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Thin-Walled Structures 46 (2008) 689–701 Concepts for morphing airfoil sections using bi-stable laminated composite structures Cezar G. Diaconu, Paul M. Weaver , Filippo Mattioni Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK Received 13 July 2007; received in revised form 19 October 2007; accepted 15 November 2007 Available online 31 December 2007 Abstract The present paper investigates the potential of using bi-stable laminated composite structures for morphing an airfoil section. The objective of the paper is to identify geometries and lay-ups of candidate configurations that offer multiple stable shapes for the airfoil section. Carbon-fiber laminated composites with non-symmetric laminate configurations are used for morphing the airfoil section. Thermal curing is used to induce residual stresses into the structure in order to achieve bi-stability. Three concepts that focus on morphing a flap-like structure and the camber and chord of an airfoil section are proposed. Several geometries and laminate configurations are investigated using finite element nonlinear static analysis. The magnitude of loads required to actuate the airfoil section between the stable shapes is evaluated. The impact of manufacturability on producing viable morphing mechanisms within the airfoil section is also discussed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Morphing compliant structures; Bi-stable laminated composites; Residual thermal stresses; Nonlinear finite element analysis; Large displacements; Unsymmetric laminates 1. Introduction Morphing structures are structures that change shape or state in order to change their operating characteristics or as a response to changes in the environment conditions. Bi- stable or multi-stable structures are good candidates to be used as morphing structures because of their ability to remain in natural equilibrium after a shape change occurs [1]. One can actuate such structures from one stable shape to another using devices such as piezoceramic [1,2] or shape memory alloy [3,4] actuators. Although multi-stability can be achieved with traditional isotropic materials [5], for morphing aerospace structures, fiber-reinforced laminated composites seem to be better suited because of their superior mechanical properties. Moreover, these laminated composites allow one to design their mechanical properties for a particular application by tailoring their laminate configuration. A non-symmetri- cally laminated composite plate can take multiple cylind- rical shapes when cooled from an elevated temperature to the room temperature. This structure can snap-through from one cylindrical shape to another by applying moments along opposite edges of the laminate. Hyer [6] showed that the bi-stability phenomenon is a coupling between the residual stresses induced by cooling and the geometric nonlinearities given by the large out-of-plane deflections that appear within the structure. It is noted that, for materials studied, bi-stability may be lost for tempera- tures greater than, say 80 1C, and that bi-stability is enhanced for colder temperatures. Thus, the study suggests that such bi-stable structures can be used in real morphing airfoils applications, where, for typical flight conditions, temperatures may range from approximately 60 to 40 1C. Various theoretical and experimental studies were carried out in order to understand and predict the behavior of the bi-stable composite structures [7–11]. Most pre- dictive models are based on using Rayleigh–Ritz mini- mization of the total potential energy in conjunction with polynomial approximations of the displacements or of the mid-plane strains. These models can only be used for simple geometries such as square or rectangular plates with ARTICLE IN PRESS www.elsevier.com/locate/tws 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.11.002 Corresponding author. Tel.: +44 117 928 7698; fax: +44 117 92 72771. E-mail address: [email protected] (P.M. Weaver).
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Concepts for Morphing Airfoil Sections Using Bi-stable Laminated Composite Structures

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Page 1: Concepts for Morphing Airfoil Sections Using Bi-stable Laminated Composite Structures

ARTICLE IN PRESS

0263-8231/$ - se

doi:10.1016/j.tw

�CorrespondE-mail addr

Thin-Walled Structures 46 (2008) 689–701

www.elsevier.com/locate/tws

Concepts for morphing airfoil sections using bi-stablelaminated composite structures

Cezar G. Diaconu, Paul M. Weaver�, Filippo Mattioni

Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK

Received 13 July 2007; received in revised form 19 October 2007; accepted 15 November 2007

Available online 31 December 2007

Abstract

The present paper investigates the potential of using bi-stable laminated composite structures for morphing an airfoil section. The

objective of the paper is to identify geometries and lay-ups of candidate configurations that offer multiple stable shapes for the airfoil

section. Carbon-fiber laminated composites with non-symmetric laminate configurations are used for morphing the airfoil section.

Thermal curing is used to induce residual stresses into the structure in order to achieve bi-stability. Three concepts that focus on

morphing a flap-like structure and the camber and chord of an airfoil section are proposed. Several geometries and laminate

configurations are investigated using finite element nonlinear static analysis. The magnitude of loads required to actuate the airfoil

section between the stable shapes is evaluated. The impact of manufacturability on producing viable morphing mechanisms within the

airfoil section is also discussed.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Morphing compliant structures; Bi-stable laminated composites; Residual thermal stresses; Nonlinear finite element analysis; Large

displacements; Unsymmetric laminates

1. Introduction

Morphing structures are structures that change shape orstate in order to change their operating characteristics or asa response to changes in the environment conditions. Bi-stable or multi-stable structures are good candidates to beused as morphing structures because of their ability toremain in natural equilibrium after a shape change occurs[1]. One can actuate such structures from one stable shapeto another using devices such as piezoceramic [1,2] or shapememory alloy [3,4] actuators.

