snap-through of unsymmetric laminates using piezocomposite actuators
TRANSCRIPT
COMPOSITES
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Composites Science and Technology 66 (2006) 2442–2448
SCIENCE ANDTECHNOLOGY
Snap-through of unsymmetric laminates using piezocomposite actuators
Marc R. Schultz a,1, Michael W. Hyer b,*, R. Brett Williams c,W. Keats Wilkie c, Daniel J. Inman d
a Composite Technology Development, Inc., 2600 Campus Dr., Suite D Lafayette, CO 80026-3359, United Statesb Department of Engineering Science and Mechanics (0219), Virginia Polytechnic Institute and State University, 219 Norris Hall,
Blacksburg, VA 24061, United Statesc Structures and Materials Technology Group, Jet Propulsion Laboratory, 4800 Oak Grove Dr., MS 299-101, Pasadena, CA 91109-8099, United States
d Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering (0261),
Virginia Polytechnic Institute and State University, 310 Durham Hall, Blacksburg, VA 24061, United States
Received 24 January 2006; accepted 31 January 2006Available online 4 April 2006
Abstract
The paper discusses the concept of using a piezoceramic actuator bonded to one side of a two-layer unsymmetric cross-ply [0/90]Tlaminate to provide the moments necessary to snap the laminate from one stable equilibrium shape to another. This concept couldbe applied to the morphing of structures. A model of this concept, which is based on the Rayleigh–Ritz technique and the use of energyand variational methods, is developed. The experimental phase of the study is discussed, including the measurement of the voltage levelneeded to snap the laminate. The voltage measurements and shapes are compared with predictions of the models and the agreementbetween measurements and the predictions are reasonable, both qualitatively and quantitatively. Suggestions for future activities arepresented.� 2006 Elsevier Ltd. All rights reserved.
Keywords: MFC actuator; Morphing; Rayleigh–Ritz technique; Stability; Multistable
1. Introduction
As is well known, a thin, unsymmetrically laminated, ele-vated temperature cure, fiber-reinforced composite lami-nate with no external loads, such as a two-layer cross-plygraphite–epoxy [0/90]T laminate, can have multiple equilib-rium shapes when cooled from the curing temperature to alower operating temperature; that is, the laminate can bemultistable. In the case of a cross-ply laminate, when thegeometry is such that the laminate is multistable, there aretwo stable cylindrical shapes and one unstable saddle shape.The laminate can be changed from one stable cylindricalshape to the other by a simple snap-through action by
0266-3538/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compscitech.2006.01.027
* Corresponding author. Tel.: +1 540 231 5372; fax: +1 540 231 4574.E-mail addresses: [email protected] (M.R. Schultz),
[email protected] (M.W. Hyer).1 Tel.: +1 303 664 0394x141.
applying moments along opposite edges of the laminate.The unstable shape, of course, is never achievable. Thismultistable phenomenon has been studied by a number ofinvestigators [1–4], and is a result of the coupling of theresidual stresses due to cooling with the geometric nonlin-earities due to the large out-of-plane deflections involved.
This study considers the use of the NASA LangleyResearch Center Macro-Fiber Composite Actuator(MFC), an orthotropic piezoceramic actuator, to effectsnap-through behavior of a [0/90]T laminate; MFC actua-tors are prepackaged and the actuation is accomplishedby applying a voltage to the actuator. The multistable char-acteristic could be applied to the design of morphing struc-tures. The geometry of the structure that was considered inthis study is shown schematically in Fig. 1; the MFC actu-ator is centrally bonded to one side of the laminate to formthe laminate-actuator combination. Predictions from amodel of the laminate-actuator combination based on the
Fig. 1. Geometry of laminate-actuator combination: cross-section view(upper); top view (lower).
M.R. Schultz et al. / Composites Science and Technology 66 (2006) 2442–2448 2443
Rayleigh–Ritz technique will be compared with experi-ments. In particular, predictions of the shapes of the lami-nate-actuator combination and of the voltage applied tothe actuator that causes snap-through are compared withexperiments.
