the mechanics of composite corrugated structures: a review...

Composite Structures 133 (2015) 358–380

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Composite Structures

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Review

The mechanics of composite corrugated structures: A review withapplications in morphing aircraft

http://dx.doi.org/10.1016/j.compstruct.2015.07.0990263-8223/Crown Copyright � 2015 Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (I. Dayyani).

I. Dayyani a,⇑, A.D. Shaw b, E.I. Saavedra Flores c, M.I. Friswell b

a Faculty of Engineering and Computing, Coventry University, Coventry CV1 5FB, UKb College of Engineering, Swansea University, Swansea SA2 8PP, UKc Departamento de Ingeniería en Obras Civiles, Universidad de Santiago de Chile, Avenida Ecuador 3659, Estación Central, Santiago, Chile

a r t i c l e i n f o a b s t r a c t

Article history:Available online 26 July 2015

Keywords:CorrugationCompositesMorphing

Corrugation has long been seen as a simple and effective means of forming lightweight structures withhigh anisotropic behaviour, stability under buckling load and energy absorption capability. This has beenexploited in diverse industrial applications and academic research. In recent years, there have beennumerous innovative developments to corrugated structures, involving more elaborate and ingeniouscorrugation geometries and combination of corrugations with advanced materials. This developmenthas been largely led by the research interest in morphing structures, which seek to exploit the extremeanisotropy of a corrugated panel, using the flexible degrees of freedom to allow a structure’s shape tochange, whilst bearing load in other degrees of freedom. This paper presents a comprehensive reviewof the literature on corrugated structures, with applications ranging from traditional engineering struc-tures such as corrugated steel beams through to morphing aircraft wing structures. As such it providesan important reference for researchers to have a broad but succinct perception of the mechanical beha-viour of these structures. Such a perception is highly required in the multidisciplinary design of corru-gated structures for the application in morphing aircraft.

Crown Copyright � 2015 Published by Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
1.1. A general description to corrugated structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3591.2. Applications of corrugated structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
1.2.1. Packaging industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3591.2.2. Civil structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3591.2.3. Marine structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3601.2.4. Mechanical engineering structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3601.2.5. Aerospace and aeronautics application concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

1.3. Corrugated structures, innovation and developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
1.3.1. Innovation based on different material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3601.3.2. Innovation based on geometric parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
2. Corrugated panel from different perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
2.1. General mechanical properties of corrugated panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
2.1.1. Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3622.1.2. Tensile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3632.1.3. Shear and compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

2.2. Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
2.2.1. Buckling of corrugated webs for beam sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3662.2.2. Buckling of corrugated shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
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2.2.3. Buckling of corrugated structures in naval applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3662.2.4. Buckling of corrugated panels in packaging application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

2.3. Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3662.4. Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3672.5. Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3682.6. Homogenization and equivalent modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3692.7. Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

3. Corrugated skin for morphing wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
3.1. Introduction, general challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3723.2. Corrugated skin, different morphing applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
3.2.1. Camber morphing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3743.2.2. Corrugated skin; application in winglet and span extension morphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

1. Introduction

1.1. A general description to corrugated structures

The term ‘‘corrugated’’ in general describes a series of parallelridges and furrows [1]. In mechanical engineering any structurewhich has a surface with the shape of corrugation either madeby folding, moulding, or any other manufacturing methods iscalled a corrugated structure. Three typical corrugated structuresmay be classified as: a corrugated pipe, a corrugated sheet and acorrugated panel. The main common feature of all corrugatedstructures is their exceedingly anisotropic behaviour; high stiffnesstransverse to the corrugation direction in contrast to the compli-ance along the corrugation direction [2]. Because of this importantfeature, these structures have been widely used in industrial appli-cations and academic research.

By adding two face sheets (also known as liners) as upper andlower surfaces to the corrugated sheet (also known as core, med-ium or fluting) a new geometry would be obtained known, as acorrugated panel [3]. By selecting the appropriate shape, dimen-sions and materials of the face sheets and corrugated core, a vari-ety of stiffness and strength at low weight of the corrugatedpanel will be achieved. The structural characteristics of this cor-rugated structure depend mainly on the lightweight corrugatedcore which separates the face sheets and provides the necessarystiffness for the panel. However by considering different materialstiffness for the face sheets and the corrugated core, differentmechanical behaviour of the identical geometry would beexpected.

If the stiffness of the material of face sheets is higher than orequal to the stiffness of the material of the corrugated core, thestructure would be recognised as a corrugated sandwich panel[4]. Such a sandwich panel demonstrates higher shear, bendingand tensile stiffness to weight ratio than an equivalent panel madeof only the corrugated core material or the face-sheet material [5].This is because the flexural stiffness of the panel is proportional tothe cube of its thickness. Hence the function of a corrugated core inthe sandwich panel is to increase the stiffness of the panel by effec-tively thickening it with a low-density corrugated core material.This results in the increase of the stiffness significantly for very lit-tle additional weight of the panel. The behaviour of such a sand-wich panel under a bending load is similar to an I-beam wherethe facings of the sandwich panel act as the I-beam flanges wherethe upper and lower face sheets are subject to the in-plane com-pression and tension, and the corrugated core material acts asthe beam’s shear web where the corrugated core is subject to shear[6]. It can be concluded that one of the most important character-istics of a corrugated core is to keep the face sheets apart and

stabilize them by resisting the out of plane deformations whichincreases the shear strength and stiffness of the panel.

1.2. Applications of corrugated structures

Corrugated structures have wide application in engineering dueto their special characteristics such as: anisotropic behaviour, highstiffness to weight ratio and high capacity of energy absorption.The applications of these structures can be classified into the fol-lowing categories in which more value is given to the special fea-tures of these structures.

1.2.1. Packaging industryCorrugated boards, either made of plastic or cardboard are used

extensively to produce rigid shipping containers of almost anyshape or size. The packaging containers are exposed to various loadconditions such as: static loads due to the compression of packagesin a stack during transport and storage and vibration loads duringtransport. The reason that the corrugated sandwich panels havereceived huge interest in the packaging industries is because oftheir stiffness and durability, lightness and cost effectiveness aswell as the recyclability and sustainability with the environment[7,8].

1.2.2. Civil structuresThe wide application of corrugated structures in civil engineer-

ing may be classified mainly as: beams with corrugated web, cor-rugated roofs and walls and corrugated pipes.

� Beams with corrugated web

The main benefit of applying corrugated web beams in support-ing roofs, floors and columns in steel structural buildings are thatthe corrugated webs increase the beam’s stability against buckling.Applying these corrugated web beams in the components of thebuilding results in a very economical design by reducing therequired web stiffeners and leads to a significant weight reductionin these beams compared with hot-rolled or welded ones [9].

� Corrugated sheets in roof and walls

Corrugated sheets are among the best candidates for applica-tion in construction elements, for roofs, claddings and walls, ofmodern industrial buildings owing to their high strength to weightratio, much lighter and lower cost than flat isotropic panels of thesame strength [10]. Corrugated metal sheets for instance are fre-quently used as the roof of buildings that have steep slopes to dis-pose of rainwater quickly. Their combination of high stiffness and

360 I. Dayyani et al. / Composite Structures 133 (2015) 358–380

lightweight nature lightens the load on the installation and theunderlying building structures.

� Corrugated tunnel and pipe

Large metal corrugated pipes or arches are frequently used intunnel structures to transport the aggregate and ore across variouspoints on their properties. The need to maximise the surface areaon such sites necessitates the use of tunnels for transporting bulkmaterials under roadways and processing these materials. Theapplication of corrugated pipes and arcs in these tunnels offersadvantages in the design, installation and operation of these pro-jects such as: reducing the design time and related costs; simplic-ity of construction which leads into the reduction of installationand maintenance costs [11].

Corrugated pipes are often used in sewerage and drainageapplications because of their light weight, high strength and com-pliance which lead into long life performance. The strength of thepipes arises from the corrugated design of the outer wall ratherthan the wall thickness, in contrast to the normal solid wall pipes.The advantages of the corrugated pipes in general can be classifiedas their lightness and flexibility. The lightness of these structuresreduces the manpower needed for installation and the costs oftransportation whereas the flexibility reduces the damages duringstorage and handling and ease the natural settlements to be toler-ated without suffering cracks or leakages [12].

1.2.3. Marine structuresThe corrugated sandwich panel has offered a wide range of

attractive design solutions to operational shipboard problems inwhich structural performance and weight are important designissues. The applications of these structures include decks, bulk-heads, helidecks and accommodation modules [13]. Another appli-cation of the corrugated sandwich panels is in the combatantdeckhouse structure of a naval ship since these structures show agood resistance to the possible blast loads [14].

1.2.4. Mechanical engineering structures

� Corrugated hoses:

Corrugated hoses are another case of important engineeringstructures which are exploiting corrugation characteristics.Because of the special properties of the corrugation structure, thesehoses can withstand very high pressure and provide maximum leaktightness. Corrugated hoses also exhibit corrosion resistance andpressure tightness under the most extreme conditions, such asaggressive seawater, extreme temperatures found in space and con-veying very hot or cold substances. The other advantage of corru-gated hoses is their flexibility which makes them a goodcandidate to connect elements where they are subjected tomovement, thermal expansion and vibrations [15]. Due to thesecharacteristics they are frequently used in hydraulic circuits, pro-tection for electrical cables and light conductors or exhaust gasinstallations.

� Corrugated gasket

Surface configuration of the corrugated gaskets enables them toadapt to rough or irregular flange surfaces without requiring exces-sive compressive load. This provides an efficient seal under varyingconditions of temperature and pressure. The substrate corrugationgeometry promotes the recovery and resilience through thermalcycles and extended service life. Hence they are excellent productsfor both standard flange and heat exchanger gaskets where lowbolt load are present or where high gasket stresses are available[16].

1.2.5. Aerospace and aeronautics application conceptsCorrugated sandwich panels are used in aerospace engineering

because of their multifunctional characteristics. These structuresoffer insulation as well as load-bearing capabilities in addition totheir lightness. These multifunctional integral thermal protectionstructures protect the spacecraft from extreme reentry tempera-tures, and possess load-carrying capabilities [17,18]. Moreover,because of their exceedingly anisotropic behaviour of these struc-tures they have been proposed as a flexible skin for the wings ofmorphing aircraft. This is due to the fact that wing structures mustbe stiff so as to withstand bending due to aerodynamic forces, andflexible so they can deform efficiently in flight due to morphingactuation [19]. This application is explained exhaustively inSection 3.

1.3. Corrugated structures, innovation and developments

As discussed so far, corrugated structures have noticeableimpacts on the engineering applications due to their superiorstructural characteristics which mainly arise from their geometricproperties. However the structural performance of these structuresis being developed further in the literature by introducing moregeometric parameters or using different material properties.Some of the concepts regarding the development of these struc-tures are reviewed in this section.

1.3.1. Innovation based on different material properties

� Elastomeric face sheets

There are some specific applications of corrugated panels suchas morphing skins in which a change in material properties ofthe corrugated sandwich panel is considered. The corrugated coreis coated with elastomeric face sheets. Although the geometry ofthis corrugated panel is similar to the corrugated sandwich panelthe function is much different. This is because the stiffness of theelastomeric coatings are significantly smaller than the stiffness ofthe material of the corrugated core. The purpose of the elastomericface sheets in this structure is not to increase the stiffness of thepanel but to provide a continuous external surface to maintainthe efficient aerodynamic performance during the flight [2]. Moredetails of the mechanical behaviour of the corrugated core withelastomeric coating are presented later in this article. Fig. 1(a)shows a corrugated core with elastomeric coating.

� Corrugated core materials

Another way to increase the design space of the corrugatedpanel is by using composite materials in the corrugated core andconsequently providing further improvement opportunitiesthrough optimising parameters such as: fibre orientations in eachlayup, curvilinear fibres and textile architecture of the plain wovencloth of the fibres. The works of Kazemahvazi et al. [20,21] inwhich they introduced a novel composite corrugation concept toprevent the core members from buckling, is highlighted. Anotherexample in this perspective is the interesting idea of combiningmulti-stable characteristics with corrugated structural perfor-mance [22] in which the multi-stability comes from the interactionbetween internal prestressed laminates and non-linear geometri-cal changes during deformation. The multi stable characteristicsenable the structure to undergo large configuration changes whichcan be sustained into the new position with no use of any lockingmechanism while the corrugation geometry provides the highstrength to weight ratio. The wide range of parameters in such astructure enables designers to tailor the stiffness properties tothe required application such as morphing wings. Fig. 1(b) illus-trates the twisted bi-stable corrugated core.

(a) Corrugated core with elastomeric coating [33]

(b) Twisted bi-stable corrugated core [22]

(c) Curved corrugated sheet and some of its global deformations

[23]

(d) Schematic of bi-directional corrugated core [26]

(e) Schematic of corrugatedbi directional core, [27]

(f) Schematics ofHierarchical corrugated core

sandwich panel [28]

(g) PMI foam filled hierarchical corrugated sheet

Kazemahvazi et al. [21]

(h) Double wall corrugated conceptPrevitali et al. [30]

(i) Schematics of extruded pyramidal lattice truss sandwich

structure, [32]

Fig. 1. Corrugated structures developments and concepts.


1.3.2. Innovation based on geometric parameters

� Curved corrugated shell

The concept of curved corrugated sheets was proposed byNorman et al. [23]. These structures which are initially manufac-tured with a curvature along the corrugation profile are capableof significant elastic shape changes, including large changes inoverall Gaussian curvature without local stretching of the surface.The detailed kinematics of the curved corrugated shells was stud-ied in this paper and it was demonstrated how the geometricalrelationship between the local and global corrugations specifythe coupling between stretching of the plate across the corruga-tions and the global curvature along the curved shell. These struc-tures may be useful to be applied in morphing structures to offerstructural integrity and large shape-change capabilities. Fig. 1(c)illustrates the curved corrugated sheet and some of its globaldeformations.

� Bi-directional corrugated-strip-core sandwich panel

A new concept in steel bi-directional corrugated-core sandwichstructures was proposed by [24] to improve the stiffness in themore flexible direction of the panel and to modify the transverse

shear stiffness of the panel. The geometry of the typical core canbe obtained by propagating the corrugated strips in both longitudi-nal and transverse directions. Depending on the pattern of propa-gation a variety of different densities and stiffnesses may beobtained. Leekitwattana et al. [25] proposed a derivation for thetransverse shear stiffness of a steel bi-directional corrugated sand-wich panel using analytical methods. Dayyani et al. [26] simulatedthe mechanical behaviour of the bi-directional corrugated corewith and without elastomeric coatings in tensile and five pointbending tests. Fig. 1(d) shows a schematic of a bi directional corru-gated core.