Although multi-stability can be achieved with traditionalisotropic materials [5], for morphing aerospace structures,fiber-reinforced laminated composites seem to be bettersuited because of their superior mechanical properties.Moreover, these laminated composites allow one to designtheir mechanical properties for a particular application bytailoring their laminate configuration. A non-symmetri-cally laminated composite plate can take multiple cylind-

e front matter r 2007 Elsevier Ltd. All rights reserved.

s.2007.11.002

ing author. Tel.: +44 117 928 7698; fax: +44 117 92 72771.

ess: [email protected] (P.M. Weaver).

rical shapes when cooled from an elevated temperature tothe room temperature. This structure can snap-throughfrom one cylindrical shape to another by applyingmoments along opposite edges of the laminate. Hyer [6]showed that the bi-stability phenomenon is a couplingbetween the residual stresses induced by cooling and thegeometric nonlinearities given by the large out-of-planedeflections that appear within the structure. It is noted that,for materials studied, bi-stability may be lost for tempera-tures greater than, say 80 1C, and that bi-stability isenhanced for colder temperatures. Thus, the study suggeststhat such bi-stable structures can be used in real morphingairfoils applications, where, for typical flight conditions,temperatures may range from approximately �60 to 40 1C.Various theoretical and experimental studies were

carried out in order to understand and predict the behaviorof the bi-stable composite structures [7–11]. Most pre-dictive models are based on using Rayleigh–Ritz mini-mization of the total potential energy in conjunction withpolynomial approximations of the displacements or of themid-plane strains. These models can only be used forsimple geometries such as square or rectangular plates with

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ARTICLE IN PRESSC.G. Diaconu et al. / Thin-Walled Structures 46 (2008) 689–701690

free boundary conditions at edges but have the advantagethat allow parametric studies to be carried out for designpurposes [12–14]. Finite element analysis (FEA) is bettersuited for predicting behavior of bi-stable or multi-stablestructures with more complex geometric configurations andboundary conditions. Due to the intrinsic nonlinearity ofthe snap-through phenomenon in bi-stable or multi-stablestructures, the finite element code have to be carefullycoaxed in order to predict behavior for each particularconcept. This makes the FEA computationally expensiveand time consuming [8].

Most of the studies carried out on bi-stable compositesfocused on structures with residual stresses induced bytemperature. However, it is known that the moisture orchemical shrinkage can be as important as the temperaturefor inducing residual stresses into the structure [2,15]. Sincethe working environment for such structures is not alwayscontrollable, some multi-stable structures were developedto be virtually free from temperature or moisture changes[16]. In these composite structures the residual stresses areintroduced by pre-stretching the fiber-reinforced laminatesduring manufacturing process.

Bi-stable and multi-stable structures are currently used fordeployable space structure applications [17–21]. A morphingconcept [1] was proposed based on bi-stable compositesactuated by devices such as piezoceramic or shape memoryalloy actuators. Also, bi-stable plates have been proposed asa high-lift device concept for UAV application and variablegeometry airfoil where the bi-stability of unsymmetricpatches is used to drive the shape change of a trailing edgemounted device [22]. However, so far, no practical applica-tion of this concept was reported in literature for morphingaeronautic structures such as wings or rotor blades.

The present paper studies the opportunity of using bi-stable laminated composite structures for morphing anairfoil section of typical aeronautic structures such as awing, helicopter rotor blade, fixed wing propeller or windturbine blade. The objective of the paper is to identifygeometry and lay-up of candidate configurations that offermulti-stability for an airfoil trailing edge. Three morphingconcepts are proposed. The aim is to assess their viability inan integrated airfoil in future work.

The first concept focuses on morphing the trailing edge ofthe airfoil section. The top and bottom surfaces of the airfoiltoward the trailing edge are made of laminated compositeswith non-symmetric laminate configurations in order tocreate a flap-like structure with multiple stable shapes.

The second concept consists of a bi-stable compositeplate inserted into the airfoil section in horizontal positionalong the chord of the airfoil. The leading edge of the bi-stable composite plate is clamped at its center to a verticalspar. On the other side, the trailing edge of the plate ishinged to a vertical web, which is also hinged to the airfoilsurfaces in order to allow relative movement of the skinsduring actuation. By actuating the bi-stable plate, thecamber of the airfoil section is morphed between twodifferent stable shapes.

The third concept consists of a bi-stable composite plate,which is inserted into the airfoil section in a verticalposition along its main spar. The bi-stable composite plateis connected to the spar. The airfoil section is composed oftwo separate parts: the leading part is connected to themain spar while the trailing part is connected to the bi-stable composite plate. The top and bottom surface of thetrailing part are allowed to slide inside the leading part ofthe airfoil. By actuating the bi-stable composite plate, thechord length or the airfoil section can be changed.For the three concepts, various geometries and laminate

configurations are investigated using nonlinear static FEA.The analysis is carried out using a multi-step approach inorder to induce residual stresses in the bi-stable laminatedcomposite structures and to analyze the behavior of theairfoil section. The magnitude and distribution of loadsrequired for actuating the airfoil section between the stableshapes is evaluated. The impact of manufacturability onproducing viable morphing mechanisms within the airfoilsection is also discussed.

