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Stiffness tailoring using prestress in adaptive composite structures
Composite Structures 106 (2013) 282–287
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Review
Stiffness tailoring using prestress in adaptive composite structures Stephen Daynes, Paul M. Weaver ⇑ Advanced Composites Centre for Innovation and Science, University of Bristol, University Walk, Bristol BS8 1TR, UK
a r t i c l e
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Article history: Available online 24 June 2013 Keywords: Stiffness tailoring Morphing aircraft Prestress Multistability Nonlinear behaviour
a b s t r a c t Adaptive aerostructures offer the potential of increasing both aerodynamic and structural efficiency compared to conventional aerospace technologies. However, there is frequently an awkward trade-off between designing for large deformations whilst being able to withstand external loads. Multistable laminates which derive their multistability via thermal expansion mismatches are of interest for adaptive structures due to their ability to demonstrate relatively high stiffness in multiple stable states whilst being able to undergo large deformations with reduced stiffness. In addition, actuators only need to work during transition between stable shapes. However, many practical problems have arisen with the application of these laminates when designing adaptive structures. Such problems include hygrothermal variability, a limited design space with regards to achievable shape change, and insufficient stiffness for many applications. Prestressing technologies offer solutions to all of these problems. This paper summarises recent developments concerning the various means by which prestress can be used for stiffness tailoring in adaptive structures. Example prestressed structures are given including camber and twist change morphing airfoils. The use of prestress for stiffness tailoring in the design of novel passive vibration isolators and adaptive air intakes is also discussed. Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved.
Contents 1. 2. 3.
4.
5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stiffness tailoring using prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In-plane prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Structures with in-plane prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Materials with in-plane prestress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Out-of-plane prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Structures with out-of-plane prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Materials with out-of-plane prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ongoing research into multistable twisting structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction The background to this research is the challenge of designing and realising adaptive structures. Such research offers the potential to create structures which have the advantages of being lighter and simpler than conventional mechanisms as well as enabling geometric changes which would not traditionally be simple to achieve. However, there is an inherent and difficult design trade-off in ⇑ Corresponding author. Tel.: +44 (0)1173315318. E-mail address: [email protected] (P.M. Weaver).
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adaptive structures, such as morphing aerodynamic control surfaces, between the need for low stiffness to enable large deformations with acceptable actuation requirements and material strain limits while still being sufficiently stiff to withstand external loading in a controlled manner [1,2]. The conventional approach to stiffness tailoring in thin-walled composite structures is via selection of the individual ply material properties, fibre orientation and laminate stacking sequence. This large design space can result in beneficial stiffness anisotropy and coupling behaviour [3,4] or tailored buckling characteristics [5,6]. A development of this technology is functionally graded
0263-8223/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.05.059
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materials where the material properties continuously vary though the thickness unlike a conventional laminate consisting of discrete plies [7,8]. Variable angle tow composites, on the other hand, offer an enhanced design space beyond conventional composite laminates by enabling fibre tow paths to be steered continuously within a given ply [9,10]. A promising solution to the conflicting stiffness requirements in adaptive structures is multistability. Multistable structures have multiple states of equilibrium where transition between states can be caused by the application of an external force or moment resulting in the structure buckling or ‘snapping through’ into a different local energy minimum. Multistable structures can be designed to have high stiffness in their stable ‘rest’ states while the transition between stable states is often characterised by lower stiffness. The simplest example of a multistable system is the case of bistability where two stable states exist. Schematic representations of mechanical systems which can demonstrate bistability are shown in Fig. 1a and b. In these mechanical systems bistability can be achieved by adding negative stiffness, via prestress, to the system which is of sufficient magnitude to overcome the positive stiffness of the monostable structure and cause a bifurcation via buckling. Between the cases of monostability and bistability a case of reduced, or even zero-stiffness, can also be realised for modest deflections where the positive stiffness of the structure is in equilibrium with the added negative stiffness from residual stresses [11,12]. Exact load paths followed depend on the particular geometry and stiffness characteristics of a system but typically, as a first approximation, they can be described using a cubic polynomial, Fig. 1c. The earliest research into multistable structures used thermal expansion mismatches upon cool down from curing temperature in unsymmetrically laminated composites [13–15]. However, the practical implementation of such laminates into adaptive structure designs has proved problematic due to their low stiffness [16], their limited design space with regards to achievable shape change [17], and their inherent hygrothermal variability [18]. A proposed solution to the limitations of using thermal expansions is to design adaptive structures which derive their multistability using prestressing technologies. This paper summarises the various research efforts made into prestressing technologies with the aim of establishing a unified taxonomy. The paper concludes with an outline of recent research by the listed authors into the design of a passive torsional vibration isolator and a morphing wing design which exhibits zero torsional stiffness about its aerodynamic centre. 2. Stiffness tailoring using prestress As with all buckling phenomena, multistable structures have at least one dimension which is substantially smaller than its other
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dimensions (i.e., two dimensional thin plates and shells) or two dimensions are much smaller (i.e., one dimensional slender columns). All of these structures derive their multistability through a combination of their geometry, internal stress states, and material properties. Stable states are formed when there is a balance between membrane and flexural effects. It is well known that for thin structures it is easier to deform through bending than it is through stretching (or compressing). Upon snap-through internal stress resultants are redistributed and equilibriate in the adjacent stable state. Stable states are conventionally sought by minimising the energy of the system with respect to potential mode shapes and their magnitudes. The elastic energy density of a general anisotropic plate in terms of the strain, curvature, and the material properties is given by [19]:
u¼
1 ðAij ei ej þ 2Bij ei jj þ Dij ji jj Þ 2
ð1Þ
where the Aij coefficients represent extensional stiffness, the Dij coefficients represent bending stiffness and the Bij terms embody the possible coupling between stretching and bending owing to material anisotropy. In (1) the first and third terms are the stretching and bending energy densities respectively while the second term describes the energy associated with the coupling between bending and stretching. The strain energy varies when there is a change in either the strain tensor e or the mid-plane curvature tensor j. Strain energy can therefore be tailored by either the modification of mid-plane strains (in-plane prestressing) or the modification of bending strains (out-of-plane prestressing). A taxonomy for multistability using prestress is presented in the following two sections based on this observation, Fig. 2. 3. In-plane prestressing 3.1. Structures with in-plane prestress Possibly the most ubiquitous example of a bistable prestressed structure is the single piece metal hair clip which becomes bistable during manufacture when its two extremities are pinned together, locking the structure into a heightened state of elastic strain energy. This causes the hair clip to buckle resulting in two stable geometries consisting of equal and opposite out-of-plane curvatures. Similar concepts have been patented [20] as means of creating multistable tabs for helicopter rotor blades where initially flat, stress free, isotropic plates are pinned in heightened states of strain energy to form bistable structures. Bistable concepts which work on the principle of buckling as a result of in-plane loads have also been proposed as a means of augmenting the performance of piezoelectric actuators [21] and also as basis for extendable chord bistable morphing airfoils [22].
