chopra_aiaa-1561-157_review of state of art of smart structures

AIAA JOURNAL

Vol. 40, No. 11, November 2002

Review of State of Art of Smart Structuresand Integrated Systems

Inderjit ChopraUniversity of Maryland, College Park, Maryland 20742

I. Introduction

A SMART structure involves distributed actuators and sensorsand one or more microprocessors that analyze the responses

from the sensors and use integrated control theory to commandthe actuators to apply localized strains/displacements to alter sys-tem response. A smart structure has the capability to respond toa changing external environment (such as loads or shape change)as well as to a changing internal environment (such as damage orfailure). It incorporates smart material actuators that allow the al-teration of system characteristics (such as stiffness or damping) aswell as of system response (such as strain or shape) in a controlledmanner. Thus, a smart structure involves four key elements: actua-tors, sensors,control strategies,and power conditioningelectronics.Many typesof actuatorsand sensors,such as piezoelectricmaterials,shape memory alloys, electrostrictive materials, magnetostrictivematerials, electro- and magnetorheological � uids and � ber optics,are being considered for various applications. These can be inte-grated with main load-carrying structures by surface bonding orembedding without causing any signi� cant changes in the mass orstructural stiffness of the system.

Numerous applicationsof smart-structurestechnology to variousphysical systems are evolving to actively control vibration, noise,aeroelasticstability,damping, shape change, and stress distribution.Applications range from space systems to � xed-wing and rotary-wing aircraft, automotive, civil structures,machine tools, and med-ical systems. Much of the early development of smart-structuresmethodologywas drivenby spaceapplicationssuch as vibrationandshapecontrolof large � exible space structures,but now wider appli-cationsare envisagedfor aeronauticaland other systems.Embeddedor surface-bondedsmart actuatorsonanairplanewingora helicopter

Inderjit Chopra is an Alfred Gessow Professor in Aerospace Engineering and Director of the Alfred GessowRotorcraft Center at the University of Maryland. Also, he was the Minta–Martin Research Professor from 1996to 2000. After receiving his Sc.D. degree from the Massachusetts Institute of Technology in 1977, he joined theNASA Ames/Stanford University Joint Institute of Aeronautics and Acoustics, where he worked for four and halfyears on the development of aeroelastic analyses and testing of advanced helicopter rotor systems. In 1981, hejoined the University Maryland as an Associate Professor, and he got promoted to Full Professor in 1986. Hehas been working on various fundamental problems related to dynamics of helicopters, including aeromechanicsstability, smart structures applications, active vibration control, modeling of composite blades, rotor head healthmonitoring, aeroelastic optimization, micro air vehicle development, and comprehensive aeromechanics analysesof bearingless, tilt-rotor, servo-� ap, teetering, and circulation control rotors. His direct graduate advising resultedin 31 Ph.D. and 55 M.S. degrees. He has been the principal investigator of three major research programs: theArmy’s University Research Initiative (URI) (1992–1997) on “Smart-Structures Technology: Innovations andApplications to Rotorcraft Systems,” the Army’s Multidisciplinary University Research Initiative (MURI) (1996–

2001) on “Innovative Smart Technologies for Actively Controlled Jet-Smooth Rotorcraft” and Army/NASA’s“Rotary-Wing Center of Excellence” (1982–2006). He acted as the Department Chairman from 1988 to 1990.An author of over 130 archival papers, Dr. Chopra has been an Associate Editor of the Journal of the AmericanHelicopter Society (1987–1991), JournalofAircraft (1987– ), andJournalof IntelligentMaterialsand Systems. (1997– ).Also,he hasbeen a member of the editorialadvisoryboardof three journalsVERTICA (1987–1991), SmartMaterialsand Structures (1994–2000) and SADHANA (1991–1995). He was awarded the 1992 UM’s Distinguished ResearchProfessorship, 1995 UM’s Presidential Award for Outstanding Service to the Schools, 2002 AIAA Structures,Structural Dynamics, and Materials Award, 2002 AHS Grover E. Bell Award, 2001 ASME Adaptive Structuresand Material Systems Prize, and 1996 AIAA/ASME Best Paper Award. He is a member of the Army Science Board(1997–). He is a Fellow of the AIAA, a Fellow of the American Helicopter Society, and a Fellow of the AeronauticalSociety of India.

Presented at the AIAA/ASME/ASCE/AHS/ASC 42nd Structures, Structural Dynamics, and Materials Conference and Adaptive Structures Forum, 16–19April 2001; received 18 July 2001; revision received 10 June 2002; accepted for publication 10 June 2002. Copyright c° 2002 by Inderjit Chopra. Publishedby the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on conditionthat the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/02$10.00 in correspondence with the CCC.

blade, for example, can induce airfoil twist/camber change that inturn can cause a variation of lift distribution and can help to controlstatic and dynamic aeroelastic problems.

Applications of smart-structures technology to aerospace andother systems are expanding rapidly.Major barriers include low ac-tuatorstroke,lackof reliablesmartmaterialcharacteristicsdatabase,unavailability of robust distributed adaptive control strategies, andreliable smart systems mathematical modeling and analysis. Theobjective of this survey is to review the state of the art of smartactuators, sensors, and integrated systems and point out the needsfor future research.These researchneeds are highlightedunder eachsection and then summarized again in the last section. In this pa-per six different topics are covered: smart material actuators andsensors and piezoelectrics, modeling of beams with induced strainactuation, modeling of plates with induced strain actuation, shapememory alloys, magnetostrictors and electrostrictors, and smart-structures applications. It is an emerging multidisciplinary � eld,and associatedmethodology/technology is in its early development(see related books1¡25 and review papers26¡41).

II. Smart Material Actuatorsand Sensors and Piezoelectrics

Piezoelectricsare the most popularsmartmaterials.They undergodeformation (strain) when an electric � eld is applied across them,and conversely produce voltage when strain is applied, and thuscan be used both as actuators and sensors. Under an applied � eldthese materials generate a very low strain but cover a wide range ofactuation frequency.Piezoelectric materials are relatively linear (atlow � elds) and bipolar,but exhibit hysteresis.The most widely usedpiezoceramics(suchas leadzirconatetitanate)are in the formof thin

2145

2146 CHOPRA

Base material with 1–2 plane electroded

Special electrode con� guration

Inherent geometrical ampli� cation

Fig. 1 Different types of actuators.

sheets that can be readily attached or embedded in composite struc-tures or stackedto form discretepiezostackactuators(Fig. 1). Thesesheets generate isotropic strains on the surface and a non-Poissonstrain across the thickness. It is possible however to generate direc-tional in-plane induced strains with piezoceramics using electrodearrangement, specially shaped piezos, bonding arrangement, andembedded � bers (Fig. 1). Piezoelectric and electrostrictivemateri-als are also available in the form of “stacks,” where many layers ofmaterials and electrodes are assembled together. These stacks gen-erate large forces but small displacements in the directionnormal tothe top and bottomsurfaces.Bimorphs or bendingactuatorsare alsoavailablecommercially, where two layers of these materials (piezo-ceramic) are stacked with a thin shim (typically of brass) betweenthem. If an opposite polarity is applied to two sheets, a bendingaction is created. Bimorphs cause larger displacement and smallerforce as compared to single piezo element.

Among other smart materials, shape memory alloys (SMA) ap-pear attractive as actuators because of the possibility of achievinglarge excitation forces and displacements.These materials undergophase transformation at a speci� c temperature. When plasticallydeformed at a low temperature, these alloys recover their originalundeformed condition if their temperature is raised above the trans-formation temperature. This process is reversible. A remarkablecharacteristic of SMA is the large change of modulus of elasticitywhen heated above phase transition temperature (typically two tofour times room temperaturevalue).The most common SMA mate-rial is nitinol (nickeltitaniumalloy),which is availablein the formofwires of different diameters. Heating of an SMA can be carried outboth internally (electrical resistance)or externally (using coils), butthe response is very slow (less than 1 Hz). It is sometime possible tospeedup the responsethroughforcedconvectiveor conductivecool-ing of material.Electrostrictivematerialsare similar to piezoelectricmaterials, with about the same strain capability. However, they arevery sensitive to temperature, have a monopolar, nonlinear relationbetween the applied � eld and induced strain, and exhibit negligiblehysteresis. Magnetostrictivematerials such as Terfenol-D elongate

when exposed to a magnetic � eld. These materials are monopolarand nonlinear and exhibit hysteresis. These materials generate lowstrainsand moderate forcesover a wide frequencyrange.Becauseofcoil and magnetic return path, these actuators are often bulky. Elec-trorheological (ER) � uid consists of suspensions of � ne dielectricparticles in an insulating � uid that exhibits controlled rheologicalbehavior in the presence of large applied electric � elds (up to 1–4 kV/mm). Application of an electric � eld results in a signi� cantchange of shear loss factor that helps alter damping of the system.Magnetorheological (MR) � uids consist of suspensions of ferrousparticles in � uid and exhibit change in shear loss factor caused bymagnetic � elds (low � elds but moderately large currents). MR � u-ids, like ER � uids, are primarily envisaged to augment damping ina system. Fiber optics is becoming popular as sensors because theycan be easily embedded in composite structures with little effecton structural integrity and also have the potential of multiplexing.Discussion on ER and MR � uids and � ber optics is not covered inthis paper.

Smart structures are becoming feasible because of the 1) avail-ability of smart materials commercially, 2) ease of embedding de-vices in laminated structures, 3) exploitation of material couplingssuch as between mechanical and electrical properties, 4) potentialof a substantial increase in performance at a small price (say weightpenalty), and 5) advances in microelectronics,information process-ing, and sensor technology.Key elements in the applicationof smartstructures technology to a system are actuators, sensors, controlmethodology, and hardware (computer and power electronics).

A. Smart ActuatorsTypical actuators consist of piezoceramics, magnetostrictives,

electrostrictives,and shape memory alloys. These normally convertelectric inputs into actuation strain/displacement that is transmittedto the host structure affecting its mechanical state. Piezoelectricsand electrostrictors are available as ceramics, whereas magne-tostrictors and shape memory alloys are available as metal alloys.Piezoelectrics are also available in polymer form as thin soft � lm.Important performance parameters of actuators include maximumstroke or strain, maximum block force, stiffness, and bandwidth.Somewhat less important parameters include linearity, sensitivityto temperature, brittleness and fracture toughness (fatigue life), re-peatability and reliability,weight density, compactness, heat gener-ation, � eld requirement, and ef� ciency. The induced strain is oftentreated like thermal strain. The total strain in the actuator is assumedto be the sum of the mechanical strain caused by the stress plus theinduced strain caused by the electric � eld. The strain in the hoststructure is obtained by establishingthe displacementcompatibilitybetween the host material and the actuator. In a piezoelectricmate-rial, when an electric � eld is applied, the dipoles of the material tryto orient themselves along the � eld causing strain in the material.This relation of strain vs voltage is linear in the � rst order. In anelectrostrictivematerial there is an interaction between the electric� eld and electric dipoles that is inherently nonlinear. The magne-tostrictive response is based on the coupling of magnetic � eld andmagnetic dipoles in the material, a nonlinear effect. Shape memoryis a result of phase transformationas a result of temperature changeof the material (caused by an thermal � eld). Phase transformationis very much a nonlinear phenomenon.

A common piezoceramic material is lead zirconate titanate(PZT), and its maximum actuation strain is about 1000 microstrain.Polyvinylidene�uoride (PVDF) is a polymer piezoelectric� lm, andits maximum actuation strain is about 700 microstrain. A commonceramic electrostrictivematerial is lead magnesiumniobate (PMN),and its maximum actuation strain is about 1000 microstrain. PZTand PMN are available in the form of thin sheets, which can beeither bonded or embedded in a structure.

The PZTs require initial polarization (with high electric � eld),whereas no such polarization is needed for PMNs. Terfenol, a rareEarth magnet-like material, can create a maximum actuation strainof about 2000 microstrain. It needs a large magnetic � eld to causethis actuation strain. Nitinol (nickel titanium alloy) normally avail-able in the form of wires can create free strain from 20,000 to 60,000microstrain(2–6%). Table 1 showscomparisonof characteristicsfor

CHOPRA 2147

Table 1 Comparison of actuators

Actuators PZT-5H PVDF PMN Terfernol DZ Nitinol

Actuation mechanism Piezoceramic Piezo� lm Electrostrictive Magnetostrictive Shape memory alloyFree strain 3, ¹ strain 1,000 700 1,000 2,000 60,000Modulus E 106 psi 10 0.3 17 7 4 for (martensite)

13 for (austenite)"max for aluminum 350 10 500 580 8,500

beam tb=tc D 10Band width High High High Moderate LowStrain-voltage linearity First-order linear First-order linear Nonlinear Nonlinear Nonlinear

Table 2 Comparison of strain sensors

Resistance Semiconductor Fiber optics 0.04-in. Piezo� lm Piezoceramicsgauge 10-V gauge 10-V interferometer 0.001-in. 0.001-in.

Sensor excitation excitation gauge length thickness thickness

Sensitivity 30 V/" 1000 V/" 106 deg/" 104 V/" 2 £ 104 V/"Localization (inches) 0.008 0.03 0.04 <0.04 <0.04Bandwidth 0 Hz acoustic 0 Hz acoustic 0 Hz acoustic 0.1 Hz GHZ 0.1 Hz GHZ

differentsmart actuators.Note tb is beamthickness,and tc is actuatorthickness. Giurgiutiu et al.42 compared the characteristics of vari-ous commercially availablepiezoelectric,electrostrictive,and mag-netostrictive actuators. The comparison was carried out in termsof output energy density. Typically, the energy density per unitmass was found to be in the range of 0.233–0.900 J/kg. There isa wide variation of performance of actuators from vendor to ven-dor. Near43 provided a good overview on the piezoelectric actuatortechnology.

B. SensorsTypical sensors consistof straingauges,accelerometers,� ber op-

tics, piezoelectric � lms, and piezoceramics. Sensors convert strainor displacement (or their time derivatives) into an electric � eld.Piezoelectric strain sensors are generally made of polymers suchas PVDF and are very � exible (low stiffness). They can be easilyformed into very thin sheets (� lms) and adhered to any surface.Keyfactors for sensors are their sensitivity to strain or displacement,bandwidth, and size. Other less important factors include tempera-ture sensitivity, linearity, hysteresis, repeatability, electromagneticcompatibility, embeddibility, and associated electronics (size andpower requirement). Typically, the sensitivity for resistor gauge is30 V per strain, for semiconductor gauge is 103 V per strain, andfor piezoelectric and piezoceramic gauges is 104 V per strain. Thesensitivity of � ber optics sensors is de� ned differently and is about106 deg per strain. Associated electronics weigh against � ber op-tics sensors. Discrete shaped sensors that apply weighting to sen-sors’ output can help increase sensitivity for a speci� c application.For example, a modal sensor can magnify the strain of a particularmode. Table 2 shows a comparison of characteristics of differentsensors.

C. Piezoelectric ActuatorsPiezoelectricity means pressure electricity and is a property of

certain crystalline materials such as quartz, Rochelle salt, tourma-line, and barium titanate that develop electricity when pressure isapplied.This is called the direct effect.Pierre and JacquesCurie dis-covered piezoelectricity in the 1880s. Soon after this, the converseeffect was discovered, that is, these crystals undergo deformationwhen an electric � eld is applied. After its discovery, it took severaldecades before this phenomenon could be used in commercial ap-plications.The � rst applicationwas perhapsduring the World War IIin the 1940s as an ultrasonic detector for submarines. With the dis-covery of piezoceramics, the domain of applications has expandedconsiderably.

Piezoceramicsarepolycrystallinein natureand do nothavepiezo-electric characteristics in their original state. Piezoelectric effectsare induced in these materials through simple poling (applicationof high dc electric � eld results in polarization). The most com-

Fig. 2 Typical PZT crystal cell.

monly used piezoceramic is PZT. These are solid solutions of leadzirconate and lead titanate, often doped with other elements toobtain speci� c properties. Mixing a powder of lead, zirconium,and titanium oxide and then heating the mixture to around 800–1000±C manufacture these ceramics. It transforms to perovskitePZT powder, which is mixed with binder and sintered into de-sired shapes. During the cooling process, the material undergoesa paraelectric to ferroelectric phase transformation.An unpoled ce-ramic consists of many randomly oriented domains with no netpolarization.

Application of high electric � eld aligns most of the tetragonaldomains in such a way that the polar axes (c axes) of unit cellsare oriented mostly parallel to the applied � eld (Fig. 2). This pro-cess is called poling, and it imparts a permanent net polarization tothe ceramic (analogous to magnetization of a ferrous material witha permanent magnet). This reorientation of domains also causesa permanent mechanical deformation. An electric � eld as well asmechanical stress can switch the crystal symmetry (that is, polaraxes). With an electric � eld it is possible to depole as well as re-pole piezoelectric ceramics, whereas mechanical stress only candepole the materials. Poled piezoceramics exhibit both direct andconverse piezoelectric effects. The actuation phase (converse ef-fect) consists of three parts. The � rst one is called intrinsic effectand covers the deformation of aligned domains. The second oneis called extrinsic effect and involves the deformation caused bynon-180-deg domains. It is believed to be the source of nonlin-earity and losses in piezoceramics.44 The third effect is caused bythe electrostriction of materials, and as a result of this effect, thedeformation is generally proportional to the square of the electric� eld. Electrostriction effects are much smaller than the other twoeffects.

2148 CHOPRA

Fig. 3 Piezoelectric sheet with polarization along axis 3.

Piezoceramic materials are relatively well behaved and linear atlow electric � elds and low mechanical stress levels; they show con-siderablenonlinearityat high values of electric � eld and mechanicalstress. The actuation strain can be modeled like an equivalent ther-mal strain. A piezoceramics sheet under surface excitation can beidealized as an orthotropic material (like composite unidirectionallaminate). The constitutive relations are based on the assumptionthat the total strain in the actuator is the sum of the mechanicalstrain induced by the stress, the thermal strain caused by temper-ature, and the controllable actuation strain caused by the electricvoltage. The axes are identi� ed by numerals: 1 corresponds to thex axis, 2 corresponds to the y axis, and 3 corresponds to the z axis.Axis 3 is assigned to the direction of the initial polarization of thepiezoceramic, and 1 and 2 axes lie in the plane perpendicular to3 axis (Fig. 3).

Coupledelectromechanicalconstitutiverelationsare as follows.45

Direct:

Di D e¾i j E j C dd

im ¾m C ai 1T

Converse:

"k D d cjk E j C SE

km¾m C ®k 1T

where vector Di of size (3 £ 1) is the dielectric displacement innewtons/millivolts or coulombs/square meter, "k is strain vector ofsize (6 £ 1), E j is the applied electric � eld vector of size (3 £ 1) involts/meter, and ¾m is stressvectorof size (6 £ 1) in newtons/squaremeter. The piezoelectricconstants are the piezoelectriccoef� cientsdd

im (3 £ 6) and dcjk (6 £ 3) in meters/volt or coulombs/newton, the

dielectric permittivity e¾i j of size (3 £ 3) in newtons/square volt or

farads/meter, and SEkm is the elastic compliancematrix of size (6 £ 6)

in square meters/newton. The coef� cient ®k is thermal coef� cientvector of size (6 £ 1) in 1/degrees Kelvin and ai is thermal con-stants vector of size (6 £ 1) in newtons/volt-meter-degreesKelvin.The superscriptsc and d refer to the converse and direct effects, re-spectively,and the superscripts¾ and E indicate that the quantity ismeasured at constant stress and constant electric � eld, respectively.The converse equation represents the actuator equation, whereasthe direct equation represents the sensor equation. Normally, theconverse effect is used to determine piezoelectric coef� cients.

Rewriting the converse equation in matrix form:8>>>>>><

>>>>>>:

"1

"2

"3

°23

°31

°12

9>>>>>>=

>>>>>>;

D

266666664

S11 S12 S13 0 0 0

S12 S11 S13 0 0 0

S13 S13 S33 0 0 0

0 0 0 S44 0 0

0 0 0 0 S44 0

0 0 0 0 066

377777775

8>>>>>><

>>>>>>:

¾1

¾2

¾3

¿23

¿31

¿12

9>>>>>>=

>>>>>>;

C

2

66666664

0 0 d31

0 0 d31

0 0 d33

0 d15 0

d15 0 0

0 0 0

3

77777775

8<

:

E1

E2

E3

9=

; C

8>>>>>><

>>>>>>:

®1

®2

®3

0

0

0

9>>>>>>=

>>>>>>;

1T

In a plane perpendicular to the piezopolarization, it has isotropicproperties, that is, transversely isotropic material in the plane 1-2. Note that, although orthotropic materials do not exhibit ther-mally induced shear strains, electric � eld components E1 and E2

can generate piezo-inducedshear strains.For piezoceramic materials the actuation strain is given by

3 D

266666664

0 0 d31

0 0 d31

0 0 d33

0 d15 0

d15 0 0

0 0 0

377777775

8<

:

E1

E2

E3

9=

;

where d33 , d31 , and d15 are called piezoelectricstrain coef� cients ofa piezoelement.The d31 characterizesstrain in the 1and 2 directionscaused by an electric � eld E3 in the 3-directionand d33 relates strainin the 3-directioncaused by � eld in the 3-direction.The d15 charac-terizes 2-3 and 3-1 shear strains caused by � eld E2 and E1, respec-tively. Thus, if an electric � eld E3 is applied to a free piezoelement,it causes direct strains "1 , "2, and "3 . This is very similar to thermalstrain. If an electric � eld E1 or E2 is applied, the material reactswith shear strain °31 and °23, respectively.For orthotropicmaterialsthere is no correspondingthermal strain. To overcome this problem,it is better to assume piezoelectricmaterials as anisotropic.

If a compressive force is applied in the polarization direction(axis 3), or tensile force is applied in the plane perpendicular topolarizationdirection (axis 2 or 3), it will result in a voltage that hasthe same polarity as the original poling direction.

For piezoelectric � lm, PVDF, the induced strain is nonisotropicon the surface of the sheet. The induced strain is expressed as

3 D

2

66666664

0 0 d31

0 0 d32

0 0 d33

0 d25 0

d15 0 0

0 0 0

3

77777775

8<

:

E1

E2

E3

9=

;

For piezo� lms the d31 is not equal to d32 , and d25 is not equal tod15 .

Piezoceramics are available commercially in the form of thinsheets (say, of thickness 0.254 mm) such as PZT-5H from MargonMatroc, and the manufacturer-suppliedcharacteristicsare shown inTable 3 (Ref. 46). Among piezoceramics PZT-5H is most widelyused because of its lower electric � eld requirement than other actu-ators for the same strain. PZT-8 requires a higher � eld than PZT-5Hbutwill need less powerbecauseof its lower dielectricconstant.Onemajor disadvantage of PZT-5H is that its dielectric and piezoelec-tric constants are very sensitive to temperature, and hence it can be

Table 3 Piezoelectric characteristics46

Coef� cient PZT-5H PZT-8 PVDF

d31 ¡274£ 10¡12 m/V ¡97 18–24d32 ¡274£ 10¡12 m/V ¡97 2.5–3d33 593£ 10¡12 m/V 225 ¡33d15 741£ 10¡12 m/V 330 ——Relative permittivity e33 3400 1000 ——Free-strain range ¡250 to C850 ¹" ——Poling � eld dc 12 kV/cm 5.5 ——Depoling � eld ac 7 kV/cm 15 ——Curie temperature 193±C 300 ——Dielectric breakdown 20 kV/cm —— ——Density 7500 kg/m3 7600 ——Open circuit stiffness E11 62 GPa 87 ——Open circuit stiffness E33 48 GPa 74 ——Compressive strength (static) >517 MPa >517 ——Compressive depoling limit 30 MPa 150 ——Tensile strength (static) 75.8 75.8 ——Tensile strength (dynamic) 27.6 MPa 34.5 ——

CHOPRA 2149

Fig. 4 Static-free strain of a PZT-5H sheet actuator.48

susceptible to self-heating problems because of its relatively largedissipation factor.

1. Static Free StrainFigure 4 shows the transverse free strain (in a plane normal to the

polarized direction) vs dc � eld. To avoid hysteresis and drift fromthe dc � eld, it is necessary bring to zero condition after each mea-surement on an initially dc cycled sheet. The curve is almost linearat low applied electric � eld levels and becomes nonlinear at high� elds, more so for negative � elds. The maximum positive � eld islimited by the breakdown of dielectric,whereas the maximum neg-ative � eld is limited by the piezoceramic depoling. To obtain highactuation authority, these materials are often operated into nonlin-ear regimes. To predict actuatorperformance accurately, it becomesnecessary to derive macroscopicmodels and analyses that are capa-ble of addressing material nonlinearities including depolarization.

2. Effect of External StressCompressive stresses tend to align the c axis of the domains per-

pendicular to the direction of stress. If the stress is acting in theplane of piezosheet (normal to polarization direction), it destroyssome of the initial polarization (that is, reduces the net polariza-tion). This changes the piezoelectric coef� cients and can cause apermanent degradation especially for soft ceramics like PZT-5H(Ref. 47). However, with applied tensile stress there is a slight in-crease in free strain. The increase in free strain is of the order of10% at 3.9 kV/cm under a tensile stress of 2500 psi (about 1

5 th ofultimate stress) (Fig. 5).48 A compressive stress along poling axisshows a slight increasein free strain and then levelsoff at high stressvalues.

3. Hysteresis and DriftThe steady strain response exhibits bias, and it takes about three

cycles to stabilize it. There is a signi� cant hysteresisfor steady � eld,and the aspect ratio (lateral width) of the hysteresis increases withthe � eld. Also, the hysteresis curve is asymmetric with respect tozero strain axes. The drift phenomenon is a slow increase of thefree strain with time after the application of the dc � eld. After theapplication of the dc � eld, the strain jumps to a certain value andthen increases slowly with time. When the � eld is switched off, thestrain falls back to some value and then slowly decreases until someresidual value is reached. This behavior of slow increase or slowdecrease can be represented by a logarithmic expression,

1" D 1"0[1 C ° .t=0:1/]

where 1"0 is the strain 0.1 s after the application of � eld and ° isa factor that depends on the material. The test data show that thepercentage of drift is about same regardless of excitation � eld.48

This drift can be as high as 15% for moderately high � elds. Thereason for this creep appears to be a gradual change in permanentpolarization of the crystal (intrinsic effect). A superimposed smallac � eld (dither � eld) appears to show a small change in the driftphenomenon.