2. The macro-fiber composite actuator
The MFC actuator consists of piezoceramic PZT mac-
rofibers embedded in a structural epoxy matrix and sand-wiched between interdigitally electroded polyimide films[5]. Two-dimensional modeling of the piezoelectricallyinduced dilatational strains of the MFC actuators used inthis study can be accomplished in a manner similar to thetypical modeling of a monolithic piezoelectric wafer,namely,
eE1 ¼ d11
DVDx1
eE2 ¼ d12
DVDx1
cE12 ¼ 0
ð1Þ
where d11 and d12 are the effective piezoelectric constantsfor the MFC actuator, DV is the voltage applied to theMFC actuator, and Dx1 is the electrode spacing. The termeffective is used to describe the piezoelectric constants be-cause the interdigitated electrodes create a nonuniformelectric field in the actuator, but by using DV/Dx1, an aver-age, or effective, electric field in the actuator is considered;as with properties of other composites, the electromechan-ical properties of the MFC actuator are ‘‘smeared’’ withinthe volume of the actuator. In the above strain notation,and throughout this paper, the principal material coordi-nate system nomenclature standard to fiber-reinforcedcomposite materials is used. That is, the material coordi-nate system is denoted by the subscripts 1, 2, and 3, andthe global coordinate system is denoted by x, y, and z, asshown in Fig. 1.
3. Model development
The analytical model that was developed for thisstudy was divided into three parts: cooling the curedlaminate, bonding the actuator to the laminate, andapplying voltage to the laminate-actuator combination.As in [1], in accordance with the requirements of theRayleigh–Ritz technique, the current model relied onusing good approximations of the displacement fields inconjunction with an extension of classical lamination the-ory by including geometric nonlinearities in the strain–displacement relations. Stationary values of the totalpotential energy of the laminate are sought. It wasbelieved that a 10-parameter approach could be usedfor predicting the shapes of the laminate-actuator combi-nation. Accordingly, in each of the three steps, the threecomponents of displacement of the geometric midsurfaceof the laminate, the reference surface in this analysis,were approximated by
u0 ¼ c5xþ 1
3c6x3 þ c7xy2 � 1
6c2
1x3 � 1
8c1c3x4 � 1
40c2
3x5
v0 ¼ c8y þ c9x2y þ 1
3c10y3 � 1
6c2
2y3 � 1
8c2d4y4 � 1
40c2
4y5
w0 ¼ 1
2c1x2 þ c2y2� �
þ 1
6c3x3 þ c4y3� �
ð2Þwhere u, v, and w are the displacements in the x-, y-, and z-directions, respectively. The 10 parameters c1 through c10
are unknown, but to-be-determined, coefficients. Becauseof symmetry requirements on the displacement, the third-order w-displacement polynomial could not be used to rep-resent the w-displacement field over the entire laminate.However, a quarter-symmetry argument was used to allowthe use of this polynomial to represent the w-displacementfield, and the total potential energy was calculated overonly one-quarter of the laminate. It should also be men-tioned that, consistent with Eq. (2), the laminate was con-sidered to be fixed at the geometric center, and with noexternal applied loads.
2444 M.R. Schultz et al. / Composites Science and Technology 66 (2006) 2442–2448
3.1. Step I: laminate cooling from cure
The first step of the model was to determine the initialcooled shape of the two-layer [0/90]T laminate. Becausethis step would not include the MFC actuator, and noforces were applied to the laminate, only thermal effectswithin the laminate needed to be considered. Accordingly,the total potential energy, P, used in Step I was the strainenergy of the cooled laminate, P1, and is given by
P ¼ P1 ¼ 41
2
Z Lx2
0
Z Ly2
0
Z z2
z0
rx � rTx
� �ex þ ry � rT
y
� �ey
h
þ rxy � rTxy
� �cxy
idxdy dz
!ð3Þ
where the rs are the stresses in the laminate, the rTs are theso-called thermally induced stresses, the es and cxy are thestrains, and Lx and Ly are the sidelengths of the laminatewhen flat as depicted in Fig. 1. Also as shown in Fig. 1,the thickness coordinates z0 and z2 are the coordinatesfor the bottom of the 0� graphite–epoxy layer and thetop of the 90� graphite–epoxy layer, respectively.