Fig. 1. Schematic representations of prestressed mechanisms subject to (a) axial load and (b) torsion. Inset graph (c) demonstrates that tailoring the magnitude of prestress can lead to either monostable, zero-stiffness or bistable mechanisms.
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Fig. 2. Organization of the review.
3.2. Materials with in-plane prestress Bistability can also be realised at the material level when two isotropic plates are bonded together, one on top of the other, while uniaxial prestress is applied to the two plates in orthogonal directions [23]. Releasing the prestress once bonded will result in stress redistribution within the structure and subsequent out-of-plane deformation to form a saddle shape. Under certain geometric conditions, material properties and prestress magnitudes, the structure can bifurcate due to nonlinear geometric effects to form two stable, near cylindrical, geometries. Unlike the hair clip example, this laminate has stable curvatures of equal magnitude and opposite direction which are in orthogonal planes to each other. The prestressing of fibres is another way of controlling the inplane strains within a material. There have been several research efforts investigating the feasibility of fibre prestressing in composites, primarily as a means of improving the mechanical performance of materials in a similar manner to prestressed steel in concrete. Prestressing techniques including stretching dry fibres as part of filament winding [24] and resin transfer moulding processes [25,26]. Once prestress is applied to the dry fibres via mechanical means the liquid matrix is added and cured at elevated temperatures while prestress is maintained. Upon curing and cooldown the prestress is released and the residual stresses within the material then redistribute. Similar techniques have also been developed for prepreg material systems with the same objective of improving the mechanical properties of the cured composite materials [27–29]. More recently, fibre prestressing has been proposed as a means of inducing buckling and multistability in symmetric laminates. Uniaxial prestress can be used as a means of manufacturing laminates which have equal and opposite curvatures (Fig. 3a) [30], analogous to the hair clip. In this example, the fibres on the outer edges of the laminate are stretched in one direction prior
to and during cure. Once cured the prestress is released. At this point, for the stresses to equilibriate within the laminate, the tensile stresses on the outer sides of the laminate decrease while the central part of the laminate, which was initially stress free, develops compressive stresses. This transverse stress distribution consisting of regions of tension and compression can cause the laminate to buckle. These laminates were used in the development of a bistable trailing edge flap for a helicopter rotor blade [31]. A stack of six glass fibre reinforced plastic (GFRP) bistable laminates were placed inside the trailing edge, Fig. 3b, with 1.1% prestrain applied to the fibres on the outer edges of the laminates. This number of laminates and amount of prestrain were chosen to provide sufficient stiffness to withstand aerodynamic loading whist still achieving a 10° flap deflection. A similar prestressing technique can be used for biaxial prestressing to manufacture bistable laminates which have stable, near cylindrical, curvatures which are opposite to each other and in orthogonal planes [32]. An example of such a laminate stacking sequence is [0P/90/90P/0] where the superscript P denotes the laminates which are prestressed. Again, this laminate has a symmetric stacking sequence for hygrothermal stability but has prestressed plies orthogonal to each other and either side of the mid-plane to induce bistability. 4. Out-of-plane prestressing 4.1. Structures with out-of-plane prestress A simple bistable structure can be created by taking a beam, applying equal and opposite end moments to cause out-of-plane displacement, and then clamping the beam at both ends in this deformed configuration. Applying a central vertical deflection can then result in a snap-through response and bistability if certain geometric conditions are satisfied [33]. A wide variety of buckling behaviours can be achieved using similar principles. For example, if the bending stiffness is tailored along the length of the beam then this can result in tristability or even higher numbers of stable states. Such an approach was used in the design of a multistable air intake [34]. This intake consists of a deformable shell, which can be approximated as a beam, with a sinusoidal variation of bending stiffness along its length using composite ply drop-offs and build-ups, Fig. 4. The shell was installed in a stiffer tube component and clamped to form a sigmoidal geometry. The air intake was shown to have a stable geometry in either its open or closed stable state, characterised by high stiffness to withstand aerodynamic loading as well as a lower stiffness intermediate stable state. The air intake was also designed to have a much lower stiffness during transition between stable states to reduce the required amount of actuation work.
Fig. 3. Uniaxially prestressed laminates; (a) application of prestress during manufacture, and (b) integration of laminates into a bistable helicopter blade flap.