Fig. 5 Static strain of PZT-5H with transverse tensile stress.48

Effect of � eld at 5 Hz

Effect of frequency

Fig. 6 Variation of strain hystereis with � eld and frequency for a freePZT-5H actuator.48

4. Dynamic BehaviorFigure 6 shows strain-� eld hysteresis loops with excitation fre-

quency and � eld level. With an increasing � eld the overall shapeof hysteresis is only slightly changed, but the mean slope increaseswith higher � eld. The area under the high frequency excitation isslightly less than the area under the low frequency showing moreenergy losses at low frequencies.A dc bias � eld increases the valueof d31 as a result of pinning of the domains (domains better aligned)and also reduces the losses.

For a free piezoceramic the strain almost remains constant withfrequency except at high frequenciesand high � elds, and the strain

2150 CHOPRA

Peak free strain

Phase angle

Fig. 7 Response of a free PZT-5H actuator.48

increases by about 15% as a result of nonlinear effects, whereasthe phase changes considerably with the excitation frequency and� eld strength (Fig. 7). For actuatorsbonded to a structure (say, to anisotropic beam), the strain increases with frequency; however, thechange of phasewith � eld strengthis small. This shows that the totalimpedanceof the actuator-beamsystem is more capacitivein nature.It is possible to predict the power consumptionof free and surface-bonded actuators satisfactorily using the impedance method, giventhe variationof the dielectricpermittivityand dissipationfactorwith� eld for a free actuator (Fig. 8). Generally, the accuracy of predic-tion deteriorates at high � eld (above 4.8 kVrms/cm) and frequency(above 100 Hz).

5. Depoling BehaviorWhen the piezoelementis exposedto a high electric � eld opposite

to the poling direction, it loses its piezoelectric property, accompa-nied by more dielectric losses and lower ef� ciency. This is calleddepoling of piezoelectric and results in a permanent deformation.For PZT-5H the dc depoling � eld is approximately 5.5 kV/cm. Un-der an ac excitation the depoling� eld of the actuatorbecomes lowerthan the dc value (Fig. 9). Figure 10 shows the effect of depolingon the actuator response. Once at or above the depoling � eld, ittakes a few seconds to depole, and the strain-� eld hysteresis looptransforms into a “butter� y loop.” It is accompanied by a rapid in-creasein currentdrawn(moreenergy loss). If a muchhighernegative� eld is kept for a long time, the material gets polarized along thenew poling direction. Thus, it is possible to repole the piezoelectricspecimen by means of exciting it with ac � eld or dc � eld. Althoughrepoling actuators bonded to the surface of a beam is possible, atensile stress is induced in the actuator, which might cause it to

Magnitude of current

Phase

Fig. 8 Current consumption for a pair of actuators surface bonded toan aluminum beam with 3 kVrms/cm excitation.48

Fig. 9 Variation of depoling � eld with frequency.48

crack. If the piezoelectric element is repoled with an ac � eld, it isrecommended to apply a dc bias � eld. This will ensure that the elec-tric � eld in the poling direction exceeds the poling threshold,whilekeeping the negative � eld below the depoling limit. To improve therepeatableperformancedata of a piezoelectricelement, it is recom-mended to apply cyclic treatment.The process involvesapplicationof slightly higher � eld (dc or ac) than needed and then shutting offthe � eld. The process is repeated several times until the residual

CHOPRA 2151

Comparison of depoled response

Repoling effectiveness

Fig. 10 Depoling strain behavior.48

strain gets stabilized. Depoling is also possible if the temperatureexceeds the Curie temperatureor if a large stress is applied.At Curietemperature the ferroelectricmaterial undergoesphase transition toparaelectric material and suddenly loses its characteristicbehavior.At high frequencies, energy from mechanical losses can generateenough self-heating that can severely affect the performanceof theactuator. Ghandi and Hagood49 developed nonlinear modeling andcharacterizationtechniquesfor phase transitionsin electromechani-cally coupledmaterials. This is especiallysuited to repolarizationinferroelectrics (PZTs). Blocks of PZT were tested under multiaxialelectrical and mechanical loading conditions, and required param-eters for modeling were determined. Lynch50 developed multiaxialphenomenological-based constitutive relations (at macroscale) forferroelectricceramics that includedhysteresis,rate dependence,andsaturation. It might be important to validate these with test data fora range of test conditions.

6. Manufacturing IssuesThere are several issues concerning building of smart structures.

These are the following:1) Electrical contact on both sides of piezo is required. One way

to overcome this problem is to use an oversize thin conductor sheetbetween actuator and substrate with one surface insulated; the sec-ond way is to drill a hole in the substructureand use nonconductingepoxy.

2) For propertransferof inducedstrain to main structure,the bondlayer thickness needs to be thin, suf� ciently stiff, and uniform. Forthis, pressure is applied during curing.

Embedding vs surface-mounting. With surface-mountedactua-tors there is a better access for fabrication, an easier access for in-

spection, and less maintenancecost. Because of exposure,actuatorsare more susceptibleto damage.Also, the functioningof actuatorsisdependent on structural surface. Embedded actuators become inac-cessible for inspection. The devices, however, are better protected,and interconnectionswith otherdevicesbecomeeasy.Also with em-bedding,the piezoelectricmust have an elastic moduluscomparableto the host structure in order to avoid structural discontinuity, andthe Curie temperature should be higher than the curing temperatureof composites. Further, piezodevicesmust be electrically insulatedfrom the host structure.This means that the piezoelementscan be di-rectly embeddedin glass/epoxy laminasbut need an insulatinglayerwith graphite/epoxy laminas. However, an insulating layer can re-duce the effectivenessof the actuator. It appears appropriateto wrapthe piezodevice in a 0.05-mm-thick Kapton � lm (Dupont) and useacrylic epoxy to reduceslippagebetween piezoand insulatinglayer.Overall, there are many issues related to integrationof piezoceramicelements in composite structure for various applications.51

Embedding electronics. To embed integrated circuits in a hoststructure, it is essential to insulate them electrically, ventilate ther-mally, and isolate them mechanically from load paths.52 For a mini-mal degradationof structure, it is important to have a minimum plyinterruption.

Current piezoceramicmaterialshave a maximumfree strainof theorder of 1000 microstrain and stiffness of the order of 50–90 GPa.Careful con� gurationsof the actuatorgeometry in conjunctionwithampli� cation/recti� cation mechanisms are needed to increase theperformance of an actuator for a speci� c application. PZT actua-tors are available in the form of thin sheets, and their d31 effect isused for actuation. Typical free strains are of the order of C250 to¡1000 microstrain (limited by depoling and dielectric breakdown,respectively).

Thus, there is a need to undertakeextensivetestingof engineeringspecimens (macrolevel) to represent most of the operating condi-tions such as stress, strain, temperature,and voltage.Building smartstructures encompassesnew fabrication methods that require expe-rience and expertise in fabricatingcomplex systems with embeddedor surface-mounted smart actuators and sensors. Sheet actuatorsare convenient for surface bonding or embedding in a laminatedstructure. The bond layer between PZT and the host structure hasa major impact on the strain transfer to the host structure and thelocal stress distribution. Piezoelectric actuators are also availablein specializedcon� gurations such as RAINBOW, THUNDER, andthe C-block actuators. To increase transverse actuation of piezoce-ramic wafers, interdigitatedelectrodepiezoelectric� ber compositesare developedwhere alternatingelectrodesare subjected to positiveand negative voltages.53;54 As a result, d33 effect is used in actua-tion. This active � ber composite (AFC) utilizes interdigitatedelec-trode poling and piezoelectric � bers embedded in an epoxy matrix,resulting in a high-performance actuator laminate. The disadvan-tages of this actuator are cost, dif� culty of processing and han-dling during fabrication, and high voltage requirement. Recently,patch actuators with interdigitated electrodes, which use the d33

effect, have become available commercially.55 Also recently, low-costmicro� ber composite(MFC) arebecomingavailableforvariousapplications.56 To increase the actuation force, multilayered stackactuatorsare used. In spite of using the d33 effect, the total displace-ment is quite small. Because of laminated structurewith alternatingbond and piezolayers, stacks typically operate only under compres-sive loads.

Piezoceramicsare widely used as actuatorsand sensors for activevibration control of beams and plates.57¡59 To increase their strainoutput, these are often operated under high electric � elds, whichresults in a signi� cant nonlinear behavior of their characteristics.Mukherjee et al.,60 Yang et al.,61 and Ren et al.62 carried out test-ing of piezoceramic sheet actuators under a range of temperature,frequency, electric � eld, dc bias, and applied stress for both soft(EC-65, EC-76) and hard (EC-64, EC-69) PZTs (manufactured byEDO Ceramics).These measuredresults showed the need of includ-ing nonlineardependenceof piezoelectricconstantson applied � eldand stress.

Asmentionedearlier,thecharacteristicsof piezoelectricmaterialschange under mechanical stress.47;61;62 Piezoceramics are normally

2152 CHOPRA

brittle, and, therefore, the effect of dynamic stress on their charac-teristics is a major concern. Recently, there have been some limitedfocusedeffortsto evaluatedtheir fatiguecharacteristicsanddynamicconditions.63¡66

Most actuators utilize direct piezoelectric effects (d31 and d33/.There have been some selected attempts to build torsional actuatorsusing shear piezoelectric effect (d51/ (Refs. 67 and 68). Bondingtogether segmented piezoceramic bars that are poled in axial direc-tion and arranging in opposite poling direction form a cylindricalactuator. A major drawback is very � eld requirement (in severalkilovolts) to achieve desired torsional de� ection.

At this time, a reliable detailed database of piezoelectric char-acteristics for a range operating conditions such as electric � eld,stress/strain level, and temperature is not readily available. It mightrequire extensive testing of standardized piezoelectric elements atmacrolevel under controlled environment in specialized test ma-chines.Simpli� ed constitutiverelationsfornonlinearhigh � eld con-ditions need to be developed. Depolarization caused by alternating� eld in terms of performancedegradationneeds to be examined.Formost applicationselectromechanicalfatigue characteristicsbecomeimportant and need to be investigated systematically. Complex ge-ometries, which result in nonuniform� elds and stress distributions,should be examined systematically through detailed modeling andre� ned testing techniques.Modeling issues related to interdigitatedelectrodepiezoelectric� ber compositesneed carefully scrutiny andvalidations.

D. Piezoelectric SensorsThe direct piezoelectric effect is the electric displacement that

is generated when a piezoelectricmaterial is mechanically stressedand can be used to sense structural deformation. Most applicationsrely on either the voltage or rate of change of voltage generated bythe sensor or the frequency spectrum of the signal generated by thesensor.A major advantageof usingpiezoelectricsensorsas opposedto conventionalfoil straingauges is their superiorsignal-to-noisera-tio and high-frequencynoiserejection.In applicationsinvolvinglowstrain levels, one requiresmuch less signal conditioningwith piezo-electric sensors, and, as such, they are less sensitive to noise. Otheradvantagesare their compactness and sensitivity over a large strainbandwidthand ease of embeddability.Most commonlyused sensorsarepiezo� lm (PVDF) becauseof its lowstiffness.Sometimes,piezo-ceramic (PZT) sensors are used for speci� c applications.For exam-ple, it might be possible to use piezoceramics for both sensing andactuation especially for collocated control strategies.69¡73 The PZTsensors exhibit high Young’s modulus, brittleness, and low tensilestrength. Also, they suffer from creep with dc � eld, depolarizationunder high stress conditions,and linearity at high strains. In spite oftheseproblems,some researchershaveusedpiezoceramicsheetsen-sors in controllablestructural systems74;75 and in health monitoringapplications.76 Giurgiutiu and Zagrai77 carried out a comparativeevaluation of different PZT sheet sensors. Because of the way thematerial is stretched during manufacturing, the PVDF behaves aspiezoelectrically orthotropic but mechanically isotropic for smallstrains.

The direct effect equations are

8<

:

D1

D2

D3

9=

; D

24

0 0 0 0 d15 0

0 0 0 d25 0 0

d31 d32 d33 0 0 0

35

8>>>>>><

>>>>>>:

¾1

¾2

¾3

¿23

¿31

¿12

9>>>>>>=

>>>>>>;

C

24

e¾11 0 0

0 e¾22 0

0 0 e¾33

35

8<

:

E1

E2

E3

9=

; C

24

a1

a2

a3

351T

The sensor equations are based on direct effect. The sensor isexposed to a stress � eld, which in turn generates an electric � eld.Monolithic PZT sensors are transversely isotropic, and as a resultd31 D d32 and d15 D d25 . Consider a case where electric � eld is zero

and there is no thermal strain:

8<

:

D1

D2

D3

9=

; D

24

0 0 0 0 d15 0

0 0 0 d25 0 0

d31 d32 d33 0 0 0

35

8>>>>>><

>>>>>>:

¾1

¾2

¾3

¿23

¿31

¿12

9>>>>>>=

>>>>>>;

A stress � eld causesan electricdisplacement,which in turn is relatedto the charge generated:

q DZ Z

[D1 D2 D3]

24

dA1

dA2

dA3

35

where dA1 , dA2 , and dA3 are, respectively,the differentialelectrodeareas in the 2-3, 1-3, and 1-2 planes. The charge q and voltage Vc

are related by the capacitance of the sensor Cp as

Vc D q=C p

Knowing the voltage, it is thus possible to measure the stress andhence the strain. Consider a sheet sensor with two faces coated withthin electrode layers. In case of uniaxial stress � eld (in direction 1),the capacitance is given as

C p D e¾33lcbc

¯tc

where lc , bc , and tc are length, width, and thickness of the sensor,respectively.Assuming strain along the 1-direction, it follows that

Vc Dd31Ycbc

C p

Z

lc

"1 dx

where Yc is Young’s modulus of the sensor and "1 is averaged overthe gauge length. The strain is calculated as

"1 DVcC p

d31Yclcbc

For this relation it is assumed that only the strain in the 1-directionexists, and there is no loss of strain in the bond layer. With theincluding of Poisson’s effect, this relation reduces to

"1 DVcC p

d31[1 ¡ º.d32=d31/]Yclcbc

where º is Poisson’s ratio. For conventional foil gauges the trans-verse sensitivityis close to zero and normallyneglected.In a generalsituation it is not possible to separate out the principal strains us-ing one piezoelectric sensor. Unless the transverse strain is knowna priori, it is not possible to obtain longitudinal strain using onesensor.

The effect of bond layer is determined assuming a uniform stainbeam theory. As a result of shear lag effect in the bond layer, theeffective length and width of sensor get reduced:

"1 DVcC p

d31[1 ¡ º.d32=d31/]Yclceff bceff

The effective length and width of a sensor are function of sensorand bond-layer characteristics.Reference 78 presents a procedureto determine these values. Because of lower thickness and stiffnessof a typical PVDF sensor, it shows a much lower shear lag loss thana PZT sensor.

CHOPRA 2153

1. Signal ConditioningTo measure accurately the output of piezoelectricsensor, it needs

to be passed through some signal conditioning system. Typically,the output impedance of piezoelectricsensor is very high while themeasuring device such as a voltmeter has comparatively low inputimpedance (order of 1 MÄ). The objective of signal conditioningsystems is to provide a signal with low output impedance whilemaintaininghigh input impedance to the sensor. One way is to shortthe electrodesof the sensor with a suitableresistorand then measurethe current passing through this resistor using an ampli� er. Thesecond way is to use a chargeampli� er to measurechargegenerated

Fig. 11 Charge ampli� er circuit to measure strain with piezoelectricsensor.

Foil strain gauge

PZT sensor

Fig. 12 Foil strain gauge and PZT sensor impulsive response infrequency domain.78

Fig. 13 Correlation of measured foil strain gauge strain and PVDF sensor strain and theoretical predictions.78

by the sensor, as shown in Fig. 11. In this circuit C p is capacitanceof sensor, Cc represents capacitance of cables, and CF is feedbackcapacitance. The value of time constant as given by RF CF can beselected to achieve the desired frequency range.

The sensitivity of piezoelectric sensor is far superior than thatof conventional foil type strain gauges, with much less signal con-ditioning required, especially for applications involving low strainsand high noise levels.Figure12 shows impulsiveun� ltered responsefrom both a conventional foil strain gauge and a PZT strain sensor,taken simultaneously after the cantilevered beam was impacted atits tip. There is signi� cant backgroundnoise in the foil gauge outputas compared to the PZT output. The spikes in PZT output are theovertones of the ac power-line frequency.

It is not advisable to use these sensors to measure strain above150–200 microstrains because of increasingnonlinear piezobehav-ior. The output of the PZT sensor requires no temperature cor-rection over a moderate range of operating temperatures (say lessthan 40±C). Figure 13 shows measured strain results from straingauge, PZT sensor, and PVDF sensor of size 7:1 £ 3:6 £ 0:056 mmattached at the root of a cantilevered aluminum beam. Correlationis generally quite satisfactory. For a constant gauge length sensi-tivity increases with increasing sensor area. Also, as the size ofsensor increases the shear lag losses as a result of bond layer de-crease, and there is a better transfer of strain from the surface ofthe structure to the sensor. However, the larger sensor measuresaverage strain over a larger area instead of local strain and alsocan add stiffness to the baseline structure,especiallywith PZT sen-sors. Even though the piezoelectric properties (dielectric permit-tivity and piezoelectric coef� cients) change with temperature, theoverall effect on the calibration of PZT sensors is small away fromCurie’s limit. On the other hand,PVDF exhibitsa signi� cant changeof pyroelectric properties with temperature in addition to change inpiezoelectric properties. Hence, PVDF sensors are relatively sen-sitive to temperature, and suitable temperature compensation mustbe included in measurements.For some speci� c measurementsspe-cially shaped distributed PVDF sensors can increase the perfor-mancesigni� cantly.79;80 Piezoelectricstrainsensorsare simple,easyto use, and a reliablealternativeto conventionalresistance-basedfoilstrain gauges.

There have been limited attempts to validate systematically thecalibration factor for piezoelectric sensors for a wide range of fre-quencies and strain levels. Doubtless, the sensitivity of sensor de-pends upon strain distribution(baselinestructuralcon� gurationandloading),sensor size, temperature,and fabricationtechniques(bondlayer thickness).There is a need to establish guidelines to optimizethe sensitivityof piezoelectricsensor under differentoperatingcon-ditions. It is equally important to develop signal-conditioningcir-cuitry to cover a wide range of frequencies.

III. Modeling: Beam with Induced Strain ActuationA one-dimensional beam with surface-bonded or embedded in-

duced strain actuators represents a basic and important element of

2154 CHOPRA

Fig. 14 Piezoelectric element.

Fig. 15 Force-strain characteristic of a piezoelement.

an adaptive structure. Many structural systems such as helicopterblades, airplanewings, turbo-machineblades, missiles, space struc-tures,and civil structuresare routinelyrepresentedas slenderbeams.With induced strain actuation it might be possible to actively con-trol aerodynamicshape for vibrationsuppression,stability augmen-tation, and noise reduction. Several beam theories have been de-veloped to predict � exural response of isotropic and anisotropicbeams with surface-bondedand embedded induced strain actuationthat range from simpli� ed models to detailed models involvinguni-form, linear, and nonlinear displacement distribution through thethickness.

An induced strain actuator such as a piezeceramic sheet actuatoris de� ned using two key parameters: free strain and block force.Let us consider a piezoelectric actuator of length `c , width bc , andthickness tc (Fig. 14), the maximum free strain 3 for an electric� eld V (voltage) is

"max D 3 D d31.V=tc/

whered31 is piezoelectricconstant(m/V). The maximumblockforce(zero strain condition) is

Fbl D d31 EcbcV

where Ec is the Young modulus of elasticity of piezo actuator.Let us imagine that an extensional free strain 3 is produced in

longitudinal direction (axis 1) caused by an electric � eld V ap-plied in the polarized direction (axis 3). Now if a compressive forceF is applied to the piezo in longitudinal direction, the net strainwill be

" D 3 ¡ F=Fbl

The strain at any compressive force F is plotted in Fig. 15.Two identical actuatorsmounted on the surface of a beam, one on

eithersurface,can producepurebendingor pureextension(Figs. 16aand 16b). For pure bending an equal but opposite potential is ap-plied to top and bottompiezos, whereas for pure extension the samepotential is applied to both piezos. Piezos experience equilibratingstresses. For example, for pure extensionalcase both piezos experi-ence compressive stress.

To modelone-dimensionalstructuressuchas slenderbeams,let us� rst start with simple beam models and then discuss re� ned models.Among the simple models, popular ones are block force model,uniform strain model, and Euler–Bernoulli model.

A. Block-Force Beam ModelThis is the most simple beam model, where each piezoactua-

tor is represented in terms of its block force and free strain. The

Fig. 16a Pure extension (same voltage to top and bottom piezos).

Fig. 16b Pure bending (equal but opposite voltage to top and bottompiezos).

piezo introduces a local strain to substrate and in turn experiencesan equilibrating force. Let us consider a case of beam with twoidentical actuators respectively surface bonded to top and bottomsurfaces.

For a pure extension case the same potential is applied to top andbottom actuators. The induced force is

F D FblE Ab

E Ab C E Ac

where Fbl is a block force of each piezoelement. The extensionalstiffness of beam and actuators are de� ned as

E Ab D Ebbbtb; E Ac D 2Ecbctc

If actuator stiffness E Ac À E Ab (beam stiffness), the actuationforce tends to zero although the actuation strain equals free strain.On the other hand, if the actuator stiffness E Ac ¿ E Ab the actu-ation strain becomes zero although actuation force equals blockforce.

For a pure bendingcase an equal and oppositepotential is appliedto top and bottom actuators, and the induced bending moment is

M D Mbl

³E Ib

E Ib C E Ic

´

where Mbl is block moment and is equal to Fbltb . The bending stiff-ness of beam and actuators are de� ned as

.E I /b D Eb Ib D 112

Ebbbt 3b ; .E I /c D 2.bc tc/.tb=2/2 Ec

Again, if E Ic À E Ib actuation moment becomes zero, and on theother hand, if E Ic ¿ E Ib , actuation strain becomes zero. Thismethod is quite insightful and is used for design studies.

B. Uniform-Strain Beam ModelThis model helps to understand and substantiate the effect of

bond layer. Let us consider two identical strain-induced actuatorsthat are bonded to an isotropic beam, one to the top surface andthe second to the bottom surface of the beam (Fig. 17). Betweenthe actuator and the beam surface, there is a � nite thickness elasticbond. The bond layer is assumed to undergo pure shear strain. Eachactuator is assumed to induce a uniform strain across its thickness.For pure bending a linear distribution of strain is assumed in thehost structure.

The surface strain on beam substructure and actuator strain arederived as

"sb

3D ®

® C 9

µ1 ¡ cosh.0 Nx/

cosh 0

¶

"c

3D ®

® C 9

µ1 C 9

®

cosh.0 Nx/

cosh 0

¶

CHOPRA 2155

De� nition of thickness and coordinate system

Transmission of shear across adhesive layer

Fig. 17 Diagram of beam mounted with symmetric actuators.

where ® D 2 is for pure extension and ® D 6 is for pure bending.Other constants are de� ned as

Nx Dx

`c=2; 02 D

G s

ts Ec

³1tc

CEc®

Ebtb

´`2

c

4

Ã D .E A/b

Ec AcD

EbtbEc tc

where ts is the thicknessand G s is the shear stiffnessof the adhesive.The parameter0 is the shear lag parameter, and the larger the value,the lower the actuation loss through the bond layer (for example, asa result of a thin bond line or high adhesive shear stiffness).

A very thin bond (0 > 30) approximately represents a perfectbond condition, and strain distributions reduce to

"sb

¯3 D "c=3 D ®=.® C 9/

This means that the induced strain on the surface of a host structureis equal to the actuator strain, and it is proportional to the productof the actuation free strain 3 and the reciprocal of one plus thestiffness ratio (beam stiffness/actuator stiffness). The second termis as a result of the impedance matching.The higher the stiffnessofthe actuator, the more effective is the strain transfer.

C. Bernoulli–Euler Beam ModelThe Bernoulli–Euler model is a consistent strain model and as-

sumes the beam, adhesive, and actuator as a continuous structureand follows the Bernoulli’s assumption.This means that a plane sec-tion normal to the beam axis remains plane and normal to the beamaxis after bending. It is assumed that there is a linear distributionofstrain in the actuator and host structure and that shearing effects arenegligible.

The strain distribution is

".z/ D "0 ¡ z·

where · is bending curvatureD ¡w;x x . The net strain is

"net D ".z/ ¡ 3.z/

The force and moment expressions can be written as»

F C F3

M C M3

¼D

µ.E A/tot .E S/tot

.E S/tot .E I /tot

¶ »"0

w00

¼

where

.E A/tot DZ h=2

¡h=2

b.z/E.z/ dz extensional stiffness

.E S/tot DZ h=2

¡h=2

b.z/E.z/z dz coupling stiffness

.E I /tot DZ h=2

¡h=2

b.z/E.z/z2 dz bending stiffness

F3 DZ h=2

¡h=2

b.z/E.z/3.z/ dz induced force

M3 DZ h=2

¡h=2

b.z/E.z/3.z/z dz induced moment

where F3 and M3 are respectivelyaxial force and bending momentas a result of induced strain. (E S)tot is equivalent to a coupling term.If the placementof actuators is symmetric, this term will be zero. Ifan actuator is attached only on one side, this term will be nonzero(extension-bendingcoupling).

For an isotropic beam with pure bending actuation,

M3 D Ecb3tc.tb C tc/

.E I /tot D Ec tcb

³2

3t2c C tbtc C

t 2b

2

´C

Ebbt3b

12

M3 D .E I /totw00; F3 D 0

" D zM3

.E I /totD

6.1 C 1=T /.2=ts/3

.9 C 6/ C 12=T C 8=T 2z

The thicknessratio T (D tb=tc) determineswhether the strainvari-ation across the piezoelement affects the analysis. Figure 18 showsthe variation of the normalized curvature with the thickness ratio.For a case with piezo stretched across the beam width, and with athickness ratio of 5, there is a 2.7% difference between the predic-tions caused by the uniform strain model and the Bernoulli–Eulermodel. For thin beams, the uniform strain model overpredictsstrain(curvature).For large thicknessratios the predictedinducedbendingcurvatures are identical using both methods. For thickness ratios ofone or less, neither of these methods works well. More details aboutthese beam models can be obtained from Refs. 81–84. Banks andZhang85 developed a curved beam analysis for a pair of surface-attached piezoelectric patches based on the Donnell–Mushtari the-ory for shell models and B-spline basis elements. They successfullyreducedthe analysis to straightbeams without any lockingproblem.

Assume two identical actuators are embedded in an isotropicbeam at an equal distance from midplane of the beam resultingin a symmetric con� guration (Fig. 19). For a very thin bond layerbetween the actuators and the beam, a perfect bond assumption ap-pears valid. An equal voltage applied to both actuators results inpure extension, whereas an equal but opposite voltage applied toboth actuators causes pure bending of the beam.