The stresses in Eq. (3) are given by
rx ¼ Q11ex þ Q12ey þ Q16cxy � rTx
ry ¼ Q12ex þ Q22ey þ Q26cxy � rTy
rxy ¼ Q16ex þ Q26ey þ Q66cxy � rTxy
ð4Þ
where the Qs are the transformed reduced stiffnesses of thegraphite–epoxy layers and the thermally induced stressesare given by
rTx ¼ Q11e
Tx þ Q12e
Ty þ Q16c
Txy
rTy ¼ Q12e
Tx þ Q22e
Ty þ Q26c
Txy
rTxy ¼ Q16e
Tx þ Q26e
Ty þ Q66c
Txy
ð5Þ
with
eTx ¼ axDT
eTy ¼ ayDT
cTxy ¼ axyDT ¼ 0
ð6Þ
The as are the coefficients of thermal deformation of thelayer and DT is the temperature change from the cure tem-perature to room temperature, here, the operating tempera-ture. Note that Q16, �Q26, and axy are retained in theformulation, even though they are zero for 0� and 90� layers.
For small strains and moderate rotations, the midsur-face strains and curvatures in the laminate are given bythe von Karman approximation to the more generalstrain–displacement relations, namely,
e0x ¼
ou0
oxþ 1
2
ow0
ox
� �2
j0x ¼ �
o2w0
ox2
e0y ¼
ov0
oyþ 1
2
ow0
oy
� �2
j0y ¼ �
o2w0
oy2
c0xy ¼
ou0
oyþ ov0
oxþ ow0
oxow0
oyj0
xy ¼ �2o2w0
oxoy
ð7Þ
According to the Kirchhoff hypothesis, the strains as afunction of the thickness location, z, are given by
ex ¼ e0x þ zj0
x
ey ¼ e0y þ zj0
y
cxy ¼ c0xy þ zj0
xy
ð8Þ
By using Eq. (2) in Eqs. (7) and (8) and, in turn, substitutingthese results into Eqs. (4) and (3), taking into account Eqs.(5) and (6), and carrying out the integrations of Eq. (3), thetotal potential energy is reduced to an algebraic equation interms of material properties, geometry, and the coefficientsc1 through c10. These coefficients are determined by solvingthe 10 simultaneous nonlinear algebraic equations that re-sult from equating to zero the first variation of total poten-tial energy with respect to these coefficients, namely,
oPoci¼ 0 i ¼ 1; n ð9Þ
Stability of the solution is determined by checking the po-sitive definiteness of the 10 · 10 matrix associated with thesecond variation of the total potential energy with respectto these 10 coefficients. The algebraic manipulations, inte-grations, and differentiations in this and the following stepswere all accomplished with the aid of Mathematica [6].