2. Thermally induced bi-stable structures

Structures with multiple stable shapes can be obtainedby using laminated composites subjected to thermal loads.Due to the difference between the thermal expansioncoefficients in the principal directions, the thermal loadsgenerate directional deformations in unidirectional singlelayers. For structures made with non-symmetric laminateconfigurations, these deformations create thermally in-duced residual stresses that generate large out-of-planedisplacements. To analyze these large displacements and totake into account the multiple stable shapes one canconsider the geometric nonlinear terms within the strains–displacements relationship using von Karman plate theory

�x ¼ �0x � z

q2w0

qx2¼

qu0

qxþ

1

2

qw0

qx

� �2

� zq2w0

qx2,

�y ¼ �0y � z

q2w0

qy2¼

qv0

qyþ

1

2

qw0

qy

� �2

� zq2w0

qy2,

�xy ¼ �0xy � z

q2w0

qxqy¼

1

2

qu0

qyþ

qv0

qxþ

qw0

qx

qw0

qy

� �� z

q2w0

qxqy,

ð1Þ

where index 0 refers to the reference middle plane of thelaminate, ex, ey, exy are the total strains, �0x; �

0y; �

0xy are the

membrane strains, and u, v, w are the displacements in x, y, z

directions. In Eq. (1) one can identify the geometricnonlinear terms 1=2ðqw0=qxÞ2, 1=2ðqw0=qxÞ2 and1=2ðqw0=qxÞ ðqw0=qyÞ that appear in the definition of themembrane strains. The multiple stable shapes, which appearwhen the structure is subjected to thermal loads, can bedetermined by minimizing the total potential energy of thestructure defined as

P ¼Z

V

1

2�T Q ���T Q aDT

� �dV , (2)

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ARTICLE IN PRESSC.G. Diaconu et al. / Thin-Walled Structures 46 (2008) 689–701 691

where V is the structural volume, � is the strains vector, Q isthe stiffness matrix, a represents the coefficients of thermalexpansion and DT is the gradient of the thermal loads.When the total potential energy of the structure is minimizedby an analytic or numeric technique, multiple local minimacan be obtained. These local minima are obtained by settingthe first variation of the potential energy with respect to fieldvariables to be zero. It is also necessary to impose a stabilitycondition for these local minima by setting the secondvariation of the potential energy to be positive. Each of thestable local minima for the total potential energy corre-sponds to a stable shape for the displacement field of thestructure. Based on this approach, many analytic solutionswere proposed for analyzing the bi-stable shapes of simplerectangular plates. However, for complex structures it isnecessary to employ complex nonlinear FEA in order todetermine the bi-stable or multi-stable shapes and to studythe snap-through from one stable shape to another.

3. Airfoil model and FEA

In the present paper, as a sample structure for analysis,we chose a NACA 23012 airfoil [23] with two stiffeningspars as shown in Fig. 1 and with a chord of 680mm. Thisstructure was selected because it exhibits similar character-istics to existing rotor blades used in practice. Typically,morphing such a structure requires modification of itsairfoil camber and/or chord in order to change itsaerodynamic characteristics. In this study, it is assumedthat the region of the airfoil between the frontal spar andthe leading edge supports the main bending and torsionloads that appear on the structure during flight conditions.Since in practice this region is usually made very stiff, in theFEA model it is considered rigid and clamped. Thus,during analysis, attention focuses on morphing the trailingedge or the region behind the spar. In these regions theaerodynamic pressure that appears on the structure isignored in the present analysis. However, in practice themorphing structure should be designed not to be activatedby ambient aerodynamic loading. As the structure to bemorphed is relatively flexible, considering such aerody-

Fig. 1. NACA 23012 airfoil.

namic pressure on it would require a detailed aeroelasticanalysis which is beyond the objective of this study.The material used for the structure is a graphite/epoxy

laminated composite with the following properties for aunidirectional lamina:

E11 ¼ 130GPa; E22 ¼ 10GPa; u12 ¼ 0:3,

G12 ¼ G13 ¼ 4:4GPa; G23 ¼ 3:2GPa,

a11 ¼ �1:8� 10�8; a22 ¼ 3� 10�5, ð3Þ

where E11 and E22 are the Young’s moduli in longitudinaland transverse directions of the lamina while G12, G13 andG23 are the shear moduli. The a11 and a22 are thecoefficients of thermal expansion in longitudinal andtransverse directions of the lamina. The thickness of eachlamina is 0.125mm. For this material, which has amanufacturing temperature of 180 1C, bi-stability appearsfor temperatures lower than a critical temperature, say80 1C depending on the geometry of the structure, and isenhanced for colder temperatures. Since for typical flightconditions, temperatures may range from approximately�60 to 40 1C, it is expected that the thermally induced bi-stability for a morphing aero-structure will vary duringservice. In this study we consider thermal gradients of 180and 135 1C which correspond to service temperatures of 0and 45 1C, respectively.Due to the complex nature of the geometry, FEA is

employed using ABAQUS finite element code [24]. Theelements selected for analysis are S4R doubly curved shellelements with 4-nodes per element and reduced integration.Various meshes are used depending on the proposedmorphing concept. The meshes are chosen in order tohave a good compromise between precision and computingtime of the analysis.The analysis is carried out using a five steps approach:

1.

