(EJlSI° + E/2S ° + EJ, k, + Ef._.k2 -GJkq_ k33+ PJ*Ok)aq¢}d_drl

I" V

P

(86)

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], the thermal expansion matrix [A], and the pyroelectric matrix [P].

The laminate density matrix is

L

10A o o f

= gllg22 E pd_ (87)

/::1

¢

L

oB= o o f

gl,g22 _-, _pd_

c

(88)

L

pD = gllg22o o E f_2od_

l=1

¢

(89)

NASA/TM--2001-210892 30
The laminate stiffness matrices are

L

Aij =gllg2 ° Z f Cijd_

l 1

(90)

for ij = 11, 12, 22, 44, 55, 66

L

Bij = gllg22 °°E f _

I:l

(91)

L

D j = gllg22 o o E f _2Cijd_

I=l

(92)

for ij = 1 i, 12, and 22. The laminate piezoelectric matrices are

L

k o o f

Eij = gllgZ2 Z j eij Vk(_)d_

1=1

(93)

for ij = 14, 15, and 24.

L

, o o f 0_k(_) d_ Eij = gllg22 Y_l:l eij O_

.

(94)

L auk({)d{

Ei) : gl°lg2°2 E f {eij 1:i a_

<.

(95)

for ij = 31, 32, and 36. The laminate dielectric permittivity matrix is

L

Gij = gl 1 g22 Z F.ij

l_l

(96)

for ij = 11 and 22, while

L av*(_) 0vm(_) d_

G37 = g ll g22 E f g33 (97)

NASA/TM--2001-210892 31
Thelaminatethermalforcevectoris

L

-77 ,, ,, f

Ai -- gllg22 Z _'i _'(_) d_ J 1 1

(98)

L

3? o o f

, = gllg22 __, _')_i _k(_) d_

l=1

(99)

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

j_

Pi k'' = g,'ig,J2 _ Pk Vk(_ ) V"(_)d_

l=l

(100)

L avk(_) _(_) d_ pkm3 = gllog22o__,f P3

l:l O_ (101)

for i --1 and 2, while where L is the number of composite plies and piezoelectric layers.

NASA/TM--2001-210892 32
CHAPTER5

FINITE ELEMENT FORMULATION

Finiteelementbasedsolutionsfor structuralproblemsareobtainedbyincorporatingadditional local in-plane approximationsas detailed in Hughes (1987) and Cook et al. (1989). For

piezoelectricmaterials,thefollowing approximationsaremadefor thegeneralizedlaminatestate

variablesintroducedin Equations(39)-(42),

M

uJ(x,Y, t) = uJi(t)N. (x, y) (102)

i=l

v (x,y,t)

M

= N_,(x,y)

i:1

(103)

wJ(x,y,t) = (104)

M

Z W ji (t) Nw(x,Y) i

i:l

g/(x,y,t) = (105)

M

Z (Pji(t) N_i(x, y)

i:1

OJ(x,y.t) (106)

M

= No(x,y)

i=l

where M is the number of in-plane shape functions N. Typically, any order of shape functions can

be used and different shape functions can be selected for each state variable, depending on the level

of detail desired in the analysis.

By implementing the in-plane approximations for the state variables into the variational form

of Equation (3) and collecting the coefficients, the following discrete matrix form can be obtained,

tKjJt {o(,)i[ : 03101j

(107)

where the submatrices I_u, K_, and I_ indicate the elastic, piezoelectric and permittivity matrices;

I_0 and K,o are the thermal expansion and pyroelectric matrices of the structure; Muu is the mass

matrix; F are the applied mechanical loads; and Q are the applied voltages.

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

NASA/TM--2001-210892 33
[KLF]J[{_PF}J

l

QF(t)}-[K_]{_pA}-[Ko]{O} t

(108)

where superscripts F and A indicate the partitioned submatrices in accordance with the sensory (free)

and active electric potential components, respectively. Thus, the left-hand side includes the unknown

electromechanical responses of the structure, u and _F (i.e. the resultant displacements and voltage

at the sensors, respectively). The right-hand includes the known excitations of the structure by the

mechanical loads, applied voltages on the actuators, applied temperature loads, and electric charges.

The matrix equations can also be condensated into the following independent equations for

the sensory electric potentials,

{cpF} = [KLF] ] ([KFA] {(pA} +[K0] {0} -[K_, FF] {u} - {Q F(t)} ) (109)

and the structural displacements,

• [KFFI 1 FF [muu]{//} +([Ku ] -[KuFF]L__q_ j [K_,]){u} =

{r(,)} -( [KJ] [K__F] ' [K_oFA]+ [K,_?] ){ q)A}+

[KuT][KFF]-IL--_, {Q F(t) } - ( [Ku_0FF ] [K_o_ FF ] _ilK 0] + [K0]){0}

(110)

5.1 Beam Element

The piezoelectric beam element formulation is obtained by incorporating the following inplane approximations to the state variables,

M

ug(x,t) = __,uJi(t) Ni(x) (111)

i=1

M

wJ(x,t) : Y_ wJi(t) Ni(x) (1 12)

i=l

M

q)J(x,t) = Z q)ji(t) Ni(x) (113)

i:1

NASA/TM--_001-21089. ,) ,9 34
Ni(x) = [1-_ _1 (114)

where M is the number of in-plane linear shape functions N, _=x/L, and L is the length of the

element. A selectively reduced integration scheme is also implemented for the second stiffness term

(containing D55) of Equation (67) in order to eliminate locking.

Incorporating Equations (111)-(114) into Equation (61) results in the following specific

forms of the submatrices defined in Equation (107) for the beam element, where the submatrices are

defined using the beam laminate matrices of Equations (62)-(71 ). The mass matrix is

[Oil] 0 I

[N] dx (115)

[M"u] = f, [Nlr 0 [9t3]

The stiffness matrices are

[Xu.]: f, [N,,.]r[Dll][N,._]+[N]T[D55][N] [N]r[B55][N,,.] t [N]rtB55][N,.,] [N',lr[Assl[N'"]] dx (116)

[Ku'_] = fx [N]r[E31] [N,,:] dx (1 17)

[Ko] : f_ [N]r[A_] dx (118)

[K ] : f,. [N]r[G33][N] dx (119)

[K,o0] : f, [N]r[P3 ] dx (120)

NASAITM--2001-210892 35
5.2 Plate Element

The piezoelectric plate element implements the following in-plane approximations to the

state variables,

M

uJ(x,y,t) = __, uJi(t) Ni(x,y) (121)

il

M

vJ(x'Y "t) = Z vii(t) Ni(x,Y) (122)

i_l

M

wJ(x,y,t) = _ wJi(t) Ni(x,y) (123)

i_l

M

q_(x,y,t) = _ _i(t) Ni(x,y) (124)

i:l

1

Ni(x,y)---_ [(1-_)(l-q) (l+_)(l-q) (l+_)(l+q) (l-_)(l+q)] (125)

where M is the number of in-plane bi-linear shape functions N, _=x/a, and rl=y/b.

Implementing Equations ( 121 )-(125) into Equation (72) leads to the following finite element

submatrices for the plate element, where the submatrices are defined using the plate laminate

matrices in Equations (73)-(85). The mass submatrices are

[Pll] 0 0 J

[Muu] = f,4 [NIT 0 [P22 ] 0 [N] dA (126)

0 0 [033]

The stiffness submatrices are

[Kll] [KI2] [KI3]]

[Kuu] = ILK12 ] [K22] [K2311

[Ex 31 x3,1]

(127)

where

[Kll ] = f {

A

[N,x]r[Atl ] [N, x]

[N]r[D55] IN]

[N,_]r[A16 ][N,v] + [N,,]TIA16 ][N,_] +

[N,y]T[A66 ] [N,y] } dA

(128)

NASA/TM--2001-210892 36
[KI2] = f { [N,x]T[AI2][N,r] + [N,x]T[A16][N,x ] + [N,v]T[A26 ][N,,,] +

A

[N]r[D45] [N] + [N,y]T[A66 ] [N, x] } dA

(129)

[K13 ] = f { [N]T[B45][N,,.] + [N]r[B55][N,x] } dA

A

(130)

[K22 ] = f { [N,,,]T[A22][N,y] + [N,x]T[A26][N,y] + [N,,.]T[Az6][N,x] +

A

[N]T[D44 ] [N] + [N,x]T[A66 ] [N, x] } dA

(131)

[K23 ] = f { [N]T[B44][N,,,] + [N]T[B45][N,x ] } dA

A

(132)

[K33] = f { [N,_]r[C44][N,,,], . + [N,,.]r[C4s][N,,]. _ , + [N,_]r[c4S][N,,.], . . +

A

[N,xlr[C_55][N,x] } dA

(133)

[x] : fa

[N,x]r[E31][N] + [N, vlr[E36][N]

[N,y]r[E32][N] + [N,_lr[E36 ][N]

dA (134)

[_o] : fa

[N,x]r[Al]

[N,y]r[A2 ]

dA (135)

NASA/TM--2001-210892 37
f { [N,xlr[Gl,l[N,x] + [N,,.]r[Gzzl[N,v] + [N]r[G331[N] } dA (136)

,4

[K_°] : fa { [N'']r[P1] + [N'v]r[P2] + [NIt[P3] } dA (137)

5.3 Shell Element

The piezoelectric shell element implements the following in-plane approximations to the

state variables,

M

uj°(_,rl,t) = _ uj°i(t) N i(_,rl) (138)

i=l

M

= _3_,(t)N i(_,q) (139)

i=1

M

(p"(_,rl,t) = _ (pmi(t) N i(4,q) (140)

i:l

M

0'n(_'l] "t)= Z omi(t) N ;(_,q) (141)

i=l

1

Ni(_,q) = -_

(1-_)(l-q) - (1-4)(1 _q2)It

!