Furthermore, Seong et al. [27] proposed another type ofbi-directional corrugated core to reduce the anisotropic character-istics of the corrugated sandwich plates which could restrict therange of the applications of these structures. The design space inthis bi-directional corrugate core was extended by introducingtwo additional geometric parameters, pass angle and corrugationlength, in order to minimize the beam buckling of the face sheetswith respect to core orientations. In this concept the continuouscorrugated channels were first fabricated by using a sectionalforming process and then the core was attached to face sheets bymeans of adhesive bonding. Fig. 1(e) shows a schematic of thisconcept.


� Hierarchical corrugated core sandwich panel concepts

Improving the transverse compression and shear collapsemechanisms of the corrugated panel, a novel concept based onthe second order hierarchical corrugation was offered by Kooistraet al. [28] as shown in Fig. 1(f). The idea was born from the fact thatmaterials with structural hierarchy can have higher stiffness toweight ratio than their single-length scale of microstructure coun-terparts. Hence the local corrugated elements were introduced topostpone the elastic buckling of the webs of the main corrugatedcore. However, the manufacturing constraints and the relative highcosts of production have limited the application of hierarchical cor-rugated cores to large sandwich structures. Kooistra et al. derivedthe analytical expressions for the compressive and shear collapsestrengths of the hierarchical corrugated cores and validated theirmodel predictions by comparing to the experimental data. Theyreported that the strength of a second order hierarchical corru-gated sandwich panel is almost ten times greater than the firstorder corrugated structure of the same relative density.

Kazemahvazi et al. [20,21] and Previtali et al. [29,30] also pro-posed two works in this regard, although with different purposes.In the first work, the local corrugated elements were replaced byPMI foam and a local sandwich panel were applied to the inclinedmembers of the global corrugated sheet. Fig. 1(g) shows a sche-matic of this concept. The idea was to improve the out of planecompressive properties and in-plane shear stiffness. However inthe second work, the local foam was removed and a double wallcorrugated sheet was obtained. Fig. 1(h) shows a schematic of thisconcept. In this concept, the deformations of local double walledelements due to the combination of shear and bending loading pro-vided further axial in-plane compliance for the application in mor-phing skin. This technique almost eliminated the rotation of theupper and lower surfaces of the corrugation when the corrugationwas strained.

� Pyramidal lattice truss sandwich structure

The concept of lattice truss structures was proposed as an alter-native to cellular core structures in the literature to furtherincrease the ratio of strength to weight of sandwich panels [31].The out-of-plane and in-plane mechanical properties of these lat-tice truss structures are dependant to the topology of the lattice,relative density and the stiffness properties of the core material.Queheillalt et al. [32] proposed a new approach for manufacturingthe uniform pyramidal lattice truss sandwich structure. In thismethod, first the solid corrugated sandwich panel was fabricatedby extruding the aluminium slabs through the moulds and thenthe corrugated core was imposed by electro discharge machining(EDM) by use of alternating pattern of triangular-shaped EDM elec-trodes normal to the extrusion direction. The result of the processwas a lattice truss sandwich panel in which the interface betweenthe core and face sheet possessed the identical metallurgical andmechanical properties. Fig. 1(i) shows the schematic of theextruded pyramidal lattice truss sandwich structure.

2. Corrugated panel from different perspectives

The design of corrugated structures for morphing technology isinherently multidisciplinary; a successful design must meet bothstructural and actuation requirements. In aerospace this must beachieved at minimum weight, and in general many other require-ments will be of importance, including but not limited to such fac-tors as vibration characteristics, fatigue life, and damage tolerance.However, multidisciplinary design depends on a strong under-standing of each discipline concerned, so this work now proceeds

to categorise literature on corrugated panels by individualperspectives.

2.1. General mechanical properties of corrugated panels

A comprehensive set of analyses about the flexural, tensile,shear and out of plane compressive strength of corrugated panelsis developed in the literature by means of experimental and finiteelement analysis. These analyses have considered mainly the non-linear effect of material properties and geometric parameters aswell as analysis of various boundary conditions and loading config-urations [3]. When possible in the literature, analytical solutionsare introduced in support of these investigations [34].

2.1.1. BendingNumerous studies have been conducted on the bending stiff-

ness of corrugated board. These investigations have incorporatedanalytical solutions, finite element simulations or experiments tofind the flexural rigidities of the board. Khalid et al. [35] investi-gated the mechanical behaviour of structural beams with corru-gated webs in three-point bending. They determined the effectsof the corrugation curvature, web thickness, material propertiesof the corrugated web, and the corrugation direction on the beam’sload-carrying capability. The experimental tests were used to val-idate the results obtained by nonlinear finite element analysis. The30% difference in the flexural stiffness which was observed inthe results highlighted the bending anisotropic characteristics ofthe composite beam with the corrugated web. It was reported alsothat increasing the radius of corrugation curvature led to higherbending stiffness and could reduce the beam’s weight by about 14%.

Chang et al. [36] presented a closed-form solution based on theMindlin–Reissner plate theory to describe the linear flexural beha-viour of the corrugated core sandwich plate with various boundaryconditions. They reduced the three-dimensional sandwich panel toan equivalent two-dimensional structurally orthotropic thick platecontinuum. They compared the numerical results of the proposedmodel by the experimental data available in the literature [37]and observed a good agreement. They investigated the effects ofseveral geometric parameters of a corrugated core sandwich panelon its rigidity and state of stress and came up with some recom-mendations for the selection of the geometric parameters ofcorrugated- core sandwich plates. These recommendations weremainly about minimizing the ratio of geometric parameters such:the ratio of the height to the thickness of the corrugated core, thick-ness of the corrugated core to thickness of the face sheet and thelength of the corrugated unit cell to the height of corrugation.However such ratios resulted in an increase of weight of the struc-ture and performing a multi objective optimization which considersboth structural rigidities and mass of the structure is important.

Yokozeki et al. [2] proposed a simple analytical model for theinitial bending stiffness of corrugated composites in both longitu-dinal and transverse directions and compared the predictions withthe experimental results. For the flexural modulus in the morecomplaint direction, they measured the deflection of one end ofthe corrugated core due to its own weight, while the other endof the corrugated sheet was clamped. Moreover, four-point bend-ing load was applied to the specimens in the longitudinal directionwhere both ends of the corrugated core were fixed. Although theapplied bending displacement was small, two modes of out ofplane flexural deformation and in-plane tensile stretching werecoupled. They highlighted the extremely anisotropic behaviour ofthe corrugated core through comparing the flexural stiffness ofthe corrugated sheet.

Seong et al., [27] performed three-point bending tests on thebi-directional corrugated sandwich panels for various core


orientations and demonstrated that this sandwich corrugatedpanel has a quasi-isotopic bending behaviour. They explained theeffect of geometric parameters of the bi-directional corrugatedcore on the buckling strength of the face sheets during large bend-ing deformations.

Dayyani et al. [38] studied the flexural characteristics of a com-posite corrugated sheet using numerical and analytical methodsand validated the results by comparing them to the experimentaldata. A good degree of correlation was observed in their workwhich evidenced the suitability of the analytical method and finiteelement model to predict the mechanical behaviour of the corru-gated sheet in the linear and nonlinear phases of deformation.The finite element simulation exploited the node to surface andfrictionless contact technique, to model the interaction betweenthe corrugated sheet and the supports. The force–displacementcurves showed three distinct phases of deformation in thethree-point bending test. Three phases of the deformations weredistinguished as: deformation due to pure bending of corrugatedsheet, deformation due to combined bending and axial forces caus-ing a step increase in the force–displacement curve and againdeformation due to pure bending of the corrugated core. Theyreported that the second phase in which the step was observedarose because of simultaneous contact of the two adjacent cornersof the corrugated unit cell with the support. Fig. 2 illustrates thecorrugated sheet in a three-point bending test and the correspond-ing force–displacement curves obtained from the experiment andsimulation results.

2.1.2. TensileNoting the extreme anisotropic stiffness properties of corru-

gated sheets [2], Thill et al. [39] investigated the effect of a varietyof materials and parameters such as number of plies and corruga-tion pitch on the overall mechanical properties of the corrugatedcomposite sheet. The output of this study was that the transversetensile elastic modulus is dependent on the squared laminatethickness and the length of corrugated unit cell length. Three yearslater, they explained the obtained results via experimental, analyt-ical and numerical analysis methods [40]. They considered trape-zoidal corrugated aramid/epoxy laminates subjected to largetensile deformations transverse to the corrugation direction andhighlighted the effect of local failure mechanisms of these speci-mens on the three stages of the tensile force–displacement graphs.They found out that the second stage, which comprised the major-ity of the displacement, occurred because of aramid fibre

(a) Bending experimental set up

Fig. 2. Three-point bending behaviour of a comp

compressive properties and delaminations in the corner regionsof the corrugated unit cell. This local phenomenon was comparedto a pseudo-plastic hinge allowing large deformations over rela-tively constant stress levels.

As an extension to this work, Dayyani et al. [38] studied the ten-sile behaviour of corrugated laminates made of plain woven glas-s/epoxy. Contrary to the literature they observed the occurrenceof delamination in all of the members of the corrugated unit cell,not only to the corner regions, and evidence that the three-stagemechanical behaviour of composite corrugated core is not confinedto aramid laminates and can be observed in other types of lami-nates. The tensile force–displacement curves in their experimentshowed three distinct phases of deformation: 1-small deforma-tions due to tension of both straight and inclined members,2-rotation of joints at the intersection of straight and inclinedmembers, 3-the tensile behaviour of the flattened panel respec-tively. These three phases are shown in Fig. 3 where the tensileforce–displacements are plotted. The plasticity was exploited infinite element simulation as a technique to model the delamina-tion, which dissipated the strain energy of the system during thetensile testing. Assigning the plasticity to all of the regionsof the corrugated unit cell resulted in a good agreement betweenthe numerical predictions and the experimental observations.The extreme sensitivity of the composite corrugated sheet to theangle of the corrugated unit cell was also demonstrated in thiswork which highlighted the importance of the precision of thedesign and manufacturing process.

2.1.3. Shear and compressionTransverse shear stiffness of the corrugated sandwich panels is

one of the important characteristics of these structures which mustbe accurately characterised in the performance analysis of thesestructures. Among the early works regarding this issue is the workof Nordstrand et al. [41] who used curved beam theory to study theshear stiffness of a corrugated cardboard. They presented a theo-retical study on how the geometry of the corrugation affects thetransverse shear moduli. Firstly by assuming rigid face sheets inthe corrugated cardboard they derived an upper limit of the trans-verse shear modulus across the corrugations and then showed howthis shear stiffness reduces if deformations of the face sheets areconsidered in the analysis. Nordstrand and Carlsson [42] experi-mentally examined the effective transverse shear moduli in theprincipal material directions of corrugated board using the blockshear test and the three-point bend test. They observed that the

(b) Force-displacement curves

osite corrugated sheet, Dayyani et al. [38].

(a) Tensile experimental set up (b) Three distinct phases of the tensile force-displacement curves

Fig. 3. The mechanical behaviour of the composite corrugated sheet in a tensile test, [38].


shear moduli obtained by the three-point bend test were almosthalf of those determined by the block shear test. This discrepancywas explained by local deformation of the face sheets of the boardwhere they were in contact with the supports in three point bend-ing test.

Isaksson et al. [43] considered a panel of corrugated paperboard as a stack of an arbitrary number of thin virtual layers withcorresponding effective elastic moduli. The elastic properties of alllayers were assembled together to analyse a corrugated board as acontinuous homogenous structure. They showed that exploitingthe shear correction factors which were derived from the equilib-rium stress field can improve the stiffness calculations. Their pro-posed model was validated by experiments on corrugated boardpanels with different geometries.

Kampner et al. [44] investigated the possibility of using a corru-gated sheet as the facings of sandwich beams to carry shear loadswhich are traditionally carried by the core. A compliant foam corewas used as a ‘‘cushion’’ between the outer skin and the internalstructure in their concept. One of the main reasons for such con-cept was improving the performance of the panel under shockloads where stiff connections between the facings were prone tolocalised failure. Finite element simulations, as well as some ana-lytical investigations were used to find out that the introductionof a corrugated face sheet improved the capability of shear carryingand reduced the weight of the panel, predominantly for heavilyloaded sandwich beams.

Leekitwattana et al. [25] took into account the concept of aforce–distortion relationship to derive a formulation for the trans-verse shear stiffness of a bidirectional corrugated sandwich panelby use of the modified stiffness matrix method. They showed theconsistency of the proposed formulation with a three-dimensionalfinite element solution. The computation time for the proposedmethod was claimed to be 40 times lower than the FE method.They assessed the effect of geometrical parameters and comparedthe performance of a bi-directional corrugated sandwich panel withother one directional corrugated sandwich panels. They realisedthat for a specific range of parameters the bidirectional corrugatedtopology shows superior performance in transverse shear stiffness.

Lu et al. [45] investigated the compressive response and failuremechanisms of a corrugated sandwich panel by use of a combinedtheoretical and experimental approach. In this work, the corru-gated specimens were modelled by use of curved beam elementsand surface contact elements. The elastoplastic material was tunedwith a bi-linear constitutive model which satisfied the J2-flow

theory and assigned to the finite element model. The effects ofboundary conditions, geometrical parameters, and material prop-erties, and geometrical imperfections on the compressive strengthof corrugated boards were studied. As a result, they found out thatthe panel has the highest compression strength when the initiallysinusoidal corrugated core deforms into a square wave pattern.Moreover, it was shown that the stress–strain curves of the corru-gated panel had an undulating behaviour in compression, whichreflects the initiation, spreading and arrest of the localised plasticcollapse mechanisms.

Rejab and Cantwell [46] investigated a series of experimentaland numerical analyses on the compression response and subse-quent failure modes of the corrugated core sandwich panels whichwere made of three different materials: aluminium alloy, glassfibre reinforced plastic and carbon fibre reinforced plastic.Particular attention in this work was paid to the effect of the num-ber of unit cells and the thickness of the cell walls in determiningthe overall deformation and local collapse behaviour of the panel.They realised that the buckling of the cell walls was the first failuremode in these corrugated structures and increasing the compres-sion loading will result in localised delamination as well asdebonding between the skins and the core. The experimentalresults were compared to finite element and analytical solutions.The predictions offered by the numerical models were in goodagreement. However the analytical model over-estimated theload-bearing capability of the corrugations due to the fact thatthe model assumed perfect bonding between the apex of the cor-rugated core and the skin and neglected the effect of initial imper-fections along the cell walls.