In the first step residual stresses are introduced by curingthe bi-stable structure from an elevated temperature toroom temperature. At the end of this step the structurewill take one stable shape that is different from the initialmanufacturing shape due to the induced residual stresses.

2.

Actuation loads are applied on the structure in thesecond step in order to make it snap-through from onestable shape to another.

3.

The loads are removed in the third step in order toensure that the new shape of the structure is stable.

4.

In the fourth step the actuation loads are applied in theopposite direction in order to make the structure snap-through and return to the first stable shape.

5.

In a similar manner with the third step, during the fifthstep the actuation loads are removed to ensure that thenew shape is stable.

During preliminary studies, three types of nonlinear FEAwere evaluated: Riks analysis, explicit dynamic analysisand static general analysis. Riks analysis has proven to berelatively unreliable because of its high dependence on the

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Fig. 3. Model of trailing edge box.

Table 1

Laminate configurations

Configuration Web Stable regions Bi-stable regions

1 [90/0]S [90/0]S [904/04]T2 [45/�452/45]AS [45/�452/45]AS [904/7454]T

a

aThe fiber distribution of this laminate configuration is shown in Fig. 4.

C.G. Diaconu et al. / Thin-Walled Structures 46 (2008) 689–701692

laminate configuration which required a cumbersome finetuning for each step in order to reach truly stable results.Explicit dynamic analysis did not show reliable resultsbecause the damping coefficients for the laminatedcomposite material are unknown. Finding such dampingcoefficients by a trial and error method proved to be timeconsuming and is beyond the purposes of this study. Staticgeneral analysis has proven to be the most robust andreliable analysis for the problem. Thus, static generalanalysis is used for further studies because of its robustnessand reliability.

4. Morphing concept for adaptive trailing edge

This concept focuses on morphing the trailing edge ofthe airfoil section using a flap-like structure with bi-stableshapes. To emulate a typical flap geometry, the structureshould rotate with an angle of 101 between the stable statesaround an axis which is at 15% of the chord from thetrailing edge. The bi-stability is thermally induced bycuring the whole structure with a gradient of DT ¼ 135 1Cbetween the elevated or manufacturing temperature andthe room or service temperature. The region between therear spar and the trailing edge is modified in such way thatthe bottom and upper surfaces are flat during manufactur-ing rather than slightly curved as is the case of a typicalNACA airfoil. This modification simplifies the manufac-turing and also the analysis. The region between theleading edge to the rear spar is considered clamped duringanalysis. Thus attention can focus on the rear area of theairfoil section, defined as a trailing edge box with thedimensions shown in Fig. 2. In this figure the width of theairfoil section is chosen to be 15% of the chord, that isW ¼ 102mm, and the rear spar is positioned at 30% of thechord from the trailing edge, that is 204mm. At the rearspar, the distances between the bottom and top surfacesand the plane containing the trailing edge normal to therear spar are given by the NACA 23012 airfoil [23].

Fig. 3 shows a more detailed model of the trailing edgebox that focuses on regions with various laminateconfigurations and on load distribution. As shown in thefigure, both the top and bottom surfaces are divided intotwo regions with equal areas halfway between the rear spar

Fig. 2. Dimensions of trailing edge box.

and the trailing edge. The stable regions are located nearthe rear spar and have symmetric laminate configurations,which are inherently stable. The stable regions should beflexible enough in order to allow the bi-stability phenom-ena to occur into the bi-stable regions. The bi-stableregions are located near the trailing edge and have laminateconfigurations that are inherently bi-stable and induce bi-stability phenomenon into the trailing edge box. A verticalweb located in the middle of the stable regions connects thebottom and top surfaces from the rear spar to the bordersof the bi-stable regions. The web is made of the samematerial as the top and bottom surfaces, is stable, and itsrole is to limit the displacement required to trigger thesnap-through from one stable shape to the other. At thelocation of the rear spar, the web, the top and the bottomsurfaces are considered clamped. Actuation loads areconcentrated loads in vertical direction applied in themiddle of the trailing edge on the top and bottom surfaceswhich are connected to which other by a rubber connector.The rubber connector allows a relative displacement of6mm between the two connecting points which is necessaryin order to allow bi-stability phenomena to occur into thestructure. During analysis, on the second step the actuationload is applied only on the top surface while on the fourthstep the actuation load is applied only on the bottomsurface. The magnitude of the actuation loads has to besufficiently large in order to trigger the snap-through fromone stable shape to another.Various laminate configurations were analyzed. Many of

these laminate configurations did not induce bi-stabilityinto the structure. The laminate configurations proposed ascandidates are given in Table 1. In this table, the subscriptsS and AS stand for laminate configurations which aresymmetric and, respectively, anti-symmetric with respect tothe middle plane of the laminate while the subscript Tstands for total through thickness laminate configurations.

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Fig. 4. Configuration [904/7454]T.

Fig. 5. Load displacement relationship for Configuration 1: (a) W ¼

102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

C.G. Diaconu et al. / Thin-Walled Structures 46 (2008) 689–701 693

Also, the bottom layer angle is the first angle shown in thelaminate configuration expressions. In Table 1, oneproposed configuration, that is Configuration 1, is a com-bination of symmetric cross-ply laminates for the web andthe stable region, and non-symmetric cross-ply laminatesfor the bi-stable regions. The other proposed configuration,that is Configuration 2, is a combination of anti-symmetricangle-ply laminates for the web and the stable regions, andnon-symmetric laminates made of +451, �451 and 901angles as shown in Fig. 4 for the bi-stable regions.

