Inflating to shape: from soft architectured elastomers to patterned fabric

Contents

CONTENTS

Introduction

What makes a salad leaf curl? And more generally, how does the living world manage to build such magnificent structures, with an incredible variety of colours, textures and shapes? Since the discovery of DNA, one common approach to answer these questions relies entirely on genetics. However, this paradigm is progressively evolving in the biology community, as numerous experiments have proved the important role of the environment and especially mechanical stresses, in gene expression [START_REF] Hamant | Developmental patterning by mechanical signals in arabidopsis[END_REF][START_REF] Uyttewaal | Mechanical stress acts via katanin to amplify differences in growth rate between adjacent cells in arabidopsis[END_REF][START_REF] Sahaf | The rheology of a growing leaf: stress-induced changes in the mechanical properties of leaves[END_REF].

The inspiration of this thesis is morphogenesis, i.e., the shape changes which occur during the life of an organism. More specifically we wish to mimic with an experimental point of view how slender 2-dimensional structures such as petals, leaves or organ walls may transform and morph onto complex 3-dimensional shapes, overcoming strong geometrical constraints. One may argue that the recent development of fast prototyping techniques, such as additive manufacturing (3D printing) and CNC micro-machining, already enables us to construct complex 3D shaped structures. Shape-changing materials present however several major advantages over these techniques:

• They are compatible with surface patterning and imprinting processes, that work primarly on planar surfaces. These advanced production techniques are crucial for developing surfaces with specific functionalities such as hydro-or oleophobicity [START_REF] Coux | Soft, elastic, water-repellent materials[END_REF][START_REF] Jiang | Biomimetic superoleophobic surfaces: focusing on their fabrication and applications[END_REF], integration of electronic circuits [START_REF] Rogers | Materials and mechanics for stretchable electronics[END_REF][START_REF] Loh | Nanoelectromechanical contact switches[END_REF], or control over cell interaction [START_REF] Dobbenga | Nanopattern-induced osteogenic differentiation of stem cells ? a systematic review[END_REF] for biomaterials.

• Flat structures are easier to manufacture, to store and to transport in the form of a roll for example.

• These structures are adaptive. They can reversibely change their shape to adapt to their environment, enabling multiple functions.

The activated material that we aim at manufacturing should continuously deform and be intrinsically soft, in contrast with classic robots made of rigid articulated structures with gears and motors. Although common robots are very successful at doing automated repetitive and precise tasks, they fail in delicate tasks, such as in minimally invasive surgery, active prosthetics, and automation tasks involving delicate irregular objects [START_REF] Cianchetti | Soft robotics technologies to address shortcomings in today's minimally invasive surgery: The stiff-flop approach[END_REF]. Interactions between human and traditional hard robots are also dangerous in unstructured environments where unforeseen motion is necessary. Safety features can be programmed in the software of classic stiff robots to impose force limits (like for automatic closing doors) and artificial compliance. However, robots made entirely out of soft materials, e.g. elastomers instead of metals, open a new path to make man-machine interaction inherently safe and to mimic soft biological systems [START_REF] Gorissen | Elastic inflatable actuators for soft robotic applications[END_REF]. This is the core concept of soft robotics [START_REF] Rus | Design, fabrication and control of soft robots[END_REF][START_REF] Shepherd | Multigait soft robot[END_REF], that is believed to have an impact on the development of collaborative robots and medical equipment. The geometric challenge when morphing an initially flat sheet into a 3D shape is the following: a thin sheet is very easy to bend in one direction but impossible to bend in two simultaneous directions. This is a consequence of Gauss Theorema Egregium, which states that the product of both principal curvatures on a point of a surface (the Gaussian curvature) only depends on the distances along the surface itself. When bending isometrically a surface, this quantity is thus conserved at every point, constraining remarkably the possibilities of shapes. For an initially flat sheet for example, one of the two principal curvatures vanishes so that the surface is always straight along one direction at least. This geometric constraint is also crucial for cartography: in order to flatten the earth (which is doubly curved) on a planisphere (which is flat), the mapmaker has to distort the actual distances. In the classical Mercator projection, these distortions are generally focused in the oceans and at the poles. One main feature of morphogenesis is growth, which changes the distances in living structures, modifying intrinsically their geometry. Growth is generally non-uniform, which induces stresses and geometrical incompatibilities, that result in instabilities in the slender structure, leading to the evolution of the shape. In this thesis we will tackle the strong but subtle coupling between local nonuniform growth in artificial structures and their global buckling.

While controlling the local growth of bioligical matter is possible, synthetic objects cannot "grow" in the biological sense. Nevertheless, it is actually possible to mimic growth since the 2000's, through various actuation strategies, such as chemical swelling [START_REF] Klein | Shaping of elastic sheets by prescription of non-euclidean metrics[END_REF][START_REF] Kim | Designing responsive buckled surfaces by halftone gel lithography[END_REF][START_REF] Huang | Ultrafast digital printing toward 4d shape changing materials[END_REF][START_REF] Naficy | 4d printing of reversible shape morphing hydrogel structures[END_REF][START_REF] Byun | Swelling-driven rolling and anisotropic expansion of striped gel sheets[END_REF][START_REF] Ge | Multimaterial 4d printing with tailorable shape memory polymers[END_REF], dielectric elastomers [START_REF] Bense | Buckling of elastomer sheets under non-uniform electro-actuation[END_REF][START_REF] Hajiesmaili | Reconfigurable shape-morphing dielectric elastomers using spatially varying electric fields[END_REF][START_REF] Hajiesmaili | Voltagecontrolled morphing of dielectric elastomer circular sheets into conical surfaces[END_REF], liquid crystal elastomers [START_REF] Warner | Topographic mechanics and applications of liquid crystalline solids[END_REF][START_REF] Modes | Gaussian curvature from flat elastica sheets[END_REF][START_REF] Mcconney | Topography from topology: Photoinduced surface features generated in liquid crystal polymer networks[END_REF][START_REF] Aharoni | Geometry of thin nematic elastomer sheets[END_REF][START_REF] Kotikian | 3d printing of liquid crystal elastomeric actuators with spatially programed nematic order[END_REF][START_REF] Ambulo | Four-dimensional printing of liquid crystal elastomers[END_REF][START_REF] Plucinsky | Programming complex shapes in thin nematic elastomer and glass sheets[END_REF][START_REF] Ware | Voxelated liquid crystal elastomers[END_REF][START_REF] Modes | Disclination-mediated thermooptical response in nematic glass sheets[END_REF][START_REF] Konya | Modeling defects, shape evolution, and programmed auto-origami in liquid crystal elastomers[END_REF] or magnetic responsive materials [START_REF] Kim | Printing ferromagnetic domains for untethered fast-transforming soft materials[END_REF][START_REF] Hu | Small-scale soft-bodied robot with multimodal locomotion[END_REF], that we will detail in a next section. Hadrien Bense, a previous student from the group, made for instance pioneering advances using dielectric elastomers [START_REF] Bense | Croissance, compaction et adhésion de plaques minces[END_REF]. His thesis has been of great inspiration for my own work.

CONTENTS

As indicated by the title of the thesis, we will introduce a different actuation method, based on the inflation of cavities inside the structure. The development and democratisation of fast prototyping technologies, such as additive 3D printing or micro-machining, opens new possibilities to engineer structures with a complex underlying architecture. This new kind of structures may benefit from the different scales to exhibit surprising and nonclassical global properties: this is what we pompously call metamaterials [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF][START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF][START_REF] Yang | Phase-transforming and switchable metamaterials[END_REF]. Said differently, the sub-structure of the object (its geometry, the materials used...) encodes specific functions in a material way (hardware), introducing a new concept of physical intelligence: the material is the machine [START_REF] Bhattacharya | The material is the machine[END_REF].

In a first part, we shall expose the coupling between geometry and elasticity in slender 2D structures and review some examples from Nature, that will be our source of inspiration throughout the entire thesis. An overview of the state-of-the-art of shape-morphing and inflated structures will be also detailed.

In a second part, we will present the baromorph, a pneumatic shape-morphing structure made of stretchable material embedding a specific network of airways that encodes the target 3D shape by means of localized, anisotropic in-plane deformation on pressurization (see Fig. 1(a)).

In a third part, we will investigate a different strategy, based on the constrained volume optimization of quasi-inextensible fabric sheets to program complex and stiff quasi-1D and 3D shapes on inflation (see Fig. 1

(b)-(c)).

Concerning the methodology, simple model experiments will be conducted in order to reduce the number of parameters and easily identify the mechanisms at play. Scaling laws and more advanced theory based on energy minimization will be qualitatively and quantitatively compared to the experimental results. Collaborations with theoreticians from various communities (solid mechanics, liquid crystal elastomers, computer graphics) will help us to deepen our approach and some preliminary applications in collaboration with craftsmen will be presented.

Chapter 1

Elasticity, Geometry and Morphing

In this first chapter, we aim at introducing the mechanical concepts on which this thesis is based and detailing the state of the art in this research area. In order to do so, we first present a simple shape-changing strategy, namely the bilayer effect, and show the strong limits of such a strategy. Studying the deformation of a slender body when mechanically sollicited, we demonstrate via energy arguments that a thin plate or shell favours bending rather than stretching/compressing. This is why they tend to buckle easily under compression. Isometries being at the core of their deformation, we then introduce some simple differential geometry concepts and, following the seminal reasoning of Carl Gauss, present the strong coupling between the curvature of a surface and its metric (Theorema Egregium). Some practical examples from the natural world, mostly from the morphogenesis of plants, will be then shortly exposed, as a source of inspiration. Growth being at the core of these examples, we will review the versatile and elegant strategies developed recently in the field to mimic these shape transformation. Finally, we will review some interesting studies on inflated structures, since we will rely on fluid pressure to actuate the transformation throughout the thesis.

First strategy to change the shape: bilayer effect

A pine cone looks very different depending on the meteorological conditions. If it rains, the pine cone is closed as shown in Fig. 1.1(right). On a sunny and dry day however, the exact same pine cone opens as if it were blooming. This opening-closing process is completely reversible and reproducible, even if the cone detaches from the tree and is technically "dead" [START_REF] Reyssat | Hygromorphs: from pine cones to biomimetic bilayers[END_REF]. What is the mechanism underlying this shape change? Humidity level plays a key role in the transformation. Each individual scale of the pine cone is indeed made of two layers with different orientation and density of cellulose fibers. The external layer is more responsive to humidity and swells in the presence of water (like a typical hydrogel), whereas the internal layer is mostly not responsive to humidity. When CHAPTER 1. ELASTICITY, GEOMETRY AND MORPHING the external layer swells, it becomes longer than the internal one. In order to accommodate this length mismatch, the whole scale bends towards the center of the pine cone (i.e., in the direction of the shorter layer). On a stadium, the runner on an outer track covers indeed a longer distance than a runner on an inner track. This is the so-called bilayer effect: it may be observed by gently depositing a strip of tracing paper on a bath of water [START_REF] Reyssat | How wet paper curls[END_REF]: the penetration of water into the sheet induces the swelling of the bottom part and hence the coiling of the strip, the direction of coiling being set by the orientation of cellulose fibers (see Fig. 1

.2).

This effect was first computed by Timoshenko [START_REF] Timoshenko | Analysis of bi-metal thermostats[END_REF] in his analysis of bi-metal thermostats. Old-fashioned thermostats are indeed made of two metallic layers welded together, which have different coefficients of expansion with temperature. The heating thus produces a length mismatch between both layers, inducing the bending of the metal strip towards the shortest layer. This bending triggers the closing/opening of an electrical circuit at a given temperature, controlling the heating. This mechanism is very efficient in slender structures to get large displacements with small strains in the material. The [START_REF] Shepherd | Multigait soft robot[END_REF] (a) (b)

.4: Bilayer plate: (a) for small bilayers effect, the plate starts bending in two directions and adopts a sperical shape. (b) for stronger bilayers effect, the structure destabilizes and bifurcates to one preferred direction along which it curls, forming a cylindrical shape. Figure adapted from Pezzulla et al. [START_REF] Pezzulla | Geometry and mechanics of thin growing bilayers[END_REF] typical strain in order to bend a strip of thickness h over a radius of curvature R scales namely as h/R (the normalized distance difference between the upper and the lower part of the strip), which converges to 0 for ultrathin strips.

This phenomenon has been extensively used in the recent development of soft robotics over the last decade [START_REF] Gorissen | Elastic inflatable actuators for soft robotic applications[END_REF][START_REF] Rus | Design, fabrication and control of soft robots[END_REF][START_REF] Yamaguchi | A robot hand using electro-conjugate fluid[END_REF][START_REF] Mosadegh | Pneumatic networks for soft robotics that actuate rapidly[END_REF][START_REF] Cianchetti | Biomedical applications of soft robotics[END_REF]. In such systems, parallel cavities are embedded in the upper part of a rubber strip. Additionnally, a thin strain-limiting material may be glued at the bottom of the strip to enhance the elongation contrast. Upon inflation, the upper part expands whereas the lower part does not (Fig. 1.3). The pneumatic strip consequently bends in a controllable manner depending on the applied pressure. It is now tempting to use the same concept for 2D plates instead of thin strips to curve the structure in two directions simultaneously. Matteo Pezzulla and coworkers investigated this possibility of shape-morphing via bilayer effect [START_REF] Pezzulla | Geometry and mechanics of thin growing bilayers[END_REF][START_REF] Pezzulla | Curvature-driven morphing of non-euclidean shells[END_REF] with the swelling of a layer of silicon rubber on a passive one: although the plates do start bending in two directions simultaneously and adopt the expected dome shape, the plate curls in one single direction for stronger bilayer effect (when the deflection of the plate is typically larger than its thickness), breaking the symmetry of the system (Fig. 1.4). No matter the boundaries of CHAPTER 1. ELASTICITY, GEOMETRY AND MORPHING Figure 1.5: A thin strip subjected to an isotropic bilayer effect bends at every point in both directions simultaneously. If the original strip shape is cleverly chosen, one may reconstruct a sphere, similarly to a unique mandarine peel. Figure adapted from Danescu et al. [START_REF] Danescu | Spherical curves design for micro-origami using intrinsic stress relaxation[END_REF] the initially flat structure, it finally coils into a cylindrical shape. The bilayer strategy is thus strongly limited in terms of producing diverse 3D shapes, as only developable shapes may be obtained for large deflections.

This strategy proves itself however very useful to produce microtubes out of functionalized flat sheets [START_REF] Kumar | Fabrication of metallic microtubes using self-rolled polymer tubes as templates[END_REF][START_REF] Gómez | Vapour processed self-rolled poly (dimethylsiloxane) microcapillaries form microfluidic devices with engineered inner surface[END_REF][START_REF] Brossard | Spontaneous curvature of polydimethylsiloxane thin films: Mechanisms and applications: A new route for the low cost fabrication of new functionalities for microfluidics[END_REF]. If the plate is cut into a thin enough strip, such that deflections perpendicular to the strip direction is of the order of the strip thickness, the whole strip does not reach the instability and bends in two directions simultaneously : it may therefore form a sphere if the initial flat pattern is cleverly chosen, as shown in Fig. 1.5 [START_REF] Danescu | Spherical curves design for micro-origami using intrinsic stress relaxation[END_REF].

In order to understand this disappointingly poor possibilities in terms of shapes, we should first describe the mechanics of slender structures and show that isometries are energetically favoured for large deflections. This result will lead us to study the geometry of surfaces in order to find a way to program the target 3D shapes.

Stretching vs. Bending

When a slender elastic body is compressed, such as a ruler for example, we observe that it spontaneously breaks the symmetry and bends in one direction to accommodate the compression instead of elastically shrinking. This can be understood by estimating the energies involved, the system spontaneously choosing the minimum of energy [START_REF] Efrati | Buckling transition and boundary layer in non-euclidean plates[END_REF][START_REF] Efrati | The mechanics of non-euclidean plates[END_REF]. Let us start by considering a ruler of length L, width w and thickness h which is compressed by an in-plane displacement Lε 0 as shown in Fig. 1.6. We may expect the solution to obey the symmetry of the problem, i.e. that the ruler remains flat and thus compresses like a spring, leading to the classical stretching/compression energy U stretch ∼ EhwLε 2 0 , where E is the elastic Young modulus of the material.

Another possibility for the system is to break the symmetry and bend in either direction to form an arc that obeys the imposed displacement of its edges while preserving the length of its neutral plane L. In order to estimate the energy of such a configuration, we can note 1.2. STRETCHING VS. BENDING that "below" the center plane, the ruler is compressed, whereas the material above the neutral plane is stretched. For slender structures, that is, when the radius of curvature -denoted as R -of the arc is much larger than the thickness h, we can assume that cross sections perpendicular to the neutral plane at rest remain perpendicular to it in the deformed state. The longitudinal strain thus depends on the coordinate z along the axis perpendicular to the neutral plane, pointing outwards, and reads ε x (z) = z/R. The elastic bending energy may be computed by integrating the energy density 1/2Eε x (z) 2 over the volume, which scales as U bend ∼ Eh 3 wL/R 2 . The radius of curvature needed for a ruler of length L to have an edge-to-edge distance of L(1 -ε 0 ) scales as R ∼ L/ √ ε 0 . The geometric strain ε 0 scales indeed as (δ/L) 2 , where δ is the typical deflection. The radius of curvature R, as a second derivative of the deflection, reads R ∼ δ/L 2 . Inputting this relationships in the bending energy, we get U bend ∼ Eh 3 wε 0 /L.

.6: Classical buckling of an elastic ruler. When an elastic plate is subject to an imposed compression ε 0 , it may elastically compress or, for a critical value of compression ε c buckle out of plane and bend rather than shrink.

We clearly see that the first pure stretching/compression energy is linear with the thickness h of the ruler but quadratic in the imposed compressive strain ε 0 . The pure bending energy is cubic in the thickness and linear with the compressive strain. For very thin plates, that is when h is very small, bending appears to be much more energetically favorable than stretching. This remarkable property of slender bodies will be extensively used throughout the thesis. Comparing both energies, we obtain that they are of the same order of magnitude when the imposed compressive strain ε 0 reads ε c ∼ h 2 /L 2 . If the compression ε 0 is smaller than ε c , stretching is energetically favorable, but beyond this critical value the ruler is energetically inclined to break the symmetry and buckle out of plane.

CHAPTER 1. ELASTICITY, GEOMETRY AND MORPHING

It is important to note that even when the ruler buckles, it is not in a pure bending mode. We indeed clearly need to apply a compressive force at the edges of the ruler to keep it bent. The ruler contains some stretching/compression energy, even if the difference is barely noticeable as soon as the deflection δ is larger than the thickness h of the ruler (At the critical compressive strain ε c , the deflection δ scales as the thickness h of the ruler: δ ∼ h.)

The buckling instability is the core instability of this thesis. We will use it in different configurations from the minimal system presented here, but the physical ingredients, i.e. the competition between bending and stretching energies will be the same. The energetic argument detailed here for a ruler may be applied to any slender plate or shell: when subjected to forces, frustration or a bilayer effect, the structure preferentially bends and keeps its original length, with very little stretch within the 2D object. Isometries are thus very important during the transformation of a slender structure. In order to better understand what this isometric limit implies, we should now introduce some basic concepts of surface geometry.

Geometry in plates

Take a sheet of paper. As it is very thin, it is very easy to roll it in any direction to form a cylinder or a conical shape. However bending it simultaneously in two directions in order to fit a spherical cap or a saddle is impossible. The sheet crumples and deformation localizes at some specific points, damaging irreversibly the sheet. In fact, when simply bending reversibly the sheet and not folding it like in origami, the shape possibilities are very limited. No matter how cleverly we may try to bend the sheet, there is always one direction that remains straight. As explained before, we may only isometrically bend the paper sheet for energetic reasons. And the isometries of the plane actually form a very limited family of shapes called developable surfaces [START_REF] Struik | Lectures on classical differential geometry[END_REF][START_REF] Audoly | Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells[END_REF].

What is then so different between a plane and a spherical cap? The answer was found in 1827 by Carl Gauss, in his Theorema Egregium, his "remarkable theorem" according to the famous mathematician himself. In order to get a first intuition, let us measure the perimeter of a circle of radius r along the surface. On a plane, we have the classical formula P = 2πr. Conversely, on a sphere of radius R, the perimeter is P = 2πr where r = R sin(r/R) = r(1 -r 2 /(6R 2 ) + o((r/R) 2 )). Perimeters are thus smaller on a sphere, which means that a human living on the surface of the earth could tell, simply by measuring perimeters of circles on the surface, that the earth is not flat (Fig. 1.7(b)). On a saddle however, the wobbling of the circle induces a perimeter longer than 2πr (Fig. 1.7(c)). Measuring distances on a surface thus gives some information about the 3D deployment of the surface in space. CHAPTER 1. ELASTICITY, GEOMETRY AND MORPHING

Differential geometry of surfaces

Let us now characterize it a bit more mathematically: a surface is a two dimensional object plunged into a three dimensional space [START_REF] Struik | Lectures on classical differential geometry[END_REF]. We thus may express its coordinates x = (x, y, z) in the 3D space as a function of two parameters (u, v) defined on a given interval.

The distance ds between two neighbouring points x and x + dx on the surface reads classically ds 2 = dx • dx. Expressing dx as a function of u and v leads to dx = x u du + x v dv, where the subscripts correspond to derivation with respect to the variable. We thus obtain the first fondamental form of a surface:

ds 2 = x u • x u du 2 + 2x u • x v dudv + x v • x v dv 2 (1.1)
This first fundamental form defines the metric tensor a :

a = E F F G (1.2)
where

E = x u • x u , F = x u • x v and G = x v • x v .
This matrix is clearly symmetric and positive definite (as ds 2 is always positive).

The second fundamental form defines the shape tensor, and gives information on how the surface is deployed in the 3D space.