As mentioned earlier, Kooistra et al. [28] analysed the trans-verse compression and collapse mechanisms of a second orderhierarchical corrugated sandwich panel. In contrast to a first ordercorrugated sandwich panel which exhibit two competing collapsemodes of elastic buckling and plastic yielding, they showed thatthe second order corrugated panel has six competing modes of fail-ure: elastic buckling and yielding of the larger and smaller struts,shear buckling of the larger struts, and wrinkling of the face sheetsof the larger struts. Fig. 4 shows the global and local failure modesof first order and second order corrugated sandwich panels in com-pression. Analytical expressions for the compressive and shear col-lapse strengths in each of these modes were derived and used toconstruct collapse mechanism maps for the second order corruga-tion models. They used these maps as a base for selecting the geo-metric parameters of second order corrugated panel to optimise

(a) First order corrugation unit cell, global elastic buckling mode

(b) Second order corrugation unit cell, local failure modes

(c) Schematics of the failure modes in the second order corrugated unit cell

Fig. 4. Global and local failure modes of first order and second order corrugated sandwich panels in compression, Kooistra et al. [28].


the ratio of collapse strength to mass and validated the proposedmodel experimentally. They discovered that increasing the levelof structural hierarchy does not lead to further enhancements inthe stiffness of the corrugated core. In other words, for a givenmass the first order corrugation exhibited slightly more stiffnessthan its second order counterpart suggesting that the hierarchicalcorrugated construction has applications in strength limitedapplications.

Likewise, the effect of hierarchy on the stiffness of corrugatedstructures in compression has been followed by other researcherssuch as Kazemahvazi et al. [20,21]. They applied sandwich panelwith PMI foam to the local inclined members of the global corru-gated core and experimentally studied a range of different failuremodes of these structures depending on their geometrical andthe material properties. In this regard, first the collapse mechanismmaps of different corrugation configurations were analyticallyobtained. The stiffness model exploited the contribution in stiff-ness from the bending and the shear deformations of the local coremembers in addition to the stretching deformation. They claimedthat the proposed hierarchical corrugated core can have more than7 times higher weight specific strength compared to its monolithiccounterpart, if designed correctly. The difference in strength arosemainly due to the increase in buckling resistance of the sandwichcore members compared to the monolithic corrugated core. Itwas observed that when the density of the core increases, themonolithic core members get thicker and more resistant to buck-ling and thus the benefits of the hierarchical structure reduces.

2.2. Buckling

Buckling occurs when a structure makes a rapid change of con-figuration due to applied load – this applied load may be compres-sion, shear or multi-axial. A structure is often said to have failedwhen buckling occurs (for example when a column collapses underaxial compression), and in these cases all that must be understood

is when the onset of buckling will occur, which is usually a linearproblem that can generally be achieved through classical analyticalmethods or using finite element analysis. However, in certain cases(such as in-plane shear buckling of a panel) some load resistanceremains after buckling, and this so called post-buckle strengthmay be exploited. The study of post-buckling is often more com-plex than that of buckling, and the complex deformations formedduring buckling often require the use of nonlinear analysis.

One of the most common methods to analyse buckling of corru-gated plate or shell structures is to use a model to homogenise thecorrugation as an orthotropic panel, and then to find global buck-ling modes using analysis similar to conventional panels. Manyexamples of this approach are presented subsequently. Althougha variety of homogenisation methods exist (see Section 2.6) anFEA unit cell may be used to derive the equivalent properties ifan analytical process is not available.

Moreover it is usually possible to make further checks on corru-gated structures especially for the configurations which have flatsections, such as trapezoidal corrugations, to ensure that localbuckling modes do not occur. However, this approach cannot beapplied in a straightforward manner to continuous profiles e.g.sinusoidal corrugations. If it is not feasible to simply check localand global modes separately, or greater accuracy is required,higher fidelity analyses must be used. Clearly, Finite Elementmethods have a broad role to play, however other higher fidelitymethods exist. Liew et al. [47] used the mesh free Galerkin methodto analyse the buckling modes of a plate. This method was an alter-native to FEA analysis, with the advantage that it could avoid cer-tain problems with element distortion. Pignataro et al. [48] used afinite strip method, with nonlinear kinematics, to study in detailthe situations where the global mode interacts with local modesto create a localised region of buckling in the post buckled shape,and reduce the critical load. The work also considered how theseinteractions led to sensitivity to initial imperfections. Few authorsuse fully analytical methods to consider both global and local


features of a corrugated structure simultaneously; although someexamples of this approach are given in Section 2.2.2.

The literature on buckling of corrugations may be separatedinto the following applications: webs of beam sections, shell struc-tures, naval structures, and the packaging industry. These applica-tions are considered in detail in the following subsections.

2.2.1. Buckling of corrugated webs for beam sectionsCorrugations have been widely used in I-beams. The purpose of

the web in an I-beam serves to resist shear force, so shear is theprimary cause of buckling in these cases. It seems that much ofthe current literature on shear buckling of corrugations has beendriven by this application. An early work in this field is given byLibove [49], where it is briefly demonstrated that shear effectsmay be important in the analysis, and that therefore homogenisedequivalent orthotropic approaches may have poor accuracy. Thework then goes on to develop a shell model approach, givingexpressions for the total potential energy that can therefore beused in variational analysis to find the buckling modes. The modeluses nonlinear Von-Karman strains in the local material, withshear effectively accounted for in the global deformation.

However, assumedly due to the complexity of the model given,later works have adopted an approach using the equivalent ortho-tropic properties. Elgaaly et al. [50] discussed the global shearbuckling of corrugated panels in terms of equivalent homogeneousproperties. Their formula for the global buckling mode’s criticalshear stress has been cited by many authors since:

sc ¼ 36bD1=4

y D3=4x

th2 ð1Þ

where Dx and Dy were the equivalent orthotropic flexural rigiditiesin the x and y directions respectively, t was the thickness and h wasthe length of the panel along the corrugation channels. b was a con-stant, between 0 and 1.9 depending on boundary conditions.

At a similar time, Luo and Edlund [51] presented numericalanalysis of both buckling and post buckling under shear forceapplied to a beam with a trapezoidally corrugated web. It wasnoted that three types of buckling may occur; local buckling (of asingle flat section within the corrugation), global buckling (wherethe entire panel fails) and ‘zonal’ buckling, which was similar toa local mode but could extend over more than one panel. The non-linear results were compared to some earlier analytical models forshear buckling, which were shown to have only approximateaccuracy.

Yi et al. [52] presented a work that focussed on the ‘zonal’mode, although they referred to it as the ‘interactive’ mode. Theycompared it to a range of previous analytical formulas in previousliterature, which were all of a form similar to

1ðsIÞn

¼ 1ðsLÞn

þ 1ðsGÞn

þ 1ðsYÞn

ð2Þ

where sI is the interactive buckling mode, sL is the local buck-ling failure stress, sG is the global failure stress and sY is the yieldstress of the material and n is an integer between 1 and 4depending on the model used. Inspecting the form of this equa-tion shows that if the critical mode of failure (as appearing inthe terms on the right) is much lower than the others, it willdominate the overall interactive mode; however if the individualmodes have similar critical loads, the resulting interactive failurewill be lower than all of them. It is shown by comparison tonumerical and previously published experimental data that thesemethods are approximately accurate, but require empirical cor-rections when operating near the yield limit of the material.Sause and Braxtan [53] extended the work of Yi et al. [52] with

further comparisons to experiment, and further suggested correc-tions to allow for empirically found areas where the derived mod-els were non-conservative.

2.2.2. Buckling of corrugated shellsCorrugations have been considered as a way of improving the

stability of shells. Semenyuk and Neskhodovskaya [54,55] providea comprehensive analysis of these problems using deep shell the-ory. It is shown that under certain conditions, the use of corruga-tion on a cylindrical shell can substantially raise the criticalbuckling load. These papers present some every intriguing possi-bilities for refined analytical approaches to corrugations, becausethey do not depend on homogenised approaches, so that localbuckling modes are calculated simultaneously with global effects.Furthermore, the shell approach used suggests that an adaptationto the curved surfaces of a true wing may be feasible, and maybea further extension to include the effects of surface pressure alongthe lines of Semenyuk et al. [56]. The constraints of this approachare a restriction to single layer geometries with smooth corruga-tion profiles.

2.2.3. Buckling of corrugated structures in naval applicationsThe American Bureau of Shipping regulations ABS [57] include

guidelines for the use of metal corrugated panels, and these aresummarised and explained by Sun and Spencer [58]. The overallapproach is to separately consider buckling cases of in plane shear(as discussed in the above section) and compressive loading inboth directions and also lateral pressure, and combine them so thata safe design is defined by

rx

grxc

� �2

þ ry

gryc

� �2

þ sgsc

� �2

< 1 ð3Þ

where g is a strength knock down factor. These results are shown tobe reasonably conservative in comparison to FEA. This work alsoincludes a formula to consider buckling in the corrugation causedby lateral pressure; at present it has not been possible to view theoriginal source of how this was derived, but the final form appearsto be that of a local buckling mode. However, accuracy is again reli-ant on empirical factors so this may not be directly applicable tomorphing applications.

2.2.4. Buckling of corrugated panels in packaging applicationMany treatments of corrugation come from the packaging

industry due to the widespread use of corrugated cardboard, andthis provides some valuable data. Indeed, one of few publicationsto discuss buckling of corrugations where the material is not iso-tropic is given by Biancolini and Brutti [59], in a study of the abilityto stack boxes on one another. This work extends equivalent stiff-ness formulae for sinusoidal corrugations found by Briassoulis [60]to allow an orthotropic source material, and uses these to calculatethe global buckling load. However, further buckling modes areneglected.

In a study on corrugated board, where the effects of the facesheets are included, Nordstrand [61] looks at global buckling andpost buckling in the presence of imperfections. The analysis is veryclear, and could easily be adapted for the purpose of morphingapplication. Johnson and Urbanik [62] provide analysis of a similarstructure in, with an approach that can be potentially be adaptedto any prismatic structure; however their focus is on local bucklingmodes with the assumption that these initiate failure.

2.3. Vibration

Vibration is an important consideration throughout engineer-ing; the demand for low weight can often result in structures with

Fig. 5. Power density versus driving frequency for a corrugated strip energyharvester, Hu et al. [67].


low damping present, which can result in destructively high ampli-tude vibrations if modes are excited at their natural frequencies. Ingeneral, structural vibration is also an issue of importance fortopics such as noise and passenger comfort in vehicles. It is alsoan important subset of the wider field aeroelasticity, which maybe a particularly crucial problem for morphing aircraft wherestructures are specifically designed to include compliance for actu-ation reasons. Relatively little work has been dedicated specificallyto the vibration of corrugated panels, although clearly many gen-eral methods such as FEA are applicable.

Many studies rely on homogenised models of the corrugation toanalyse vibration; these have the advantage of simplicity, but havethe disadvantage that they may not account for ‘local’ effectswithin the structure of the corrugation or behaviour that lies out-side the assumptions of the fitted model. However, in many casesthese deficiencies may simply be addressed with a further check.Hui and Huan-ran [63] presented a linear analysis of a simply sup-ported sandwich plate with a corrugated core; many of the mod-elling assumptions were directly applicable to corrugations withcovering skins (with the exception being that the skins were notconsidered to be under tension, as an elastomer would be). Themodel considered first order shear effects, however shear deforma-tion was assumed to be negligible along the longitudinal directionof the corrugation. This assumption led to an elegant derivationwith closed form solutions that may be readily used in optimisa-tion. Liew et al. [64] used a mesh free method to understand thevibration of a stiffened corrugated plate. The approach used ahomogenisation technique that accounted for first-order sheareffects in the panel; the mesh free method was a numericalmethod but had some advantages to FEA when applied to optimi-sation problems, because there was no requirement to regeneratemeshes after geometry changes.

Other works consider the effect of corrugation on the vibrationof shells. Semenyuk et al. [65] used deep shell theory and theHamilton principal to provide a comprehensive study of the vibra-tions of a corrugated cylindrical shell, in manner that captures bothlocal and global effects simultaneously. It was shown that homoge-nised approaches capture the first vibration mode only over a lim-ited range of the corrugation pitch; if the pitch is too long, localmodes will occur as the first mode, and if the pitch is too fine thenin-plane (or ‘accordion’ style) vibrations occur. In another analyti-cal survey based on deep shell theory, Gulgazaryan andGulgazaryan [66] established the geometric conditions that couldlead to the presence of Rayleigh waves along the free edge of a cor-rugated cylindrical shell.

Hu et al. [67] discussed an energy harvester, using a corrugatedplate as the vibrating element; where the purpose of the corruga-tion was to allow deformations that alter the natural frequency tomatch the most prominent ambient vibration frequency. Thisstudy highlighted an important phenomenon that will affect thevibration of corrugated skins; that their natural frequencies willbe strongly influenced by their states of deformation. Fig. 5 showsan example of this; power density, which is largely determined bythe natural frequency, was shown to vary significantly with thespan length of the corrugated strip.

Experimental studies concerning the vibrations of corrugatedplates seem to be somewhat rare; however two such examplesare given found in Mandal [68] and Mandal et al. [69], examiningrigid trapezoidal plates. The first work concerns the vibrationtransmitted through in plane vibration, and finds that the presenceor size of corrugations has little conclusive effect. The latter consid-ers the loss factors of different modes of flexural vibration of theplates, and shows that corrugations cause slightly higher loss fac-tors for the first mode, with the effect increasing with increasingcorrugation depth. Recently, Yang et al. [70] published a numericaland experimental study of the modal responses of shells made

from CFRP corrugated core sandwich. The study considered theinfluence on modal properties of material thickness, corrugationdepth, corrugation angle and also whether corrugations ran aroundthe circumference or along the length of the cylinder. However,there are many different option of corrugation geometry thatremain to be considered by these works; indeed neither workcan be considered as directly relevant to morphing corrugationsbecause they both consider rigid corrugations only.

Not all work is dedicated to rectangular planforms; in Ren-Huaiand Dong [71], the authors consider the flexural stiffness andvibration of circular corrugated plates, with the primary motiva-tion being components of precision machinery. This paper is inter-esting as it uses an energy based approach to reformulate theresponse of the corrugations in a radial coordinate system, and alsoallows for large deflections. However, few practical morphing air-craft components will meet the symmetry requirements of themodel developed.

2.4. Impact

There appears to be no literature that is directly concerned withimpact loads on corrugated skins within the context of morphingaircraft. However, the low-velocity impact performance of corru-gated panels and sandwich cores has been widely studied in otherapplications. In general, this interest is due to the complex defor-mations that can occur in a corrugation as it is impacted andpotentially crushed; internal buckling, contact friction and plasticdeformation can all occur, and these may be seen as energy absorb-ing processes, that therefore may protect other elements in thestructure from damage.