Parametric studies for various widths of the airfoilsection were carried out in order to evaluate and compareshape changes and predicted loads and to evaluate thepotential of the proposed candidate topologies andlaminate configurations. For the parametric studies, thevalues for the width of the airfoil section vary between 91.8and 306mm.

Figs. 5 and 6 show the relationship between theconcentrated actuation loads applied to make the structuresnap-through and the vertical displacement at the locationwhere the loads are applied for the width of the airfoilsection taking values of W=102, 204 and 306mm. Thecontinuous lines show the relationship between loads anddisplacements for the top skin while the dashed lines showthe relationship between loads and displacements for thebottom skin. When the actuation loads are zero, onecan identify the vertical displacements for the two stableshapes from the initial manufacturing position. Forconvenience, we define the first and the second stableshapes as having negative and, respectively, positive valuesfor the vertical displacements. One can understand thesnap-through phenomena between the two stable shapes byfollowing the lines in clockwise direction. The actuationloads are acting in a positive direction to make thestructure snap-through from the first to the second stableshape and in negative direction for opposite actuation. Thesnap-through loads can be identified as the loads for whichlarge displacements occur. In the figures, these largedisplacements appear as the regions where the lines arehorizontal.

In the figures, it can be seen that the curves relating loadsto displacements are different depending on the width ofthe airfoil section and on the laminate configuration. Also,the loads required to actuate the structure are differentdepending on the width value, the laminate configuration

and on the direction of the actuation. However, a certainpattern can be observed for the two laminate configura-tions. Thus, the required load to actuate the structure fromfirst stable shape to the second stable shape has a largerabsolute value than the required load for actuation inopposite direction. Also, the actuation load is higher forthe first stable shape than for the second stable shape. Thissuggests that first shape is more stable, that is, it has lessstrain energy than the second shape.

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Fig. 6. Load displacement relationship for Configuration 2: (a) W ¼

102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

Fig. 7. Actuation loads for given airfoil section widths.

Fig. 8. Lateral view of first stable shape for trailing edge box with

Configuration 1: (a) W ¼ 102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

C.G. Diaconu et al. / Thin-Walled Structures 46 (2008) 689–701694

When the two laminate configurations are compared, itcan be observed from the figures that the actuation loadsare larger for Configuration 1 than for Configuration 2.This suggests that, for Configuration 1, the structureis more bi-stable, that is, the strain energy required toactuate the structure is larger than that required forConfiguration 2.

Fig. 7 shows the relationship between the width of theairfoil section and the actuation loads. The actuation loadsare shown with continuous lines for Configuration 1 and

dashed lines for Configuration 2. It can be observed thatfor both Configurations 1 and 2 the actuation loads havemaximum absolute values for the width W ¼ 306mm inthe positive direction and for width W ¼ 204mm in thenegative direction. From this figure, a good compromisefor design seem to be obtained for width W ¼ 204 becauselarge actuation loads are obtained on positive and negativedirection for a rather small relative difference betweenthem.Regarding the shape changes for the trailing edge box,

Figs. 8–15 show the stable shapes for Configurations 1 and2 for the parametric studies regarding changes in the widthW of the airfoil section. Lateral and also isometric viewsare shown in the figures. For reference purposes, in thelateral view, the stable shapes are shown together with theinitial manufacturing shape. Moreover, for the sake ofsimilarity with traditional flapping structures we define the

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Fig. 9. Isometric view of first stable shape for trailing edge box with

Configuration 1: (a) W ¼ 102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

Fig. 10. Lateral view of second stable shape for trailing edge box with

Configuration 1: (a) W ¼ 102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

C.G. Diaconu et al. / Thin-Walled Structures 46 (2008) 689–701 695

flap angle a as the angle between the chord and the lineconnecting the trailing edge with the point where the chordmeet the vertical web as shown in Fig. 8a.

For Configuration 1, in Figs. 8 and 9, it can be seen thatthe first stable shape is almost identical for all the widthsand makes a flap angle of around 91 with the initial

manufacturing shape. Moreover, it can be observed thatfor the first stable shape the trailing edge of the airfoilsection is straight. On the other hand, for the second stableshape shown in Figs. 10 and 11, the trailing edge is curveddepending on the width of the airfoil section. Thus, thedisplacements of the trailing edge are much larger on thesides of the section than in the middle. Also, in the middleof the airfoil section, the flap angle between the initialmanufacturing shape and the second stable shape variesdepending on the airfoil section width between �0.31 forW=91.8mm, and �3.81 for W=306mm.For Configuration 2, in Figs. 13 and 15, the curvature of

the trailing edge is in the opposite direction whencompared with Configuration 1. Also, the curvature ismostly localized in the middle of the trailing edge.Moreover, this curvature can be observed for both stableshapes. However, the flap angle between the initialmanufacturing shape and the first stable shape takes valuesbetween around 7.91 for W=102mm, and 6.81 forW=306mm. On the other hand, for Configuration 2, inFigs. 14 and 15 it can be observed that the curvature alongthe trailing edge for the second stable shape is almost zeroand more pronounced curvatures are observed along theseparation line between the stable and bi-stable regions.Also, in the middle of the airfoil section, the flap anglebetween the initial manufacturing shape and the secondstable shape varies depending on the airfoil section widthbetween around 1.51 for W=102mm, and �4.31 forW=306mm.