In order to derive this form, we consider on the surface of interest a curve C passing through a point P . At this point, the unit tangent t to the curve C belongs to the tangent plane and its curvature dt/ds may be projected onto the normal N = x u ∧ x v / x u ∧ x v of the surface at the point P . We call this quantity the normal curvature κ n , in contrast with the component in the tangent plane, the geodesic curvature κ g which reflects how much the curve C is curved along the surface (how much you turn your wheel when driving a car along this curve on the surface). The normal curvature, which corresponds to the actual curvature of the surface along one direction, thus reads:

κ n = N • dt/ds (1.3)
As N and t are orthogonal (N • t = 0) we retrieve by differentiation:

κ n = -t • dN/ds = -dx/ds • dN/ds = - dx • dN ds 2 (1.4)
By differentiation, we trivially have:

dx = x u du + x v dv (1.5) dN = N u du + N v dv (1.6)
(1.7) Inputting these expressions in equation 1.4, we get

GEOMETRY IN PLATES

κ n = - x u N u du 2 + (x u N v + x v N u )dudv + x v N v dv 2 Edu 2 + 2F dudv + Gdv 2 (1.8)
The second fundamental form is simply the numerator of this expression:

II = edu 2 + 2f dudv + gdv 2 (1.9)
where e = x uu N, f = x uv N and g = x vv N. and may be written in a matrix form b, known as the shape tensor:

b = e f f g (1.10)
In order to get a geometric interpretation of this equation, let us consider a point Q near the point P and compute its distance D to the tangent plane at P , or more exactly the projection of the vector PQ on N (Fig. 1.8).

The coordinates x of the point Q may be obtained by a Taylor expansion around the point P of coordinates x 0 :

x = x 0 + dux u + dvx v + 1/2(du 2 x uu + 2dudvx uv + dv 2 x vv ) + o(du 2 , dv 2 ) (1.11)
The coordinate D along N thus reads:

D = (x -x 0 ) • N = 1/2(du 2 x uu • N + 2dudvx uv • N + dv 2 x vv • N) + o(du 2 , dv 2 ) (1.12)
The "distance" to the tangent plane is thus half the second fondamental form up to the second order (which is positive or negative depending on which side of the tangent plane Q lies). It can be easily shown (see lectures on Classical Differential Geometry from Dirk Struik for the details [START_REF] Struik | Lectures on classical differential geometry[END_REF]) that the curvature κ n is found to have two orthogonal directions in which it obtains an extreme value, one being a maximum the other a minimum (we do not consider here special cases called ombilic points, on which the curvature is constant, and hence the two fundamental forms are proportional). These directions are called the CHAPTER 1. ELASTICITY, GEOMETRY AND MORPHING directions of principal curvature. If one chooses the parametric lines along the principal curvature directions, the principal curvatures simply read:

κ 1 = e/E , κ 2 = g/G (1.13)
The product of both principal curvature K = κ 1 κ 2 is called the Gaussian curvature and it can be shown that it reads in terms of the coefficients of both fundamental forms:

K = eg -f 2 EG -F 2 (1.14)
i.e., the ratio of the determinants of shape and metric tensors.

Gauss Theorem

We will not go here through all the computational details of the demonstration of Gauss' Theorema Egregium. The key question behind this theorem is wether we can find any relation between the coefficients of the first and the second fundamental form. We saw above that a circle on a dome and on a flat surface do not have the same perimeter: it means that the metric of a surface is not completely decorrelated from its 3D deployment. It can be indeed shown that the Gaussian curvature may be expressed solely as the coefficients of the metric tensor and their first and second derivatives. The Gaussian curvature is thus fully determined by the metric: it is a bending invariant. This is the reason why bilayer structures destabilize and eventually curl in one direction, thus preserving the distances along the plate. In order to morph onto complex 3D shapes, it is thus crucial to change the distances in the flat plate to prescribe a metric as close as possible to the metric of the target shape, as we know that isometries are energetically favoured in slender structures. Conversely and somewhat surprisingly, simply prescribing distance changes in the plane may spontaneously induce the bending of the structure in multiple directions, if this new metric is incompatible with the flat state. Such metrics are called non-Euclidean.

It can be shown that the Gaussian curvature is the second order correction of small circles perimeter length on a doubly curved surface, in a generalization of the expression found on spheres [START_REF] Audoly | Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells[END_REF]:

P = 2πr(1 -6Kr 2 + o(r 2 )) (1.15)
Perimeters are thus relatively longer on a negative Gaussian curvature surface (saddle) and shorter on a positive Gaussian curvature surface (dome).

Non-Euclidean plate theory

Metric prescription is thus very important in order to program in general the shape of an initially flat plate. However, real plates have a finite bending stiffness. Moreover, the prescribed target metric tensor, that we call a may be discontinuous and may not have embeddings in the 3D space (i.e., continuous 3D shapes with the target metric a may not exist). This is typically the case for a growing patch in a passive membrane. At the 1.4. EXAMPLES FROM NATURE boundary between the two domains, elasticity plays a role in order to ensure continuity and kinematic compatibility of the solution [START_REF] Amar | Growth and instability in elastic tissues[END_REF]. In recent years, Dervaux and collaborators [START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF][START_REF] Dervaux | Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the föppl-von kármán limit[END_REF] and Efrati et al. [START_REF] Efrati | Elastic theory of unconstrained non-euclidean plates[END_REF][START_REF] Efrati | Buckling transition and boundary layer in non-euclidean plates[END_REF][START_REF] Efrati | The mechanics of non-euclidean plates[END_REF] developed the theory of non-Euclidean plates and shells, introducing new concepts and computational tools to predict the shape that a plate adopts depending on its target metric a and target curvature b. A target curvature may indeed also be programmed with a bilayer effect. Assuming the plate is linearly elastic and isotropic, the theory shows, similarly to the buckling instability introduced before, that the actual geometry is selected by minimizing the elastic energy U , sum of stretching and bending energies:

U (a, b) = E 1 -ν 2 [h(a -a) 2 + h 3 (b -b) 2 ]dS (1.16)
where E, ν are respectively the Young's modulus and the Poisson ratio of the material, h is the thickness of the sheet, a and b are the actual metric and curvature tensor of the plate that we are computing by minimization.

The target metric and curvature tensors are in general geometrically incompatible: no surface with the target metric can be formed in 3D space obeying the shape tensor. This is for example the case for a non-Euclidean target metric and a flat target curvature (b = 0). In such cases, the elastic energy does not vanish and the actual geometry emerges from a competition between bending and stretching. The correspondence between the target and actual geometries is very subtle and highly non-trivial: multi-scale solutions and refinement with thickness may indeed occur [START_REF] Klein | Experimental study of shape transitions and energy scaling in thin non-euclidean plates[END_REF]. When the thickness h of the plate is very small, the bending energy term is negligible with respect to the stretching one. The plate thus searches for geometries following the target metric (which is not always possible), and eventually selects the shape with the smallest bending energy among the possible isometries. We may also note that the selected shape does not depend on the stiffness of the material since bending and stretching energies both scale linearly with the Young's modulus E of the material. However, when taking into account gravity forces, the stiffness of the material is of course crucial to allow the slender structure to withstand its own weight, as we shall see later.

Examples from Nature

We saw that the opening and closing of the scales of pine cones are induced by a bilayer effect. However, in order to explore more complex shapes, we know from Gauss' Theorem that the metric must evolve in a non uniform or non isotropic fashion. Indeed, if every element in the plate grows to twice its original length, we simply have the same flat structure, but twice as large.

Plants [START_REF] Liang | Growth, geometry, and mechanics of a blooming lily[END_REF][START_REF] Liang | The shape of a long leaf[END_REF][START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF][START_REF] Armon | Geometry and mechanics in the opening of chiral seed pods[END_REF][START_REF] Rebocho | Formation and shaping of the antirrhinum flower through modulation of the cup boundary gene[END_REF][START_REF] Nath | Genetic control of surface curvature[END_REF][START_REF] Fratzl | Cellulose fibrils direct plant organ movements[END_REF] change their shape during morphogenesis by controlling precisely their local growth rate and direction. In plants, cell division stops at an early stage of the development, the number of cells being more or less constant during the CHAPTER 1. ELASTICITY, GEOMETRY AND MORPHING Figure 1.9: Shape changes in the vegetal world. (a) Blooming lily: petals change shape (and Gaussian curvature) during the blooming, starting with a almost zero (slightly positive) Gaussian curvature and finally ending with a clear negative Gaussian curvature [START_REF] Liang | Growth, geometry, and mechanics of a blooming lily[END_REF]. Photo credit: David de los Santos Gil. (b) Curly salad, with negative Gaussian curvature [START_REF] Liang | The shape of a long leaf[END_REF]. Source: http://mazwo.com (c) Acetabularia acetabulum: during the life of this surprisingly complex one-cell organism, the cap evolves from a dome shape (positive Gaussian curvature), to a flat (zero Gaussian curvature) and finally saddle shape (negative Gaussian curvature) [START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF]. (d) Opening of chiral seed pods [START_REF] Armon | Geometry and mechanics in the opening of chiral seed pods[END_REF].

1.4. EXAMPLES FROM NATURE (a) (b)
Figure 1.10: (a) When a plastic bag is torn, the sheet strongly and plastically deforms near the crack tip, yielding a complex buckling pattern in the wake of the fracture path. The length near the edge is indeed longer than far from the edge, this new metric being incompatible with the flat initial state. [START_REF] Roman | Buckling cascades in free sheets[END_REF][START_REF] Roman | Geometrically driven wrinkling observed in free plastic sheets and leaves[END_REF] (b) Crochet coral reef, by the artists Margaret and Christine Wertheim. Increasing the number of stitches in every row, the crochet spontaneously buckles out of plane in a similar fashion as the torn plastic bag or the curly salad [START_REF] Henderson | Crocheting the hyperbolic plane[END_REF]. More details on https://crochetcoralreef.org. large majority of its life [START_REF] Sahaf | The rheology of a growing leaf: stress-induced changes in the mechanical properties of leaves[END_REF]. The migration of cells is also hindered by the cell walls primarily made of cellulose fibrils. Plants cells are under high internal pressure that pushes the cell membrane towards the stiff cell walls. This internal pressure of typically 5 bar, called turgor pressure, is due to the osmotic flow through the semi-permeable membrane inside the cell, caused by a gradient of solute concentration between outside and inside the cell [START_REF] Dumais | Vegetable dynamicks: The role of water in plant movements[END_REF][START_REF] Forterre | Slow, fast and furious: understanding the physics of plant movements[END_REF]. The water transport and subsequent internal pressure lead to slow conformations in the stiff cell walls. These conformations are not isotropic but are determined by the orientational anisotropy of the highly organised cellulose fibrils embedded in the pectin matrix of the cell wall. This is the simplified growth process in the vegetal world. Inspired by this shaping strategy based on two key components -fluid under pressure and conforming cell walls-we will aim at mimicking these beautiful shape evolutions. Nature indeed overflows with examples of geometrically complex and evolving thin objects, such as petals, leaves or organs walls [START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF].

For instance, the deregulation of the growth rate in an initially flat leave leads to curly shapes with negative Gaussian curvature [START_REF] Nath | Genetic control of surface curvature[END_REF]. This is what we observe on curly salad, the outer edges growing faster than the center part of the leaves (see Fig. 1.9(b)). The increase of length near an edge leads to hyperbolic surfaces. Tearing a plastic bag induces an excess length near the edge and a beautiful buckling cascade may be observed in the wake of the crack [START_REF] Roman | Buckling cascades in free sheets[END_REF][START_REF] Roman | Geometrically driven wrinkling observed in free plastic sheets and leaves[END_REF] (Fig. 1.10(a)). Invented by the mathematician Daina Taimina, hyperbolic crochet is based on the same idea [START_REF] Henderson | Crocheting the hyperbolic plane[END_REF]. The crocheter increases the number of stitches at every row, leading to an excess length and thus a hyperbolic shape reminiscent of curly salads or coral reef. This concept was then used and deepened by the artists Margaret Similarly, the elegant shape of flowers petals is due to differential growth [START_REF] Rebocho | Formation and shaping of the antirrhinum flower through modulation of the cup boundary gene[END_REF]. During the blooming, lily petals change their shape from a one-directional curved surface (similar to a cylinder) to a doubly curved surface (similar to a saddle), as shown in Fig. 1.9(a). While the growth may be spatially homogeneous, the orientation of cellulose fibres, as explained before, induces anisotropic growth, which for instance leads to the hygroscopic actuation of wheat awns [START_REF] Fratzl | Cellulose fibrils direct plant organ movements[END_REF] or to the chiral shape of some seed pods [START_REF] Armon | Geometry and mechanics in the opening of chiral seed pods[END_REF] (see Fig. 1.9(d)). The algae Acetabularia acetabulum [START_REF] Serikawa | An analysis of morphogenesis of the reproductive whorl of acetabularia acetabulum[END_REF] is a very good and simple example of shape change induced by anisotropic growth (see Fig. 1.9(c)): after initiation, the cap of this unicellular algae evolves from a cone to a flat and eventually to a saddle shape: biological growth in the cap is stronger in the circumferential direction than along the radius. A cone is indeed a disc with a missing sector (see Fig. 1.11(a)-(b)), and growing azimuthally makes up for it, leading to a flat disc. Additional azimuthal growth yields an excess sector, as if one would add a slice in an already full pizza (see Fig. 1.11(c)-(d)). Perimeters are thus relatively larger than 2πr, which corresponds to negative Gaussian curvature (see Equation 1.15). The obtained shape is called an e-cone (excess angle cone) or anticone, with negative Gaussian curvature concentrated at the apex.

Synthetic shape morphing

Inspired by biological morphogenesis, pioneering experiments have been carried out during the last decade to make the first shape-morphing structures using several strategies to mimic the growth process. Klein et al. [13] proposed a first method in 2007 to achieve this non-Euclidean shape morphing. Varying the cross-link density of NIPAM hydrogels, one may control the isotropic swelling rate of the hydrogel when immersed in a bath of hot water. Playing with spatially non uniform swelling rate, shapes with positive and negative Gaussian curvature could be obtained (Fig. 1.12(a)). This technique was however constrained to axisymmetric reticulation pattern (which does not mean axisymmetric shape). This idea was then deepened by Kim et al. [START_REF] Kim | Designing responsive buckled surfaces by halftone gel lithography[END_REF] with a halftone technique obtained by lithography, in which highly cross-linked dots embedded in a lightly cross-linked matrix provide access to nearly continuous (coarse-grained) patterns of isotropic swelling. The method has been recently widely explored and enriched in numerous studies, taking advantage of spatially varying UV insulating times or intensities to directly prescribe the local swelling rate when immersed in water [START_REF] Huang | Ultrafast digital printing toward 4d shape changing materials[END_REF][START_REF] Naficy | 4d printing of reversible shape morphing hydrogel structures[END_REF][START_REF] Byun | Swelling-driven rolling and anisotropic expansion of striped gel sheets[END_REF][START_REF] Ge | Multimaterial 4d printing with tailorable shape memory polymers[END_REF]. Adding stiffer elements such as fibres in the hydrogel matrix may also induce anisotropic swelling of the plate and may be used to distort the metric [START_REF] Erb | Self-shaping composites with programmable bioinspired microstructures[END_REF][START_REF] Jeon | Shape-morphing materials from stimuli-responsive hydrogel hybrids[END_REF]. The orientation of small fibers may be induced by the flow during the printing process of the hydrogel plate [START_REF] Gladman | Biomimetic 4d printing[END_REF] (Fig. 1.12(c)).

Swelling hydrogels

Swelling gels presents however some inherent drawbacks. The objects are very soft and floppy, and may be actuated only in water. The process is also slow, based on the diffusion of water through the plate. In a new study, which is still under review but available on A structure with spikes is programmed upon heating [START_REF] Ware | Voxelated liquid crystal elastomers[END_REF]. (b) Inverse problem in order to program an arbitrary shape (here a face), finding the optimal director field leading to the target shape on heating. [START_REF] Aharoni | Geometry of thin nematic elastomer sheets[END_REF][START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF] arXiv, Levin et al. investigate a faster and autonomous shape transforming sheet made from a hydrogel that shrinks and swells in response to an oscillatory chemical reaction [START_REF] Levin | Self-powered shape-transforming membranes: an active matter approach to soft robotics[END_REF].

Liquid crystal elastomers Liquid crystal elastomers (LCEs) are slightly crosslinked liquid crystalline polymer networks. They thus combine the entropy elasticity of an elastomer with the self-organization of the liquid crystal phase [START_REF] Warner | Liquid crystal elastomers[END_REF].

At room temperature, the orientation of the crystal forces the polymer chains into a stretched conformation. Heating the material above the clearing temperature, the orientation is lost and the polymer backbone can relax, yielding a macroscopic anisotropic deformation. Controlling the local orientation of the director field thus prescribes the direction of the anisotropic contraction or elongation when heated, distorting the metrics in a potentially non-Euclidean way [START_REF] Warner | Topographic mechanics and applications of liquid crystalline solids[END_REF][START_REF] Modes | Gaussian curvature from flat elastica sheets[END_REF][START_REF] Mcconney | Topography from topology: Photoinduced surface features generated in liquid crystal polymer networks[END_REF]. The spectrum of possible shapes and the improvement of manufacturing processes have been highly investigated in multiple studies [START_REF] Aharoni | Geometry of thin nematic elastomer sheets[END_REF][START_REF] Kotikian | 3d printing of liquid crystal elastomeric actuators with spatially programed nematic order[END_REF][START_REF] Ambulo | Four-dimensional printing of liquid crystal elastomers[END_REF][START_REF] Plucinsky | Programming complex shapes in thin nematic elastomer and glass sheets[END_REF]. Numerous studies are centred on the effect of defects (called disclinations) in the nematic director field on the shape of the surface (Fig. 1.13(b) [START_REF] Ware | Voxelated liquid crystal elastomers[END_REF][START_REF] Modes | Disclination-mediated thermooptical response in nematic glass sheets[END_REF][START_REF] Konya | Modeling defects, shape evolution, and programmed auto-origami in liquid crystal elastomers[END_REF]. The inverse problem for anisotropic deformations of constant amplitude has been recently analytically solved for axisymmetric configurations [START_REF] Mostajeran | Encoding gaussian curvature in glassy and elastomeric liquid crystal solids[END_REF][START_REF] Warner | Nematic director fields and topographies of solid shells of revolution[END_REF]. More recently, an optimization based method has been developed by Aharoni et al. to find the director that leads to an arbitrary shape [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF] (Fig. 1.13(b)).

Dielectric elastomers

Dielectric elastomers are another strategy to change in-plane distances. In order to do so, opposite sides of a sheet of elastomer are coated with compliant electrodes (typically carbone powder). Applying a high voltage V to this soft capacitor tends to squeeze the membrane across its thickness, positive and negative charges being attracted. By volume conservation of such materials, the sheet thus stretches isotropically his surface [START_REF] Pelrine | High-speed electrically actuated elastomers with strain greater than 100%[END_REF][START_REF] Keplinger | Röntgen?s electrode-free elastomer actuators without electromechanical pull-in instability[END_REF]. In contrast with piezoelectric ceramics, dielectric elastomers can undergo very high strains of up to 500% [START_REF] Huang | Giant, voltage-actuated deformation of a dielectric elastomer under dead load[END_REF]. When immersed in a passive and common elastomer sheet, a voltage-induced growing dielectric patch buckles out of plane leading to secondary buckling patterns reminiscent of inflated mylar ballons [START_REF] Bense | Buckling of elastomer sheets under non-uniform electro-actuation[END_REF] (Fig. 1.14(a)). During his thesis in the laboratory, Hadrien Bense investigated the possibility of controlling the local in-plane "growth" by varying the size and orientation of textures on a surface [START_REF] Bense | Croissance, compaction et adhésion de plaques minces[END_REF]. The local control of isotropic in-plane deformation has also been recently completed with spatially varying electric fields by Hajiesmaili et al. [START_REF] Hajiesmaili | Reconfigurable shape-morphing dielectric elastomers using spatially varying electric fields[END_REF], leading to the shaping of the multilayered sheets into saddles and domes (Fig. 1.14(b)). Cones and anticones were also very efficiently programmed inserting stiff fibers in the elastomer to induce anisotropic deformations [START_REF] Hajiesmaili | Voltagecontrolled morphing of dielectric elastomer circular sheets into conical surfaces[END_REF] (Fig. 1.14(c)).

All strategies introduced so far present the advantage to rely on a simple uniform input, respectively, humidity, temperature, UV-light or electric field. However, their fabrication relies on relatively complex processes: control of reticulation rate in hydrogels, precise control of the orientation of the nematic director field in LCE, multilayered structures for dielectric elastomers. These objects are moreover inherently soft (typical Young's modulus under 1MPa), hindering large scale fabrication and thus architectural and spatial application. At the meter scale such slender soft structures indeed collapse under their weight.

Bending-based shape shifting structures. All shape-shifting manufactured structures do not rely on the non uniform in-plane growth. Even though we saw that the spectrum of reachable shapes is highly reduced, combining one directional bending or folding with complex pattern enables very interesting transformations. Another way to play with the apparent metric of a sheet without directly significantly stretching the material of the sheet [START_REF] Callens | From flat sheets to curved geometries: Origami and kirigami approaches[END_REF] is indeed to make cuts (kirigami) [START_REF] Dias | Kirigami actuators[END_REF][START_REF] Castle | Making the cut: Lattice kirigami rules[END_REF][START_REF] Sussman | Algorithmic lattice kirigami: A route to pluripotent materials[END_REF] or folds (origami) [START_REF] Dudte | Programming curvature using origami tessellations[END_REF][START_REF] Zhao | Approximating 3d surfaces using generalized waterbomb tessellations[END_REF].

Using this trick, the local metric of the structure is still flat since no significant stretch Figure 1.15: Shape shifting strategies based on bending, folding and cutting. (a) Origami tessellations programmed to shape onto shapes with apparent Gaussian curvature [START_REF] Dudte | Programming curvature using origami tessellations[END_REF]. (b) Kirigami strategy to make 3D surfaces [START_REF] Sussman | Algorithmic lattice kirigami: A route to pluripotent materials[END_REF]. Origami at the microscale: (c) micropatterned photo-crosslinkable polymer trilayers [START_REF] Na | Programming reversibly self-folding origami with micropatterned photo-crosslinkable polymer trilayers[END_REF] and (d) graphene-based bimorphs [START_REF] Miskin | Graphene-based bimorphs for micron-sized, autonomous origami machines[END_REF]. (c) Self-folding ferromagnetic soft robots responding to a magnetic field [START_REF] Kim | Printing ferromagnetic domains for untethered fast-transforming soft materials[END_REF] 1.6. INFLATION appears within the material. However, looking at the scale of the objects, distances between elements undergo significant changes due to the bending and folding of the structure. The smoothed overall shape may thus be doubly curved, as shown in Fig. 1.15(a) and (b). Much stronger materials may be used (Young modulus of the order of 1GPa) with these techniques. However, the actuation of such structures is tedious: folding complex origami patterns may take days... Recent advances have been made in the actuation of large scale kirigami structures [START_REF] Konaković-Luković | Rapid deployment of curved surfaces via programmable auxetics[END_REF] deployed by a pneumatic balloon. Small scale origami ( [START_REF] Na | Programming reversibly self-folding origami with micropatterned photo-crosslinkable polymer trilayers[END_REF][START_REF] Miskin | Graphene-based bimorphs for micron-sized, autonomous origami machines[END_REF]) may also spontaneously fold using bi-or trilayers techniques but this actuation method remains complex and prone to errors, since the sequence of folding steps is not controlled (Fig. 1.15(c) and (d)). Recently, structures made of ferromagnetic material selffold in response to a magnetic field, aligning the magnetic polarity of each domain with the magnetic field [START_REF] Kim | Printing ferromagnetic domains for untethered fast-transforming soft materials[END_REF][START_REF] Hu | Small-scale soft-bodied robot with multimodal locomotion[END_REF] (see Fig. 1.15(e)).