3.2. Step II: bonding the MFC actuator to the laminate
The MFC actuator was to be bonded to the laminate byplacing the laminate and actuator, with adhesive, inside avacuum bag, and using a vacuum pump to evacuate thebag to pull the laminate and the actuator into contact untilthe adhesive cured. The result was that the contacting sur-faces of the actuator and the laminate would develop thesame curvature during the bonding process, the laminatelosing some curvature and the actuator gaining curvature.This common curvature would initially be unknown. StepII of the model was developed to simulate this vacuumbonding technique, i.e., to determine the shape of the lam-inate-actuator combination resulting from the vacuumbonding process. This resulting shape of the laminate-actu-ator combination when the adhesive had cured will bereferred to as the first actuator-added shape. With the actu-ator bonded to the laminate, there would be anotherunknown shape to which the laminate-actuator combina-tion could be snapped. This other shape will be found inStep III, and will be referred to as the second actuator-
added shape.The total potential energy of the laminate and MFC
actuator bonded to it in Step II was the sum of the strainenergy from Eq. (3), P1, and the strain energy of the actu-ator, which is given as
P2¼41
2
Z LMFCx
2
0
Z LMFCy
2
0
Z z3
z2
ra2x ea2
x þra2y ea2
y þra2xy c
a2xy
h idxdydz
0@
1A
ð10Þ
M.R. Schultz et al. / Composites Science and Technology 66 (2006) 2442–2448 2445
where ra2x , ra2
y , and ra2xy and ea2
x , ea2y , and ca2
xy are the stres-ses and strains in the MFC actuator due to the bondingprocess. As shown in Fig. 1, the inplane dimensions ofthe actuator when flat are LMFC
x and LMFCy . The thickness
coordinates z2 and z3 define the thickness of the actuator.Until the adhesive began to cure, the actuator could sliprelative to the laminate, so the stresses created in theactuator due to the bonding process would be just bend-ing stresses. The stress–strain relations for the bondedactuator are
ra2x ¼ Q11e
a2x þ Q12e
a2y þ Q16c
a2xy
ra2y ¼ Q12e
a2x þ Q22e
a2y þ Q26c
a2xy
ra2xy ¼ Q16e
a2x þ Q26e
a2y þ Q66c
a2xy
ð11Þ
For simplification, it is assumed that the curvatures of themidsurface of the laminate and the midsurface of the actu-ator (rather than the curvatures of the contacting surfaces)are the same. The strains in the actuator may then beapproximated as
ea2x ¼ z� z3 � z0
2
� �� �j0
x
ea2y ¼ z� z3 � z0
2
� �� �j0
y
ca2xy ¼ z� z3 � z0
2
� �� �j0
xy ¼ 0
ð12Þ
where the js are not known, but will be the same in theactuator and the laminate. The displacement fields of theform of Eq. (2) are again assumed to be valid. For StepII, the total potential energy of the laminate-actuator com-bination is given as
P ¼ P1 þP2 ð13ÞIt should be noted in Eq. (13) that, in the total potentialenergies of the laminate and actuator, the work termsdue to the stresses between the laminate and actuator can-cel each other. The coefficients c1 through c10 for this stepare again determined by finding stationary values of P, asin Eq. (9), and stability is checked by examining the secondvariation.
It is important to again delineate at this point the con-sequences of bonding the actuator to the laminate. Theanalysis shows that when cooled, the two-layer [0/90]Tcross-ply laminate considered could have either one stableequilibrium shape, or it could have an unstable saddleshape and two stable cylindrical shapes, depending onthe dimensions of the laminate. In the latter case, thetwo cylindrical shapes would have equal and oppositeradii of curvature and perpendicular generating axes.The actuator could be bonded to the cooled laminatewhen the laminate was in either of these two cylindricalshapes. As a result of bonding the actuator to the laminatewith the laminate in a particular shape, that particularshape would change, and the shape to which it could besnapped to would also change. The shape resulting fromStep II, the first actuator-added shape, would be the initialshape for Step III.
3.3. Step III: actuation of the MFC
Step III of the model considered the laminate-actuatorcombination with no slipping between the laminate andthe actuator, and included the deformation due to voltageapplied to the actuator. The initial shape was to be speci-fied by the coefficients c1 through c10 associated with theshape determined from Step II; for Step III, these knowncoefficients were renamed ci
1 through ci10, respectively.
The superscript ‘i’ denotes ‘initial.’ The total potentialenergy of the model was again broken into the laminatecontribution, again P1, and the actuator contribution,P3. The contribution to the total potential energy fromthe MFC actuator in Step III is written as
P3 ¼ 41
2
Z LMFCx
2
0
Z LMFCy
2
0
Z z3
z2
�ra3
x � rEx
�ea3
x þ ra3y � rE
y
� �ea3
y
h0@
þ ra3xy � rE
xy
� �ca3
xy
idxdy dz
1A ð14Þ
where the ea3s and ca3xy are the strains in the actuator and are
to be discussed below. The midsurface of the graphite–epoxy laminate remains the reference surface. The ra3sare the stresses in the actuator and are given by
ra3x ¼ Q11e
a3x þ Q12e
a3y þ Q16c
a3xy � rE
x
ra3y ¼ Q12e
a3x þ Q22e
a3y þ Q26c
a3xy � rE
y
ra3xy ¼ Q16e
a3x þ Q26e
a3y þ Q66c
a3xy � rE
xy
ð15Þ
The rEs are piezoelectrically induced stresses and are givenby
rEx ¼ Q11e
Ex þ Q12e
Ey þ Q16c
Exy
rEy ¼ Q12e
Ex þ Q22e
Ey þ Q26c
Exy
rExy ¼ Q16e
Ex þ Q26e
Ey þ Q66c
Exy
ð16Þ
where the eEs and cExy are piezoelectrically induced strains.