5. Morphing concepts for adaptive airfoil section

For morphing the airfoil section, attention was paid onchanging the camber of the airfoil section and on changing

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Fig. 11. Isometric view of second stable shape for trailing edge box with

Configuration 1: (a) W ¼ 102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

Fig. 12. Lateral view of first stable shape for trailing edge box with

Configuration 2: (a) W ¼ 102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

C.G. Diaconu et al. / Thin-Walled Structures 46 (2008) 689–701696

the length of its chord. For the concepts described below,the residual stresses in the bi-stable structure wereintroduced by a thermal gradient DT=180 1C in order toincrease the actuation loads supported by the structure.

5.1. Camber change for airfoil section

The camber change for the airfoil section is realized byinserting a square bi-stable composite plate with the in-plane dimensions 234� 234mm into the airfoil section inhorizontal position along its chord as shown in Fig. 16.The width of the airfoil section is W ¼ 234mm in order tomatch the dimensions of the square bi-stable plate. The

leading edge of the bi-stable composite plate is clamped atits center to the spar of the airfoil section. In the FEAmodel, this clamped condition is modeled using a weldconnector. On the other side, the trailing edge of the plateis hinged to a vertical web, which is also hinged to theairfoil surfaces in order to allow relative movement of theskins during actuation. The top skin is allowed to slide overthe bottom skin during actuation. In the FEA model, thestructure is considered clamped ahead the spar because thisregion is usually filled with foam and very rigid. For thefirst 32mm behind the spar, the bottom skin is alsoclamped in order to ensure a proper second stable shape forthe airfoil section.The selection of the laminate configuration for the given

structure should be compliant in order to ensure bi-stability and also stiff in order to support the aerodynamicloads. Also, orthotropic laminate configurations werechosen in order to ensure that twisting of the airfoilsection due to anisotropy is eliminated. Thus, for the skinsabove and below the bi-stable plate, that is the regionbetween the spar and the web, an anti-symmetric angle-plyconfiguration with only eight layers, that is [45/�452/45]AS,is used to ensure flexibility for the structure. For all ofthe other parts of the structure, these are the spar, the web,and the skins ahead spar and behind web, an anti-symmetric configuration with 16 layers, that is [45/�452/45/90/02/90]AS, is used to ensure high stiffness. For the bi-stable plate, three non-symmetric cross-ply laminateconfigurations, [904/04]T, [906/06]T, and [908/08]T, werechosen in order to sustain high loads while keeping thelarge displacements of the plate within a desirable range.By actuating the bi-stable plate, the airfoil section is

morphed between two different stable shapes. The two

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Fig. 13. Isometric view of first stable shape for trailing edge box with

Configuration 2: (a) W ¼ 102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

Fig. 14. Lateral view of second stable shape for trailing edge box with

Configuration 2: (a) W ¼ 102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

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stable shapes are shown in Fig. 17 for the [08/908]T laminateconfiguration of the bi-stable plate. The first and secondstable shapes are superposed for comparison purposes. Thefirst stable shape is the original airfoil with the bi-stableplate inserted along the chord. For the second stableshapes, it can be observed that the top and bottom skinsbend above and below the bi-stable plate. The top skin also

slides over the bottom skin along the trailing edge.Moreover, the web behind the bi-stable plate is slightlymoved from its original vertical position. Similar stableshapes are obtained for the other laminate configurations.In the model, the loads used for actuating the structure

between the stable shapes are two concentrated loadsapplied vertically on the two leading corners of the bi-stable plate near the spar. Fig. 18 shows the relationshipbetween the actuation loads and the vertical displacementsfor the two corners of the bi-stable plate obtained from theFEA for each of the three laminate configurations. In thefigure, the continuous line with diamonds corresponds toeight layer [04/904]T, the dashed line corresponds to 12layers [06/906]T, and the continuous line corresponds to 16layers [08/908]T laminate configuration. In the figure, thesnap-through between the two stable shapes can beidentified in the form of almost horizontal regions for thethree curves. In these regions, for a very small increase inthe magnitude of the actuation loads can be observed alarge increase in the magnitude of displacements of thecorners of the plate. It can be seen that the plate snapsbetween the stable states for loads and strokes that dependon the laminate configuration of the plate. Thus, for thelaminate configurations [04/904]T, [06/906]T and [08/908]T,the actuation loads necessary to snap-through the platefrom the first stable shape to the second stable shape areapproximately �16.7, �30 and �42.2N, respectively. Alsofor these laminate configurations, the strokes or verticaldisplacements required to trigger the snap-trough areapproximately �27, �20 and �14mm, respectively. Theplate snaps back into the first stable state for actuation loadsand strokes that also depend on its laminate configuration.Thus, for the laminate configurations [04/904]T, [06/906]T and[08/908]T, the snap-through actuation loads are approximately

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Fig. 15. Isometric view of second stable shape for trailing edge box with

Configuration 2: (a) W ¼ 102mm; (b) W ¼ 204mm; and (c) W ¼ 306mm.