In Toccalino et al. [72], the authors proposed a variety of corru-gated forms for use in an impact energy absorber. One of theseforms includes an interesting arrangement of two layers of corru-gation in close proximity, so that frictional contact between thelayers dissipates yet further impact energy. Jiang et al. [73] consid-ered impact on corrugations made from bamboo fibre based com-posites. This paper finds that the laminate stacking sequence andcorrugation direction have a complicated effect on the impactenergy absorbed by the impact, with a unidirectional layup withfibres oriented along the corrugation channels being optimal.However, the choice of different layups led to completely differentfailure modes in response to impact, with the mode varyingbetween tensile fibre failure, delamination or local buckling. A dif-ferent approach to the impact of corrugations is found inKhabakhpasheva et al. [74], which discusses corrugations impacted


by waves of liquid, and the complex dynamics that arise due totrapped gas bubbles that become pressurised as a result of thewave impact.

More literature is dedicated to corrugations when used as coresin sandwich panels, and while a full review of sandwich panelsunder impact is beyond the scope of this section, a few examplesare listed here. Zangani et al. [75] described a sandwich panel witha phenolic foam core, reinforced by an FRP corrugation, with sim-ulations of an impacting ball. Kiliçaslan et al. [76] studied impactson a multi-layer stack of aluminium honeycomb sandwiches.Russell et al. [77] showed experimental data for impacts of differ-ent speeds on an e-glass composite with an integrated corrugatedcore, both with and without a foam filling to stabilise the corruga-tion webs, and highlighted the difference between buckling andcompressive failure modes. Further examples can be found in thepackaging industry, for example Garcia-Romeu-Martinez et al.[78].

However none of these studies address structures that aredesigned to have the flexibility required for morphing aircraft oradaptive structural elements. In this context the idea of exploitingthe complex deformation of corrugations under impact as means ofenergy absorption is feasible, only if the damage caused by impactis reversible (buckling with no permanent deformation), or limitedto an extent that does not impinge on the structural and actuationrequirements of the corrugation. However, if these requirementsare met, it should be possible to develop corrugated skins withbenign characteristics under impact. There remains much scopefor new studies that consider the more flexible types of corruga-tions considered for morphing applications.

2.5. Fatigue

In the context of fatigue of corrugated plates, very little researchhas been carried out. The few works published in this area focusmainly on the investigation of girders with corrugated steel webs,whose main applications can be found in highway bridges. Here,the use of corrugated plates in girder webs represents an alterna-tive to achieve considerable out-of-plane stiffness and bucklingresistance without the need to use stiffeners or thicker web plates.A typical configuration of a corrugated web consists of folds paral-lel and inclined to the longitudinal direction of the beam, as shownin Fig. 6.

In this context, Wang and Wang [79] studied experimentallythe fatigue assessment of welds joining corrugated steel webs toflange plates. The results of the study revealed that most of thecracks initiated at the weld toe of the external weld line of thetransition curvature and propagated through the main plate thick-ness. The fatigue strength of the test joints was improved with thedecrease of the angle hc as shown in Fig. 6. Such an improvement

Fig. 6. Schematic representation of a welded joint with corrugated plate, Wang andWang [79].

was found to be less significant when hc increased over 45�.Wang and Wang [80] also investigated analytically and experimen-tally the carbon fibre-reinforced polymer strengthened weldedjoints with corrugated plates. The authors showed that the fatiguecrack generally occurred in the region of the transition curvaturebetween the longitudinal fold and the inclined fold of the corru-gated plate. The authors also reported that the joints with transi-tion curvature region reinforcement and single sidereinforcement produce slightly lower rigidity but longer fatiguelife in contrast to those with full width reinforcement on the dou-ble side of the main plate.

Anami et al. [81] investigated experimentally and analyticallyfatigue performance of the web-flange weld of steel girders withtrapezoidal corrugated webs using large-scale girder specimens.By analysing the fatigue cracks in corrugated web girder speci-mens, the authors were able to determine the corresponding fail-ure modes.

The fatigue strength of the web-flange weld was also examinedby Anami and Sause [82] by means of finite elements and crackpropagation analysis. The authors concluded that the fatiguestrength is affected negatively by the existence of the longitudinalfolds of the corrugated web. They also found that it is necessary tohave a large bend radius to eliminate the influence of the longitu-dinal folds.

Sause et al. [83] studied eight large-scale girders subject tofour-point bending fatigue tests. In this study, fatigue cracks initi-ated in the tension flange at the web-to-flange fillet weld toe alongthe inclined web folds and adjacent bend regions and propagatedin the flange. It was found that steel corrugated web I-girders exhi-bit a fatigue life longer than that of conventional steel I-girderswith transverse stiffeners.

Henderson and Ginger [84] investigated the low-cycle fatigueresponse of corrugated metal roof cladding to fluctuating windloads. They demonstrated that the initiation and propagation ofcracks, the type of cracks and the number of cycles to failure weresimilar in several cladding specimens subject to static, cyclic andsimulated cyclonic wind loads. The authors also concluded thatthe peak load during a cycle and its amplitude mainlygovern crack initiation and growth, more than the cycling rateand form.

Kövesdi and Dunai [85] studied experimentally the fatiguebehaviour of trapezoidally corrugated web girders. Six large-scaletest specimens were investigated under static and repeated load-ing. They determined that the combined loading condition on thecorrugated web girders has a significant influence on the fatiguelife. Furthermore, the results highlighted the importance of theweld size from the point of view of fatigue design. By using a smal-ler weld size, the authors concluded that the fatigue life of the gir-der was longer and therefore, they recommended the usage of theminimal required weld size for design purposes.

Ibrahim et al. [86] showed that the fatigue life of plate girderswith corrugated webs is 49–78% higher than the conventionallystiffened plate girders with full-depth stiffener when subjected tothe same stress range. They also concluded that the fatigue lifecan be improved with the trapezoidal waveband without a signif-icant decrease in the static capacity.

Takeshita et al. [87] investigated the dynamic behaviour of newtypes of shear connectors between a corrugated web and a con-crete flange in a composite girder. The shear connectors consistedof studs welded to the top flange; holes with penetrating reinforce-ment placed on the top of the corrugated web; and holes with pen-etrating reinforcement and wire net. Two-point fatigue tests wereconducted using the above three types of shear connectors.Experimental results revealed that, holes with penetrating rein-forcement were more effective than studs in the case of compositegirders with corrugated web.

Fig. 9. Six basic deformation mechanisms, Kress and Winkler [99].


2.6. Homogenization and equivalent modelling

Over the last two decades, homogenisation-based modellingtechniques have attracted considerable attention within the com-putational mechanics community [88–92]. The importance andincreasing interest in this area stems mainly from the ability ofthese techniques to capture the effective response of complexmicrostructures under a wide range of conditions. In such cases,the structural response has to be approximated to avoid the com-putational modelling of the corrugations and thus, circumvent themajor drawback of excessive computing times. Often the loads arewell distributed and only the overall deflections are required. If thedimensions of the whole corrugated panel are much larger thanthe period of the corrugations, then a suitable approach is theuse of homogenisation techniques, in which the corrugated panelis replaced by an orthotropic plate with equivalent stiffness prop-erties [2,40,93,94]. Fig. 7 illustrates further details on this concept.

In homogenisation-based equivalent models, the effectiveresponse is calculated by means of a representative volume ele-ment (RVE) of material or structure. The RVE is such that itsdomain X has a characteristic length much smaller than that ofthe macroscopic continuum and, at the same time, is sufficientlylarge to represent the macroscopic mechanical behaviour in anaveraged sense. Fig. 8 shows the choice of a typical RVE of a corru-gated structure.

Periodic boundary conditions are often adopted to model corru-gated sheets. Periodic boundary conditions are typically associatedwith the modelling of periodic media. In this particular case, theRVE is a so-called unit cell whose periodic repetition generatesthe entire heterogeneous macrostructure [92]. Here, the funda-mental assumption consists of prescribing identical displacementvectors u for each pair of opposite points, y+ and y�, of the RVEboundary domain @X, such that, u(y+) = u(y�).

In the context of modelling of corrugated panels, Briassoulis[60] and McFarland [96] investigated the equivalent flexural stiff-ness of sinusoidal and rectangular corrugations. Briassoulis [60]studied corrugated shells on the assumption of thin and uniformthickness, and proposed new expressions for their equivalentorthotropic properties. McFarland [96] investigated the static sta-bility of corrugated rectangular plates loaded in pure shear.Dayyani et al. [38] proposed numerical and analytical solutionsfor the modelling of composite corrugated cores under tensileand three-point bending tests. Their results revealed that the

Fig. 7. Schematic representation of a fully modelled corrugated she

Fig. 8. Typical RVE chosen for the modelling of

mechanical behaviour of the core in tension is sensitive to the vari-ation of core height.

Kress and Winkler [97] and Winkler and Kress [98] derived ana-lytical expressions for an equivalent orthotropic plate with circularcorrugations. Later, Kress and Winkler [99] studied corrugatedlaminates by solving a set of six load cases under the assumptionof generalised plane-strain. The first three cases correspond toin-plane loading states which are associated with extension alongthe x-direction (Nx) and y-direction (NY ), and shear in the xy-plane(Nxy). Here, the x and y-axes are assumed to define the plane of thecorrugated sheet. The other three cases correspond to out-of-planeloading states represented by bending along the x-direction (Mx)and y-direction (My), and twist out of the xy-plane (Mxy). Thesesix cases are independent and can be combined linearly to formany generic loading state as long as the superposition principleholds. This is generally true for linear analyses. Fig. 9 shows theschematic representation of these six basic cases.

By assuming that the mechanical response of the corrugatedsheet can be established in terms of these six independent cases,the constitutive equation of the equivalent orthotropic plate isdetermined as (Xia et al. [93])

et and its equivalent orthotropic model, Wennberg et al. [95].

a corrugated structure, Dayyani et al. [121].


Nx

Ny

Nxy

Mx

My

Mxy

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>;

A11

A12

0

000

A12

A22

0

000

00A66

000

000

D11

D12

0

000

D12

D22

0

000

00D66

2666666666666664

3777777777777775

ex

ey

�cxy

kx

ky

kxy

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

ð4Þ

where ex, ey, cxy, kx, ky and kxy are the strain components in the glo-bal coordinate system of the corrugated sheet associated with theircorresponding force and moment components Nx, Ny, Nxy, Mx, My

and Mxy, respectively. In the above equation, the coupling termsbetween in-plane strains and out-of-plane loads have been ignored.

For the definition of each of the components Aij and Dij (with i,j = 1, 2, 6) in Eq. (4), several authors have proposed analyticalexpressions for different geometries of corrugation. Refer, forinstance, to those expressions proposed by [2,60,93,100–103],among many others.

One approach to implement the above constitutive laws Eq. (4),within a standard commercial finite element software is to uncou-ple the in-plane and out-of-plane mechanical responses. That is, forin-plane loading conditions we have the following constitutiveequation

Nx

Ny

Nxy

8>>>><>>>>:

9>>>>=>>>>;¼

A11 A12 0A12 A22 00 0 A66

26664

37775

�ex

�ey

�cxy

8>>><>>>:

9>>>=>>>;

ð5Þ

and for out-of-plane conditions we have

Mx

My

Mxy

8>>><>>>:

9>>>=>>>;¼

D11 D12 0D12 D22 00 0 D66

26664

37775

�kx

�ky

�kxy

8><>:

9>=>; ð6Þ

For an orthotropic sheet subject to in-plane loads, the followingconstitutive equation applies:

�ex

�ey

�cxy

8><>:

9>=>; ¼

1Ex

� mxy

Ex0

� mxy

Ex

1Ey

0

0 0 1Gxy

266664

377775

�rx

�ry

�sxy

8><>:

9>=>; ð7Þ

By comparing Eq. (5) with the inverse relation of Eq. (7), it isstraightforward to obtain the following expressions for thein-plane equivalent mechanical properties. The Young’s modulusalong the x-direction (according to Fig. 7), that is, along the corruga-tion direction, is given by

Ex ¼A11A22 � A2

12

tA22ð8Þ

and the Young’s modulus along the y-direction is

Ey ¼A11A22 � A2

12

tA11ð9Þ

where the parameter t is the thickness of the equivalent plate.Furthermore, the in-plane Poisson’s coefficient is

txy ¼A12

A22ð10Þ

and the corresponding in-plane shear modulus is

Gxy ¼A66

tð11Þ

On the other hand, for an orthotropic sheet subject to out-of-planeloading conditions, the following stress–strain expression applies:

�rx

�ry

�sxy

8>>><>>>:

9>>>=>>>;¼ 12

t3

D11 D12 0D12 D22 00 0 D66

26664

37775

�ex

�ey

�cxy

8>>><>>>:

9>>>=>>>;

ð12Þ

By comparing Eq. (7) with the inverse relation of Eq. (12) weobtain the following equivalent mechanical properties. TheYoung’s modulus along the x-direction is

Ex ¼12ðD11D22 � D2

12Þt3D22

ð13Þ

and the Young’s modulus along the y-direction is

Ey ¼12ðD11D22 � D2

12Þt3D11

ð14Þ

In addition, the in-plane Poisson’s coefficient is

mxy ¼D12

D22ð15Þ

and the corresponding in-plane shear modulus is

Gxy ¼12D66

t3 ð16Þ

With the above equivalent mechanical properties at hand, it isstraightforward to introduce them in a commercial finite elementprogram and model the corrugated geometry via a standardshell-type finite element without resorting to more computation-ally expensive modelling techniques.

As commented earlier, the previous modelling strategy relies onthe linear combination of six independent cases, which is generallyvalid for linear problems. However, non-linear effects have alsobeen investigated. Samanta and Mukhopadhyay [100] performednon-linear geometric analyses on trapezoidal corrugated panels.They proposed an equivalent orthotropic model by taking intoaccount both extensional and bending rigidities. Peng et al. [104]investigated the equivalent elastic properties of sinusoidal andtrapezoidal corrugated plates by means of a mesh-free Galerkinmethod. Liew et al. [105] used this method for the geometricallynonlinear analysis of stiffened and unstiffened corrugated plates.Large deflection von Karman theory was adopted in the nonlinearanalysis of the orthotropic plate. Both the equivalent flexural andextensional properties were employed in the analyses.

In the particular context of morphing wings, corrugated lami-nates represent an ideal solution for the design of morphing skinsin adaptive aircraft structures. Yokozeki et al. [2] developed a sim-ple model for the in-plane stiffness of corrugated composites. Theymanufactured carbon fiber plain woven fabrics as candidates forflexible structural components. They tested them under tensileand bending loads in both in-plane longitudinal and transversedirections and their results were compared with the analytical pre-dictions. Thill et al. [40,106] compared the homogenised plateproperties for candidate morphing aircraft skins to experimentalresults by adopting the same procedure proposed by Samantaand Mukhopadhyay [100].