(1 +_)(1 -q) (1 +_)(1 _q2)

(l+_)(l+rl) - - (1-_2)(l+q)

(1 -_)(l +q) - - (l -_)(1 _q2)

(1 -_2)(1 -1])

(1 -_:) (1 -q)

(l +_)(1 -¢)

(1 -42)(1 +q)

(l -_2) (1 -q)

(1 +_)(1 _q2)

(1 -_2) (1 +n)

(1 -¢)(1 _q2)

(142)

where M is the number of in-plane shape functions N for an eight-noded Serendipity element.

NASA/TM--2001-210892 38
By introducingEquations(138)-(142)intoEquation(86),thefollowing finite elementsubmatricesfor theshellelementarecalculatedin accordancewith theshelllaminatematricesdefined

in Equations(87)-(101).Themasssubmatrixis

[PA] [PB]] [M ] = fa [N]r [PB] [PDIJ [N] d_dr 1 (143)

The stiffness submatrices are

where

[KIj]

[K12]

: [K13]

[KI 4]

[KIs]

[Kt2] [Kt3] [KI4]

[K:31 [K33] [K_, 4]

[K241 [K34] [K,4]

tx'2,J tx35 K45J

[Kls]]

[K241

[K35]

[K45]

[K55]

(144)

[Kll] : f { _[N,a]r[All][N,a] 1 + 2g22,_ _ [N]T[AI,][N,_] +

O _ - - , 0,, 2 O _ -

a (gll)" tgllP g22

o 2

(g22,_) [N]T[A22][N] + I____[N]T[Assl[N] +

(g_] g22) 2 (R1) 2

O O

---_1 (IN, q] - g_l'____3_[NJlr[A66]([N,n ] - glJ'._____3_ [N]) } dgdrl

0 0

(g2_) 2 gli g_J

(145)

NASA/TM--2001-210892 39
[K12 ]

O

: f { gll,q [N,a]r[A1 ][N] + 1 [N,e.]r[Al,][N,n ]

o ", o 1 o o -

A (gll)-g22 gllg22

o o o

gll'n g22"_.[N]r[AI2][N] + gz2'n [N]V[A22][N,,_] +

(gl'i gO)2 ._ gll,, (g22)- ,,_

o O

1 ([N,n] _ glp,_.___![N])T[A66]([N,_ ] _ gz2,____[N] ) } d_dr 1

() o o O

gJl g22 gil g22

(146)

[Kt3 ] = f { --[N,_]V[AII][N] 1 +--[N,e]r[AI2][N] 1 +

a gl°l RI gl°l R2

o o

g22,_ [N]r[Al2] [N] + "- _ [N]T[A22 ][N] +

0 0 0 0

gll g22 Rl gll g22 R2

_____1 ( [N,n ] -gll,q [N] )Via55 ] [N,J } d_dr 1

O O

glt g22

(147)

(glOl)2[N,a]r[Btl][N,_ .] +

A " " (gll)2 g22

[N,¢]r[Bl2 ][N] +

O 9

(g22,_)" [N]r[B,,] [N] - 1 [Nlr[Assl[N] +

o o )2 -- (gll g22

O O

1 ([N,n] _ gll,q [N])T[B66]([N,q] _ gll,q [N]) } d_dr 1

0 0 0

g22 gl I gl!

(148)

NASA/TM--2001-210892 40
o

[Kts]- -- f { g"' o -, o [N'_-]:"[BII][N] + ol

A (gll)" g22 gll g22

O O (I

g_2 _

gll.q g22._ [N]T[BI2] [N] + " ' [N]r[B,,]__ [N, n] +

O O ") O

(gll g22)- gll (g2 °12

o o

g,,,_

1 ( [N,q] - g_ t"------2 [N] )T[B66] ( [N,J - _'- _ [N] ) } d{dq

O O O - O

gll gz2 gll g22

o [N'_-]r[B12] [N,q] +

(149)

[ K22 ]

,,.) O

= f { (gl]'q)2oo , [N]r[AII][N] + "gll'noo , [N]r[A12][N'q] +

A (gll g22)" gll (g22)"

I----_[N,q]r[A22][N,n ] + 1 [N]r[A44][N] +

(g2°2)2 (R2)_

o O

__.._.L([N,j gz2,g [N])r[A66]([N,O _ g22,_ [N]) } d{dr 1

o O

gll - g-. g22

(150)

[ K23 ]

A

1

O

g22 Rl

O

o gl l,q

glt'n [N]r[All][N] +

O O

gll g22 R1 gll g22 R2

1

_[N,q]T[Alz][N] +

0

g22 R2

1 [N]r[A441 [N] } d_drl

O

g22 R2

[N]T[AI2 ][N] +

[N,n ]r [A22 ] [N] - (151)

NASA/TM--2001-210892 41
[K24] = f {

,4

o O O

gll'q g'_'_ d.

gll., [N]r[Bll][N._] + .... [N]r[BI2][N] +

(gll o )2 g22o (gll g22o )2

o

l g22,_

o o [N'q ]T[Btz][N'_] + - [N,n]T[Bz2 ][N] +

gll g22 gllo (g22o )2

o 0

i g,,,,,_

([N,J--"_[N])r[B66]([N,n]-gll'n[N]) } d_dq

0 0 - 0 0

gll g22 g22 gll

(152)

f ll.n [N]T[Bll][N] + 2gll,n [N]T[Bp][N,q] +

[K25] = { (gO)2

A (glt o g22)o 2 gll o (g22)-o 9 "

(g22°)2 [N'n]r[B221 IN'n] - . [N]T[A44 ] [NI +

O O

1 ([N,J - g2v-mA[N])r[B661([N,_] - g2v¢ [N]) } d_dr 1

(gl'] g.))2 _. g22.. g22..

(153)

[K33] : f { 1-[N]r[A,, ][N] + 2--_[N]T[A,2][N] +

a R: Rl R2

[N]T[A22][N] + 1 [N,n]r[A44][N,n] +

Rg o ,, (g22)"

1 [N,_]r[Ass][N,_] } d_dr 1

o 9

(gll)-

(154)

NASAfI'M--2001-210892 42
O

[K34 ] = f { 1 [N]rIB_j[N,_] + g22.4 [N]T[BI2][N] +

O O O

A (gll) Rl gll g_': Rj

O

_ [N]r[BL2] [N,_] + g22,_ [N]T[B22] [NI +

0 0 0

g_ R2 g_ g2,_ R,

1

---_) [N, JT[A551[ N] } d_drl

gll

(155)

g o

ta:,,]: f ¢ o ll,no [N]r[Bll][N] +

a gli g22 R_

O

g11._ [N]r[Bz,][N] + I_____[N]r[B,,][N,n ] +

0 0 _ 0 _

gz_ g22 R2 g22 R2

--_) [N,q]T[A44 ] [N] } d_dq

g22

1 [N]r[B12][N,_] +

0

g22 R1

(156)

[K44 ] = f {[N]r[A_5][N] + _[N,¢]r[Dll][N,__] 1 +

A (all)2

2 o o 2

g"" _ (g"2,_)

.... [N,_]T[DI2 ] [N] + - " [N]T[D22 ] [AT] +

(glOl)2 o o o 2 g22 (gl I g22 )

1

o [ o _([N'q]-gll,,1 N])T[D66]([N, rl]-glI,_[N]) } d_dr]

(g22) 2

(157)

NASA/TM--2001-210892 43
fx 51

- ' _ : l][N ] + 1

" g,7g/ " -

0 0

o

gll'll g"_,[Njr[Ol2i[N] +

.... g2,_._

o o "_ 0 [N]r[D22] [As] +

(gii g22)- gl! (g22)2

o o ([m,,iJ - glla----Jl[lVJ)r[D66]([m,_ ] _ g22{ [NJ) } d{dq

O " 15'

gll g22 gii g22

(158)

z o 9

ix,,] : f {[NiT[A44J[N] + _ tglI,n)'_[N]T[DIIJ[N j +

a (gil g22)" "

o

g,l._ [N,4ir[Dt2][N] +

g(_ (g2O)2 (g2_)2 [N,_]r[D22][N,q] +

g22o d O- g,_,_

([N'qi-_fN])SfD661(fN, n]--'"_[NJ) ] d{dq

gll g22 g2';

(159)

[E3oi ] + [Njr_g2_,: [E-_'32J

[N,{] r IN] o o [N]-[N] r [Els] IN]

gtl gll g:'2 gl_ RI

[xirgi °l,,,[ _'"3i! E_3o2 [ ] o IN] + [N,o ]r [N] - [N] r [E24] [N]

gll G2° g22 g22 R2

[Njr ( [E,j] [E3, ]

