phd thesis - mark schenk.pdf

Folded Shell Structures

A thesis submitted for the degree of

Doctor of Philosophy

31 August 2011

Mark Schenk

Clare College

University of Cambridge

Supervisor: S.D. Guest

Declaration

I declare that, except for commonly understood and accepted ideas, or

where specific reference is made to the research of other authors, this

dissertation is the result of my own original work and includes nothing

which is the outcome of work done in collaboration. I further state

that this dissertation has not been previously or is currently being

submitted, either in part or as a whole, for any degree, diploma, or

other qualification at any other university. The thesis presented is 148

pages, containing approximately 32000 words and 91 figures, and does

thus not exceed the limit of length prescribed by the Degree Commitee

of the Faculty of Engineering.

Acknowledgements

I would like to extend a warm thank you to my supervisor Simon Guest

for his time, advice and insights throughout my PhD. A few further

words of thanks are due. To Keith Seffen for many interesting discus-

sions on the mechanics of folded sheets, which helped crystallise several

ideas on unit cell kinematics. To Julian Allwood for his infectious en-

thusiasm when presented with the challenge of manufacturing a Miura

sheet from metal, which opened up a whole new direction of study.

ii

Abstract

A novel type of shell structure was analysed, folded shell structures. These shell

structures have a distinct structural hierarchy: globally they can be regarded as

thin-walled shells, but at a meso scale they consist of tessellated unit cells, which

in turn are composed of thin-walled shells joined at distinct fold lines. It is this

structural hierarchy that imbues the folded shell structures with their interest-

ing mechanical properties. The global sheet deformations are a combination of

bending along the folds, and deformation of the interlying material. The former

is primarily a kinematic problem, with a parallel in the flexibility of hinged plate

structures. A review of the mathematics of rigid origami provides the necessary

background to develop non-trivial geometries of these folded shells that still exhibit

a soft deformation mode.

Two example folded shell structures are introduced, the Miura and Eggbox sheet.

Both consist of a tessellation of parallelogram facets; the first is developable, while

the other has points of positive and negative Gaussian curvature. The first prop-

erty of interest is their increased in-plane flexibility, by virtue of the opening and

closing of folds. The Miura and Eggbox sheet respectively have an effective nega-

tive and positive in-plane Poissons ratio. Secondly, both sheets can modify their

global Gaussian curvature, with no stretching at the material level. Thirdly, both

sheets exhibit an oppositely signed Poissons ratio for in-plane and out-of-plane

deformations; e.g. when bending the Miura sheet it exhibits a negative Poissons

ratio behaviour and deforms anticlastically.

The salient global deformations of the sheets were analysed in terms of the kine-

matics of the constituent unit cells. The characteristic in-plane and out-of-plane

properties of the sheets followed directly from developable deformations of the

tessellated unit cells. A more holistic top-down numerical approach modelled the

sheets as an array of unit cells. The sheets were represented by a pin-jointed

bar framework, and additional planarity constraints between facets enabled the

inclusion of a bending stiffness for the facets and fold lines. A modal analysis of

the sheets stiffness matrix showed that the characteristic deformation modes are

among the dominant eigenmodes of the sheets for a wide range of geometries and

material properties.

Many folded shell structures can be folded from flat sheet material, with only

minimal material deformations. The manufacturing processes must overcome the

intrinsic kinematics of the sheets, whereby the sheet contracts in two directions

simultaneously. Existing methods were reviewed, and classified into synchronous

folding, gradual folding and pre-gathering techniques. A novel cold gas pressure

manufacturing method was introduced, and it was shown that a simple plastic

hinge model cannot yet fully account for the total required forming energy.

iii

Contents

1 Introduction 1

1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background and Concepts 5

2.1 Textured Shell Structures . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Texture Bending Stiffness . . . . . . . . . . . . . . . . . . 5

2.1.2 Texture Flexibility . . . . . . . . . . . . . . . . . . . . . . 12

2.1.3 Texture Impact Absorption . . . . . . . . . . . . . . . . . 15

2.1.4 Folded Shell Structures . . . . . . . . . . . . . . . . . . . . 15

2.2 Origami Folding and Foldability . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Curvature and Creases . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Rigid Foldability . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3 Fold Pattern Design . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Folded Shell Structures 32

3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.1 Example Sheets . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Kinematic Analysis 47

4.1 Planar Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 Miura Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.2 Eggbox Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Out-of-Plane Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Miura Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.2 Eggbox Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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CONTENTS

4.3 Conclusion & Discussion . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Numerical Analysis 73

5.1 Mechanical Model: Bar Framework . . . . . . . . . . . . . . . . . . 73

5.1.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 74

5.1.2 Kinematics: Compatibility . . . . . . . . . . . . . . . . . . 74

5.1.3 Stiffness: Material Stiffness . . . . . . . . . . . . . . . . . . 75

5.1.4 Extensions to Framework Analysis . . . . . . . . . . . . . . 78

5.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.1 Symmetry Analysis . . . . . . . . . . . . . . . . . . . . . . . 87

5.2.2 Curve Veering & Imperfections . . . . . . . . . . . . . . . . 87

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Manufacture of Folded Sheets 94

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 Review of Manufacturing Methods . . . . . . . . . . . . . . . . . . 95

6.2.1 Synchrononous Folding Processes . . . . . . . . . . . . . . . 97

6.2.2 Gradual Folding Processes . . . . . . . . . . . . . . . . . . . 102

6.2.3 Pre-Gathering Processes . . . . . . . . . . . . . . . . . . . . 107

6.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Cold Gas-Pressure Folding . . . . . . . . . . . . . . . . . . . . . . . 117

6.3.1 Process Description . . . . . . . . . . . . . . . . . . . . . . 117

6.3.2 Process Calculations . . . . . . . . . . . . . . . . . . . . . . 117

6.3.3 Manufacturing Trials . . . . . . . . . . . . . . . . . . . . . . 126

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7 Conclusions & Future Work 133

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

References 137

v

Chapter 1

Introduction

Traditionally, thin-walled shells have been designed with the purpose of carry-

ing external loads efficiently and rigidly, by an appropriate design of their global

geometry. This thesis forms part of ongoing work to extend the capabilities of

conventional shell structures, and expand the applications beyond their tradi-

tional remit. For example, using a combination of geometry, material anisotropy

and pre-stress it is possible to design multi-stable shells. These shells can switch

repeatedly and reliably between multiple states, adapting to different functional

requirements. Examples include bistable deployable cylindrical booms (Guest &

Pellegrino, 2006), and a bistable helicopter blade that changes shape during hover

and vertical lift (Daynes et al., 2009). In both cases the bistability is introduced

by tailoring the fibre layup of a composite laminate shell.

Alternatively, the mechanical properties and the range of applications can be ex-

tended by means of textured shell structures. By introducing a local texture (such

as corrugations, dimples, folds, etc.) to otherwise isotropic thin-walled sheets, the

global mechanical properties of the sheets can be favourably modified. The local

texture has no clearly defined scale, but lies somewhere between the material and

the structural level. A classic example is a corrugated sheet, where the bending

stiffness is increased through the local geometry; a more advanced extension is a

pre-stressed bistable corrugated sheet, where the local texture locks in the stored

strain energy (Norman et al., 2008b).

The research into textured thin-walled shells branched into the development of

compliant shell mechanisms, where the texture provides a distinct kinematic layer

to the mechanical properties (Seffen, 2011); Figure 1.1. The folded shell structures

described in this thesis are examples of such sheets. Globally they can be regarded

as thin-walled shells, but at meso scale they consist of a tessellation of unit cells,

which in turn are composed of thin-walled facets joined at distinct fold lines. These

folds may be straight or curved, and the surface may have points or lines of non-

zero Gaussian curvature where the facets meet. It is the hierarchical interaction

between the articulation at the fold lines and the deformation of the interlying

facets, which imbues these shells with unconventional mechanical properties.

1

Figure 1.1: The morphing curved corrugated shell investigated by Norman et al.

(2009) is an example of a compliant shell mechanism. Under uni-axial tension the

shell coils up tighter (top-right), and under bending it can assume global shapes

with both positive or negative Gaussian curvature (bottom).

The characteristic behaviour of folded shell structures straddles that of structures

and mechanisms. Firstly, they have the ability to undergo large displacements

within soft mechanistic deformation modes. Consequently, there is an anisotropy

in deformation modes, as the fold lines enable motion in some, and inhibit de-

formation in other modes. There may also be a non-trivial interaction between

different modes, such as in-plane and out-of-plane coupling. A common thread

further appears to be the ability of the shells to change their global Gaussian

curvature, i.e. the curvature of an equivalent mid-plane, with only developable de-

formations at material level. Our focus is on the mechanistic deformations of the

folded sheets, and we generally consider the deformation of the interlying material

as developable (i.e. no stretching of the material).