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Tape springs (known commonly as carpenter’s tape measures) can also be elastically deformed and used as a basis for creating multistable structures. The tape spring is a cylindrical shell structure which conventionally has only one stable, stress free, geometry when manufactured. They have highly nonlinear momentrotation characteristics often with a linear elastic response for small deformations followed by a sudden snap-through to a much lower stiffness. Structures can be created using assemblies of such tape springs which can exploit these nonlinearities to achieve multistability. Examples of which are energy storage devices [35] and reconfigurable tetrahedra [36]. Attaching two tape spring together, one on top of the other, can also result in twisting bistability [37]. This is done by taking two tape springs and placing them together to form a tube. At this point the tape springs are clamped flat and their two ends are pinned together. Upon releasing of the clamp the assembled structure then demonstrates bistability through axial twist. Suggested applications are variable twist rotor blades and morphing wings. A later work developed this idea further by off-setting the two curved shells using multiple rigid spokes and investigating a wider variety of composite laminate stacking sequences [38]. In this work the curvature of the shells was in the lengthwise rather than the transverse direction. The spokes were allowed to rotate enabling the structure to fully coil in either direction. Analytical and finite element models were developed and revealed a wide variety of potential deployment characteristics. It was shown that by varying the stacking sequence, assembled geometry, and prestress the structure could be made to be stable in a variety of configurations. A set of conditions was also found to create a zero-stiffness structure which could be partially twisted without a tendency to coil up or deploy. 4.2. Materials with out-of-plane prestress Permanent residual stresses can be induced in a ductile material when it is taken past its yield limit. Such residual stresses can be used as a basis for designing multistable structures. This is most easily achieved in metallic plates and shells through bending deformations. Beryllium Copper alloy is particularly well suited for this task since it has a very high ratio of yield strength to Young’s modulus. A plastically deformed multistable structure can be manufactured from a simple tape spring. If the tape spring is coiled tightly back on itself (opposite-sense prestressing) so it passes its yield point then bistability can be induced in the structure [39]. This is the mechanism by which ‘slap bracelets’ work. Conversely, if a tape spring is coiled tightly in on itself (same-sense prestressing) then a case of neutral stability can be achieved in isotropic materials [40,41]. In this case, the tape spring can twist about its longitudinal axis without any change in internal strain energy occurring. Corrugated isotropic shells have also been prestressed by coiling them past their yield point about an axis which
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is perpendicular to the direction of the corrugations [42]. This results in a tristable shell which has one coiled stable geometry and two equal magnitude and opposite sense twisted geometries. Tape springs can also be bonded together to create shells with unusual stiffness characteristics. For example, it has been shown that a neutrally stable fibre reinforced plastic composite tape spring can be constructed using a 0/90 cross-ply stacking sequence where each ply is individually pre-curved in one direction and cured prior to assembling the laminate [43]. A consequence of this is that the tape spring can be partially rolled and neither have a tendency to unroll or to roll up. The ability to undergo large elastic deformations and large axial extensions with diminishingly small forces is particularly desirable in deployable space structures. Sheets with dimples plastically depressed into the surface have been shown to demonstrate multistability [44]. The ability to add multiple dimples substantially increase the number of potential stable states a structure can have. If there are N dimples which are all individually reversible then there can be a maximum of 2N stable configurations. Sheets with non-reversible dimples can also demonstrate multistability [45]. Such sheets have been named ‘doubly corrugated’ since the grid of dimples creates a sinusoidal pattern in two orthogonal directions. Depending on how the dimples are applied to an initially stress free metallic sheet a bistable structure can be created which displays cylindrical curvature changes in either same-sense or opposite-sense bending. Another way of inducing plastic deformation in thin shells is to add local, discrete hinge lines through folding, which articulate and deform to achieve shape change. This is the principle behind origami. Compliance follows from elastic deformation of the material coupled to mechanistic articulation about the hinge lines. The distribution of the fold lines then determines the shell’s kinematic behaviour. It has been shown that concentric square or circular patterns of alternating ridge and valley folds can result in a bistable shell which twists to form two hyperbolic paraboloid geometries [46]. 5. Ongoing research into multistable twisting structures Ongoing research by the listed authors concerns the development of an elastically prestressed adaptive twisting structure which does not require the articulations or moving parts used in previous studies [37,38]. The structure consists of carbon fibre reinforced plastic (CFRP) strips which are assembled to form a grid. Each strip is initially manufactured in a ‘stress free’ configuration using a curved mould which can be used to create an initial prebend and/or an initial pre-twist, Fig. 5a. The strips are then forced flat during the assembly of the structure using steel clips to align the strips and then epoxy resin is used to bond the CFRP strips and steel clips together. The increase in strain energy due to prestressing during assembly can cause a twist to develop in the assembled structure as it moves to a stable, lower energy,
Fig. 4. Multistable air intake: (a) closed state, (b) open state, and (c) bottom view in open state.
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Fig. 5. Manufacture of multistable twisting structures; (a) curved mould used for inducing pre-curvature, and (b and c) stable states of structure B.