Fig. 18 Single actuatornormalized curvature comparisonfor a perfectbond condition.83

2156 CHOPRA

Fig. 19 Beam with embedded piezoelements.

Combine bending-extensionrelations into the matrix

»F C F3

M C M3

¼D

µE Atotal E Stotal

E Stotal E Itotal

¶ 8<

:

"0

d2w

dx2

9=

;

where the stiffness terms gets modi� ed. The matrix form of thesegoverning equations is similar to that for surface-bondedactuators,except that the stiffness terms need to be modi� ed to account for theembedded location for the actuators.

Ghiringhelliet al.86 developeda semianalyticalformulationfor anarbitrary cross-sectionbeam with embedded piezoelectricelementsand showed good agreement of their predicted results with three-dimensionalmodel results.

D. Energy FormulationUsing the samebasicassumptionsas used in the forceequilibrium

formulation, the principle of virtual work can provide the govern-ing equationsand boundary conditionsthat can be easily adapted todynamic systems. Assuming that the only allowable modes of de-formation are actuator extension,adhesiveshear, and beam bendingand extension, the strain energy relations can be directly written.

Beam extension:

U 0b D 1

2

Z lc=2

¡lc=2

Eb Ab

³@u0

b

@x

´2

dx

Beam bending:

U ·b D 1

2

Z lc=2

¡lc=2

Eb Ib

³@2w

@x2

´2

dx

Substrate shear:

Us D1

2

Z`c=2

¡`c=2

Gs As.°s/2 dx

Actuator extension:

Uc D1

2

Z`c=2

¡`c=2

Ec Ac

³@uc

@x¡ 3

´2

dx

De� ne

Ib D 1

12Abt 2

b ; U ·b D 1

2

Z lc=2

¡lc=2

Eb Ab

3

¡"s

b ¡ "0b

¢2dx

Us D 1

2

Z `c=2

¡`c=2

G s As

t 2s

¡uc ¡ us

b

¢2dx

The principle of virtual work for static behavior is mathematicallystated as

±WE D ¡±U

It will result in governing equations and boundary conditions, asobtained directly with the uniform-strain model.83

Fig. 20 Piezoaxis offset from the beam axis.

Fig. 21 Analytical strains and de� ections (Ãb = 38 ) (Ref. 83).

E. Extension-Bending-Torsion Model (Skewed Piezo)If a slender actuator is oriented at an angle with respect to the

beam axes (Fig. 20), it results in a coupled extensional, bending,and torsion response. Despite two-dimensionallocal induced straindistribution, the large-aspect-ratioactuator primarily acts along itsmajor axis becauseof dominantshear lag effects in the lateraldirec-tion. If two identical actuators are oriented at §¯ deg at the top andbottom surfaces, respectively, the same potential will induce puretwisting, whereas an opposite potential will cause pure bending.For the maximum induced twist the piezos should be oriented at§45 deg, respectively, on the top and bottom surfaces. Park et al.83

and Park and Chopra84 developed uniform-strain and Bernoulli–Euler beam models for an isotropic beam with a surface-bondedsingle piezoceramicactuator at an arbitrary orientationwith respectto the beam axis. The shear lag effectsas a result of a � nite thicknessof adhesive layer were included. The actuator was assumed to be aline element and only permitted to induce strain in its lengthwisedirection.

Using the uniform-strain theory, an expression for beam twist Áwith a single actuator oriented at angle ¯ is obtained as83

Á

3. Nx/ D 3

4

lc

tb

Eb

Gb

cos.¯/ sin.¯/

Ã C ® cos3.¯/

µsinh.0 Nx/ C sinh.0/

0 cosh.0/¡ . Nx C 1/

¶

where Gb is the shear modulus of beam. Figure 21 shows variationof strains with skew angle. The maximum twist that occurs for ¯equals 45 deg.

CHOPRA 2157

F. Shear-Based (d15) Piezoelectric ActuationThere have been a few selected works on shear actuation, where

piezoelectricactuators are normally poled in longitudinal directionand subjectedto transverseelectric � eld. As a result, the actuatorun-dergoes shear deformation through d15 effect. Theoretical analysesof beams with surface-bondedshear actuators have been developedby Trindade et al.,87 Benjeddouet al.,88;89 Zhang and Sun,90 and Caiand Gao.91 It appears that the shear actuation mechanism is bettersuited for active damping control.

G. Validation of Beam TheoriesExperimentationwas performedon rectangularaluminumbeams

with surface-bondedG-1195 piezoceramicactuators to validate thebeam theories.81;83;84 Crawley and de Luis81 previously correlatedthe analysis for a pair of piezoelements surface bonded to an alu-minum beam. In Ref. 83 static tests were performed to validatebending results for a pair of piezos with differentwidth ratios, a sin-gle piezo aligned with the beam axis, and a single piezo not alignedwith the beam axis. A necessary ingredient for the analysis is thefree piezostrainas a functionof the appliedelectric � eld (see Fig. 22for G-1195 material).Strain data for an unconstrainedpiezoceramicelement were obtained vs applied voltage and zeroed for each datapoint. This was done in order to minimize the effects of hysteresisand creep.

The test specimens were 116

and 132

in. thick 2 £ 16 in. aluminumbeams (1 in.D 25.4 mm) with 9.5 mil G-1195 piezoceramic ele-ments.Con� gurationsincludedactuatorsthat spannedthe fullwidthand half-width of the beam. The piezoceramics were bonded to thespecimensusinga cynoacrylateadhesiveto minimize the bond layereffects. Bending slope and twist data were obtained by measuringthe relative horizontal travel of a laser beam re� ected off a mirrorat the tip of the beam. Predictions for bending slope show goodagreement with test data in Fig. 23.

Fig. 22 G-1195 piezoceramic element free behavior.83

Fig. 23 Bending slope test data and predictions for 132 -in. (0.79-mm)-

thick aluminum beam.83

Fig. 24 Typical torsion test specimen.

Fig.25 Effects of actuatororientationon beam bendingslopeand twistangle Ãb = 38; Ãs = 0.0020 (Ref. 82).

Experimental torsion and bending results for cantilevered beamswith actuators bonded to one surface and oriented with an angle¯ relative to the beam axis (Fig. 24) were compared with ana-lytical predictions from the combined extension-bending-torsionuniform strain model.84 The test specimens were for 1

32 -in. thickaluminum beams with three 2 in. £ 1

4 in. £ 7.5 mil G-1195 piezo-ceramic elements distributed along the 16-in. beam in 4-in. inter-vals. Because actuators of aspect ratio eight were used to approx-imate the theoretical line element assumption, three piezos wererequired to obtain measurable de� ections using optical measure-ment system. Superposition of the analytical results is assumed forcomparison with the experiment and independentlyveri� ed for thepiezospacing.

As shown in Fig. 25, the bending slope predictions are within20% of the experimentalvalues up to ¯ D 45 deg. Beyond this pointthe theory signi� cantly diverges from the test results. The torsionanalytical results follow the experimental trend over the full rangeof ¯ but overpredict the experimentalmagnitudeby 35–100% in themidrange .15 < ¯ < 75/. Beam torsion and bending are adequatelymodeled one-dimensionally,but the mechanism that produces twistin this system is inherently two-dimensional. The primary sourceof discrepancy can be attributed to the one-dimensional approxi-mation of the strain state. Effects of chordwise and lateral bendinghave been neglected and might constitute signi� cant error depend-ing upon the beam cross-section aspect ratio. Overall, the resultsindicate that a one-dimensionalmodel is not satisfactory to predictcombined torsion, bending, and extension of beams with surfacebonded induced strain actuators.

To assess the effect of bond thickness on actuator performance,simple beam specimens were built and tested under a static � eld.92

The bond thickness was varied from 0.0025 in. (close to perfectbond condition) to 0.02 in. (not too abnormal), and the orientationof actuators was varied from 0 (aligned with beam axis) to 65 deg.Also, uniform-strain beam theory for embedded skewed actuatorsincluding effect of bond layer was developed and compared withtest data. The effect of reducing the bond thickness can be seen inFig. 26. The maximal torsional and bending de� ections increasedby 60 and 90% respectively when the bond thickness was reducedfrom 0.020 to 0.0025 in. A minimal bond layer thickness results inthe most ef� cient shear transfer, which in turn results in maximumtorsional and bending response. An optimum blade twist actuationwill result for perfectly bonded actuators oriented at skew angles of§45 deg.

2158 CHOPRA

Table 4 Comparison of smart beam models

PiezoelectricModeling type Actuator coupling Beam type Validation Reference

Block force Surface and embedded Uncoupled Isotropic —— ——Euler–Bernoulli Surface and embedded Uncoupled Isotropic Cantilevered aluminum Crawley and de Luis81

Straight patches Park et al.83

Skewed patches Isotropic Cantilevered aluminum Park et al.83

Park and Chopra84

Uniform strain Surface and embedded Uncoupled Isotropic Cantilevered aluminum Crawley and de Luis81

Straight patches Park et al.83

Skewed patches Isotropic Cantilevered aluminum Park et al.83

Park and Chopra84

Timoshenko Surface and embedded Uncoupled Isotropic —— Shen108

(FSDT)Vlasov with Surface-bonded Uncoupled Isotropic and Cantilevered composite Chandra and Chopra102

chordwise bend Straight patched composite coupled Bernhard and Chopra106

and ShearEuler–Bernoulli Surface Coupled Isotropic Cantilevered aluminum Hagood et al.94

coupledLWSD theory Surface Coupled Isotropic and —— Robbins and Reddy110

composite Saravanos and Heyliger111

Bond thickness 0.020 in. (0.508 mm)



Fig. 26 Effect of bond thickness and actuator orientation on can-tilevered beam tip twist.92

The preceding subsections examined several one-dimensionalstructural models that can predict the behavior of different beamcon� gurations with induced strain actuators. Modeling of shearlag effects of the � nite bond layer appears important especiallyfor beams with thick and/or soft bond layer. The single actuatoruniform-strain model governing equations can be also formulatedusing an energy approach that can be easily adapted to dynamicsystems. A one-dimensional treatment of a strain actuated beam incoupledextension,bending,and torsionwas shown to be inadequateto predict its structural behavior. Because the torsion trend is pre-dicted, analytical accuracy can be improved by integrating a localtwo-dimensional model of the actuation mechanism with a globalone-dimensional system model.

H. Assessment of Beam TheoriesTable 4 lists different smart beam models. Crawley and de Luis81

formulated the uniform-strain model for a beam with surface-bonded piezoceramic actuators (patched and aligned with beamaxis).The model calculated� exural responseincludingshear lag ef-fects of the adhesive layer between the piezoceramicand the beam.It was shown that the strain transfer from the piezoelectric to thesubstructure takes place over a small zone near the ends of actuatorand there is maximum shear stress in this region. As the adhesivelayer becomes thinner and/or stiffer (shear modulus), it approachesa perfect bond condition (shear concentrated at two ends of actu-ator). The dynamic model was experimentally veri� ed for the � rsttwo bending modes of a cantilevered aluminum beam. They alsopresented a uniform-strain model for an isotropic beam with em-bedded actuators and satisfactorilyvalidated the dynamic responseat resonancefor aluminum, glass-epoxyand graphite-epoxybeams.Im and Atluri93 developed a nonlinear analysis of a piezoactuatedbeam with � nite thickness bond layer including the effects of trans-verse shear and axial forces in addition to the bending moment onthe beam. Again, it was shown that the maximum shear stress oc-curs near the two ends of the piezoelectric element and is also afunction of externally applied axial and shears forces. Crawley andAnderson82 formulated the Euler–Bernoulli model for a beam withsurface-bonded or embedded induced strain actuators (symmetricactuation)and compared it with a uniform-strainmodel, a � nite ele-ment model, and an experiment.For the uniform beam theory a uni-form shear stress is assumed through the thickness of the adhesive,and a uniformaxial stress is assumed though the thicknessof the ac-tuator. In spite of these gross assumptions, the uniformstrain modelwas generally found satisfactory except for low beam-to-actuator-thickness ratios (<4). The Bernoulli–Euler model was quite satis-factory to predict bending and extensional response even for lowthickness ratios. There is no doubt that for thickness ratio of oneor less (such as the case with bimorphs) a re� ned model including

CHOPRA 2159

three-dimensionaleffects might be needed.The linear model (usinglinearpiezoelectriccharacteristics)is accurateonlyfor small strains.To predict reliable � exural results with high � eld conditions, non-linear � eld-strainsrelationsshould be included.The precedinganal-yses neglected the couplingof piezoelectricson mechanicalproper-ties (uncoupledanalyses). Hagood et al.94 formulated a completelycoupled piezoelectric-mechanical model for a beam with surface-bonded actuators. Predicted dynamics were found to be in goodagreement with experimental data obtained with a cantileveredalu-minumbeam. Park et al.83 developedcoupledbendingand extensionanalysisfor an isotropicbeamwith isolatedsurfacebondedactuator.A � nite thicknessadhesive layer between actuatorand beam was in-cluded.The convergencepoint of the Bernoulli–Euler and uniform-strain predictions was shown to be a function of beam-to-actuator-width ratio in addition to thickness ratio. Satisfactory validation ofpredicted bending slope with measured values was carried out forseveral different aluminum beams. Benjeddou et al.88 developed auni� ed beam� nite elementmodel for extensionand shearpiezoelec-tric actuationmechanism.This is especiallysuitablefor sandwichedbeams.The modelusedBernoulli–Euler theoryfor the surfacelayersand Timoshenkobeam theory for the core. It was shown that the pre-dicted induceddeformationwas lower with the shear-actuatedbeamtheory.

Park and Chopra84 developed coupled extension, bending, andtorsionanalysisfor an isotropicbeam with surface-bondedactuatorsat an arbitrary orientation with respect to the beam axis. Piezoce-ramicactuatorswere representedas lineactuators.Systematicexper-imental tests with cantileveredaluminumbeamswere carriedout forinducedbendingand twist at differentorientationanglesto check theaccuracy and limitation of models. Comparison of predicted resultswith test data showed that the modelswere satisfactoryin predictingtrends for bending slope and twist with different orientation angle.The predicted bending slope deviated signi� cantly from measuredvalue for orientation angles greater than 45 deg (Fig. 25), more sofor piezoceramicswith moderate aspect ratios. Detailed strain mea-surements showed that the local strains are quite two-dimensional.Therefore, the inclusion of effects of transverse actuation might benecessary to re� ne the analysis.

Jung et al.95;96 made an assessment of the state of the art in mod-eling of thin- and thick-walled composite beams with a view toemphasize the special characteristics of composite materials. Thereview encompasses modeling nonclassical effects such as out-of-plane warping, warping restraints, and transverse shear. Compos-ite beam modeling ranges from simple analytical models to de-tailed � nite element models and has been validated using limitedtest data from simple tailored specimen.97¡101 The anisotropic na-ture of composite materials makes the structural properties direc-tion dependent. Using special ply layup, structural couplings suchas bending-torsionand extension-torsioncan be introduced. Thesecouplings can be exploited with induced strain actuation to activelycontrol aerodynamic shape such as in helicopter blades or an air-plane. In modeling of a composite beam with induced stain actu-ation as a one-dimensional structure, it is important to encompassall of the important effects caused by bending and shear de� ec-tions and twist of reference axis of the beam and warping defor-mations of the cross section. Normally, the warping deformationsare much smaller than the � exural deformations.This helps to sim-plify the complexity of inherently three-dimensional problem intotwo parts: a two-dimensional local deformation � eld of the crosssection that is used to calculate the section properties and a one-dimensional global deformation � eld to predict the response of thebeam. The � rst level of idealization of the global deformation in-cludes the Bernoulli–Euler model for bending and the St. Venantmodel for torsion. In the next levels torsion-related warping, trans-verse shear strain, and cross-sectiondeformation(in-planewarping)effects are included. For composite thin-walled beams it is possibleto model the shell wall either as a membrane or as a thick laminateincluding the effect of transverse shear as well as bending distribu-tion. Chandra and Chopra102 developed a formulation for coupledcomposite thin-walled open- and closed-section beams with dis-tributed induced strain actuation (surface-mounted or embedded)and then validated the analysis with experimental data. Beam mod-

eling was based on Vlasov theory, where two-dimensional stressand strain distributions associated with any local plate (shell) ele-ment of the beam are reducedto one-dimensionalgeneralizedforcesand moments. Effects of transverse shear and warping restraintswere included. Comparison with experimental data from bending-twist and extension-twist coupled graphite-epoxy composite solidbeams with surface-mounted piezoceramic actuators showed thatthe inclusion of chordwise (lateral) bending is essential to predicta beam’s coupled response accurately. Also, Kaiser103 carried outa similar type of study with thin-walled, open- and closed-section,coupled composite beams with piezoelectric actuation. Cesnik andShin104 developed a re� ned multicell composite beam analysis foractive twist rotor with embedded AFC actuators. The approach isbased on a two-step asymptotic solution: a linear two-dimensionalcross-sectional analysis and a global nonlinear one-dimensionalanalysis.Subsequently,the analysiswas successfullyvalidatedwithtest data for different blade con� gurations and load conditions.105

Ghiringhelli et al.86 developed a re� ned � nite element analysis foranisotropic beam with embedded piezoelectric actuators and suc-cessfully compared their results with three-dimensionalresults. Asa part of smart-tip rotor development, Bernhard and Chopra106 de-veloped Vlasov-type beam analysis for a tailored composite cou-pled beam with induced strain actuation. It consisted of a num-ber of spanwise segments with reversed bending-twist couplingsfor each successive segment. Each segment acts like a bimorph,and the polarity of successive surface-bonded piezoceramics is re-versed. Because of the � ip-� opping excitation, the beam de� ectsinto a sinusoidal bending wave, whereas the induced twist is addi-tive spanwise.Predictionswerevalidatedsatisfactorilywith testdatafor several different beam con� gurations. For accurate predictionsit became necessary to include nonlinear measured characteristicsof piezoceramics and modeling of chordwise bending. It is nowwell established that effects of transverse shear can be very impor-tant at both local and global level for the response of compositebeams because of low values of shear modulus G compared withthe direct modulus E (G/E ratio). A � rst-order shear deformationmethod is Timoshenko’s model, which assumes a constant shearstress across the cross section. This is called � rst-order shear de-formation theory (FSDT).107;108 This model, however, violates thetraction-freeboundaryconditionon the top and bottom surfaces.Tocompensate this anomaly, a shear correction factor is applied. Tocapture the nonlinear distribution of transverse shear strain acrossthe cross section, higher-order shear deformation theories (HSDT)are used. These theories however are unable to capture accuratelya dramatic change of properties at a local ply level. A further re-� nement to HSDT is to use layer-wise shear deformation theory(LWSDT),109 which models shear distribution for each layer sepa-rately. Robbins and Reddy110 carried out static and dynamic anal-ysis of piezoelectricallyactuated beams using LWSDT. Saravanosand Heyliger111 developed coupled layer-wise analysis of compos-ite beams with embedded piezoelectric actuators and sensors. Itwas shown that consistentand more detailed stress distributionses-pecially near the end of the actuator are obtained with layer-wisetheory. For prediction of higher modes of vibration and/or thickercomposite structures, it might be more appropriateto use layer-wisetheory.

It is clear from testing of simple isotropic beams with surface-attachedpiezoelectricelementsthat the localstraindistribution(atornear the actuator) is two-dimensional,84 and, therefore, beam mod-elingwith inducedstrain actuationshould re� ect such a distribution.Simple beam theories often give erroneous results for beams withlow actuator-to-beam-thickness ratio (such as the case with piezo-bimorphs). Detailed three-dimensionalmodels [say, � nite elementmethod (FEM) models] should be used to establish the strain ac-tuation mechanism. Most beam theories have either neglected theshearing effect of bond layer (by assuming perfect bond condition)or have incorporated a highly approximate shear model (for ex-ample, uniform shear stress within bond thickness); however, testresults92 showed that the bond thickness has a dominant effect onthe induced strain transfer from the actuator to beam. It shouldbe important to examine systematically the shearing effect of bondlayer usinga higher-ordersheardeformationtheorysuch as LWSDT

2160 CHOPRA

and establish the limits of simple beam models (uniform-strain andBernoulli–Euler models). There have been only limited studies onthe validation of predictions for composite coupled beams withsurface-attached or embedded piezoceramics; these should be ex-panded to cover more beam con� gurations and tailored couplingsfor static and dynamic loads. Such studies can be very important forshape control of aerospacesystems. Most predictionshave incorpo-rated linear piezoelectriccharacteristicsthat are strictly true for low� eld conditions.To covermoderate to high electric � elds, it is worth-while to include nonlinear characteristics of piezoelectrics. It willbe important to examine systematically the effect of piezoelectric-mechanical couplings on actuation strain for a range of isotropicand laminated beams.

IV. Modeling: Plate with Induced Strain ActuationOne of the basic elements of adaptive structures is a thin compos-

ite plate with surface-inducedor embedded sheet actuators. With atailored laminated plate, induced strain actuation can control its ex-tension,bending,and twisting.Plates with distributedinducedstrainactuators can be used to control pointing of precision instrumentsin space; to control structural borne noise; and to change aerody-namic shape for vibration reduction, � utter suppression, and gustalleviation.

Several plate theories have been developed to predict � exuralresponse of laminated plates with surface-bondedor embedded in-duced strain actuators that include classical laminated plate theory(CLPT), FSDT, HSDT, and LWSDT. All of these theories assumethat the actuatorsand substrateare integratedas plies of a laminatedplate undergoingconsistentdeformation.They,however,differfromeach other in terms of displacement distribution through the thick-ness of plate (modeling of transverse shear).

Among these plate theories, the CLPT is most widely used. Itis based on the Kirchoff–Love hypothesis that is quite similar tothe Bernoulli–Euler beam theory. It assumes a linear variation ofbending strain across thickness, and the effects of transverse shearare neglected. (A line originally normal to midplane of plate re-mains normal to midplane after bending deformation.) It implies aperfect bondedconditionbetween actuatorsand substrate.This the-ory is truly applicablefor thin plates (length/thickness> 30), wheretransverse shear effects are negligible.

The constitutive relation for any ply of a laminated plate is

¾ D NQ." ¡ 3/

where stress vector

¾ D

8<

:

¾x

¾y

¿x y

9=

;

For a generic coupled laminated plate with surface-bondedor em-bedded induced strain actuators (piezoceramic sheets) placed atarbitrary locations, the strain distribution is expressed as

² D ²0 C z·

where ²0 is the midplane strain and · is the bending curvature. Co-ordinates x and y are in the in-plane directions,and z represents theout-of-planedirection (Fig. 27). The u, v, and w are displacements

Fig. 27 Plate coordinate system.

in the x, y, and z directions.The strains are de� ned as

"0 D

8><

>:

"0x

"0y

"0x y

9>=

>;D

8>>>>>><

>>>>>>:

@u

@ x

@v

@y

@u

@yC @v

@x

9>>>>>>=

>>>>>>;

· D

8<

:

·x

·y

·x y

9=

; D

8>>>>>>><

>>>>>>>:

¡@2w

@ x2

¡@2w

@y2

¡2@2w

@x@y

9>>>>>>>=

>>>>>>>;

and actuation strain vector

K D

8<

:

3x

3y

3x y

9=

;

Matrix NQ is the transformed reduced stiffness of the plate. By sub-stituting the assumed deformation into the stress strain equationsand integrating through the thickness t of plate with N plies for netforces and moments,

8>>>>>>><

>>>>>>>:

Nx

Ny

Nx y

Mx

My

Mxy

9>>>>>>>=

>>>>>>>;

D

2

666666664

24

A11 A12 A16

A12 A22 A26

A16 A26 A66

35

24

B11 B12 B16

B12 B22 B26

B16 B26 B66

35

24

B11 B12 B16

B12 B22 B26

B16 B26 B66

35

24

D11 D12 D16

D12 D22 D26

D16 D26 D66

35

3

777777775

£

8>>>>>>>><

>>>>>>>>:

"0x

"0y

° 0x y

·x

·y

·xy

9>>>>>>>>=

>>>>>>>>;

¡

8>>>>>>><

>>>>>>>:

Nx3

Ny3

Nx y3

Mx3

My3

Mx y3

9>>>>>>>=

>>>>>>>;

where extensional stiffness

A DZ

t

NQ dz; Ai j DNX

k D 1

. NQ i j /k .hk ¡ hk ¡ 1/

coupling stiffness

B DZ

t

NQz dz; Bi j DNX

k D 1

. NQ i j /k

³h2

k ¡ h2k ¡ 1

2

´

bending stiffness

D DZ

NQz2 dz; Di j DNX

k D 1

. NQ i j /k

³h3

k ¡ h3k ¡ 1

3

´

actuator forces and moments

N3 DZ

t

Q3 dz; M3 DZ

t

Q3z dz

CHOPRA 2161

A. Symmetric LaminatesThese are laminates with ply layups that are symmetric with re-

spect to the midplane. Coupling stiffness terms Bi j become identi-cally zero, and bendingand actuationequationsbecome uncoupled.

Extensional actuation is

24

A11 A12 A16

A12 A22 A26

A16 A26 A66

35

8>><

>>:

"0x

"0y

° 0x y

9>>=

>>;D

8><

>:

Nx3

Ny3

Nx y3

9>=

>;

where8><

>:

Nx3

Ny3

Nx y3

9>=

>;D 2Ec tc3

1 ¡ º

8<

:

1

1

0

9=

;

There is no induced shearing strain.Bending actuation is

24

D11 D12 D16

D12 D22 D26

D16 D26 D66

35

8<

:

·x

·y

·x y

9=

; D

8<

:

Mx3

My3

Mx y3

9=

;

where8<

:

Mx3

My3

Mx y3

9=

; DEc tc.tb C tc/3

1 ¡ º

8<

:

1

1

0

9=

;

There is no direct twisting moment with piezos. There are how-ever extension-shear and bending-twist couplings with symmetriclaminates.

B. Antisymmetric LaminatesThese are laminates with ply layups that are antisymmetric with

respect to midplane.Behaviorof such laminates can be signi� cantlydifferent from that of symmetric laminates. For an antisymmetriclaminate, the extensional-shear couplings and bending-twist cou-plings are zero. There are, however, extension-twist couplings andbending-shearcoupling with antisymmetric laminates.