In this thesis, we present two new approaches to make shape-shifting structures based on pneumatic inflation. Let us thus review some typical examples of inflated structures and point out several asymptotic regimes.

Inflation

Inflated structures are widely used in applications because they present several advantages such as high stiffness to weight ratio, efficient storage, quick deployment and cost efficiency. Modern lightweight tents, in which rigid poles are replaced by air beams, scientific ballooning [START_REF] Pagitz | The future of scientific ballooning[END_REF], stent deployment in angioplasty [START_REF] Serruys | A comparison of balloon-expandable-stent implantation with balloon angioplasty in patients with BIBLIOGRAPHY coronary artery disease[END_REF], ultralight deployable structures [START_REF] Pellegrino | Deployable structures[END_REF] for space exploration and soft robotics are, among others, fields in which inflatables are believed to have a strong impact. Inflated structures may be separated into two regimes depending on the magnitude of membrane strains induced by the pressure. In a first regime, when p ∼ Et/L, (where p is the pressure difference between inside and outside the structure, E the Young's modulus of the envelope material, t the typical envelope thickness and L the smallest typical size of a structure unit) the envelope is strongly stretched. This may only occur with elastomers, that can accommodate finite stretching without failure. Common examples from everyday life are rubber balloons or inner tubes (Fig. 1.16(a)-(b)). This regime has also been recently investigated in numerous studies, in soft robotics (Fig. 1.16(c)), in the propagation of bulges [START_REF] Chater | On the propagation of bulges and buckles[END_REF][START_REF] Kyriakides | The initiation and propagation of a localized instability in an inflated elastic tube[END_REF], or in the shape and stability of toroidal membranes [START_REF] Roychowdhury | Inflating a flat toroidal membrane[END_REF][START_REF] Tamadapu | Finite inflation analysis of a hyperelastic toroidal membrane of initially circular cross-section[END_REF][START_REF] Roychowdhury | Symmetry breaking during inflation of a toroidal membrane[END_REF]. We note that these elastomer structures remain however intrinsically soft and, if they are slender, cannot sustain their own weight at large scale. Balancing the gravity induced moment and the elastic bending moment, we get a typical elasto-gravitational length L g ∼ (Eh 2 /ρg) 1/3 above which a structure of total thickness h collapses under its own weight. For typical rubber Young's modulus E ∼ 10 MPa and h ∼ 10 mm we get L g ∼ 0.5 m.

Large inflated structures thus belong to a second regime p Et/L, and can be separated in two other subcases: when p Et 3 /L 3 , the pressure forces may not bend strongly the envelope, and the displacements remain small: this is the case for pressurized engineer-CHAPTER 1. ELASTICITY, GEOMETRY AND MORPHING p Et/L: in this doubly asymptotic regime, the sheet may be considered as quasi-inextensible but with vanishing bending stiffness. Mylar balloons (e) and numbers (f), scientific ballooning (g), or large inflatable architectures (h) (source: buildair.com) are typical examples from day life. In Chapter 6, we will investigate a shaping strategy in this regime (i).

INFLATION

ing structures (tank and shells) [START_REF] Kiefner | Failure stress levels of flaws in pressurized cylinders[END_REF], which must be designed to sustain pressure without failing. In our quest for shape transforming materials, this regime appears of course to be irrelevant.

The other regime is Et 3 /L 3 p Et/L, where the envelope is quasi-inextensible but highly bendable. In this regime, the envelope can accommodate compression at almost no cost by forming wrinkles [START_REF] King | Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities[END_REF][START_REF] Paulsen | Wrapping liquids, solids, and gases in thin sheets[END_REF]. Popular examples are the mylar balloons at funfairs or birthday parties (Fig. 1.16(e)-(f)). Architectural sized inflatable structures, as shown for example in (Fig. 1.16(h), also belong to this doubly asymptotic regime. Theoretical work has been dedicated to the shape of inflated circular disks, starting with the shape of parachutes (Fig. 1.16(g)) rationalized by Taylor [START_REF] Taylor | On the shapes of parachutes[END_REF][START_REF] Paulsen | What is the shape of a mylar balloon? The American mathematical monthly[END_REF][START_REF] Ligaro | Equilibrium shapes of inflated inextensible membranes[END_REF], polyhedral surfaces [START_REF] Pak | Inflating polyhedral surfaces[END_REF], instabilities in scientific balloons [START_REF] Deng | Clefted equilibrium shapes of superpressure balloon structures[END_REF][START_REF] Pagitz | Buckling pressure of ?pumpkin? balloons[END_REF]. Recent progress has been done in solving numerically the inverse problem, that is programming the envelope, as a collection of flat panels seamed together, in order to morph onto a target shape upon inflation [START_REF] Skouras | Designing inflatable structures[END_REF].

During this PhD, we investigated two different strategies to make shape-morphing inflatables, corresponding to the two interesting regimes described above. In part I, we will take advantage of the anisotropic stretching of elongated rubber balloons to program anisotropic distortions of a rubber plate embedding elongated airways (see Fig. 1.16(d)). In part II, we observe that flat structures made of quasi-inextensible but highly bendable sheets go out of plane in order to maximize their inner volume (by bending), inducing an anisotropic net in-plane contraction, that we harness to program coarse-grained non-Euclidean metric. (see Fig. 1.16(i)). These two strategies are of very different natures, since the same pattern, namely concentric circular channels, codes for very different shapes on inflation, as one can see in Fig. 1 One of the simplest and most common geometries we may think of when inflating are the popular cylindrical balloons at funfairs, with which clowns make dog sculptures. In this short chapter, we revisit this very classical problem [START_REF] Kyriakides | Mechanics of offshore pipelines: volume 1 buckling and collapse[END_REF][START_REF] Kyriakides | The initiation and propagation of a localized instability in an inflated elastic tube[END_REF] first in the linear regime, at small deformation, to show that the deformation of the structure is highly anisotropic. We will employ this anisotropy in 2D plates to distort the metric in a controllable manner in the next chapter (Chapter 3). We will then extend our analysis to large strains, and thus non-linear response, where a surprising "phase separation" occurs. In Chapter 4, we will extend these results to 2D plates, where the phase change impacts the metric of the plate and thus its geometry, in contrast with quasi-one dimensional cylindrical balloons.

At small deformations

Let us consider a long cylindrical shell of inner radius R and thickness e, its length L being much greater than the radius. The shell is made of an elastomer, that we will first model as a linear Hookean material. A pressure p is applied inside the shell (see Fig. Balancing the forces and using Hooke's law, the strains may be computed as a function of the dimensionless pressure pR/(Ee). Interestingly, no deformation occurs along the tube direction e x .

AT SMALL DEFORMATIONS

Balancing the force perpendicular to the membrane leads us to the famous Laplace's law

σ θθ = pR/e (2.1)
Considering an imaginary cut in the e r , e θ plane, the pressure applies a force pπR 2 that is balanced by the stress induced force in the membrane σ xx 2πRe (if the balloon is free of external load). The stress in the x direction thus reads:

σ xx = pR/2e = σ θθ /2 (2.2)
Applying Hooke's law and assuming that σ rr = 0 one gets the strains:

ε rr = - ν E (σ θθ + σ xx ) ε θθ = 1 E (σ θθ -νσ xx ) ε xx = 1 E (σ xx -νσ θθ ) (2.3)
where E is the Young modulus of the material and ν the Poisson ratio. Inputting the actual values of the stresses and knowing that, for elastomers, the Poisson ratio is equal to 1/2, we get:

ε rr = - 3νpR 2Ee ε θθ = (1 -ν/2)pR Ee ε xx = (1 -2ν)pR 2Ee = 0 (2.4)
Remarkably (and somewhat surprisingly!), the strain along the tube direction e x is zero in this small strain elasticity limit. A trivial example from daily life is the pumping of the inner tube of a bicycle: the section increases a lot whereas the perimeter remains very close to its initial value, that is of the size of the wheel. Arteries are another (way more crucial) example [START_REF] Gordon | Structures: or why things don't fall down[END_REF]: when the heart pumps blood towards the extremities of the body, an overpressure pulse propagates (this is what one senses when manually measuring the heart rate after a jog) and thus deforms the artery wall. Luckily, this deformation is purely in the section of artery and not along its length: an overlength may indeed induce the buckling and more catastrophically the rupture of the artery. Elongated tubes thus deform anisotropically and we will benefit from this very phenomenon to program shape changes in the next chapter. However, at finite deformation, although the anisotropy remains, the predicted zero strain along the tube is clearly not valid anymore. Fully inflate an elongated balloon or a bicycle inner tube, and you will observe that the length of the structure eventually strongly increases. We also observe the coexistence of two "phases" [START_REF] Chater | On the propagation of bulges and buckles[END_REF][START_REF] Kyriakides | The initiation and propagation of a localized instability in an inflated elastic tube[END_REF] (see Fig. 2.2(a)). When one starts to inflate a balloon, it is at first quite difficult, the pressure needed is high, until a bulge (that we shall call second phase) appears, associated with a drop in pressure and CHAPTER 2. INFLATION OF A CYLINDRICAL RUBBER BALLOON a facilitated inflation. The bulge then expands at constant pressure when air is added in the system.

At finite deformation

In order to understand this phenomenon, we need to shift away from Hooke and turn to Gent and its more recent work on nonlinear elasticity [START_REF] Gent | A new constitutive relation for rubber[END_REF]. Gent model is a phenomenological model that takes into account the material non-linearities (strain stiffening), based on the concept of limiting molecular chain extensibility. In this model, the strain energy density function is designed such that it has a singularity when the first invariant

(J 1 = λ 2 ii -3) of the left Cauchy-Green deformation tensor B (B ij = ∂x i ∂X K ∂x j
∂X K ) reaches a limiting value J m . Physically, it means that when the molecular chains are fully stretched, there is no possibility for further stretch: in practice, rupture arises before this theoretical limit. We have the following constitutive law:

σ = -p r I + µJ m J m -J 1 B (2.5)
where p r is a bulk pressure to be determined, I is the identity matrix, µ = E/2(1 + ν) is the shear modulus of the material. Incompressibility imposes λ xx λ rr λ θθ = 1. The stresses computed above via force balance are still correct if we infer the actual dimensions in the deformed state:

     σ xx = pRλ θθ /(2eλ rr ) σ rr = 0 σ θθ = pRλ θθ /(eλ rr ) (2.6)
The actual radius R in the deformed state indeed reads Rλ θθ whereas the thickness e is eλ rr . Inputting these values in the constitutive law 2.5, we obtain this system of three equations:

             pRλ θθ /(2eλ rr ) = -p r + µJ m J m -J 1 λ 2 xx 0 = -p r + µJ m J m -J 1 λ 2 rr pRλ θθ /(eλ rr ) = -p r + µJ m J m -J 1 λ 2 θθ (2.7)
After eliminating p r using the first equation in the last two others and λ xx using incompressibility, the system simplifies to two equations

     pRλ θθ /(2eµλ rr ) = J m J m -J 1 ((λ θθ λ rr ) -2 -λ 2 rr ) pRλ θθ /(eµλ rr ) = J m J m -J 1 (λ 2 θθ -λ 2 rr ) (2.8)
where J 1 = λ 2 θθ + λ 2 rr + (λ θθ λ rr ) -2 -3, λ θθ and λ rr are the two unknowns and pR/(eµ) and J m are two parameters, respectively the dimensionless pressure applied inside the shell and the maximum value for the first invariant. This non linear system of two equations Deformations as a function of the non dimensional pressure pR/(µe) using a Gent model for the elastomer for various values of the first invariant limit J m (respectively 10, 20 and 30 from lighter to brighter colors). For long enough polymer chains, we obtain a S-shape for the strain-pressure curve. It implies that for one given pressure, two very different states are possible, as observed in the experiments.

.

CHAPTER 2. INFLATION OF A CYLINDRICAL RUBBER BALLOON

is solved iteratively on Matlab using the function fsolve and giving the last computed solution as a starting point for the solver. The strain-pressure curves are presented in Fig. 2.2(b) for three different values of the limiting first invariant value J m [START_REF] Gorissen | Elastic inflatable actuators for soft robotic applications[END_REF][START_REF] Hajiesmaili | Reconfigurable shape-morphing dielectric elastomers using spatially varying electric fields[END_REF][START_REF] Modes | Disclination-mediated thermooptical response in nematic glass sheets[END_REF] from light to bright colors. For long enough chains, i.e. for high enough J m , we obtain an S-shape for the strain-pressure curve. It implies that for one given pressure, two very different states (one barely and one highly stretched) are possible, as observed in the experiments. In fact, there is a third solution, which appears to be unstable. In order to get this instability, called superelasticity [START_REF] Jirasek | Inelastic analysis of structures[END_REF][START_REF] Latorre | Active superelasticity in three-dimensional epithelia of controlled shape[END_REF], we first need a strain softening effect. In our case, it comes from the geometry: as the balloon is inflated, the radius increases whereas the thickness decreases: the stresses increase nonlinearly with the pressure, up to a point where, if the elastomer had no stiffening behaviour at large strain, the bulge would easily deform further in an unbounded fashion and the balloon would suddenly explode. That is why superelasticity requires, besides strain softening, re-stiffening at large strains to confine the high-strain phase. In our case, the limitation comes from the material properties and the strain stiffening of the elastomers when the stretch reaches the chain length limit.

Conclusion

We have seen in this short chapter that when an elongated cylindrical rubber balloon is slightly inflated, its section increases linearly with pressure whereas its length remains unchanged. Upon inflation, a rectangle made of an array of parallel cylindrical balloons would thus expand perpendicular to the cylinder direction and would conserve its length along it (It would also expand in the third direction perpendicular to the plate, but we are mostly interested in in-plane expansions that distort the metric.). We will use this anisotropy in the next chapter (Chapter 3) to change the in-plane distances in a controllable fashion and thus program shape transformations.

When the same balloon is highly inflated, we showed that a highly stretched bulge suddenly appears, whereas the rest of the balloon is barely stretched. Two "phases" thus coexists and the bulge expands, when air volume is added inside the balloon, at constant pressure. In chapter 4, we will make superelastic plates, extending the "two-phases" instability to 2D structures. The coexistence of these two phases will distort the metric, making the plate buckle in a surprising fashion. In this chapter, we present a shape-morphing strategy where mesostructured elastomer plates undergo fast, controllable and complex shape transformations under applied pressure. Similar to pioneering techniques based on soft hydrogel swelling, these pneumatic shape-morphing elastomers, termed as 'baromorphs', overcome geometric restrictions by controlling precisely the local growth rate and direction through a specific network of airways embedded inside the rubber plate. We show how arbitrary three-dimensional shapes can be programmed using an analytic theoretical model, propose a direct geometric solution to the inverse problem, and illustrate the versatility of the technique with a collection of configurations.

I thank Cyprien Blanquart, a one-month intern in the lab, for developing the 3D scanning techniques based on the work (and with the support) of Pablo Cobelli. The 3D mould have been printed both at the technical platform from IPGG with the help of Guillaume Laffite and at the Institute Jean le Rond d'Alembert thanks to Mathias Lebihain and Laurent Ponson.

Introduction and idea

We have seen in the introduction that shape morphing into doubly curved surfaces requires a distortion of the metric, i.e. a change in the distances along the surface. Here we present a new approach in order to create such metric distortions using an architected elastomer. We know that the inflation of an elongated tube leads to an anisotropic deformation: the section increases whereas the length barely elongates. Similarly to liquid crystal elastomers (LCE), we can benefit from this anisotropy to change the distances in the elastomeric plate in a nonuniform way.

Our approach bridges two emerging fields, bio-inspired shape morphing (as discussed in section 1.5) and pneumatic soft robotics (section 1.6), with a new easy-to-build, easyto-control object, referred to as a baromorph. Baromorphs consist in elastomer plates embedding a network of airways, which give rise to a programmed family of shapes upon air inflation or suction. The unusual shape changes and buckling instabilities that we will study in this chapter do not come from any particular material property such as LCE, since any rubber may be used. The surprising mechanics of these baromorph plates solely come from the inner airways architecture and geometry, that "code" for the shape shifting upon inflation. In that sense, our structures may be viewed as pneumatic metamaterials [START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF].

When the inner pressure is increased (or decreased), the elongated channels tend to inflate (or deflate) anisotropically [START_REF] Timoshenko | Theory of plates and shells[END_REF]. Below, we show that the length of an inflated channel remains almost unchanged, while its width increases. This anisotropy results in a controllable modification of the effective rest-lengths of the coarse-grained baromorph, mimicking the anisotropic growth of a biological tissue, with a magnitude that depends on the geometry of the inner channels and on the applied pressure. The plate deforms

TUTORIAL -MANUFACTURING BAROMORPHS

according to the new target metric imposed by the network of airways, and may buckle out of plane to reach an equilibrium 3D shape that minimizes the total elastic energy -sum of stretching and bending energies. In Fig. 3.1(c), we illustrate the deformation of a baromorph plate with radial channels obtained by casting the 3D printed template shown in Fig. 3.1(b): the target expansion is mainly circumferential. Upon suction and consequent azimuthal contraction, the plate adopts a bowl shape (with positive Gaussian curvature). Conversely, inflation induces an azimuthal expansion and leads to an excess angle in the plate, which destabilises into a surface of negative Gaussian curvature.

These transformations mimic the morphing evolution of Acetabularia as it grows [START_REF] Serikawa | An analysis of morphogenesis of the reproductive whorl of acetabularia acetabulum[END_REF][START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF]. After initiation, the cap of this unicellular alga evolves from a bowl to a flat and eventually to a saddle shape, essentially for the same reason as our baromorph: biological growth in the cap is stronger in the circumferential direction than along the radius (Fig. 3.1(d)).

Tutorial -Manufacturing baromorphs

This section is an independent tutorial whose purpose is to present in details the manufacturing process of baromorphs structures from a very practical point of view, ranging from the mould design to the actuation of the object. Even if the protocol is quite simple and straightforward, there are some subtleties that I painfully encountered during my PhD and that I will detail here to save time for the curious reader who may want to build baromorphs. Baromorph plates are made of silicone rubber (or any rubber) by mixing "catalyst" and "base" liquids in proper proportion. The mixture is then poured into a 3D printed mould (see Fig. 3.2(a)) and cures. At the same time, a sheet of controlled thickness of the same elastomer is spread on a flat surface and cured. The structure removed from the mould is finally closed by "gluing" the flat sheet on top of the moulded sheet using a thin layer of uncured mixture of the same material. After sealing, a pattern of interconnected channels remains embedded in the sheet. These channels are connected to an external air pressure controller (e.g. a syringe).

Design of the mould

Each channel can be locally described by its height h, its width d, a wall thickness d w and a top and bottom membrane thickness e, as presented in Fig. 3.2(b) In order to design an efficient mould, a few conception rules have to be followed:

• One wants to keep the cross sections size of the channels more or less constant everywhere in the plate, that is h and d remain more or less constant. If one channel is for example significantly wider that the others ones, it will form one bulge at a pressure which is too low for the others to deform, and a local protuberance will appear on a flat sheet. • It is important to have the channels at least twice higher than wide (h > 2d).

We indeed seek in-plane expansion upon inflation, and a high aspect ratio, that is, channels cross sections looking like a vertical slit, help orienting the deformation in plane.

• The thickness of the top and bottom membranes should be more or less equal to the channels width (e ∼ d) in order to avoid significant local bending of the membrane upon pressurization.

• The top and bottom membrane should have the same thickness e. Small differences will induce a bilayer effect, that might be interesting for several purposes, and can be related to common soft robotics techniques (PneuNets [START_REF] Shepherd | Multigait soft robot[END_REF][START_REF] Rus | Design, fabrication and control of soft robots[END_REF][START_REF] Yamaguchi | A robot hand using electro-conjugate fluid[END_REF][START_REF] Gorissen | Elastic inflatable actuators for soft robotic applications[END_REF][START_REF] Mosadegh | Pneumatic networks for soft robotics that actuate rapidly[END_REF][START_REF] Cianchetti | Biomedical applications of soft robotics[END_REF]), where one membrane is made of a strong strain-limiting material. However, strong bilayer effects hinder in-plane metrics changes and thus doubly curved surfaces.

The mould may be designed using any 3D solid CAD modeller. I personally use OpenScad, a free software compatible with Linux, Windows and OS. The modeling is generated by a code, which is very convenient for repetitive patterns and parametrisation. An online documentation may be found at the following URL: https : //en.wikibooks.org/wiki/OpenSCAD_U ser_M anual. An example of mould for programming a cone is provided in Appendix A.

The mould is thus made of the bottom plate patterned with the channels network in positive of height h. A rim of height h + e defines the contour of the molded structure (Fig. 3.2).

Printing of the mould

The mould may be made using either 3D printers or CNC machining (or even microlithography techniques for small scale structures). It is important to ensure that the precision of the machine/printer is sufficient. In some cases, depending on the printer, the resin and the elastomer, post-curing or careful washing of the mould are necessary in 3.2. TUTORIAL -MANUFACTURING BAROMORPHS order to ensure proper curing of the elastomer. We used mostly the 3D printer Form2 by FormLabs and insulated the printed mould with UV light for post curing.

Materials

The baromorph technology relies on the architecture of the airways network inside the plate and not on the material itself. However, three criteria have to be fulfilled to ensure proper shape morphing:

• the material should allow strains of typically 100% without plasticity nor rupture.

• the material should be able to be strongly bonded with itself.

• the material may be obtained by curing, that is, by mixing initially liquid phases poured into the mould.

During this thesis, we have successfully made baromorphs out of silicone elastomers (Ecoflex 0050, Dragonskin 10, Dragonskin 20 from Smooth On, Elite Double 8, 16 and 22 from Zhermack) and polyurethane elastomers (Vytaflex 20 from Smooth-On), this list being far from exhaustive. In the following, we detail the protocole followed for making baromorphs out of Ecoflex 0050.