Table 1 Pre-bend and pre-twist of members in structures A to D prior to assembly. Structure
Stability characteristics
Pre-bend radius (mm)
Pre-twist rate (°/mm)
A B C D
Monostable Bistable Zero-stiffness Zero-stiffness + offset
1 100 164 164
0 0 0 0.1
equilibrium due to the existence of geometric nonlinear effects. Details of the analytical and finite element modelling of this nonlinear behaviour are given in [47]. Prestressing can therefore lead to a change in the torsional stiffness of the structure. Four sample structures were manufactured, each having the same assembled grid geometry and material properties, but with different prestress built in during manufacture. Each structure consists of two sets of three composite strips which are orthogonal to each other, Fig. 5b and c. The assembled structures have external dimensions of 100 mm by 100 mm by 10 mm. Each strip has a thickness of 1 mm and is manufactured from unidirectional Hexcel 8552/IM7 CFRP with the fibre direction aligned with the length of each strip. The various pre-curvatures built into the four structures are detailed in Table 1. Structure A has no prestress added and is therefore monostable while structure B consists of CFRP strips which initially have a radius of curvature of 100 mm. The prestress in structure B results in bistability with a rate of twist of approximately ±0.7°/mm. The two stable twisted geometries are shown in
Fig. 5b and c. The special case of zero-stiffness occurs in structure C when the radius of curvature is 164 mm. The required magnitude of prestress to achieve zero-stiffness depends upon a combination of the bending stiffness properties of the members within the structure and their distance from the twist axis, details of this are given in [38]. The influence of adding pre-twist on the other hand is demonstrated in structure D, Fig. 6. Pre-twist has the same effect as preloading the structure with an externally applied twisting moment. An immediate application for this technology is vibration isolation devices where stiffness reduction using non-linear springs can be used as a means of eliminating unwanted resonant frequencies [48]. Another application is twist-morphing airfoils. A morphing wing proof-of-concept model has been developed using this prestressing technology [12] which has prestressed spars to suppress the wing’s torsional stiffness. The model consists of a grid of unidirectional CFRP 0.5 mm thick ribs and 1 mm thick spars, Fig. 7. There are three spars situated at 0.125%, 0.25% and 0.75% chord. The spars are positioned so the wing’s shear centre is at 25% chord. A NACA 0012 airfoil section was selected since its aerodynamic centre is also at 25% chord for a wide range of aerodynamic conditions. A 124 mm radius of pre-curvature was added to the forward and aft spars to create state of zero-torsional stiffness similar to structure C in Fig. 6. Finally a silicone outer skin was added which was reinforced in the spanwise direction with nylon fibres. The low shear modulus of the silicone ensures it does not significantly contribute to the torsional stiffness of the wing while the fibre reinforcement can be used to withstand aerodynamic loading. The result is a neutrally stable system where aerodynamic and
Fig. 6. Moment-rotation characteristics demonstrating the influence of prestress in structures A to D.
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Fig. 7. Morphing wing construction with skin attached to lower surface.
torsional stiffnesses are minimised to enable the wing to be actuated about its 25% chord. This could be an enabling technology for the widespread use of smart material actuators such as piezoelectric patches which often provide insufficient forces to deform adaptive structures while working against external loads. Integration of an actuator into this concept remains an area of future work at this stage. Similarly, future work also includes a more detailed investigation concerning the aerodynamic and aeroelastic assumptions made as well as a more detailed investigation concerning the stiffness tailoring of the morphing skin. 6. Conclusion There is considerable interest from both academia and industry into adaptive structures due to the potential to create systems that are lighter, simpler, and more efficient than their conventional mechanical equivalent, or in some way have additional functionality. Following the advances outlined in this paper, adaptive structures which have tailored stiffness and stability using prestressing techniques are a rapidly maturing technology. Prestressing in adaptive structures provides solutions to many previous problems associated with using thermal expansion effects to induce structural multistability. Industrial applications such as morphing wings and vibration suppression are apparent and deserve further investigation. References [1] Campanile LF. Initial thoughts on weight penalty effects in shape-adaptable systems. J Intell Mater Syst Struct 2005;16(1):47–56. [2] Daynes S, Weaver PM. Design and testing of a deformable wind turbine blade control surface. Smart Mater Struct 2012;21(10):105019. [3] Lemanski S, Weaver P. Optimisation of a 4-layer laminated cylindrical shell to meet given cross-sectional stiffness properties. Compos Struct 2006;72(2):163–76. [4] Weaver P. On beneficial anisotropic effects in composite structures. In: 43rd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Denver, CO; 2002. [5] Coburn BH, Pirrera A, Weaver PM, Vidoli S. Tristability of an orthotropic doubly curved shell. Compos Struct 2013;96:446–54. [6] Weaver P. On superior buckling performance of flat plates through anisotropy. In: 44th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Norfolk, VA; 2003. [7] Jha D, Kant T, Singh R. A critical review of recent research on functionally graded plates. Compos Struct 2012;96:833–49. [8] Ferreira A, Batra R, Roque C, Qian L, Martins P. Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Compos Struct 2005;69(4):449–57. [9] Raju G, Wu Z, Kim BC, Weaver PM. Prebuckling and buckling analysis of variable angle tow plates with general boundary conditions. Compos Struct 2012;94:2961–70. [10] Wu Z, Weaver PM, Raju G. Postbuckling optimization of variable angle tow composite plates. Compos Struct 2013;103:34–42. [11] Barrett RM, Barnhart R. Solid state adaptive rotor using postbuckled precompressed, bending-twist coupled piezoelectric actuator elements. Smart Mater Res 2012;2012:1–10.