Extensional actuation is

24

A11 A12 0

A12 A22 0

0 0 A66

35

8>><

>>:

"0x

"0y

° 0x y

9>>=

>>;C

2

640 0 B16

0 0 B26

B16 B26 0

3

75

8<

:

·x

·y

·x y

9=

;D

8<

:

Nx3

Ny3

Nx y3

9=

;

Bending actuation is

24

0 0 B16

0 0 B26

B16 B26 0

35

8>><

>>:

"0x

"0y

° 0x y

9>>=

>>;C

2

64D11 D12 0

D12 D22 0

0 0 D66

3

75

8<

:

·x

·y

·x y

9=

;D

8<

:

Mx3

My3

Mx y3

9=

;

C. Approximate Solution (Energy Approach)For many problems it is not possible to obtain an exact solution.

Thus, it becomes necessary to obtain an approximatesolution usingan energy approach such as the Rayleigh–Ritz method. The strainenergy of the system is

U D 12

Z

A

Z ©"0T

·Tª µ

A B

B D

¶ »"0

·

¼dA

¡Z

A

Z[N3 M3]

»"0

·

¼dA

where

»"0

·

¼D [D]

»u

Q

¼D

2

666666666666666666664

@

@x0 0

0@

@y0

@

@y

@

@x0

0 0 ¡@2

@x2

0 0 ¡ @2

@y2

0 0 ¡ 2@2

@ x@y

3

777777777777777777775

8<

:

u

v

w

9=

;

The Rayleigh–Ritz approximate solution is

u DMX

i D 1

Áui qi ; v DNX

j D 1

Áv j q j C M

w DPX

k D 1

Áwk qk C M C N ;

24

u

v

w

35 D [H ]q

where

H D

24

[Áu1 ; Áu2 ¡ Áu M ] [0] [0]

[0] [Áv1 ; Áv2 ¡ ÁvN ] [0]

[0] [0] [Áw1 ; Áw2 ¡ ÁwP ]

35

3 £ r

where

r D M C N C P;

µ"0

·

¶D [D][H ]q

Substitution in strain energy reduces to

U D 12qT K q ¡ Q3qK q D Q3

where

K DZ

A

Z.DH /T

µA B

B D

¶.D H / dA

Q3 DZ

A

Z.D H /T

»N3

M3

¼dA

The matrix K is of size r £ r , and the actuation-forcingmatrix Q3

is of size r £ 1.Applying Lagrange’s equation, it is possible to solve for modal

amplitudes:

K q D Q3

For more details, see Refs. 112 and 113.Let us considera cantileveredrectangularcompositeplate (length

L and chord c) with two surface-mounted piezoceramic sheets.Assume a simple three-term solution:

u D .x=L/q1; v D 0; w D .x=L/2q2 C .x=L/.y=c/q3

This results in2

6666664

A11 ¡ 2L

B11 ¡2c

B16

¡2L

B114L2

D114

cLD16

¡2c

B164

cLD16

4c2

D66

3

7777775

8<

:

q1

q2

q3

9=

; D

8<

:

Nx3L

¡2Mx3

0

9=

;

2162 CHOPRA

Table 5 Comparison of smart plate models

Validation withPlate theory Actuators Piezoelectric coupling Plate type test data References

CLPT Surface-bonded full Uncoupled Composite, Cantilevered Crawley and Lazarus112

surface nonlinear aluminumpiezochar. and composite

Surface-bonded Uncoupled Composite, linear Cantilevered Lee114

patches piezochar. compositeModi� ed CLPT with Surface and embedded Uncoupled Composite, Cantilevered Hong and Chopra113

transverse shear Dicrete patches nonlinear compositepiezochar.

Reissner–Mindlin Surface and embedded Coupled Nonlinear —— Carrera124

FSDT Dicrete patches Karmananalysis, thickisotropic

LWSD Surface and embedded Coupled Isotropic and —— Mitchell and Reddy129

theory Piezoply composite Robbins and Reddy130

Chattopadhyay et al.131

Zhou et al.132

Higher-order Surface and embedded Coupled Isotropic, thick —— Sun and Whitney128

(kth-order) Piezoply Batra and Vidoli135

three-dimensional Ha et al.136

thick plate theory

For a symmetric plate with bending-torsion coupling (B11 DB16 D 0),

q3 DcL

2

D16

D11 D66 ¡ D216

2D16

cLMx3

This shows that induced twist is caused by bending actuation (op-posite � eld on top and bottom piezos), and it depends on the ratioof the couplingstiffness D16 to the bending stiffness D66 times D11.

Bending-twist coupling ÃT D D16=p

.D11 D16/. For an antisym-metriccompositeplatewith extension-torsioncoupling(B11 D 0 andD16 D 0),

q3 D 1

2cL

µB16

A11 D66 ¡ B216

¶Nx3

L

This shows that induced twist in an antisymmetricplate is causedbyextensional actuation (same � eld on top and bottom piezos), and itdependson the ratio of couplingstiffness B16 to the extensionalstiff-ness A11 times torsionalstiffness D66. The extension-twistcouplingparameter is

ÃB DB16p

A11 D66

D. Review of Plate ModelingTable 5 lists differentsmart platemodels.Crawley and Lazarus112

systematically developed the CLPT formulation and a Rayleigh–Ritz analysis for anisotropicplates and validated it with test data ob-tained by testing cantilevered aluminum and composite plates withsurface-bondedpiezoceramic actuators (attached at top and bottomsurfaces fully). Nonlinear piezo characteristics(d31 with � eld) weremeasured experimentallyand included in analysisusing an iterativeapproach.Results demonstrated the validity of analysis for selectedplate con� gurationsand showed the potential for shape control withinduced strain actuation.Also, Lee114;115 developeda CLPT formu-lation for a composite plate using linear actuation characteristicsof piezoelectric laminas. A limited validation study was carried outwith test data obtained from a thin composite plate actuated withpiezoelectricpolymer � lm (PVDF and PVF2). Wang and Rogers116

applied CLPT to determine the equivalent force and moment in-duced by � nite-length surface-attached piezoelectric actuator to alaminate. They used linear characteristicsof piezoelectricsand de-veloped a simpli� ed analysis to calculate bending and extensionof the plate. Hong and Chopra113 developed a consistent � nite ele-ment formulation for coupled composite plates including modelingof transverse shear and nonlinear piezoelectric characteristics.Theanalysis is applicable to a generic anisotropic plate with a numberof piezoactuators of arbitrary size, surface bonded or embedded atarbitrary locations. Composite cantilevered plates with extension-twist and bending-twistcouplingswith two rows of surface-bonded

Fig. 28 Cantilevered plate with surface-mounted piezoceramics.113

piezoceramics on both top and bottom surfaces were tested exten-sively, and data were used to validate analysis (Fig. 28). Predictionsagreed satisfactorily with test data for most con� gurations, the ex-ception being stronglybending-twistcoupled plates, where the pre-dicted induced twist caused by bendingwas underestimatedby 20%(Figs. 29 and 30). The use of an iterative procedure with the incor-poration of nonlinear piezoelectric characteristics (as suggested byotherresearchers)was foundto beunnecessary.Heyliger117 obtainedexact solutions for some idealized plate con� gurations.

For CLPT formulationthe transverseshear stiffnessof plate is as-sumed to be in� nite.As a result, there is a zero transverseshearstrainacross the complete thickness.For CLPT the in-planedisplacementsare caused by rotationof midplane normal. However, with the pres-ence of transverse shear strains the in-plane displacements will getmodi� ed as a result of rotations of cross-sectionalplanes relative tomidplanes. For re� ned plate models different levels of shear mod-eling are introduced. A generic displacement distribution is

u.x; y; z/ D u0.x; y/ C zÁx .x; y/ C z2&x .x; y/ C z3’x .x; y/

v.x; y; z/ D v0.x; y/ C zÁy .x; y/ C z2&y .x; y/ C z3’y.x; y/

w.x; y; z/ D w0.x; y/

where Áx ; Áy represent rotations of the cross section and&x ; &y; ’x ; ’y correspond to higher-order terms in the Taylor-series

CHOPRA 2163

Fig. 29 Longitudinal bending and twist distribution at midchordof a bending-twist coupled [+302 /0]s graphite/epoxy plate caused bypiezobending excitation at 100 V.113

Fig. 30 Twist distribution at midchord of an extension-twist coupled[+453 /¡ ¡ 453] graphite/epoxy plate caused by piezoexcitation at 100 V(same voltages to top and bottom piezos).113

expansion. The u0, v0 , and w0 are midplane displacements.For theCLPT &x ; &y ; ’x ; ’y are assumedto be zero, and Áx ; Áy are gradientsof out-of-plane displacements. References 118–121 reviewed dif-ferent plate theories for induced strain actuation. First-order sheardeformable-plate theory (FSDT) is based on the Reissner–Mindlinplate model and is quite similar to Timoshenko’s beam theory. It re-laxes theassumptionof the normalityof the cross-sectionplaneafterdeformation.Transverse shear strains are assumed uniform throughthe thickness of the plate. It fails to account for changes in shearstrains caused by the variation of material properties of each layer.For FSDT, &x ; &y; ’x ; ’y are assumed to be zero, and Áx ; Áy are as-sumed constant through thickness (independentof w0). This theoryestimates lower � exural stiffness than that from the CLPT theory.Another anomaly with this theory is that there is nonzero shearstrain at top and bottom free surfaces which violates the physicalboundarycondition.Normally, a shear correctionfactor is applied tocompensate for nonzero shear strain at free lateral surfaces.Bisegnaand Maceri122 and Bisegna and Carusa123 developed a Reissner–Mindlin type � nite element formulation(locking-freequadrangularelements) for the analysisof a plate with surface-bondedthin piezo-electric sheet actuators. It was shown to be suitable and effective forsome vibration-controlanalyses.Carrera124 extended the Reissner–Mindlinplatemodel to themultilayeredstructuresthroughthe inclu-sion of both the zigzag in-plane displacements and the interlaminaequilibriumof transverse shear stresses.The theory is normally de-notedby the acronymRMZC (Reissner–Mindlinzigzagcontinuity).For the calculationof results, a quadratic distributionof the voltage� eld along the thickness direction and von Karman-type nonlinearplate analysis were used including the effect of electromechanicalcoupling. It was shown that RMZC effects become important forthick anisotropic plates, especially in the evaluation of transverseshear stresses.

A highersheardeformabletheory(HSDT) developedbyReddy125

models a general distribution of transverse shear strain through thelaminate thickness. For HSDT &x ; &y; ’x ; ’y are assumed nonzero.This represents a cubic variation of displacements (u; v) throughthe thickness resulting in a quadratic variation of shear strain. Thisdistribution satis� es the traction-free boundary condition on topand bottom surfaces but lacks accurate representationof layer-wisevariation of shear strain caused by different material properties oflaminas. In general, it is expected that the HSDT should give a bet-ter prediction of � exural stiffness than that with FSDT, but it is notensured for all plate con� gurations.126 To model the variations ofmaterial stiffness from layer to layer, it appears appropriate to uselayerwisesheardeformabletheory (LWSDT) attributedto Reddy127

as well as Sun and Whitney.128 For this theory the laminate is di-vided into a number of sublayers that are perfectly bonded, and ineach layer the in-plane displacement is assumed piecewise linearlyalong the z direction. There is a signi� cant increase in degrees offreedom of the model:

u.k/.x; y; z/ D u0.x; y/ C zÁ.k/x .x; y/

v.k/.x; y; z/ D v0.x; y/ C zÁ.k/y .x; y/

w.k/.x; y; z/ D w0.x; y/

where Á.k/x ; Á.k/

y represent rotations of the cross section of the kthlayer. For a case of single layer, LWSDT reduces to FSDT. Eventhough the shear strain is assumed uniform in each layer, there is avariation from layer to layer. Mitchell and Reddy129 used LWSDTto model smart composite laminates with embedded piezoelectricsheets using linear piezoelectriccharacteristics.Also, this model in-cluded the coupling between mechanical deformation and electro-static charge equations. Robbins and Reddy130 formulated a linearglobal-local analysis based on layer-wise shear deformation the-ory to determine local shear � elds and global response in surface-mounted piezoelectricactuated plates. Using a variable-order� niteelement discretization, interlaminar stresses in the adhesive layerwere determined. It was shown that the highest transverse normalstressoccursat the interphasebetweenbond layer and beamnear thefreeedgesthat canbe the likelysourceof debonding.Chattopadhyayet al.131 and Zhou et al.132 used LWSDT to calculate static anddynamic response of composite plates with surface-bondedpiezo-electric actuators using a completely coupled thermo-piezoelectric-mechanical model. Most researchershave neglected these couplingeffects. They have shown that to model the behavior of smart com-posite laminates accurately, it is important to model transverseshear of each layer using LWSDT and incorporate piezoelectric-mechanical biway coupling effects. Vel and Batra133 developed athree-dimensionalanalytical solution using Eshelby–Stroh formal-ism to calculate static response of thick multilayered piezoelectricplates. Only linear piezoelectric characteristics are incorporated.Using a three-dimensionalmixed variationalprinciple,134 Batra andVidoli135 derivedhigher-order(kth order) anisotropichomogeneouspiezoelectric plate theory. The electric potential, mechanical dis-placement, and in-plane stresses were expressed as a � nite seriesof order k in the thickness coordinate using Legendre polynomi-als as the basis functions. The boundary conditions on the top andbottom surfaces were exactly satis� ed. Results were obtained forbendingof cantileveredthickplatewith surface-bondedPZT sheets.It was shown that the seventh-order plate theory captured well theboundary-layereffects near the free and clamped edges. Ha et al.136

used a three-dimensionalcomposite brick element to analyze staticand dynamic response of a laminated plate with distributed piezo-ceramic actuators. Even though such an analysis can increase thecomputationalinvolvementenormously, it has the � exibility to ana-lyzegenericplate con� gurationsincludingthickplateswith surface-bonded or embedded patched actuators.

Most current plate analyses assume a perfect bond condition be-tween actuator and bond surface (that is, neglected shearing effectof adhesive).This assumption is too restrictiveand therefore shouldbe examined systematically especially for discrete actuators. Sim-ple plate theories such as CLPT are routinely used to analyze platestructures. It might be important to examine its limits for differ-ent plate con� gurations and actuation � elds with the help of either

2164 CHOPRA

higher order shear deformation theories (such as LWSDT) or de-tailedFEMs (such as three-dimensionalsolid elements).There havebeen limited studies to validate predictions with experimental testdata for coupledcompositeplateswith surface-bondedor embeddedpiezoelectric elements. These should be expanded to cover a rangeof plate con� gurations including strongly coupled bending-torsioncoupled plates.

V. Shape Memory AlloysShape-memory-alloy(SMA) actuators are � nding increasingap-

plications in aerospace, civil, mechanical, medical, and other sys-tems. They have a special characteristicof “memorizing” a certainstretched or bent shape and recovering that shape at a higher tem-perature.Buehler et al.137 discovereda nickel-titaniumalloy in 1961called NITINOL that exhibited much higher shape memory effectsthan previous materials. This alloy demonstrates 100% recovery ofstrain up to a maximum of 8% extensionalprestrain,which makes itattractive for use in low-frequency(less than 1 Hz) actuators. SMAactuators have high work density ratios, very high strokes, directactuation capabilities, and embeddable characteristics.Another in-teresting feature is a two- to fourfold increase in Young’s modulusabove the critical phase transition temperature. The phase changeis also accompanied by a large change in resistivity and release (orabsorption) of latent heat. Restraining the recovering strain resultsin large recoverystress (several times more than the initial stress re-quired to prestrain at room temperature). These characteristics areexploitedto tune thepropertiesof structuresforvariousapplications.Figure 31 explains the shape memory effect schematically.

The shape that an SMA memorizes can be assignedor reassignedthrough an annealing (say, above 500±C). At room temperature theSMA is generally in martensite phase, and its undeformed crystalstructure is twinned (less symmetric). In this phase it is easy to movetwin boundariesin the directionof applied stress, and thus the mate-rial modulus of elasticity and yield stress are very low. Applicationof extensional stress to the material above its yield stress causesdetwinning (change orientation of crystal twins) and hence a plas-tic deformation.Through an applicationof heat to this material, theplastic prestraincan be completelyrecovered.At a high temperaturethematerial is in austenitephase(parentphase),and its crystalsresultin a right-angleorderedlattice. In the austenitephase it is more dif� -cult to deformthe twin boundariesthanmartensitephase,resultingina higheryield stressandmodulusof elasticity.Figure 32 sketchestheshapememory process.Typically,a plastic strainof more than 8% atlow temperature (fully detwinned martensite phase) can introducepermanentor irrecoverableplastic strain as a result of the formationof dislocations. If an SMA is constrained with a spring (such as thecase with embedding in a host structure), the alloy is preventedfromreturning to the original shape upon heat activation. This results inthe generationof a large recovery stress. On the other hand, if SMAis not constrained there will be no recovery stress (free recovery).For an SMA embedded in a host structure, the recovery stress willdecrease on cooling, and if this stress is still higher than martensite

Fig. 31 Schematic of shape memory effect in stretching.

Fig. 32 Process of shape memory effect.

a) Shape memory effect b) Pseudoelasticity effect

Fig. 33 Stress-strain behavior of SMA.

yield stress, it will again result in a plastic strain (two-way motionactuator).

The thermomechanical behavior of SMA material depends ontemperature, stress, and history of the material. In the heating cyclefor temperaturesbelow As (austenitestart temperature),the materialis in the 100% martensitephase,whereas for temperaturesabove A f

(austenite � nish temperature), the material is in the 100% austenitephase. Higher stress increases As because more energy is needed tomove the lattice structureunder imposed stress. In the cooling cyclefor temperaturesabove Ms (martensite start temperature), the mate-rial is in 100% austenite phase, whereas for temperature below M f

(martensite � nish temperature), the material is in 100% martensitephase.Under an imposed stress the materialwill start to transitiontomartensite phase at a higher temperature.Hysteresis can be viewedas the friction associated with the movement of twin boundaries.At any other temperature between A f and M f , the material can bepartly in the martensite phase and partly in the austenite phase. Attemperature below As application of stress causes a transformationfrom “twinned” martensite to the stress-preferred or “detwinned”martensite, resulting in a large strain at a nearly constant stress. Onthe removalof stress, a signi� cant amount of strain stays, which canbe completely recovered by heating the material above A f . This iscalled the shape memory effect (SME) (Fig. 33a). The ability ofthe shape memory alloys to recover large strains comes from thereversible phase transformation characteristics.

At a temperatureabove A f , the applicationof stresscausesa trans-formation from austeniteto the stress-preferredmartensite state. Onthe removalof stress, the strain is completelyrecoveredand is calledpseudoelasticity(Fig. 33b).

The state of the material is characterizedby the volume fractionof the martensite phase » . Figure 34 schematically illustrates thechange in martensitevolume fractionwith temperature.For a mixedstate the value of » varies between 0 and 1.

A. Constitutive ModelsMany constitutive models have been developed to describe the

thermomechanical behavior of SMA materials. Some models arebased primarily on thermomechanics, and others are based ona combination of thermomechanics and SMA phenomenology

CHOPRA 2165

Fig. 34 Schematic of phase transformation.

and/or statistical mechanics. Most of these constitutive models arephenomenological-based macroscalemodels that are developed forquasi-static loading only. One of the most popular one-dimensionalmodels is Tanaka’s model138 and is based on thermomechanics. Inthis model the second law of thermodynamics is written in terms ofthe Helmholtz free energy,followed by derivationof the rate form. Itis assumed that the uniaxial strain, temperature, and martensitevol-ume fraction » are the only state variables. An exponential expres-sion of » is developed in terms of stress and temperature.Liang andRogers139 presented a model, which is based on the rate form of theconstitutiveequation developedby Tanaka. In their model, a cosinerepresentationwas used to describe the martensite volume fraction.A major drawback of these two models is that they only describethe stress-induced martensite transformation and do not considerstrain-induced martensite transformation (shape memory effect).Hence, they cannot be applied to model the detwinning of marten-site that is responsible for the SME at low temperatures. To over-come this de� ciency,Brinson140 developeda constitutivemodel thatseparates out the martensite volume fraction into two parts: stress-induced martensite and temperature-induced martensite. The � rstpart describes the amount of detwinned or stress-preferred variantof martensite present, and the second part describes the fractionof martensite caused by the reversible phase transformation fromaustenite phase.

To cover both the shape memory and pseudoelasticity effects,the coef� cients of the constitution equation are assumed to be non-constants.Another constitutivemodel is the thermodynamicmodelby Boyd and Lagoudas,141;142 which is based on free energy anddissipationpotential.This model is derived from the Gibbs free en-ergy instead of the Helmholtz free energy used in Tanaka’s model.This model can cover three-dimensionalstates and nonproportionalloading. Another model that uses a thermodynamic formulation isby Ivshin and Pence.143 There are other models that are not basedon these two approaches. For example, Graesser and Cozzarelli144

developed a three-dimensional model based on evolutionary plas-ticity. Sun and Hwang145 derived a micromechanical model takinginto account thermodynamics,microstructure,and micromechanicsand coveringboth SME and pseudoelasticity.Matsuzakiet al.146 de-velopeda generalone-dimensionalthermomechanicalmodel takinginto accounteffectsof energydissipation,latentheat,and heat trans-fer during phase transformation.They introduceda general form ofphase interaction energy function that was determined form qua-sisteady experimental stress-strain data. The model showed goodagreement with test data at different strain rates. Barrett147 devel-oped a one-dimensional constitutive model for SMA that includesphase change hardening,hysteresiswith partial transformation,andtension-compressiontransformations.The model however differen-tiates between stress-induced and temperature-induced martensitevolume fractions.

The � rst four models are described in more detail. Tanaka138

assumedthat strain² , temperatureT , and martensitevolumefraction» , are the state variables to describe one-dimensional behavior ofthe shape memory alloys (such as SMA wires). The constitutiveequation is as follows:

.¾ ¡ ¾0/ D E.» /." ¡ "0/ C 2.T ¡ T0/ C Ä.» /.» ¡ »0/

where subscript 0 represents the initial condition of SMA. Thisequation shows that the stress consists of three parts: the mechan-

ical stress, the thermoelastic stress, and the stress caused by phasetransformation. Note that the Young’s modulus E and the phasetransformationcoef� cient Ä are functionsof the martensitevolumefraction. These are normally expressed as

E .» / D E A C ».EM ¡ E A/; Ä.»/ D ¡"L E.» /

where "L is the maximum recoverable strain. The E A and EM re-spectively represent Young’s modulus at austenite and martensitestates. Tanaka developed an evolutionary equation for the marten-site volume fraction. The evolutionary equation determined fromdissipation potential was resolved to have the following form:

» D 4.¾; T /

This equationimplies that the martensitevolume fraction» is a func-tion of stress and temperature. Tanaka models » as an exponentialfunction.During the M Ã A transformation,the martensitevolumefraction is

» D 1 ¡ expfaM .Ms ¡ T / C bM ¾ gand during the M ! A transformation it is de� ned as

» D exp[aA.As ¡ T / C bA¾ ]

where the material constants are de� ned as

aA D .0:01/

.As ¡ A f /; bA D

aA

CA

aM D.0:01/

.Ms ¡ M f /; bM D

aM

CM

The coef� cients used in the preceding constitutive relations, E ,2, and Ä, and the parameters Ms , M f , As , A f , CA , and CM are de-termined normally throughtestingof the SMA wires (Table 6). Notethat the transformation temperatures Ms , M f , As , and A f are deter-mined in a stress-freecondition.The stress in� uencecoef� cientsC A

and CM are the slope of the critical stress-temperatureplots for theaustenite and martensite transformationboundaries, respectively.

Liang and Rogers139 utilized the same constitutive relation butdeveloped a new form of the evolutionary equation for the marten-site volume fraction. The differencebetween the two models arisesin the modeling of the martensite volume fraction. In this model »is modeled as a cosine function. Both models however lack properrepresentation of shape memory effect (detwinning at room tem-perature). Both models describe transformation from martensite toaustenite and vice versa. They are, however, valid only for fullydetwinned martensite state.

Unlike the preceding two models, Brinson’s model140 representsthe transformation from the detwinned martensite to the stress-preferred variant, which leads to a description of shape memoryeffect below As . This model uses a similar representation for con-stitutive equationwith some modi� cations.The modi� ed relation isas follows:

¾ ¡ ¾0 D E.» /" ¡ E.»0/"0 C Ä.» /»s ¡ Ä.»0/»s0 C 2.T ¡ T0/

In this model the martensite volume fraction is divided into twoparts:

» D »S C »T

where »S is the portionof the detwinned(or stresspreferred)marten-site present at low temperature and »T is the portion of twinned (or

Table 6 Thermomechanical properties of SMA wires tested(Dynalloy 15 mil NiTi binary alloy)158

Coef� cient/ Coef� cient/property Unit Value property Unit Value

¾ scr Pa 1.38eC6 As

±C 52¾

fcr Pa 1.72eC6 A f

±C 65CA Pa/±C 8eC6 Ms.Tanaka/

±C 43.5CM Pa/±C 12eC6 M f .Tanaka/

±C 40.7E A Pa 45eC9 Ms.Brinson/

±C 55EM Pa 20.3eC9 M f .Brinson/

±C 42

2166 CHOPRA

randomly oriented) martensite that comes from reversible thermalphase transformation from austenite. Also, a modi� ed cosine rep-resentation for the martensite volume fraction is used.

The fourth model by Boyd and Logoudas141;142 is a more generalthree-dimensionalmodel. In this model the total speci� c Gibbs freeenergy is determined by summing the free energy of each phaseof shape memory materials plus the free energy of mixing. A con-stitutive relation satisfying the second law of thermodynamics isdeveloped. The total strain consists of two parts: the mechanicalstrain "i j and the transformationstrain "tr

i j , which is a functionof themartensite volume fraction:

¾i j D Ci jkl

£"kl ¡ "tr

kl ¡ ®kl.T ¡ T0/¤

An evolutionary equation for the martensite volume fraction is de-rived from the dissipation potential:

¾ effi j C d1T ¡ ½b1» D Y ¤¤ C d3

i jkl ¾i j ¾kl C d4i j ¾i j 1T

where Y ¤¤ is the threshold stress value, d1T is related to the en-tropy reference state, d3

i jkl and d4i j are parameters that are related

to changing elastic moduli, ½ is the mass density, b1 is a mate-rial constant, and 1T is the temperature difference. Brinson andHuang148 showed that this model reduced to one-dimensional for-mat becomes identical to Tanaka’s model. There is, however, oneexception, that unlike Tanaka’s model this model is applicable torepresent the martensite transformationfrom twinned to detwinnedstate at a low temperature.