The presented study is a first analysis of folded shell structures and their mechan-

ical properties. The aim is to describe the kinematic and structural properties,

and find suitable modelling methods to capture their salient behaviour. Low-level

effects such as material fatigue at the fold lines are not currently considered. A re-

view of the mathematics of rigid origami provides a solid theoretical background for

the design of folded shell structures, which ensures the presence of a soft kinematic

mechanism. Furthermore, a potential impediment to their practical applications

is removed by investigating suitable manufacturing techniques. This study has

generally been agnostic towards any specific applications; instead, a broad foun-

dation was established for the design, analysis and manufacture of these types of

structures. Nonetheless, potential applications of folded shell structures certainly

exist and will briefly be highlighted.

2

Morphing Skins An emerging field of research is concerned with the technical

challenges of designing morphing aircraft; the ultimate aim is to modify the wings

aerodynamic profile during flight, to adapt to changing conditions. An important

aspect is the development of a suitable morphing skin. The requirements are de-

manding and contradictory: the skin should be light-weight, have a low membrane

stiffness (small actuation forces), high flexural stiffness (to withstand aerodynamic

loads) and be able to attain large strain rates (Olympio, 2009). Thill et al. (2008)

provides an extensive review of proposals, but notes that the level of maturity

for morphing skins is low and the existing concepts are very early in develop-

ment. Yokozeki et al. (2006) explored a promising solution using highly anistropic

corrugated skins; this provides the desired combination of mechanical properties,

but morphing is limited to one direction. Olympio (2009) explores the concept

of morphing honeycombs, whose Poissons ratio can be tailored to requirements.

Those honeycombs either support a closed elastic skin, or are filled with a low

modulus material to form a closed surface. Ramrakhyani et al. (2005) proposed

several skin concepts, including a folded skin mounted on a load bearing variable

geometry truss; however, this concept has not been further pursued. The folded

shell structures provide an unexplored area for morphing skins, with a promising

combination of mechanical properties. They can be designed to have low in-plane

moduli, whilst maintaining their bending stiffness; they can expand biaxially, and

attain doubly-curved configurations. Furthermore, some folded shells can be tai-

lored for a desired coupling between extension and bending. Many issues, however,

are unresolved. For example, the maximally attainable in-plane strains and re-

sulting fatigue at the fold lines are currently unknown, and the shells strongly

non-linear Poissons ratio may be undesirable.

Compliant Mechanisms Compliant mechanisms are designed to transmit both

motion and force mechanically, relying on elastic deformation of its elements rather

than conventional joints and hinges. Advantages include high displacement accu-

racy, reduced hysteris, zero backlash and wear, and ease of manufacture without

assembly. The design of such mechanisms treads a fine line between achieving

adequate stiffness in order to exert sufficient force at the end-effector, and yet be

flexible enough that the required motion due to applied loads is realised. The

folded shell structures may extend this concept beyond the current linkage-based

compliant mechanisms towards compliant surfaces; the shells stiff deformation

modes may then be employed to actively transmit forces, rather than merely pas-

sively resist external loads. A possible application may be found in supportive

exoskeletons. For example, the X-arm project by InteSpring (2011) investigates

the potential applications of a passive exoskeleton, to support a users upper body

weight and thereby reduce health problems for people in professions with heavy

lifting and long uncomfortable postures, such as healthcare workers and mechan-

3

1.1 Outline

ics. Compliant shell mechanisms may provide the capability to conform to the

users movements, but preserve the ability to transmit forces.

Architectural Facades Much of recent theoretical work on rigid origami fold-

ing was motivated by the desire to design deployable architectural structures, such

as facades, roofs and shelters. The folded shell structures may serve a different

architectural purpose, providing a flexible building envelope that can adjust to

changes in shape, whether for structural, functional or visual purposes. Further-

more, the textured sheets may also be used as static cladding material, for their

ability to conform to doubly-curved surfaces.

1.1 Outline

Chapter 2 establishes the necessary context and concepts for the study of folded

shell structures. The state of the art in textured shell structures is described, and

examples from the literature were classified by their dominant structural purpose.

The mathematics of curvature and creases was summarised to provide essential

concepts for the deformations of these shells. The distinct fold lines in the shell

surface invite a strong link to origami folding, and the recent developments in

the mathematics of rigid origami are presented as a means to design folded shell

structures with soft internal deformation modes.

Chapter 3 introduces two representative example folded shell structures. Their

characteristic mechanical properties are described and illustrated with some demon-

strative experiments, and potential modelling methods are discussed.

Chapter 4 describes the global deformations of folded shell structures, by analysing

the kinematics of its constituent unit cells. Their characteristic in-plane and out-

of-plane properties were described in terms of developable deformations of the unit

cells. This analysis provides the compatibility equations, as a first step towards

an equivalent continuum model. The numerical analysis in Chapter 5 takes a

more holistic approach to the global sheet deformations. The folded shells were

modelled as pin-jointed trusses with additional planarity constraints to provide

bending stiffness for the folds and facets. A modal analysis establishes the domi-

nant deformation modes for a range of geometries and material properties.

Chapter 6 describes the manufacturing challenges of developable folded sheets.

Existing manufacturing processes are reviewed and classified. A novel cold gas

pressure manufacturing method is proposed, and the required forming pressures

are derived analytically and compared with experimental data.

4

Chapter 2

Background and Concepts

2.1 Textured Shell Structures

A versatile and promising means of extending the mechanical properties of thin-

walled shells is to introduce a local texture pattern, with surface features at a scale

intermediate to the material and global structural scale. The shells base material

will generally be isotropic, and the change in mechanical properties is entirely due

to the local geometry. Paradoxically, the main structural applications for these

types of shell structures either exploit the texture for its increase in stiffness, or

its increase in flexibility.

2.1.1 Texture Bending Stiffness

The primary application of local texture in thin-walled shells has been to increase

the shells bending stiffness, by increasing the second moment of area. The classic

example is corrugated sheets. Invented in 1829 by Henry Palmer, they enabled

the construction of large-span enclosures such as the great railway stations of

the Victorian era (Mornement & Holloway, 2007), and have been ubiquitous ever

since. Exploiting their increased stiffness at minimal expense of weight, Junkers

(1929) used corrugated sheets as the outer skin for several aircraft designs. The

mechanical properties of corrugated sheets are now well established; Briassoulis

(1986) provided the equivalent shell properties of smoothly corrugated sheets,

which Samanta & Mukhopadhyay (1999) extended to folded plate corrugations.

Miura (1969) introduced a category of textured shell structures, Pseudo-Cylindrical

Concave Polyhedral (PCCP) shells, inspired by the stable inextensional post-

buckling geometry of axially compressed cylinders, known as the Yoshimura pat-

tern. Rather than considering the buckled sheets as failed, Miura (1969) exploited

the increased circumferential bending stiffness. These globally curved shells con-

sist of planar facets joined at straight fold lines, and their local geometry increases

the shells bending stiffness, but at the expense of its in-plane stiffness. Proposed

applications included undersea pressure hulls due to the increased buckling resis-

5


tance (Knapp, 1977); a stress analysis of PCCP shells under both axial load and

hydrostatic pressure was therefore performed by Tanizawa & Miura (1975). A

commercial application was found in textured drinking cans, which enable the use

of thinner material (Ishinabe et al., 1992). Another proposed application of PCCP

shells is their ability to cover large unsupported spans, and is very closely related

to the use of folded plates in architecture (Salvadori & Heller, 1963; Engel, 1968;

Buri, 2010). For the textured shells, the features are generally less pronounced

and the appearance of a globally thin-walled shell is therefore stronger.

Many textured sheets aim to increase the second moment of area along any bend-

ing axis; see Figure 2.1. Ewald (1948) describes a sheet where the texture consists

of alternating protrusions, whose tessellation pattern is staggered to avoid any di-

rect fold lines in the core. The dimpled sheet described by Pfistershammer (1952)

aims to minimise the variability in bending stiffness by a suitable choice of dim-

ple pattern and their up/down orientation. Farmer & Spangler (1962) studied

the bending properties of such Collattan sheets, and observed remaining preferred

bending axes. The stiffness of sheets with interlocking embossed patterns was in-

vestigated by Tewes (1966) and measurements confirmed an increased, but highly

anistropic bending stiffness. A great variety of such interlocking embossing pat-

terns is possible (de Swart, 1955), but no studies of preferred patterns have been

found. Mirtsch et al. (2006) describe a vault-structured material, where the mate-

rial is locally buckled into a three-dimensional shape under hydrostatic pressure.