Manufacturing protocol

Safety Read carefully the Safety Data Sheet (SDS) and technical bulletin provided with the material before using.

Mixing In a container, weigh equal quantities of part A and part B and mix thoroughly during approximately 1 minute, making sure that you scrape your container several times. It is important to keep time, as the pot life of Ecoflex 0050 is 18 minutes at room temperature.

Degassing Vacuum degassing is strongly recommended to remove entrapped air bubbles in the viscous mix. When degassing, the volume of the bubbles (and thus the volume of the mixture) increases strongly, before "boiling" and collapse of the volume. Make sure the container volume is at least twice larger than the mixture volume. Depending on your pump, 2 to 7 minutes of degassing should be sufficient to eliminate the large majority of trapped bubbles.

Molding The degassed mixture is then poured into the clean mould (Fig. 3.3(a)).

In order to minimize entrapped air, pour it at a constant flow far from intersections of channels. Let the viscous liquid slowly penetrate the textured mould. If a few large bubbles are trapped, a needle may be used to bring them to the surface and pop them. If too much air is entrapped, the mould may be placed in the vacuum chamber again. A flat plate (typically made of PVC) is then carefully placed on top of the mould to ensure the In parallel, a thin layer of the same elastomer, newly mixed, is spread on the flat membrane and will serve as a glue. (e) The moulded part is placed on the flat membrane and gently pressed in order to make sure that there is contact everywhere. (f) Once the thin layer is cured, the exceeding area of the membrane may be cut.

CHAPTER 3. BAROMORPHS (a) (b) (c) (d) (e) (f)

TUTORIAL -MANUFACTURING BAROMORPHS

proper membrane thickness e above the channels, with weights on top of it (Fig. 3.

3(b)).

No air should be entrapped at that moment. The flat membrane of thickness e, that will be assembled next with the molded part, is made by pouring the rest of the mixture on a flat surface and spread with a scraper leaving a controlled thickness of the fluid (Fig. 3.3(a)-(b)). The liquid film thickness does not match the height between the spreader and the flat surface due to flow conservation, so it is important to first make some calibration tests to ensure that the correct thickness e is spread on the surface.

Removal and assembling

Once the mixture is cured (typically after 3 hours at room temperature for Ecoflex 0050), the structure may be carefully removed from the mould. Mix a small amount of the same material and spread a thin film (of typically 100µm) onto the cured flat membrane (Fig. 3.3(d)): it will be the glue for closing the airways.

Place the moulded part on the membrane, making sure that there is contact everywhere by gently pressing on the structure with the fingertips (Fig. 3.3(e)). After 3 more hours, the total object is finally cured (Fig. 3.3(f)). We recommend to put the whole structure during 1 hour at 100 • C for attaining maximum physical performance. Let the structure cool to room temperature before using.

Inflating the baromorph Connect the airways to an air supplier. We penetrate the closed structure with a needle that we connect to an inflating bulb or a syringe. At this stage, any pressure activation process and measurement may be developed. The typical maximum pressure that the structure can sustain scales as 2Ee/h, where E is the Young's modulus of the material.

Possible failures and mitigation strategy

Cure inhibition Due to the channels, the moulded part presents a high surface to volume ratio. The mould must thus be cleaned carefully and post-cured to ensure proper curing of the moulded elastomer. Uncured material remaining from the printing process may indeed hinder the rubber curing.

Delamination upon inflation

The most common failure is the partial delamination of the structure upon inflation. The main reasons for that are:

• the glue layer is too thin

• the membrane or the moulded part were not clean before the assembling

Unpredicted shape upon inflation

The main reason for unpredicted shape change upon inflation is either non-homogeneous thickness of the membranes or too large differences between the top and bottom membranes. The local deformation is indeed strongly linked to the local thicknesses of the membranes, as shown analytically in section 3.4. The deformation is tracked using DIC program CorreliQ4 on Matlab [START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF]. The outer diameter of the ring is 8 cm.

Conclusion

This tutorial is far from being exhaustive and details one simple fabrication strategy. Many others strategies may be considered, such as direct 3D printing of the structure, which could be very promising in order to make shell baromorphs, i.e. non flat structures at rest, or to control more precisely the shape of the airways: our strategy limits ourself to rectangular cross sections. At smaller scale, soft lithography is the natural candidate in order to make precise moulds.

Experimental target strain measurement

In order to understand how the plate will shape in 3D, the first step is to understand the local deformation upon pressurization. We thus need an object that will freely deform without geometrical constraint, or said differently, a Euclidean metrics distortion. One solution could be parallel stripes in a rectangular plate, where the side perpendicular to the channels will "grow" whereas the distances along the channels should remain more or less constant according to our computations on inflated tubes. However, this solution presents the drawback to have strong boundary conditions that may affect the deformation. The channels have indeed to be sealed at both ends with an elastomeric wall, adding thus an extra stiffness to the structure perpendicular to the channels. We thus propose to study the deformation in a slightly different configuration, free of boundary: rings of three channels, as shown in Fig. 3 In order to experimentally measure the deformation on the ring, we use a digital image correlation (DIC) technique, developed by Hild and Roux [START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF], with the program CorreliQ4 on Matlab. A speckled pattern is generated on the surface of the ring by spraying paint. Top view pictures of the structure are taken at different pressures (as shown in Figure 3.4(a)-(b)) and the programm tracks in-plane displacements with respect to the reference image before inflation.

We then track the center of symmetry (see Fig. 3.5(a)) of the structure. We know from the symmetry of the object that the displacement will be purely radial. We call it u(r) and compute it in a cylindrical coordinates system for all values of θ and r. In Fig. 3.5(b), we show the radial displacement u as a function of the radial distance r, both expressed in pixel. We know that the radial strain (or said differently the strain perpendicular to the channels) reads ε ⊥ = du/dr, which corresponds to the local slope of the curves plotted in Fig. 3.5(b). We observe small fluctuations in the slope of the curve, which looks like a stairway. These alternating slope changes correspond to the location of channels (large slope) and walls (smaller slope) below the top membrane. However, here, we are not interested in these small details, as we seek to describe the plate as a homogenized structure that expands in average by a specific amount in some particular directions. We thus extract the mean slope by fitting the curves in Fig. 3.5(b) (black solid lines) and repeat the operation for each angle (Fig. 3.5(c)). We then plot the average strain over all angles as a function of the applied pressure, the errorbar corresponding to the standard deviation (Fig. 3.6). We proceed similarly for the azimuthal strain, which reads ε = u/r ( Fig. 3.5(d)). We do observe a strong anisotropy in the response of the ring, as shown in Fig. 3.6, that we will model in the next section.

Theoretical target strain

Naive analytical approach

We aim at modelling the in-plane target strains induced by the pressurization of the baromorph. We thus need to determine the whole stress distribution in the material. Here, we propose a simple approach to compute analytically the homogenized deformation of the plate. In order to do so, we define the relevant parameters in Fig. 3.7 and consider a local basis with unit vectors in-plane and perpendicular to the channel directions e ⊥ , parallel to the channel e and normal to the plane e z . Let us respectively define:

Ψ = h/(h + 2e) (3.1) Φ = d/(d + d w ) (3.2)
as the relative channel height with respect to the total sheet thickness and the in-plane channel density (Fig. 3.7). In our experiments, the aspect ratio of the channels cross sections h/d, as said in the manufacturing tutorial, is kept almost constant, around 3. We define three regions (1), ( 2) and (3) (See Fig. 3.7) in which we make the crude hypothesis that the strains and stresses are constant. This assumption is valid when the wall (region (2)) is slender enough: the boundary condition (the strain continuity at the top and bottom boundaries of the wall) in this case does not affect much the bulk stresses that are uniform due to the uniform pressure in the channels. It also requires that the membrane thickness e is of the same order as the channel width d, to ensure that the membrane does not significantly bend when the inner pressure is applied. A detailed examination of these assumptions is part of a current collaboration with Corrado Maurini and Marcello Rubino from the Jean Le Rond d'Alembert Institute, who derive a rigorous linear homogenization of our system (see subsection 3.4.2 for more details).

(2)

(3) For one given pressure we seek to compute the mean deformation in the material:

ε ⊥ = Φε (1) ⊥ + (1 -Φ)ε (3) ⊥ ε = ε (1) = ε (2) (3.3)
We thus need to compute the strains in the different regions. In the membrane region at the top and at the bottom of the plate, denoted with the subscript (1), a simple force balance perpendicular to the imaginary cut AA' yields:

ph = σ (1) ⊥ 2e (3.4)
Hence, we get:

σ (1) ⊥ = p Ψ 1 -Ψ (3.5)
We moreover assume, as in the case of the cylindrical balloon, that σ

z ∼ 0.

The wall (2) is compressed by the pressurized fluid in the two adjacent channels. Far from the top and bottom boundaries, we thus retrieve:

σ (2) ⊥ = -p (3.6)
The force balance perpendicular to a cut along the mid-plane leads to:

pd = σ (2) z d w (3.7)
Hence, we retrieve

σ (2) z = p Φ 1 -Φ (3.8)
To derive the average stress σ mean in the direction e , we balance the force resulting from the pressure hd p in one channel with the stress integrated on the complementary area CHAPTER 3. BAROMORPHS d w (h + 2e) + 2ed, which results into:

σ mean = p ΦΨ 1 -ΦΨ (3.9)
However, the distribution of σ in both regions remains undetermined. Nevertheless, the continuity of deformation along the direction of the channels imposes:

ε (1) = ε (2)
(3.10) Following Hooke's law, we obtain:

1 E (σ (1) -ν(σ (1)
⊥ + σ (1) z )) = 1 E (σ (2) -ν(σ (2)
⊥ + σ (2) z )) (3.11)
where E and ν are the Young modulus and the Poisson ratio, respectively. The material properties of our silicone elastomer are E = 250 ± 15 kPa and ν = 1/2. We now assume that stresses along the channels in regions (1+3) (noted (1)) and ( 2) are distributed according to their corresponding surface fraction, which leads to:

(1 -Ψ)σ (1) + (1 -Φ)Ψσ (2) = (1 -ΨΦ)σ mean (3.12)
We finally obtain a system of two equations and two unknowns that can be easily solved:

       σ (1) -σ (2) = ν(σ (1)
⊥ + σ (1)
z -σ (2)
⊥ -σ (2) z ) (1 -Ψ)σ (1) + (1 -Φ)Ψσ (2) = (1 -ΨΦ)σ mean (3.13)
In terms of strain in the parallel direction, we obtain:

       ε (1) -ε (2) = 0 (1 -Ψ)ε (1) + (1 -Φ)Ψε (2) = p E ΨΦ(1 -2ν) (3.14)
It is remarkable to notice that for incompressible elastomers, the Poisson ratio is equal to 1/2. The right hand side of the second equation is then equal to zero and we end up with a homogeneous linear system of two equations and two unknowns, whose only solution is the pair (0, 0). Hence, we have ε 1) and σ (2) , we can now determine the strain in the other directions. Following Hooke's law, we get for region (i):

(1) = ε (2) = 0. Knowing σ (
                     ε (i) ⊥ = 1 E (σ (i) ⊥ -ν(σ (i) + σ (i) z )) ε (i) = 1 E (σ (i) -ν(σ (i) ⊥ + σ (i) z )) = 0 ε (i) z = 1 E (σ (i) z -ν(σ (i) ⊥ + σ (i) )) (3.15)

THEORETICAL TARGET STRAIN

Shear is finally not accounted for in our simplified model. Assuming that the strain in the upper part of the walls (region (3)) is the same as in the channel part (region (1)) is not reasonable. We thus propose to make the following assumption:

ε (3) ⊥ ∼ Φε (1) ⊥ (3.16)
Although this assumption is kinematically incompatible, in the two extreme cases, when Φ tends to 0, ε

⊥ also approaches 0 and when Φ tends to 1, ε

⊥ tends to ε

⊥ . The macroscopic homogenized radial strain is thus:

ε ⊥ = Φε (1) ⊥ + (1 -Φ)ε (3) ⊥ (3.17)
The homogenized strains parallel and perpendicular to the local channel direction thus read:

           ε t = p E ΨΦ (1 -ΨΦ) (1 -2ν) = 0 ε t ⊥ = p E Φ(2 -Φ) Ψ 1 -Ψ - νΨΦ 1 -ΨΦ 1 + ν 1 -Φ Φ(1 -Ψ) -1 (3.18)
Note that both Ψ and Φ evolve with the pressure:

               Φ = d(1 + ε (1)
⊥ ) d(1 + ε (1) ⊥ ) + d w (1 + (ε (2) ⊥ + ε (3) ⊥ )/2) Ψ = h(1 + ε (2) z ) h(1 + ε (2) z ) + 2e(1 + ε (1) z ) (3.19)
This dependence of the geometry with the pressure constitutes a key feature in inflation: as the pressure increases, the cross-section of the plate resisting the load is reduced, which results in concentrated stresses. This behaviour explains the non-linearity of the amplitude of the deformation with the pressure. We limit so far our simplified model to linear elasticity and do not account for the strain stiffening of actual elastomers. Such stiffening tends to regularize the divergence of the strain at large pressures, as we will show in the next chapter. The resulting non-linear prediction for the target strain is in very good agreement with experimental data for moderate pressures, without any fitting parameter as illustrated in Fig 3 .8. In the domain of interest, that is for channels of aspect ratio h/d > 2 and for membrane thickness e of the same order of magnitude as the channel width d, our model captures the variation in slope and in the onset of non-linearity depending on the channel density Φ and the relative channel height Ψ. The larger the channel density Φ and the relative channel height Ψ are, the more the plate will stretch perpendicular to the channels at a given pressure. Both geometrical parameters may thus be used in order to tune the target homogenized strain value at one given pressure.

Linear homogeneization

Although this simple analytical approach gives very satisfactory results (surprisingly, given the crude hypotheses made) in the regime of the parameter space experimentally investigated, it might be useful to have more general and rigorous framework in which the channel geometry could be effortlessly changed or even optimized to achieve specific target pre-strain. It is indeed obvious that for low aspect-ratio channels ((h, e) < d) the top and bottom membranes will significantly bend and our model will fail, since it does not take into account such deformations.

To that end, we started a collaboration with Marcello Rubino and Corrado Maurini from the Jean le Rond d'Alembert Institute. They are applying standard linear homogenization methods to this periodic pressurized mesostructure (Fig. 3.9(a)). The pre-strain (ε , ε ⊥ ), when a pressure p is applied, is obtained from a FEM calcuation on a unit cell with periodic boundary conditions (Fig. 3.9(b)). First results and the comparison with our model are shown in Fig. 3.9. We see that for the prestrain along the channels (ε ), their numerical results match perfectly our linearized predictions and the dependence on Ψ and Φ. This prestrain ε is here non-zero, since they use a material Poisson coefficient ν = 0.45 (and not 0.5) in order to avoid numerical complications. In the direction perpendicular to the channels however, deviations are observe, a direct consequence of the crude assumption made in equation 3.16. In the regime of experimental interest however (0.3 < Ψ < 0.9 and 0.3 < Φ < 0.6), the mismatch remains modest.

Marcello's method presents the great additional advantage to give the complete stiffness orthotropic matrices of the homogenized structure, and the effective Poisson ratio ν ef f , which will be very helpful to compute with precision the actual shape of the shell subjected to pre-strains. This method will also provide the region of validity for our approximated expression 3.18. The approach is however linear, which is clearly not the case for the typical finite deformations obtained experimentally. The generalization of this approach to finite strains would be in that sense of high practical interest. For more details on the method and the results obtained, please refer to Marcello's master thesis [START_REF] Rubino | Homogenized Finit Element Method for pneumatically actuated shells[END_REF].

Shape changes

We now know the local anisotropic response of a plate embedding rectangular air channels under pressure. The aim of this section is to understand which orientation and geometry of the channels, i.e. which pre-stress, or equivalently which new target metric, should be embedded in the plate in order to obtain a target shape upon inflation. In comparison to other shape-morphing methods, our strategy presents the advantage of providing one additional degree of freedom in order to distort the metric. Indeed in swelling hydrogels, only the isotropic swelling rate may be controlled locally, via a halftone technique [START_REF] Kim | Designing responsive buckled surfaces by halftone gel lithography[END_REF] or radially in the set up of Klein et al. [START_REF] Klein | Shaping of elastic sheets by prescription of non-euclidean metrics[END_REF]. One simple scalar can thus be tuned to change the metric. In the same manner as in conformal mapping [START_REF] Konaković | Beyond developable: Computational design and fabrication with auxetic materials[END_REF], the angles are preserved during the transformation. For liquid crystal elastomers, the CHAPTER 3. BAROMORPHS orientation of the director field is the only knob one may tune to control the direction of anisotropic contraction (along the director) and expansion (perpendicular to the director). In baromorphs, both the orientation of the channels (similar to the perpendicular dual of the director field in LCE) and the channel density, which sets the local expansion rate perpendicular to the channels at one given pressure, may be adjusted to play with metric. We will show that the programming of arbitrary target shapes (i.e. solving the inverse problem) is highly facilitated by the addition of one degree of freedom.

Starting with a cone

Let us start simply with a cone. A cone can be made with a disk with a missing sector, as shown in the introduction. It has Gaussian curvature concentrated at the tip. If one draws a circle of radius r centred on the apex, the perimeter of the circle is P = 2πr cos α where tan α is the slope of the cone. In order to get this angular deficit 2π(1 -cos α), one way to proceed is simply to stretch the radii by an amount

(1 + ε t r ) = 1/ cos α (3.20)
while preserving the perimeter as shown in Figure 3.10(a). In this 3D mapping, all material points of the disc simply move along the z-axis. Such distance distortion may be reproduced in a baromorph with a configuration made of concentric and regularly spaced circular air channels: a uniform radial target strain ε t r is induced, while the azimuthal target strain remains null. For high applied pressures, we do observe conical shapes with a tip regularised by the finite bending stiffness of the plate, as shown in Figure 3.10(b). Moreover, our simple geometric argument (Equation 3.20) does predict quite well the angle of the cone at large pressure (Fig. 3.11(h)).

However, for small pressure and hence small deformation, the plate remains flat. The variation of α as a function of the applied pressure exhibits a buckling transition. (See Figure 3.11)

The buckling obtained is somewhat surprising, if we forget about our geometrical argument, since the baromorph is free and does not have any force or torque applied at its boundary. Moreover, the structure is subject to internal pressure, that should tend to set the structure under tension. However, the plate being symmetric with respect to its mid plane, there must be some frustration that drives the classical symmetry-breaking buckling. It comes from the new metric (or prestress) imposed by the geometry and orientation of the channels, which is incompatible with the flat configuration. The structure may thus buckle and bend equally one way or the other, in contrast to previous pneumatic soft robotics design. In practice, small defects in the fabrication tend to bias the direction of popping out of plane.

As in traditional buckling of slender structures, the finite bending stiffness of the plate prevents its out-of-plane buckling for small strains [START_REF] Dias | Programmed buckling by controlled lateral swelling in a thin elastic sheet[END_REF]. In order to understand this transition, we need to express the two elastic energies involved in the structures, namely CHAPTER 3. BAROMORPHS stretching and bending energies.

(c) (d) (e) (f) (b) (c) (d) (e) (f) 0 (g) (h)

Scaling law for buckling threshold.

The buckling threshold leading to out-of-plane deformations can indeed be roughly predicted by comparing stretching and bending energies in the flat and bent plate, respectively. We expect the transition to occur when bending gets energetically more favourable than in-plane strain. In terms of scaling, the stretching energy U stretch is proportional to:

U stretch ∼ Eε 2 r HR 2 (1 -ΦΨ) (3.21)
where H = h + 2e is the total thickness of the plate. Indeed if the plate remains flat, the azimuthal strain has to be of the same order of magnitude than the target radial strain ε r (homothetic growth). Due to the presence of channels, the effective Young modulus in the azimuthal direction is now E(1 -ΦΨ).

We expect the bending energy per unit length U bend to scale as:

U bend ∼ EI R 2 κ 2 (3.22)
where EI is the effective bending stiffness of the homogenised plate and κ is the typical induced curvature. Considering the presence of the channels, the effective bending stiffness in the azimuthal direction scales as

I = I f ull -I channels ∼ H 3 (1 -ΦΨ 3 ) (3.23)
At the transition, radial strain and curvature are simply related by:

ε r ∼ κ 2 R 2 (3.24)
The buckling transition should correspond to U bend ∼ U stretch , which leads for the critical strain:

ε c ∼ H 2 R 2 1 -ΦΨ 3 1 -ΦΨ (3.25)
We remark that we obtain the typical critical buckling strain (H/R) 2 corrected by the geometry of the plate. Similarly to I beams, our plates are indeed relatively stiffer in bending than in compression in comparison with full plates, since material is concentrated far from the neutral plane. As shown in the previous section (Equation 3.13), the strain dependence on pressure reads:

ε r ∼ ΦΨ(2 -Φ) 1 -Ψ p E (3.26)
which finally leads to the critical buckling pressure: Figure 3.12 shows the cone angle α as a function of p/p c for various plates. We see that all the buckling thresholds are close to p/p c ∼ 6. We can nevertheless do better and compute both energies in the homogenized plate with the prefactors and minimize their sum, in order to get slightly modified plates equations.

p c ∼ E H 2 R 2 1 -Ψ ΦΨ(2 -Φ) 1 -ΦΨ 3 1 -ΦΨ (3.
Bending and elongation stiffness of inflated baromorphs. Before computing precisely the shape of an inflated baromorph it is crucial to estimate the bending and stretching stiffnesses of the structure and their potential dependence on the applied pressure. We will show here using simple arguments, that, surprisingly, both stiffnesses are barely affected by the pressure. First, when the baromorph is deflated, the stretching homogenized stiffness parallel Y to the channel direction respectively read:

Y = Y f ull -Y holes = EH(1 -ΨΦ) (3.28)
Y ⊥ is more complex to compute, since the role of the wall depends on the geometry. For very thin walls (compared to the membrane thickness) we expect Y ⊥ ≈ EH(1 -Ψ) which is significantly smaller than Y . The dependence of Y ⊥ on the geometric parameters was part of a collaboration with Marcello Rubino and Corrado Maurini. Similarly, the bending stiffness along the direction of the channels, as discussed in the previous section, reads:

EI = EH 3 (1 -ΦΨ 3 )
Since bending a slender beam conserves the volume, this deformation does not involve any work of the inner pressurized fluid. As a consequence, the inner pressure only induces a pre-stretch that postpones the local buckling of the membrane and should not affect the linear response (i.e. the bending stiffness). However, inflating the structure tends to thicken it, which slightly increases the bending stiffness. The mere role of the pressure in the bending stiffness is thus to slightly change the dimension and geometry of the object. We measured the shape of baromorph beams sagging under their own weight for different pressures (Fig. 3.13(a) and (b)). The dependence on the pressure remains modest, as expected and predicted by the model. Similarly, a stretching deformation conserves the volume in the solid (ν = 1/2) as well as in the inner the fluid. As a result, the stretching stiffness should not change when the structure is stretched. A minor deviation from the ideal behaviour is however observed in Fig. 3.13(c) due to material stiffening at large strains.