Another model developed by Ivshin and Pence143 was derivedfrom a thermodynamic consideration of the kinetic relations forthe hysteresis of the phase fraction. Instead of using the martensitevolume fraction » as the primary variable, they used the austenitevolume fraction ®

® D 1 ¡ »

Total strain is

" D .1 ¡ ®/"m C ®"a

where "m and "a are strains in martensite and austenite regions.Since the formulationof the originalTanaka model,138 it has been

updated to cover the in� uence of the SME.149¡154 The primary up-date of the model has been carried out through the introduction ofnew effect called the rhombohedral-phase transformation (RPT).The RPT is a thermally reversible transformation that takes placeat low strains and low temperatures, much like martensite detwin-ning. At low temperatures (below As) RPT takes place followed byregular martensite transformation. In repeated stress-strain cyclingat room temperature (strain> 1%), the RPT is noticeable only inthe � rst cycle and disappears thereafter. This is attributed to two-way shape memory effect observed through heating and coolingof the specimen. At high temperatures (between As and A f ) theRPT is noticeable in the form of small nonlinearity of the linearregion of the pseudoelastic curve. The RPT or R-phase effect isrepresented by a volume fraction coef� cient ´. Assuming the � rstcycle under zero stress/strain condition (´ D 0, » D 0), an applica-tion of strain below 1% results in R-phase transformation (´ D 1,» D 0). On heat activation we recover R-phase. In all other caseswe do not recover R-phase. Using this concept, the authors wereable to correlate predictions with test data especially at low tem-peratures. Naito et al.154 and Sittner et al.153 represented the RPTand martensite transformation in a uni� ed way using energy func-tion, with appropriate “switching functions” to handle differenttransformations.

Using the constitutive models in conjunction with the modelsfor the martensite volume fraction develops stress-strain curves.Because » is a function of stress in all models, an iterative methodsuch as the Newton–Raphson method is used to solve for the stress.

B. Critical Transformation RegionsFor constitutive models the required constants are obtained from

the critical stress-temperaturediagram. Figure 35 is a plot that de-lineates the regions of transformation. The regions marked witharrows are those in which the material exists in pure form. All otherregions could have a mixture of phases, and the exact content of

a) Original Tanaka model

b) Brinson model

Fig. 35 Critical stress-temperature pro� les in constitutive models.

the mixture depends on the thermomechanicalhistory of the mate-rial. There are, however, differences in the de� nition of the materialconstants in different models. Both the Tanaka model and Liangand Rogers model assume a straight-line stress-temperature rela-tionship, and Ms and M f correspond to zero stress condition. Thestress-temperaturerelationship is somewhat different for Brinson’smodel; the critical stresses divide the transformation regions. TheMs and M f are de� ned as the temperature above which the appli-cation of stress does not cause a pure transformation from twinnedto detwinned martensite.

Because of these differences, the interpretation of the transfor-mation temperatures in the two types of models differs from eachother. In Brinson’s model the parameters Ms and M f are de� ned asthe temperature above which the martensite transformationstressesare a linear function of temperature, as shown in the Fig. 35b. Inthe Tanaka and Liang and Rogers models, however, these parame-ters are de� ned at zero stress and are the temperatures for marten-site start and � nish obtained by cooling from austenite withoutthe application of stress (Fig. 35a). Therefore, when calculatingthese constants from the experimental critical points, the numer-ical values used for Tanaka and Liang and Rogers model for Ms

and M f are obtained from extrapolating the martensite start and� nish lines to zero stress, whereas those used by Brinson modelare obtained at the critical stresses ¾ s

cr and ¾f

cr . These different val-ues for the models must be used in order to obtain a fair com-parison between these models and to match them to experimentalobservations.

Prahlad and Chopra155 presented critical stress-temperaturedatafrom different tests on the same curve and demonstrated the linearvariation of the critical stresses with temperature that is assumedin all of the models (Fig. 36). Data corresponded to three differenttests that, respectively,used heat � ow measurements with differen-tial scanningcalorimeter(DSC), constant temperature,and constantstrain (constrainedrecovery)conditions.All of these tests were car-ried out at a very low heat rate and as such represented quasistaticconditions. Tests at constant strain and constant temperature werecarried out using MTS 810 tensile test machine with a controllablethermal chamber (environment heating). Two separate thermocou-ples were used to monitor temperature, one in the thermal chamberand anotherK-type thermocoupledirectlymountedon the wire. Therecovery stress data points lie in the transformationregion betweenthe austenite start and � nish states for heating phase and martensitestart and � nish states for cooling phase. Thus the two straight linesrepresent martensite start and help determining Ms and As . The

CHOPRA 2167

a) Environmental heating

b) Resistive heating

Fig. 36 Experimental critical stress-temperature plot.155

martensite � nish and austenite � nish follow nearly the same slopesand help determine M f and A f . The critical stress ¾ s

cr is the stressat which the transformation from twinned to detwinned martensitestarts, and thecriticalstress¾

fcr representsthe stresswhere this trans-

formation is nearly complete.These critical stresses are determinedfor a temperaturebelow Ms . As shown in Fig. 35b, the region above¾

fcr line represents pure detwinned martensite state, and the region

below ¾ scr line is a twinned state. The interpretation of Ms and M f

can be differentby two approaches.Strain at which detwinninggetscompleted is referred to as "l . The parameters CA and CM are theslopes of these two curves. It is assumed that below Ms the criticalstresses ¾ s

cr and ¾f

cr associatedwith the detwinning are independentof temperature (a small anomaly from test data). The critical stress-temperature data obtained at low stresses and temperatures werefound to be more sensitive to test conditions and in fact were moredif� cult to obtain. To determine the transformation temperatures(Ms , M f , As , and A f ) of the SMA wire under no stress condi-tion, Perkin–Elmer DSC is normally used. This instrumentuses thechange in heat capacity of the material to determine the start and� nish transformation temperatures. When the material undergoesphase transformation, it absorbs (or emits) large amount of heatwith a relatively small change of temperature, and the DSC cap-tures such a change. Two different temperature control methods areused; the � rst involves the use of a thermal chamber (environmentalheating), and the second involves internal heating (resistive heat-ing). Even though similar trends are observed with both methods ofheating, transformationstresses differ as a result of small variationsin temperature. For example, the transformation from martensiteto austenite state takes place at lower temperatures in the case ofresistive heating.

To stabilize the SMA for a repeatablebehavior,it becomesneces-sary to “cycle” the materialmechanically.One approach is to stretch

the wire (to, say,4% strainwith a strain rate of 0.0005/s) at a constanttemperature (well above A f ) and then release to zero stress condi-tion. After 20–30 cycles the pseudoelasticbehavior of the wire getsstabilized.156 In the second approach the SMA wire is mechanicallystrained at room temperature (in martensite phase) followed by athermal cycle under no stress condition by heating the wire aboveA f and then cooling down below M f . Again, it requires about 20–30 cycles to stabilize the material.156;157 Both methods are equallyeffective to stabilize the material, except the � rst method is simplebecause it does not require thermal cycling.

C. Comparison of Stress-Strain ResultsThe stress-strainplots at constant temperature show shape mem-

ory effect at low temperaturesand pseudoelasticityat high tempera-tures (Fig. 37).At lowtemperature(say35±C, below As ) thematerialis in martensite state.Once critical stress ¾ s

cr is reached, the transfor-mation to stress-preferredvariant starts. The Tanaka model and theLiang and Rogers model are not valid for temperatures below As ,whereas the Brinson model is applicable.The points 1 represent thestart of transformation to detwinned martensite and points 2 repre-sent the start of transformation to austenitephase and points 3 showthe � nish of transformation to austenite phase. Figure 38 shows agood comparison of the Brinson model with test data. At high tem-perature (say 85±C, above A f ) the material is fully austenitic to startwith and gets transformed to the stress-preferredmartensite variantat critical stress. This transformation gets reversed on unloading,causing a complete pseudoelasticrecovery.For temperatures aboveAs , all threemodelsare applicable.Becauseboth theBrinsonand theLiang and Rogers models use a cosine formulation to describe themartensitic volume fraction, they yield nearly identical predictions

Fig. 37 Stress-strain plots.158

Fig. 38 Comparison of models with test data.158

2168 CHOPRA

in this regime. It is clear that as long as coef� cients of models areidenti� ed properly from test data all constitutive models can pre-dict stress-strainbehavior satisfactorilyat high temperatures(aboveAs).158 Note that the Tanaka model uses exponential representa-tion for martensite volume fraction, which can lead to numericalconvergenceproblem during the iterative solution process.

D. Resistive Heating vs Internal HeatingDuring the phase transformation of SMA, there is a large vari-

ation of internal resistance of material.159 As a result of this, it isquite dif� cult to hold constant temperature with resistive heatingespecially during phase transformations. It is however possible toachievea temperaturecontrolof theorderof 2–3±C byveryslowheatactivation of wire. Figure 36a shows the critical stress-temperatureplot from experimental data points obtained using environmentalheating. Figure 36b shows the correspondingcritical stress plot forresistive heating. By comparing the two, it appears that the entireplot appears shifted towards lower temperatures in case of resistiveheating. However, the slopes of critical stresses seem to remain un-changed. In a polycrystal structure involving different phases, theresistance level can be quite different in each phase. Passing thecurrent through the wire can result in a differential heating of ma-terial (surface temperature different from internal temperature). Asa result, it might not be prudent to mix two sets of data. It might beimportant to use different values of Ms and M f for modeling twocases of heat activation.

E. Constrained Recovery BehaviorLarge recovery stresses are developed when the SMA wire is

constrained during the heating process. Figure 39a shows the com-

a)

b)

Fig. 39 Restrained recovery behavior with prestrain.155

plete constrained stresses developed at different temperatures fordifferent prestrains. The � nal constrained stress is independent ofprestrainas long as it is above a threshold value of 2%. Below about2% prestrain, the stress path followed, and the � nal stresses are de-pendent on prestrain. Most of the models satisfactorily predict theconstrained recovery stress, as long as the values are low (less thanplasticyield stress).Lower prestrainsoffer theadvantageof less per-manent plastic deformation and fatigue with repeated cycles.159;160

Figure 39b shows the stresshistoryofSMA wire in repetitivecycles.After three cycles the material is stabilized, and the following cy-cle repeats the results of previous cycle. This clearly shows thatthe SMAs can be used as active force generators under repetitiveloading.

F. Nonquasistatic LoadingIf the material is strainedat a faster rate, the materialdoesnothave

time to relax and as a result attains a higher stress value. The risein stresses might be a result of local temperature changes that occurimmediately if the wire undergoes nonquasistatic loading. After ahigh strain rate, if the material is relaxed it returns to its quasistaticstress value. (Local temperatures might settle down to equilibriumvalues.)This is called stress relaxationphenomenon.159¡162 Shaw163

pointedout that some of the self-heatingeffect arises from the orig-ination of local nucleation sites with temperature differencesalongthe wire. It has been shown that the transformationstresses increasesigni� cantly as a function of strain rates (Fig. 40). If the loading isnoncontinuous,theSMA appearsto fall back into its thermodynamicequilibrium state (stress relaxation). This phenomenon also mani-fests itself as a dependence of the material behavior on the loadingpattern (Fig. 41). For dynamic loading it is necessary to develop

Fig. 40 Effect of loading rates on stress-strain loading pattern onstress-strain behavior.155

Fig. 41 Effect of loading pattern on stress-strain behavior.155

CHOPRA 2169

Stress pro� les

Temperature pro� les

Fig. 42 Comparison of nonquasisteady model prediction with experi-mental data; strain stepped from 0.01/s to 0.0005/s at 3% strain.162

suitable constitutive models. Lexcellent and Rejzner164 determinedthe change in temperature caused by strain rate through the inte-gration of heat equation.Predicted results agreed satisfactorilywithtest data. Prahlad and Chopra162 used the rate form of the Brinsonequation coupled with an energy equilibrium analysis to obtain si-multaneously both temperature and stress as a function of strainfor a given strain rate. The predictions showed good qualitativeagreement for both stress and temperature evolution under load-ing involving an instantaneous change in the strain rate during theloading cycle (Fig. 42). However, more careful temperature mea-surements are required to validate the model quantitatively.Potapovand Silva165 developed a simple time response model of Ni-Ti al-loys, which takes into account latent heat and thermal hysteresis oftransformation under conditions of free and forced air convection.The actuation frequency of shape memory actuators was seen to becontrolled primarily by the cooling time while increasing the inputpower can reduce the heating time. The calculated time responseshowed good agreement with test data.

G. Torsional Characteristics of SMA Rods and TubesIn addition to the use of SMA wires under uniaxial loading as

actuation elements, SMA rods and tubes have also been used fortwist actuation.The principleof operation is the same as in the caseof SMA wires—the rod (or tube) is � rst pretwistedand then tends torecover its original, untwisted state when heat activated. One of the� rst torsionalmodels was proposedby Davidson et al.166 for the tor-sional actuationof SMA rods uses the Liang and Rogers constitutivemodel in conjunctionwith standard relationshipsfor pure shear de-formations. The model parameters were obtained by curve � ttingthe parametersrequired for Liang and Rogers model to the torsionalexperimental data. Keefe and Carman167 proposed an exponential

a) Constant temperature predictions

b) Actuation against torsional springs

Fig. 43 Comparison of torsional model prediction with experimentaldata.162

model for the relationship between shear stresses and strains in theSMA alloy. However, this relies more on � tting the model parame-ters to torsional data over a prescribed range of thermomechanicaldata and does not include the modelingof the SMA phenomenologyover the entire thermomechanicalrange.Prahladand Chopra168 pro-posed a torsional model based on the Brinson model for the SMAconstitutive behavior. The model parameters were obtained in ex-tensionaltestingand then applied to the torsionalcase.Good overallagreement was obtainedboth in the constant temperature (Fig. 43a)case and while actuating a torsional spring (Fig. 43b). In contrast tothese models that model only shear deformationsof the SMA, thereare several models that are fully three-dimensional and thereforetheoretically capable of handling loading in any arbitrary direction(includingcombined tension-torsionloading).An example of thesemodels is the Boyd and Lagoudas141;142 model. However, the iden-ti� cation of model parameters and implementationfor these modelsare complex.

H. Damping CharacteristicsAbove a critical temperature, the shape memory alloys exhibit

pseudoelasticstress-strainhysteresisover a large range of strain (6–8%). Cyclic variationin stress and strain can result in a largedissipa-tion of strain energy that translates into damping augmentation.169

Wolons et al.170 carried out experimental investigationto determinedamping characteristicsof NiTi alloy wires under uniaxial loadingand systematically examined the effects of cycling, frequency ofoscillation,strain amplitude, temperature,and static-strainoffset ondamping. It was shown that the shape of the hysteresis loop changessigni� cantly with frequency; the energy dissipation decreases withfrequencyand reachesa saturationvalueat about10Hz.Also, theen-ergydissipationdecreaseswith highertemperature(say,above90±F)and increaseswith lower staticstrainoffset.171 Also,LammeringandSchmidt172 examineddamping capacityof NiTi in the pseudoelastic

2170 CHOPRA

range and showed that the area of hysteresis loop decreases withincreasing strain rate. Ju and Shimamoto173 developed compos-ite beam with embedded SMA � bers to augment its damping.Damping was shown to be a function of both temperature andcurrent.

I. Composite Beams with Embedded SMA WiresSMA wires were used to alter the natural frequencies of com-

posite beams. Rogers and Barker174 showed an increase in the nat-ural frequencies of a composite beam because of the activation ofSMA wires embedded directly in the structure. In this experimen-tal study, the beam and SMA wires were independently clamped.When the SMA wires were heated, the beam was subjected toan axial force caused by the shape memory effect. They demon-strated a 200% increase in the natural frequency of graphite-epoxybeams by using a 15% volume fraction of SMA wires. Correlationof experimental results with predictions was not carried out. Bazet al.175 conducted a study on the active vibration control of � ex-ible beams. Experiments were conducted on � exible beams withSMA wires mechanically constrained on the exterior of the struc-ture. The recovery force caused by mechanically constrained, pre-strainedSMA wires at higher temperatureswas used to demonstrateactive vibration control of � exible beams. In such an application,external access to the substructure becomes essential in order toclamp SMA wires to an independent support. For many aerospacestructures like rotor blades or airplane wings, it might not be pos-sible to follow this scheme. In another study Baz et al.176 insertedSMA wires into � exible beams with sleeves to control their buck-ling and vibration behavior. They used a � nite element method tocorrelate with their experimental results. Overall, predictions cor-related satisfactorily with test data. They showed that the bucklingload of a � exible � berglass composite beam could be increasedthree times when compared to the buckling load of an uncontrolledbeam.

Using a single surface-attached SMA wire, an active controlof beam de� ection with temperature activation was investigatedby Brinson et al.177 Also, predictions were successfully comparedwith test data. The potential of shape control with SMA wires waspointed out. Lagoudas et al.178 used a layer-wise shear deforma-tion theory to show the shape control of plate structures with em-bedded SMA strips using thermal activation. Turner179 developeda � nite element formulation for predicting the thermomechanicalresponse of SMA hybrid composite structures with constrained orfree boundary conditions. The model captures the material non-linearity with temperature for composites with embedded SMAactuators.

Building composite structures with embedded SMA wires is achallenging task. First, the surface of wires needs to be treated toachieve a good bond condition.Second, the loss of prestrain duringthe curing process needs to be prevented. There are three possibleapproachesto overcomethisproblem.One approachis to use specialreinforcedcompositeplies (Hercules)that cureat roomtemperature.The second approach is to clamp each one of the SMA wires sep-arately during the formal curing process in an autoclave. The thirdapproachis to use fused silica tubes � lled with “dummy” steel wiresduringcuringprocess.Once the curingprocess is complete, the steelwires are replaced with prestrained SMA wires. To ensure a goodbond between the silica tubes and the graphite-epoxy material, a� lm adhesive is used. Figure 44 shows a fabricatedcomposite beamwith three SMA wires in sleeves.

Fig. 44 Graphite-epoxy beam with SMA wires.

Fig. 45 Fundamental frequency of clamped-clamped graphite-epoxybeam activated by one 20-mil-diam SMA wire. Beam dimensions:length =18 in. (45.72cm), width = 0.25 in. (6.35mm), thickness = 0.068in.(0.173 mm).180

Epps and Chandra180 tested graphite-epoxy solid beams withsleeve-inserted SMA wires for their bending frequencies underclamped boundary conditions.The natural frequenciesof this com-posite beam depend not only on the beam properties but also onSMA characteristics. An important SMA-related characteristic isthe recovery force in wires, which in turn dependson prestrain,me-chanical properties of SMA and temperature. Figure 45 shows the� rst bending frequency of a graphite-epoxy composite beam acti-vated by one 20-mil-diam SMA wire. It shows a 22% increase inthe � rst bending frequency with an SMA volume fraction of 2%.For analysis, the SMA wires were treated as an elastic foundationfor the composite beam, and the stiffness of foundation dependedon the constraint recovery force. Good correlation between theoryand experiment was achieved.

Furuya181 discussed design and material evaluation for the de-velopment of shape memory composites. Two types of conceptswere proposed: TiNi � ber/aluminum matrix and TiNi � bers andparticles/plaster matrix. The � rst composite showed increase oftensile strength (yield stress) and fatigue resistance, and the sec-ond showed increase of fracture resistance and vibration damping.Birman182 presented micromechanics analysis of composite struc-ture with embedded SMA wires. Most analyses are in early stage ofdevelopment and require systematic validationsunder a wide rangeof load conditions.

Shape memory alloys such as nitinol have large force and strokeand therefore have enormous potential for low frequency (steady)applications. The materials are highly nonlinear function of tem-perature, stress, and strain history, and it requires a � ne tuning (say,using an adaptive feedback controller) to achieve the desired state.Also, incorporating a locking mechanism to maintain the desiredstate of the actuator becomes important. Most of the constitutivemodels are developed for quasistatic loading, and these should beexpanded to cover low-frequency dynamics. There has been goodsuccess to validate constitutive models for nitinol wires (isolated)in extensional loading with test data. To cover expanding applica-tions, the material constitutivemodels should be validated for otherload conditions such as torsional loading. In the next phase it mightbe important to validate predictive models with test data for simplecompositebeamsand plateswith directlyembeddedSMA wires.Forsuch structures local stress/strain distributionsusing either detailed� nite element analysis(suchas three-dimensionalsolid elements)orhigher-ordershear deformation theory might reveal the mechanismof actuation as well as help to establish the integrity of structure.

VI. Magnetostrictives and ElectrostrictivesA. Magnetostrictives

Magnetostrictivematerialsconsist of alloys of iron and rare Earthelements such as terbium and dysprosium, which undergo defor-mation when exposed to magnetic � eld. The commercial mag-netostrictive material is Terfenol-D (Terbium-Ferrous-Naval Ord-nance Laboratory-Dysprosium) and is normally available in theform of rods of different diameters. With no magnetic � eld oblongmagnetic domains in the material are randomly oriented (mostlyperpendicular to rod’s longitudinal axis). With the application of

CHOPRA 2171

compressivestress,most of the domainsget orientednormal to rod’saxis. In the presence of a magnetic � eld along the longitudinalaxis,these domains rotate and becomemostly parallel to this longitudinalaxis causingan induced strain. As the intensityof the magnetic � eldincreases,moredomainsrotate,and longitudinalstrainincreasesand� nally saturates(at about0.2%) at high � eld levels.Because magne-tostrictionis a molecularaction, themechanicalresponseis very fast(kHz). The magnetic � eld is producedeither by a permanentmagnetor by a magnetic coil surrounding the rod. Normally, a permanentmagnet is used to create a steady bias � eld, and an ac current in thesurrounding coil is used to control the time-varying magnetic � eldand, in turn, the nonsteady induced strain in the Terfenol-D rod. Anextensional strain is induced in the direction of magnetic � eld. Ifthe � eld is reversed, the domains reverse direction,but again inducean extensional strain. On the macroscopic level a magnetostrictivematerial conserves volume, and as a result the diameter shrinks asa result of Poisson’s effect. For Terfenol-D, Curie temperature isabove 380±C. At high operating regimes hysteresis and nonlineari-ties are intrinsic to magnetostrictivebehavior.

James Joule discovered the magnetostrictiveeffect � rst in nickelin 1840. Later, cobalt, iron, and their alloys were shown to have sig-ni� cant magnetostrictive effects like nickel. The maximum strainswere of the order of 50 ppm (parts per million, 0.005%). In early1970s Arthur Clark and his research group at Naval Ordnance Lab(later known as NSWC) discovered Terfenol-D, which produced asigni� cantly larger magnetostrictionresulting in a maximum strainof the order 2000 ppm. This is almost twice the maximum strainproduced by piezoceramics. The strains and forces produced byTerfenol-D are more than those generated by piezoelectric andelectrostrictivematerials.Magnetostrictives� nd applicationsin ma-chine tools, servo-valves,hybrid motors, sonar and tomography,au-tomotive brake systems, micropetitioners,and particulate-actuatorsand sensors. As a result of the Joule effect, an application of themagnetic � eld results in longitudinal extensional strain accompa-nied by transversecompressivestrain.The reciprocaleffect is calledVillari effect, where an application of stress (that is, strain) resultsin a change in its magnetic � eld. The Joule effect is used in actua-tors, whereas the Villari effect is used in sensors. The Joule effecttransfersmagnetic energy to mechanical energy,whereas the Villarieffect transforms mechanical energy to magnetic energy. Using ahelical magnetic � eld around the magnetostrictivematerial, a twist-ing action is produced, which is called the Wiedemann effect. Theinverse effect in which application of torque results in a change ofmagnetization is called the Matteusi effect. The manufacturing ofTerfenol-D is carried out by melting the material and then castingand directionally solidifying to provide the crystalline microstruc-ture needed to produce large strains.Magnetostrictivematerials areavailable in the form of rods, thin � lms, and powder. Normally, toproduce magnetic � eld a wound wire solenoid that converts electricenergy to magnetic energy is used. As per Maxwell’s equation, theintensity of magnetic � eld is proportion to the current in solenoid.

The strain increases quadratically with magnetic � eld intensityH (Fig. 46). There is no hysteresis because only a static condition

Fig. 46 Strain for a Terfenol-D rod.

Fig. 47 dc bias circuit for magnetostrictive transducers.

a) B-H b) S or "-B c) S or "-H

Fig. 48 Hysteretic behaviors for magnetostriction.

is considered:

Magnetic intensity H D nI .A/m or Oe/

where I is the current though the surrounding coil of length L andN turns and n is the number of turns per unit length,

n D N=L

The magnetic � eld H can be expressed either in electromagneticunits (emu) or in MKS units. In magnetic units it is expressed as Oe(oersted), whereas in MKS units it is de� ned as A/m (amperes permeter):

1.A/m/ D 45

1000Oe D 79:58Oe

The dynamic strain amplitude is limited by the maximum avail-able strain. To create oscillating strain, a bias dc � eld is applied.Figure 47 shows a possible circuit to create an oscillatory strain.The capacitor C is used to block the dc supply from entering acpower source, and inductance L is selected to be large enough toprevent the ac current from entering the dc power supply.Typically,the impedance of additional inductance L is larger than that of theactuator, which in turn is larger than that of the capacitor. For dy-namicconditionthere is hysteresisbetweenmagnetic� eld andstrainproduced. Figure 48a illustrates magnetic � ux density B (internal)vs magnetic � eld intensity H (external), and there is hysteresis.Figure 48b shows strain " vs internal � ux density B , and there isno hysteresis. Figure 48c is obtained combining these two graphs,and a butter� y curve is obtained between strain and magnetic in-tensity H . For ETREMA Terfenol-D, this hysteresis effect is smalland is often ignored for large strains. Similar to electrostrictives,Terfenol-D has a very low hysteresis.

Magnetostrictives are capable of providing both actuation andsensing capabilities. This reciprocal magnetomechanical transduc-tion is often referred to as bidirectional energy exchange betweenthe magnetic and elastic regimes.

Constitutive relations are as follows.Strain:

" D SH ¾ C d H

Flux:

B D d¾ C ¹¾ H

where d is the magnetostrictive constant (magneto-mechanicalcross-coupling coef� cient) that corresponds to the slope of linearpart of " ¡ H curve and ¹¾ is the free permeability (at constantstress) that corresponds to the slope of the B ¡ H curve in the � rst

2172 CHOPRA

quadrant. SH is the compliance of the rod at nominal operating� eld (bias � eld). The � rst equation shows that the strain of a mag-netostrictive element is function of mechanical stress and appliedmagnetic � eld. The second equation shows that the magnetic in-duction B varies with stress and applied � eld. This model appearsa generalization of two phenomenological relations: linear Hook’slaw (" D S¾/ and magnetic constitutive law (B D ¹H ). Thus thetotal strain consists of two parts: mechanical strain and magne-tostrictive strain. In a similar way, the magnetic induction consistsof two parts: constant-stressmagnetic componentand magnetoelas-tic interactioncomponent. The magnetic and structural regimes arecoupled throughmagnetizationas a result of both externallyappliedmagnetic � eld and stress-induced � eld. This linear model providesadequate characterization of magnetostrictive material at the lowoperating regime. Note that at high operating regimes hysteresisand nonlinearitiesbecome important.