The described manufacturing process aims to minimise material deformation as

well as required forming forces (Behrens & Ellert, 2005).

An important application of textured sheets is as a stiffening element in a combined

panel. Rauscher et al. (2008) describe the manufacture of hump plates, whereby

two sheets are connected through humps which were formed into one or both of

the cover sheets. More commonly, face sheets are added on either side of the

textured sheet to form a sandwich panel; in contrast to conventional honeycomb

cores, these textured cores not only provide compressive and shear strength, but

also significantly contribute to the bending stiffness of the panel. Multiple sheets

may be stacked and joined together to form a core material (e.g. Pfistershammer,

1952). Countless variations of textured cores have been found in the literature,

and a detailed comparison is beyond the scope of this review. Two types of core are

noteworthy. Schott (1975) describes a core of a square array of alternating conical

protrusions, with an important feature that the core material can be deformed into

both single and compound curvatures with minimal loss of load carrying capacity

of the core; see Figure 2.2. Another important textured core material was first

introduced by Rapp (1960), based on a doubly-corrugated sheet which can be

folded from a flat sheet without stretching of the base material: a folded core.

6


(a) Ewald (1948). (b) Pfistershammer (1952).

(c) de Swart (1955).

(d) Mirtsch et al. (2006).

Figure 2.1: Examples of textured sheets with an increased stiffness along any bend-

ing axis, with (a) a staggered tessellation of protrusions, (b) an array of dimples,

(c) interlocking embossing patterns, and (d) a locally bulged vault-structured

material.

7


(a) Schott (1975) describes a core material with conical protrusions,

which can be adjusted to fit curved surfaces.

(b) a core material that can be folded from flat sheet material, with only bend-

ing operations (Rapp, 1960).

Figure 2.2: Two types of textured sandwich panel cores, with (a) designed to

conform to double curvatures, and (b) the first known example of a folded sandwich

panel core.

8


(a) the Zeta core provides increased bond-

ing contact area, at the expense of no longer

being developable (Miura, 1975b).

(b) first known example of curved folded

core material (Sehrndt, 1960).

Figure 2.3: Examples of fold core geometries.

Folded Core A similar core structure, termed Zeta core was analysed by Miura

(1972), who derived analytical predictions for the core shear strength, with an

optimal efficiency found for a fold angle of 55. Among its advantages were listedthe high shear modulus and strength, an isotropic or controllable shear modulus,

and improved buckling resistance due to the ruled surface along the facets. By

providing bonding surfaces on the crests of the corrugations, the Zeta core sheet

is no longer developable, and therefore cannot be folded from sheet material.

This decision was further motivated by the absence of suitable mass-production

methods. The Zeta cores were instead manufactured by vacuum forming plastics,

or progressive press forming of aluminium sheets (Miura, 1975b).

In recent years there has been a revived interest in folded core material for

lightweight sandwich panels for aircraft fuselages (Heimbs et al., 2010). An im-

portant motivation is the fact that fold cores possess open ventilation channels,

solving the problem of moisture condensation in the core material. Furthermore,

high-performance materials such as resin-impregnated aramid paper (e.g., Nomex

or Kevlar) can be folded, and the unit cells can be designed with respect to specific

mechanical requirements, as they offer a large design space for tailored properties.

The application in the aerospace industry also spurred the development of contin-

uous manufacturing techniques for fold cores; a review is given in Chapter 6.

Extensive work has taken place on experimental and numerical testing of the me-

chanical properties of fold cores, with a specific focus on impact protection. Heimbs

et al. (2010) provide a good overview of recent work, and identify two major issues

in the virtual testing simulations. The first point is a reasonable implementation

of imperfections in the numerical models, which are inevitably present in cellular

structures resulting from the manufacturing process, affecting the buckling and

9


strength properties of the foldcore. A range of methods to introduce appropri-

ate imperfections to the finite element models is discussed by Heimbs (2009) and

Baranger et al. (2011b). The second point is a correct constitutive modelling of

the cell wall material; especially for the resin-impregnated aramid paper fold cores

this requires extensive correlation with experimental data (Fischer et al., 2009).

Lebee & Sab (2010) derived analytical and numerical upper and lower bounds for

the shear stiffness of the chevron folded core materials in a sandwich panel. For

some existing patterns, both the numerical and analytical bounds were found to

be too loose. While for honeycomb cores the discrepancy between the bounds was

determined to be a skin effect, in the case of the folded core the large discrepancy

still has no explanation and requires more refined models, to take into account

interaction between skin and core.

Klett & Drechsler (2009) identified three design scales for folded cores: macro-level,

concerned with the global geometry of the core material; meso-level, which tunes

the unit cell geometry to desired mechanical requirements, and micro-scale, at

the level of the sheet material. Especially the ability to attain globally non-planar

geometries without distortion of the core material is an important benefit of folded

cores; see Figure 2.4. The possibilities include single curvature panels (Sehrndt,

1960; Khaliulin & Desyatov, 1993), doubly-curved cores (Talakov, 2010), as well

as helical core material designed to wrap around a cylindrical fuselage (Akishev

et al., 2010). The development of computational design tools has further enabled

the design of more freeform geometries, with varying curvatures, densities and fold

depths (Klett et al., 2007).

Bending Anisotropy

The introduction of a texture to thin-walled shells will generally introduce an

anistropy in its bending stiffness. While often considered undesirable, this anisotropy

may be also be positively exploited, for example in bi- and multistable structures.

Composite shells with anistropic bending stiffness have been used to enable bista-

bility in cylindrical shell structures (Guest & Pellegrino, 2006). A combination

of pre-stress and simple corrugations was used to design multi-stable corrugated

shell structures (Norman et al., 2008b), where the introduced geometry locks in

the stored strain energy. Greater control over the bistability can be obtained by

exploring texture patterns; for example, Norman et al. (2008a) describe a doubly-

corrugated pattern, which introduces the necessary anistropy for a bistable cylin-

drical shell. Another example of the combination of pre-stress and texturing was

found in a study into hierarchical multi-stable dimpled shells: an array of bistable

dimples provides local control over the release of pre-stress, and thereby determines

the global geometry of the sheet (Seffen, 2006). When all dimples are oriented the

10


(a) transversely curved core. (b) longitudinally curved core.

(c) freeform core geometry, with varing den-

sity, fold depth and curvature; the fold pat-

tern is found by reflection in the two pre-

scribed bounding curves.

(d) helical fold core, enclosing a cylindrical

fuselage (Akishev et al., 2010).

Figure 2.4: The versatility of folded cores is illustrated by the ability to design

globally curved geometries, whilst folding from flat sheet material with minimal

material deformation. Images by Klett et al. (2007) and Akishev et al. (2010).

11


same way, there is a uniform state of self stress, but the sheet will have preferred

axes of cylindrical bending orthogonal to the directions of least packing of dimples;

Seffen (2007) provides a simplified homogenisation of the dimpled sheet by assum-

ing cylindrical beam-like bending and integrating the second moment of area. An

alternative approach to controlling the bending anistropy of thin-walled shells, is

the use of perforation patterns (Duncan & Upfold, 1963; Fort, 1970).

While the ability to control the anistropy of laminated composite shells is well

established (e.g. Halpin, 1992; Nettles, 1994) the possibilities and limitations of

texture patterns remain largely unknown.

2.1.2 Texture Flexibility

Another application of textured sheets is for their in-plane flexibility, for example

to provide stress relief for applications with large changes in temperature, such as

linings for cryogenic storage vessels. Various patents have been found describing

an expandable sheet with a number of parallel corrugations set at an angle to each

other, thereby allowing expansion of the sheet in multiple directions; see Figure 2.5.

The challenge in those cases is finding a satisfactory solution for the point where

the corrugations intersect. Dunajeff (1939) used a perforation at the vertices of

crossing corrugations to avoid stress concentrations. Arne (1973) places expandible

conical frusta at the intersection, whereas Girot (1964) describes expansion joints

that can be folded from the sheet material along with the corrugations. Similarly,

French & Petty (1965) describe expansion joints which are formed by twist-folding

the material at the point where corrugations meet.

Whereas the corrugated sheets concentrate the expansion along specific locations

in the sheet, examples exist of sheets which distribute their in-plane flexibility

throughout the surface; see Figure 2.6. Dunajeff (1941) describes several concepts

for resilient sheets, including the tessellation of expansible hexagonal units, as

well as a zigzag corrugation that enables extension along two orthogonal axes.