+ _)[N] + [N,q] r [E2a] IN] + [N,_] r [EI5] [N] d_dr I

ei (g2_)2 (gi_)--_2

O

[N] r NJ + [N, flr [N] + [N] r 22,{

o o N]

o o

gli gil gil g22

[N]r [ o ,, [c#32] E24] [N] +IN] r [gll,q E31![N][N,q]r [N] o o

g22 gii g22 g2°

(160)

NASA/TM--2001-210892 44
[/%] =fA

m o

[ A__]o[ g 22" [ A _] [N] r N,_] + [N] v _ "

() O

gll gll g22

IN]

m m

rA 1

rt_,Jr _,n

IN] Tg('l''[All [N] + IN, n] -_--t_,J

0 o

gll g22 g22

[N]T([AI] + [A,]

R l R e

--)[N]

^ 0

[N] r __[ N,_] + [N] r g'_' -"_ [/_ 2]

o O

gll gll g22

IN]

[N] r gl AI][N] + [N,q] r [N]

o O

gll g22 g22

d_dq (161)

[K ] : f {[N]T( [Gll]o* + _[G22]o

A (gll)- (g22)-

+ [ G33 ] ) IN] } d_ drl (162)

[x_o] =f_{[N]r[P']o [N,_]

gll

+ [N]T [P2] [N,,1] + [N]T[P3] [N] } d_dr I

0

g22

(163)

NASA/TM--2001-210892 45

CHAPTER 6

CASE STUDIES AND DISCUSSION OF RESULTS

Results from representative problems for a variety of different beam, plate, and shell

structures are presented in this section. The accuracy of the current analytical formulation is verified

with comparisons from published experimental data and other analytical models. Additional

numerical studies are also conducted to demonstrate the capabilities of the current formulation to

model the sensory and active behavior of piezoelectric composite materials.

6.1 Comparisons with Room Temperature Experimental Results

Results from previously published room temperature experimental studies are used to verify

the developed analytical formulation. The problems examined consist of a cantilevered piezoelectric

bimorph beam examined by Lee and Moon (1989) and a clamped composite plate with attached

discrete piezoelectric patches studied by Crawley and Lazarus (1991 ).

6.1. l Piezoelectric Bimorph Beam

The cantilevered piezoelectric bimorph beam problem examined by Lee and Moon (1989)

is shown in Figure 2. The bimorph is created by bonding two layers of a piezoelectric polymer

together such that the poling directions are opposite in each layer. In this configuration, the

application of an external voltage will result in a pure bending deformation. The beam has a length

(L) of 8 cm, a width of 1 cm, and a height of 220 _tm. The material properties of the polyvinylidene

fluoride (PVDF) piezoelectric polymer used in the experiment are listed in Table 1. A finite element

analysis of the bimorph beam problem is performed using the developed beam, plate, and shell

elements using sixteen elements along the length of the beam with two discrete layers through-thethickness. All three elements produced similar results, so only one set of predictions are presented.

Figure 3 illustrates the predicted free end (at x/L = 1) displacement obtained by applying active

voltages of between 0-500 Volts, along with the experimental results of Lee and Moon (1989). Good

agreement is observed between the predicted results of the current analytical formulation and the

experimental data.

6.1.2 Composite Plate with Piezoelectric Patches

The cantilevered composite plate problem examined by Crawley and Lazarus (1991) is

shown in Figure 4. The plate is constructed using AS4/3501 Graphite/Epoxy with a length (L) of

29.2 cm, a width (b) of 15.2 cm, and a thickness of 0.83 mm. Two laminate configurations, [0/___45]_

and [30/02] s, are examined with fifteen G-1195 piezoceramic patches with a thickness of 0.25 nun

bonded on both the top and bottom surfaces of the plate. Figure 4 also depicts the finite element

mesh used in the analysis: a 16 by 9 mesh with 8 discrete layers (one for each composite and

piezoelectric layer through-the-thickness) is used for both the plate and shell elements. Only one set

of results is presented since similar results are obtained using each element. The material properties

of the composite and piezoelectric material are listed in Table 2. The purpose of the experimental

NASA/TM--2001-210892 47
studieswasto investigatetheresultingdeflectionsof theplatewhenactivevoltagesareappliedto

thepiezoceramicpatches.Theexperimentaldisplacementsalongthecenterline(y/b=0.5)of theplate

areshownin Figure5 for the [0/_+45]_ plate with an applied voltage of 394 V/mm and in Figure 6

for the [30/0__]_ plate with an applied voltage of 472 V/mm. Good agreement is observed between the

predictions of the current analytical formulation and the experimental results.

PVDF

PVDF

( 8 cm

_ ____/__20_ m

1 cm

Figure 2: Cantilevered piezoelectric PVDF bimorph beam (Lee and Moon, 1989).

NASA/TM--2001-210892 48
Table 1"Materialpropertiesof PVDF piezoelectricpol_zmer(LeeandMoon, 1989).

Propert_ PVDF

ElasticModulus,E(10'_Pa): 2.0

Poisson'sRatio,u: 0.29

ShearModulus,G (10°Pa): 0.8

PiezoelectricChargeConstants(10 q2 m/V):

d31 23.0

d32 3.0

4

B

o 3

.,i4

<D

2

•,-. 1

[--

0

7

!

.......... Current Formulation

+_ Lee & Moon (1989) _-'+'

/

_ _ /

+..-

/

{}//

/

I I t

0 100 200 300 400 500

Electric Voltage (volts)

Figure 3: Comparison of tip deflections of a PVDF bimorph beam under applied

active electric voltages of 0-500 Volts at room temperature.

NASA/TM--2001-2 10892 49
i

I

2.5 cm

Piezoceramic Graphite/Epoxy

-- I

0.25 mm

29.2 cm

• !

"! 5.1 cm

X

0.83 mm

\

:_ I-- " X

-TI__ i

Figure 4: Cantilevered AS4/3501 composite plate with 30 attached discrete

G- 1195 piezoelectric patches (Crawley and Lazarus, 1991).

NASA/TM--2001-210892 50
Table2:Materialpropertiesof piezoceramic(G-1195)and

Graphite/Epoxy(AS4/3501)(CrawleyandLazarus,1991).

Property' Piezoceramic Graphite/Epox]¢

ElasticModuli (109Pa):

Ell 63.0 143.0

E22 63.0 9.7

E33 63.0 9.7

Poisson's Ratio:

DI2

1)23

1)31

0.3 0.3

0.3 0.3

0.3 0.3

Shear Moduli (109 Pa):

GI 2 24.2 6.0

G23 24.2 2.5

G3I 24.2 6.0

Piezoelectric Charge Constant (10 -12m]V) •

d3! 254.

d32 254.

Electric Permittivity (10 .9 f/m):

ell 15.3

e22 15.3

E33 15.0

NASA/TM--2001-210892 51
_D

cD

• _,,,q

° ,,,,,,q

_D

,4,,,a

0.)

r,.)

0.060

0.048 i

i

i

0.036

0.024 -

0.012

Current Formulation

Crawley & Lazarus (1991 )

/,

/

O /

/

/

tD///

//

/

0.000

0.00 0.20 0.40 0.60 0.80 1.00

Axial Length (x/L)

Figure 5: Comparisons of centerline displacement of an AS4/3501 [0/_+45]_ plate with

attached G-1195 piezoelectric patches under 394 V/mm at room temperature.

NASA/TM--2001-210892 52
0.060 r

/:

= 0.048

0.036

I

•- 0.024 _

--'- 0012. -

0.000

0.00

g

Current Formulation

Crawley & Lazarus (1991 )

0/ ̧

/

®

/

/

O/

/

i

_.---0- --II-_ I I i

0.20 0.40 0.60 0.80 1.00

Axial Length (x/L)

Figure 6: Comparisons of centerline displacement of an AS4/3501 [30/0 2]_ plate with

attached G-1195 piezoelectric patches under 472 V/mm at room temperature.

NASA/TM--2001-210892 53
6.2 Comparisons with Analytical Results for Temperature Effects

Results from previously published high temperature analytical studies are used to verify the

developed analytical formulation. The problems examined consist of a cantilevered piezoelectric

bimorph beam examined by Sunar and Rao (1997) and a simply supported composite plate with

attached discrete piezoelectric patches investigated by Ha et al. (1992). These problems present slight

variations on the room temperature experimental studies of Lee and Moon (1989) and Crawley and

Lazarus (1991 ) to analytically quantify the influence of temperature effects.

6.2.1 Piezoelectric Bimorph Beam

The cantilevered piezoelectric bimorph beam problem examined by Sunar and Rao (1997)

is shown in Figure 7. The bimorph has a length (L) of 100 mm, a width of 5 mm, and a height of 1

ram. The material properties of the polyvinylidene fluoride (PVDF) piezoelectric polymer at 27 °C

and at 77°C are listed in Table 3. An applied electric voltage of 250 Volts was applied to the

bimorph beam at 27°C to induce a bending deformation. The impact of thermal effects on the

deflection of the bimorph is subsequently examined by applying a uniform 50°C temperature

increase. The current finite element analysis is performed using 20 elements along the length of the

beam with two discrete layers through-the-thickness. All three elements (i.e. beam, plate, and shell)

produced similar results, so only one set of predictions are presented. Figure 8 shows the predicted

deflection of the bimorph along the length of the beam under an applied active voltages of 250 Volts

at 27 °C and 77 °C, along with the corresponding analytical results of Sunar and Rao (1997). Good

agreement is observed between the two sets of results, which demonstrate the impact of temperature

effects on the deflection of the bimorph beam.