Fig. 16. Assembly for camber change.

Fig. 17. Lateral view of the stable shapes for camber change of airfoil

section.

Fig. 18. Loads relationship with displacements at actuation points.

Fig. 19. Actuation loads relationship with trailing edge displacements.

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3.2, 17.4 and 35.6N, respectively, and the strokes or verticaldisplacements required to trigger the snap-trough areapproximately 4.5, 9 and 10mm, respectively. These actua-tion loads have smaller absolute values than the actuationloads in opposite direction due to the fact that the plate isalready loaded with stresses resulting from the deformationof the rest of the structure.Fig. 19 shows the relationship between actuation load on

one of the leading corners of the bi-stable plate and thevertical displacement of the trailing edge for each of thethree laminate configurations. As in the previous figure, thecontinuous line with diamonds corresponds to eight layers[04/904]T, the dashed line corresponds to 12 layers [06/906]T,and the continuous line corresponds to 16 layer [08/908]Tlaminate configuration. Once again, the snap-through is

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Fig. 20. Assembly for chord length change.

Fig. 21. Loads relationship with displacements at actuation point.

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accompanied by a change of slope for the three curves and alarge increase of the trailing edge vertical displacement. Itcan be seen that the change of slope occurs for the actuationloads identified in Fig. 19. In Fig. 20, the curvescorresponding to the bi-stable plates with 12 and 16 layers,these are [06/906]T and [08/908]T, respectively, are ratherclose to each other which suggest similar behavior forthe two configurations. However, there are large differencesbetween these curves and the curve corresponding to the bi-stable plate with eight layers, that is [04/904]T. For the secondstable state, when the actuation loads are zero, the trailingedge displacements corresponding to bi-stable plates with[04/904]T, [06/906]T and [08/908]T laminate configurations are�46.15, �68.7 and �67.8mm, respectively.

5.2. Chord length change for airfoil section

The chord length change for the airfoil section is realizedby inserting a rectangular bi-stable plate into the airfoilsection in vertical position along its main spar as shown inFig. 20. The bi-stable plate has the length L ¼ 300mm, thewidth W ¼ 70mm and is connected at its corners tothe spar. During actuation between the two stable statesthe corners of the plate are allowed to slide horizontallyalong the spar. The airfoil section is composed of twoseparate parts: the leading part is fixed to the main sparwhile the trailing part is fixed to the center of the bi-stableplate. The top and bottom surface of the trailing part areallowed to slide along the chord inside the leading part ofthe airfoil.

Two cross-ply non-symmetric laminate configurationswith four and, respectively, six layers were consideredfor the bi-stable plate. These are [02/902]T and, respectively[03/903]T. These laminate configurations were chosen toensure that bi-stability occurs for the plate which has awidth of only 70mm. The configuration of the remainingstructure is less important for this analysis since, in thismodel, no other parts of the structure contribute elasticallyto its bi-stability.

By applying an actuating transverse load at the center ofthe bi-stable composite plate, the chord length or the airfoilcan be changed between two stable shapes. Fig. 21 showsthe relationship between the actuation load and thedisplacement of center of the bi-stable plate for the twolaminate configurations. For the four layer [02/902]Tlaminate configuration, the relationship is shown with a

continuous line while for the six layer [03/903]T laminateconfiguration, the relationship is shown with a dashed line.The displacements at the center of the plate are consideredzero when the structure is in its first stable shape and noloads are applied. When the load is applied toward thetrailing edge, the displacement increases little until a loadof approximately 10N for the four layer [02/902]T laminateconfiguration, and 8N for the six layer [03/903]T laminateconfiguration. For greater load values, the displacementsincreases with the loads are much larger because the platebegins to snap-through toward the second stable state. Inthe FEA model a maximum load of 12N was applied. InFig. 21 it can be seen that when the load is removed, theplate continues to snap-through based on its residualstresses and stabilizes into the second stable shape for amaximum displacement of approximately 107.5mm for thefour layer [02/902]T laminate configuration, and 87.5mmfor the six layer [03/903]T laminate configuration. When theload is applied in the opposite direction, the displacementreduces drastically due to the low bending stiffness of theplate. In the case of the four layer [02/902]T laminateconfiguration, when the load reaches its minimum of�12N most of the plate has buckled locally and is alreadyin the first stable state apart from the localized boundarylayers at the longitudinal sides of the plate. When the loadis removed, the plate continues to snap-through andstabilizes itself into the first stable state, which is the initialposition at 0mm, due to the larger residual stresses at thecenter of the plate than at the boundary layers. In the caseof the six layer [03/903]T laminate configuration, when theload reaches its minimum of �12N the plate had snappedcompletely. In this case, when the load is removed, theplate stabilizes itself into the first stable state, which is theinitial position at 0mm.The two stable shapes of the structure are shown for the

four layer [02/902]T laminate configuration in Fig. 22 froma lateral view and in Fig. 23 from an isometric perspective.Similar stable shapes but with a smaller relative distancebetween their trailing edges are achieved for the six layer[03/903]T laminate configuration. In Fig. 22b the second

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Fig. 22. Lateral shape view for chord length change: (a) first stable shape

and (b) second stable shape.