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[31] Daynes S, Nall SJ, Weaver PM, Potter KD, Margaris P, et al. Bistable composite flap for an airfoil. J Aircraft 2010;47(1):334–8. [32] Daynes S, Diaconu CG, Potter KD, Weaver PM. Bistable prestressed symmetric laminates. J Compos Mater 2010;44(9):1119–37. [33] Brinkmeyer A, Santer M, Pirrera A, Weaver PM. Pseudo-bistable self-actuated domes for morphing applications. Int J Solids Struct 2012;49(9):1077–87. [34] Daynes S, Weaver PM, Trevarthen JA. A morphing composite air inlet with multiple stable shapes. J Intell Mater Syst Struct 2011;22(9):961–73. [35] Santer MJ, Pellegrino S. An asymmetrically-bistable monolithic energy-storing structure. In: 45th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, 19–22 April 2004, Palm Springs, CA. [36] Santer M, Pellegrino S. Compliant multistable structural elements. Int J Solids Struct 2008;45(24):6190–204. [37] Schultz MR. A concept for airfoil-like active bistable twisting structures. J Intell Mater Syst Struct 2008;19(2):157–69. [38] Lachenal X, Weaver PM, Daynes S. Multistable twisting structure for morphing applications. Proc Roy Soc A-Math Phys 2012;468(2141):1230–51. [39] Kebadze E, Guest SD, Pellegrino S. Bistable prestressed shell structures. Int J Solids Struct 2004;41(11–12):2801–20. [40] Seffen KA, Guest SD. Prestressed morphing bistable and neutrally stable shells. J Appl Mech 2011;78(1):1–6. [41] Guest SD, Kebadze E, Pellegrino S. A zero-stiffness elastic shell structure. J Mech Mater Struct 2011;6(1–4):203–12. [42] Norman AD, Seffen KA, Guest SD. Multistable corrugated shells. Proc Roy Soc A-Math Phys 2008;464(2095):1653–72. [43] Murphey TW, Pellegrino S. A novel actuated composite tape-spring for deployable structures. In: 45th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, 19–22 April 2004, Palm Springs, CA. [44] Seffen KA. Hierarchical multi-stable shapes in mechanical memory metal. Scripta Mater 2007;56(5):417–20. [45] Norman AD, Golabchi MR, Seffen KA, Guest SD. Multistable textured shell structures. Adv Sci Technol 2008;54:168–73. [46] Seffen KA. Compliant shell mechanisms. Philos T Roy Soc A 2012;370(1965):2010–26. [47] Daynes S, Lachenal X, Weaver PM. Twisting structures with tailored stability and their application to morphing wings. In: 23rd International conference on adaptive structures and technologies (ICAST), 11–13 October 2012, Nanjing, China. [48] Shaw AD, Neild SA, Wagg DJ. Dynamic analysis of high static low dynamic stiffness vibration isolation mounts. J Sound Vib 2013;332(6):1437–55.
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Review
Stiffness tailoring using prestress in adaptive composite structures Stephen Daynes, Paul M. Weaver ⇑ Advanced Composites Centre for Innovation and Science, University of Bristol, University Walk, Bristol BS8 1TR, UK
a r t i c l e
i n f o
Article history: Available online 24 June 2013 Keywords: Stiffness tailoring Morphing aircraft Prestress Multistability Nonlinear behaviour
a b s t r a c t Adaptive aerostructures offer the potential of increasing both aerodynamic and structural efficiency compared to conventional aerospace technologies. However, there is frequently an awkward trade-off between designing for large deformations whilst being able to withstand external loads. Multistable laminates which derive their multistability via thermal expansion mismatches are of interest for adaptive structures due to their ability to demonstrate relatively high stiffness in multiple stable states whilst being able to undergo large deformations with reduced stiffness. In addition, actuators only need to work during transition between stable shapes. However, many practical problems have arisen with the application of these laminates when designing adaptive structures. Such problems include hygrothermal variability, a limited design space with regards to achievable shape change, and insufficient stiffness for many applications. Prestressing technologies offer solutions to all of these problems. This paper summarises recent developments concerning the various means by which prestress can be used for stiffness tailoring in adaptive structures. Example prestressed structures are given including camber and twist change morphing airfoils. The use of prestress for stiffness tailoring in the design of novel passive vibration isolators and adaptive air intakes is also discussed. Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved.
Contents 1. 2. 3.
4.
5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stiffness tailoring using prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In-plane prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Structures with in-plane prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Materials with in-plane prestress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Out-of-plane prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Structures with out-of-plane prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Materials with out-of-plane prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ongoing research into multistable twisting structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction The background to this research is the challenge of designing and realising adaptive structures. Such research offers the potential to create structures which have the advantages of being lighter and simpler than conventional mechanisms as well as enabling geometric changes which would not traditionally be simple to achieve. However, there is an inherent and difficult design trade-off in ⇑ Corresponding author. Tel.: +44 (0)1173315318. E-mail address: [email protected] (P.M. Weaver).
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adaptive structures, such as morphing aerodynamic control surfaces, between the need for low stiffness to enable large deformations with acceptable actuation requirements and material strain limits while still being sufficiently stiff to withstand external loading in a controlled manner [1,2]. The conventional approach to stiffness tailoring in thin-walled composite structures is via selection of the individual ply material properties, fibre orientation and laminate stacking sequence. This large design space can result in beneficial stiffness anisotropy and coupling behaviour [3,4] or tailored buckling characteristics [5,6]. A development of this technology is functionally graded
0263-8223/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2013.05.059
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materials where the material properties continuously vary though the thickness unlike a conventional laminate consisting of discrete plies [7,8]. Variable angle tow composites, on the other hand, offer an enhanced design space beyond conventional composite laminates by enabling fibre tow paths to be steered continuously within a given ply [9,10]. A promising solution to the conflicting stiffness requirements in adaptive structures is multistability. Multistable structures have multiple states of equilibrium where transition between states can be caused by the application of an external force or moment resulting in the structure buckling or ‘snapping through’ into a different local energy minimum. Multistable structures can be designed to have high stiffness in their stable ‘rest’ states while the transition between stable states is often characterised by lower stiffness. The simplest example of a multistable system is the case of bistability where two stable states exist. Schematic representations of mechanical systems which can demonstrate bistability are shown in Fig. 1a and b. In these mechanical systems bistability can be achieved by adding negative stiffness, via prestress, to the system which is of sufficient magnitude to overcome the positive stiffness of the monostable structure and cause a bifurcation via buckling. Between the cases of monostability and bistability a case of reduced, or even zero-stiffness, can also be realised for modest deflections where the positive stiffness of the structure is in equilibrium with the added negative stiffness from residual stresses [11,12]. Exact load paths followed depend on the particular geometry and stiffness characteristics of a system but typically, as a first approximation, they can be described using a cubic polynomial, Fig. 1c. The earliest research into multistable structures used thermal expansion mismatches upon cool down from curing temperature in unsymmetrically laminated composites [13–15]. However, the practical implementation of such laminates into adaptive structure designs has proved problematic due to their low stiffness [16], their limited design space with regards to achievable shape change [17], and their inherent hygrothermal variability [18]. A proposed solution to the limitations of using thermal expansions is to design adaptive structures which derive their multistability using prestressing technologies. This paper summarises the various research efforts made into prestressing technologies with the aim of establishing a unified taxonomy. The paper concludes with an outline of recent research by the listed authors into the design of a passive torsional vibration isolator and a morphing wing design which exhibits zero torsional stiffness about its aerodynamic centre. 2. Stiffness tailoring using prestress As with all buckling phenomena, multistable structures have at least one dimension which is substantially smaller than its other