The strains in the actuator, ea3s and ca3xy , are given by
ea3x ¼ ex � es
x
ea3y ¼ ey � es
y
ca3xy ¼ cxy � cs
xy
ð17Þ
where the es and cxy are Kirchhoff-like strains that are zeroat the reference surface (i.e., at the midplane of the lami-nate) and vary linearly with thickness. The ess and cs
xy areconsidered shift strains and account for the discontinuousthrough-thickness distributions of the strains due to theroom-temperature bonding of the actuator to the laminate,and allow the strain in the actuator to be defined by refer-ence-surface strains and curvatures. To explain, after theactuator is bonded to the laminate, the strains in the lam-inate would be due to cooling and due to bending fromadding the actuator. The strains in the actuator would bedue only to bending. As a result, the strain distributionsthrough the thickness of the laminate and actuator would
2446 M.R. Schultz et al. / Composites Science and Technology 66 (2006) 2442–2448
not be continuous. When the laminate-actuator combina-tion is snapped to the second actuator-added shape, thelaminate and actuator would both have extensional andbending strains; the strain distributions would continue tobe discontinuous between the laminate and actuator. Withactuation, it was assumed, in the spirit of the Kirchhoffhypothesis, that the strain increments in the laminate andactuator would be continuous through the thickness. As aresult, the profiles of strains through the laminate and actu-ator would remain discontinuous. To account for this dis-continuity, yet have one set of coefficients, i.e., c1 throughc10, to define the strains within the laminate and the actuatordue to the application of voltage, the shift of strain mea-sures was necessary. The shift strains are thus given by
esx ¼ ei0
x þz3 � z0
2
� �ji0
x
esy ¼ ei0
y þz3 � z0
2
� �ji0
y
csxy ¼ ci0
xy þz3 � z0
2
� �ji0
xy
ð18Þ
where the ei0s and ci0xy are the initial strains and the ji0s are
the initial curvatures, i.e., the strains and curvatures at thereference surface from Step II. These initial strains and cur-vatures are known.
The total potential energy for the actuation portion ofthe model is thus
P ¼ P1 þP3 ð19ÞThe shapes, actuated or unactuated, of the laminate-actua-tor combination are determined by solving for c1 throughc10 by equating to zero the first variation of the total poten-tial energy of Eq. (19). Stability is checked by examining atthe second variation.
4. Comparison of model with experiments
For this study, a [0/90]T laminate, fabricated from AS4/3502 graphite–epoxy prepreg, was cured flat and thencooled to room temperature. A parameter study using the
Table 1Material properties and other parameters for analytical model
Property AS4/3502(layer 1)
E1 (GPa) 132.0E2 (GPa) 9.798G12 (GPa) 5.112m12 0.2990a1 (1/�C) �0.04156 · 10�6
a2 (1/�C) 23.77 · 10�6
d11 (le/(kV/mm))a
d12 (le/(kV/mm))a
Thickness (m) 133.8 · 10�6
Lx (m) 0.15Ly (m) 0.15Dx1 (m)DT (�C) �11
a Determined by calibration with cantilevered aluminum plate.
developed analytical model was conducted to determinethe dimensions of the laminate, i.e., Lx and Ly, that couldbe snapped by the actuator. (The performance of the MFCactuator had been calibrated in a separate step using a can-tilevered aluminum plate.) As determined by the parameterstudy, the laminate was cut to 150 · 150 mm. The dimen-sions of the laminate and curvatures of the two cylindricalshapes were measured. An MFC actuator was centrallybonded to one side of the laminate to form the [0/90/0MFC]T laminate-actuator combination.