2.7. Optimization

The design of corrugated structures can be improved signifi-cantly by implementing the optimization techniques in which


the geometrical parameters and material properties of the struc-ture play a dominant role. Some of the goals for the optimizationmay be represented as: minimizing the weight of the structure,maximising the structural stiffness of the structure, postponingthe buckling load of the structure as much as possible, and max-imising the distance between the natural frequencies of the struc-ture and the excitation frequencies and maximising the impactenergy absorption. However the key question in the optimal designof these structures is the measure of what is desirable about adesign. In practical applications, the design process is often mea-sured with respect to multiple objectives which are often conflict-ing. In other words achieving the optimal value for one objectiverequires compromise on other objectives. In this case the goalmay be to find a representative set of Pareto optimal solutions,and/or quantify the trade-offs in satisfying the different objectives,and/or finding a single solution that satisfies the subjective prefer-ences of a human decision maker. In this section a review of the lit-erature for the optimization of the mechanical behaviour of thecorrugated structures is presented.

Liang et al. [14] investigated the optimum design of metallictrapezoidal corrugated core sandwich panels subjected to blastloads by using a combined algorithm in which the FeasibleDirection Method [107] was coupled with the Backtrack Programtechnique [108]. A simple-beam theory and small-deflection platetheory were adopted to model the behaviour of the corrugated coresandwich panels. They considered a simply supported boundarycondition for the panel and estimated the blast load by an equiva-lent static pressure. Geometric parameters such as: the corrugationleg, corrugation angle, face sheet thickness, core thickness and cor-rugation pitch were selected as design variables. The optimizationproblem was performed with the objective of minimizing theweight of the corrugated sandwich panel with regards to explicitconstraints arising from structure buckling and failure behaviour.Manufacturing limitations on the sizes or shape of corrugationswere considered as implicit constraints. They realised that the cor-rugation leg, corrugation angle, core thickness and corrugationpitch were important design parameters for the core componentwith respect to buckling strength. For the face sheet with respect

(a) Geometric parameters of optimization (b) Sy

(c) The original configuration (grey), the optimization for isotropic material (black), bes

Fig. 10. Optimum design of corrugated board un

to axial stress, the corrugation leg and face sheet thickness werekey design parameters.

Rathbun et al. [109] implemented a general methodology forthe design of the geometry of light weight sandwich panels. Theyconsidered several core topologies, such as: square-section trussmembers in pyramidal and tetrahedral configurations, square hon-eycombs and triangular corrugated sheets. Closed-form analyticalsolutions were proposed for the optimal design when the numberof design parameters was up to three; however for a furtherparameter which was needed to fully characterise the corrugationgeometry, they used numerical techniques. The geometric param-eters they considered were: face sheet thickness, core thickness,height of corrugation and corrugation length. The four possiblefailure modes of the structure including face yielding, face buck-ling, core yielding and core buckling were represented as con-straints in the optimization problem. Minimizing the weight ofthe structure for a prescribed load, a numerical proceduredescribed by Wicks and Hutchinson [110] was used. The resultsrevealed that the four parameter optimization was kind of mar-ginal benefit; i.e. leading to a design weight slightly lighter thanthat of the three parameters optimization. It was shown that thevariations in the weights of the optimised panels were quite smallin almost all cases, specifically in contrast to the weight of a solidpanel. However for the case of a transverse corrugated core panel,the optimised weight was reported almost twice as heavy as theothers. They concluded that other objectives or constraints, repre-senting the manufacturing costs and prospect of multifunctionalitymust be considered in the optimization problem. The structuralanalysis, however should have been developed further to take toaccount the coupled effects arising from the anisotropy of thestructure; i.e. different characteristics of longitudinal and trans-verse orientations.

Daxner et al. [111] optimised the corrugated paper board withthe purpose of attaining the maximum reduction in thearea-specific weight of the structure in the presence of bendingstiffness constraints as well as local and global stability constraints.They considered only the geometrical parameters of the sinusoidalcorrugated profile in the optimization (as shown in Fig. 10(a)) and

stematic variation of the unit cell length and its influence on the critical compressive membrane force

optimized geometry by the semianalytical t three designs for orthotropic material model

der buckling constraints, Daxner et al. [111].


studied the effective mechanical behaviour of the structure withinthe limits of linear shell theory by application of appropriate peri-odicity boundary conditions. Both isotropic and orthotropic mate-rial models were considered in their analysis; the corrugated boardwith isotropic material was analysed by a semi-analyticalapproach and for orthotropic material, the optimization algorithmwas wrapped around a finite element model. Calculation of thecritical loads with respect to local instabilities involved a mini-mization scheme within the optimization loop in order to findthe critical buckling wavelength and the unit cell size was adjustedaccordingly (Fig. 10(b)). In their analysis, the function ‘NMinimize’in Mathematica was used with different optimziation techniques:Downhill-Simplex method and a genetic algorithm to solve theproblem. The optimised results (as shown schematically inFig. 10(c)) introduced a set of parameters describing a new designof corrugated board with the same buckling strength, but anarea-specific weight which was reduced by more than 18% withrespect to the original design.

Bapanapalli et al. [17] proposed the corrugated sandwich panelas a candidate of the integral thermal protection system of the spacevehicle from extreme aerodynamic heating, due to their heat trans-fer and load bearing capabilities. They developed an optimizationproblem as a part of the design process with the objective of mini-mizing the mass per unit area of the trapezoidal corrugated sand-wich panel. The constraints were represented in terms of heattransfer and structural mechanics, such as: sustaining the facesheets temperature below certain limit and maintaining the struc-ture far from the global and local buckling due to mechanical andthermal forces. The response surface approximations (Roux et al.,[112]) of transient heat transfer and buckling analyses were con-structed by use of finite element software, ABAQUS. The FE toolboxwas a function called in the optimization process which was per-formed by use of fmincon command in Matlab. The optimised resultsshowed that the corrugation webs should be as thin as possible inorder to reduce the mass of the structure as well as the amount ofheat entering the corrugated sandwich panel. In other words, theyrealised that the material for the corrugation webs should have avery low conductivity and high Young’s modulus. They suggestedthat the conductivity of the webs can be decreased further by remov-ing some material from the corrugation webs which will also reducethe weight of the corrugated sandwich panel. They claimed that thebuckling capacity will also improve in this case, contrary to intu-ition. However, a disadvantage could be the shear stiffness of thesandwich panel which reduces considerably.

Hou et al. [113] investigated the effects of the shape (trape-zoidal and triangular configurations) and the corresponding geo-metrical parameters on the crash behaviour of corrugatedsandwich panels. Two different types of impact loading were con-sidered in their analysis: the low-velocity local impact and the glo-bal planar impact which were simulated using ANSYS ParametricDesign Language (APDL) and LS-DYNA finite element software.The surrogate modelling method (Forrester et al. [114]) was usedas an approximate modelling technique to mimic the behaviourof the finite element simulations as closely as possible, whilereducing the cost of computations. For the case of low velocitylocal impact, the corrugated core with both trapezoidal and trian-gular corrugation profiles was optimised with the purpose of max-imising the energy absorption (internal energy) of the structure.However for the case of global planar impact, a multi-objectiveparticle swarm optimization (Mostaghim et al. [115]) was per-formed with the objectives of maximising the internal energyand minimizing the peak crushing force. The comparison of twooptimal configurations with the identical face sheet thicknessand core density showed that the triangular configuration has bet-ter crashworthiness performance.

3. Corrugated skin for morphing wing

Improving the performance of an aircraft is significantly impor-tant for a variety of reasons such as: reducing the energy consump-tion, decreasing the toxic emissions and noise pollution orincreasing the maneuverability of the aircraft (Argüelles et al.,[116]; European Commission, [117]; Barbarino et al. [118]). Theproblem with the current traditional aircraft is that they cannotbe optimised for every single point of the flight envelope [118]. Inother words the wings of an aircraft are a compromise that limitsthe flight to a range of conditions where the performance of the air-craft at each condition is sub-optimal. Hence, a new generation ofaircraft, known as morphing aircraft, are needed for furtherimprovement of the aircraft performance without unacceptablepenalties in terms of cost, complexity and weight. The morphingaircraft have the ability to adapt their shape in flight so as to alwaysmatch the optimal configuration. Three major types of the morph-ing deformations which can be envisaged for an aircraft wingmaybe classified as: planform alteration involving span, chord,and sweep changes; out-of plane transformation including twist,dihedral/gull and spanwise bending; as well as airfoil adjustmentsuch as camber and thickness alteration. However, the require-ments of morphing aircrafts are conflicting. For instance the designof a proper skin is a huge challenge and a key issue. The skin mustbe stiff to withstand the aerodynamic loads, but flexible to enablethe expected large shape changes. Among all possible structuresfor morphing skins such as segmented structures, reinforced elas-tomers or flexible matrix composite tubes embedded in a low mod-ulus membrane, corrugated skins has attracted more attention inthe literature. This is because the corrugated skins have exceedinglyanisotropic behaviour; they are stiff along the corrugation direc-tion, but flexible in the transverse direction. In addition, corrugatedskins have other remarkable characteristics, such as high ratio ofstrength to density, good energy absorption and easy fabrication.

3.1. Introduction, general challenges

A variety of possible materials and structures such as: elas-tomer matrix composites and auxetic materials are suggested asmorphing skins in the literature (Thill et al., [119]). Among thepotential candidates mentioned in the literature, corrugated coreshave exceedingly anisotropic behaviour; they are stiff along thecorrugation direction, but flexible in the transverse direction.Yokozeki et al. [2] demonstrated the use of corrugated compositesheets as a candidate material for flexible wing skins. As men-tioned earlier, the mechanical properties of the corrugated sheetsobtained by experiments were presented and followed by a simpleanalytical model. They proposed the use of stiff rods with corru-gated skin in the longitudinal direction to increase further the ani-sotropy of the skin and suggested the use of elastomeric coating formaintaining the smooth aerodynamic surface of the skin.

However the optimal design of these structures requireshigh-fidelity models of the coated corrugated skin to be incorpo-rated into multi-disciplinary system models. Hence numericaland experimental investigations were required which maintainthe dependence on the nonlinear static and dynamic behaviourof these structures. Dayyani et al. [33] noticed this necessity andstudied the nonlinear effects of the elastomeric coating and themechanism of deformation of the composite corrugated core.They performed a series of experiments such as: static and cyclictensile tests on the standard samples of composite laminates andelastomeric strips; static and cyclic tensile tests on the compositecorrugated skin with and without elastomeric coating andthree-point bending test on the coated corrugated skin. The linearstress–strain curves of the glass fibre laminates demonstrated the


high stiffness of corrugated core. The large elastic deformationcapacity of the elastomers was also demonstrated; the elastomericspecimens stretched 1.5 and 3.3 times over their original length inlongitudinal and transverse directions respectively. The experi-ments and FE simulations of the coated corrugated skin in tensileand bending tests are shown in Fig. 11. The tensile behaviour ofthe coated corrugated skin, i.e. the three stages of the deformationmechanism with respect to the function of the elastomer coatingwas discussed before and after the failure of the corrugated core.In terms of the flexural behaviour of the coated corrugated skin,they realised that the elastomer coating functioned as springs thatundergo only tension. In other words the elastomer coatingresisted the gap opening between two adjacent corners of eachunit cell of the corrugated core. Because of the membrane functionthe elastomer coating showed, the other elastomer coating, whichwas subjected to the compressive forces, wrinkled and causednon-smooth surface for the skin during bending. This was maybethe main drawback of the corrugated skin for maintaining theaerodynamic performance as high as possible. However they

(a) Tensile test of the composite corrugated core with elastomeric coatings

(c) Coated corrugated skin during the three-point bending test

Fig. 11. Experiments and FE simulations of the coated corruga

suggested two concepts to deal with this drawback:pre-stretching the elastomer coating and a reentrant corrugatedconfiguration.

Structural components which are subjected to cyclic loadingmay yield due to fatigue, causing them to fail at stress levels lowerthan static mechanical loading. The cyclic loading behaviour ofcomposite corrugated skin for morphing applications, where thesource of cyclic stresses may be the aerodynamic flow over the skinor the actuator, is important. In this regard, Dayyani et al. [33]investigated the cyclic loading behaviour of the corrugated skinwith and without elastomeric coating before and after the elasticlimit and compared the energy dissipation in each case. It mustbe mentioned that the authors of this paper interpreted the limitin which the delamination cracks initiated and started dissipatingthe strain energy of the structure as the elastic limit. They observedthe Mullins effect (Diani et al., [120]) in the elastomer coatingseven before the elastic limit of the corrugated skin, because ofthe microscopic damage which resulted in considerable energydissipation in the elastomeric coatings.

(b) Tensile properties of the corrugated corewith and without elastomeric coatings

(d) Bending force-displacement behaviour; experiment and simulation

ted skin in tensile and bending tests, (Dayyani et al. [33]).


However the optimal design of these corrugated skins requiredsimple models of the structure to be incorporated intomulti-disciplinary morphing system models. Therefore equivalentstructural modelling and homogenizations techniques wererequired to retain the dependence on the geometric parametersand material properties of the coated corrugated skin, while reduc-ing the cost of computations. Dayyani et al. [121] investigated ananalytical homogenisation model which took into account thefunction of the elastomer coatings based on Castigliano’s secondtheorem. They proposed two analytical solutions for calculatingthe equivalent tensile and bending flexural properties of a coatedcomposite corrugated core in the longitudinal and transversedirections of the skin. The effect of combined loading and the num-ber of unit cells on the mechanical behaviour of the coated corru-gated skin were investigated in their work and were verified withnumerical simulations and experimental analysis in the literature.The comparison demonstrated the suitability of the proposedequivalent model for application in further design investigations.

Dayyani et al. [122] investigated the mechanical properties ofthe morphing corrugated skin with regard to different corrugationshapes. They took into account three typical configurations asre-entrant, rectangular and trapezoidal corrugation and modelledtheir mechanical behaviour in ABAQUS to realise which configura-tion leads into higher out-of-plane stiffness, lower in-plane stiff-ness and mass. The sinusoidal corrugation shape was notincluded in their analysis, because of the presence of a single lineof contact between the elastomer coating and corrugated core, pro-viding insufficient area for bonding as opposed to other counter-parts. They found out that although the uncoated reentrantcorrugated core had maximum bending stiffness and minimumtensile stiffness, adding the elastomeric coating to the corrugatedcore resulted in minimum bending and maximum tensile stiffness.This story was reversed for the trapezoidal corrugation shape.Hence they evidenced the suggestion of the coated trapezoidal cor-rugated core for the application of the morphing skin, which hadthe highest ratio of the bending stiffness to tensile stiffness. Thecalculation revealed also that the coated reentrant and coated rect-angular corrugated cores were 30% and 11% heavier than thecoated trapezoidal corrugated core.