6.2.2 Composite Plate with Piezoelectric Patches

The simply supported composite plate problem examined by Ha et al. (1992) is shown in

Figure 9. The [0/_+45]_ plate has a length (L) of 37.2 cm, a width (b) of 22.8 cm, and a thickness of

0.75 mm. Fifteen equally sized G-1195 piezoceramic patches with a length of 6 cm, a width of 6 cm,

and a thickness of 0.13 mm are bonded on both surfaces of the plate. Figure 9 also depicts the finite

element mesh used in the analysis: a 16 by 10 mesh with 8 discrete layers (one for each composite

and piezoelectric layer through-the-thickness) used for both the plate and shell elements. Only one

set of results is presented since similar results are obtained using each element. The material

properties of the composite and piezoelectric material are listed in Table 4. Analytical studies were

performed to investigate the capability to achieve thermal distortion management by applying active

voltages to the piezoelectric patches to eliminate the bending deflection induced by applying a

100 °C linear thermal gradient through-the-thickness of the plate (50°C on the top surface and -50°C

on the bottom surface). The predicted displacements along the centerline (y/b=0.5) of the plate along

with corresponding applied active voltages to produce this deflection are shown in Figure 10, along

with the corresponding results from Ha et al. (1992). Good agreement is observed between the two

predictions. The results demonstrate the capability to achieve shape control of the thermally induced

bending deformation by the application of increasing active voltages to the piezoelectric actuators.

NASA/TM--2001-210892 54
-250

250

PVDF

PVDF

100mm

Figure 7: Cantilevered piezoelectric PVDF bimorph beam (Sunar and Rao, 1997).

NASA/TM--2001-210892 55
Table 3: Material properties of PVDF piezoelectric polymer

(Sunar and Rao, 1997).

Propert_ 27 °C 77 °C

Elastic Compliance, c,_ (109 Pa): 3.8 1.8

Thermal Expansion, a (10 .6 m/m°C): 150. 215.

Piezoelectric Constant, e31 (C/m2): 0.046 0.049

Electric Permittivity, _ (10 .9 f/m): 0.1026 0.1553

Pyroelectric Constant, P (10 .3 C/m-_°C): 0.04 0.055

Reference Temperature, TI_(°C): 27.0 27.0

0.25

0.20

o 0.15

0.10

cD

.._ 0.05

[..

0.00

-- Current Formulation

Sunar and Rao (1997) _*

o

i

77 °C

e

/

L e

....... _::_:-5_ r

-e_ -_'_ 27 °C

0 2 4 6 8 10

Axial Distance (cm)

Figure 8: Comparison of tip deflections of a PVDF bimorph beam under an applied

active electric voltages of 250 V at 27 °C and 77°C.

NASAITM--2001- 210892 56
!

Piezoceramic Graphite/Epoxy 0.75 mm

I __

0.13 mm

37.2 cm

X

X

Figure 9: Simply supported AS4/3501 composite plate with 30 attached discrete

G-1195 piezoelectric patches (Ha et al., 1992).

NASA/TM--2001-210892 57
Table 4: Material properties of piezoceramic (G- 1 195) and

Graphite/Epoxy (AS4/3501) (Ha et al., 1992).

Property, Piezoceramic Graphite/Epoxy

Elastic Moduli (10 ° Pa):

Ell 63.0 150.0

E,, 63.0 9.0

E33 63.0 9.0

Poisson's Ratio:

!)12

1)23

1)31

0.3 0.3

0.3 0.3

0.3 0.3

Shear Moduli (10 ° Pa):

GI2 24.2 7.1

G23 24.2 2.5

G31 24.2 7.1

Piezoelectric Charge Constant (10 q2 m]g) •

d31 254.

d32 254.

Electric Permittivity ( 10 -9 f/m):

etl 15.3

_,, 15.3

_33 15.0

Thermal Expansi on ( 10 .6 m/m °C):

(111 0.9

(1_ 0.9

(133 0.9

1.1

25.2

25.2

Reference Temperature, T O(° C): 20.0 20.0

NASA/TM--2001-210892 58
2.50 E

E

= 2.00

©

°¢,,_

1.50

1.00

• w...*

i

i

i

F

I

!

i

Current Formulation

Ha et al. (1992)

0V _/

31V

/

//

N f ,

/

/

" * 61 V

0.50 _ _,\,

/ /

0.00 .... _................................... ................

0.00 0.20 0.40 0.60 0.80 1.00

Axial Distance (x/L)

Figure 10: Comparisons of centerline displacement of an AS4/3501 [0/_+45]_ plate with

attached G-1195 piezoelectric patches under a 100 °C thermal gradient through-thethickness (50°C on top, -50°C on bottom) with 0, 31, and 61 V applied active voltages.

NASA/TM--2001-210892 59
6.3 Comparisons with a Commercial Finite Element Analysis Program

Evaluations of the developed analytical formulation for both the sensory and active responses

are verified with corresponding results from a commercial finite element analysis program,

ABAQUS (1996). In order to represent the location of piezoelectric layers, the standard laminate

notation is expanded, such that the piezoelectric layer is indicated by the letter p. The problem

examined consists of a Carbon/Epoxy composite plate with an attached layer of piezoceramic

material. The [0s/p] plate has a length (a) of 50.8 mm, a width (b) of 25.4mm, and a height (h) of

5.08 mm, as shown in Figure 11. The plate is clamped on one side and is subjected to a uniform

thermal load of 50°C. The material properties used are listed in Table 5. A 20 x 10 mesh with two

discrete layers (one for the Carbon/Epoxy and one for the piezoelectric layer) is used for both the

plate and shell elements from the current formulation, while a 10 x 5 x 2 mesh is used for the 20

noded continuum elements in ABAQUS. Only one set of results is presented for the current

formulation, since similar results were obtained for both the plate and shell elements. Pyroelectric

effects are intentionally neglected (i.e. set to zero) in the current formulation, since ABAQUS

neglects such effects.

6.3.1 Sensory Mode

The piezoelectric layer can be utilized in either a sensory or active mode. In the sensory

mode, the piezoelectric layer is free to develop a corresponding electric potential in response to the

thermally induced deformation of the plate through the direct piezoelectric effect. The resulting

displacements along the centerline of the plate (along y/b = 0.5) are shown in Figure 12, while the

corresponding electric potentials which develop in the piezoelectric layer are shown in Figure 13.

There is good overall agreement between the current formulation and the ABAQUS results. The

differences in the sensory electric potentials arise due to the consideration of the through-thethickness Poisson's effect in ABAQUS, which are neglected in the current formulation. In typical

health monitoring applications, these sensory voltages would be used to provide the essential

feedback necessary to infer the current state of deformation in the structure.

6.3.2 Active Mode

The piezoelectric layer can also be operated as an actuator by applying a voltage differential

to the piezoelectric layer and utilizing the converse piezoelectric effect to produce a corresponding

structural deformation. Figure 14 shows the centerline deflection (along y/b = 0.5) for three different

applied electric potentials (0, 250, and 500 Volts). Good agreement is observed between the current

formulation and the ABAQUS predictions. The results also demonstrate the potential of piezoelectric

actuators to completely alter the initial thermally induced deformation of the plate by the application

of increasing electric potentials in order to achieve thermal shape control.

NASA/TM--2001-210892 60
Y

/

/

/

/

/

/

/

, ,

b = 25.4 mm

Piezoceramic Carbon/Epoxy

a = 50.8 mm >

h = 5.08 mm

X

m

Figure 11: Clamped Carbon/Epoxy composite plate with an attached

piezoceramic layer.

NASA/TM--2001-210892 61
Table 5: Material properties of piezoceramic and Carbon/Epoxy.

Property Piezoceramic Carbon/Epoxy

(Daniel and Ishai, 1994)

Elastic Moduli (109 Pa):

Ell 69.0 142.0

E_ 69.0 10.3

__

E33 69.0 10.3

Poisson's Ratio:

DI2

1)23

1)31

0.3 0.27

0.3 0.27

0.3 0.20

Shear Moduli ( 109 Pa):

Gl2 26.5 7.20

G23 26.5 4.29

G31 26.5 7.20

Piezoelectric Charge Constant (10 -_z m/V) •

d31 - 154.

Electric Permittivity (10 9 tim):

_1_ 15.05

_,, 15.05

_33 15.05

Thermal Expansion (10 -6 m/m°C):

_tH 1.2

a22 1.2

Reference Temperature, T O(°C): 20.0 20.0

NASA/TM--2001-210892 62
O

O

0.00

-0.10

-0.20

-0.30

-0.40

0.00

k . •

 o uaion

i I I

0.20 0.40 0.60 0.80 1.00

x/a

Figure 12: Comparison of centerline sensory deflections of a clamped [08/p]

Carbon/Epoxy plate with a piezoceramic layer under a uniform 50°C load.

NASA/TM--2001-210892 63
-125r

o _ Current Formulation

_=-100 .....