The measurements of the laminate affected the parame-ters for the model as follows: when the curvatures of thelaminate were measured, the major curvatures of the twostable shapes were not of the same magnitude. It wasassumed that uneven resin bleed during manufacture wasresponsible for the unequal curvatures. In the model, itwas found that by increasing the thickness of one layerand decreasing the thickness of the other layer a corre-sponding amount, the slightly different curvatures couldbe predicted quite accurately. Since the total thickness ofthe laminate was a measured quantity, the assumption ofthe layers having an equal but opposite thickness changewas justified. It was also found that, over time, the laminatelost some curvature, presumably due to moisture absorp-tion. The DT used in the model, in Eq. (6), was reducedto account for this loss of curvature. The numerical valuesof the specific properties that were used in the analyticalmodel are given in Table 1.
To demonstrate the snap-through event, actuationexperiments were performed. The experimental setup con-sisted of a voltage supply, voltage amplifier, strain gageamplifier/conditioner, voltmeter, a LabVIEW data acquisi-tion system on a personal computer, and the laminate-actu-ator combination with a strain gage bonded to the sideopposite the actuator. The output from the strain gagewas used to determine when the snap through eventoccurred. To perform the experiment, the laminate-actua-tor combination was snapped, by hand, to the second actu-ator-added shape, Fig. 2b. Since the analyses of the model
AS4/3502(layer 2)
MFC actuator(active portion)
128.0 299.608 184.895 5.30.3009 0.285.538 · 10�9
24.78 · 10�6
281�111
138.2 · 10�6 290. · 10�6
18 0.0857318 0.05715
1.0668 · 10�3
7
Fig. 2. Experimental and predicted stable shapes of laminate-actuator combination with zero applied voltage: (a) experimental as-bonded shape; (b)experimental second stable shape; (c) predicted as-bonded shape; (d) predicted second stable shape.
0 200 400 600 800 1000 1200 1400
-2.0
-1.5
-1.0
-0.5
0.0
0.5
c 1 (m
-1)
Applied Voltage, ΔV (V)
stable unstable
snap through
0 200 400 600 800 1000 1200 1400
-6
-5
-4
-3
-2
-1
0
1
c 2 (m
-1)
Applied Voltage, ΔV (V)
stable unstable
snap through
0 200 400 600 800 1000 1200 1400
-20
0
20
40
60
80
100
120
c 3 (m
-2)
Applied Voltage, ΔV (V)
stable unstable
snap through
0 200 400 600 800 1000 1200 1400
-50
-40
-30
-20
-10
0
10
c 4 (m
-2)
Applied Voltage, ΔV (V)
stable unstable
snap through
Fig. 3. Coefficients c1–c4 vs. applied voltage, showing existence of snap through.
M.R. Schultz et al. / Composites Science and Technology 66 (2006) 2442–2448 2447
2448 M.R. Schultz et al. / Composites Science and Technology 66 (2006) 2442–2448
assumed the edges of the laminate-actuator combinationwere free of any specified force resultants or displacements,the laminate-actuator combination was suspended, like apendulum, by the strain gage wire when conducting thesnap-through experiments. The voltage was increased fromzero until the laminate-actuator combination snapped to ashape similar to that of Fig. 2a; the voltage was thendecreased to zero and the laminate-actuator combinationassumed the shape given by Fig. 2a. This procedure wasrepeated six times with good repeatability and an averagesnapping voltage of 1695 V.