Dayyani et al. [123] proposed a general super element for a cor-rugated core unit cell with elastomeric coating for the applicationof morphing skin. The direct stiffness method and Castigliano’s sec-ond theorem, together with appropriate boundary conditions, wereapplied to obtain the stiffness matrix of the generic super elementwhich captured the small deformation of 2D thin curved beamswith variable curvatures. Numerical and symbolic analytical mod-els were investigated to validate the accuracy and efficiency of thepresented super element model. The super element used the geo-metric and mechanical properties of the panel as variables thatcould be applied for further topology optimization studies. Theyused the ratio of the bending stiffness of a panel to the in-planestiffness as a performance metric of the morphing skin for a varietyof corrugation profiles such as: trapezoidal and polynomial curva-ture corrugation as shown in Fig. 12, where the bending stiffnessdetermined the out-of plane displacement due to aerodynamicloadings, and the in-plane stiffness determines the required actua-tion forces. They showed that for the trapezoidal corrugationshape, the stiffness ratio increased with the height of the corruga-tion and the angle of the sloping side panel and for the polynomialcorrugation shape, the stiffness ratio increases when the corruga-tion shape became narrower. Low fidelity system level optimiza-tion of morphing aircraft, can be one of the main application ofthe developed model, where the in-plane and out-of-plane stiff-ness of a corrugated skin may be estimated accurately and effi-ciently from its geometry rather than a detailed finite elementmodel.

3.2. Corrugated skin, different morphing applications

To the knowledge of the authors the investigations for theapplication of the corrugated skin is so far limited to the cambermorphing, winglet morphing and span wise morphing extension.Although each of these applications have their own specific bound-ary conditions, structural and aerodynamic loading configurationas well as the geometric and manufacturing constraints; the corru-gated skin shows these two main characteristics in common:highly anisotropic behaviour and lightness. In addition the aerody-namic performance in all these applications is highly dependent onthe corrugation geometry and Reynolds number. The aerodynamicperformance however can be improved further if the corrugatedskin is coated by pre stretched elastomeric face sheet, a segmentedskin or used with foam slices filling the empty surface betweenthe corrugated unit cells. On the other hand these solutions leadto the further mass of the structure. Hence any concept formodifying the performance of the corrugated skin should passthrough the multi-objective optimization and multidisciplinarysystem level design.

Low fidelity analysis in the first steps are highly necessary foruse in these system level analyses and optimizations, since theyprovide a feasible estimate of each design objective while keepingthe computations efficient in terms of time and cost. This sectionreviews the literature of the corrugated skin with the given focusto camber morphing, winglet morphing and span extensionmorphing.

3.2.1. Camber morphingOne of the effective ways to change the aerodynamic forces

generated by a wing is through actively varying the airfoil’s cam-ber, which allows for control of the aircraft flight path and opti-mization of the aerodynamic performance over different flightregimes. In addition to the fixed-wing applications, rotary aircrafthave also followed the concept of the change in camber to post-pone effectively the occurrence of stall and vibration control ofthe rotary blades. However, the concept of camber variation isnot a recent novel achievement and has been traditionally pro-posed and followed through the use of discrete trailing edge flaps.Nevertheless, a smooth and continuous change in camber is still aninterest in the aerospace industries so as to reduce significantly thedrag and noise penalty associated with rigid flap deflections. Asmooth and continuous change in the camber through a highly ani-sotropic skin can also lead to a further reduction in the systemcomplexity which decreases the mass of the structure and produc-tion and maintenance cost as well. In this section, the applicationsof the corrugated skin for the camber morphing structures whichare proposed in the literature are discussed.

Thill et al. [19] used the composite corrugated sheets for theskin of the trailing edge of a NACA 0024 aerofoil section. Both cordwise extension and camber morphing deformations were consid-ered for the trailing edge section which was about 35% of the 1meter length cord. They used a simple scissor mechanism toextend the length of the cord and adjusted the manual camberdeflections by use of locating pins. The skin was manufacturedfrom attaching and filling foam into two corrugated laminates.Considering the manufacturing limits, the dimension of a corru-gated unit cell was between 5–10 mm, which represented about1% of the chord length for acceptable aerodynamic performance.Low speed wind tunnel testing and open source code XFOIL as wellas a two-dimensional computational fluid dynamics (CFD) panelmethod with viscous effects (Derla, [124]) were carried out toexplore the limitations of these concepts. Reynolds number andMach number which were used in the analysis were about2 * 106 and 0.1 respectively, over a range of angles of attack forthe NACA 0024 with morphing trailing edge. Fig. 13 shows the

(a) Different variations of the trapezoidal corrugated unit cell (b) Distribution of corresponding out of plane to in-plane stiffness for trapezoidal corrugation

(c) Corrugated unit cell with geometry of polynomial functions

(d) Distribution of corresponding out of plane to in-plane stiffness for polynomial functions

Fig. 12. A variety of the corrugation geometry of the unit cell obtained by use of super element of a curved beam, (Dayyani et al., [123]).

Fig. 13. The trailing edge section of the morphing NACA 0024 airfoil with corrugated skin, Thill et al. [19].


trailing edge section of the morphing NACA 0024 airfoil with cor-rugated skin, as it is stretched up to 4% and deflected up to 12�.The main problem of this concept was the lack of internal structure

to support the skin against the aerodynamic and structural load-ings. This problem caused the global buckling of the lower skinduring the actuation, as it is subjected to compressive strains.


The wind tunnel results highlighted a major increase in drag gen-eration, when the morphing arifoil was compared to the conven-tional NACA 0024. Improving the aerodynamic surface of theskin, they suggested the use of discontinuous segmented compos-ite laminates on top of the corrugated sheets.

Yokozeki et al. [125] proposed a camber morphing airfoil inwhich the corrugated sheet was used as internal structure in thetrailing edge section. The morphing section consisted of a circularcorrugated structure and upper thin skin, both of which were madeof carbon fibre reinforced plastic. The lower surface of the corru-gate region was coated by a thin plastic sheet. The morphing sec-tion consisted almost 30% of the chord length, as shown inFig. 14. Nonlinear finite element analysis was used to confirmthe feasibility of the proposed morphing system in terms of actua-tion force and displacement. They developed a low speed (30 m/s)wind tunnel test for the camber morphing airfoil (WortmannFX63-137 baseline), inside which two servomotors were used asthe actuation system to control the airfoil shape by thechord-wise tension of the connected wires. The aerodynamic per-formances of the morphing and hinged mechanism wing werecompared together and it revealed that the morphing model exhib-ited superior properties in lift coefficients. However the proposedconcept had some restrictions as it limited the morphing deforma-tion to downwards trailing edge displacement. Although this mightbe useful for ‘‘flaps’’ and low speed maneuvers such as take-off andlanding, but is not suitable for higher speed maneuvers. In additionthe unsmooth lower surface of the morphing trailing edge has con-siderable impact on increasing the generated drag.

As mentioned before the role of internal structure for the sup-port of corrugated skin in the camber morphing deformation is veryimportant. This is because the internal structure supports the skinagainst the aerodynamic loadings, structural shear and bucklingloads and vibration. In addition the internal structure has a mainrole in enforcing the morphing trailing edge to move on the rightpath from the start point of the actuation to the desired destination.A further point of discussion of the role of the internal structure isthe influence of the structure on the actuation energy required tomorph, as the correct design of the internal structure can lead tominimizing the power required for actuation.

Probably Dayyani et al. [126] were the first authors in the liter-ature who noticed the importance of investigating the mechanicalbehaviour of the corrugated skin in interaction with the internalstructure, mainly in terms of aero-elastic effects and the boundary

(a) Schematic representation

Fig. 14. Camber morphing airfoil with corrugated structure in trailin

Fig. 15. The geometric design parameters of the NACA0012 camber morphing airfoil

conditions arising from the internal wing structure. They proposedthe design of an elastomer coated corrugated skin for the cambermorphing airfoil which exploited the FishBAC internal structure(Woods and Friswell, [127]), as shown in Fig. 15. Considering theboundary conditions and the number of FishBAC stringers, the geo-metric parameters of the coated corrugated skin were optimised tominimize the in-plane stiffness and the weight of the skin and tomaximise the flexural out-of-plane stiffness of the skin. In thisregard a finite element code for thin beam elements was used withthe aggregate Newton’s technique for performing themulti-objective optimization. The optimization was repeated fora variety of weight distribution in the aggregate method optimiza-tion corresponding to the dominant objective functions and thenormalised optimum Pareto surface was attained. Collecting allof the best compromise points corresponding to all of the configu-rations of FishBAC stringers and corrugation unit cells, the designdecision was made by repeating the process of finding the bestcompromise point among the collection in the normalised spaceof objective functions. Taking to account the cord length of305 mm, the height and the length of the corrugation unit cell cor-responding to decision point were about 4.6 mm and 11 mm. Inaddition, they discussed in detail the advantages of the corrugatedskin over the elastomer skin for the FishBAC morphing airfoil basedon finite element comparison studies. Some of the advantagemight be classified as: almost 18% decrease in weight of the wholemorphing structure, almost five times more flexibility for stretch-ing the skin in the morphing actuation process and about 4.5 timesmore resistance to out-of plane deformation due to aerodynamicloads and buckling deformations caused by actuation. These bene-fits were in addition to the structural anisotropy of the corrugatedskin, which helps it to withstand more aerodynamic loads in thespanwise direction. Considering the specific characteristics of thecorrugated skin and the pressure distribution over the airfoil, theysuggested variable distance between FishBAC stringers; more dis-tance between the stringers in regions exposed to lower pressurewhich resulted in a further reduction of the mass of the structure.

The literature on the buckling of the corrugated skins in morph-ing applications is fairly light, specifically when the effect of theinternal structure should be considered. Many of the key publica-tions on morphing analysis or experimental work have briefly dis-cussed buckling but give little detail on the methods used, for anunusual corrugation incorporating rigid stiffener sections. Shawet al. [128] continued the proposed design of skin in Fig. 15 and

(b) Wind tunnel testing

g edge, Yokozeki et al. (Images courtesy of Tomohiro Yokozeki).

with FishBAC internal structure and coated corrugated skin, Dayyani et al., [126].


investigated the optimization problem for the buckling loads alongthe spanwise direction of the wing. Weight, buckling performance,and actuation compliance of the corrugated skin were three objec-tives they tried to optimise. Classical buckling models with homo-genised plate properties were used for deriving the equivalentplate properties of the skin to obtain approximate estimates ofthe buckling loads. Fig. 16(a) shows the trend of max loads without-of-plane stiffness for optimal solutions of a variety of corruga-tion geometry. In addition to the global and local out-of-planebuckling modes of the skin, they observed the existence of a fur-ther buckling mode which occurs entirely in-plane. This uniquein-plane mode was due to extreme anisotropic behaviour of theskin. As shown in Fig. 16(b), this mode affected deep, finely pitchedcorrugations where the transverse in-plane stiffness became lessthan the out of plane stiffness. Finite element analysis was usedto evaluate the accuracy of the results obtained by the optimiza-tion method which exploited the equivalent methods.

The aerodynamic performance of the skin is perhaps the mainchallenge of the morphing wing. In this regard, Xia et al. [129]investigated the 2D aerodynamic performance, particularly liftand drag characteristics of the NACA0012 airfoil with unsmoothrigid corrugated surface. The effects of corrugation size andReynolds number were analysed and quantified experimentallyand numerically in compare to the standard NACA0012 airfoil.They evidenced that the increase in the roughness of the surfacei.e. increasing the corrugation size, had negative impact on both liftand drag. In addition, the aerodynamic performance of corrugatedairfoils at low angles of attack i.e. the lift-to-drag ratio decreasedslightly as the Reynolds number was increased. They showedthrough CFD simulations how the local flow maintained anattached flow as eddies filled the corrugation troughs andsmoothed the shape of the corrugated airfoil, similar to flowaround streamlined airfoils. In particular, for a deformable trailingedge, the increased drag from the camber morphing skin is in tradeoff with the reduced drag obtained from avoiding the sharpchanges in camber at the hinge mechanism which lead to flowseparation.

3.2.2. Corrugated skin; application in winglet and span extensionmorphing

In addition to camber morphing, there are some other effectiveways to change the aerodynamic forces generated by a wing, such

(a) Trend of max loads with out-of-plane stiffness for optimal solutions of a corrugation

Fig. 16. Multi objective optimization of the corrugated skin

as: through actively varying winglet and wing span. Both thesemethods can control the aircraft performance over different flightregimes and can increase the range and endurance of the aircraft.For example, the increase in the wingspan leads to a higher aspectratio and wing surface; hence it results in a higher lift and higherlift to drag ratio in some aerodynamic configurations(Beaverstock et al., [130]). As a result, the range or endurance ofthe aircraft increases although the wing-root bending momentcan increase considerably due to the larger span. Therefore bothaerodynamic and structural characteristics of the aircraft shouldbe investigated in the design of winglet and span extension morph-ing wings. Corrugated skin may be a good candidate for the appli-cation in both concepts, when the corrugation profile is sweptalong the airfoil, as shown in Fig. 17. The extreme anisotropicbehaviour of the corrugated skin provides sufficient compliancefor the actuation and increases the out of plane stiffness of thewing to withstand more aerodynamic and inertial loads. This isbecause of the combination of the global airfoil curvature alongthe wing span and the local curvature of the corrugations.Furthermore, since the corrugation lines are along the flow direc-tion the aerodynamic performance is less influenced than the cam-ber morphing application. However from a structural perspective,the combination of the global airfoil curvature along the wing spanand the local curvature of the corrugation makes the mechanicalbehaviour of the structure much more complex in contrast to thecamber morphing application. Although the concept of the applica-tion of corrugated skin in winglet and span extension morphing isattracting, not too many efforts have been devoted to these con-cepts in the literature, maybe because of the complexity of themanufacturing method and required 3D structural and aerody-namic analysis.

Ursache et al. [131] presented a preliminary demonstration ofmorphing wingtip with corrugated skin in span wise direction, asshown in Fig. 17(a). One of the key challenges in the demonstrationof the corrugated skin in the span wise direction is the specialmanufacturing process required to hand lay the composite lami-nates around the corrugated mould with air foil cross section.They preferred the plain woven Kevlar over other type materialsbecause of its ease of manufacture into corrugations and its espe-cial characteristics such as: lightness, good durability and impactresistance as well as high stiffness and strength. Finite elementsimulation was performed to assess the mechanical behaviour of

(b) In plane buckling mode of a corrugated skin subjected to compressive load in the vertical direction

with regards to buckling constraints, Shaw et al., [128].