•-_ ',_ : ABAQUS

= \

_' 75 O

.r-

•_ -50

_' -25

O

\• _!;ii:i i_! ii!i_ iiiili; !_i_ ,iiiii! _iiiii!_i_i_i iiiii _ii!iii ii!ii :i_ii!_:_i_ii_iiii;i ....

i ........ I i

0.00 0.20 0.40 0.60 0.80 1.00

x/a

Figure 13: Comparison of centerline sensory electric potentials of a clamped [0s/p]

Carbon/Epoxy plate with a piezoceramic layer under a uniform 50°C load.

NASA/TM--2001-210892 64
..¢:

O

O

0.00 ,

{

-O.lO

-0.20

-0.30 -

OA=OV

! . ABAQUS

....................... Current Formulation

-0.40 • ....

0.00 0.20 0.40 0.60 0.80

x/a

1.00

Figure 14: Comparison of centerline deflections of a clamped [08/p] Carbon/Epoxy

plate with a piezoceramic layer under a uniform 50°C load with applied

active voltages of 0, 250, and 500 V.

NASA/TM--2001-210892 65
6.4 Additional Numerical Studies

Results from numerical studies are presented to demonstrate additional capabilities of the

current formulation to model the behavior of piezoelectric composite structures. The capabilities

investigated include: incorporating pyroelectric effects and temperature dependent material

properties into the analysis and evaluating the significance of these effects on the thermal response;

demonstrating capabilities to achieve selective shape control of thermally induced bending and

twisting deformations; assessing the improved accuracy of layerwise models in predicting thermal

stresses; and determining the influence of curvature on the sensory and active responses.

6.4.1 Pyroelectric Effects

Pyroelectric effects arise in piezoelectric material due to temperature dependent changes in

the electrical properties. This behavior is analogous to the change in mechanical strains observed in

elastic materials due to thermal expansion. In order to demonstrate the effect of incorporating

pyroelectric effects, the clamped plate problem shown in Figure 11 is re-examined incorporating

pyroelectric effects. The problem details remain the same as before except for the consideration of

non-zero pyroelectric constants in the piezoelectric material properties as listed in Table 6. The

influence of the pyroelectric effects on the centerline deflection of the plate when the piezoelectric

layer is operating in a sensory mode is shown in Figurel 5 along with the corresponding result when

the pyroelectric constant is neglected. The corresponding results for the sensory electric potential

which develops is shown in Figure 16. Both figures demonstrate that pyroelectric effects can have

a significant impact on both the displacements and electric potentials of piezoelectric composite

structures in thermal environments.

NASA/TM--2001-210892 66
Property

Table 6: Material properties of piezoceramic and Carbon/Epox]¢.

Piezoceramic C arbon/Epox y

(Daniel and Ishai, 1994)

Elastic Moduli (109 Pa):

ELi 69.0 142.0

E_ 69.0 10.3

E33 69.0 10.3

Poisson's Ratio:

!)12

1)23

1)3l

0.3 0.27

0.3 0.27

0.3 0.20

Shear Moduli (109 Pa):

GI2 26.5 7.20

G23 26.5 4.29

G31 26.5 7.20

Piezoelectric Charge Constant (10 -12 m/V) :

d3t -154.

Electric Permittivity (10 -9 f/m):

I_11 15.05

_22 15.05

_33 15.05

Thermal Expansion (10 6 m/m °C):

511 1.2

oh2 1.2

Pyroelectric Constant (10 -3 C/m 2 eC):

P3 -0.2

Reference Temperature, To (°C): 20.0 20.0

NASA/TM--2001-210892 67
0.00

-0.10

-0.20

x/a

Figure 15: Pyroelectric effects on the centerline sensory deflections of a

clamped [08/p] Carbon/Epoxy plate with an attached piezoceramic layer

under a uniform 50°C load.

NASA/TM--2001-210892 68
©

>

e03

©

"t"

¢3

03

O

03

-600

-450

Pyroelectric Effects

-300 -

i

- 150 No Pyroelectric Effects

o

0.00 0.20 0.40 0.60 0.80 1.00

x/a

Figure 16: Pyroelectric effects on the centerline sensory electric potentials of a

clamped [08/p] Carbon/Epoxy plate with an attached piezoceramic layer

under a uniform 50°C load.

NASA/TM--2001-210892 69
6.4.2 Temperature Dependent Material Properties

The various elastic, dielectric, piezoelectric, and pyroelectric material properties of

piezoelectric materials are influenced differently by temperature variations. In addition, piezoelectric

materials also possess a characteristic limiting temperature, called the Curie temperature, beyond

which the material loses its piezoelectric properties. Thus, in typical applications the operating

temperature of piezoelectric materials must be considerably lower than the Curie temperature. In the

current study, the temperature dependent variations of an AS4/3501-6 Carbon/Epoxy composite and

a PZT-5A piezoceramic were implemented using a piecewise linear representation irr the constitutive

equations (Equations 23-26). The temperature dependence of the AS4/3501-6 moduli and ct22 are

depicted in Figure 17, while the variation of the PZT-5A properties d3L and _33 are shown in Figure

18, while ctL_, and P3 are shown in Figure 19. All properties are nondimensionalized with their

respective room temperature values shown in Table 7. The simply supported plate with discrete

piezoelectric patches examined in Figure 9 is used to demonstrate the effects of temperature

dependent material properties. Linear thermal gradients of different values are applied through-thethickness of the plate with the bottom surface (z/h =-0.5) fixed at 20°C and the upper surface (z/h

= 0.5) varying from 45 ° to 170°C.

The first case utilizes a combined active/sensory configuration of the piezoceramic patches,

in which the upper patches are utilized as sensors and the lower patches are used as actuators with

zero volts applied. All the piezoceramic patches are grounded on the surface in contact with the

carbon/epoxy plate. The resulting thermally induced deflection at the center of the plate is shown in

Figure 20 for different applied thermal gradients. The deflections are nondimensionalized with

respect to the total laminate thickness (H = 1.01 mm). The results show that when material properties

are assumed to remain constant with temperature, a linear variation in deflection occurs. In contrast,

a nonlinear variation in the deflection is observed when the temperature dependent properties are

incorporated. Larger discrepancies occur between the two cases as larger thermal gradients are

applied. The corresponding sensory voltages at the central piezoceramic patch on the upper surface

of the plate is shown in Figure 21. As with the deflections, the assumption of constant material

properties leads to a linear variation of the voltages, while nonlinearities are introduced when

temperature dependent properties are modeled. Closer agreement is observed in this case.

A fully active configuration of the piezoceramic patches is used to investigate the effect of

temperature dependent material properties on thermal shape control. Equal voltages are applied to

all the piezoceramics. The applied voltages required to equally minimize the deflection at the center

of the plate to zero for different applied voltages are shown in Figure 22. Once again, a nonlinear

variation is predicted by the temperature dependent case and an increasing discrepancy is observed

as the thermal gradient increases. This discrepancy corresponds to the trend observed for the center

deflection of the plate (Figure 20). Since larger deflections occur for the temperature dependent case,

this in part justifies the larger applied voltages required to minimize the deflections.

NASA/TM--2001-210892 70
¢;

©

N

°,,N

O

_D

E

O

Z

1.50 ? ........

1.20

0.90

0.60

0.30

!

I

EI_

..... _{

...-4t

I E22 , E33, G's

0.00 i I

0 40 80 120 160 200

Temperature (°C)

Figure 17: Temperature dependence of AS4/3501-6 Carbon/Epoxy

properties (Daniel andlshai, 1994).

NASA/TM--2001-210892 71
_D

O

_D

_q

°_,_

O

r/3

E

°_

O

Z

3.00

2.40

1.80

1.20

0.60

!

i

IN

J_

.,_ .............

.;iff

___ E33

0.00 i , l

-300 -200 - 100 0 100 200 300

Temperature (°C)

400

Figure 18: Temperature dependence of PZT-5A properties d31 and c33.

NASA/TM--2001-210892 72
ca,

©

N

• v_,,(

©ovil

r/b

E

5

I-

!

31

I

I

I

I

i

2

_

O I

Z o L

-300

D

P3 /'

II ....... m .... --t_7_'- -It i "

IIc

i I

-200 - 100 0 100 200 300

Temperature (°C)

400

Figure 19: Temperature dependence of PZT-5A properties a_ and P3.

NASA/TM--2001-210892 73
Table 7: Material properties of piezoceramic (PZT-5A) and Carbon/Epoxy

(AS4/3501-6) composite.

Piezoceramic Carbon/Epoxy

(Daniel and Ishai, 1994)

Elastic Moduli (109 Pa):

Ell 69.0 142.0

E,, 69.0 10.3

E33 53.0 10.3

Poisson's Ratio:

u_2 0.31 0.27

u23 0.44 0.20

u31 0.38 0.02

Shear Moduli (109 Pal):

Gl2 23.3 7.2

G23 21.1 4.29

G31 21.1 7.2

Thermal Expansion ( 10 -6 m/m°C):

all 1.2 -0.9

a_, 1.2 27.0

Piezoelectric Charge Constant (10 -12m/V) :

d31 -154.

Electric Permittivity (10 -9 f/m):

E33 15.05

Pyroelectric Constant (10 3 C/m 2 °C):

P3 -2.0

Reference Temperature, T o, (°C): 20. 20.

Curie Temperature, T c, (°C): 365.