For an unloaded rod, strain and � ux are as follows.Strain:

" D d H

Flux:

B D ¹T H

For longitudinal induced strain

d33 D³

d"33

dH3

´

¾

D³

dB3

d¾3

´

H

; S33 D³

d"3

d¾3

´

H

D 1

Y H3

¹¾33 D

³dB3

dH3

´

¾

The elastic modulus Y H3 and magnetomechanical coef� cients d33

vary from material to material and often with operating conditions.For ETREMA Terfenol-D rods

¹¾33 D 9:2 .emu/

D 11:56 £ 10¡6.MKS units/

d33 D 1:6 £ 10¡6 Oe¡6

D 20 £ 10¡9 m/A

The nominal open circuit Young’s modulus is

Y H3 D 2:65 £ 1010 N/m2

SH33 D 1

¯Y H

3 D 0:377 £ 10¡10 m2/N

The properties of a magnetostrictive element depend on level ofstress, magnetization, and temperature distribution. Butler183 andButler et al.184 provided a comprehensive introduction to the mag-netostrictivematerialsand especially to the ETREMA’s Terfenol-D.EngdahlandSvensson185 presenteda simple,uncoupled� nitediffer-ence analysis to predict steady responseof magnetostrictiverod as aresultof appliedsinusoidalmagnetic� eld usinglinearmaterial char-acteristics. Kvarnsjo and Engdahl186 developed a two-dimensional� nite difference transient analysis caused by magnetic � eld usingnonlinearmaterial characteristics.The � nite differencemethods areless versatile to deal with structures constituting dissimilar materi-als such as the case with smart structures. Claeyssen et al.187 de-veloped a three-dimensional,coupled, linear � nite element analysisto establish the effective dynamic coupling constants of a magne-tostrictive actuator. They used an empirical representationof mate-rial characteristics. Carman and Mitrovic188 formulated a coupledone-dimensional nonlinear � nite element analysis incorporating aphenomenologicalconstitutivemodel for magnetostrictiveactuator.The model showed good agreement with test data at high preloads.However this model is unable to represent saturation effects. Fol-lowing the work of Hom and Shanker,189 Duenas et al.190 developeda more comprehensive constitutive model of magnetostrictive ma-terial that includes magnetization saturation and thermal effects.

Dapino et al.191 developeda coupled nonlinear and hysteretic mag-netomechanicalmodel for magnetostrictives.The magnetostrictiveeffect is modeledby taking into account the Jiles–Athertonmodel offerromagnetichysteresis in combinationwith quartic magnetostric-tion law. This model provides a representation of the bidirectionalcouplingbetween themagneticandelasticstates.The modelappearsto represent accurately the magnetic hysteresis in the material. An-janappa and Bi192 and Anjanappa and Wu193 presented a simpleone-dimensional model to simulate the quasistatic response of amagnetostrictiveminiactuator (they developed)as a result of the ap-pliedmagnetic� eld.Also,Wu andAnjanappa194 andKrishnamurthyet al.195 developed a simple rule-of-mixture model to calculate theresponse of magnetostrictive particulate composite. Flatau et al.196

discussed magnetostrictive particle composites in terms of un-derlying physical processes that occur during fabrication, mate-rial characterization, design considerations, and structural healthsensing.

Kannan197 provided a continuum level quasistatic, three-dimensional� nite elementanalysisusingnonlinearbehaviorof bulkmagnetostrictive materials and particulate magnetostrictive com-posites. Two alternate possibilities for a nonlinear incremental con-stitutive model are explored: characterization in terms of magnetic� eld (normally used) or in terms of magnetization. The analysiswas validated with available experimental data on structures incor-porating Terfenol-D. To model particulate magnetostrictive com-posites, interactions between particles are captured by combin-ing a numerical micromechanical analysis with the Mori–Tanakahomogenization approach. Pradhan et al.198 developed � rst-ordershear deformation theory (FSDT) to study vibration control of lam-inated composite plate with embedded magnetostrictivelayers. Theeffects of material properties and placement of magnetostrictivelayers on vibration suppression were examined. It was found thatthe maximum suppression is obtained when the magnetostrictivelayers were relatively thin and placed far away from the neutralaxis.

Calkins et al.199 and Dapino et al.200 provided an overview ofmagnetostrictive sensor technology. Magnetostrictive sensors takeadvantage of the coupling between the elastic and magnetic statesof a material to measure motion, stress, and magnetic � eld. Sen-sors are classi� ed into three categories: passive, active, and hybrid.Passive sensors are based on Villari effect and measure change inmagnet � ux in a coil surrounding the sensor caused by an exter-nally imposed stress. Active sensors use an internal excitation ofthe material (such as with coil) to facilitate the measurement ofpermeability (often with another coil) caused by an external forc-ing. Hybrid or combined sensors rely on the use of magnetostrictiveelement to actively excite another material (say, � ber optic) thatallows measurement of change in its properties as a result of ex-ternal change. Many different sensors based on their applicationsare discussed and contrasted with conventional sensors in terms ofsensitivity and implementation issues. Flatau et al.201 developed ahigh-bandwidth-tuned vibration absorber using Terfenol-D actua-tor and showed a signi� cant change of modulus from demagnetizedstate to magnetic saturation.202 Simple experimentswere conductedto demonstrate proof of concept. Kellogg and Flatau203 developedan analysis of the noncontactnature of sensing using magnetostric-tives. Kellogg and Flatau carried out systematic measurement ofelastic modulus of Terfenol-D under controlled thermal, magnetic,and mechanical loading conditions and showed dramatic changeof modulus with the dc applied magnetic � eld. Because the mag-netostrictive materials, especially Terfenol-Ds, are brittle in ten-sion (tensile strength» 28 MPa, compressive strength» 700 Mpa),they are normally placed under a mechanical compressiveprestress.Also, a prestress improves the magnetic state of the material andhence the magnetomechanical coupling. However, a large com-pressive prestress can overpower the elastic de� ections caused bymagnetostriction. Under dynamic conditions the performance ofmagnetostrictivematerial is affected by eddy currents that producemagnetic � ux opposite to externally applied magnetic � eld (skineffect). Pratt et al.204 exploited the nonlinear transduction of non-biased Terfenol-D actuators to design an autoparametric vibrationabsorber.

CHOPRA 2173

Overall, there is a general lack of detailed database for magne-tostrictive actuators for a wide range of test conditions.

More in-depthinvestigationsare needed to understandthe behav-ior of magnetostrictive materials under a wide range of controlledoperating conditions. For modeling, the least well-de� ned compo-nent is magnetic state of magnetostrictivecore, which is functionofoperating conditions. It is important to develop reliable modelingof magnetization using either micromagnetic representationof ma-terial or Preisach model or ferromagnetic hysteresis model. Thereis a need to develop a comprehensive three-dimensional constitu-tive model of magnetostrictivematerials that includenonlinear ther-mal effects,magnetizationsaturation,eddy current losses,prestress,hysteretic behaviors, and dynamic effects and then systematicallyvalidate with test data.

B. ElectrostrictivesMaterials such as relaxor ferroelectrics undergo strain when an

electric � eld is applied.Under this category of materials, lead mag-nesium niobate (PMN) alloys have a suf� ciently large dielectricpermittivity that helps to generate signi� cant polarizationand hencestrains. In the absence of electric � eld, small electric domains arerandomly oriented in these materials. With the application of elec-tric � eld, these domains rotate, resulting in strain. The variationof strain with electric � eld is quadratic (independent of polarityof � eld) (Fig. 49). At a suf� ciently high � eld, the induced straingets saturated.Unlike piezoelectrics,unchargedelectrostrictivesareisotropic and are not poled. With an applicationof � eld, the materi-als get instantlypolarizedand become anisotropic.For example, thetransversematerial stiffnessof PMN-PT decreasesby about 20% asthe electric � eld becomes 1300 V/mm. On the removal of � eld, thematerials become depolarized.An electric � eld produces an exten-sional strain in the directionof � eld and contractionin the transversedirection. If the � eld is reversed, the domains reverse direction, butit again induces an extensional strain in the directionof � eld (thick-ness direction). To produce an oscillatory (bidirectional) strain, itbecomes necessary to apply a bias dc � eld. Hence, electrostric-tives are used as actuator in a wide range of applications.205;206 Themaximum strain is of the order of 0.1%. Because no permanent po-larization is needed for electrostrictives, these are not subjected toelectric ageing. They are characterizedby very low hysteresis (lessthan 1%) but are very sensitive to surrounding temperature.207

In the absenceof an electric � eld, the material is not polarized.Asa result, an application of stress does not change the electric � eld.Hence, electrostrictives are not normally used as sensors. Becausethese materials are very sensitive to temperature (thermal environ-ment), most applications of electrostrictives are focused to under-water operationsor in vivo, ranging from ultrasonicmotors to med-ical probes. Because of the nonhysteretic nature of this material,it is used in micropositioners.Superior characteristicsare obtainedwhen PMN is doped with lead titanate (PT) in low ratio such as0.9PMN-0.1PT (Ref. 208).

The constitutive relations for electrostrictivematerial at constanttemperature is209 as follows.

Converse:

"i j D Si jkl ¾kl C m pqi j E p Eq

Fig. 49 Strain-electric � eld distribution for an electrostrictive element.

Direct:

D j D eTj n En C 2m j nkl En¾kl

where D is electric displacement, S is material compliance, E iselectric � eld, m is electrostrictive-coupling parameter similar topiezoelectriccoef� cient d, and eT is dielectricpermittivity.A factorof two is needed in the electromechanical coupling in the chargeequation caused by stress, and this factor is not needed in the strainequationcausedby the � eld. These equationscontainmoderatenon-linear terms and are valid for low � elds (less than 300 V/mm).

Expanding the converse relation,8>>>>>><

>>>>>>:

"1

"2

"3

°23

°31

°12

9>>>>>>=

>>>>>>;

D

2

6666664

m33 m31 m31

m31 m33 m31

m31 m31 m33

0 0 0

0 0 0

0 0 0

3

7777775

8>><

>>:

E21

E22

E23

9>>=

>>;

The electrostrictive coef� cients depend upon the � eld. Let us sayonly voltage is applied to top and bottomof electrostrictivesurfaces:

"1 D "2 D "3 D m33.V=tc/2

The strain per � eld is 2m33.V=tc/.Electrostrictives exhibit variable dielectric characteristics with

strain (decrease with strain) resulting in a nonquadraticvariation ofstrain with � eld. On the other hand, the electrostrictive materialsshow very low hysteresis and creep characteristics. Unlike piezo-electric materials, the performance of electristrictivesdoes not de-grade with time. Because their capacitance is high (an order ofmagnitude more than piezoceramics), internal heating is low.

Fripp and Hagood210 presented a comprehensive set of consti-tutive equations for electrostrictive materials and developed a dy-namic analysis for an electromechanical system with distributedelectrostrictivecouplings. A Raleigh–Ritz analysis was formulatedfor a cantilevered beam actuated with surface-bondedelectrostric-tive wafers and satisfactorily validated it for static and dynamicresponse with experimental test data. Hom and Shankar211;212 pre-sented a fully coupled, two-dimensional, quasistatic � nite elementanalysis for electroceramics and applied it to electrostrictive stackactuators. This formulation incorporates the effect of body forcesof dielectric origin, but ignores the body moments of dielectric ori-gin. Recently, Pablo and Petitjean208 carried out stress free electricbehavior (in transversedirection) of electrostrictivepatches experi-mentally at a macrolevelfor a range of excitation� elds and frequen-cies and temperatures. Piquette and Forsythe213 covered nonlinearmodeling of PMN materials. Most of the existing studies are quiterestrictive is scope and applications.

Moredetailedinvestigationsare requiredto understandtheperfor-mance of these actuatorsundera wide rangeof operatingconditions.Simpli� ed constitutive models of materials need to be developedcovering a range of � elds, strains, and temperatures and validatedwith experimentaldata obtainedunder controlledtest environments.

VII. ApplicationsApplications of smart-structures technology to various physi-

cal systems are focused to actively control vibration, performance,noise, and stability.Applicationsrange from space systems to � xed-wing and rotary-wing aircraft, automotive, civil structures, marinesystems, machine tools, and medical devices. Early applications ofsmart-structures technology were focused to space systems to ac-tively control vibration of large space structures31 as well as forprecision pointing in space (telescope, mirrors, etc.214) The scopeand potential of smart-structures applications for aeronautical sys-tems are expanding. Embedded or surface-bonded smart materialactuators on an airplane wing or helicopter blade can induce alter-ation of twist/camber of airfoil (shape change), which, in turn, cancause variation of lift distribution and might help to control staticand dynamic aeroelastic problems. For � xed-wing aircraft, appli-cations cover active control of � utter,215¡219 static divergence,220;221

2174 CHOPRA

Table 7 Comparative test evaluation of commercially available piezostack actuators232

Operating Maximum strain, Block force Normalized block Strain-force Energy density,Part/material no. voltage, V ¹ strain (BF), lb force, ksi index ft-lb-slug

MM 8M (70018) 360 254 128 1.05 0.133 1.27MM 5H (70023-1) 200 449 101 0.83 0.180 1.80MM 4S (70023-2) 360 497 143 1.17 0.291 2.78PI P-804.10 100 1035 1133 7.31 3.783 36.72PI PAH-018.102 1000 1358 1505 9.71 6.593 62.95XI RE0410L 100 468 95 5.16 1.207 11.52XI PZ0410L 100 910 70 3.58 1.629 15.55EDO 100P-1 (98) 800 838 154 2.00 0.838 8.00EDO 100P-1 (69) 800 472 50 0.66 0.156 1.49SU 15C (H5D) 150 940 266 7.48 3.516 33.57SU 15C (5D) 150 1110 274 7.70 4.274 40.80

panel� utter,222 performanceenhancement,223 and interiorstructure-borne noise.224 Compared to � xed-wing aircraft, helicopters ap-pear to show the most potential for a major payoff with the ap-plication of smart-structures technology. Given the broad scope ofsmart-structures applications, developments in the � eld of rotor-craft are highlighted in this section. Although most of current ap-plications are focused on the minimization of helicopter vibration,there are other potential applications such as interior/exterior noisereduction, aerodynamic performance enhancement including stallalleviation, aeromechanical stability augmentation, rotor tracking,handling qualities improvement, rotor head health monitoring, androtor primary controls implementation (swashplateless rotors) (seereview paper30). For aerospace systems two types of actuation con-cepts have been incorporated. One approach uses active materialsdirectly, surface-bonded or embedded, to actively twist or cambercontrol of lifting surface. Another approach actively controls auxil-iary lifting devices such as leading-/trailing-edge � aps using smartmaterial actuators.Currently, a major barrier is the limited stroke ofsmart actuators, requiring a large ampli� cation.

A. Stroke Ampli� cationFor most applicationsthere is a need for compact,moderateforce,

moderate bandwidth, and large displacement actuators. Most actu-ators, in particular piezoceramic actuators, are low force and lowstroke devices. Typically, piezoceramic sheet actuators generatefree displacements from 1 to 5 ¹m and block forces from 2 to20 lb (1 lb D 453.6 g) and frequencies up to 20 kHz. Individualpiezoelectric sheet actuators can be combined in series to obtainhigher actuation displacement. The actuation force is however un-affected, and also, there is a limit on increasing the length of thinsheet actuators (buckling constraint).Another approach to increasethe actuationdisplacement is by building piezoelectricbimorphs. Abimorph or bending actuator consists of two or more even layers ofpiezoelectric sheets bonded on either side of a thin metallic shim(main load carrying member). By applying an opposite potential totop and bottom sheets, a pure bending actuation is generated. In acantileveredarrangement the tip displacement can be used for actu-ation of a system. With piezobimorphone can obtain displacementsfrom 5 to 10 ¹m and forces up to 0.5 lb. Using more layers canincrease the actuation force, but the displacement is reduced. Toincrease actuationforce, multilayeredactuators such as piezostackscan be used. However, the stroke of piezostacks is quite small. Akey challenge is to amplify the stroke of these actuators. Large me-chanical ampli� cation using a compact leverage system often leadsto substantial losses at hinges and slippage at knife edges. Replac-ing mechanical hinges with � exure can overcome some of theseproblems, but requires large effort and experience to perfect suchsystems. Also, the actuation ef� ciency is reduced. To amplify thestroke of piezodevices, specially shaped actuators are being built.Typicalexamplesincludereducedand internallybiasedoxidewafers(RAINBOW) actuators, thin-layer composite unimorph ferroelec-tric driver and sensor (THUNDER) actuators, Moonie actuators,and C-block actuators. RAINBOW are dome-shaped actuators thatare built by bonding piezoceramic layer and a chemical reducedlayer (acts like a shim).225 The piezoelectric layer is on the convexside, and an electric � eld changes the curvature of the actuator.Theprojected free displacements are of the order of 1000 ¹m, forces

up to 100 lb, and actuation frequency up to 10 KHz. By stackingRAINBOWs in a clamshell con� guration, it is possible to obtainhigher stroke.226;227 THUNDER actuator is a curved shaped devicecomposed of a metallic layer bonded to a prestressed piezoelectriclayer.228 Displacement is achieved via the induced d31 contraction.A cantilevered1 £ 0.5 in. actuator can generate the displacementof10 mils and a block force of 8 lb. The Moonie actuator consists ofmetal-ceramiccomposite, composedof a piezoelectricceramicdisksandwichedbetween two metal end caps. The end caps act as strokeampli� ers of lateral displacementof the PZT sheet.229 C-block actu-ators are multilayeredarched bimorph piezoelectricactuators,and alarge axial displacementcan be achievedusing a series arrangementof C-actuators.230;231 So far, most of these specially shaped actua-tors have not been exploited into challenging applications. Manyof these actuators are in their early state of development and lackrigorous modeling and database.

For many practical applications it is necessary to develop largestrokes. To increase actuation force, multilayeredpiezostackactua-tors are used. These consist of a large number of thin piezoelectricsheets stacked in a series arrangement, separated by electrodes thatmake use of induced strain in thickness direction (d33 actuation).Thesedevices inducesmall freedisplacementsbutmuch larger actu-ation force than sheetactuators.Nominal performanceof piezostackactuatorsrange in free displacementfrom 15 to 25 ¹m, block forcesup to 1000 lb, and frequencies up to 20 kHz. Combined with suit-able ampli� cation mechanism,piezostackshavebeenused in a widerangeof applications.Therehavebeenseveralstudiesto characterizethe electromechanicalbehavior of peieostacks.232¡235 For example,Lee232 evaluated the characteristics of 11 different stack actuatorsincluding maximum free strain, maximum block force, operatingvoltage, and energy density (Table 7). These actuators were testedsystematically using specially built test apparatus under different� eld levels, operating frequencies, and preloads.

To increase the stroke of piezostackactuators, some form of am-pli� cationmechanismis needed.Current ampli� cationmechanismscan be divided into two types: � uidic and mechanical.Typically,� u-idic approaches use two cylinders of different diameters to obtaindesiredampli� cationof stroke.Although� uid ampli� ers might pro-vide higher ampli� cation than mechanicaldevices, they suffer from� uid losses, weight penalty, and complexity of system.236;237 Sofar, there has been limited success with these devices. Mechanicalampli� cation can be categorized into two types: rigid lever/frameampli� ers238¡241 and elastic � extensional ampli� ers.242;243 Theseampli� cationdevices tradeforcewith displacement,buthavea detri-mental effect on power transfer ef� ciency and energy density.244;245

Several single-stage mechanical ampli� cation devices that includelever-fulcrummechanism(Fig. 50) and triangularframe mechanismhave been built to actuate a trailing-edge � ap of a rotor blade. Incomparison to � uidic devices, these provide simple, lightweight,and ef� cient ampli� cation systems. However, they suffer from thedegradation of performance caused by losses (stiffness and power)especially with high ampli� cation factor. From the stiffness pointof view, the triangle frame system is normally more ef� cient thanthe lever-fulcrum system because its structural members experi-ence mostly extensional loads in contrast to bending loads forlever-fulcrum ampli� ers (Table 8). However, the stiffness of thelever-fulcrum can be maintained at much higher value but with

CHOPRA 2175

Table 8 Comparison of mechanical ampli� cation concepts246

Concept Lever-fulcrum Triangle frame

Ampli� cation factor <10 <10Existing actuators Physik instruments SatCon � ap actuator

Boeing biaxial actuator MIT X-Frame actuatorUM leverage ampli� cationUM L-arm ampli� cationUM L-L ampli� cation

Scaling up Constant lever ratio Constant lever ratioSpecial features Design � exibility Small chordwise sizeDisadvantage Elastic deformation Lateral vibration of

caused by bending piezostacksDesign Multiple-stage Different frame material

enhancements ampli� cation

Fig. 50 Leverage ampli� cation.

Fig. 51 X-Frame actuator schematic.244

a weight penalty. For design re� nements including scale-ups, thelever-fulcrum appears superior to the triangle frame system (freefrom geometrical constraints). To achieve higher ampli� cation fac-tors (say, over 10), multistage ampli� cation is incorporated suchas in L-L ampli� cation mechanism.246 It is a combination of lever-fulcrumsand elastic linkages.The stroke of piezostacksis ampli� edby an inner lever with a low ampli� cation factor (<6) and then am-pli� ed again by the outer lever.Two lever-fulcrumsare connectedinseries using an elastic linkage that also provides returning force aswell as piezostackpreload.To achievebidirectionalfunctionality,itbecomes necessary to build dual-stage ampli� ers, respectively im-plemented in full-scale X-frame actuator (Fig. 51) and L-L actuator(Fig. 52). There are other � extensional ampli� cation devices, suchas oval-shaped actuators, Moonies, and oysters,232 that might bemore bene� cial in speci� c applications.A typical stroke ampli� ca-tion factor of about � ve is achieved by sacri� cing the load-carryingcapacity.

To overcome losses with pin-jointed ampli� cation mechanism,� exure hinges or fully complaint mechanisms are used. Often, ad-hoc designproceduresare used to developthese devices.As a result,the � nal design might not be optimal.Frecker and Can� ed247 formu-lated a systematic topology optimization approach to the design ofcomplaint mechanical ampli� ers for piezoceramic stack actuators.In this approach, any direction of force and motion transmissionfrom the active material can be chosen. This methodology appearsto show potential to build devices with precise motions.248;249

For actuation of trailing-edge � aps of wings/rotor blades andother application, it is necessary to convert the linear displacementinto angular displacement. A mechanical conversion can signi� -

Actuator layout

Fabricated actuator

Fig. 52 L-L ampli� cation mechanism.232

Fig. 53 Active piezo� ber composites.

cantly reduce the effectivenessof the device. Bothwell et al.250 usedextension-torsioncoupling of a thin-walled composite tube to con-vert linearmotion of magnetostrictiveactuator into torsionalmotionto actuate a trailing-edge� ap. This device appears less meritoriousbecause of low actuation ef� ciency. Bernhard and Chopra251 usedbending-torsion coupling of composite beam in conjunction withsurface-bonded piezoelectric elements and voltage phase to con-vert bending into twisting to actuate blade tip. This concept appearspromising for scaled rotor models. Giurgiutiu and Rogers234 usedthe twist-warping concept of the thin-walled open tube to convertlinear motion of PZT stacks into rotary motion. This device suffersfrom low actuation ef� ciency and low load-carryingcapability.

One can achieve a large induced strain in the d15 shear mode.For example, for PZT-5H, d31, d33 , and d15 are ¡274, C593, andC741 pm/V. However, it is extremely dif� cult to build a practicalactuator in this mode. Recently, a segmented torsional tube actuatoroperating in the d15 mode was developed and tested.252;253 Thereare two major drawbacks with this actuator. It requires high voltage(several kilovolts), and it requires special conductors to apply � eld.One revolutionarydevelopmentin piezoceramicshas beentheemer-gence of piezo� bers in the form of active plies by Hagood and hisgroup (Fig. 53).254¡256 The piezo� bers are actuated in the d33 modeusing interdigitated electrodes. The piezo active � bers have beenused successfully in the development of active twist rotor.257¡261

There are other potential high-stroke actuators that need furtherinvestigation to examine their projected ampli� cations.232;262

B. Actuator Performancelooseness-1Irrespective of ampli� cation mechanism, the actuator

output energy is less then the available active material strain energybecause of several different losses. Consider a piezostack of lengthlc with external spring load Ke (Fig. 54). The support structure isassumed as rigid. The stack is characterizedwith two important pa-rameters: maximum free displacement ufree (or 3lc) and maximumblock force Fbl.

2176 CHOPRA

Fig. 54 Rigid support.

Fig. 55 Flexible support.

The elastic de� ection is

ue D u free ¡ .keue=Ec Ac/lc

where Ec Ac is extensional stiffness of stack actuator. Rewriting

ue Dufree

1 C kelc=Ec Ac

De� ning stiffness ratio r as

r Dkelc

Ec AcD external stiffness

internal actuator stiffness

Output energy U0 is

U0 D 12Keu

2e

D [r=.1 C r /2]£

12 .Ec Ac=lc/u

2free

¤

D [r=.1 C r /2]Umax

where Umax represents the maximum strain energy possible from anactuator. Actuator ef� ciency ´ is de� ned as

´ D U0=Umax D r=.1 C r/2

where r D 0 represents a free condition and actuator ef� ciency iszero. The parameter r approaching in� nity represents a blockedcondition, and again the actuator ef� ciency is zero. Maximum ef� -ciencyis achievedwith ´ D 1,which representsthematchedstiffnesscondition (external stiffness equals actuator stiffness). For this casea maximum output energy is achieved, and actuator ef� ciency ´becomes equal to 1

4 (Ref. 236).Let us next consider an elastic support structurewith stiffness Ks

(Fig. 55).The elastic de� ection is

ue D u free ¡ .keue=Ec Ac/lc ¡ Keue=Ks

Rewriting results in

ue D u free=.1 C kelc=Ec Ac C Ke=Ks /

Let us de� ne support stiffness ratio rs :

rs Dkslc

Ec AcD support stiffness

actuator stiffness

Thus, elastic de� ection becomes

ue D u free=[1 C r .1 C 1=rs/]

Output energy is

U0 D 12Keu

2e

D©r¯

[1 C r.1 C 1=rs/]2ªUmax

Actuator ef� ciency is

´ D r¯

[1 C r .1 C 1=rs/]2

Optimum actuation ef� ciency is achieved by setting d´=dr D 0,which results in

´opt D 14 [r=.1 C rs/]

Now the actuation ef� ciency depends on both output stiffness ratior and support stiffness rs . The rs D 10 represents a case where thesupport stiffness is 10 times the actuator stiffness, and it is quiteclose to the rigid support case. For a � exible support case (say,rs D 1) there is not only reduction of actuator ef� ciency, but there isalso reduction of r at which maximum ef� ciency takes place. Forthis case there is a reduction of maximum output energy by 50%.It is clear that now half of the available energy is dissipated by thesupportsystem.To improvetheactuationef� ciency,it is importanttoincreasethe stiffnessof supportingstructure,which in turn increasesthe weight of the system. Another important and practical index ofef� ciency should be to consider the mass of supporting and activestructures. Let us de� ne active material energy density ratio as

´mass D .U0=Umax/.Mact=Mtot/

where Mact is the mass of actuator and Mtot is the total mass ofstructure including frame, supporting structure,and active systems.This ef� ciency helps to evaluate different actuation mechanisms,especially under static conditions.