Brunner (1968) describes expansible sheets, consisting of parallelograms joined at

their edges such that each parallelogram is inclined to the midplane of the sheet;

this includes both the Eggbox and Miura sheet described in this thesis. Lueke

(1994) provides a brief review of flexible linings for cryogenic storage vessels, and

suggests that strains of about 1-2% are necessary. Furthermore, sheets with a

negative Poissons ratio are considered desirable, in part because they are also

flexible at 45 to the primary axes. The proposed sheet is a smoothed versionof the doubly-corrugated fold pattern to avoid stress concentrations at the folds,

and the depth of the pattern is limited for weight efficiency. Lastly, Fritz et al.

(1996) describe a modification of an Eggbox pattern, with a corrugation within

the parallelogram facets, enabling expansion along all axes.

12


(a) Arne (1973). (b) Girot (1964).

(c) French & Petty (1965).

Figure 2.5: Examples of plane expansible surfaces with expansion joints at the

intersection of crossing corrugations.

13


(a) Dunajeff (1941).

(b) Brunner (1968).

(c) Fritz et al. (1996).

Figure 2.6: Examples of plane expansible sheets, including (a) the first known

example of the Miura sheet for engineering applications, (b) the first known ex-

ample of a flexible Eggbox sheet, and (c) a modified Eggbox pattern that enables

in-plane expansion along all axes.

14


Most flexible textured materials only consider in-plane flexibility. Yokozeki et al.

(2006) explore the idea of ultra-anistropic corrugated surfaces of composite ma-

terials. For the intended application in morphing aircraft wings, it is desirable

for the skin to be stiff in span direction, but flexible in chord. Analytical expres-

sions were found for the bending stiffness along and across the corrugations, and

the anisotropy in bending was further increased by inserting rigid rods into the

corrugations.

2.1.3 Texture Impact Absorption

A further application for textured sheets is for their ability to absorb impact

through plastic deformation of the texture pattern. Deshpande & Fleck (2003)

looked at the collapse mechanism of a doubly-corrugated Eggbox sheet, and de-

scribe feasible geometries that induce a travelling plastic knuckle, rather than

tearing of the material. Zupan et al. (2003) further detail the collapse mecha-

nisms, and shows the dependency on the in-plane kinematic constraints. Basily &

Elsayed (2004) investigated the impact absorption of folded core material; com-

pared to equivalent honeycombs the folded core provides a more consistent force-

displacement profile during impact, and the folded sheets have the ability to absorb

energy in all directions of impact.

2.1.4 Folded Shell Structures

The folded shell structures described in this thesis are a type of textured shell struc-

ture. The shell consists of a tessellation of piecewise developable facets, bounded

by distinct fold lines. These fold lines may be curved or straight, and the sheets

may be developable or may have points and lines of non-zero Gaussian curvature.

The distinguishing feature of the folded shell structures are the distinct fold lines

which enable flexibility, but also provide a texture with an increased second mo-

ment of area. This combination creates an anistropy in the shells deformation

modes, which is more complex than for example for the highly orthotropic cor-

rugated material of Yokozeki et al. (2006). Furthermore, as described by Seffen

(2011), these structures display unusual properties due to the hierarchical interac-

tion of the deformation along the fold lines and the interlying material; these are

described in Chapter 3.

Several textured shells discussed previously fall within the definition of folded

shell structures. However, they were either exclusively employed for their bending

stiffness (e.g., Miura, 1969) or their planar expansibility (e.g., Brunner, 1968), and

where any coupling between in-plane and out-of-plane properties was observed this

was neglected (Lueke, 1994).

15

2.2 Origami Folding and Foldability


A key feature of the folded shell structures is the distinct fold lines in the surface,

which begs a strong link to origami folding. Many folded shells can also be folded

from flat sheet material, further solidifying this connection. The mathematics

of origami is a rich and thriving field (Demaine & ORourke, 2007); the topics

presented here form only a small subset, and are selected for their application

to the analysis and design of folded shell structures. We shall first discuss the

intrinsic geometry of surfaces and folds, before exploring the rigid foldability of

tessellated fold patterns, and resulting design methods.

2.2.1 Curvature and Creases

Any introduction to the mathematics of origami benefits from the discussion of

the curvature of surfaces, and specifically the concept of Gaussian curvature. Dif-

ferential geometry describes the properties of surfaces; an excellent introduction is

given by Hilbert & Cohn-Vossen (1952), with more detailed information in Struik

(1961). For sufficiently smooth surfaces (C2, smooth in second derivative), the

principal curvatures 1 2 are defined for each interior point as the maximumand minimum (signed) curvatures for the geodesics through the point. The Gaus-

sian curvature K is then given as the product of the two principal curvatures:

K = 12 (2.1)

By virtue of Gauss Theorema Egregium1 (Gauss, 1828) the Gaussian curvature

remains invariant under bending of the surface; a bending is any deformation

for which the arc lengths and angles of all curves drawn on the surface remain

invariant. The invariability means the Gaussian curvature is thus an intrinsic

property of the surface, independent of any external coordinate system. Moreover,

it enables a description of curvature at folds and vertices where the conventional

notion of the radius of curvature no longer holds.

An intrinsic description of Gaussian curvature considers a small area F on the

surface, and the solid angle G it subtends. The Gaussian curvature is then given

as

K = limF0

G

F(2.2)

which only involves information locally on the surface. In order to find the sub-

tended angle G, we introduce the spherical representation of a surface; see Fig-

ure 2.7(a). Consider a closed contour k oriented counterclockwise on the surface

1Latin: Remarkable Theorem. A translation of Gauss treatise on geometry of surfaces is

published as Gauss (1902).

16


and enclosing a point on the surface; each point on the contour has an associated

unit vector normal to and oriented away from the surface. Now transfer this set

of normal vectors to the centre of a unit radius sphere, the Gaussian sphere. The

ratio K of the area G enclosed by the resulting closed contour k (the trace of thecontour k) on the sphere, to the area F enclosed by the contour k has a definite

limit as k shrinks to a point, and is the Gaussian curvature. When k is orientedcounterclockwise, the calculated area and therefore the curvature is positive (and

vice versa).

This approach may now be extended to folds and polyhedral vertices, as shown in

Figure 2.7(b). At a fold line the normal vector is not uniquely defined and the arc

on the trace covers all possible directions; the arc length therefore corresponds to

the dihedral fold angle between the two planes. Any closed curve around a point on

the fold line will map onto a single arc on the unit sphere, which therefore has zero

Gaussian curvature. Using results from spherical trigonometry it can be shown

that at the vertex the area G within the trace is equal to 2pi minus the sum of the

sector angles of the corresponding surface (Huffman, 1976; Calladine, 1983); this

is called the spherical excess, or angular defect. It follows that the spherical excess,

and thereby the Gaussian curvature is independent of the fold angles beween the

plane sectors, and it provides an intrinsic measure of the curvature at the vertex.

When the sum of sector angles falls short of 2pi the vertex has positive Gaussian

curvature, when it exceeds 2pi it is negative, and when they add up to 2pi the

vertex has zero Gaussian curvature.

Of interest to origami folding is the concept of developable surfaces. These surfaces

locally have Gaussian curvature everywhere zero, and are globally isometric to the

plane. A developable surface may alternatively be defined as a ruled surface (there

exists a line segment through any point on the surface) for which the tangent

plane is the same at any point along a line embedded in the surface. Under any

bending or folding deformation, the zero Gaussian curvature everywhere on the

sheet is preserved, which limits its smooth deformations to cylindrical, conical and

tangent surfaces.

The presence of creases (or folds) in the surface greatly affects the attainable

geometries. Demaine et al. (2009) characterise the facets of a crease pattern,

which are by definition regions folded without creases; the creases are defined

as discontinuities of the first derivative of the surface. One result was that for

polygonal facets (bounded by straight lines) of the crease pattern, the fold lines

fold to straight line segments in space, and the facets must therefore stay planar

(provided they have no boundary edges). The geometry of fold lines and vertices

was first investigated in a landmark paper by Huffman (1976); using the trace and

the condition that its net surface must be zero, an algebraic relationship between

the dihedral fold angles of degree-4 vertices was found. Miura (1989) showed that

17


(a) the spherical image of a closed contour around a point on a surface

is produced by tracing the contour and transferring the unit normals

to the centre of a unit sphere. The Gaussian curvature is defined as

the limit of the ratio of the area G of the trace over the area F of

the curve on the surface, as F shrinks to a point. If the contour k is

traversed counterclockwise, a clockwise enclosed area on the spherical

image is considered to be negative, as is the case for the saddle shape.