Axisymmetric plate equations for circular channels

Consider a flat circular baromorph, embedded with circular concentric channels of constant relative height Ψ. The channel density Φ(r) may vary along the radial direction.

The elastic energy stored in the plate corresponds to the sum of the stretching and bending energies:

U = U stretch + U bend (3.29) with U stretch = 1 2 σ : (ε -ε t )dS (3.30) U bend = 1 2 D(κ 2 1 + 2νκ 1 κ 2 + κ 2 2 )dS (3.31)
where D is the bending stiffness of the plate (assumed here isotropic for simplicity), (κ 1 , κ 2 ) the principal curvatures of the plate and ε t the target radial deformation. In the weakly non-linear regime (within the limit of small slopes), the strain can be expressed at the second order in w as:

         ε r = u + 1 2 w 2 ε θ = u r (3.32)
where u and w correspond to the in-plane radial displacement and to the out-of-plane deflection, respectively. Using Hooke's law, we derive the stress distribution:

           σ r = E(1 -ΦΨ) 1 -ν 2 ε r -ε t r + ν(ε θ -ε t θ ) σ θ = E(1 -ΦΨ) 1 -ν 2 ε θ -ε t θ + ν(ε r -ε t r ) (3.33)
where, E is the Young modulus of the elastomer, ν the Poisson ratio of the material, ε t r and ε t θ are respectively the radial and azimuthal target strains computed in the previous section. For the sake of simplicity, we have assumed that the effective stretching stiffness of the plate is isotropic, which is not strictly true, as explained in the last section. The elastic stretching energy stored in the plate finally reads:

U stretch = Eπh 1 -ν 2 R 0 (1 -ΨΦ) (u + 1 2 w 2 -ε t r ) 2 + 2ν(u + 1 2 w 2 -ε t r ) u r + ( u r ) 2 rdr (3.34)
We now focus on the bending energy. For an axisymmetric surface parametrised by z = w(r), the two principal curvatures are determined by:

           κ 1 = w (1 + w 2 ) 3/2 κ 2 = w r √ 1 + w 2 (3.35)
At the second order in w , the bending energy in the plate is given by:

U bend = Eπh 3 12(1 -ν 2 ) R 0 (1 -ΦΨ 3 ) rw 2 + 2νw w + w 2 r dr (3.36)
The total energy can be minimized with traditional variational methods and we obtain the following set of equilibrium equations for the shape of the plate which includes the incompatible target strain from Equation 3.18 within the Föppl-Von Karman approximation [START_REF] Efrati | Elastic theory of unconstrained non-euclidean plates[END_REF] :

                   w = Ψ 3 Φ 1 -Ψ 3 Φ - 1 r w + w r 2 + 12 H 2 1 -ΨΦ (1 -Ψ 3 Φ) w u -ε t r + 1 2 w 2 + ν u r u = ε t r + ΨΦ 1 -ΨΦ - 1 r u + u r 2 -w w + ΨΦ 1 -ΨΦ - 1 -ν r 1 2 w 2 -ε t r +νε t θ + ν ΨΦ 1 -ΨΦ - 1 -ν r ε t θ (3.37)
Free boundary conditions in plates are subtle [START_REF] Mansfield | The bending and stretching of plates[END_REF] but emerge naturally from the variational derivation and read:

                                   u(0) = 0 w (0) = 0 u (R) + 1 2 w (R) 2 -ε t r (R) + ν u(R) R -ε t θ (R) = 0 w (R) + ν w (R) R = 0 (3.38)
Equations 3.37 together with the boundary conditions 3.38 can be solved using the bound- ary value problem function bvp4c from Matlab. We tested this theoretical approach with experiments on cones made with uniformely spaced circular concentric channels and thus constant radial target strain ε t . Results of the integration of these equations for uniform target strain configurations corresponding to various channel densities Φ and relative channel heights Ψ are presented in Fig. 3.14. The theoretical cone angle was extracted by fitting the slope of the obtained shape (see Fig. 3.15) far from the apex. Despite the numerous simplifying assumptions (isotropic property of the plate, Hookean material, weakly non-linear model, target strain..) the model does an impressive job at predicting both the buckling threshold and the post buckling angle evolution, without any fitting parameters. Not only the angle of the cone, but also the regularized apexes are well captured by the model, as shown in Fig. 3.15.

Using equations 3.37 and 3.38, we can also predict the shapes obtained with gradients of the channel density Φ. Nevertheless, the assumption that the plate has isotropic mechanical properties here affects the prediction, since the structure is also bent in the radial direction (in contrast with the cone configuration). Our model does capture qualitatively the shape (see Fig. 3.16) and the buckling threshold of axisymmetric baromorphs, but the match is not as good as in the case of cones. Moreover, this approach is limited to small slopes, which is not always true in experiments (even if the slope is bounded by the maximal stretch the material can sustain). Numerically inverting the system of equations (Equations 3.37 and 3.38 together with the target metric equations 3.18 and 3.19) is also tedious. All together, the complexity of this approach for programming axisymmetric shapes is not worth the effort, since the programming is not very precise. Full finite elements methods or homogeneization-based nonlinear shell models should be (in principle) more predictive. However, we should recall that far from the buckling threshold, isometries are energetically highly favoured: simple geometrical approaches, neglecting the mechanical properties of the plate, are in this case relevant.

Geometric programming of axisymmetric shapes

Let us now neglect the bending energy of the baromorphs and geometrically program the metric distortion in order to transform a flat disc onto an arbitrary axisymmetric shape. This shape can be described by its profile equation z = h(r). The geometric argument for the cone still holds locally, having a varying slope angle tan α(r) = h (r). The target azimuthal and radial strains, following the same mapping strategies according to which all material points move along the z-axis, read:

     ε t θ (r) = 0 ε t r (r) = 1 + h 2 r (r) -1 (3.39)

SHAPE CHANGES

This can be achieved with a purely azimuthal channel pattern with varying channel geometry. Remembering indeed the fact that ε t is a growing function of Ψ, Φ and p, and Φ being experimentally much simpler to tune, we can easily invert numerically the equation:

ε t r (Φ(r), Ψ, p) = 1 + h 2 r (r) -1 (3.40)
using equation 3.18 for the expression of ε t . For instance, in order to get a portion of a sphere of radius R as illustrated in Fig. 3.16, the target radial strain should follow:

ε t r (r) = 1 1 - r R 2 -1 (3.41)
Once again, this simple geometrical analysis completely neglects the finite bending stiffness of the plate and will poorly predict the shape close to the buckling threshold.

Programming negative Gaussian curvature shapes

In our quest to explore the possibilities of the baromorph approach, the natural following step is to tackle non axisymmetric shape. One of the simplest of this kind is the anti-cone [START_REF] Müller | Conical defects in growing sheets[END_REF][START_REF] Dervaux | Morphogenesis of growing soft tissues[END_REF], the shape that one obtains when inserting a slice of pizza in an already full pizza. We thus need an extra angular sector, which is equivalent to uniform azimuthal extension at constant radius. This may be easily obtained in a baromorph plate with purely radial channels at constant density (already shown in the introduction of the chapter, in Fig. 3.1 and in Fig. 3.17).

In order to produce a surface of constant negative Gaussian curvature K, one should look closer at the geometry: The first fundamental form of such a surface reads in cylindrical coordinates, ds 2 = dr 2 -K -1 sinh(r √ -K))dθ 2 . Starting from a flat plate of first fundamental form ds 2 = dr 2 + r 2 dθ 2 , a straightforward solution to transform a circular plate in a saddle consists in keeping the distances in the radial direction unchanged and adapting the azimuthal expansion to the target metrics, ε t θ = -r 2 /6K + o(r 4 /K 2 ) (at the first order in r 2 /K). In our baromorphs, this strategy involves purely radial channels. For the sake of simplicity, we approximate the target azimuthal expansion by a constant piecewise function and compute the corresponding channel density required to achieve the target at one given pressure. Despite such approximations, the actual shape obtained is visually satisfying as shown in Fig. 3.17, mainly because elasticity tends to smooth the programmed target deformations.

Particular attention should however be paid to the boundary effects in the plate: the additional wall necessarily present at the outer edge of the baromorph in order to seal the airways induces an additional resistance to expansion. It is thus necessary to design this wall as thin as possible in order to minimize its influence. In order to characterize and document the non axisymmetric shape obtained, measuring the topography, and then compute the Gaussian curvature of the surface (See Fig. 3.17) was required : we thus developed our own simple 3D scan, together with Cyprien Blanquart, a very efficient one-month intern in July 2017.

3D scanning and computation of Gaussian curvature. with Cyprien Blanquart

The surface topography of inflated baromorph structures was measured with a 3D scanning system developed in the laboratory and based on the work of Cobelli et al. [START_REF] Cobelli | Global measurement of water waves by fourier transform profilometry[END_REF]. Basically, the 3D shape is inferred from the distorsion of a pattern of stripes projected on the structure ( Fig. 3.18(a)(b)). The local height is thus deduced from the phase shift of the periodic pattern. Although Fourier transform is generally used to extract the phase, the lack of periodic boundaries prevented us from using this method. We instead used a phase shifting profilometry technique, detailed by Van der Jeught et al. [START_REF] Der Jeught | Real-time microscopic phase-shifting profilometry[END_REF]. Four patterns are successively projected and recorded with the camera, each time shifted by π/2. The local phase φ(x, y) can thus be directly computed as:

φ(x, y) = arctan I 4 -I 2 I 3 -I 1 (3.42)
where I 1-4 is the local fringe intensities at pixel (x,y) for the different phase-shifted patterns. The local phase is defined at each pixel modulo π. It is then unwrapped using a 2D unwrap Matlab code written by Muhammad F. Kasim (2D Weighted Phase Unwrapping) based on the work of Ghiglia and Romero [START_REF] Ghiglia | Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods[END_REF]. The local surface height can be deduced from phase shift with respect to reference image ∆φ(x, y) using basic geometrical optics:

h(x, y) = ∆φL ∆φ -2πD/t (3.43)
where ∆φ = φ(x, y) -φ 0 (x, y), D is the distance between the video projector and camera, L is the height of both instruments with respect to the flat surface of reference and t is the spatial wavelength of the fringe pattern (Fig. 3.18(c)). The Gaussian curvature is finally deduced from a local quadratic fit of the surface (Fig. 3.18(d)).

Other examples of lateral expansion: truncated cones and helicoids.

Truncated cone. Using the lateral expansion of radial channels is not limited to programming negative Gaussian curvature. We employed the same strategy to program truncated cones. Consider a circular plate with a central hole and radial channels (Fig. 3.19(a)).

Applying an azimuthal strain while conserving radial distances also results in a new shape.

In the geometric limit of negligible bending stiffness, we obtain the local angle from ele- The predicted shape is thus a truncated cone of angle α if the azimutal strain is of the form ε θ = cos α -1 + A/r. In practice, we approximate this form by a piece-wise function (fixed number of radial channels of uniform width in the inner part of the plate followed by an annular region without channels). The resulting shapes are in qualitative agreement with the prediction.

CHAPTER 3. BAROMORPHS 2cm (a) 2cm (b) (c) (d) (e) (f)

Helicoid

Let us now consider a helicoidal shape of pitch Λ and width 2b. All the points at a distance b from the mid line form a helix of length √ π 2 b 2 + Λ 2 per pitch. This result can be easily shown by unwrapping the cylinder of radius b, as shown in Fig. 3.19(f). A natural way to go from a plane (e y , e z ) to a helicoid is thus to stretch the lines of equation y = r by an amount

λ z = 4π 2 r 2 Λ 2 + 1 1/2 (3.45)
Following this simple idea, we made long rectangular plates with channels following a fishbone design (Fig. 3.19(d)). Applying pressure thus tends to expand the lines perpendicular to the channels by an amount set by the local in-plane channel density. We chose properly the gradient in channel density to approximate the metric distortion described in equation

3.6. INVERSE PROBLEM, A GENERAL PROCEDURE
3.45 and do obtain a helicoid upon inflation (Fig. 3.19(e)). Once again, the symmetry of the object with respect to its mid-plane implies that the structure may buckle equally into a right or left-handed helicoid.

Inverse problem, a general procedure

So far, we developed a strategy to program simple shapes, knowing the target anisotropic deformation. We managed to make structures with positive gaussian curvature (dome, cone), negative gaussian curvature (saddle, anti-cone, helicoid) but also with both positive and negative Gaussian curvature (spiky structure). It is thus tempting to claim that, as every surface has locally either positive, negative or zero Gaussian curvature, we can program any shape with our method.

Programming an arbitrary shape involves however a non-trivial inverse problem. There are indeed an infinity of possibilities to project a 3D surface onto a plane. Most of them are completely irrelevant in our practical case, since they involve very large stretching factors, or isotropic deformation (as in conformal mapping). Inverting the problem is thus tricky since there is no one to one correspondence between a 3D shape and a distortion metric of the plane. It is also important to point out that the metric does not in general define a unique 3D shape. The surface may be indeed additionally extrinsically bent and several embeddings may be found: for a flat metric for example, any cylinder has the same metric as the flat plane.

So far, the inverse problem in practical cases (direction of the anisotropic target growth (in nematic elastomers [START_REF] Aharoni | Geometry of thin nematic elastomer sheets[END_REF][START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF]), or the isotropic growth factor in swelling gels [START_REF] Kim | Designing responsive buckled surfaces by halftone gel lithography[END_REF] or auxetic materials [START_REF] Konaković | Beyond developable: Computational design and fabrication with auxetic materials[END_REF]) corresponding to a given target shape) was only solved through an optimisation numerical procedure with no formal guarantee for the existence of a solution. In the specific case of baromorphs, the possibility to select both the orientation and the density of the channels enables us to tune at each point both the direction and intensity of the local expansion (similarly to the theoretical work of Plucinsky et al. [START_REF] Plucinsky | Programming complex shapes in thin nematic elastomer and glass sheets[END_REF]). Taking advantage of this additional degree of freedom, we propose a straightforward and intuitive analytical recipe for programming a smooth surface that can be parametrised as z = h(x, y). This recipe is geometrically exact, but does not involve the mechanics of the baromorph, and will be relevant only far from the buckling threshold. In this procedure, each point of the baromorph is moving along the z axis during activation, in a simple generalisation of the axisymmetric case (Fig. 3.16).

Consider a given smooth surface parametrized by z = h(x, y), that we want to reproduce, starting from a flat sheet, at a given pressure p. The first fundamental form reads:

ds 2 = (1 + h 2 x )dx 2 + 2h x h y dxdy + (1 + h 2 y )dy 2 , (3.46)
where h x and h y respectively correspond to ∂h/∂x and ∂h/∂y. This first fundamental CHAPTER 3. BAROMORPHS form can be written in the matrix form:

a = 1 + h 2 x h x h y h x h y 1 + h 2 y (3.47)
Diagonalising this matrix leads to the eigenvalues λ 1 , λ 2 and corresponding eigenvectors u, v:

λ 1 = 1 u = 1 h 2 x + h 2 y -h y h x λ 2 = 1 + h 2 x + h 2 y v = 1 h 2 x + h 2 y h x h y (3.48)
Interestingly, the fact that the first eigenvalue is equal to unity implies that there is no extension along isodepth lines with respect to a reference flat sheet. A natural idea is thus to draw channels along the contour lines projected on the reference plane, since no expansion parallel to the airways is predicted by our model. The second step consists in matching the target orthogonal strain to the metrics in the direction of the gradient:

ε t ⊥ = λ 2 -1 (3.49)
According to our model, ε t ⊥ is a function of Ψ, Φ and p/E. Since airways are interconnected, p/E is uniform through the structure. In our manufacturing process, it is practically easier to tune the channel density Φ (i.e. the local channel width d) than the relative channel height Ψ. Equations 3.18 and 3.19 are thus solved using Matlab to find the relevant channel width to match the metrics along the gradient for fixed values of p/E and Ψ. Following this concept, a simple code can automatically generate the airways path for any target 3D shape of the form z = h(x, y). Nevertheless, the lateral expansion of the channels is limited by material properties (ε t ⊥ max ∼ 0.5). As a consequence, only surfaces with slopes smaller than typically 45 • can be reproduced with this procedure. Figure 3.20 shows the programming and the realisation of a face following the method described above. The results are qualitatively in good agreement with the target shape, except for the finest details, as the eyes, that are smoothed out by finite bending stiffness. Indeed, the size of the eyes is of the same order as the thickness of the sheet, and bending rigidity cannot be neglected at this scale (these features are not far from the buckling threshold). One way to bypass this limitation would be to make the object thinner, in order to be closer to the geometric limit. Forgetting the potential fabrication complications that would surely be caused, the structure most importantly would be less stiff and may not sustain its own weight (see section on scalability).

We did not tackle at all the full inverse problem taking into account mechanical aspects of the baromorph. However, we know that relatively small features have to be "exagger- ated" since they will be smoothed out by elasticity. We thus can design the pattern, without any quantitative rationale but using our intuition based on the geometric strategy, to get qualitatively a target shape, such as a mask, by magnifying the small patterns on the plate, as shown in Fig. 3.21. Additionally, we may take advantage of cuts in the plane, such as in kirigami [START_REF] Lamoureux | Dynamic kirigami structures for integrated solar tracking[END_REF][START_REF] Dias | Kirigami actuators[END_REF][START_REF] Isobe | Initial rigid response and softening transition of highly stretchable kirigami sheet materials[END_REF], to release some geometric constraints and enlarge the shape possibilities. In the mask example, a cut under the nose enables discontinuous height changes.

The new baromorph technique thus appears as a very simple, adaptative and versatile tool to program efficiently 3D shapes out of an initially flat elastomer plate. One of its main advantages in the shape programming quest relies on the two degrees of freedom to distort the metric, namely the orientation of the channels and the channel density, that correspond to a coarse-grained average anisotropic target strain perpendicular to the channels. In the next section, we will discuss some practical aspects of this new strategy, its properties, its potential applications and limitations.

Practical aspects

Family of shapes In the last section, we only focused on the programming of one single target shape. When one inflates a baromorph, it is however manifest that the object continuously deforms with the pressure: a continuous family of shapes is thus obtained when adjusting the pressure, as shown in Fig. 3.22. Each shape corresponds to an equilibrium state and can be easily actuated on demand, in contrast to previous solutions based on hydrogel swelling or LCE sheets, where the transition between states is rather sharp. Moreover, the channels tend to shrink upon suction leading to a negative value of ε t r . As a consequence, for the concentric pattern depicted in Fig. 3.10(a) the structure adopts a saddle shape with negative Gaussian curvature upon suction. We did not insist on this particularity in the last section since the depressurization of airways is not as smooth and reproducible as inflation (the airways indeed tend to collapse).

Dynamics

Another key advantage of baromorphs relies on their fast actuation, as simple pneumatic structures. Few works have been dedicated to the dynamics of inflated architectured elastomer structures, and the few studies do not give a physical rationale for the maximum actuation rate [START_REF] Mosadegh | Pneumatic networks for soft robotics that actuate rapidly[END_REF]. In the example described in Fig. 3.23 a reversible transformation of the plate could be achieved with a frequency of 3 Hz. The structures presented in this study have an initial diameter of the order of 10 cm and channels of a section close to 1 mm 2 . Provided a sufficient power input, the actuation velocity is, in this case, limited by the natural frequency of the plate, ω ∼ h(E/ρ) 1/2 /R 2 , where ρ is the elastomer density, leading to a typical frequency of 10Hz. For small scale structures, a poroelastic time scale involving the viscosity of the fluid could limit the actuation as in the context of water transport in plants [START_REF] Dumais | Vegetable dynamicks: The role of water in plant movements[END_REF]. Conversely, the finite compressibility of air may be a limiting issue for large baromorph structures. A careful study of this dynamical aspect has not been conducted during the three years of the thesis, it is indeed not easy to find a minimal system, involving simple channel paths and 3D shape shifting, in a controlled fashion. In a very recent work, Matia and Gat [START_REF] Matia | The effect of connections between fluid-filled cavities on the dynamics of solid-liquid composite beams[END_REF] investigates the coupling of fluid transport and deformation in solid-liquid composite beams, restricting their study to small Reynolds numbers, not very relevant for the scale of our structures.

Scale From the static mechanical point of view, when neglecting gravity forces, our system is scale free. As shown in the previous sections, stresses and deformations in the plate only depend on geometric parameters, Φ, Ψ -ratios between lengths-and a normalized pressure p/E. In order to highlight this property, we show in Fig. 3.24(a) three baromorphs with the same design at three different scales allowed by the precision of the 3D printer, connected to only one pump. As the pressure is increased, all structures adopt the same shape, as predicted by our model.

Nevertheless, gravity forces do not have the same impact on all three objects. Indeed, the normalised deflection δ/R of a clamped baromorph scales as δ/R ∼ ρgR/(ES 2 ) in the small amplitude limit, where R is the radius of the baromorph and S = h/R the slenderness ratio, kept constant. The larger the object, the less it can sustain its own weight, as highlighted in Fig. 3.24(b). The maximum size of a baromorph which would sustain its own weight is given by a balance between the bending energy in the structure and the moment due to gravity forces: R max ∼ ES 2 /(ρg). This maximum size would be of the order of one meter for baromorphs made of elastomer of Young's Modulus of a few MPa and with a slenderness ratio of 1:10. The actuation principle is moreover largely material-independent. Indeed, everything depends on the architecture inside the plate, the only ingredient needed from the material is its elasticity and a high failure strain (so typically any elastomer). Relatively stiff -allowing meter-sized structures to resist their weight-, tough and wear resistant rubber can thus be used depending on the functionality sought. With the silicone rubber used in our experiments we could perform more than 1000 cycles without any observable degradation of the actuation : we are currently testing the life time of these objects further, since one baromorph will be featured at the 130th anniversary Eiffel tower exhibition this fall.