Fig. 23. Isometric shape view for chord length change: (a) first stable

shape and (b) second stable shape.

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stable shape is plotted over the first stable shape forcomparison purposes. One can see the gain in chord lengthand, respectively, the aerodynamic surface that can beachieved with this morphing concept. Also, in Fig. 23b itcan be observed that, in the second stable shape, the bi-stable plate has a new role, that is, the bi-stable plate actsas an internal stiffener for the airfoil section because half ofits fibers are perpendicular to the chord and in the directionof the aerodynamic forces transverse to the airfoil. Thus,this morphing concept can offer an advantage over otherexisting solutions for increasing the chord of the airfoilbecause of the dual functionality of the bi-stable plate thatacts as an actuator and also as a stiffener.

6. Summary and discussion

Three morphing concepts for an airfoil section wereinvestigated. These concepts take advantage of the bi-stability phenomenon that is induced through thermalresidual stresses in thin walled laminated compositestructures. The results show the performances and thelimitations of these bi-stable composite structures when thebi-stability is induced by residual stresses due to thermalcuring. Using this approach, the displacements necessaryfor many practical applications are achievable. However,these composite structures are highly compliant and theloads requested by most of the practical applications mightnot be achievable solely by the composite structures andadditional locking mechanisms might be required. Also,since the residual stresses are induced by thermal curing,the performance of these structures is highly affected byenvironmental factors such as temperature and moisture.Considering temperature effects first, the bi-stability isenhanced for cold conditions (due to large temperaturechange from the gelling temperature) whilst for hottemperatures it is possible that the structure’s bi-stablecharacteristics will diminish and in the extreme, maydisappear. Note that moisture absorption effects alsocounter the effect of colder conditions and diminish theeffects of bi-stability. Nevertheless, the concepts proposedin this study can be also applied to bi-stable structureswhich are virtually free from temperature or moisturechanges. In such bi-stable structures, the residualstresses are introduced by pre-stretching the fiber-rein-forced laminates during manufacturing and not by thermalcuring [16].For morphing the bi-stable flap-like structure, two stable

shapes, which are aerodynamically optimal, can beachieved with difficulty because of the limitations givenby the laminate configurations required to retain bi-stability. Also, the aerodynamic loads that can besupported by the bi-stable flap-like structure might besmaller than the requirements for typical aerodynamicstructures met in practice. In this case an internal structuremight be required to sustain the aerodynamic loads.Nevertheless the locking mechanism offered by the bi-stable skins will reduce the requirements for a very stiff andinherently very heavy internal structure.The morphing concept regarding the camber change for

the airfoil section seems to be the most suitable if optimizedaerodynamic airfoil shapes are desired because the morph-ing mechanism, that is the bi-stable plate, is separated fromthe aerodynamic surfaces. Various parameters such asgeometry or laminate configurations can be changed inorder to achieve the desired aerodynamic airfoils. How-ever, an optimization for optimal shape and for aeroelastictailoring of the airfoil section can be difficult because of thenonlinear structural analysis required. Also, as for theprevious concept, the aerodynamic loads that can besupported by the structure might be smaller than therequirements for typical aerodynamic structures met in

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practice. In this case, again, an internal structure will berequired.

If chord length change is required for the airfoil section,the concept proposed in this study may be advantageousbecause the bi-stable plate is positioned transversely to thechord and can support aerodynamic loads transverse to theskin. Thus, the bi-stable plate has a dual role and acts bothas an actuator and also as a stiffener. However, thisconcept is regarded to be aerodynamically less efficientthan changing the camber of the airfoil.

From a manufacturing point of view, the concept of thebi-stable flap-like structure at the trailing edge seems to bethe simplest and the easiest to build. By contrast, the othertwo concepts, that morph the camber of the airfoil section,are assemblies of various pieces that are in relativemovement with each other. At a closer look, however,for these two morphing concepts, the bi-stable plate has asimple rectangular shape and can be manufacturedseparately from the rest of the structure. This aspect canbecome an advantage if the residual stresses in the bi-stableplate will be introduced during manufacturing by pre-stretching the laminae rather than by thermal curing.

These three concepts show sufficient merit that the nextstage of design can be addressed and that is how themorphing function can be realized and how it compromisesstructural integrity, aerodynamic efficiency and dynamicstability. Obviously, the morphing function is expected toincrease overall aerodynamic performance. The morphingfunction realized herein is done without mechanisms butdoes rely on compliant components. Although structuralproperties such as bending stiffness will not be compro-mised significantly, torsional stiffness locally may be andneeds to be further examined in more detail. The conceptsthat rely on sliding skins do transmit aerodynamic loadseffectively but their overall structural integrity requiresfurther assessment. Many potential actuation devices canbe incorporated such as shape piezoelectric devices [2],shape memory alloys [4], and electrical drives [25]. Theintegration of the power supply and actuator will affectlocal dynamic performance so that attention must be paidto local center of gravity, mass and stiffness distributions.All these issues are to be addressed in further work.

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