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dimensions (i.e., two dimensional thin plates and shells) or two dimensions are much smaller (i.e., one dimensional slender columns). All of these structures derive their multistability through a combination of their geometry, internal stress states, and material properties. Stable states are formed when there is a balance between membrane and flexural effects. It is well known that for thin structures it is easier to deform through bending than it is through stretching (or compressing). Upon snap-through internal stress resultants are redistributed and equilibriate in the adjacent stable state. Stable states are conventionally sought by minimising the energy of the system with respect to potential mode shapes and their magnitudes. The elastic energy density of a general anisotropic plate in terms of the strain, curvature, and the material properties is given by [19]:
u¼
1 ðAij ei ej þ 2Bij ei jj þ Dij ji jj Þ 2
ð1Þ
where the Aij coefficients represent extensional stiffness, the Dij coefficients represent bending stiffness and the Bij terms embody the possible coupling between stretching and bending owing to material anisotropy. In (1) the first and third terms are the stretching and bending energy densities respectively while the second term describes the energy associated with the coupling between bending and stretching. The strain energy varies when there is a change in either the strain tensor e or the mid-plane curvature tensor j. Strain energy can therefore be tailored by either the modification of mid-plane strains (in-plane prestressing) or the modification of bending strains (out-of-plane prestressing). A taxonomy for multistability using prestress is presented in the following two sections based on this observation, Fig. 2. 3. In-plane prestressing 3.1. Structures with in-plane prestress Possibly the most ubiquitous example of a bistable prestressed structure is the single piece metal hair clip which becomes bistable during manufacture when its two extremities are pinned together, locking the structure into a heightened state of elastic strain energy. This causes the hair clip to buckle resulting in two stable geometries consisting of equal and opposite out-of-plane curvatures. Similar concepts have been patented [20] as means of creating multistable tabs for helicopter rotor blades where initially flat, stress free, isotropic plates are pinned in heightened states of strain energy to form bistable structures. Bistable concepts which work on the principle of buckling as a result of in-plane loads have also been proposed as a means of augmenting the performance of piezoelectric actuators [21] and also as basis for extendable chord bistable morphing airfoils [22].
Fig. 1. Schematic representations of prestressed mechanisms subject to (a) axial load and (b) torsion. Inset graph (c) demonstrates that tailoring the magnitude of prestress can lead to either monostable, zero-stiffness or bistable mechanisms.
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Fig. 2. Organization of the review.
3.2. Materials with in-plane prestress Bistability can also be realised at the material level when two isotropic plates are bonded together, one on top of the other, while uniaxial prestress is applied to the two plates in orthogonal directions [23]. Releasing the prestress once bonded will result in stress redistribution within the structure and subsequent out-of-plane deformation to form a saddle shape. Under certain geometric conditions, material properties and prestress magnitudes, the structure can bifurcate due to nonlinear geometric effects to form two stable, near cylindrical, geometries. Unlike the hair clip example, this laminate has stable curvatures of equal magnitude and opposite direction which are in orthogonal planes to each other. The prestressing of fibres is another way of controlling the inplane strains within a material. There have been several research efforts investigating the feasibility of fibre prestressing in composites, primarily as a means of improving the mechanical performance of materials in a similar manner to prestressed steel in concrete. Prestressing techniques including stretching dry fibres as part of filament winding [24] and resin transfer moulding processes [25,26]. Once prestress is applied to the dry fibres via mechanical means the liquid matrix is added and cured at elevated temperatures while prestress is maintained. Upon curing and cooldown the prestress is released and the residual stresses within the material then redistribute. Similar techniques have also been developed for prepreg material systems with the same objective of improving the mechanical properties of the cured composite materials [27–29]. More recently, fibre prestressing has been proposed as a means of inducing buckling and multistability in symmetric laminates. Uniaxial prestress can be used as a means of manufacturing laminates which have equal and opposite curvatures (Fig. 3a) [30], analogous to the hair clip. In this example, the fibres on the outer edges of the laminate are stretched in one direction prior
to and during cure. Once cured the prestress is released. At this point, for the stresses to equilibriate within the laminate, the tensile stresses on the outer sides of the laminate decrease while the central part of the laminate, which was initially stress free, develops compressive stresses. This transverse stress distribution consisting of regions of tension and compression can cause the laminate to buckle. These laminates were used in the development of a bistable trailing edge flap for a helicopter rotor blade [31]. A stack of six glass fibre reinforced plastic (GFRP) bistable laminates were placed inside the trailing edge, Fig. 3b, with 1.1% prestrain applied to the fibres on the outer edges of the laminates. This number of laminates and amount of prestrain were chosen to provide sufficient stiffness to withstand aerodynamic loading whist still achieving a 10° flap deflection. A similar prestressing technique can be used for biaxial prestressing to manufacture bistable laminates which have stable, near cylindrical, curvatures which are opposite to each other and in orthogonal planes [32]. An example of such a laminate stacking sequence is [0P/90/90P/0] where the superscript P denotes the laminates which are prestressed. Again, this laminate has a symmetric stacking sequence for hygrothermal stability but has prestressed plies orthogonal to each other and either side of the mid-plane to induce bistability. 4. Out-of-plane prestressing 4.1. Structures with out-of-plane prestress A simple bistable structure can be created by taking a beam, applying equal and opposite end moments to cause out-of-plane displacement, and then clamping the beam at both ends in this deformed configuration. Applying a central vertical deflection can then result in a snap-through response and bistability if certain geometric conditions are satisfied [33]. A wide variety of buckling behaviours can be achieved using similar principles. For example, if the bending stiffness is tailored along the length of the beam then this can result in tristability or even higher numbers of stable states. Such an approach was used in the design of a multistable air intake [34]. This intake consists of a deformable shell, which can be approximated as a beam, with a sinusoidal variation of bending stiffness along its length using composite ply drop-offs and build-ups, Fig. 4. The shell was installed in a stiffer tube component and clamped to form a sigmoidal geometry. The air intake was shown to have a stable geometry in either its open or closed stable state, characterised by high stiffness to withstand aerodynamic loading as well as a lower stiffness intermediate stable state. The air intake was also designed to have a much lower stiffness during transition between stable states to reduce the required amount of actuation work.