C. Leverage Ampli� cationA leveragesystem is used to amplify the actuator stroke (Fig. 56).

Let us imagine that the output displacement at point B is ue andexternal load is represented by a linear spring of stiffness constantKe .

The effective displacement at point A is

ui D ufree ¡ Flc=Ec Ac

where Ec Ac is the axial stiffness of actuator and F is the actuationforce at point A. The external de� ection at point B is

ue D u i .l2=l1/

D .l2=l1/[ufree ¡ .Kelc=Ec Ac/.l2=l1/ue]

Let us de� ne the leverage ampli� cation by G

G D l2=l1

and stiffness ratio by r

r D Kelc=Ec Ac

The output energy U0 is

U0 D 12Keu

2e

D 12 [Ke=.Ec Ac=lc/][G

2=.1 C G2r/2].Ec Ac=lc/u2free

The actuation energy ef� ciency is

´ D12Keu2

e12 .Ec Ac=lc/u2

free

DrG2

.1 C rG2/2

Fig. 56 Leverage ampli� cation.

CHOPRA 2177

The actuation energy ef� ciency depends upon stiffness ratio r andleverage ampli� cation ratio G. An optimal value is obtained bysetting d´=dr D 0, which results in

ropt D 1=G2; ´opt D 14

The maximum value of energy ef� ciency can be 14 . Because G is

greater than 1, the maximum energy transferoccurs when the outputstiffness is lower than actuator stiffness. For G D 1 (no stroke am-pli� cation) ropt becomes equal to 1.0 (matched stiffness condition).The optimal value of r decreases rapidly as G increases.

To cover the effect of mass, the active material energy densityratio is de� ned as

´mass DU0

Umax

Mact

Mtot; ´mass D

rG2

.1 C rG2/2

Mact

Mtot

D. Comparison of ActuatorsThere is a wide variationof characteristicsamong different smart

material actuators. Hence, it becomes important to make a compar-ative evaluation of their characteristics.Many applications requiremoderate force and large displacement at low frequencies (say lessthan100Hz). Strokeampli� cationdevicestrade forcewith displace-ment; however, energy per cycle remains constant. If the actuatoris operating at the same frequency, the total work per cycle mightnot be suf� cient for a speci� c application.One way to increase thepower output of the actuation device is to operate it at a higher fre-quency than the operation frequency and harness more energy percycle. However, this requires frequency recti� cation to achieve thedesired actuator output frequency independent of piezodrive fre-quency. Thus, the amount of work from an actuator is maximizedthrough impedance matching with load, whereas the power is max-imized by increasing its frequency. Devices like inchworm motors,ultrasonic motors, micropulse actuators, and piezohydraulic pumpare all based on this principle. The piezohydraulicpump appears toshow the most potential among these devices and is discussed next.

There have been attempts to develop hybrid hydraulic actuationsystems to amplify the stroke of piezostack actuators.263¡265 It is astepwise actuation concept to provide moderately large force andstoke.Typically,it consistsof two parts:a pumpdrivenbypiezostackactuators that pumps � uid from a low-pressure accumulator to ahigh-pressureaccumulator and an output hydraulic actuator drivenby Whetstone bridge network of valves that use the pressure dif-ferential created in the accumulators. The high-frequency energyof the piezoelectric stack is exploited by using valves. Sirohi andChopra266;267 built a prototype pump driven by two piezostack ac-tuators (Fig. 57). The stack displaces a piston of diameter 1 in.

Schematic arrangement

Valve arrangement

Fig. 57 Actuation schematic of piezoceramic pump.

(25.4 mm), which is bonded to a thin steel diaphragm that providesa leakproof seal and a restoring spring force. Two passive ball-typecheckvalveswere installedon thepumpingheadto direct� ow to andfrom high-capacityaccumulators.The piezostackswere actuatedatfrequenciesof 10–250 Hz by a sinusoidalvoltageof amplitude50 V.The results showed that the differentialpressurecreatedbetweenac-cumulatorsincreaseswith actuationfrequencyand reachesthemaxi-mum at 150Hz. Beyondthis frequencythepressuredifferentialfalls.A maximum temperature of 55±C was measured on the piezostack.There are two major challenges with the successful developmentof this device, operation of control valves at high frequencies, andtemperature control of piezostack at high � elds and frequencies.

E. Piezoactuation Power ReductionThe piezoelectricand electrostrictiveactuators exhibit highly ca-

pacitive electrical characteristics. If a sinusoidal voltage is appliedto an ideal capacitor, the charge during the positive half-cycle ofexcitation equals the discharge during the negative half-cycle, andhence the net energy dissipation in one cycle is zero. For a nonidealcapacitor (capacitor and resistance in series) there is a net energydissipationby the resistanceas a result of ohmic heating.As a result,a typical ampli� er requires a large heat sink. This power is calledactive power:

Pactive D V £ I £ cos Á W

where V is voltage, I is the currentdrawn in amperes, Á is the phaselag of current with respect to voltage, and cos Á is called the powerfactor of the circuit. To make the most ef� cient use of power supply,the power factor should be close to unity. The size and weight ofthe ampli� er are determined by the amount of reactive power itgenerates. The power factor correction is achieved by adding aninductancein the circuit.This can be achievedin two differentways:in series and in parallel with the capacitance (Fig. 58).

The net impedance of the series con� guration is

NZ total D R Cj .!2 LC ¡ 1/

!C

and the impedance of the parallel con� guration is

NZ total D R Cj!L

1 ¡ !2LC

a) Series

b) Parallel

Fig. 58 Series and parallel resonant circuits.

2178 CHOPRA

where ! is frequency of excitation. The resonant frequency isde� ned as

!0 D 1¯p

LC

At the resonant frequency, the impedance of series circuit be-comes purely resistive, and the power factor equals unity. At thiscondition energy is continuously transferred between the inductor(energy storageas a magnetic � eld) and capacitor(energy storageasan electric � eld). The only energy dissipationis throughthe resistivelosses. In a parallel circuit, the impedanceapproaches in� nity at theresonant condition; there is no voltage drop across the resistor. Thevoltage across the actuator is the same as the power supply voltage,and the net current drawn from the power supply is nearly zero.This appears more attractive because the power supply can be quitecompact.

In reality, the piezoelectric actuator is a nonideal capacitanceand can be modeled as an ideal capacitor in series with a resistor.Normally, the resistancefor a PZT actuator is function of excitationfrequency and decreases with higher frequency.268 Even though theresistance is negligible as compared to the impedance of capacitor,it has a signi� cant effect on system performance.

Following the parallel circuit for a nonideal capacitor and induc-tor, its performance is evaluated.As shown in Ref. 268, to make theimpedance of this circuit totally resistive, the frequency has to be!01:

!01 DL ¡ R2

l C¡L ¡ R2

c C¢LC

Note that the capacitance of a typical PZT sheet actuator (size2 £ 1 £ 0:010 in.) (50.8£ 25.4 £ 0.254 mm) is of the order of150 nF, whereas the capacitance of a typical piezostack actuatoris of the order of 7 mF. There are three possible cases to achieve thiscondition.For a general case of Rl 6D 0, Rc 6D 0, there is a maximumallowable value of Rl :

Rl · 1=2C!0

The required value of inductance increases rapidly for the lowvalue of actuator capacitance and excitation frequency. In practice,it is dif� cult to obtain physical inductance in the range of tens ofhenries. Physical inductance of such a magnitude tends to be bulkyand heavy and makes it impractical to integrate in a smart system.Such inductors also tend to have a high value of resistance.Further,the requirement of inductance changes with frequency and as aresult requires continuous tuning of the circuit. To overcome theseproblems, the concept of pseudoinductanceis introduced.

Pseudoinductance is an active inductor created using an opera-tional ampli� er. It is compact, and an inductance value of the orderof 100 H can be achieved. The ampli� er changes the phase of thecurrent � owing through the circuit so that the equivalent impedanceis inductive. Also, the value of inductance can be easily adjustedby changing the values of the capacitance or the resistance in thecircuit using a potentiometer (Fig. 59). The major disadvantage ofpseudoinductanceis that thepower ampli� er requiresanotherpowersupply (dc) for its functioning.However, it should be noted that thereductionof theacpower requirementgenerallywill result in a loweroverall power and weight of the combined reduced ac power supplyand the pseudoinductance dc power supply. Sirohi and Chopra268

implemented a pseudoinductance system by connecting it parallelto a stack actuator. Figure 60 shows test results of power savingobtained by connecting a 100-H pseudoinductor in parallel with a

Fig. 59 Nonideal reso-nant drive circuit withpseudoinductor.

Fig. 60 Power savings obtained by continuous tuning, for a 7uFactuator.268

stack actuator of capacitance 7 mF by continuously tuning the cir-cuit. Two differentexcitationvoltagesare shown:0–50and0–100 V,for which the circuit supply voltages are 75 and 125 Vdc, respec-tively. A current saving of 70% has been obtained under a normaloperating condition at 20 Hz. The power saving decreases with fre-quency. At higher frequencies and excitation � elds, although theactuator power saving is still signi� cant, the overall power savingmight become negative.

Another approach to save actuation power is to use switchingampli� ers.269¡271 Simulations have shown enormous potential ofweight saving with this scheme.

F. Rotorcraft ApplicationsMore than any other system, the structural,mechanical, and aero-

dynamic complexity and the interdisciplinary nature of rotorcraftoffer numerousopportunitiesfor the applicationof smart-structurestechnologieswith the potential for substantial payoffs in system ef-fectiveness.Comparedto � xed-wingaircraft,helicopterssufferfromsevere vibration and fatigue loads, more susceptibility to aerome-chanical instability,excessivenoise levels,poor � ight stabilitychar-acteristics,andweak aerodynamicperformance.The primarysourcefor all of these problems is the main rotor, which operates in an un-steady and complex aerodynamic environment leading to stalledand reversed � ow on the retreating side of the disk, transonic � owon the advancing blade tips, highly yawed � ow on the front andrear part of the disk, and blade-vortex interactions in certain � ightconditions.Currently,considerableresearch is focusedon the appli-cation of smart-structures technology to rotor systems to improvetheir performanceand effectiveness.Three typesof smart-rotorcon-cepts are under development: leading- and trailing-edge � aps actu-ated with smart material actuators,controllablecamber/twist bladeswith embedded piezoelectric elements/� bers, and active blade tipsactuated with tailored smart actuators. The performance of theseactuation systems degrades rapidly at high rotational speeds be-cause of large centrifugal force, dynamic pressure, and frictionalmoments. For � ap actuation, actuators range from piezobimorphs,piezostacks, and piezoelectric/magnetostrictive-induced compositecoupled systems. These conceptshave been demonstratedon scaledrotor models such as Froude and Mach scaled (Fig. 61) and arecurrently being incorporatedin full-scale rotor systems. Most smartmaterial actuators are moderate force and extremely small strokedevices, and hence some form of mechanical/� uidic/hybrid ampli-� cation of stroke is needed to achieve practicable � ap de� ections.Because of compactness and weight considerations,the stroke am-pli� cation mechanism and high-energydensity actuatorshave beenkey barriers for application to rotor blades.

Koratkar and Chopra272¡274 built 6-ft-diam (1.83-m-diam) dy-namically scaled rotor models with trailing-edge� aps actuatedwithmultilayered piezobimorphs. Initially Froude-scaled rotor modelswere built and successfully tested in a vacuum chamber and on ahover tower, and � nally, Mach-scaled rotor models were demon-strated in closed-loop testing in the wind tunnel. The � aps spannedabout 10% of rotor radius and were centered at 75% of blade lengthand showed over §5-deg de� ection at 4/rev excitation using 3:1 ac

CHOPRA 2179

a) Piezobimorph-actuated� ap: 6-ft-diam rotor model test in Glenn L.Martin wind tunnel; successfully tested in both open- and closed-Loopstudies

b) Blade-tab-actuated with shape memory alloys actuator, wingsection tested in open-jet wind tunnel; produced tab de� ections ofover 20 deg

c) Piezostacks-actuated � ap: full-scale wing section model tested inopen-jet wind tunnel; produced §§ 6-deg � ap de� ection

d) Active twist blade with embedded piezoactuators 6-ft-diamrotor test in Glenn L. Martin wind tunnel; produced §§ 0.5-degblade twist up to 5/rev excitation

e) Smart tip rotor model (diam 6 ft) on hover stand, actuatedwith composite bending-torsion coupling and piezos; produced§§ 2 deg tip de� ections up to 5/rev

f) Froude-scale rotor model (6 ft diam) on hover tower withpiezobimorph-actuated � aps

Fig. 61 Smart rotor development at Alfred Gessow Rotorcraft Center.

bias at an rpm of 2150.Using a neural-network-basedadaptivefeed-back controller,individualblade control resultedin over 80% reduc-tion in vibratory hub loads in the Glenn L. Martin wind tunnel. AFroude-scaled rotor model was also tested successfully in an open-loop investigationby Fulton and Ormiston.275

Lee232 and Lee and Chopra238;246;276 builta model of blade sectionof length 12 in. and chord 12 in. (1 in. D 25.4 mm) with trailing-edge � ap (span 4 in. and chord 3 in.) actuated with piezostacks inconjunctionwith double-lever (L-L) ampli� cation mechanism. Themodel was tested in a vacuum chamber to simulate the full-scalecentrifugal � eld (600 g) and showed a desired stroke-ampli�cationfactor of about 20 at all rotor harmonics (up to six). The model wastested in an open-jetwind tunnel and successfullydemonstrated� apperformanceof about §10 deg at 120 ft/s. To improve bidirectional

performance of this actuation device, a dual L-L ampli� cation sys-tem was built and successfully tested in vacuum chamber and inwind tunnel. This new actuation system showed a signi� cant im-provement in � ap performance at different operating conditions.276

Straub et al.277 are building a full-scale smart rotor system for theMD-900 Explorer (� ve-bladed, 34-ft-diam) with piezostack actu-ated � aps to actively control its vibration and noise. To amplify thestrokeof piezostacks,a biaxialX-frame mechanismis incorporated.The system will be tested in both open- and closed-loop� ight inves-tigations.Hall and Prechtl278 built 1

6 th Mach-scale rotor model withtrailing-edge� aps actuatedwith X-frame actuatorsand successfullytested on a hover stand.Flap de� ections of §2.4 deg were achieved.Also, Janker et al.242 developed a novel piezostack-based � exuralactuator for actuation of trailing-edge � aps.

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Bernhard and Chopra279¡281 built 6-ft-diam Mach-scaled smartactive blade-tip (10%) rotor actuated with piezoinduced bending-torsion coupled composite beam. A novel spanwise variation in plylayup of the composite beam and phasing of surface-mountedpiezoceramic actuators is used to convert the bending-torsioncoupledbeaminto a pure twist actuator.At 2000 rpm in hover,blade-tip pitch de� ections of 1.7–2.9 deg were achieved at the � rst fourharmonics (for an excitation of 125 Vrms). The associated changesin blade lift corresponded to an aerodynamic thrust authority upto 30%. This concept appears promising as an auxiliary device forpartial control of noise and vibration.

Chen and Chopra282;283 built a 6-ft-diam Froude-scaled rotormodel with controllable twist blades. For this concept banks ofspecially shaped (large aspect ratio) multilayeredpiezoceramicele-ments were embeddedat §45 deg relative to blade axis respectivelyover the top and bottom surfaces; an in-phase activation resultedin pure twist in the blade. The model was successfully tested ona hover stand and in the Glenn L. Martin wind tunnel. A tip twistof the order of §0.4 deg at 4/rev was obtained in both hover andforward � ight (¹ D 0:33) that amounted to over 10% rotor thrustauthority. Although the oscillatory twist amplitudes attained in theforward � ight experiments were less than the target value (1 degof tip twist for complete vibration suppression), it showed the po-tential for partial vibration suppression. Rogers and Hagood257;258

and Derham and Hagood259 built a controllable-twistMach-scaledrotor by embedding active � ber composite (AFC) and tested on ahover stand. Even though it did not achieve the projected tip twistof §2 deg, it showed enormous potential for full-scale rotor appli-cations. Cesnik et al.284 and Cesnik and Shin285 further improvedthis technology and successfully tested a Mach-scaled rotor modelwith embedded active � bers in the transonic dynamics wind tunnelin both open-loop and closed-loop investigations. They have alsore� ned analytical tools related to this rotor system.

Shape memory alloys (SMA) show enormous potential in pro-viding large induced strains (up to 6%), but are limited to low-frequency (less than 1 Hz) applications such as tab adjustment forrotor tracking. Epps and Chopra,286;287 systematically investigatedthe development of an SMA-actuated trailing-edge tab for in-� ightblade tracking. They built a model of blade section of span andchord of 12 in. with a tab of span 4 in. and chord 2.4 in. actuatedwith two to � ve nitinol wires of diameter 0.015 in. respectivelyboth on top and bottom surfaces. To lock the tab at a desired angle(in power-off condition), a gear-locking mechanism consisting ofspur gears, pulling solenoid, and pawl was built. A displacementfeedback controller was developed to � ne tune the tab de� ectionin about 10 s. This wing section was tested in the open-jet windtunnel, and tab de� ections of the order of 20 deg were obtained at aspeedof 120 ft/s. This conceptappearspromisingfor full-scalerotortracking. Recently, Singh and Chopra288 improved this design andsuccessfully tested it in the wind tunnel for a repeatable open-loopand closed-loop performance.

There are other potentialapplicationsof smart-structurestechnol-ogy to rotary-wing systems that might result in enormous payoff interms of performance improvement and cost saving. These includeexternalnoise suppression,289 internalnoise suppression,290 primaryrotor controls,291;292 performance enhancement including dynamicstall delay,293 active transmission mounts,294;295 and active/passivedamping augmentation.296¡298

VIII. SummaryThe summary of the state of the art as presented in the paper

together with recommendations for future work are as follows:

A. Development of Large Stroke Smart ActuatorsAt this time most of the commerciallyavailablesmartmaterial ac-

tuators (such as piezoceramicsand magnetostrictors)are low strokeand low force devices. For most applications there is a need forcompact,moderate force (about100 lb), moderatebandwidth(about100 Hz), and large displacement (in millimeters) actuators. A keychallenge is to amplify the stroke of existing actuators by tradingforce with displacement. Large mechanical ampli� cation using acompact leverage system often leads to substantial losses at hinges

and slippageat knife edges.The goal shouldbe to increasedisplace-ment capability of smart materials actuators by 300–500% and/orbuild ef� cient stroke-ampli�cation devices. Even though there arenew revolutionary smart materials such as single crystal piezoce-ramics and magnetic shape memory alloys are emerging, their com-mercial viability and applications are at an embryonic stage.

B. Database for Smart Materials CharacteristicsFor design development of smart structures, it is essential to

have a reliable database of characteristics of smart materials. Atthis time such a database for a wide range of test conditions isnot available. Thus, there is need to develop a reliable database ofsmart material characteristicsthroughextensive testing of engineer-ing specimens (standardized at macrolevel). For example, electro-mechanical-thermal fatigue characteristics that are key to productreliability are lacking and need focused effort.

C. Piezoelectric Strain SensorsPiezoelectricsensors have enormouspotential to measure signals

with high noise levels; however, their calibration factors for a widerange of operating conditions and con� gurations are not available.Hence, it is important to establish sensitivity of these sensors fordifferent sensor size, temperature, bond layer, and strain level.

D. Beam ModelingBecause the local strain distribution near a piezoactuator is two-

dimensional, it is important to re� ne one-dimensionalbeam modelsthat capture this effect. Simple models for beams with actuators notaligned with the beam axis are inadequate and should be improved.Because the bond layer plays an important role in the induced straintransfer from actuator to beam, higher-ordershear deformation the-ory should be used to examine this effect and also to establish thelimits of widely used simple beam theories. Beam analyses need tobe re� ned to include nonlinear piezocharacteristics,piezoelectric-mechanical couplings, and layerwise shear variations. Validationstudies should be expanded to cover coupledcomposite beams withsurface-mounted or embedded actuators. Modeling of active com-posite plies that use AFC/MFC type � bers need carefully scrutinyand systematic validations.

E. Plate ModelingThere is a general lack of validation studies with experimental

data or with detailed analyses. Most plate models assume a perfectbond condition that is too restrictive. There is a need to developdetailed three-dimensional � nite element analyses or higher-ordershear deformationformulationsincludingnonlinearactuationstrainand piezoelectric-mechanical couplings to check the limitations ofexisting laminated plate analyses (such as CLPT) as well as to un-derstand the diffusion of actuation strain. Validation studies withexperimental test data should be expanded to cover a range of com-posite coupledplate con� gurationsincludingactive � ber compositeplies.

F. Shape Memory AlloysSystematicvalidationof constitutivemodels ofSMA for different

temperaturesand strains (extensionaland shear) is not readily avail-able. Building of composite structureswith embeddedSMAs is stilla challenge and needs a focused effort. Analysis of structures withembeddedSMA has so far shown mixed success in terms of correla-tion of predicted results with measured data. Re� ned analysesusinghigher-ordersheardeformationtheoryor detailedthree-dimensional� nite element analysis should be developed to investigate the actu-ation mechanism as well as the structural integrity of system. Toexploit the potentialof SMA to various applications, it is importantto re� ne and simplify analysesand carry out systematicvalidations.Most current constitutive models are developed for uniaxial qua-sisteady loadingand require reformulationsand extensions to covertorsionaland three-dimensionalstressconditionsforboth steadyandtransient thermal and mechanical loadings. Innovations are neededto increase frequency response of SMA actuators. Exploitation ofSMA to augment structural damping needs further examination.

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G. Magnetostrictives and ElectrostrictivesAt this time most of the smart-structures studies are focused

on piezoelectric and SMA actuators. Simpli� ed constitutive re-lations for magnetostrictive and electrostrictive actuators are notreadily available. To exploit these actuators for some special ap-plications, it is important to develop simpli� ed constitutive rela-tions and validate them with experimental data. Detailed measurednonlinear electromechanical/magnetomechanicalcharacteristicsofelectostrictives and magnetostrictives are not readily available andshould obtained for a wide range of load conditions.

H. Application to Realistic SystemsCurrently, this emerging technology is applied to models/

simulations that are often unrealistic. For a proper assessment ofthis technology,it is necessary to apply it to realistic structures.Ma-jor barriers are actuator stroke, unavailability of robust distributedparameter control strategies, and nonexistent mathematical model-ing of the smart system. The objective should be to build and testdynamically scaled models and evaluate performance of actuatorsand feedback control algorithms under different operating condi-tions. Using these test data, validate comprehensive analyses andthen carry out multidisciplinary optimization studies to develop asmart actuation system.

I. Expand Smart-Structures Applications to ImprovePerformance and Minimize Acoustics

Currently, most applications of smart-structures technology toaeronautical systems such as wings and rotors are focused on vi-bration minimization. There is an enormous payoff to exploit thistechnology to improve system performance, primary controls im-plementation, and minimize external/internal noise. It is necessaryto build dynamically scaled models and test them in different oper-ating conditions.Key barriersare largebandwidth/large stroke actu-ators, reliable active noise control analyses, robust acoustic sensors,well-de� ned objectives,and unavailabilityof distributedparameterscontrolalgorithms.The objectiveshouldbe to build and test dynam-ically scaled models and evaluate performance of smart actuatorsand feedback controllers under different � ight conditions. Usingthese test data, validate comprehensive analyses and then carry outmultidisciplinaryoptimization studies to developa smart-structuressystem to simultaneously increase system performance and mini-mize noise and vibration.The challenge is to replace servoactuatorswith smart actuators for primary controls in a cost-effectivemannerand with no degradation in product reliability.

J. Smart-Structures Application to Increase DampingThere are systems that are inherently low damped and require

some form of damping augmentation. For example, to overcomeaeromechanical instability of helicopter rotors, they are installedwith either complex mechanical dampers or expensive elastomericdampers. To reduce the maintenanceand procurementcost substan-tially, there is a potential applicationof smart-structurestechnologyto augment rotor damping. Hybrid active/passive constrained layerdamping in conjunction with electrorheologicaland magnetorheo-logical � uid dampers as well as shape control devices (such as � aps)can be used to augment blade lag mode damping. Current barriersare reliable analytical models, test data to cover a wide range ofoperating conditions, and limited stroke of actuators. Road mapsshould be established to develop and validate different analyticalelements of damping augmentation, as well as to apply these todynamically scaled models and evaluate their performance underdifferent operating conditions.

K. Structural Health Monitoring of SystemsAerospace systems are highly susceptible to damage because of

severe vibratory and fatigue loads. The problem is addressed byfrequent inspection of damage-sensitive parts and, if required, re-placement of such parts. This contributes to higher operating andmaintenance costs. Health monitoring of aerospace systems offerthe potential to lower operating costs and enhance � ight reliabil-ity. At this time there have been only limited experimental studieson this topic, and analytical tools are primitive. Major barriers areunavailability of robust sensors, reliable system identi� cation al-

gorithms, proper analog/digital � lters, and data multiplexing andwireless transmission. Keeping in view the future goal to reducemaintenance costs of aerospace systems by at least 25–50%, thereis a need of focusedfundamentalresearchactivitiesrelated to healthmonitoring of aerospace and other systems using smart sensors.

L. Revolutionary Smart MaterialsRecently, there have been some exciting developments in smart

materials such as the emergence of magnetic shape memory alloys,single crystal piezoceramics, and piezoelectric and electrostrictivepolymeric actuator materials that have a stroke of an order of mag-nitude larger than current piezoceramics. These materials requirecareful scrutiny by researchers. There is every potential that theserevolutionarymaterials might open up many new frontiers of appli-cations and result in “quantum-jump” in system performance andreliabilityand dramatic reduction in acquisitionand operatingcosts.That will be the dawn of smart materials.

AcknowledgmentsMost of these research activities were supported by the U.S.