Smaller baromorphs could also be made using the classical microfluidic techniques, such as soft lithography, down to the millimeter scale and beyond.

Applications

Arca Ebenisterie During my first year of PhD, Benoît discovered at an exhibition"Airwood", the work of Arca Ebenisterie, a small french woodworking company. Among other beautiful realizations, one of unique specialities of Arca is to develop "inflatable" wood by finely cutting very thin wood sheets -typically 0.2 mm thick -glued on a latex membrane. The result is astonishing (Fig. 3.25): the apparently mundane wood plate suddenly comes to life and inflates, exhibiting a tessellation of small cut wooden tiles. We met Steven Leprizé, the founder of the company, a few weeks later, to show him the baromorphs and discuss with him the potential collaboration we could develop together. This technology could indeed seem limited to common balloon inflation, governed by Laplace law, where the single degree of freedom comes from the choice of the boundaries (Fig. 3.25). This is however not completely true, since the design of the cuts in the wood sheet strongly affects the stiffness (and most importantly the anisotropy) of the composite membrane, made of hard quasi-inextensible wood elements on top of a soft elastic substrate. The exact same ingredients were used recently in a Science research article by Pikul et al. [START_REF] Pikul | Stretchable surfaces with programmable 3d texture morphing for synthetic camouflaging skins[END_REF] to design morphing textures. It was mesmerising to discuss with Steven, and see how experience and everyday practice on the production of shapes give a very strong geometric and mechanical intuition, using however a completely different terminology.

Practically, Steven was limited to bas-relief on a flat surface. The natural idea was thus to cover our baromorph structure with his wooden tessellation in order to make shape morphing of apparent wooden plate, opening the way for shape morphing furniture. Wood would thus regain its organic growth property in an intriguing fashion. Although the idea was very simple, we faced a lot of practical complications: in our experiments, we used silicon elastomers, that are very difficult to bond with anything, and we needed to look for another material. Using natural latex was appealing, but the moulding step requires vulcanisation, which could not be conducted in the laboratory. We finally found polyurethane (Vytaflex 20 from Smooth-On) as a potential solution, but it was very hard to remove it from the machined mould made of plexiglas. To circumvent this difficulty, we 3.8. APPLICATIONS prepared a negative mould in silicon rubber (see Fig. 3.26(a)-(b)) It appeared nevertheless to be very difficult to bond the elastomer with itself, which is required in the making process of baromorphs structures (see section 3.2). We then discovered a flexible glue for polyurethane elastomers (UreBond II from Smooth On), and we were able to make our first prototypes (Fig. 3.26(c)-(f)). In May 2019, a first piece of furniture was finished and exposed at the exhibition "Révélations" in the Grand Palais: it is a low game table, named Ludum, that turns into an funnel with a central trap to tidy up efficiently the game pieces into one of the four storage places below the table (see Fig. 3.27). Additionally, we project to make a dresser without any hinge nor handle, that would just look like a wooden cube, and that would open up -just like a blooming-when the user will stand on a footboard, acting like a pump (see Fig. 3.28).

This collaboration is still in its infancy, Steven having many ideas of surprising and clever designs exploiting the possibilities of this new material. It is for us very enriching to exchange, discuss and produce objects with him, with all the small but time-demanding complications that one faces when applying a concept to real structures, that should be reliable and capable of sustaining many inflation cycles. It is frequently performed in critically injured, ill, or anesthetized patients, when they do not have the swallowing reflex anymore, to facilitate ventilation of the lungs, assisted by a medical ventilator. After the trachea has been intubated, a balloon cuff is typically inflated just above the far end of the tube to help secure it in place, to prevent leakage of respiratory gases, and to protect the lung from contamination from stomach acid, swallowing being blocked by the tube. These inflating cuffs are very simple (typically balloons see Fig 3 .29(b)) but not perfectly efficient as they do not completely seal the trachea, leading to lung infections. The cross section of the trachea is indeed not simply circular but has a horse-shoe shape, making the fitting of the cuff more difficult (see Fig 3 .29(c)). Moreover, long intubations are often associated with injuries or necrosis of trachea mucosa, due to stress concentration: the stiff balloon membrane indeed tends to wrinkle in the confined trachea. For soft membranes, the balloon adopts a nearly spherical shape, leading to a localized line contact with the inner wall of the trachea and thus poorer sealing. We thus intend to apply the baromorph technique to make a cylindrical shapemorphing cuff, that would fit closely the geometry of the trachea (see Fig. 3.30(a)). The practical realization of initially non-flat baromorph shells is however challenging, and moulding techniques are not very satisfactory. We managed to make one first cylindrical prototype (see Fig. 3.30(b)), that expands upon inflation thanks to longitudinal channels in the membrane. It does not have yet the target horse-shoe shape, for which direct 3D printing of the structure is required. The printing of soft and highly stretchable materials is now possible [START_REF] Tibbits | Additive manufacturing in gel-supported environment[END_REF] and should soon be available on the market, allowing us to explore with ease the design possibilities. Another difference of the baromorph technique comes from the fact that, in contrast with simple cuffs, the pressure inside the cavities does not match the pressure applied by the structure on the trachea walls (it is higher).

Medical equipment

Conclusion and perspectives on channel patterns

We have presented the baromorph concept : air channels are embedded in an elastomer plate and their geometry and orientation "codes" for the 3D shape upon pressurisation. We computed the target anisotropic deformation in an unconstrained plate and harnessed it to program non-Euclidean metrics distortion, leading to the out-of-plane buckling of the structure in a programmable fashion. We also presented a simple and intuitive geometric inverse recipe to design channel patterns shifting to a target arbitrary shape. Baromorphs thus constitute an efficient and versatile tool to transform 2D sheets into complex 3D structures reversibly with fast actuation. With this architectured and simply processed active material, shape-morphing technology may enter the engineering scene thanks to the ease of use, material independence (e.g. bio-compatible or stiff elastomers), and pressure controllability.

However, there is plenty of room for future investigations, that we will succinctly list now.

So far, we limited ourselves to channels centred with respect to the mid-plane of the structure. Displacing them slightly upwards or downwards would break the up-down symmetry and induce a bilayer effect, similarly to current pneumatic soft robotic strategies. In this academic study, we made an effort to design objects as symmetric as possible, in order to prove that the 3D shape-morphing is not due to a programmed bending, but rather due to metric incompatibilities. For practical realisations, it is however convenient to bias the system in order to favour one bending direction.

Additionally, several intertwined networks may be embedded in one plate to program various shapes on demand. One of the simplest configuration to start with is to make a bilayer composed of two independent networks of airways stacked on top of each other, as shown in Fig. 3.31. Using such a strategy, the target curvature tensor can also be programmed to match the curvature of the target shape [START_REF] Efrati | Elastic theory of unconstrained non-euclidean plates[END_REF][START_REF] Van Rees | Growth patterns for shapeshifting elastic bilayers[END_REF]. In practice, such curvature programming is not trivial and should require a careful study. With a purely anisotropic deformation, it is indeed only possible to program curvature tensors of negative Gaussian curvature. If the curvature tensor could be programmed, the homogenized sheet would be free of constraint and the actuation would be continuous (no buckling threshold).

Snapping instabilities [START_REF] Pandey | Dynamics of snapping beams and jumping poppers[END_REF][START_REF] Forterre | How the venus flytrap snaps[END_REF][START_REF] Gomez | Dynamics of viscoelastic snapthrough[END_REF][START_REF] Urbach | Delayed instabilities in viscoelastic solids through a metric description[END_REF] can also be triggered with out-of-phase actuation of the layers. In the example depicted in Fig. 3.31(b), two identical circular concentric patterns are stacked on top of each other. Pressurizing the top channel network induces the shape of a dome shell of length L and height h, its typical radius of curvature R being a function of the applied pressure p 1 . Increasing now incrementally the pressure p 2 in the lower network, the bilayer effect is reduced and the structure adopts a nearly conical shape when p 1 = p 2 , as in the case of single network cone baromorphs. Reducing the pressure p 1 in the upper layer induces a bilayer effect opposed to the present configuration. The structure will eventually snap and not smoothly go from one curvature direction to the other, if the typical strain necessary to flatten the shell is larger than the critical buckling strain ε c ∼ (h/L) 2 . This strain may be easily estimated as (L/R) 2 . The relevant snapping factor α = L 4 /(hR) 2 [START_REF] Forterre | How the venus flytrap snaps[END_REF] indicates whether the structure will snap (α > 1) or not (α < 1). The prefactor, found to be close to 1, and many more details may be found in the relevant bibliography [START_REF] Pandey | Dynamics of snapping beams and jumping poppers[END_REF][START_REF] Forterre | How the venus flytrap snaps[END_REF][START_REF] Gomez | Dynamics of viscoelastic snapthrough[END_REF][START_REF] Urbach | Delayed instabilities in viscoelastic solids through a metric description[END_REF]. Our system presents however the advantage to span various value of α with a single object, by adjusting the pressure.

The preferred target curvature may also be chosen by playing with the pressure difference in both layers. Moreover, we focused so far only on uniaxial "growth", where the orientation and the magnitude of expansion can be controlled. A possible extension of this work is to embed in the plate a connected array of controlled cavities with different sizes and shapes, rather than channels. As the same pressure is applied to the whole network, regions with a higher density of bubbles tend to expand more than regions with lower densities, and elongated cavities will expand anisotropically. It should then be possible, in principle, to impose all three components of the target growth tensor, in contrast with current morphing techniques [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF][START_REF] Klein | Shaping of elastic sheets by prescription of non-euclidean metrics[END_REF][START_REF] Kim | Designing responsive buckled surfaces by halftone gel lithography[END_REF][START_REF] Gladman | Biomimetic 4d printing[END_REF]. The natural simple system to start to investigate is thus a configuration of cavities in which the target deformation is locally isotropic but whose expansion may vary spatially. In this short chapter, we present the basic rationale to make isotropic baromorphs. While trying to calibrate the target deformation depending on pillar or bubbles densities and relative height, we observed a propagative instability in the plate with two phases, one highly stretched and the other barely stretched, which is reminiscent of the instability in inflated cylindrical balloons (see Section 2.2). This instability induces a nonuniform distortion of the metric, leading to the buckling of the plate in an intriguing fashion.

Chapter 4

Isotropic baromorphs and propagative instability Contents

We are very grateful to Corrado Maurini and Marcello Rubini for the discussions and the insights that made us discover this effect.

Introduction

Another possibility to achieve shape-morphing with pneumatic architectured elastomers thus relies on non-uniform but isotropic growth: a hexagonal lattice of cylindrical bubbles interconnected by small channels, or conversely, a network of cylindrical pillars embedded in the elastomer plate. Similarly to Kim et al. halftone swelling technique [START_REF] Kim | Designing responsive buckled surfaces by halftone gel lithography[END_REF], we take advantage of different local sizes of the regularly spaced cavities to induce a controlled non-uniform isotropic expansion upon pressure: the higher the density of cavities, the larger the local "growth". In Fig. 4.1, we highlight an example were cylindrical bubbles are larger in the centre than at the periphery of the plate, giving rise to positive Gaussian curvature once inflated. The center part of the plate wants to grow more than its surroundings and is thus frustrated in its expansion, leading to the buckling of the structure.

In the following, we study a configuration schematically represented in Fig. 4.2 where two membranes are separated by a lattice of pillars. Aiming at understanding the relation between the geometry of the pillars, the applied pressure and the target homogenized deformation, we first make a plate made of a regular hexagonal lattice of same sized pillars. Such a plate should, in principle, grow in a homothetic manner and remain flat (see Fig. 4.2). We however observe very surprising shape changes for large pressures, the structure buckles out of plane and adopts a bowl shape (see Fig. 4.3), in apparent contradiction with our theory : we indeed obtain a large Gaussian curvature although the encoded target metric is Euclidean (that is, compatible with a flat state). These shape changes are not the results of default in the fabrication of the structure, since a pure bilayer effect cannot explain such large amplitude doubly curved surfaces. Moreover, the structure is bistable, thus in contradiction with the bilayer hypothesis. Another hypothesis is that the additional presence of an elastomer wall around the structures frustrates the in-plane extension of the structure, acting like an additional rubber ring, and causes the plate to buckle out of plane.

MODEL AND ASSUMPTIONS

Model and assumptions

In the following, we thus aim at modelling the target deformation as a function of pressure of a baromorph plate made of a hexagonal lattice of pitch a of pillars of diameter d and height h, as schemed in Fig. 4.2. The top and bottom membranes have the same thickness e. The ratio h/a is fixed equal to 1. We define the two remaining relevant geometric parameters, Ψ = h/(h+2e) the relative height of the pillars and Φ = π

2 √ 3 ( d a )
2 the in-plane pillar density (the corresponding airways density thus reads 1 -Φ). Concerning the material constitutive law, we use the phenomenological Gent model, presented in section 2.2. The incompressible elastomer material may be described by two parameters, its shear modulus µ and the limit value of the first invariant J m .

For the sake of simplicity, we assume that the top and bottom membranes are subject to spatially homogeneous equibiaxial extension in the e x and e y directions (see Fig. 4.4): the principal stretches are λ x = λ y = λ. From incompressibility, λ x λ y λ z = 1. Hence λ z = 1/λ 2 . Therefore,

I 1 = λ 2 x + λ 2 y + λ 2 z = 2 λ 2 + 1 λ 4 (4.1)
The left Cauchy-Green deformation tensor can then be expressed as B = λ 2 e x ⊗ e x + λ 2 e y ⊗ e y + 1 λ 4 e z ⊗ e z (4.2)

Assuming that σ zz = 0 in the membrane, we get the following expression:

σ xx = σ yy = λ 2 - 1 λ 4 µJ m J m -J 1 . (4.3)
Moreover, we get through simple force balance, that

σ xx = σ yy = p ψ 1 -ψ (4.4) where ψ = Ψλ p Ψλ p + (1 -Ψ)/λ 2 (4.5)
is the actual relative height of the pillars in the deformed configuration. Pillars are indeed stretched by an amount λ p and the top and bottom membranes are thinner by an amount 1/λ 2 due to material incompressibility. It is thus critical to compute the deformation λ p of the pillar in the vertical direction. Deformation and stresses in the pillars are denoted with the superscript p . Pillars are compressed in the horizontal plane by the pressure inside the chamber: hence, σ p xx = σ p yy = -p. In the vertical direction, balancing the forces perpendicular to a cut in the xy-plane, yields:

σ p zz = p 1 -φ φ (4.6)
where

φ = Φ λ 2 λ p (4.7)
is the actual pillar density in the deformed configuration. The pillar cross section is indeed reduced whereas the elementary size of the lattice increases, both effects leading to a reduction of the pillar density. The symmetries of the pillars yield λ p z = λ p , λ p x = λ p y . From incompressibility, we get λ p2

x = λ p2 y = 1/λ p . Therefore,

I 1 = λ p2 x + λ p2 y + λ p2 z = λ p2 + 2 λ p (4.8)
The left Cauchy-Green deformation tensor can then be expressed in the pillar as

B p = 1 λ p (e x ⊗ e
x + e y ⊗ e y ) + λ p2 e z ⊗ e z (4.9)

Hence, we have where p r is a bulk pressure coming from the incompressibility that we shall now determine. As σ p xx = σ p yy = -p, we have

σ p zz = -p r + λ p2 µJ m J m -J 1 ; σ p xx = -p r + µJ m λ p (J m -J 1 ) = σ p yy (4.10)

Gent model fit Experiment

p r = -p + µJ m λ p (J m -J 1 (4.11)
Therefore,

σ p zz = -p + λ p2 - 1 λ p µJ m J m -J 1 (4.12)
The actual in-plane pillar density φ in the deformed state reads thus φ = Φ/(λ 2 λ p ).

To summarize, we propose a very simple model, in which the physical link of the pillars on the membrane (i.e. boundary conditions on the pillar and actual stiffness of the membrane due to the presence of pillars) is completely overseen. This model is believed to be more accurate when both Ψ → 1 and Φ → 0, that is when the pillars are slender structures and do not affect much the membranes stretching (see Figure 4 The system to be solved is thus the following:

                                                           σ xx = p ψ 1 -ψ = λ 2 - 1 λ 4      µJ m J m -2 λ 2 - 1 λ 4 + 3      σ p zz = p 1 -φ φ = -p + λ p2 - 1 λ p      µJ m J m -λ p2 - 2 λ p + 3      φ = Φ λ 2 λ p ψ = Ψλ p Ψλ p + (1 -Ψ)/λ 2 (4.13)
where the unknowns are λ and λ p , µ and J m are material properties, Ψ and Φ are geometric parameters of the structures and p the applied pressure.

Experimental realisation and quantitative comparison with the model

A tensile test is first performed at low speed to measure the characteristics of the silicone elastomer as shown in Fig. 4.5 for the case of Elite Double 8 from Zhermack (see Table 4.1 for the typical values measured during this thesis). Fitting the experimental stress-deformation curve with the theoretical constitutive law of Gent model for incompressible hyperelastic materials, one infers the shear modulus of the material µ and the limiting value of the first invariant J m for each elastomer.

The experimental deformation in the structure is measured by taking top view pictures at various pressures, as shown in Fig. 4.6(a). The evolution of the local mean distance between two pillars is then tracked to extract the stretching factor λ as a function of pressure (Fig. 4.6(b)). Solving the system of equations 4.13 with Matlab, the computed theory obtained is in good quantitative agreement with the experimental data points 4.4. CONCLUSION without any fitting parameter (see Fig. 4.6(a)). Both the linear response at small pressure and the strong non-linearities at larger pressure are well predicted by the model. Similarly to the cylindrical balloon (see section 2.2), we obtain, for long enough chains (i.e. for large J m ), an S-curve. It means that two phases may coexist for one given pressure, one highly stretched, the other barely stretched. In contrast with the cylindrical balloon, we have in addition to the dimensionless pressure p/µ two geometric parameters, namely Φ and Ψ, that may impact this bulge instability (in the case of the cylindrical balloon we had a single dimensionless parameter of interest pR/(µe)). In Figure 4.6(b), we show for each possible pair (Ψ, Φ), the minimum limiting first invariant value J m in order to get this instability. It appears that the in-plane relative stiffness, measured by Ψ, and the vertical stiffness, measured by Φ, must typically have the same value in order to get the instability at low J m .

Nearly one-dimensional objects, such as roots, that grow non-uniformly are free to deform according to the growth distribution and therefore do not develop internal stresses. Conversely, non-uniform growth of two-dimensional sheets can be geometrically incompatible, leading to the accumulation of stresses within the sheet. For example, if one portion of the sheet grows more rapidly than its surrounding, it may buckle out of plane, as studied by Bense et al. [START_REF] Bense | Buckling of elastomer sheets under non-uniform electro-actuation[END_REF] in the case of an expanding patch of dielectric elastomer in a passive elastomer sheet. In the cylindrical balloon configuration, the apparition of a bulge, which is much more stretched than the rest of the balloon, is not affected by the barely stretched rest of the ballon. There is a smooth transition between the two phases, but no internal stresses build up in the balloon, as a nearly 1D structure. In the 2D baromorph sheet, conversely, the apparition of a bulge is geometrically incompatible with its barely stretched surrounding. The structure thus locally buckles out of plane, as shown in Fig. 4.7, leading to a complex topology of the very simple and initially regular internal structure. The two phases corresponding to the same pressure, the structure picks the proportion of the highly stretch phase depending on the air volume inserted in the structure.

Although the target strain-pressure curve and the subsequent bulge apparition are well understood and captured by our minimal model, the induced 3D shapes are beyond the scope of this thesis. They involve the theory of non Euclidean incompatible plates at finite deformation with a specific superelastic response.

Conclusion

In the last two chapters, we investigated a first strategy to create shape-morphing pneumatic structures in the regime where the pressure forces induce a significant stretch of the material (p ∼ Et/L where L is the typical size of the cavities and t the thickness of the membranes). The choice of the material is thus limited to elastomers, which have an elastic rubber-like response and a high failure strain. Their Young's modulus is rather weak, ranging from typically 100 kPa to 10 MPa. The structure will be thus inherently soft, with all the advantages (safe, harmless, controlled in force and not in position), but also the main drawback, that is the lack of structural stiffness, associated with this property. We also showed that the stiffness is barely affected by the applied internal pressure. It seams hopeless with this strategy to create strong and rigid shape-morphing structure.

The ideal scenario would be to have an initially soft material that would stretch like an elastomer and then, at a programmable stretch, would suddenly become orders of magnitude stiffer. This is the case of most elastomers, with their limiting chain extensibility, but at such stretches they tend to fail. Human skin, and more generally biological tissues, have a strongly non linear strain-stiffening response [START_REF] Fratzl | Fibrillar structure and mechanical properties of collagen[END_REF][START_REF] Motte | Strain stiffening in collagen i networks[END_REF][START_REF] Storm | Nonlinear elasticity in biological gels[END_REF], which remains robust in the stiff configuration. The reason behind this strain-stiffening is the alignment of collagen fibres upon stretching, inducing a bending to stretching transition in the fibres [START_REF] Vader | Strain-induced alignment in collagen gels[END_REF]. Male mammals dispose remarkably of a quasi one-dimensional inflatable structure that strongly stiffens when inflated with blood, from which we could get inspiration. In a series of scientific studies [START_REF] Kelly | The functional morphology of penile erection: tissue designs for increasing and maintaining stiffness[END_REF][START_REF] Kelly | Turtle and mammal penis designs are anatomically convergent[END_REF][START_REF] Kelly | Penises as variable-volume hydrostatic skeletons[END_REF], Kelly noticed that inflatable penises have evolved independently at least four times in amniotes, specifically in mammals, turtles, squamates, and the archosaurs. "Males in these lineages therefore share the functional problem of building a penis out of soft and flexible tissues that can increase its flexural stiffness and resist bending during copulation. Research on penile erectile tissues in mammals and turtles shows that these two taxa have convergently evolved an axial orthogonal array of collagen fibres to reinforce the penis during erection and copulation; in both lineages, the collagen fibres in the array are crimped and folded in the flaccid penis. Collagen fibre straightening during erection increases the stiffness of the tissue and allows changes in penile radius that increase its second moment of area: both of these changes increase the flexural stiffness of the penis as a whole. And once erect, axial orthogonal arrays have the highest flexural stiffness of any fibre arrangement. The high degree of anatomical convergence (to the level of microanatomical features) within mammals and turtles suggests that the stiffness requirements for copulation produce an extremely restrictive selective regime in organisms that evolve inflatable penises." From a biomimetic point of view, this programmable strain-stiffening property may be engineered with composite materials: the matrix would still be made out of a soft rubber, embedding an arrangement of crimped fibres, the local direction and amount of looseness coding for the stretching factor at which the strain-stiffening will occur. Although the principles are simple, the practical fabrication of such a material is challenging, since common industrial textile machines work at a nominal tension. Programmable looseness is indeed not something sought in the fabric industry. We propose a strategy to obtain local equivalent looseness in thin plastic sheets, using kirigami [START_REF] Dias | Kirigami actuators[END_REF][START_REF] Lamoureux | Dynamic kirigami structures for integrated solar tracking[END_REF][START_REF] Isobe | Initial rigid response and softening transition of highly stretchable kirigami sheet materials[END_REF]. The plastic sheets are embedded inside the elastomer matrix with a network of cuts enabling programmable deformation via local bending in the structure (see Fig. 4.8)). This idea will be investigated by Maïka Saint-Jean in the next three years in the lab. She is however not using a pneumatic system to trigger the extension. She embeds the kirigami sheets inside an emulsion of ethanol and elastomer: upon heating, ethanol experiences a liquid-vapour transition, causing a significant swelling of the material [START_REF] Miriyev | Soft material for soft actuators[END_REF].