Fig. 3. Uniaxially prestressed laminates; (a) application of prestress during manufacture, and (b) integration of laminates into a bistable helicopter blade flap.
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Tape springs (known commonly as carpenter’s tape measures) can also be elastically deformed and used as a basis for creating multistable structures. The tape spring is a cylindrical shell structure which conventionally has only one stable, stress free, geometry when manufactured. They have highly nonlinear momentrotation characteristics often with a linear elastic response for small deformations followed by a sudden snap-through to a much lower stiffness. Structures can be created using assemblies of such tape springs which can exploit these nonlinearities to achieve multistability. Examples of which are energy storage devices [35] and reconfigurable tetrahedra [36]. Attaching two tape spring together, one on top of the other, can also result in twisting bistability [37]. This is done by taking two tape springs and placing them together to form a tube. At this point the tape springs are clamped flat and their two ends are pinned together. Upon releasing of the clamp the assembled structure then demonstrates bistability through axial twist. Suggested applications are variable twist rotor blades and morphing wings. A later work developed this idea further by off-setting the two curved shells using multiple rigid spokes and investigating a wider variety of composite laminate stacking sequences [38]. In this work the curvature of the shells was in the lengthwise rather than the transverse direction. The spokes were allowed to rotate enabling the structure to fully coil in either direction. Analytical and finite element models were developed and revealed a wide variety of potential deployment characteristics. It was shown that by varying the stacking sequence, assembled geometry, and prestress the structure could be made to be stable in a variety of configurations. A set of conditions was also found to create a zero-stiffness structure which could be partially twisted without a tendency to coil up or deploy. 4.2. Materials with out-of-plane prestress Permanent residual stresses can be induced in a ductile material when it is taken past its yield limit. Such residual stresses can be used as a basis for designing multistable structures. This is most easily achieved in metallic plates and shells through bending deformations. Beryllium Copper alloy is particularly well suited for this task since it has a very high ratio of yield strength to Young’s modulus. A plastically deformed multistable structure can be manufactured from a simple tape spring. If the tape spring is coiled tightly back on itself (opposite-sense prestressing) so it passes its yield point then bistability can be induced in the structure [39]. This is the mechanism by which ‘slap bracelets’ work. Conversely, if a tape spring is coiled tightly in on itself (same-sense prestressing) then a case of neutral stability can be achieved in isotropic materials [40,41]. In this case, the tape spring can twist about its longitudinal axis without any change in internal strain energy occurring. Corrugated isotropic shells have also been prestressed by coiling them past their yield point about an axis which
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is perpendicular to the direction of the corrugations [42]. This results in a tristable shell which has one coiled stable geometry and two equal magnitude and opposite sense twisted geometries. Tape springs can also be bonded together to create shells with unusual stiffness characteristics. For example, it has been shown that a neutrally stable fibre reinforced plastic composite tape spring can be constructed using a 0/90 cross-ply stacking sequence where each ply is individually pre-curved in one direction and cured prior to assembling the laminate [43]. A consequence of this is that the tape spring can be partially rolled and neither have a tendency to unroll or to roll up. The ability to undergo large elastic deformations and large axial extensions with diminishingly small forces is particularly desirable in deployable space structures. Sheets with dimples plastically depressed into the surface have been shown to demonstrate multistability [44]. The ability to add multiple dimples substantially increase the number of potential stable states a structure can have. If there are N dimples which are all individually reversible then there can be a maximum of 2N stable configurations. Sheets with non-reversible dimples can also demonstrate multistability [45]. Such sheets have been named ‘doubly corrugated’ since the grid of dimples creates a sinusoidal pattern in two orthogonal directions. Depending on how the dimples are applied to an initially stress free metallic sheet a bistable structure can be created which displays cylindrical curvature changes in either same-sense or opposite-sense bending. Another way of inducing plastic deformation in thin shells is to add local, discrete hinge lines through folding, which articulate and deform to achieve shape change. This is the principle behind origami. Compliance follows from elastic deformation of the material coupled to mechanistic articulation about the hinge lines. The distribution of the fold lines then determines the shell’s kinematic behaviour. It has been shown that concentric square or circular patterns of alternating ridge and valley folds can result in a bistable shell which twists to form two hyperbolic paraboloid geometries [46]. 5. Ongoing research into multistable twisting structures Ongoing research by the listed authors concerns the development of an elastically prestressed adaptive twisting structure which does not require the articulations or moving parts used in previous studies [37,38]. The structure consists of carbon fibre reinforced plastic (CFRP) strips which are assembled to form a grid. Each strip is initially manufactured in a ‘stress free’ configuration using a curved mould which can be used to create an initial prebend and/or an initial pre-twist, Fig. 5a. The strips are then forced flat during the assembly of the structure using steel clips to align the strips and then epoxy resin is used to bond the CFRP strips and steel clips together. The increase in strain energy due to prestressing during assembly can cause a twist to develop in the assembled structure as it moves to a stable, lower energy,
Fig. 4. Multistable air intake: (a) closed state, (b) open state, and (c) bottom view in open state.