Army Research Of� ce under Grants URI: DAAL 03-92-G-0121and MURI: DAAH-04-96-10334 with Gary Anderson and TomDoligalski as Technical Monitors. The author acknowledges manyuseful discussions and enthusiastic input from Friedrich Straub,Boeing, Mesa, Arizona; Yung Yu, National Rotorcraft TechnologyCenter; Alfred Gessow, Norman Wereley, Darryll Pines, and V. T.Nagaraj,UniversityofMaryland;and many graduatestudents(PeterChen, Andy Bernhard, Nikhil Koratkar, Jeanette Epps, Taeoh Lee,Jayant Sirohi, Harsha Prahlad, Mike Spencer, and Judah Milgram).

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210Fripp,M., and Hagood, N., “Comparison of Electrostrictive and Piezo-ceramic Actuators for VibrationSuppression,”Proceedingsof the 1995Soci-ety ofPhoto-OpticalInstrumentationEngineers’ConferenceonSmartStruc-tures and Materials, Society of Photo-Optical Instrumentation Engineers(International Society for Optical Engineering), Bellingham, WA, 1995.

211Hom, C. L., and Shankar,N., “A Fully Coupled ConstitutiveModel forElectrostrictive Ceramic Materials,” Journal of Intelligent Material Systemsand Structures, Vol. 5, No. 6, 1994, pp. 795–801.

212Hom, C. L., and Shankar,N., “A ConstitutiveModel forRelaxorFerro-electrics,”Proceedingsof the 1999Society of Photo-OpticalInstrumentationEngineers’ North American Symposium on Smart Structures and Materials,Society of Photo-Optical Instrumentation Engineers (International Societyfor Optical Engineering), Bellingham, WA, 1999.

213Piquette, J. C., and Forsythe, S. E., “A Nonlinear Material Model ofLead Magnesium Niobate (PMN),” Journal of the Acoustical Society ofAmerica, Vol. 101, No. 1, 1997, pp. 289–296.

214Wada, B. K., Fanson, J. L., and Crawley, E. F., “Adaptive Structures,”Journal of Intelligent Material Systems and Structures, Vol. 1, No. 2, 1990,pp. 157–174.

215Lin, C. Y., Crawley, E. F., and Heeg, J., “Open- and Closed-Loop Re-sults of a Strain-Actuated Active Aeroelastic Wing,” Journal of Aircraft,Vol. 33, No. 5, 1996, pp. 987–994.

216Kudva, J., Appa, K., Martin, C., Jardine, P., and Sendeckji, G., “De-sign, Fabrication and Testing of the DARPA/WL ‘Smart Wing’ Wind TunnelModel,” AIAA Paper 97-1198, April 1997.

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217Kudva, J., Martin, C. A., Scherer, L. B., Jardine, A. P., McGovan,A. R., Lake, R. C., Sendeckji, G., and Sanders, B., “Overview of theDARPA/AFRL/NASA Smart Wing Program,” Proceedings of the 1999 So-ciety of Photo-Optical Instrumentation Engineers’ North American Sympo-sium on SmartStructuresandMaterials, Vol. 3674,Society ofPhoto-OpticalInstrumentation Engineers (International Society for Optical Engineering),Bellingham, WA, 1999, pp. 230–248.

218Becker, J., and Schroder, W., “Advanced Aircraft Structures Program:An Overview,” Proceedings of the 1999 Society of Photo-Optical Instru-mentation Engineers’ North American Symposium on Smart Structures andMaterials, Vol. 3674, Society of Photo-Optical Instrumentation Engineers(International Society for Optical Engineering), Bellingham, WA, 1999,pp. 2–12.

219Durr, J. K., Herold-Schmidt, U., and Zaglauer, W., “On the Integrationof Piezoceramic Actuators in Composite Structures for Aerospace Appli-cations,” Journal of Intelligent Material Systems and Structures, Vol. 10,No. 11, 1999, pp. 880–889.

220Ehlers, S. M., and Weisshaar, T. A., “Static Aeroelastic Control of anAdaptive Lifting Surface,” Journal of Aircraft, Vol. 30, No. 4, 1993,pp. 534–540.

221Lazarus, K. B., Crawley, E. F., and Bohlmann, J. D., “Static AeroelasticControl Using Strain Actuated Adaptive Structures,” Journal of IntelligentMaterial Systems and Structures, Vol. 2, No. 3, 1991, pp. 386–410.

222Frampton, K. D., Clark, R. L., and Dowell, E. H., ”Active Control ofPanel Flutter with Piezoelectric Transducers,” Journal of Aircraft, Vol. 33,No. 4, 1996, pp. 768–774.

223Bein,T.,Hanselka, H., and Breitbach, E., “An AdaptiveSpoiler to Con-trol the Transonic Shock,” Smart Materials and Structures, Vol. 9, No. 2,2000, pp. 141–148.

224Fuller, C. R., Hansen, C. H., and Snyder,S. D., “Experimentson ActiveControl of Sound Radiation from a Panel Using a Piezoceramic Actuator,”Journalof the Acoustical Society of America, Vol. 91, No. 6, 1992,pp. 3313–3320.

225Hooker, M. W., “Properties and Performance of RAINBOW Piezo-electric Actuator Stacks,” Proceedings of the Society of Photo-Optical In-strumentation Engineers’ 4th Annual Symposium on Smart Structures andMaterials, Industrial and Commercial Applications, 1997, pp. 413-420.

226Li, G., Furman, E., and Haertling, G. H., “Fabrication and Proper-ties PSZT Antiferroelectric Rainbow Actuators,” Ferroelectrics, Vol. 188,No. 1–4, 1996, pp. 223–236.

227Li, G., Furman, E., and Haertling, G. H., “Finite Element Analysis ofRainbow Ceramics,” Journalof IntelligentMaterial Systems and Structures,Vol. 8, No. 5, 1997, pp. 434–443.

228Mossi, K. M., and Bishop, R. P., “Characterization of Different Typesof High-Performance THUNDER Actuators,” Proceedings of the Societyof Photo-Optical Instrumentation Engineers’ 6th Annual Symposium onSmart Structures and Materials, Smart Materials Technologies Conference,Vol. 3675, Society of Photo-Optical Instrumentation Engineers (Interna-tional Society for Optical Engineering), Bellingham, WA, 1999, pp. 43–52.

229Onitsuka, K., Dogan, A., Tressler, J. F., Xu, Q., Yoshikawa, S., andNewnham, R. E., “Metal-Ceramic Composite Transducer, The Moonie,”Journal of Intelligent Material Systems and Structures, Vol. 6, No. 4, 1995,pp. 447–455.

230Moskalik, A., and Brei, D., “Parametric Investigation of the De� ectionPerformance of Serial Piezoelectric C-Block Acutaors,” Journal of Intelli-gent Material Systems and Structures, Vol. 9, No. 3, 1998, pp. 223–231.

231Moskalik, A., and Brei, D., “Quasi-Static Behavior of IndividualC-Block Piezoelectric Acuators,” Journal of Intelligent Material Systemsand Structures, Vol. 8, No. 7, 1997, pp. 571–587.

232Lee, T., “Design of High Displacement Smart Trailing Edge Flap Ac-tuator Incorporating Dual-Stage Mechanical Stroke Ampli� ers for Rotors,”Ph.D. Dissertation, Dept. of Aerospace Engineering, Univ. of Maryland,College Park, Dec. 1999.

233Giurgiutiu,V., Rogers, C. A., and Chaudhary,Z., “Energy-Based Com-parison of Solid State Induced-Strain Actuators,” Journal of Intelligent Ma-terial Systems and Structures, Vol. 7, No. 1, 1996, pp. 4–14.

234Giurgiutiu, V., and Rogers, C. A., “Large Amplitude Rotary Induced-Strain (LARIS),” Journal of Intelligent Material Systems and Structures,Vol. 8, No. 1, 1997, pp. 41–50.

235Mitrovic, M., Carman, G. P., and Straub, F. K., “ElectromechanicalCharacterization of Piezoelectric Stack Actuators,” Proceedings of the Soci-ety of Photo-Optical Instrumentation Engineers’ 6th Annual Symposium onSmart Structures and Materials, Smart Materials Technologies Conference,Vol. 3668, Society of Photo-Optical Instrumentation Engineers (Interna-tional Society for Optical Engineering), Bellingham, WA, 1999, p. 589.

236Giurgiutiu, V., and Rogers, C. A., “Stiffness Issues in the Design ofISA Displacement Ampli� cation Devices: Case Study of a Hydraulic Dis-placement Ampli� er,” Proceedings of the Society of Photo-Optical Instru-mentation Engineers’ 2nd Annual Symposium on Smart Structures and Ma-terials, Smart Materials Conference, Vol. 2443, Society of Photo-OpticalInstrumentation Engineers (International Society for Optical Engineering),Bellingham, WA, 1995.

237Garcia-Bonito, J., Brennan, M. J., Elliott, S. J., and Pinnington, R. J.,“A Novel High-Displacement Piezoelectric Actuator for Active VibrationControl,” Smart Materials and Structures, Vol. 7, No. 1, 1998, pp. 31–42.

238Lee, T., and Chopra, I., “Design of Peizostack-Driven Trailing-EdgeFlap Actuator for Helicopter Rotors,” Smart Materials and Structures,Vol. 10, No. 1, 2001, pp. 15–24.

239Prechtl, E. F., and Hall, S. R., “Design of a High Energy DiscreteServo-Flap Actuator for a Helicopter Rotor Control,” Proceedings of the1997 Society of Photo-Optical Instrumentation Engineers’ North Ameri-can Symposium on Smart Structures and Materials, Vol. 3041, Society ofPhoto-Optical Instrumentation Engineers (International Society for OpticalEngineering), Bellingham, WA, 1997, pp. 158–182.

240Straub, F. K., “Active Flap Control for Vibration Reduction and Per-formance Improvement,” Proceedings of the 54th Annual Forum, Vol. 1,American Helicopter Society, Alexandria, VA, 1998.

241Fenn, R. C., Downer, J. R., Bushko, D. A., Gohalekar, V., and Ham,N. D., “Terfenol-D Driven Flaps for Helicopter Vibration Reduction,”SmartMaterials and Structures, Vol. 5, No. 1, 1996, pp. 49–57.

242Janker, P., Kloppel, V., Hermle, F., Lorkowski, T., Storm, S.,Christmann, M., and Wettemann, M., “Development and Evaluation of aHybridPiezoelectric Actuator for Advanced Flap Control Technology,”25thEuropean Rotorcraft Forum, Rome, Sept. 1999.

243Xu, W., and King, T., “Flexural Hinges for Piezoelectric Displace-ment Ampli� ers: Flexibility,Accuracy, and Stress Concentration,”PrecisionEngineering, Vol. 19, No. 1, 1996, pp. 4–10.

244Prechtl, E. F., and Hall, S. R., “Design of a High Ef� ciency, LargeStroke,Electromechanical Actuator,”SmartMaterialsandStructures, Vol. 8,No. 1, 1999, pp. 13–30.

245Paine, J. S. N., and Chaudhry, Z., “The Impact of Ampli� cation onEnergy Ef� ciency and Energy Density of Induced Strain Actuators,” Pro-ceedings of ASME Aerospace Division, AD-Vol. 52, American Society ofMechanical Engineers, 1996, pp. 511–516.

246Lee, T., and Chopra, I., “Design Issues of a High-Stroke, On-BladePiezostack Actuator for Helicopter Rotor with Trailing-Edge Flaps,” Jour-nal of Intelligent Material Systems and Structures, Vol. 11, No. 5, 2000,pp. 328–342.

247Frecker, M., and Can� ed, S., “Optimal Design and Experimental Val-idation of Compliant Mechanical Ampli� ers for Piezoceramic Stack Actua-tors,” Journal of Intelligent Material Systems and Structures, Vol. 11, No. 5,2000, pp. 360–369.

248Frecker, M., Ananthsuresh, G. K., Nishiwaki, S., Kikuchi, N., andKota, S., “Topological Synthesis of Complaint Mechanisms Using Multi-Criteria Optimization,”JournalofMechanicalDesign, Vol. 119,No. 2, 1997,pp. 238–245.

249Edinger, B., Frecker, M., and Gardner, J., “Dynamic Modeling of anInnovative Piezoelectric Actuator for Minimally Invasive Surgery,” Jour-nal of Intelligent Material Systems and Structures, Vol. 11, No. 10, 2000,pp. 765–770.

250Bothwell, C. M., Chandra, R., and Chopra, I., “Torsional Actuationwith Extension-Torsional Composite Coupling and Magnetostrictive Actu-ators,” AIAA Journal, Vol. 33, No. 4, 1995, pp. 723–729.

251Bernhard, A. P. F., and Chopra, I., “Hover Testing of Active RotorBlade-Tips Using a Piezo-Induced Bending-TorsionCoupled Beams,” Jour-nal of Intelligent Material Systems and Structures, Vol. 9, No. 12, 1998,pp. 963–974.

252Glazounov, A. E., Zhang, Q. M., and Kim, C., “New Torsional Actu-ator Based on Shear Piezoelectric Response,” Proceedings of the Society ofPhoto-Optical Instrumentation Engineers’ 5th Annual Symposium on SmartStructuresandMaterials, Smart Materials Conference, Vol. 3044,Society ofPhoto-Optical Instrumentation Engineers (International Society for OpticalEngineering), Bellingham, WA, 1998.

253Centolanza, L. R., Smith, E. C., and Morris, A., “Induced Shear Piezo-electric Actuators for Rotor Blade Trailing Edge Flaps and Active Tips,”AIAA Paper 2001-1559,April 2001.

254Bent, A. A., and Hagood, N. W., “Anisotropic Actuation with Piezo-electric Fiber Composites,” Journal of Intelligent Material Systems andStructures, Vol. 6, No. 3, 1995, pp. 338–349.

255Bent, A. A., and Hagood, N. W., “Piezoelectric Fiber Composites withInterdigitatedElectrodes,” Journalof IntelligentMaterialSystems andStruc-tures, Vol. 8, No. 11, 1997, pp. 903–919.

256Hagood, N. W., and Pizzochero, A., “Residual Stiffness and ActuationProperties of Piezoelectric Composites: Theory and Experiment,” Journal ofIntelligentMaterialSystems andStructures, Vol. 8,No. 9, 1997,pp.724–737.

257Rogers, J. P., and Hagood, N. W., “Preliminary Mach-Scale HoverTesting of an Integral Twist-Actuated Rotor Blade,” Proceedings of the1997 Society of Photo-Optical Instrumentation Engineers’ North Ameri-can Symposium on Smart Structures and Materials, Vol. 3041, Society ofPhoto-Optical Instrumentation Engineers (International Society for OpticalEngineering), Bellingham, WA, 1997.

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258Rogers, J. P., and Hagood, N. W., “Design and Manufactureof an Integral Twist-Actuated Rotor Blade,” Proceedings of the 38thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materi-als Conference and Adaptive Structures Forum, AIAA, Reston, VA, 1997.

259Derham, R. C., and Hagood, N. W., “Rotor Design Using Smart Ma-terials to Actively Twist Blades,” Proceedings of the 52nd Annual Forum,Vol. 2, American Helicopter Society, Alexandria, VA, 1996, pp. 1242–1252.

260Wilkie, K. W., Wilbur, M. L., Mirick, P. H., Cesnik, C. E. S., and Shin,S. J., “Aeroelastic Analysis of the NASA/Army/MIT Active Twist Rotor,”Proceedings of the 55thAnnualForum, Vol. 2, American Helicopter Society,Alexandria, VA, 1999, pp. 533–544.

261Wikkie, W. K., Bryant, R. G., High, J. W., Fox, R. L., Hellbaum, R. F.,Jalink, A., Little, B. D., and Mirick, P. H., “Low-Cost Piezocomposite Ac-tuator for Structural Control Application,” Proceedings of the 2000 Societyof Photo-Optical Instrumentation Engineers’ North American Symposiumon Smart Structures and Materials, Vol. 3985, Society of Photo-OpticalInstrumentation Engineers (International Society for Optical Engineering),Bellingham, WA, 2000.

262Pokines, B. J., and Garcia, E., “A Smart Material Microampli� cationMechanism Fabricated UsingLIGA,” SmartMaterialsandStructures, Vol. 7,No. 1, 1998, pp. 105–112.

263Mauck, L. D., and Lynch, C. S., “Piezoelectric Hydraulic Pump De-velopment,” Journal of Intelligent Material Systems and Structures, Vol. 11,No. 10, 2000, pp. 758–764.

264Nasser, K., and Leo, D. J., “Ef� ciency of Frequency-Recti� ed Peizo-hydraulic and Piezopneumatic Actuation,” Journal of Intelligent MaterialSystems and Structures, Vol. 11, No. 10, 2000, pp. 798–810.

265Roberts, D., Hagood, N., Carretero, H., Li, H., Mlcak, R., and Su,Y., “Piezoelectrically Driven Hydraulic Ampli� cation Microvalve for HighPressure High Frequency Applications,” Proceedings of the 2000 Societyof Photo-Optical Instrumentation Engineers’ North American Symposiumon Smart Structures and Materials, Vol. 4327, Society of Photo-OpticalInstrumentation Engineers (International Society for Optical Engineering),Bellingham, WA, 2000, pp. 366–380.

266Sirohi, J., and Chopra, I., “Development of a Compact Hybrid Ac-tuator,” Proceedings of the 2001 Society of Photo-Optical InstrumentationEngineers’ North American Symposium on Smart Structures and Materials,Smart Materials Technologies Conference, Vol. 4327, Society of Photo-Optical Instrumentation Engineers (International Society for Optical Engi-neering), Bellingham, WA, 2001, pp. 401–412.

267Sirohi, J., and Chopra, I., “Design and Testing of a High-PumpingFre-quency Piezoelectric-Hydraulic hybrid Actuator,” Proceedings of the 2002Society of Photo-Optical Instrumentation Engineers’ North American Sym-posium on Smart Structures and Materials, Vol. 4701, Society of Photo-Optical Instrumentation Engineers (International Society for Optical Engi-neering), Bellingham, WA, 2002, pp. 552–569.

268Sirohi, J., and Chopra, I., “ActuatorPower ReductionUsing Tuned L-COscillator Circuits,” Journal of Intelligent Material Systems and Structures,Vol. 12, No. 12, 2000, pp. 867–878.

269Clingman, J. D., “Drive Electronics for Large Piezoactuators,” Pro-ceedings of the Society of Photo-Optical Instrumentation Engineers’ 4thAnnual Symposium on Smart Structures and Materials, Industrial Applica-tions Conference, Vol. 3044, Society of Photo-Optical Instrumentation En-gineers (International Society for Optical Engineering), Bellingham, WA,1997, pp. 459–467.

270Main, J. A., Garcia, E., and Newton, D. V., “Precision Position Controlof Piezoelectric Actuators Using Charge Feedback,” Journal of Guidance,Control, and Dynamics, Vol. 18, No. 5, 1995, pp. 1068–1073.

271Chandrasekaran, S., Linder, D. K., and Smith, R. C., “Optimized De-sign of Switching Ampli� ers for Piezoelectric Actuators,” Journal of Intel-ligent Material Systems and Structures, Vol. 11, No. 11, 2000, pp. 887–901.

272Koratkar,N. A., andChopra, I., “Developmentof aMach-Scaled Modelwith Piezoelectric Bender Actuated Trailing-EdgeFlaps for Helicopter Indi-vidual Blade Control (IBC),” AIAA Journal, Vol. 38, No. 7, 2000, pp. 1113–1124.

273Koratkar, N. A., and Chopra, I., “Wind Tunnel Testing of a Mach-Scaled Rotor Model with Trailing-Edge Flaps,” Smart Materials and Struc-tures, Vol. 10, No. 1, 2001, pp. 1–14.

274Koratkar, N. A., and Chopra, I., “Wind Tunnel Testing of a Mach-Scaled Rotor Model with Trailing-Edge Flaps,” Proceedings of the 57thAnnual Forum of the American Helicopter Society, Vol. 2, American Heli-copter Society, Alexandria, VA, 2001, pp. 1069–1099.

275Fulton, M., and Ormiston, R. A., “Small Scale Rotor Experimentswith On-Blade Elevons to Reduce Blade Vibratory Loads in Forward Flight,”Journalof the American Helicopter Society, Vol. 46,No. 2, 2001,pp. 96–106.

276Lee, T., and Chopra, I., “Wind Tunnel Test of Blade Sections withPiezoelectric Trailing-Edge Flap Mechanism,” Proceedings of the 57th An-nualForum of the American Helicopter Society, Vol. 2, American HelicopterSociety, Alexandria, VA, 2001, pp. 1912–1923.

277Straub, F. K., Ngo, H. T., Anand, V., and Domzalski, D. B., “Devel-opment of a Piezoelectric Actuator for Trailing-Edge Flap Control for FullScale Rotor System,” Smart Materials and Structures, Vol. 10, No. 1, 2001,

pp. 25–34.278Hall, S. R., and Prechtl, E. F., “Preliminary Testing of a Mach-Scaled

Active Rotor Blade with a Trailing Edge Servo-Flap,” Proceedings of the1999 Society of Photo-Optical Instrumentation Engineers’ North Ameri-can Symposium on Smart Structures and Materials, Vol. 3668, Society ofPhoto-Optical Instrumentation Engineers (International Society for OpticalEngineering), Bellingham, WA, 1999, pp. 14–21.

279Bernhard, A. P. F., and Chopra, I., “Trailing Edge Flap Activated by aPiezo-Induced Bending-Torsion Coupled Beam,” Journal of the AmericanHelicopter Society, Vol. 44, No. 1, 1999, pp. 3–15.

280Bernhard, A. P. F., and Chopra, I., “Analysis of a Bending-TorsionCou-pled Actuator for a Smart Rotor with Active Blade Tips,” Smart Materialsand Structures, Vol. 10, No. 1, 2001, pp. 35–52.

281Bernhard, A. P. F., and Chopra, I., “Hover Test of an Active-Twist,Mach-Scale RotorUsing a Piezo-InducedBending-TorsionActuatorBeam,”National Specialists Meeting,American Helicopter Society, Alexandria, VA,Oct. 2000.

282Chen, P. C., and Chopra, I., “Wind Tunnel Test of a Smart Rotor withIndividualBlade Twist Control,”Journalof IntelligentMaterial Systems andStructures, Vol. 8, No. 5, 1997, pp. 414–425.

283Chen, P. C., and Chopra, I., “Hover Testing of Smart Rotor withInduced-Strain Actuation of Blade Twist,” AIAA Journal, Vol. 35, No. 1,1997, pp. 6–16.

284Cesnik, C. E. S., Shin, S. J., and Wilbur, M. L., “Dynamic Response ofActive Twist Rotor Blades,” Smart Materials and Structures, Vol. 10, No. 1,2001, pp. 62–76.

285Shin, S. J., Cesnik, C. E. S., and Hall, S. R., “Control of Integral Twist-Actuated Helicopter Blades for vibration Reduction,” Proceedings of the58th American Helicopter Society Annual Forum, Vol. 2, American Heli-copter Society, Alexandria, VA, 2002, pp. 2122–2134.

286Epps, J. J., and Chopra, I., “In-Flight Tracking of Helicopter RotorBlades Using Shape Memory Alloy Actuators,”Proceedings of the 56thAmerican Helicopter Society Annual Forum, Vol. 1, American HelicopterSociety, Alexandria, VA, 2000, pp. 328–343.

287Epps, J. J., and Chopra, I., “In-Flight Tracking of Helicopter RotorBlades Using Shape Memory Alloy Actuators,” Smart Materials and Struc-tures, Vol. 10, No. 1, 2001, pp. 104–111.

288Singh, K., and Chopra, I., “Design of an Improved Shape MemoryAlloy Actuator for Rotor Blade Tracking,” Proceedings of the 2002 Societyof Photo-Optical Instrumentation Engineers’ North American Symposiumon Smart Structures and Materials, Vol. 4701, Society of Photo-OpticalInstrumentation Engineers (International Society for Optical Engineering),Bellingham, WA, 2002, pp. 244–266.

289Chen, P. C., Baeder, J. D., Evans, R. A. D., and Niemczuk, J., “Blade-Vortex Interaction Noise Reduction with Active Twist Smart Rotor Technol-ogy,” Smart Materials and Structures, Vol. 10, No. 1, 2001, pp. 77–85.

290Sampath, A., and Balachandran, B., “Active Control of MultipleTonesin an Enclosure,” Journal of the Acoustical Society of America, Vol. 106,No. 1, 1999, pp. 211–225.

291Shen, J., and Chopra, I., “Actuation Requirements for a Swashplate-less Helicopter Control System with Trailing-Edge Flaps,” Proceedings ofthe 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics andMaterials Conference and Adaptive Structures Forum, AIAA, Reston, VA,2002.

292Shen, J., and Chopra, I., “A Parametric Design Studyfor a Swashplate-less Helicopter Rotor with Trailing-Edge Flaps,” Proceedings of the 58thAmerican Helicopter Society Annual Forum, Vol. 2, American HelicopterSociety, Alexandria, VA, 2002, pp. 2134–2148.

293Kube, R., and Kloppel, V., “On the Role of Prediction Tools for Adap-tive Rotor System Developments,” Smart Materials and Structures, Vol. 10,No. 1, 2001, pp. 137–144.

294Pelinescu, I., and Balachandran, B., “Analytical Study of Active Con-trol ofWave Transmission ThroughCylindrical Struts,” Smart Materials andStructures, Vol. 10, No. 1, 2001, pp. 121–136.

295Mahapatra, D. R., Gopalkrishnan, S., and Balachandran, B., “ActiveFeedback Controlof MultipleWaves in Helicopter Gearbox SupportStruts,”Smart Materials and Structures, Vol. 10, No. 5, 2001, pp. 1046–1058.

296Kamath, G. M., Wereley, N. W., and Jolly, M. R., “Characterizationof Magnetorheological Helicopter Lag Dampers,” Journal of the AmericanHelicopter Society, Vol. 45, No. 4, 1999, pp. 234–248.

297Kim, S. J., and Yun, C. Y., “Performance Comparison Between Piezo-electric and Elastomeric Lag Damper on Ground Resonance Stability ofHelicopter,” Journal of Intelligent Material Systems and Structures, Vol. 12,No. 4, 2001, pp. 215–222.

298Konstanzer, P., Grohmann, B., and Kroplin, B., “Decentralized Vibra-tion Control and Coupled Aeroservoelastic Simulation of Helicopter RotorBlades with Adaptive Airfoils,” Journal of Intelligent Material Systems andStructures, Vol. 12, No. 4, 2001, pp. 209–214.

A. M. BazAssociate Editor

chopra_aiaa-1561-157_review of state of art of smart structures

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