Another idea to address this stiffness issue is to change the material, using a stronger and stiffer one. Applying pressure cannot reasonably induce a significant material stretch (p Et/L), that would anyways probably lead to catastrophic failure. As seen in the 4.4. CONCLUSION introduction, another (often preferred) deformation mode for thin membranes is bending. We will investigate in the next chapter this possibility of engineering a flat textile pneumatic structure, controlling locally its bending to program global shape shifting of much stiffer structures. We will start, as in the present part, by studying the deformation upon inflation of a simple straight and then curved path (like the mylar balloons letters) and then extend it to 2D surfaces.

Part II

Inflating textiles and quasi-inextensible thin sheets

In this second part, we present another strategy to produce shape-morphing inflatables, made of much stiffer material, such as textiles or thin mylar sheets. These structures can be fabricated at a large scale and may thus find applications in architecture. We will consider the material as quasi-inextensible (p Et/L) but with vanishing bending stiffness (p Et 3 /L 3 ). It might seem to the reader that it is a strange idea, since we saw in the first part that thin sheets tend to bend rather than stretch, such that only isometries of the plane are reachable. Isometries of the plane are the developable surfaces, and we aim at producing much richer shapes, with positive and negative Gaussian curvature. The key ingredients that will allow us to change the average metric of the structure are the local bending of the sheet and its wrinkling, enabling an average contraction of the surface without any stretch.

In order to understand the basic physics of quasi-inextensible flat inflatables, we shall first look in chapter 5 at the trivial inflation of a balloon made by two identical elongated rectangles sealed on their edges, and then study the much more subtle case of curved paths. We observe indeed that a curved path surprisingly overcurves upon inflation and presents a radial wrinkling pattern.

In our quest to change distances in a plane, we shall note that the width of the paths tend to contract in plane, as they bend in the third direction to maximize the volume. The length of the paths remains however constant. We will then use this anisotropic contraction in chapter 6 to distort the average in-plane metric of 2D structures made of two superimposed flat sheets, by means of a specific patterns of locally parallel seam lines, perpendicular to which the contraction will occur.

Wrinkling

The wrinkling instability will be at the core of this second part of the thesis and we will now present its physics in a simple way. To that end, let us consider a ruler, glued to a thick soft elastic substrate and then put under compression. This system has been historically a problem of high practical interest for the engineers in the XIX th century: it is indeed the simplest model of railways tracks we may think of : a series of works analysed thus the stresses and deflections in rails [START_REF] Winkler | Die Lehre von der Elasticitaet und Festigkeit: mit besonderer Rücksicht auf ihre Anwendung in der Technik für polytechnische Schulen, Bauakademien, Ingenieue, Maschinenbauer, Architecten, etc[END_REF][START_REF] Biot | Bending of an infinite beam on an elastic foundation[END_REF][START_REF] Timoshenko | History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures[END_REF] modelling them has beams grounded to a flexible support. More recently, wrinkling instability and its behaviour far from the threshold has been intensively studied to understand pattern formation in biological tissue or the wrapping of structures with thin sheets [?, [START_REF] Amar | Growth and instability in elastic tissues[END_REF][START_REF] Huang | Nonlinear analyses of wrinkles in a film bonded to a compliant substrate[END_REF][START_REF] Audoly | Buckling of a stiff film bound to a compliant substrate?part i:: Formulation, linear stability of cylindrical patterns, secondary bifurcations[END_REF][START_REF] Hure | Stamping and wrinkling of elastic plates[END_REF].

Coming back to our problem, the ruler cannot freely bend anymore upon compression since it means deforming the substrate. In order to simplify this system, we consider that the ruler is much stiffer than the substrate and that it is very thin, such that bending will be the dominant deformation mode, and the ruler may be considered as quasi-inextensible. The ruler will thus bend to accommodate the compression ε 0 L with a characteristic amplitude A and wavelength λ (see Fig. 4.9). The mean slope A/λ is set by the amount of compression (ε 0 ∼ (A/λ) 2 ). The bending energy may be computed in a similar way as in the last section and reads U bend ∼ Eh 3 κ 2 wL, where κ is the typical curvature of the Figure 4.9: Wrinkling of a thin elastic sheet anchored on a soft elastic substrate. When the structure is set under compression, the thin sheet buckles out of plane. As it is attached to the substrate, the buckling induces stretching in the substrate that prevents large out of plane deformations. Smaller deformations however implies larger curvature and thus higher bending energy for the thin elastic sheet. The wave length chosen is thus a compromise between both energies. ruler. It may be estimated easily as the second derivative of the deflection, and scales thus as A/λ 2 . Putting everything together we get the following expression for the bending energy: U bend ∼ Eh 3 wLε 0 /λ 2 . From the bending point of view, the larger the wavelength, the better it is energetically, as in the case of pure simple bending. A large wavelength means however large amplitude A which results in large strains in the substrate, at the cost of a strong stretching energy U sub that we aim at estimating. For the sake of simplicity, we model the substrate as a continuously distributed set of springs, whose stiffness is defined by a "modulus of the foundation" K. The stretching energy classically scales as: U sub ∼ KA 2 wL or expressing it as a function of λ : U sub ∼ Kλ 2 ε 0 wL. As predicted, the stretching energy of the foundation increases with the amplitude of the oscillations, and thus favors a configuration of vanishing wavelength λ, This would mean a diverging curvature resulting in an infinite bending energy which is not acceptable. In order to solve this problem, we cannot, as in free buckling, find asymptotic regimes where one energy may be neglected with respect to the other. We need to construct our solution by balancing both energies. Doing so, we get, for the typical wavelength:

λ ∼ Eh 3 /K (1/4) (4.14)
And for the typical amplitude:

A ∼ √ ε 0 λ (4.15)
Remarkably, the state of compression ε 0 does not impact the wavelength selected by the elastic ruler. A slightly different but analogous system, that will be the one of interest in this thesis, is an elastic sheet, not bounded to any substrate, but subjected to a tension T perpendicular to the direction of imposed compression. Considering the elastic sheet as inextensible, the bending energy term does not change, but a vertical displacement of the sheet of an amplitude A corresponds to a displacement δ of the order of A 2 /w in the direction of the tension, resulting in a work W ∼ T δ which has the same quadratic dependence on the amplitude A as the substrate energy in the last configuration. For a very thin sheet set under high tension, the wrinkling wave length and amplitude will thus be very small and one may neglect this decoration on the surface and consider the sheet as a flat homogenised sheet with a new apparent length. In this chapter, we experimentally and theoretically investigate the shapes of initially flat straight ribbons, annuli, and more generally, any curved curvilinear path, like typically the mylar balloons letters or numbers that are popular at surprise parties (Fig. 5.1). These balloons exhibit non-intuitive features that we wish to rationalize. When comparing the outline of the flat piece (Figure5.1(a) left) and that of the inflated structure (Figure 5.1(a) right), one notices that inflation tends to over-curve the outline, which now reaches self-contact at its end. It may seem contrary to intuition that inflation of a featureless elongated structure leads to overcurvature. One may also note the characteristic and puzzling wrinkling pattern that appears on the envelope (Fig. 5.1(b)). Wrinkles are radial, along the geodesics linking the inner and outer boundaries, indicating an excess of material that could be consistent with the compression induced by curvature of the outline (compression on the inner part and tension in the outer part, as in classical bending). However several observations contradict this naive interpretation : first it is easy to see that the inner outline is under strong tension, and secondly, if wrinkles are absent of a band between the inner and outer boundaries, the most outer region of the curved inflated structure is again subject to wrinkling. In this chapter we aim at understanding the inflation-induced coiling of the outline, the shape of the cross sections, and the location of radial wrinkles.

Chapter 5

Inflating single paths

Straight ribbons

Let us first consider a mylar balloon made of two superimposed rectangular sheets of length L and width w -with L w -sealed together along their boundary (see Fig. 5.2). Upon inflation, the sheets bend and form a tube, which maximizes the volume given the inextensibility constraint. As L is much larger than w, the structure bends along its width, whereas the length remains straigth. Far from the extremities of the ribbon, the problem thus reduces to the maximisation of the cross-section area given a fixed perimeter P = 2w. This is the famous Dido problem : Dido is the legendary founder and first queen of Carthage in Tunisia. When she arrived after many adventures on the coast of Tunisia, she asked the local Berber king Iarbas for a small parcell for a temporary refuge until she could continue her journeying, only as much land as could be encompassed by an oxhide. He agreed. Dido cleverly cut the oxhide into the finest strips she could and then encircled an entire hill. This land became Carthage. The maximum area that can be enclosed in a given perimeter is of course a circle and may be solved using classical variational methods. The cross section of the inflated ribbon is thus circular, as expected by our intuition. The in-plane lengths remain thus unchanged along the ribbon and the width of the inflated tube is now 2/πw

CURVED PATHS

Curved paths

We now generalize the problem of tube inflation to a curved ribbon. The system becomes much more subtle, as we shall now see.

Closed rings

Let us first consider two identical superimposed flat annuli, sealed along their edges, of inner radius R and outer radius R+w (see Fig. 5.4). We are interested here in the resulting inflated overall shape (and not on the wrinkles morphology decorating the surface). This structure is assumed to remain axisymmetric upon inflation, and typically adopts a donut shape, like swim rings. The surface thus evolves from a flat metric to a non-Euclidean one, with positive (respectively negative) Gaussian curvature in the inner (respectively outer) part of the donut. As the material is nearly inextensible, the change of metric associated with the shape change can only be accommodated by the compression of the overall sheet, i.e. by the presence of wrinkles.

We first assume that the structure remains symmetric with respect to the e r axis upon inflation. The section is described by s, the curvilinear coordinate along a meridian, z(s) the vertical coordinate, R+r(s) the radial coordinate, and ϕ the angle of the tangent to the meridian line with respect to e r (see Fig. 5.4). The angle α measures the longitude. What is the shape of an inflated ring? To address this question, two methods may be followed: maximizing the volume given the inextensibility constraint [START_REF] Ligaro | Equilibrium shapes of inflated inextensible membranes[END_REF][START_REF] Pak | Inflating polyhedral surfaces[END_REF][START_REF] Barsotti | Approximated solutions for axisymmetric wrinkled inflated membranes[END_REF][START_REF] Pak | Profiles of inflated surfaces[END_REF] or balancing the forces applied on the membrane [START_REF] Taylor | On the shapes of parachutes[END_REF][START_REF] Deng | Clefted equilibrium shapes of superpressure balloon structures[END_REF]. We will present both approaches in the following. In order to get some intuition about the shape selected by energy -pV minimization

CHAPTER 5. INFLATING SINGLE PATHS

CURVED PATHS

(and thus volume maximization), we note that the volume may be computed as the product of the area of the cross section times the mean radius, the mean radius being the distance from the center of symmetry to the centroid of the cross section (V = 2πR mean ×Area, see Fig. 5.5). When the deflated ring is highly slender (R/w 1) the mean radius R mean ∈ [R, R(1 + w/R)] is nearly independent of the cross section. The volume optimization thus reduces to the maximization of the cross section. However, for non slender rings, when R/w → 0 (R mean ∈ [0, w]), the position of the centroid is crucial for the volume optimization problem: the system may pay a loss in the area of the cross section in order to push the centroid away from the axis of revolution, leading to an asymmetric cross section shape (see Fig. 5.5(c)).

Let us now look more mathematically into this problem. In the framework presented above, the volume V of the toroidal shape obtained by rotational symmetry reads

V = 4π w 0 [R + r(s)]z(s) cos ϕ ds (5.1)
under inextensibility condition.

The inextensibility in the radial direction is simply ensured by the boundaries of the integral (because we assume that radial lines are not wrinkled). In the azimuthal direction this condition imposes the following inequality ∀s, P (s) ≤ P 0 (s),

where we have defined the apparent perimeter P (s) = 2π(R + r(s)) of the circle passing through a point s of the section, and P 0 (s) = 2π(R + s) its initial perimeter. Here this continuous inequality can be greatly simplified in the following way: radial inextensibility trivially imposes that ∀s, r (s) = cos φ ≤ 1, and thus P (s) ≤ P 0 (s). Therefore equation 5.2 is satisfied if and only if P (0) ≤ P 0 (0), or equivalently if and only if r(0) ≤ 0.

An optimal solution must equalize the inequality constraint for at least one curvilinear coordinate, and from the previous equation it must be at s = 0. The inextensibility condition 5.2 thus reduces to the boundary condition:

r(0) = 0 (5.3)
Nondimensionalizing by w, the Lagrangian for the optimization problem may be written in terms of dimensionless variables and parameters denoted by * :

L = 4π 1 0 (R * + r * )z * cos ϕ + A(cos ϕ -r * ) + B(sin ϕ -z * ) ds * , (5.4)
where ( ) stands for derivative with respect to s * . z, r and ϕ are chosen to be independent variables of the minimization problem. A and B are two Lagrange multipliers enforcing the geometrical relations (r * = cos ϕ and z * = sin ϕ). Using classical variational methods,

CHAPTER 5. INFLATING SINGLE PATHS

we get the following system of equations:

               A = -z * cos ϕ B = -(R * + r * ) cos ϕ r * = cos ϕ z * = sin ϕ [A + z * (R * + r * )] sin ϕ = B cos ϕ (5.5)
together with the boundary conditions A(1) = 0 because r * (0) = 0 is fixed as commented above (equation [5.3]), whereas r * (1) is free. Symmetry also imposes the boundary conditions z * (0) = z * (1) = 0 . We do have the four needed boundary conditions to solve the system of four first order ODEs:

           A(1) = 0 r * (0) = 0 z * (0) = 0 z * (1) = 0 (5.6)

Alternative method: Derivation of the shape using force balance

To compute the shape of the inflated structure, we may also follow the pioneer reasoning of Sir G. I. Taylor in 1919 who computed the optimal shape of parachutes [START_REF] Taylor | On the shapes of parachutes[END_REF] from a force balance. Parachutes should be indeed as light as possible : wrinkles, which mean an excess of material, should therefore be avoided. Taylor computed the flattest possible shape for a parachute (as shown in Fig. 5.6), thus saving the most material, since the drag force is proportional to the projected area of the parachute. The hoop stress should therefore be zero, just at the onset of wrinkles, assuming that the pressure inside the parachute is constant. It is equivalent to the shape that would adopt a mylar balloon made of two flat circular disks sealed together on their edges as the thin sheet accommodates any compression at zero cost by forming wrinkles, the overall shape should therefore be the same (overseeing the wrinkles decoration).

CURVED PATHS

In the deformed state, one can consider an infinitesimal surface dS (Fig. 5.6(b)), which is subjected to forces at equilibrium: radial and hoop tensions per unit length in the inflated wrinkled configuration, denoted by T ϕ and T α respectively and the normal pressure p. From our experimental observations (radial wrinkles may be observed upon inflation), we assume that there is no tension in the hoop direction : T α = 0 Force balance in the radial direction gives

∂((R + r)T ϕ ) ∂ϕ = 0 (5.7)
which leads, after integration, to:

T ϕ = T 0 R + r (5.8)
where T 0 is a constant, the tension per unit angle dα, that will be determined by the boundary conditions, independent of α by symmetry and of r by force balance in the radial direction. Following Laplace law (force balance in the normal direction), with T α = 0, we get: p = κT ϕ (5.9)

where κ = -dϕ/ds is the local curvature in the radial direction. Combining, the two last equations in order to eliminate T ϕ , we obtain that the curvature κ varies linearly with the distance r to the axis of symmetry.

dϕ ds = - p T 0 (R + r) (5.10)
Differentiating with respect to s and recalling that dr/ds = cos ϕ, we retrieve the classical non linear oscillator ODE:

d 2 ϕ ds 2 = - p T 0 cos ϕ (5.11)
The boundary conditions found in the last section may also be expressed as function of ϕ(s). The condition z(0) = z(w) = 0 simply imposes that w 0 sin ϕds = 0, and the condition r(0) = 0 inserted in 5.9 leads to ϕ (0) = -pR/T 0 (5.12)

Finally, evaluating the last equation in the set of equation 5.5 at s = w, and knowing that B = T 0 /p sin ϕ, z * (1) = 0 and A(1) = 0, we obtain that

ϕ(w) = -π/2. (5.13)
It is interesting to show that the equations derived with both methods are equivalent. Differentiating the last equation in the set of equation 5.5 and using the other relations to simplify, one gets the equation:

dϕ ds = - (R + r) sin ϕ B (5.14)

CHAPTER 5. INFLATING SINGLE PATHS

It can easily be shown by direct differentiation that the quantity sin ϕ/B is a constant (using again [5.5] and [5.9]). Hence equation [5.9] shows that dϕ/ds is strictly proportional to (R + r), and we have recovered equation 5.10 where the constant sin ϕ/B corresponds in physical terms to p/T 0 .

We may also interpret Lagrange parameters A and B that appear in the variational equations by separating an angular sector of the system with an imaginary cylindrical cut of radius R + r 0 and considering the force per unit length exerted by the portion r > r 0 on the part r < r 0 . A and B are simply the horizontal and vertical projection of this force per unite length non-dimensionalized by pw. B = T 0 /p sin ϕ, (5.15) is the vertical projection of the dimensionless membrane tension per unit angular sector. The total dimensionless horizontal force

A = T 0 /p cos ϕ -z * (5.16)
also includes a contribution of the pressure on the section in addition to the projected membrane tension.

Experimental and theoretical cross-sections

We can solve either boundary value problems using Matlab function bvp4c varying the only dimensionless parameter of the system, the ratio of the inner over the outer radius S = R * /(R * + 1), that we call the slenderness ratio. To compare with actual experiments, Figure 5.7: Schematic of the set up: a soldering iron is mounted on an XY-plotter: the desired sealed path is simply printed on two superimposed flat heat-sealable sheets.

CURVED PATHS

we fabricate the ring balloons by laying two thin sheets made of the same heat-sealable material (TPU (thermoplasticurethane) impregnated nylon fabric, mylar or polypropylene), covered by a sheet of greaseproof paper to physically isolate the melting plastic from the heatig head, in the working area of an XY-plotter. A soldering iron with controllable temperature (PU81 from Weller) is then mounted on the tracing head of the plotter (Fig. 5.7). We then "print" the desired path designed with any vector graphics software. Playing with the temperature, the pressure and the displacement speed of the head, one can find the right set of parameters to seal effectively along the path. The envelopes obtained are connected to the compressed air supply of the laboratory and inflated. The pressure is set at typically 0.1 bar, to ensure that we remain in the regime of interest (quasi-inextensible, compression modulus negligible) for our decimetric structures, of thickness t of typically 10µm and of Young modulus E of the order of the GPa. Cross sections are measured by drawing a radial line on a transparent version of the balloon, a photograph from the side is then taken and the line extracted (see Fig. 5.8(a)). As shown in Fig. 5.8(b), the theoretical predictions (solid lines) are in remarkable quantitative agreement with the experimental measurements (triangles) without any adjustable parameter. For slender rings, that is when S → 1, the cross section tends as expected to the trivial cross section of a straight inflated path, a circle. However, when S → 0, the cross section strongly deviates from a circle and a singularity appears at the inner point s * = 0. The asymmetric cross section enables to increase the mean radius (at the cost of a smaller cross section area) and thus the volume, as explained in the introduction.

The toroidal structure is, as predicted by the geometrical model, decorated with alternating wrinkles and crumples [START_REF] King | Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities[END_REF][START_REF] Paulsen | Geometry-driven folding of a floating annular sheet[END_REF], everywhere except at the inner edge of the structure, as shown in Fig. 5.8(a). All material points, by going in the third direction to maximize the volume, have indeed a net radial displacement towards the axis of symmetry of the structure, leading to hoop compression.

Experimental realizations shown in Fig. 5.9 exhibit nonetheless surprising features. Contrary to our axisymmetric hypothesis, slender rings buckle out of plane, forming a smooth outline with fabric envelope or displaying localized kinks with polypropylene sheets. In Fig. 5.8(a), we prevented this instability using confining plates. This instability is the consequence of geometric frustration of the inflated ring, reminiscent of the warping of curved folds [START_REF] Dias | Geometric mechanics of curved crease origami[END_REF][START_REF] Dias | The shape and mechanics of curved-fold origami structures[END_REF] or the buckling of elastic growing rods with incompatible intrinsic curvature [START_REF] Moulton | Morphoelastic rods. part i: A single growing elastic rod[END_REF] (see Fig. 5.10) . In order to release this frustration, we cut the ring radially and seal the edges (Fig. 5.11), thus removing the closing condition. We observe a significant coiling of the structure upon activation, that we now aim at rationalising.

Inflating open rings

First approach

With the additional degree of freedom of open rings, we observe that the structures remain in-plane, but the curvature of their outline increases, which results into an over- is entirely supported by the inner seam, as all others points of the membrane are under hoop compression and bear no azimuthal load. On a single cross section, the pressure force induces a residual torque with respect to the inner seam. For an open ring, having a free end and no external loading imposes a vanishing internal torque in any cross section of the structure. The initially unbalanced pressure torque induces an increase of the curvature of the structure until two symmetric lines of tension appear and provide internal torque balance (Fig. 5.12(c)). Counter-intuitively, pressurizing curved structures increases their curvature.