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Fig. 5. Manufacture of multistable twisting structures; (a) curved mould used for inducing pre-curvature, and (b and c) stable states of structure B.
Table 1 Pre-bend and pre-twist of members in structures A to D prior to assembly. Structure
Stability characteristics
Pre-bend radius (mm)
Pre-twist rate (°/mm)
A B C D
Monostable Bistable Zero-stiffness Zero-stiffness + offset
1 100 164 164
0 0 0 0.1
equilibrium due to the existence of geometric nonlinear effects. Details of the analytical and finite element modelling of this nonlinear behaviour are given in [47]. Prestressing can therefore lead to a change in the torsional stiffness of the structure. Four sample structures were manufactured, each having the same assembled grid geometry and material properties, but with different prestress built in during manufacture. Each structure consists of two sets of three composite strips which are orthogonal to each other, Fig. 5b and c. The assembled structures have external dimensions of 100 mm by 100 mm by 10 mm. Each strip has a thickness of 1 mm and is manufactured from unidirectional Hexcel 8552/IM7 CFRP with the fibre direction aligned with the length of each strip. The various pre-curvatures built into the four structures are detailed in Table 1. Structure A has no prestress added and is therefore monostable while structure B consists of CFRP strips which initially have a radius of curvature of 100 mm. The prestress in structure B results in bistability with a rate of twist of approximately ±0.7°/mm. The two stable twisted geometries are shown in
Fig. 5b and c. The special case of zero-stiffness occurs in structure C when the radius of curvature is 164 mm. The required magnitude of prestress to achieve zero-stiffness depends upon a combination of the bending stiffness properties of the members within the structure and their distance from the twist axis, details of this are given in [38]. The influence of adding pre-twist on the other hand is demonstrated in structure D, Fig. 6. Pre-twist has the same effect as preloading the structure with an externally applied twisting moment. An immediate application for this technology is vibration isolation devices where stiffness reduction using non-linear springs can be used as a means of eliminating unwanted resonant frequencies [48]. Another application is twist-morphing airfoils. A morphing wing proof-of-concept model has been developed using this prestressing technology [12] which has prestressed spars to suppress the wing’s torsional stiffness. The model consists of a grid of unidirectional CFRP 0.5 mm thick ribs and 1 mm thick spars, Fig. 7. There are three spars situated at 0.125%, 0.25% and 0.75% chord. The spars are positioned so the wing’s shear centre is at 25% chord. A NACA 0012 airfoil section was selected since its aerodynamic centre is also at 25% chord for a wide range of aerodynamic conditions. A 124 mm radius of pre-curvature was added to the forward and aft spars to create state of zero-torsional stiffness similar to structure C in Fig. 6. Finally a silicone outer skin was added which was reinforced in the spanwise direction with nylon fibres. The low shear modulus of the silicone ensures it does not significantly contribute to the torsional stiffness of the wing while the fibre reinforcement can be used to withstand aerodynamic loading. The result is a neutrally stable system where aerodynamic and
Fig. 6. Moment-rotation characteristics demonstrating the influence of prestress in structures A to D.
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Fig. 7. Morphing wing construction with skin attached to lower surface.
torsional stiffnesses are minimised to enable the wing to be actuated about its 25% chord. This could be an enabling technology for the widespread use of smart material actuators such as piezoelectric patches which often provide insufficient forces to deform adaptive structures while working against external loads. Integration of an actuator into this concept remains an area of future work at this stage. Similarly, future work also includes a more detailed investigation concerning the aerodynamic and aeroelastic assumptions made as well as a more detailed investigation concerning the stiffness tailoring of the morphing skin. 6. Conclusion There is considerable interest from both academia and industry into adaptive structures due to the potential to create systems that are lighter, simpler, and more efficient than their conventional mechanical equivalent, or in some way have additional functionality. Following the advances outlined in this paper, adaptive structures which have tailored stiffness and stability using prestressing techniques are a rapidly maturing technology. Prestressing in adaptive structures provides solutions to many previous problems associated with using thermal expansion effects to induce structural multistability. Industrial applications such as morphing wings and vibration suppression are apparent and deserve further investigation. References [1] Campanile LF. Initial thoughts on weight penalty effects in shape-adaptable systems. J Intell Mater Syst Struct 2005;16(1):47–56. [2] Daynes S, Weaver PM. Design and testing of a deformable wind turbine blade control surface. Smart Mater Struct 2012;21(10):105019. [3] Lemanski S, Weaver P. Optimisation of a 4-layer laminated cylindrical shell to meet given cross-sectional stiffness properties. Compos Struct 2006;72(2):163–76. [4] Weaver P. On beneficial anisotropic effects in composite structures. In: 43rd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Denver, CO; 2002. [5] Coburn BH, Pirrera A, Weaver PM, Vidoli S. Tristability of an orthotropic doubly curved shell. Compos Struct 2013;96:446–54. [6] Weaver P. On superior buckling performance of flat plates through anisotropy. In: 44th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Norfolk, VA; 2003. [7] Jha D, Kant T, Singh R. A critical review of recent research on functionally graded plates. Compos Struct 2012;96:833–49. [8] Ferreira A, Batra R, Roque C, Qian L, Martins P. Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Compos Struct 2005;69(4):449–57. [9] Raju G, Wu Z, Kim BC, Weaver PM. Prebuckling and buckling analysis of variable angle tow plates with general boundary conditions. Compos Struct 2012;94:2961–70. [10] Wu Z, Weaver PM, Raju G. Postbuckling optimization of variable angle tow composite plates. Compos Struct 2013;103:34–42. [11] Barrett RM, Barnhart R. Solid state adaptive rotor using postbuckled precompressed, bending-twist coupled piezoelectric actuator elements. Smart Mater Res 2012;2012:1–10.
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