(a) Demonstration of morphing wingtip with corru-gated skin in span wise direction,

(Ursache et al. [131])

(b) Structural and aerodynamic mesh of a half ofa corrugated unit cell in span wise direction,

(Image courtesy of James Fincham).

Fig. 17. The application of corrugated skin in the span wise direction, winglet and span extension morphing.


the corrugated skin while winglet morphing deformation. The FEresults and the trends of the deformation showed some local insta-bilities of the skin and highlighted the need for updating the modelto enhance shape adaptability.

Xia et al. [129] studied the feasibility of the concept of usingcorrugated skin for the application in span extension morphingwing. The geometric parameters of the corrugated skin were opti-mised to minimize the axial stiffness of the skin with respect toout-of-plane deformation constraints. The equivalent structuralproperties of the skin were calculated by ANSYS APDL finite ele-ment simulation based on the global and local geometric parame-ters. They estimated the aerodynamic performance of the skin byuse of low fidelity aerodynamic solvers: Tornado Vortex LatticeMethod (Melin, [132]) and XFOIL in which the effect of camberthickness and airflow in corrugated channels were neglected. Theresults showed that the constraint on maximum out of plane dis-placement forced the optimizer to choose fewer corrugations inthe span direction.

Fincham et al. [133] noted the sharp leading edges generated inthe span wise application of the corrugated skin and examined thelow pass filtering and arc fitting for rounding of the leading edge toimprove the aerodynamic performance. They examined the effectof corrugation wavelength and corrugation depth on the aerody-namic performance, in both 2D and 3D CFD simulations. They rea-lised that that corrugation wavelength has minor effect in theaerodynamic efficiency of the wing in contrast to the corrugationdepth which incurred a significant performance penalty. Theyemphasised that the aerodynamic penalty of corrugated surfacewould be too large to be practical without some kind of compliantskin covering the corrugated surfaces.

4. Conclusion

Corrugated structures have noticeable impacts on the engineer-ing applications due to their superior structural characteristicssuch as extreme anisotropic behaviour and high stiffness to weightratio, mainly arising from their geometric properties. In this papera detailed review of the literature on corrugated structures waspresented through three sections. The first section described differ-ent types of corrugated structures, their specific characteristics andtheir categorised applications. Extending their applications, inno-vation and developments of these structures were discussed interms of introducing further geometric parameters and combiningdifferent material properties.

In the second section, a comprehensive set of analyses about themechanics of these structures was presented. The in-plane and outof plane stiffness of the corrugated panels were reviewed in detailsthrough experimental, numerical and analytical homogenisationanalysis. The most common methods to analyse buckling andvibration of corrugated sheets were discussed such as FEM andcombination of their homogenised models with the classic shell

and plates methods. Although the latter methods had the advan-tage of simplicity, they could not account for local effects andhigher modes, within the structure of the corrugation. The complexdeformations occurring in corrugated structures during impactsuch as: internal buckling, contact friction and plastic deformationwere discussed in terms of energy absorbing capability of thesestructures. The fatigue assessment of the corrugated structuralcomponents subjected to cyclic loading was also reviewed. Theresults revealed that the crack initiations and delamination weretwo main factors for the fracture of these structures. Finally theoptimization problem as an important tool in the design of corru-gated structures was discussed. A variety of optimization objec-tives were represented, where some of them had potentialconflicting nature. This highlighted the importance of finding thebest compromise situation in the multidisciplinary designproblems.

The third section of the paper reviewed the use of corrugatedstructures in morphing applications. It highlighted the importanceof a study which ensures that all likely aspects of the morphingcorrugated skin will be considered and integrated into a completeanalysis. The applications of corrugated skin in camber morphing,winglet morphing and span wise morphing extension was dis-cussed in detail in terms of specific boundary conditions, structuraland aerodynamic loading configuration as well as the geometricand manufacturing constraints that each application poses. Theaerodynamic performance in all these applications was reportedhighly dependent on the corrugation geometry and Reynolds num-ber. This could be improved further if the corrugated skin wascoated by a pre stretched elastomeric face sheet, a segmented skinor used with foam slices filling the empty surface between the cor-rugated unit cells, although these potential solutions could lead tothe further mass of the structure. In fact any concept for modifyingthe performance of the morphing corrugated skin should passthrough the multi-objective optimization and multidisciplinarysystem level design. The low fidelity analysis were found highlynecessary for use in these system level analyses and optimizations,as they provided a feasible estimate of each design objective whilekeeping the computations efficient in terms of time and cost. Thehigh fidelity analysis of the corrugated skins was required in vali-dation of the initial results and modelling complex deformations.This review shows that the level of maturity for morphing corru-gated skins is yet low and for most of the concepts analyses suchas: vibration and aeroelasticity, buckling, impact, fatigue and frac-ture and chemical resistance, have not been so far noticed.

Acknowledgements

The research leading to these results has received funding fromthe European Research Council under the European Union’sSeventh Framework Programme (FP/2007-2013)/ERC GrantAgreement n. [247045].


References

[1] Corrugate. Merriam-Webster.com; 2014. <http://www.merriam-webster.com>.

[2] Yokozeki T, Takeda SI, Ogasawara T, Ishikawa T. Mechanical properties ofcorrugated composites for candidate materials of flexible wing structures.Compos A Appl Sci Manuf 2006;37(10):1578–86.

[3] Gilchrist AC, Suhling JC, Urbanik TJ. Nonlinear finite element modeling ofcorrugated board. ASME Appl Mech Div Publ AMD 1998;231:101–6.

[4] Carlsson LA, Nordstrand T, Westerlind B. On the elastic stiffnesses ofcorrugated core sandwich. J Sandwich Struct Mater 2001;3(4):253–67.

[5] Mallick PK, Boorle R. Sandwich panels with corrugated core-a lightweightingconcept with improved stiffness (No. 2014-01-0808). SAE Technical Paper;2014.

[6] Patel P, Nordstrand T, Carlsson LA. Local buckling and collapse of corrugatedboard under biaxial stress. Compos Struct 1997;39(1):93–110.

[7] Twede D, Selke SE. Cartons, crates and corrugated board: handbook of paperand wood packaging technology. DEStech Publications Inc; 2005.

[8] Singh SP, Chonhenchob V, Singh J. Life cycle inventory and analysis ofre-usable plastic containers and display-ready corrugated containers usedfor packaging fresh fruits and vegetables. Packag Technol Sci 2006;19(5):279–93.

[9] Dubina D, Ungureanu V, Gîlia L. Cold-formed steel beams with corrugatedweb and discrete web-to-flange fasteners. Steel Construct 2013;6(2):74–81.

[10] Ng CF, Zheng H. Sound transmission through double-leaf corrugated panelconstructions. Appl Acoust 1998;53(1):15–34.

[11] Jim Noll, Steve Tysl, Matt Westrich. The use of corrugated metal pipe andstructural plate for aggregate tunnel and conveyor enclosure applications,professional development advertising section. CONTECH ConstructionProducts Inc; 2009.

[12] Corrugated pipes for sewage and drainage applications; 2011, Corma.com.[13] Knox EM, Cowling MJ, Winkle IE. Adhesively bonded steel corrugated core

sandwich construction for marine applications. Marine Struct 1998;11(4–5):185–204.

[14] Liang CC, Yang MF, Wu PW. Optimum design of metallic corrugated coresandwich panels subjected to blast loads. Ocean Eng 2001;28(7):825–61.

[15] Hachemi H, Kebir H, Roelandt JM, Wintrebert E. A study of the braidedcorrugated hoses: Behavior and life estimation. Mater Des 2011;32(4):1957–66.

[16] Brown W. The suitability of various gasket types for heat exchanger service.In: ASME 2002 pressure vessels and piping conference. American Society ofMechanical Engineers; 2002. p. 45–51. January.

[17] Bapanapalli SK, Martinez OM, Gogu C, Sankar BV, Haftka RT, Blosser ML.Analysis and design of corrugated core sandwich panels for thermalprotection systems of space vehicles. AIAA Paper 2006;1942:2006.

[18] Martinez OA, Sankar BV, Haftka R, Bapanapalli SK, Blosser ML.Micromechanical analysis of composite corrugated-core sandwich panelsfor integral thermal protection systems. AIAA J 2007;45(9):2323–36.

[19] Thill C, Etches JA, Bond IP, Potter KD, Weaver PM. Composite corrugatedstructures for morphing wing skin applications. Smart Mater Struct2010;19(12):124009.

[20] Kazemahvazi S, Zenkert D. Corrugated all-composite sandwich structures.Part 1: modeling. Compos Sci Technol 2009;69(7):913–9.

[21] Kazemahvazi S, Tanner D, Zenkert D. Corrugated all-composite sandwichstructures. Part 2: failure mechanisms and experimental programme.Compos Sci Technol 2009;69(7):920–5.

[22] Norman AD, Seffen KA, Guest SD. Multistable corrugated shells. Proc R Soc A2008;464(2095):1653–72.

[23] Norman A, Seffen K, Guest S. Morphing of curved corrugated shells. Int JSolids Struct 2009;46(7):1624–33.

[24] Ray H. U.S. Patent No. 5,543,204. Washington, DC: U.S. Patent and TrademarkOffice; 1996.

[25] Leekitwattana M, Boyd SW, Shenoi RA. Evaluation of the transverse shearstiffness of a steel bi-directional corrugated-strip-core sandwich beam. JConstr Steel Res 2011;67(2):248–54.

[26] Dayyani I, Ziaei-Rad S, Nonlinear finite element analysis of compositecorrugated boards with elastomeric coatings [Master dissertation]. IsfahanUniversity of Technology, Iran; 2011.

[27] Seong DY, Jung CG, Yang DY, Moon KJ, Ahn DG. Quasi-isotropic bendingresponses of metallic sandwich plates with bi-directionally corrugated cores.Mater Des 2010;31(6):2804–12.

[28] Kooistra GW, Deshpande V, Wadley HN. Hierarchical corrugated coresandwich panel concepts. J Appl Mech 2007;74(2):259–68.

[29] Previtali F, Arrieta AF, Ermanni P. Double-walled corrugated structure forbending-stiff anisotropic morphing skins. Journal of Intelligent MaterialSystems and Structures 2014. 1045389X14554132.

[30] Previtali F, Delpero T, Bergamini A, Arrieta AF, Ermanni P. Extremelyanisotropic multi-functional skin for morphing applications. In: Proceedingsof 23nd AIAA/AHS adaptive structures conference; 5–9 January 2015,Kissimmee, Florida.

[31] Wadley HN, Fleck NA, Evans AG. Fabrication and structural performance ofperiodic cellular metal sandwich structures. Compos Sci Technol2003;63(16):2331–43.

[32] Queheillalt DT, Murty Y, Wadley HN. Mechanical properties of an extrudedpyramidal lattice truss sandwich structure. Scripta Mater 2008;58(1):76–9.

[33] Dayyani I, Ziaei-Rad S, Friswell MI. The mechanical behavior of compositecorrugated core coated with elastomer for morphing skins. J Compos Mater2013. 0021998313488807.

[34] Luo S, Suhling JC, Considine JM, Laufenberg TL. The bending stiffnesses ofcorrugated board. Mechanics of cellulosic materials, AMD-Vol. 145/MD-Vol.36. ASME; 1992.

[35] Khalid YA, Chan CL, Sahari BB, Hamouda AMS. Bending behaviour ofcorrugated web beams. J Mater Process Technol 2004;150(3):242–54.

[36] Chang WS, Ventsel E, Krauthammer T, John J. Bending behavior of corrugated-core sandwich plates. Compos Struct 2005;70(1):81–9.

[37] Tan KH, Montague P, Norris C. Steel sandwich panels: finite element, closedsolution, and experimental comparisons on 6 m � 2.1 m panel. Struct Eng1989;67(9):159–66.

[38] Dayyani I, Ziaei-Rad S, Salehi H. Numerical and experimental investigationson mechanical behavior of composite corrugated core. Appl Compos Mater2012;19(3–4):705–21.

[39] Thill C, Etches JA, Bond IP, Potter KD, Weaver PM. Corrugated compositestructures for aircraft morphing skin applications. In: 18th Internationalconference of adaptive structures and technologies, Ottawa, Ontario, Canada;2007, October.

[40] Thill C, Etches JA, Bond IP, Potter KD, Weaver PM, Wisnom MR. Investigationof trapezoidal corrugated aramid/epoxy laminates under large tensiledisplacements transverse to the corrugation direction. Compos A Appl SciManuf 2010;41(1):168–76.

[41] Nordstrand T, Carlsson LA, Allen HG. Transverse shear stiffness of structuralcore sandwich. Compos Struct 1994;27(3):317–29.

[42] Nordstrand TM, Carlsson LA. Evaluation of transverse shear stiffness ofstructural core sandwich plates. Compos Struct 1997;37(2):145–53.

[43] Isaksson P, Krusper A, Gradin PA. Shear correction factors for corrugated corestructures. Compos Struct 2007;80(1):123–30.

[44] Kampner M, Grenestedt JL. On using corrugated skins to carry shear insandwich beams. Compos Struct 2008;85(2):139–48.

[45] Lu TJ, Chen C, Zhu G. Compressive behaviour of corrugated board panels. JCompos Mater 2001;35(23):2098–126.

[46] Rejab MRM, Cantwell WJ. The mechanical behaviour of corrugated-coresandwich panels. Compos B Eng 2013;47:267–77.

[47] Liew KM, Peng LX, Kitipornchai S. Buckling analysis of corrugated plates usinga mesh-free Galerkin method based on the first-order shear deformationtheory. Comput Mech 2006;38(1):61–75.

[48] Pignataro M, Pasca M, Franchin P. Post-buckling analysis of corrugated panelsin the presence of multiple interacting modes. Thin-walled Struct2000;36(1):47–66.

[49] Libove C. On the stiffness, stresses and buckling analysis of corrugated shearwebs. In: Second specialty conference on cold-formed steel structures.Missouri S&T (formerly the University of Missouri-Rolla); 1973.

[50] Elgaaly M, Hamilton RW, Seshadri A. Shear strength of beams with corrugatedwebs. J Struct Eng 1996;122(4):390–8.

[51] Luo R, Edlund B. Shear capacity of plate girders with trapezoidally corrugatedwebs. Thin-walled Struct 1996;26(1):19–44.

[52] Yi J, Gil H, Youm K, Lee H. Interactive shear buckling behavior of trapezoidallycorrugated steel webs. Eng Struct 2008;30(6):1659–66.

[53] Sause R, Braxtan TN. Shear strength of trapezoidal corrugated steel webs. JConstr Steel Res 2011;67(2):223–36.

[54] Semenyuk NP, Neskhodovskaya NA. On design models in stability problemsfor corrugated cylindrical shells. Int Appl Mech 2002;38(10):1245–52.