With the adjustments to the layer thicknesses and DT,which were discussed above, the model was used to predictthe behavior of the laminate as the voltage was varied. Thevalues of c1, c2, c3, and c4 were computed as a function ofthe voltage increase. These relationships are shown inFig. 3, the solid lines represent stable equilibrium shapesand the dashed lines represent the unstable equilibriumshapes. Recall, from Eq. (2), that these coefficients describethe out-of-plane displacement of the laminate-actuatorcombination. As shown in Fig. 3, for a given voltage level,each of these coefficients has either three solutions, or onesolution. At voltage levels below 1262 V there are threesolutions. For voltages greater than 1262 V, there is justone solution for each of the coefficients. Consider the lam-inate-actuator combination at zero applied voltage in theshape similar to Fig. 2b, i.e., c1 � � 1 m�1 andc3 � +95 m�2, while c2 and c4 are both approximately zero.As the voltage is increased from zero, both c1 and c3
increase in magnitude, while c2 and c4 remain close to zero.At 1262 V, the solution exhibits limit-point behavior, and afurther increase in voltage results in the solution jumpingto a configuration with a large curvature in the y-directionand little curvature in the x-direction, i.e., c1 � 0,c2 � �5.5 m�1, c3 � 0, c4 � �45 m�2. With a return ofthe voltage to zero, the room-temperature equilibriumshape similar to Fig. 2a results.
The two predicted room-temperature shapes of the lam-inate-actuator combination are shown along with theactual shapes in Fig. 2. Though this figure illustrates onlya qualitative comparison, which is quite good, the quanti-tative comparison is good also. Considering the simplicityof the model, the predicted snapping voltage of 1262 V alsocompares fairly well with the snapping voltage measured inthe experiments of 1695 V.
5. Concluding remarks
A new concept for morphing structures using active con-trol of the snap-through behavior of a bistable unsymmet-ric crossply graphite–epoxy laminate was presented herein.The principal advantage of the active bistable morphing
concept is that statically stable shapes are the operationalshapes, and power to the actuation systems is only requiredwhen transforming the structure from one shape toanother. In this study, a Rayleigh–Ritz method was devel-oped to model the behavior of a representative active bista-ble laminate, and used to design an experimental testspecimen. The experimentally measured stable shapes ofthe laminate-actuator combination were predicted verywell using the model, and the predicted actuation voltageneeded to initiate the snap-through behavior was reason-ably close to the experimental value. This demonstratesthat the analytical approach used here suitably capturesthe switching behavior of simple active bistable compositelaminates. More complex, and interesting, geometries maybe studied by extending the present work in several ways.The most immediate extension is to include the presenceof an additional actuator on the opposite side of the lami-nate. This second actuator could be used to reset the lam-inate to its initial stable shape. The number of appliedactuators, actuator dimensions and shapes, and actuatorpositioning can also be studied. Changing the scale of theconcept, e.g., changing laminate side length dimensionsand number of layers, is also worth pursuing. Laminateswith fiber angles and actuator orientations other than 0�and 90�, which can permit twist curvature shape change,should also be investigated. Many of these alternate geom-etries can be examined using variants of the simple meth-ods presented here, although more detailed modelingtechniques, such as finite-element analysis, may be neededto capture important local effects that occur in these morecomplicated active structures.
Acknowledgments
Mr. Robert Simonds, Prof. Dwight Viehland, and Dr.Jie-Fang Li of Virginia Tech are to be thanked for helpingwith experimental aspects of this study.
References
[1] Hyer MW. Calculations of the room-temperature shapes of unsym-metric laminates. J Compos Mater 1981;15:296–310.
[2] Hyer MW. The room-temperature shapes of four-layer unsymmetriccross-ply laminates. J Compos Mater 1982;16:318–40.
[3] Dano M-L, Hyer MW. Thermally induced deformation behavior ofunsymmetric laminates. Int J Solids Struct 1998;35(17):2101–20.
[4] Hufenbach W, Gude M. Analysis and optimisation of multistablecomposites under residual stresses. Compos Struct 2002;55(3):319–27.
[5] Wilkie WK, Bryant RG, High JW, Fox RL, Hellbaum RF, Jalink A,Jr., et al. Low-cost piezocomposite actuator for structural controlapplications. In: SPIE’s 7th annual international symposium on smartstructures and materials, Newport Beach (CA), March 5–9, 2000.
[6] Wolfram S. The mathematica book. New York: Wolfram-Media,Champaign-Urbana and Cambridge University Press; 1999.