(b) spherical image for polyhedral vertex. The fold angle between

facets is reflected by the arc length on the spherical image. The

enclosed area on the trace is given as the sum of the sector angles

minus 2pi, and is independent of the fold angles.

Figure 2.7: The spherical representation provides an intrinsic view of the Gaussian

curvature, both for (a) smooth and (b) polyhedral surfaces. Images from Hilbert

& Cohn-Vossen (1952).

18


the simplest origami fold is a vertex with four folds, one valley and three mountain

folds. Huffman (1976) also considered curved folds; curved folding is beyond the

scope of this review, and the reader is referred to Duncan & Duncan (1982) and

Fuchs & Tabachnikov (1999) for the necessary differential geometry.

2.2.2 Rigid Foldability

Rigid foldability considers an origami crease pattern as freely hinged flat rigid

panels, and explores whether a continuous folding can take place without any

deformation of the facets. Unlike the study of flat-foldability, which is concerned

with the final state only, rigid foldability describes the continuous route from flat

to fully folded state.

The investigation of rigid foldability has largely been motivated by the design

of deployable structures, both developable and non-developable. For example,

Dureisseix et al. (2011) describe deployable structures for architectural applica-

tions, which maintain their flat-foldability for compact stowage, but are purposely

non-developable to improve the structures load-bearing capacity in its fully de-

ployed state. For the design of folded shell structures, the mathematical concepts

of rigid foldability and the resulting design methods are also of great interest.

Namely, if the tessellated fold pattern is rigid-foldable there exists at least one

soft internal mechanism by virtue of the bending along all fold lines. Further-

more, rigid folding provides the ability to manufacture a textured surface from

flat sheet material with minimal material deformation.

First several modelling methods for rigid origami are described, followed by an

overview of recent developments in describing the rigid foldability of 1 DOF multi-

vertex fold patterns. In recent years significant progress has been made in the

understanding of rigid foldable patterns, providing an unprecedented freedom of

design.

Origami Modelling

The deformation of developable surfaces can be represented in a variety of ways,

with a suitable choice depending on the application (Balkcom, 2004). The overview

given here is restricted to rigid origami, where the material does not stretch and

facets do not bend.

Nodal Coordinates A convenient choice of representing the folding process of

rigid origami, is by the position and motion of its vertices. The vertices are then

modelled as pin-joints and the fold lines as bars (Schenk & Guest, 2011). For

non-triangular facets, additional bars and planarity constraints must be added to

19


avoid deformation of the facets. This is the simplest mathematical representation

of origami folding, as the Jacobian of the bar length constraints is linear in the

nodal coordinates. Alternatively, the facets can be modelled by hinged plane stress

elements in a finite element analysis (Resch & Christiansen, 1971).

Fold Angles Representing the rigid folding process in terms of the fold angles

of the crease lines is not only more intuitive, but also fosters the development of

a more fundamental understanding of rigid foldability.

Belcastro & Hull (2002) describe the modelling of non-flat origami using piece-

wise affine (flat facets) and isometric (no stretching) transformations around each

vertex; this results in a series of rotation matrices that include both the sector

and fold angles around the vertex. The loop closure constraint (no cutting of

material) then results in the identity of those rotation matrices (Kawasaki, 1994).

This condition is a necessary, but not sufficient condition for rigid foldability as

it does not take into consideration any self-intersection (Belcastro & Hull, 2002).

For single vertex foldings Streinu & Whiteley (2005) further showed that the 3D

folded state can be attained by continuous motion without bending of the facets,

by analogy with spherical polygonal linkages. Wu & You (2010) also exploit the

analogy with spherical linkages, and describe the loop closure in terms of the ro-

tations of normal vectors to the panels, using quaternions. Balkcom (2004) used

mechanism theory with forward/inverse kinematics, by virtual cutting along some

creases, and much of the geometric work by Stachel (2010a) relies on the coupling

between multiple spherical 4-bar linkages around facets.

For the numerical simulation of rigid origami, the non-linear loop closure equations

must be continuously satisfied. Tachi (2009c) describes such a numerical method,

based on a modified version of the equations of Belcastro & Hull (2002). If the

surface is a topological disk, the closure of any loop around vertices can be reduced

to the combination of local constraints around interior vertices. For each interior

vertex and its incident fold lines Li with fold angles 0, . . . , n (see Figure 2.8) the

rotation matrix identity condition is written as

F (0 . . . n) = 0 . . . n1n = I (2.3)

where i represents the rotation around each of the fold lines

i =

1 0 00 cos i sin i0 sin i cos i

cos i sin i 0sin i cos i 0

0 0 1

(2.4)with i the sector angle between fold lines i and i+1. When deriving the Jacobian

of the constraint equation it is useful to realise that the partial derivative with

20


L0

L1

L2

L3

1

0

2

3

0

1

2

3

Figure 2.8: At each vertex the incident fold lines Li have an associated fold angle

i, and sector angles i between Li and Li+1.

respect to a fold angle i represents an instantaneous rotation around the direction

vector of its fold line; this is a skew symmetric matrix

F

i=

0 a ca 0 bc b 0

(2.5)where a, b, c constitute the direction cosines for the fold line; the number of

independent constraints per vertex thereby reduces to three. The constraints for

every vertex in the sheet can subsequently be combined into a 3M N matrix

J =

J1...

JM

1...

N

=

0...

0

(2.6)with N the number of creases and M the number of vertices. If there are holes

in the surface, additional loop constraints must be added (Tachi, 2010c), but

are here neglected for simplicity. The nullspace of the Jacobian J provides the

possible motions that satisfy the constraints. Next configurations are found using

an Euler integration and projecting each step onto the constraint space using the

Moore-Penrose pseudo-inverse, in combination with a Newton-Raphson iteration

to eliminate errors.

For a multi-vertex pattern the Jacobian J will generally be overconstrained with

3M > N . The nullity of the matrix, and thereby the DOF of the fold pattern,

must therefore follow from Ns redundant constraints: DOF = N 3M + Ns.Finding multi-vertex crease patterns that supply that singular configuration is the

challenge of rigid foldability.

21


Multi-Vertex Rigid Foldability

Firstly, as the number of vertices, facets and edges are related by the Euler-

Poincare characteristic of the surface, the number of degrees of freedom for the

overall system is limited. Specifically, a crease pattern has at most N0 degrees of

freedom (assuming that all facets are triangulated), where N0 is the number of

vertices on the boundary of the surface (Tachi, 2010c).

In the case of quadrilateral mesh origami, with degree-4 vertices, the number of

fold lines is generally smaller than the number of constraints. Let us consider a

tessellation of degree-4 vertices, with an n n array of quadrilateral facets:

N = 2n(n 1) (2.7)M = (n 1)2 (2.8)

DOF = N 3M = (n 2)2 + 1 (2.9)

where the DOF become negative for n > 2. A similar derivation was given by

Dureisseix (2011) using recursion of mobility equations. Alternatively, consider

that in a multi-vertex pattern with degree-4 vertices, an assigned set of fold angles

will generally conflict; see Figure 2.9. The flexibility of a quadrilateral mesh is

therefore due to redundancy of constraints, such as parallel fold lines or an intrinsic

symmetry at the vertices. The best known example of a rigid foldable degree-4

vertex pattern is the Miura-ori (Figure 2.10) and it will be used throughout this

thesis.

Watanabe & Kawaguchi (2009) described a test for infinitesimal rigid-foldability of

a crease pattern with mountain-valley assignment, using the Jacobian and Hessian

of the rotation matrix introduced by Belcastro & Hull (2002), but a more funda-

mental understanding of the necessary conditions for rigid foldability of multi-

vertex patterns is desirable.

Rigid foldability of a general planar quadrilateral mesh is still an open question,

but the conditions for several classes of fold patterns are now understood. We

shall describe some recent developments in generalising rigid origami folding, and

point towards a direction by which these may be extended further.

Plane Tessellation A first type of rigid foldable mesh is described by Huffman

(1976) and Kokotsakis (1933, Fig. 15). It is the plane tessellation of a degree-

4 vertex whereby the vertex is rotated in order to avoid contradiction of the

fold angles; see Figure 2.11. This was considered self-evidently rigid-foldable by

Huffman (1976), but a more rigorous geometric proof was given by Stachel (2009).

22


0 0

0 0

Figure 2.9: Conflicting fold angle assignment for a mesh of general degree-4 ver-

tices. When a single incident fold angle 0 is prescribed, all fold angles at the

vertex are subsequently defined. When looping around the central facet, this re-

sults in a conflict at the last vertex, as two incident fold angles are prescribed. The

pattern will therefore generally not be rigid-foldable. Figure after Tachi (2010a).