[55] Semenyuk NP, Neskhodovskaya NA. Timoshenko-type theory in the stabilityanalysis of corrugated cylindrical shells. Int Appl Mech 2002;38(6):723–30.

[56] Semenyuk NP, Zhukova NB, Ostapchuk VV. Stability of corrugated compositenoncircular cylindrical shells under external pressure. Int Appl Mech2007;43(12):1380–9.

[57] ABS; American bureau of shipping, Buckling and ultimate strengthassessment for offshore structures; 2004.

[58] Sun HH, Spencer J. Buckling strength assessment of corrugated panels inoffshore structures. Marine Struct 2005;18(7):548–65.

[59] Biancolini M, Brutti C. Numerical and experimental investigation of thestrength of corrugated board packages. Packag Technol Sci 2003;16(2):47–60.

[60] Briassoulis D. Equivalent orthotropic properties of corrugated sheets. ComputStruct 1986;23(2):129–38.

[61] Nordstrand T. Analysis and testing of corrugated board panels into the post-buckling regime. Compos Struct 2004;63(2):189–99.

[62] Johnson MW, Urbanik TJ. Analysis of the localized buckling in composite platestructures with application to determining the strength of corrugatedfiberboard. J Compos Tech Res 1989;11(4):121–8.

[63] Hui W, Huan-ran Y. Natural frequency for rectangular orthotropiccorrugated-core sandwich plates with all edges simply-supported. ApplMath Mech 2001;22(9):1019–27.

[64] Liew KM, Peng L, Kitipornchai S. Vibration analysis of corrugated reissner–mindlin plates using a mesh-free galerkin method. Int J Mech Sci2009;51(9):642–52.

[65] Semenyuk NP, Babich IY, Zhukova N. Natural vibrations of corrugatedcylindrical shells. Int Appl Mech 2005;41(5):512–9.

[66] Gulgazaryan G, Gulgazaryan L. Vibrations of a corrugated orthotropiccylindrical shell with free edges. Int Appl Mech 2006;42(12):1398–413.

[67] Hu H, Zhao C, Feng S, Hu Y, Chen C. Adjusting the resonant frequency of a pvdfbimorph power harvester through a corrugation-shaped harvesting structure.IEEE Trans Ultrason Ferroelectr Freq Control 2008;55(3):668–74.

http://www.merriam-webster.com

http://www.merriam-webster.com

http://refhub.elsevier.com/S0263-8223(15)00645-5/h0010
























http://www.Corma.com

































































































































[68] Mandal NK. Experimental studies of quasi-longitudinal waves power flow incorrugated plates. J Sound Vib 2006;297(12):227–42.

[69] Mandal NK, Rahman RA, Leong M. Experimental study on loss factor forcorrugated plates by bandwidth method. Ocean Eng 2004;31(10):1313–23.

[70] Yang J, Xiong J, Ma L, Feng L, Wang S, Wu L. Modal response of all-compositecorrugated sandwich cylindrical shells. Compos Sci Technol 2015;115:9–20.

[71] Ren-Huai L, Dong L. On the non-linear bending and vibration of corrugatedcircular plates. Int J Non-Linear Mech 1989;24(3):165–76.

[72] Toccalino E, Maurer M, Vogel G, Katakkar L, Shembekar P, Velusamy S. Impactabsorption structure. US Patent 7,513,344; 2009.

[73] Jiang Z, Chen F, Wang G, Shi SQ, Yu Z, Cheng H-T, et al. Bamboo bundlecorrugated laminated composites (bclc). Part II. Damage analysis under lowvelocity impact loading. BioResources 2013;8(1):923–32.

[74] Khabakhpasheva T, Korobkin A, Malenica S. Fluid impact onto a corrugatedpanel with trapped gas cavity. Appl Ocean Res 2013;39:97–112.

[75] Zangani D, Robinson M, Gibson A. Energy absorption characteristics of web-core sandwich composite panels subjected to drop-weight impact. ApplCompos Mater 2008;15(3):139–56.

[76] Kılıçaslan C, Güden M, Odaci _IK, Tas�demirci A. The impact responses and thefinite element modeling of layered trapezoidal corrugated aluminum coreand aluminum sheet interlayer sandwich structures. Mater Des2013;46:121–33.

[77] Russell B, Malcom A, Wadley H, Deshpande V. Dynamic compressive responseof composite corrugated cores. J Mech Mater Struct 2010;5(3):477–93.

[78] Garcia-Romeu-Martinez M, Sek MA, Cloquell-Ballester V. Effect of initial pre-compression of corrugated paperboard cushions on shock attenuationcharacteristics in repetitive impacts. Packag Technol Sci 2009;22(6):323–34.

[79] Wang Z, Wang Q. Fatigue assessment of welds joining corrugated steel websto flange plates. Eng Struct 2014;73:1–12.

[80] Wang Z, Wang Q. Fatigue strength of CFRP strengthened welded joints withcorrugated steel plates. Compos B Eng 2015;72:30–9.

[81] Anami K, Sause R, Abbas HH. Fatigue of web-flange weld of corrugated webgirders: 1. Influence of web corrugation geometry and flange geometry onweb-flange weld toe stresses. Int J Fatigue 2005;27:373–81.

[82] Anami K, Sause R. Fatigue of web-flange weld of corrugated web girders: 2.Analytical evaluation of fatigue strength of corrugated web-flange weld. Int JFatigue 2005;27:383–91.

[83] Sause R, Abbas HH, Driver RG, Anami K, Fisher JW. Fatigue life of girders withtrapezoidal corrugated webs. J Struct Eng 2006;132:1070–8.

[84] Henderson D, Ginger J. Response of pierced fixed corrugated steel roofingsystems subjected to wind loads. Eng Struct 2011;33:3290–8.

[85] Kövesdi B, Dunai L. Fatigue life of girders with trapezoidally corrugated webs:an experimental study. Int J Fatigue 2014;64:22–32.

[86] Ibrahim S, El-Dakhakhni W, Elgaaly M. Behavior of bridge girders withcorrugated webs under monotonic and cyclic loading. Eng Struct2006;28:1941–55.

[87] Takeshita A, Yoda T, Sato K, Sakurada M, Shiga H, Nakasu K. Fatigue tests of acomposite girder with corrugated web. J Jpn Soc Civil Eng 2001;668:55–64.

[88] Miehe C, Schotte J, Schröder J. Computational micro–macro transitions andoverall moduli in the analysis of polycrystals at large strains. Comput MaterSci 1999;16:372–82.

[89] Miehe C, Schotte J, Lambrecht M. Homogenization of inelastic solid materialsat finite strains based on incremental minimization principles. Application tothe texture analysis of polycrystals. J Mech Phys Solids 2002;50(10):2123–67.

[90] Pellegrino C, Galvanetto U, Schrefler BA. Numerical homogenization ofperiodic composite materials with non-linear material components. Int JNumer Meth Eng 1999;46:1609–37.

[91] Suquet P. Overall potentials and extremal surfaces of power law or ideallyplastic materials. J Mech Phys Solids 1993;41:981–1002.

[92] Saavedra Flores EI, de Souza Neto EA. Remarks on symmetry conditions incomputational homogenisation problems. Eng Comput 2010;27(4):551–75.

[93] Xia Y, Friswell MI, Saavedra Flores EI. Equivalent models of corrugated panels.Int J Solids Struct 2012;49:1453–62.

[94] Mohammadi H, Ziaei-Rad S, Dayyani I. An equivalent model for trapezoidalcorrugated cores based on homogenization method. Compos Struct2015;131:160–70.

[95] Wennberg D, Wennhage P, Stichel S. Orthotropic models of corrugatedsheets in finite element analysis. ISRN Mech Eng 2011;2011:9. Article ID979532.

[96] McFarland DE. An investigation of the static stability of corrugatedrectangular plates loaded in pure shear [Ph.D. thesis]. University of Kansas,Lawrence, KS; 1967.

[97] Kress G, Winkler M. Corrugated laminate homogenization model. ComposStruct 2010;92(3):795–810.

[98] Winkler M, Kress G. Deformation limits for corrugated cross-ply laminates.Compos Struct 2010;92(6):1458–68.

[99] Kress G, Winkler M. Corrugated laminate analysis: a generalized plane-strainproblem. Compos Struct 2011;93(5):1493–504.

[100] Samanta A, Mukhopadhyay M. Finite element static and dynamic analyses offolded plates. Eng Struct 1999:277–87.

[101] Ahmed E, Badaruzzaman WH. Equivalent elastic analysis if profiled metaldecking using finite element method. Steel Struct 2003;3:9–17.

[102] Ye Z, Berdichevsky VL, Yu W. An equivalent classical plate model ofcorrugated structures. Int J Solids Struct 2014;51(11–12):2073–83.

[103] Seydel E. Shear buckling of corrugated plates. Jahrb Dtsch Vers Luftfahrt1931;9:233–45.

[104] Peng L, Liew K, Kitipornchai S. Analysis of stiffened corrugated plates basedon the FSDT via the mesh-free method. Int J Mech Sci 2007:364–78.

[105] Liew KM, Peng LX, Kitipornchai S. Nonlinear analysis of corrugated platesusing a FSDT and a meshfree method. Comput Methods Appl Mech Eng2007;196(21–24):2358–76.

[106] Thill C, Etches JA, Bond IP, Weaver PM, Potter KD. Experimental andparametric analysis of corrugated composite structures for morphing skinapplications. In: 19th International conference on adaptive structurestechnology, 6–9 October 2008, Ascona, Switzerland; 2008.

[107] Rao SS. Engineering optimization theory and practice. 3rd ed. John Wiley &Sons; 1996.

[108] Bitner JR, Reingold EM. Backtrack programming techniques. Commun ACM1975;18(11):651–6.

[109] Rathbun HJ, Zok FW, Evans AG. Strength optimization of metallic sandwichpanels subject to bending. Int J Solids Struct 2005;42(26):6643–61.

[110] Wicks N, Hutchinson JW. Optimal truss plates. Int J Solids Struct2001;38:6165–83.

[111] Daxner T, Flatscher T, Rammerstorfer FG. Optimum design of corrugatedboard under buckling constraints. In: Proceedings 7th world congress onstructures and multidisciplinary optimization, BMD Co., Seoul; 2007, May.pp. 349–358.

[112] Roux WJ, Stander N, Haftka RT. Response surface approximations forstructural optimization. Int J Numer Meth Eng 1998;42(3):517–34.

[113] Hou S, Zhao S, Ren L, Han X, Li Q. Crashworthiness optimization of corrugatedsandwich panels. Mater Des 2013;51:1071–84.

[114] Forrester A, Sobester A, Keane A. Engineering design via surrogate modelling:a practical guide. John Wiley & Sons; 2008.

[115] Mostaghim S, Teich J. Strategies for finding good local guides in multi-objective particle swarm optimization (MOPSO). In: Swarm intelligencesymposium, 2003. SIS’03. Proceedings of the 2003 IEEE. IEEE; 2003. p. 26–33.April.

[116] Argüelles P, Bischoff M, Busquin P, Droste BAC, Evans R, Kröll W. Wittlöv A.European aeronautics: a vision for 2020. Report of the group of personalities,The European Commission, 12; 2001.

[117] European commission. Flightpath 2050. Europe’s vision for aviation. Reportof the high level group on aviation research, Publications Office of theEuropean Union, Luxembourg; 2011.

[118] Barbarino S, Bilgen O, Ajaj RM, Friswell MI, Inman DJ. A review of morphingaircraft. J Intell Mater Syst Struct 2011;22(9):823–77.

[119] Thill C, Etches J, Bond I, Potter K, Weaver P. Morphing skins. Aeronaut J2008;112(1129):117–39.

[120] Diani J, Fayolle B, Gilormini P. A review on the Mullins effect. Eur Polymer J2009;45(3):601–12.

[121] Dayyani I, Friswell MI, Ziaei-Rad S, Saavedra Flores EI. Equivalent models ofcomposite corrugated cores with elastomeric coatings for morphingstructures. Compos Struct 2013;104:281–92.

[122] Dayyani I, Friswell MI, Khodaparast HH, Woods BK. The design of acorrugated skin for the FishBAC compliant structure. In: AIAA SciTechconference, Maryland, United States; 2014, January. pp. 13–17.

[123] Dayyani I, Friswell MI, Saavedra Flores EI. A general super element for acurved beam. Int J Solids Struct 2014;51(17):2931–9.

[124] Drela M. Xfoil: an analysis and design system for low Reynolds numberairfoils low Reynolds number airfoil aerodynamics. IN: Notre Dame; 1989.

[125] Yokozeki T, Sugiura A, Hirano Y. Development and wind tunnel test ofvariable camber morphing wing. In: 22nd AIAA/ASME/AHS adaptivestructures conference. National Harbor, Maryland: AIAA SciTech; 2014.http://dx.doi.org/10.2514/6.2014-1261. 13–17 January.

[126] Dayyani I, Khodaparast HH, Woods BKS, Friswell MI. The design of a coatedcomposite corrugated skin for the camber morphing airfoil. J Intell Mater SystStruct 2014. 1045389X14544151.

[127] Woods BKS, Friswell MI. Preliminary investigation of a fishbone activecamber concept. In: ASME 2012 Conference on Smart Materials, AdaptiveStructures and Intelligent Systems. American Society of MechanicalEngineers; 2012. p. 555–63. September.

[128] Shaw AD, Dayyani I, Friswell MI. Optimisation of composite corrugated skinsfor buckling in morphing aircraft. Compos Struct 2015;119:227–37.

[129] Xia Y, Ajaj RM, Friswell MI. Design and optimisation of composite corrugatedskin for a span morphing wing. In: 22nd AIAA/ASME/AHS adaptive structuresconference. Maryland: National Harbor; 2014.

[130] Beaverstock CS, Ajaj RM, Friswell MI, De Breuker R, Werter NPM. Optimisingmission performance for a morphing mav. In: 7th Ankara internationalaerospace conference; 2013.

[131] Ursache NM, Melin T, Isikveren AT, Friswell MI. Technology integration foractive poly-morphing winglets development. In: ASME 2008 Conference onSmart Materials, Adaptive Structures and Intelligent Systems. AmericanSociety of Mechanical Engineers; 2008. p. 775–82. January.

[132] Melin T. A vortex lattice MATLAB implementation for linear aerodynamicwing applications [Master’s Thesis]. Department of Aeronautics, RoyalInstitute of Technology (KTH), Stockholm, Sweden; 2000.

[133] Fincham JH, Ajaj RM, Friswell MI. Aerodynamic performance of corrugatedskins for spanwise wing morphing. In: 14th AIAA aviation technology,integration, and operations conference, 16–20 June 2014, Atlanta, USA; 2014.


























































































































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the mechanics of composite corrugated structures: a review...

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