NASA/TM--2001-210892 74
4

3 Temperature Dependent

2

tant

i I

0

0 50 100 150 200

Applied Gradient (°C)

Figure 20: Influence of temperature dependent material properties on the deflections

at the center of a [0/-+45] SCarbon/Epoxy plate with 30 attached piezoceramic

patches in an active/sensory mode with varying applied thermal gradients

through-the-thickness (25 °C to 150°C).

NASA/TM--2001-210892 75
300

>

¢=

©

>

©

200

lO0

0

Constant

_" ._ : ii

// _ Temperature Dependent

0 50 100 150 200

Applied Gradient (°C)

Figure 21: Influence of temperature dependent material properties on the electric

potentials at the center of a [0/+45]sCarbon/Epoxy plate with 30 attached

piezoceramic patches in an active/sensory mode with varying applied

thermal gradients through-the-thickness (25 °C to 150°C).

NASA/TM--2001-210892 76
;>

e_0

4_a

©

>.

,,,,,a

<

300

200

Temperature Dependent

100 - _

. " /O f

I

0

Constant

0 5 0 100 150 200

Applied Gradient (°C)

Figure 22: Influence of temperature dependent material properties on the active

voltages to minimize deflections at the center of a [0/_+45LCarbon/Epoxy plate

with 30 attached piezoceramic patches in an active/active mode with varying

applied thermal gradients through-the-thickness (25 °C to 150°C).

NASA/TM--2001-210892 77
6.4.3 Selective Shape Control of Bending and Twisting

The potential of piezoelectric actuators to selectively minimize thermally induced

deformations is demonstrated on a clamped [453/-453] plate with attached piezoelectric patches. The

geometry and material properties remain the same as in Figure 9 and Table 4, respectively. The plate

is clamped along x=0 and a 30°C thermal gradient is applied through-the-thickness with the

piezoelectric patches operating as actuators. The temperature dependence of material properties is

neglected in this study. The resulting initial deformation is shown in Figure 23 and shows a

combined bending and twisting deformation induced by the thermal gradient. The bending

deformation can be selectively minimized by applying the same voltages to the top and bottom

piezoceramic patches. The resulting deformation when 40 V is applied is shown in Figure 24. The

twisting deformation remains unaffected and is now more apparent due to the removal of the

bending. In a similar manner, the twisting deformation can be minimized without affecting the

bending behavior. This is accomplished by applying voltages of opposite polarities on the top and

bottom surface piezoceramic patches. The deformation produced by the application of 145 V on the

top piezoceramic patches and -145 V on the bottom piezoceramic patches is shown in Figure 25. The

bending deformation is more apparent due to the removal of the twisting. In order to minimize both

the bending and twisting deformations simultaneously, the previous two cases can be superimposed.

Thus, by applying 185 V on the top actuators and -105 V on the bottom actuators, all deformations

in the plate can be eliminated as shown in Figure 26.

NASA/TM--2001-210892 78
15 -

10 "

5

-5

-10

-15 --- - 0.60

0.20 0.40 .... - _ -_ -- -- 0.80

0.60

0.80------ -_ 1.00 Y/b 1.oo

L--~ _ _ _

.... - 0.40

Y

Figure 23: Initial bending and twisting deformation of a clamped [453/45_3]

AS4/3501 plate.with 30 attached discrete G-1195 piezoelectric patches

under a 30°C thermal gradient through-the-thickness.

NASA/TM--2001-210892 79
15 -

10

-5

-10

-15

- _f

oO OO

0.40

0.60 _¢'

-_ - -_ ..... _ 0.80 0.20- 0.40 0.60 ..... -

Y/b o.so ........ 1.00 1.00

Figure 24: Selective shape control of the bending deformation of a clamped [453/45_3]

AS4/3501 plate.with 30 attached discrete G-1195 piezoelectric patches

under a 30°C thermal gradient through-the-thickness by applying

40 V to all piezoceramic patches.

NASA/TM--2001-210892 80
15 -"

I

i

10

I

5 _

Figure 25: Selective shape control of the twisting deformation of a clamped [453/45.3]

AS4/3501 plate.with 30 attached discrete G-1195 piezoelectric patches

under a 30 °C thermal gradient through-the-thickness by applying

145 V to top patches and -145 V to bottom patches.

NASA/TM--2001-210892 81
15

i

10 _

0

-5

-10

-15

"=: _ _ .... 0.20

i ......... -_ -- _--- _ .... 0.40

-- _ - -_- 0.60 ,,_'_

0.20_" 0.40- 0.60-.... _ --- - -'-0.80

Y/b o.8o-- ...... 1.00 1.00

Figure 26: Shape control of both bending and twisting deformation of a clamped

[453/45.3] AS4/3501 plate.with 30 attached discrete G-1195 piezoelectric patches

under a 30°C thermal gradient through-the-thickness by applying

185 V to top patches and -105 V to bottom patches.

NASA/TM--2001-210892 82
6.4.4 Thermal Stresses

The increased accuracy gained from implementing a layerwise laminate theory is

demonstrated by examining the thermal stresses in a cantilevered [08/p] beam. The beam is depicted

in Figure 27 and has a length (L) of 25.4 cm and a width (b) of 2.54 cm. Two thicknesses (H) are

examined; a thin beam (H = 0.1143 cm) with an aspect ratio (L/H) of -200 and a thick beam (H =

11.43 cm) with an aspect ratio (L/H) of -2. The beam is composed of eight unidirectional EGlass/Epoxy plies and one piezoelectric layer. The material properties are listed in Table 8. The

beam element is used to conduct the analysis using 30 elements along the length of the beam, with

one discrete layer modeled through-the-thickness for each piezoelectric and E-Glass/Epoxy ply. A

uniform temperature rise of 100°C is applied with the piezoelectric layer grounded (i.e. 0 V applied).

The predicted normal stress, g_1, near the middle of the beam (x/L = 0.5) is shown in Figure 28 for

both the thin and thick beams, along with the corresponding results predicted by classical laminate

theory (CLT). The normal stress is nondimensionalized using the equivalent laminate modulus, EH_

= 42.2 109 Pa. The CLT predicts the same stress regardless of the thickness of the beam. At this

location of the beam, there is good agreement between the current formulation and CLT regardless

of the beam thickness. The predicted normal stress, ¢_, at the free end of the beam (x/L = 1.0) is

shown in Figure 29 and shows that CLT is only accurate for the thin beam. The predicted shear

stress, ¢Y_3,at the free end of the beam (x/L = 1.0) is shown in Figure 30. The shear stress is

nondimensionalized using the equivalent laminate modulus, GL_3 = 6.3 109 Pa. The results show that

the CLT assumption of neglecting shear stresses is only accurate for the thin beam case. These

results help demonstrate the improved accuracy in stress predictions achieved by using the layerwise

laminate theory, especially for thick laminates.

NASA/TM--2001-210892 83
Piezoelectric Layer

\

[08 ] E-Glass/Epoxy

25.4 cm

Figure 27: Cantilevered [0p/p] E-Glass/Epoxy and

piezoelectric beam

NASA/TM--2001-210892 84
Table 8: Material properties of piezoelectric and E-Glass/Epoxy.

Elastic Moduli (109 Pa):

Ett

E33

Piezoelectric E-Glass/Epoxy

(Daniel and Ishai, 1994)

68.0 39.0

68.0 8.6

Poisson's Ratio:

1)13

0.30 0.28

Shear Modulus (100 Pa):

Gl3 26.2 3.8

Thermal Expansion ( 10-6 m/m °C):

(111

(133

3.8 7.0

3.8 21.0

Piezoelectric Charge Constant (10 12 m/V) •

d31 -125.

Electric Permittivity (10 9 f/m):

E33 11.06

Pyroelectric Constant (10 .3 C/m 2 °C):

P3 -2.5

Reference Temperature, T o, ( °C): 20. 20.

NASA/TM--2001-210892 85
O

0.40 ....

0.24

• CLT

0.08 -0.08

Thin Beam i

Thick Beam _-_*_

-0.24

!

I

-0.40

-0.50-0.30-0.10 0.10 0.30 0.50

z/H

Figure 28: Comparison of normal stresses at the middle (x/L=0.5) of a [0s/p]

E-Glass/Epoxy and piezoelectric beam under a uniform 100°C load for two

aspect ratios (thin=200 and thick=2).

NASA/TM--2001-210892 86
0.40

Thin Beam

0.24 _ Thick Beam

+ CLT

0.08 ..

_ -0.08

-0.24

-0.50-0.30-0.10 0.10 0.30 0.50

z/H

Figure 29: Comparison of normal stresses at the free end (x/L=l.0) of a [0s/p]

E-Glass/Epoxy and piezoelectric beam under a uniform 100°C load for two

aspect ratios (thin=200 and thick=2).

NASA/TM--2001-210892 87
.d

m

4

3

2

0

-1

Thin Beam

Thick Beam

CLT

'l

t

!

I

..... I

I

I

i !

' 1

,!

J ........ , I

_,4"_t *,*,_*_ *J-"'*_,_'__°i.* • ,_,

! k

-0.50-0.30-0.10 0.10 0.30 0.50

z/H

Figure 30: Comparison of shear stresses at the free end (x/L=l.0) of a [0s/p]

E-Glass/Epoxy and piezoelectric beam under a uniform 100°C load for two

aspect ratios (thin=200 and thick=2).