Figure 2.10: The classic Miura-ori pattern is both rigid-foldable and flat-foldable;

the redundancy of its constraints follows both from intrinsic symmetry at its ver-

tices, as well as global symmetry with parallel fold lines. The Miura pattern forms

the basis for the generalisations of rigid-foldable developable origami.

23


Figure 2.11: Rigid foldable tessellated fold pattern described by Huffman (1976)

and Stachel (2009). The plane tessellation of any convex quadrilateral, by iterated

180 rotations about the midpoint of the sides, is rigid foldable as all fold anglesautomatically connect without contradiction. Image by Stachel (2009).

Flat Foldable and Developable Tachi (2009a) described the first general

extension for rigid-foldable flat-foldable degree-4 vertices, to form a generalised

freeform Miura-ori surface. The consideration of flat-foldability came from a de-

sire for compact stowage of deployable structures. The Kawasaki-Justin theorem

provides a necessary condition for local flat-foldability (Belcastro & Hull, 2002;

Demaine & ORourke, 2007). The theorem states that for a vertex where n is

even,

ni=1

(1)ii {0, 2pi,2pi} (2.10)

holds for flat-foldable vertices. For the case of a developable degree-4 vertex this

reduces to

0 = pi 2 and 1 = pi 3 (2.11)

The relationship between the fold angles i and j incident to the vertex can be

found using spherical trigonometry, and is given as follows (Tachi, 2010b):

tani2

=

{Aij tan

j2 (i j = 1 or 3)

tan j2 (i j = 2)(2.12)

where the latter represents that pairs of opposing fold lines have an equal abso-

lute folding angle, and Aij is the coefficient between these two equivalent pairs,

determined by 0 . . . 3. In other words, it is an intrinsic measure of the crease

24


pattern. Specifically, if |0| = |2| > |1| = |3|, then

|A01| =

1 + cos(0 1)1 + cos(0 + 1)

(2.13)

This relationship is a special case of the general formula for an origami vertex by

Huffman (1976). The formulation by Huffman (1976) provides further insight into

the degree-4 vertex, including conditions for local self-intersection.

Tachi (2009a) uses a modified version of Equation 2.12 to map between the al-

ternating pairs of fold angles, in terms of a conversion coefficient. Using a loop

argument around each facet he provides a necessary condition for the existence

of rigid-foldable flat-foldable quadrilateral mesh origami. More importantly, it is

shown that if there exists a partially folded state (where every fold line is semi-

folded, i.e. 6= 0, 6= pi,pi), it is finitely rigid foldable. An alternative way ofderiving this result is to note that Equation 2.12 is linear in tan i2 . Therefore, if

an arbitrary semi-folded configuration{

tan i(t0)2

}is found that satisfies all fold

angles, there must exist a finite path:{tan

i(t)

2

}=

{tan

i(t0)

2

}tan t2tan t02

(2.14)

where t (0 t pi) is the parameter that defines the amount of folding. Thisdirectly leads to a numerical design method: starting from a partly folded con-

figuration of a known rigid foldable pattern such as the Miura-ori, the nodes in

the crease pattern can be incrementally modified whilst satisfying a number of

constraints: developability, flat-foldability and planarity of facets. This provides

an effective means to design freeform rigid-foldable surfaces. A useful corollary

from Tachi (2009a) states that if the conversion coefficient of every inner vertex

is constant in each row or each column, the pattern is rigid foldable.

Flat Foldable and Non-Developable The extension to non-developable sur-

faces made use of recent results on the finite integrability (continuous motion) of

a discrete Voss surface (Schief et al., 2007). This is a planar quadrilateral mesh

surface composed of degree-4 vertices, each of which satisfies 1 = 3 and 0 = 2;

this includes the Eggbox pattern described in Chapter 3. By introducing the con-

cept of complementary fold angles, i = pi i, Tachi (2010b) showed that theflat-foldable Miura and Voss vertices share an intrinsic symmetry, and their kine-

matics are identical. The resulting hybrid Miura-Voss patterns are bi-directionally

flat-foldable (Tachi, 2010b). Dureisseix et al. (2011) also describe a generalisation

of Miura-ori into doubly-curved non-developable surfaces, but it retains a plane

of reflective symmetry. It is interesting to note that in its generalised form the

Miura vertex will always expand in all directions simultaneously, whereas the Voss

vertex expands in one and contracts in the orthogonal direction.

25


Kokotsakis Meshes Recent work has shown that both the generalised Miura

and Voss patterns are a subset of flexible Kokotsakis meshes (Kokotsakis, 1933;

Stachel, 2010a,b). This connection unifies and further extends the families of

possible rigid-foldable quadrilateral mesh patterns.

In discrete differential geometry there is an interest in polyhedral structures com-

posed of quadrilaterals, i.e. quadrilateral surfaces. When all quadrilaterals are

planar, the edges are geodesics and form a discrete conjugate net. When each

quadrilateral is seen as a rigid body and only the dihedral angles can vary, the

question arises under which conditions such structures are flexible. Stachel (2010a)

states a theorem due to Schief et al. (2007), that a discrete conjugate net in gen-

eral position is continuously flexible if and only if all its 3 3 complexes, i.e. allincluded Kokotsakis meshes, are continuously flexible. A Kokotsakis mesh is a

polyhedral structure consisting of an n-sided central polygon P0 surrounded by a

belt of polygons; see Figure 2.12. Each vertex Vi is the meeting point of 4 facets;

each facet is a rigid body and only the dihedral angles can vary. The lengths

of the sides of the central polygon have no bearing on the connectivity, and its

kinematics are therefore represented by a coupling of spherical linkages.

Stachel (2010b) presents several known categories of continuously flexible Kokot-

sakis meshes (see Figure 2.12):

I plane-symmetric : reflection in the plane of symmetry of V1 and V4 maps

each horizontal fold onto itself while the vertical ones are exchanged.

II translational : there is a translation V1 7 V4 and V2 7 V3 mapping thethree faces on the right-hand side of the vertical fold through a2 onto the

triple on the left-hand side of the vertical fold through a4.

III isogonal : at each vertex the fold angles are congruent.

IV orthogonal : here the horizontal folds are located in parallel (say: horizontal)

planes, and the vertical folds in vertical planes. P0 is a trapezoid.

V line-symmetric : a line-reflection maps the linkage at V1 7 V4 and V2 7 V3.

Case V is newly derived by Stachel (2010a), and includes the earlier example of

the plane tessellated quadrilateral shown in Figure 2.11. It is emphasised that a

complete classification of continuously flexible Kokotsakis meshes has not yet been

achieved.

Stachel (2010b) rephrases case III, such that a Kokotsakis mesh is flexible if at

each vertex Vi opposite angles are either equal or complementary: i = i, i = i,

or i = pi i, i = pi i. This describes the intrinsic symmetry that unitesthe generalised Miura-ori pattern and Voss surface, and shows why both are

26


(a) Kokotsakis mesh with n = 4, with sector angles i, i, i, i at vertex Vi.

The polygons need not be planar, i.e. it can be non-developable.

(b) I - plane-symmetric. (c) II - translational.

(d) IV - orthogonal. (e) V - line-symmetric.

Figure 2.12: For a Kokotsakis mesh with n = 4 (a), there exist 5 known classes

that ensure continuous flexibility (b-e). Images from Stachel (2010b).

27


continuously flexible. It is noted that the traditional Miura-ori pattern is of types

II, III and IV simultaneously, providing its redundant flexibility.

Concluding from the analysis of continuously flexible Kokotsakis meshes, future

extensions of generalised rigid-foldable degree-4 meshes will have to consider condi-

tions around facets, rather than on individual vertices to establish finite flexibility.

The restriction to flat-foldable meshes can then be relinquished, opening up the

possibility of new types of rigid-foldable origami patterns.

Topological Extensions

In the previous discussion of rigid foldability, it has been assumed that the quadri-

lateral mesh is homeomorphic to a disk. Several of the concepts, however, have

also been extended to cylindrical surfaces in order to design rigid-foldable tubes

and architectural coverings (Tachi, 2009b, 2010b).