NASA/TM--2001-210892 88
6.4.5 Influence of Curvature on Active and Sensory Response

The problem examined consists of a [09/p] (where 0 ° is along the _ axis) circular cylinder

with a radius (R) of 0.76 m, a length (L) of 1.524 m, and a thickness (h) of 7.62 mm as depicted in

Figure 31. The cylinder is composed of nine unidirectional Carbon/Epoxy plies and one

piezoceramic layer. The material properties are listed in Table 6. Due to the symmetry of the

problem, only 1/8of the cylinder is modeled using a 10x 10 mesh with two discrete layers (one for the

Carbon/Epoxy and one for the piezoceramic). A clamped-clamped configuration in which both ends

of the cylinder (z/L = 0 and z/L = 1) are fixed is examined. The cylinder is subjected to three

different types of thermal loads (a uniform thermal load and two types of sinusoidally varying

temperatures along the hoop direction) as shown in Figure 32. Results from the current shell element

are presented to investigate the influence of the curved structure on the sensory and active response

under the different applied thermal loads. All displacements (w) and electric potentials (q_)presented

in this section are along the hoop direction at z/L = 0.5 and r/R =1.0 of the cylinder.

The influence of pyroelectric effects on the displacements and sensory electric potentials of

the cylinder under the three different applied thermal loads are presented in Figures 33-38. Each

figure contains results for two cases to show the effects of incorporating the pyroelectric constant

in the analysis (the p3=0 case neglects pyroelectric effects). Figures 33 and 34 show, respectively,

the displacement and sensory electric potential induced from applying a uniform thermal load of

50 °C. A uniform deflection of the cylinder is achieved, which maintains the original circular shape.

The sensory electric potential also shows a uniform response that corresponds to the displacement.

The incorporation of pyroelectric effects produces larger displacements and electric potentials. The

results from applying a cosine varying temperature are shown in Figures 35 and 36. The deflection

of the cylinder now shows a sinusoidal pattern that translates into either an oval-shaped or figure

eight shaped circumferential displacement, depending on whether pyroelectric effects are modeled.

The electric potential only displays a sinusoidal pattern when pyroelectric effects are included and

becomes almost zero when the pyroelectric effect is neglected. Figures 37 and 38 show the results

of applying a double cosine varying thermal load. A sinusoidally varying displacement and electric

potential pattern is produced which corresponds to an oval-shaped deflection of the cylinder. Once

again, incorporating the pyroelectric effect leads to increased displacements and electric potentials.

The active response of the cylinder under the three applied thermal loads is examined in

Figures 39-41. Each figure shows the displacement of the cylinder for three configurations of the

piezoelectric layers: a sensory mode, a grounded configuration (zero electric potential applied), and

an active mode. The magnitude of the applied electric potential remains the same for all three cases,

although the form of the electric potential varies to correspond to the type of thermal load. Figure

39 shows the deflection of the cylinder under a uniform thermal load (50°C). Application of

increasing uniform applied electric potentials produces a noticeable decrease in the thermally

induced sensory deflection. Figure 40 depicts the displacement of the cylinder under a cosine

varying thermal load and shows that although some changes in the overall shape of the cylinder can

be obtained by applying active electric potentials, no significant overall reduction of the thermal

deflection is achieved. Figure 41 illustrates the deflections produced under a double cosine variation

of the thermal load. For this case, the double cosine applied electric potential effectively minimizes

the thermal deflection.

NASA/TM--2001-210892 89
X

_-_-- L = 1.$24 m

_ _;___zoceramic (0.7 mm thick)

_l _ (_ Carbon/Epoxy (7.55 mm thick)

___ 1/8 of Cylinder

Figure 31 : Geometry of a [09/p] Carbon/Epoxy circular cylinder with an

attached piezoceramic layer.

NASA/TM--2001-210892 90
T=50 oC

i/'

L_

T=50 °C

- _% - ,

Y

T=25 o C

(a) Uniform T (b) T cos y (c) T cos 2 y

• i

Y

Figure 32: Different types of applied thermal loads applied to the cylinder:

a) 50°C uniform thermal load; b) 50°C cosine varying load; and

c) 25 °C double cosine varying load.

NASAfI'M--2001-210892 91
0.20 F...........

0.10 i

I

0.00

010

__ A A _ A A A A A

p3=-2e - 4

P3=0

-0.30

0 30 60 90

7 (degrees)

Figure 33: Influence of pyroelectric effects on the displacements of a [09/p]

Carbon/Epoxy circular cylinder with a piezoceramic layer under

a 50°C uniform thermal load.

NASA/TM--,.001-21089_ 92
0.20

0.10

0.00

=_ -0.10

-e-

-0.20

-0.30

0 30 60 90

y(degrees)

Figure 34: Influence of pyroelectric effects on the sensory electric potentials of a [09/p]

Carbon/Epoxy circular cylinder with a piezoceramic layer under

a 50°C uniform thermal load.

NASA/TM--2001-210892 93
0.20

0.10

0.00

-0.30

i

_- _:. 5 _ '_

i- °°..° °°°_

0 30 60 90

_/(degrees)

p3=-2e-4

P3 =0

Figure 35: Influence of pyroelectric effects on the displacements of a [09/p]

Carbon/Epoxy circular cylinder with a piezoceramic layer under

a 50°C cosine varying thermal load.

NASA/TM--2001-210892 94
0.20

0.10

0.00

v-¢

:_ .0.10 I

-e- l

-0.20

-0.30

0 30 60 90

T (degrees)

Figure 36: Influence of pyroelectric effects on the sensory electric potentials of a [09/p]

Carbon/Epoxy circular cylinder with a piezoceramic layer under

a 50°C cosine varying thermal load.

NASA/TM--2001-210892 95
0.20 ...............

0.10

-- 0.00

-0.10

v-q

-0.20

-0.30 '

0 30 60 90

p3=-2e-4

P3=O

7 (degrees)

Figure 37: Influence of pyroelectric effects on the displacements of a [09/p]

Carbon/Epoxy circular cylinder with a piezoceramic layer under

a 25 °C double cosine varying thermal load.

NASA/TM--2001-210892 96
"O

-e0.20

0.10

0.00

-0.10

-0.20

-0.30

0 30 60 90

V (degrees)

Figure 38: Influence of pyroelectric effects on the sensory electric potentials of a [09/p]

Carbon/Epoxy circular cylinder with a piezoceramic layer under

a 25 °C double cosine varying thermal load.

NASA/TM--2001-210892 97
0.20

0.10

0.00 L

I

I

020 * ............ *

_ .... @ .,..... v I. .,. @ @ @ @ _. _.

" . J

i

I i

!

-0.30

0 30 60 90

(degrees)

Sensory

..... _ .....OV

-500 V

Figure 39: Comparison of displacements of a [09/p] Carbon/Epoxy circular cylinder

with a piezoceramic layer in sensory, grounded, and active modes

under a 50°C uniform thermal load.

NASA/TM--2001-210892 98
0.20 r

i

i

i

0.10

_ A + . ._.+ " i :

_ _

-0.20 _ JI

-0.30

0 30 6O 90

Sensory

....._ .... OV

--i-- cos2 _,

T (degrees)

Figure 40: Comparison of displacements of a [09/p] Carbon/Epoxy circular cylinder

with a piezoceramic layer in sensory, grounded, and active modes

under a 50°C cosine varying thermal load.

NASA/TM--2001-210892 99
0.20 r

0.10 _ i

0.00 _ _ Sensory

_+ __ ...._ .... OV

_ -0.10 _

-0.20 i

-0.30

0 30 60 90

-cos2 7

7 (degrees)

Figure 41: Comparison of displacements of a [0Jp] Carbon/Epoxy circular cylinder

with a piezoceramic layer in sensory, grounded, and active modes

under a 25 °C double cosine varying thermal load.

NASA/TM--2001-210892 100
CHAPTER 7

CONCLUSION AND SUGGESTIONS FOR FUTURE WORK

New analytical formulations and mechanics are developed to characterize the coupled

mechanical, electrical, and thermal behavior of piezoelectric composite materials. The coupled

thermopiezoelectric behavior is captured at the material level through the constitutive equations and

the displacements, electric potential, and temperature are introduced as state variables in the

formulation. The equations of motion are developed in both cartesian and curvilinear coordinates

and a layerwise laminate theory is incorporated to provide more accurate predictions of interlaminar

and intralaminar effects, especially for thick laminates. Thermal effects arising from thermal

expansion mismatch, pyroelectric effects, and temperature dependent material properties are

incorporated in the formulation. Corresponding finite element equations are developed and

implemented into stand alone computer programs for beam, plate, and shell elements. The accuracy

of the formulation is verified with comparisons from published experimental data and other

analytical methods. Additional numerical studies are also conducted to demonstrate additional

capabilities of the formulation to predict the sensory and active behavior of piezoelectric composite

materials. The verification and numerical studies have helped to demonstrate the effectiveness and

accuracy of the developed mechanics and finite elements. However, future experimental studies will

help to complement this work by contributing to additional verification studies.