2.2.3 Fold Pattern Design

Developments in the mathematics of rigid foldability, as well as the emergence of

computational design tools have enabled great freedom in the design of origami

patterns. For architectural applications the ability to design a complex freeform

surface geometry is desirable, but the increased complexity comes at the cost

of difficulty of manufacture. Most technical applications such as folded cores

therefore use relatively simple fold patterns, as the manufacturing processes rely

on the repetition of a small set of operations. Klett & Drechsler (2009) further

note that the relative simplicity of most technically relevant fold patterns is also a

consequence of optimised structural and functional efficiency. In general, multiple

overlapping folds and material layers do not add much performance to the resulting

folded core but result in higher mass and material consumption. We here briefly

describe freeform origami design methods, as well as tools to design row-tessellated

fold patterns for engineering applications.

Freeform Surfaces The minimal requirement for the design of origami fold

patterns is developability, i.e. the sector angles at each vertex add up to 2pi.

Tachi (2009a) describes a freeform origami design methodology, where a partly-

folded sheet can be interactively modified by moving individual vertices, whilst

preserving the necessary developability constraints. Further constraints may be

added, such as local flat foldability and global symmetries, as well as the ability for

quadrilaterals to collapse into triangular facets when nodes merge (Tachi, 2010a).

Kling (2007b) developed an Aspect-Shaping-Floating algorithm, where an initial

28


Figure 2.13: Overview of available modifications of a Miura-ori unit cell. Possibil-

ities include transverse and longitudinal curvature (mod 1 and mod 6), increased

bonding area (mod 5) and an increase in local density (mod 4). Image from

Khaliulin (2005).

partly-folded surface is transformed to a desired global geometry, before solving

the developability constraint for each of the nodes by floating on the surface.

Modular Unit Cells For many structural applications of folded sheets, modu-

lar unit cells are tessellated in orthogonal rows and columns. By tailoring the local

geometry of the unit cells, a wide range of mechanical properties can be achieved.

Khaliulin (2005) describes the synthesis of folded cores by modifying existing row-

arranged patterns. The modifications are parametric (modifying sector angles at

the vertices), structural (adding additional facets and creases), or a hybrid of both;

see Figure 2.13. The available modifications provide the ability to introduce global

(double) curvature, local increase in core density, and increased contact area with

face sheets. This synthesis method offers a large scope for pattern design, without

straying from simple repetitive unit cells that can be manufactured continuously.

A computational design methodology for Doubly Periodic Folded (DPF) surfaces is

proposed by Kling (2005). The method relies on prescribing desired row/column

cross sections, with intersection points, by which the interlying surface can be

constructed; see Figure 2.14. The resulting DPF surfaces are rigid foldable, as a

result of the underlying mathematics described in Kling (1997). The local isometry

used to design the doubly periodic sheets can be regarded as reflection of the rows

in mirroring planes, which thereby implicity introduces the local flat-foldability

conditions around the degree-4 vertices.

29


Figure 2.14: A Doubly Periodic Folded surface can be constructed by specifying

a row cross section and a column reflection scheme. Image from Kling (2007b).

Figure 2.15: A modified Miura fold pattern which is not flat foldable, and thereby

produces a rigid cellular core material (McKay, 1984).

Self-Locking Patterns By purposely introducing non-flat-foldable vertices to

a fold pattern, a folded structure can be designed to lock into a prescribed con-

figuration due to facet-to-facet contact. For example, Akishev et al. (2010) add

self-locking to the helical fold core pattern to fix the configuration and provide flat

bonding areas between adjacent helical curves; see Figure 2.4(d). McKay (1984)

describes folded core material which self-locks, and thereby introduces vertical

separating walls which provide additional compressive strength; see Figure 2.15.

Lastly, self-locking patterns are of great interest when considering self-assembly of

folded sheets, for example for MEMS applications.

30

2.3 Conclusions

2.3 Conclusions

In textured shell structures, a local texture pattern (e.g. corrugations, dimples,

folds) is introduced to a thin-walled structure, in order to modify its global me-

chanical properties. Existing applications either exploit the increased in-plane

flexibility (e.g. to accommodate large thermal strains) or the increased out-of-

plane bending stiffness (e.g. corrugated sheets, or folded sandwich panel cores)

provided by the texture patterns.

The folded shell structures described in this thesis exploit the anisotropy of de-

formation modes introduced by the fold pattern. The distinct fold lines increase

the shells second moment of area, whilst simultaneously providing compliant de-

formation modes. The compliance is driven by the hierarchical interaction of

mechanistic articulation around the fold lines, in combination with the inexten-

sional deformations of the thin-walled shells. The mechanics of these folded shell

structures has not previously been studied, and forms the topic of Chapters 4

and 5.

A review of origami mathematics provides a background for the study of folded

shell structures. In rigid origami a fold pattern is modelled as rigid facets con-

nected by frictionless hinges. For tessellated fold patterns, such as folded shell

structures, the resulting mechanism rapidly becomes overconstrained and a fold-

ing motion is only possible under specific geometric conditions. Recent work in

origami mathematics has revealed several rigid-foldability conditions. In the case

of the folded shell structures, this enables the design of soft kinematic modes where

the shells deform through articulation around the fold lines. It also has important

consequences for the manufacture of folded sheets, as described in Chapter 6.

31

Chapter 3

Folded Shell Structures

3.1 Description

The distinguishing feature of the folded shell structures described in this thesis,

is the presence of distinct fold lines in the shell surface, which provides the sheet

with an inherent degree of flexibility. Globally the folded shell structures can be

regarded as thin-walled shells, built up of a tessellation of unit cells, which are in

turn composed of thin-walled facets joined at the distinct fold lines.

This description covers a wide range of possible configurations, and we shall limit

ourselves to a subset of the folded shell structures. The folded sheets need not

necessarily be developable, and may have points or lines of non-zero Gaussian

curvature. The fold lines, and thereby the facets, may in general be curved, but

here we only consider straight fold lines and planar facets. Furthermore, the

sheets considered here consist of regular tilings of degree-4 vertices, where four

fold lines meet. If constructed of rigid panels with hinges, these sheets would form

a mechanism with a single degree of freedom (DOF). Higher-order vertices would

provide additional flexibility in the sheet, but thereby also reduce its load-carrying

capacity. Lastly, although global curvatures can be introduced by modifying the

tessellation pattern, we only consider initially planar folded sheets.

3.1.1 Example Sheets

Two representative examples of folded shell structures will be used throughout

this thesis: the Miura and Eggbox sheet, shown in Figure 3.1. Both sheets con-

sist of a regular tessellation of identical parallelogram facets, but the Miura sheet

is developable and can therefore be folded from flat sheet material, whereas the

Eggbox sheet consists of alternating apices and saddle points with equal and op-

posite angular defect; see Figure 3.2. These two patterns provide representative

examples of both developable and non-developable folded shell structures.

The Miura sheet is named after Koryo Miura who first introduced this fold pat-

tern to engineering applications, and it has remained the most commonly studied

32

3.1 Description

(a) overview of Miura sheet.

(b) overview of Eggbox sheet.

(c) close-up of unit cells

Figure 3.1: photographs of (a) the Miura, and (b) the Eggbox sheet. The models

are made of standard printing paper, and the parallelograms in both sheets have

sides of 15mm and an acute angle of 60. The Miura sheet is developable, whereasthe Eggbox sheet has an (equal and opposite) angular defect at its apices and

saddle points.

33

3.2 Mechanical Properties

Figure 3.2: The Miura sheet is folded from a single flat sheet of paper (left); in

contrast, the Eggbox sheet (right) is made by joining individual strips of paper,

which introduces the angular defects at the vertices.

fold pattern. It can be straightforwardly modified to create sheets with a global

(double) curvature, varying densities and tapering fold depths, and helical config-

urations (e.g., Klett & Drechsler, 2009; Akishev et al., 2010; Talakov, 2010); some

examples of modified Miura patterns are shown in Figure 3.3. In the literature

the pattern goes by many monikers, such as chevron or herringbone pattern,

Z-crimp and zigzag corrugation. The Eggbox pattern was simply named after

its resemblance to boxes used for the storage of eggs. The pattern is much less

common in engineering literature, although it has previously been proposed as an

expansible sheet material (Brunner, 1968), and a deployable structure for archi-

tectural applications (Tachi, 2010b).


The first interesting property of the folded sheets is their ability to undergo rel-

atively large deformations, by virtue of the folds opening and closing. Moreover,

the fold patterns enable the sheets to locally expand and contract and thereby

change their global Gaussian curvature without any stretching at material level.

Our interest lies with the macroscopic behaviour of the sheets, and we therefore

consider the global Gaussian curvature of an equivalent mid-surface of the folded

sheet. Both the Eggbox and Miura sheets are initially flat, and thus have a zero

global Gaussian curvature. Now, unlike conventional sheets, both folded sheets can

easily be twisted into a saddle-shaped configuration which has a globally negative

Gaussian curvature see Figure 3.4(a) and Figure 3.5(a).

The sheets most intriguing property, however, relates to their Poissons ratio.

Both sheets have a single in-plane mechanism whereby the facets do not bend and

the folds behave as hinges; by contrast, facet bending is necessary for the out-of-

plane deformations. As shown in Figure 3.4(b) and Figure 3.5(b), the Eggbox and

34


Figure 3.3: A selection of folded Miura geometries. Left to right: angled sheet;

continuously transversely curved; planar sheet; stacked set of sheets that expand

in a coordinated manner.

the Miura sheet respectively have a positive and a negative Poissons ratio in their

planar deformation mode. A negative Poissons ratio is fairly uncommon, but

can for instance be found in foams with a reentrant microstructure (Lakes, 1987),

chiral honeycomb lattices (Prall & Lakes, 1997) and materials with hinged rotating

units (Grima et al., 2005). Conventionally, materials with a positive Poissons ratio

will deform anticlastically under bending (i.e., into a saddle-shape) and materials

with a negative Poissons ratio will deform synclastically into a spherical shape. As

illustrated in Figure 3.4(c) and Figure 3.5(c), however, both folded textured sheets

behave exactly opposite to what is conventionally expected, and their Poissons

ratio is of opposite sign for in-plane stretching and out-of-plane bending. This

remarkable mechanical behaviour has only been described theoretically for auxetic

composite laminates (Lim, 2007) and specially machined chiral auxetics (Alderson

et al., 2010), but is here observed in folded sheets made of conventional materials.

35


(a)

(b)

(c)

Figure 3.4: mechanical behaviour of the Miura sheet; it can be twisted into a

saddle-shaped configuration with a negative global Gaussian curvature (a). Sec-

ondly, the Miura sheet behaves as an auxetic material (negative Poissons ratio)

in planar deformation (b), but it assumes a saddle-shaped configuration under

bending (c), which is typical behaviour for materials with a positive Poissons

ratio.

36


(a)

(b)

(c)

Figure 3.5: mechanical behaviour of the Eggbox sheet. Firstly, it can change

its global Gaussian curvature by twisting into a saddle-shaped configuration (a).

Secondly, the Eggbox sheet displays a positive Poissons ratio under extension (b),

but deforms either into a cylindrical or a spherical shape under bending (c). The

spherical shape is conventionally seen in materials with a negative Poissons ratio.

37


Demonstrative Experiments

To demonstrate the unexpected and contrasting bending properties of the Miura

and Eggbox sheets, simple three-point bending tests were carried out; Figures 3.7

3.11. The Miura and Eggbox sheets consisted of 5 5 unit cells, composed ofparallelograms with sides of 15mm and an acute angle of 60. The Miura sheetwas folded from standard 80gr printing paper; the Eggbox sheet was constructed

by glueing together two orthogonal sets of strips of parallelograms of the same

paper. The sheets were simply supported, and the top vertex of the central unit

cell was displaced downwards by 10mm from its rest configuration; see Figure 3.6.

The coordinates of the top vertices were subsequently measured using a three-

axis coordinate measuring machine, to within 0.5mm. In order to characterise the

out-of-plane bending, the curvature 1 along, and 2 transversely to, the bending

line were calculated by fitting a circle to the measured points. Of main interest

is the ratio 1/2, which describes the coupling between the curvatures along the

principal axes. The results are given in Figures 3.7(c)3.11(c).

The experiments affirmed the remarkable bending properties observed in the Miura

and Eggbox sheets, under minimal loading and boundary conditions. Several

observations are of interest. For the Miura sheet, the deformed configuration

differed when bending along or transversely to the corrugations, whereas one would

expect the ratio of the curvatures to be inverted for the two orientations. This

can be attributed to the difficulty of imposing appropriate boundary conditions,

and this simple experiment will therefore not excite a precise bending mode, but

rather a combination of bending and stretching. Secondly, the symmetry of the

Eggbox sheet would suggest a spherical deformation mode with equal principal

curvatures, when bending along the principal axes; this is not the case, and can

be attributed to an induced in-plane strain along the bending axis. Furthermore,

unlike the Miura sheet, when bending at 45 we do not obtain a twisting mode,but rather a cylindrical deformation mode, as in this configuration uninterrupted

bending lines extend across the sheet.

38


d = 10mm

Figure 3.6: Experimental set-up for the three-point bending of the folded sheet.

The sheet is simply supported on two blocks, and the top node of the central unit

cell is displaced downwards by 10mm from its rest configuration.

39


(a) overview

(b) side views

(c) measured nodal coordinates, with circles of curvature.

Figure 3.7: Three-point bending experiment of the Miura sheet, longitudinally

along the corrugations. The ratio 1/2 2.4, with therefore a relatively weakcoupling between the two bending axes.

40


(a) overview

(b) side view


Figure 3.8: Three-point bending experiment of the Miura sheet, transversely

across the corrugations. The fitted circles of curvature provide the ratio 1/2 1, indicating a strong coupling between the two axes.

41


(a) overview

(b) side views


Figure 3.9: Three-point bending experiment of the Miura sheet, with bending axis

at 45 to the corrugations; this is effectively identical to the twisting mode. Thefitted circles of curvature showed that the ratio of 1/2 1.3.

42


(a) overview

(b) side views


Figure 3.10: Three-point bending experiment of the Eggbox sheet, into the spher-

ical deformation mode. The ratio 1/2 1.6, which can be attributed to com-pression along the bending line.

43


(a) overview

(b) side views


Figure 3.11: Three-point bending experiment of the Eggbox sheet, along the

double-corrugation. The ratio of 1/2 6.5, and the deformed configurationis therefore almost cylindrical.

44

3.3 Structural Analysis


The folded shell structures can be regarded as a type of compliant structure, where

the dominant deformation modes can be considered as mechanisms with a non-

zero stiffness. Their mechanical properties therefore straddle that of mechanisms

and structures, and a suitable modelling method must capture this behaviour.

For the numerical analysis of folded sandwich panel cores, non-linear Finite El-

ement Analysis is used, as it provides the necessary detailed modelling of local

buckling and crushing (e.g., Heimbs, 2009; Baranger et al., 2011a). Resch &

Christiansen (1971) used a simple folded plate finite element model to analyse

both the kinematics and stiffness of a triangulated folded sheet. An analytical ap-

proach was employed by Norman (2009) for (curved) corrugated sheets, where the

global behaviour was described using an equivalent mid-surface. For the analysis

of compliant shell mechanisms which are closely related to the folded sheets

discussed in this thesis Seffen (2011) places a stronger emphasis on the hier-

archical nature of these types of structures, and the proposed modelling method

describes the global geometry in terms of unit cell kinematics. This is also the

approach taken in this thesis, as it best elucidates the interrelationship between

the various mechanical scales of the folded shell structures. In our analysis the

sheet material of the folded shell structures is assumed to have zero thickness.

This implies that fold lines are infinitely sharp, whereas in reality there must be

a finite (but large) curvature. Furthermore, the stiffness analysis of the fold lines

must necessarily be simplified, as both the curvature and material thickness are

singular. These simplifications, however, remove many of the clouding details to

gain a conceptual understanding of the kinematics of the folded shell structures.

First the kinematics of the unit cells are described in Chapter 4, followed by a

more holistic stiffness matrix approach in Chapter 5 where the sheets are given a

simplified material model. Both approaches provide complementary insights into

the deformations of the sheets.

Moving Vertices Some deformations observed in experimental models chal-

lenge the underlying assumption for the mechanical models used thus far: the fold

pattern changes during the sheets deformation. Figure 3.12 shows a model of a

planar Miura sheet manufactured by vacuum forming High Impact Polystyrene

(HIPS) onto a mould. When bending the sheet, it can attain a single (trans-

verse) curvature; this is made possible by virtue of the vertices moving plastically

through the material, thereby modifying the fold pattern. The shifting of fold ver-

tices through the material has previously been employed for the energy dissipation

in automobile crash boxes (Ma & You, 2011), but not for its ability to introduce

additional flexibility to fold patterns.

45


Figure 3.12: A model of a planar Miura sheet [left], made by vacuum forming High

Impact Polystyrene (HIPS). When bending, the fold pattern changes slightly as

the vertices move plastically through the material, enabling the sheet to attain a

single curvature [right].

46

Chapter 4

Kinematic Analysis

4.1 Planar Kinematics

The planar kinematics of the folded textured sheets can be described algebraically,

by assuming that the facets bounded by the fold lines remain rigid and the folds

behave as frictionless hinges.

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