7.1 Suggested Plan of Future Experimental Studies

In order to verify the accuracy of the developed analytical formulation in more detail, the

following plan of experimental studies are outlined. As illustrated in the previous sections, previous

researchers have conducted numerous room temperature static experimental studies along with some

limited high temperature studies. However, dynamic studies, and in particular high temperature

dynamic experiments, have received less attention. Thus, the development of an experimental setup

for conducting high temperature dynamic studies will help to more fully characterize the behavior

of piezoelectric composite materials and determine the feasibility of applying these materials in

extreme temperature environments.

A generic dynamic experimental setup is shown in Figure 39. A beam specimen with an

attached piezoelectric patch will be excited by an electromagnetic shaker and the corresponding

response of the beam measured by an accelerometer. For high temperature tests, the beam will be

placed inside an oven or furnace. The time varying responses of the beam and piezoelectric patch

are passed through various test measurement devices including signal conditioners, amplifiers, and

oscilloscopes to increase the signal strength and gain, as well as to monitor and measure the dynamic

waveforms. The time signals are then passed on to a PC based dynamic data acquisition system

which contains hardware to acquire the signals and software to calculate the frequency response of

the data. The resulting output will include the natural frequencies of the beam, the mode shapes, and

the sensor output.

A series of tests will be performed to investigate the individual sensory and active dynamic

responses of piezoelectric composite materials at high temperature. The first test configuration will

consist of a beam with a single piezoelectric patch as shown in Figure 40. A series of tests will be

NASA/TM--2001-210892 101
performedaslistedin Table9 to determinethesensoryandactiveresponsesindividually.A room

temperaturebaselinetestwill beperformedto determinethenaturalfrequenciesof thebeamwith

thepiezoelectricpatchgrounded.Thepiezoelectricpatchwill thenbetestedin a sensorymodeto

demonstratethe capability to monitor modalfrequenciesthroughmeasurementof the sensory

voltages,andto determinethe effect of the sensorymodeon the naturalfrequencies.Next, the

piezoelectricpatchwill beconfiguredasanactuatorandelectricvoltagesappliedto measurethe

capabilityto activelycontrolthevibrationof thebeam.Theseexperimentswill thenberepeatedat

highertemperaturetoassessthermaleffectsonthesensoryandactiveresponses.A seriesof different

beammaterialsfrom metalsto compositescanbestudied,aswell asdifferenttypesof piezoelectric

materialsfrompiezopolymerstopiezoceramics,in ordertodeterminethemosteffectivesensorand

actuatormaterials.In addition,thesizeandlocationof thepiezoelectricpatchcanbevariedto assess

theinfluenceof coverageareaandlocationon sensingandactuation.

Thenexttestconfigurationwill consistof abeamwith two piezoelectricpatches,asshown

inFigure41,to investigatecombinedsensory/activeandactive/activemodesof operation.A series

of testswill beperformedasshownin Table10to determinecapabilitiesfor combinedsensoryand

actuatoroperations.Thebaselinecasewill consistof aroomtemperaturetestwithbothpiezoelectric

patchesgroundedto determinethe naturalfrequenciesof the beam.A combinedsensory/active

configurationwill thenbestudied,in which variousvoltageswill beappliedto theactuatorpatch

to reducethenaturalfrequencies,while thesensorpatchmonitorstheresponse.Thenextcasewill

involve both patchesoperatingas actuatorsto determinewhethermultiple actuatorscanmore

effectively reducethe beamvibrations than a singleactuator.Theseexperimentswill thenbe

repeatedat highertemperatureto determinethermaleffectson thecombinedsensory/activeand

active/activeconfigurations.Additional caseswith differentbeamandpiezoelectricmaterials,as

well as different sized patchesand locationswill also be studiedto determinethe optimal

combinationsfor combinedsensory/activeandactive/activeoperations.

Additionalstudieswill alsobeconductedona flat platewith multiplepiezoelectricpatches

(Figure42) andacurvedplatewith multiplepiezoelecuicpatches(Figure43).Theflat platewill

provideadditionalinformationontwo-dimensionalgeometryeffectson sensingandactuation,as

well asto determinecapabilitiesto utilizepatchesat specificlocationsto senseandactuatecertain

naturalfrequencies.A similarstudywill alsobeperformedonacylindrical shellto determinethe

effectof curvatureonactuationandsensing.As inpreviousproblems,thebaselinecasewill involve

determiningtheroomtemperaturenaturalfrequencieswith thepiezoelectricpatchesinoperative.A

variety of different sensorand actuatorconfigurationswill be examinedat room and high

temperatures,for differentbeamandpiezoelectricmaterials.

Throughouttheexperimentalstudies,the developedfinite elementcodeswill beusedto

provide designguidelinesto reduce the numberof experiments.Even for the simple beam

configurationsstudiedin Figures40and41,therea largenumberof differentexperimentsrequired

to determinetheinfluenceof differentmaterials,patchsizes,andlocations.As morecomplicated

geometriesareconsideredin Figures42 and43, the numberof experimentaltestswill increase

substantially.Thus,in orderto keepexperimentaltestingtime andcostsreasonable,a combined

analyticalandexperimentalapproachis essential.

NASA/TM--2001-210892 102
Test Specimen Dynamic Measurement

• Natural Frequencies

• Dynamic Deflections

• Sensor Output

f'---Accelerometer

I

Time Signals

. ,_I1ii ii _.

_ _ t i _ : f i i _ _i_

II

• Data Acquisition

• Spectral Analyzer

Figure 39: Dynamic experimental setup.

NASA/TM--2001-210892 103
Testing Configuration

Piezoceramic fAcceler

Shaker

Oven or

Furnace

Figure 40: High temperature test configuration for a cantilevered beam

with one piezoelectric patch to examine sensory and

active modes individually.

NASA/TM--2001-210892 104
Table 9: Series of tests for a cantilevered beam with one piezoelectric patch.

Beam and Piezoelectric Materials

Aluminum Beam [08] Graphite/Epoxy [02/90_]s

Beam Graphite/Epoxy Beam

Temp. Piezo PZT PVDF PZT PVDF PZT PVDF

Mode Patch Patch Patch Patch Patch Patch

Room Grounded

Room Sensory

Room Active

Elevated Grounded

Elevated Sensory

Elevated Active

NASA/TM--2001-210892 105
Testing Configuration

Piezoceramic

ccele

Shaker

Oven or

Furnace

Figure 41: High temperature test configuration for a cantilevered beam

with two piezoelectric patches to examine combined sensory/active

and active/active modes of operation.

NASA/TM--2001-210892 106
Table 10: Series of tests for a cantilevered beam with two piezoelectric patches.

Temp. Piezo.

Mode

Room Grounded

Room Active/

Sensory

Room Active/

Active

High Grounded

High Active/

Sensory

High Active/

Active

Aluminum Beam

PZT/ PVDF/ PZT/

PZT PVDF PVDF

Beam and Piezoelectric Materials

[08] Graphite/Epoxy

Beam

[0,_/902] s Graphite/Epoxy

Beam

PZT/ PVDF/ PZT/ PZT/ PVDF/ PZT/

PZT PVDF PVDF PZT PVDF PVDF

NASA/TM--2001-210892 107
Piezoelectric patch

I I

Figure 42: Test configuration of a flat plate with attached piezoelectric

patches to investigate two-dimensional effects on combined

sensory and active operations.

NASA/TM--2001-210892 108
Piezoelectric Patch

Figure 43: Test configuration of a curved cylinder with attached piezoelectric

patches to investigate curvature effects on combined

sensory and active operations

NASA/TM--2001-210892 109

REFERENCES

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NASA/TM--2001-210892 117
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May 2001 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Finite Element Analysis of Active and Sensory

Thermopiezoelectric Composite Materials

6. AUTHOR(S)

Ho-Jun Lee

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135 -3191

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National Aeronautics and Space Administration

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WU-708-87-13-00

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E-12768

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NASA TM--2001-210892

11. SUPPLEMENTARY NOTES

This report was submitted as a dissertation in partial fulfillment of the requirements for the degree Doctor of Philosophy

to Northwestern University, Evanston, Illinois, December 2000. Responsible person, Ho-Jun Lee, organization code

5930. 2 ! 6-433-3316.

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This publication is available from the NASA Center for AeroSpace Information, 301--621-0390.

13. ABSTRACT (Maximum 200 words)

Analytical formulations are developed to account for the coupled mechanical, electrical, and thermal response of

piezoelectric composite materials. The coupled response is captured at the material level through the thermopiezoelectric constitutive equations and leads to the inherent capability to model both the sensory and active responses

of piezoelectric materials. A layerwise laminate theory is incorporated to provide more accurate analysis of the displacements, strains, stresses, electric fields, and thermal fields through-the-thickness. Thermal effects which arise from

coefficient of thermal expansion mismatch, pyroelectric effects, and temperature dependent material properties are

explicitly accounted for in the formulation. Corresponding finite element formulations are developed for piezoelectric

beam, plate, and shell elements to provide a more generalized capability for the analysis of arbitrary piezoelectric

composite structures. The accuracy of the current formulation is verified with comparisons from published experimental

data and other analytical models. Additional numerical studies are also conducted to demonstrate additional capabilities

of the formulation to represent the sensory and active behaviors. A future plan of experimental studies is provided to

characterize the high temperature dynamic response of piezoelectric composite materials.

14. SUBJECT TERMS

Composite; Piezoelectric; Temperature; Smart materials;

Sensors; Actuators; Finite element method

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