As a first attempt to describe quantitatively this phenomenon, we consider an open annulus of inner radius R and width w in the flat configuration. The overlap results into a new inner radius R 1 = R/λ, with the strain λ = α/α 0 (see Fig. 5.12(a)). We assume that, far from the ends of the open annuli, the family of profiles calculated for closed rings remains valid. However, the current shape profile r 1 (s) corresponds to the new aspect ratio R * 1 = R 1 /w. The local projected perimeter of the structure at the curvilinear coordinate s is thus equal to P λ (s) = 2πλ[R 1 + r 1 (s)]. Due to inextensibility condition, this perimeter is shorter than in the flat configuration P λ (s) ≤ P(s) = 2π(R + s). In Fig. 5.12(b), we represent the normalized difference as a function of the dimensionless abscissa s * , imposing λ for the case R * 1 (circular cross sections). As described previously, u * is always negative for λ = 1, that is, all material points are under azimuthal compression except for the inner point s * = 0 (Fig. 5.12(b)). As λ is increased, the curve u * (s * ) presents a secondary maximum which increases. This maximum eventually reaches 0 at a position s * tens for a particular value λ tens (Fig. 5.12(b)). The corresponding perimeter is not under compression anymore. Beyond this point, u * is partly positive, which breaks the inextensibility condition. As the open structure is inflated, we thus expect λ to take the value λ tens , for which mechanical equilibrium is satisfied with two additional up-down symmetric lines of tension along the membrane. Although in Eq. (5.17) the profile r * 1 (s * ) depends, in principle, on λ, we assume here that this dependence remains modest. If we approximate this profile as the closed axisymmetric

u * = [P λ (s) -P(s)]/2πw = λr * 1 (s * ) -s * (5.

CURVED PATHS

profile before additional curving, that is r * 1 (s * ) r * (s * ) in Eq.5.17, the position of the line under orthoradial tension can be readily determined. Searching for the condition when the maximum of u * vanishes leads to:

1 λ = max s * r * (s * ) s * (5.18)
As an illustration, this value can be directly computed in the limit R w, where the section is almost circular and the profile follows r * = 1 π [1 -cos(πs * )]. Searching for the maximum of the function r * /s * leads to the transcendental equation πs * sin(πs * ) = 1-cos(πs * ). The numerical solution gives s * tens 0.74 and consequently λ 1.38, i.e. ∆α tens 137 • . The curvature varies accordingly from 1/R to λ/R 1.38/R. In Fig. 5.12(c), we compare the experimental measurement of λ conducted with polymer sheets with the theoretical predictions from equation (5.18), and find a very good agreement with experimental data for S > 0.5. The predicted position for the region under tension (red crosses in Fig. 5.12(d)) also matches the observed region free from wrinkles. Nevertheless, this region is actually not limited to a line but presents a finite width and the coiling prediction is poor for thick rings (R < w). We may interpret this difference as a consequence of the finite stiffness of the sheet, as described in a seminal paper by King et al. [START_REF] King | Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities[END_REF] in a simpler geometry. But most importantly, we made the crude assumption that the families of shapes in the open and closed ring configurations are the same. That is, that the azimuthal inextensibility condition is met for one single material point, whereas experiments suggest, as shown in Fig. 5.12(d), that a whole band is under tension.

A more subtle optimization

The model must be slightly updated in order to predict quantitatively the coiling factor for rings of small slenderness (S 1) and to understand the unwrinkled band on the balloon. Due to the coiling of the structure, the azimuthal condition of inextensibility is no longer satisfied if and only if it is satisfied at the inner radius (see equation 5.3). In fact, the so-called perimeter now reads P(s) = 2πλ(R/λ+r(s)) and might exceed the rest perimeter P 0 (s) = 2π(R + s). We should thus now enforce the inequality ∀s ∈ [0, w], u * (s) < 0 at every point. Nevertheless, the condition u(0) = 0 still holds, for the same reason as in the closed ring configuration. To take into account the azimuthal inextensibility condition, we add in our optimization a soft inextensibility condition, exp βu , where β is a large numeric number. This means that the energy that we want to minimize has a new regularization term that strongly penalizes any azimuthal extension (i.e. any u * > 0) The Lagrangian now reads:

L = 4π 1 0 R * + s * + u * λ z * cos ϕ + A(cos ϕ - 1 + u * λ ) + B(sin ϕ -z * ) -exp βu * ds * (5.19) CHAPTER 5. INFLATING SINGLE PATHS
where ( ) stands for derivative with respect to s * . We get the following system of equations: This set of equations is numerically solved (using the function bvp4c from Matlab) for varying values of λ, as presented in Fig. 5.13 for S = 0.1. For sufficiently large values of β, the results are of course independent of the numerical extensibility penalization β. The profiles (Fig. 5.13(a)) does not significantly evolve with the increasing coiling factor λ until compression at a point s * > 0 vanishes (i.e u * (s * ) = 0 as shown in Fig. 5.13(b)). A flat tilted region under tension appears and grows, forming an increasing angle γ = arccos (1/λ) with the horizontal axis. The same angle emerges when a flat cut circular ring of paper is over-curved. Each solution maximizes the volume for a given imposed coiling factor λ. Plotting the evolution of the volume as a function of the coiling factor, a maximum appears at a given λ max . Upon inflation, the structure naturally selects this optimal solution and coils by this predicted amount. Predicted cross sections profiles for varying slenderness S are plotted in solid lines in Fig. 5.14. The model quantitatively reproduces the experimental cross sections with good precision, and even the small discrepancies between open rings of varying slenderness. The cross sections in open configuration significantly deviate from the ones in close configuration (Fig. 5.15(a)), specifically for thick rings. They appear to be much closer to the circle for any slenderness (see Fig. 5.14). In the case of thick open annuli (S → 0), the inner radius R is small compared to the width w. The cross section of the structure near both ends of the structure is affected by their presence, and, in the case of thick annuli, boundary effects may propagate through the entire structure, since such structures are not slender. In order to make sure that boundary effects do not influence the experimental measurements, flat rigid panels of shapes close to the theoretically predicted cross section are sealed on both ends of the open ring, as shown in Fig. 5.15(b).

                   A = -z * cos ϕ B = -u * cos ϕ -1 r * = cos ϕ + β exp βu * z * = sin ϕ A + z * R * + s * + u * λ sin ϕ = B cos ϕ

Wrinkles

The coiling factor λ predicted by the new optimization problem is plotted against the slenderness of the open ring and compared to experimental data in Fig. 5.16: it varies monotonically from 1.28 to 1.38. In contrast with our previous approach, the new optimization with an extensibility regularization term exhibits a quantitative match with the experiments over the whole spectrum of slenderness. For annuli with S 1, the optimal volume is found for an extended uncompressed region, thus impacting significantly the cross section shape. For slender annuli, the previous approach is still valid.

Commercially available mylar balloon letters are empirically designed to compensate for this over-curvature. For example the letter "O" has , before inflation, a missing angular

CURVED PATHS

sector, and rather looks like a "C" [START_REF] Pak | Inflating polyhedral surfaces[END_REF] (see Fig. 5.17(a). It is not made out of a flat annulus since it would buckle out of plane, as shown earlier. The function u * is a measure of the local azimuthal compression, i.e. of the presence (or absence) of wrinkles. Wrinkles appear orthogonal to the main compressive direction [?, [START_REF] King | Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities[END_REF][START_REF] Aharoni | The smectic order of wrinkles[END_REF]. In our case, wrinkles are indeed radial. The experimental extent of the unwrinkled band should correspond to the region where the function u * is found to be close to zero. Figure 5.17(b) shows a remarkable agreement between both features. However, a few details of these inflated curved paths remain misunderstood: the alternation of smooth wrinkles and sharp crumples is not captured by our model (see Fig. 5.17). We did not investigate neither the selected wrinkle wavelength nor the impact of curvature on those. Numerous recent studies have tackled these challenges in slightly different configurations [START_REF] Paulsen | Geometry-driven folding of a floating annular sheet[END_REF][START_REF] Paulsen | Wrapping liquids, solids, and gases in thin sheets[END_REF]. We shall now look at these structures from a more global point of view and program the outline shape.

Inverse problem

Having rationalized the change in curvature upon inflation, we propose to use our geometrical model to inverse the problem, i.e. to determine the path of width w leading to an inflated structure of arbitrary desired 2D shape. Let us start with a smooth target 5.2. CURVED PATHS path after inflation, a lemniscate (Fig. 5.18 (a)). We first numerically calculate the target curvature κ tar (v * ), where v * denotes the curvilinear coordinate along the path to be programmed normalized by the width w. The same parametrization may be used in the flat state since the inner edge does not stretch nor contract upon inflation and the tube is chosen slender (κ tar 1). The normalized radius of curvature R * (v * ) of the corresponding flat ribbon is then obtained by numerically solving the relationship λ(R * ) R * = κ tar .

(5.21)

where the function λ(R * ) was computed above and plotted in Fig. 5.16. In Fig. 5.18(b), we show the computed curvature 1/R * and the target curvature of a lemniscate κ tar as a function of the curvilinear coordinate v * . The curvature in the flat state 1/R * is typically smaller than the target curvature κ tar , by a factor λ varying between 1.28 and 1.38 depending on the target curvature. This relationship is rigorously valid only in the case of slowly varying curvatures (dκ tar /dv * 1), i.e. when the outline of the path may be locally seen as a path of constant curvature. However, the fact that the cross sections of open rings are barely dependent of the normalized radius of curvature R * (as shown in Fig. 5.14) implies that the inverse programming should not be too sensitive of sharp variations in the target curvature κ tar , as we shall show experimentally.

The contours of the balloons are then plotted with the correct curvature κ(v) = 1/(wR * (v)) (Fig. 5.18(c)). Upon inflation, we do obtain with great precision the target lemniscate shape (Fig. 5.18(d)).

The same protocol may be expanded to arbitrary hand written 2D lines (Fig. 5.19(a)&(e)), using the program freehanddraw from Brett Shoelson on Matlab. Despite the fact that the numerical curvatures are very noisy (Fig. 5.19(b)&(f)), the inflated structures closely fit the target paths (Fig. 5.19(d)&(h)) If overlap occurs, as in the case of the "Hello" curve (Fig. 5.19(c)), the path is printed in several non overlapping distinct parts, that are bonded together using tape. Depending on the initial curves, the inflated structures may expand ("Hello") or, conversely contract (waving man). This offers a path for a new kind of strong lightweight actuators with programmable shapes. Harnessing geometrical non-linearities, one can predict the complex deformation path to displace objects with mere pressure input. In Fig. 5.19, the octopus-like arm lifts a mug weighting several times its own weight, and carries it to a platform behind an obstacle. Large workload, with particularly large stroke may thus be reached with a very simple object.

Miscellaneous

In this short conclusive section, we list some observations on these curved inflated paths and present the idea that will be the core of the last chapter of this thesis, i.e. shape morphing of welded textiles structures.

CHAPTER 5. INFLATING SINGLE PATHS

Sharp path

We also experimentally investigate the shape of inflated sharp corners. It corresponds to the limit when R/(R + w) → 0 and we observe an angle change λ close to 1.28, that is the limit observed experimentally for open rings. More surprisingly, it appears to be independent of the outer path geometry (either sharp our rounded). Nevertheless, as predicted by our model, these structures are prone to failure because of the stress concentration at the inner sharp angle.

CURVED PATHS

Swim rings

The following question naturally arises: why do inflated swim rings remain in plane and do not buckle when inflated (Fig 5 .21(a))? To answer this existential question, I bought a simple swim ring and realized that at rest, the ring is not flat, but exhibits a strong radial slope near the inner radius R (Fig. 5.21(b)). The azimuthal lengths at rest and upon inflation, extrapolated from the profile lines in red, are thus very close up to a region near the outer radius, where the inflated perimeter becomes significantly smaller than the one at rest : this is the reason why wrinkles appear solely at the outer edge (see Fig 5 .21(a)). It is however highly unlikely, that these rest shapes are precisely manufactured. Indeed, inflating flat samples with high enough pressures leads to the same result. The inner seam, under diverging hoop stress in our quasi-inextensible model, stretches. Using a plastic material (in our case polyethylene), it is thus possible to fabricate axisymmetric swim rings with one initial large pressure load, in order to plastically deform the initially flat rings (see Fig. 5.22). The behaviour of hyperelastic toroidal structures was studied in recent studies [START_REF] Roychowdhury | Inflating a flat toroidal membrane[END_REF][START_REF] Roychowdhury | Symmetry breaking during inflation of a toroidal membrane[END_REF] and could be adapted in our case to quantify precisely for which typical pressure the structure returns in-plane.

Conclusion

In this chapter we have shown that the physics and geometry of apparently mundane flat sealed inflatables such as "mylar balloons" are far richer than expected: the shape of their section includes singularities and a non-trivial distribution of wrinkles; the outline of an inflated curved balloon with free ends overcurves under inflation. Our model based on the assumption of perfectly inextensible and infinitely bendable membranes does capture quantitatively this coiling for any slenderness S = R/(R + w), as well as the shape of the cross sections and the positions of wrinkles. A remaining challenge is to rationalize the mechanical properties of such structures: how does the complex stress pattern revealed by regular folds and wrinkles impact the bending stiffness of the inflated device [START_REF] Vandeparre | Wrinkling hierarchy in constrained thin sheets from suspended graphene to curtains[END_REF][START_REF] Vella | Wrinkling reveals a new isometry of pressurized elastic shells[END_REF][START_REF] Géminard | Wrinkle formations in axisymmetrically stretched membranes[END_REF]? This mechanical question will be crucial in the next chapter of this thesis, in which we present the 2-dimensional counterpart of these quasi-1-dimensional structures. In this chapter, we propose a simple method to reversibly program a large diversity of stiff and thus potentially large scale 3D shapes. Two flat superimposed sheets (typically made of thermoplastic coated fabric) are sealed together along a specific pattern. Upon inflation between the two layers, volume maximisation in this constrained structure leads to controlled in-plane contraction through the local bending of the sheets. In contrast with the manual folding of origami, the structure morphs spontaneously onto a target 3D shape. We will thus study the link between the drawing of seam lines in 2D and the actual 3D geometry on inflation. The resulting object has typically the same rigidity as strong inflated structures, such as playground castles, balloons [START_REF] Pagitz | The future of scientific ballooning[END_REF], fabric air beams [START_REF] Wielgosz | Deflections of inflatable fabric panels at high pressure[END_REF] and can sustain its weight at the meter scale or more. We shall study the stretching and bending rigidity of such structures in section 6.7. In contrast with common inflatables, they are however not made of a complex assembly of flat patches stitched together which approximate by parts the target metric of the inflated structure [START_REF] Skouras | Designing inflatable structures[END_REF], but on two simply superimposed fabric sheets. The key advantage of such structures is therefore the ease of fabrication and their flat metric at rest. This study has been a great opportunity for us to collaborate with theoreticians: the results presented here bear their mark and style. The spiral patterns (Section 6.4) and the non-isometric origami (Section 6.5) have been done with Mark Warner (Cavendish Laboratory, Cambridge University) : he was giving a seminar at ESPCI about liquid crystal elastomers (LCE) at the end of 2018 and we were stricken by the connections between LCE and the inflatables that we were developing. It was for us a way to discover the LCE literature and our discussions were very inspiring and fruitful. The "zigzag" pattern strategy (Section ??) is now part of a collaboration with Antonio DeSimone (MathLab, SISSA), and this strategy is deepened by Tian Gao, a PhD student in the group. Lastly, the general inverse problem (Section 6.8) is the work of Julian Panetta and Mark Pauly (Laboratory Geometry and Graphics, EPFL), bringing their impressive computer graphics skills in the game.

FABRICATION

Fabrication

Fabrication of these flat inflatables structures is straightforward and simple, since the manufacturing process is completely 2D. We present here briefly the different steps and strategies investigated during this PhD.

Material

Any thermoplastic material may be potentially used, including among others Mylar, polypropylene, polyethylene. Nevertheless, nylon textiles coated with thermoplasticurethane (TPU) matrix are best suited for the material choice. They indeed present the advantage to be easily and strongly sealed at relatively low temperature (around 200 • C). At such temperatures, only the TPU matrix melts and not the nylon textile, thus avoiding unwanted piercing of the sheet while welding. They are also stiff in the direction of nylon fibers (average Young's modulus of typically 1 GPa) while permitting some shear [START_REF] Guyon | Du merveilleux caché dans le quotidien, la physique de l'élégance[END_REF]. This slight shear prevents the apparition of kinks and localization in the structure, as we shall later see.

Welding

The inflatable structure is made primarly of two superimposed thermoplastic sheets, that should be sealed together along a specific pattern. Various simple techniques may be used to weld the two sheets.

Heat printing A soldering iron may be mounted on an XY-plotter or a CNC-machine to directly "print" the heat-sealed seams pattern on the two superimposed sheets (Fig. 6.2(a)).

During my PhD project, we used the XY-plotter from Makeblock and a CNC Workbee from Openbuilds. The temperature, pressure and displacement velocity of the heating head are tuned in order to obtain a strong bonding between the two layers. These technical settings are however sensitive to the thickness and the material of the sheets. The rationale behind the choice of parameters is the following: the temperature should be high enough to melt the TPU, the speed should be slow enough to ensure the heat diffusion through the thickness of the sheet and the pressure should be high enough to make a strong bonding (but too much pressure may damage the sheets).

If the sheet is coated on both sides with thermoplastic material, a sheet of baking paper should be placed on top of the structure during the welding in order to protect the structure and to avoid the adhesion of the melted material on the printing head.

Heat press

Alternatively, the pattern of air channels may be laser cut in baking paper so that the designed seam lines are cut out from the baking paper sheet. The resulting pattern is then placed between both thermoplastic fabric sheets; the whole sandwich is then heat-pressed between two additional baking paper sheets to prevent the adhesion of the melted thermoplastic on the heat press (Fig. 6.2(b)). The two fabric sheets are sealed together only where the baking paper has been cut out. The patterned baking paper remains trapped in the structure but does not play any mechanical role upon inflation.

Heat press mould

The last proposed method is best suited for serial production of the same structure. A metallic plate is engraved except at the target location of the seam lines. It serves as a waffle iron in the heat press, welding only the appropriate lines on the surface. This fabrication technique could be directly used for industrial application, enabling a large and cost-efficient throughput.

Metric change

Upon inflation, the obtained structure tends to maximize its volume, as in the case of curvilinear paths, given the constraints imposed by the seam lines and the inextensibility of the material. Although the direct volume optimisation of the whole structure is complex and involves numerical finite element tools, that have been developed by our collaborators from the Laboratory Graphics and Geometry in Lausanne (see section 6.8), we propose here a simple framework that describes quantitatively the main features. In our approach, the pattern is chosen to be locally made of parallel stripes of width w. The seam lines have a thickness e (Fig. 6.3(a)). Upon inflation, similarly to the curvilinear paths studied in the last chapter, the sheet bends perpendicular to the seam lines to form a quasi-circular cross section for sufficiently large pressure (p Et 3 /w 3 ). Owing to the quasi-inextensibility of the fabric sheets, this cross-section change implies an in-plane contraction perpendicular to the channels, whose direction is denoted by n, of magnitude The unit repeat length has a fraction ξ of weld and 1 -ξ of free sheet. A section before and after inflation. The direction of contraction, n is equivalent to a nematic director in a liquid crystal solid. The contraction of the 1 -ξ fraction is by a factor of 2/π in the ideal case, or by a factor > 2/π if there is length taken up by the bend near the seam.

λ = 2/π (6.1)
in an idealized scenario (Fig. 6.3(c)). Taking into account the thickness e of the seam line which do not contract, one can easily derive a more accurate contraction:

λ = 2/π(1 -ξ) + ξ (6.2)
where ξ = e/w is the relative seam width, in very good agreement with the experimental measurements (Fig. 6.3(b)). The actual width e of the seam line being impossible for us to measure precisely, we use it as a fitting parameter in the curve 6.3(b). The result found is 1.4 mm for the soldering iron head used, which is completely plausible. The finite bend near the welded seam adds to the value found for the experimental width e.

CHAPTER 6. INFLATABLE MAZES

In practice, the experimental actuation ratio typically amounts to 0.7 in our structures, quite larger than the idealized 2/π = 0.63 scenario. In the other direction, along the seam lines, no length change is observed, as the material is nearly inextensible. The direction perpendicular to the seam lines may thus be seen as a director field n for the anisotropic metric distortion of magnitude λ and 1 parallel to the welded lines. The in-plane, 2 × 2 deformation gradient F and the associated metric tensor a = F T • F are:

F = (λ -1)n ⊗ n + δ (6.3) a = (λ 2 -1)n ⊗ n + δ, (6.4)
where δ denotes the identity operator on R 2 .

For the sake of simplicity, we completely disregard in this chapter the dependence of the contraction and of the cross-section on the actual curvature of the seam, and we shall not look at the wrinkling pattern. Moreover, the structures are made of thermoplasticurethane(TPU)-coated textiles and not of thin polypropylene or polyethylene sheets as in the last chapter. The detailed mechanics of fabrics is much more complex, since the sheet is softer in a 45 • -tilted direction than along the fibres. Some shear may thus be induced by the inflation, changing slightly the metric at the level of the sheet. Here, we do not take into account these details and simply assume that the structure experiences contractions of constant magnitude λ perpendicular to the local seam direction, independently from its local curvature.

We can vary the orientation of the seam lines, i.e. the orientation of the director n on the flat 2D surface. If the director makes locally an angle α with respect to an arbitrary reference direction, the metric reads:

a(u, v) = R(α(u, v)) t λ 2 0 0 1 R(α(u, v)) (6.5)
where (u, v) is a parametrization of the plane, R the matrix of rotation and λ the contraction rate perpendicular to the channels. This framework is exactly the same as in liquid crystal elastomers (LCE) [START_REF] Aharoni | Geometry of thin nematic elastomer sheets[END_REF][START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF][START_REF] Mostajeran | Encoding gaussian curvature in glassy and elastomeric liquid crystal solids[END_REF][START_REF] Modes | The activated morphology of grain boundaries in nematic solid sheets[END_REF] with a Poisson ratio ν = 0. However, in contrast to nematic elastomers, the contraction rate, λ may be varied in the range [2/π, 1] by changing the relative thickness of the seam line ξ = e/w, at the cost of structural stiffness. This method is also reminiscent of curved crease origami [START_REF] Dias | Geometric mechanics of curved crease origami[END_REF][START_REF] Callens | From flat sheets to curved geometries: Origami and kirigami approaches[END_REF], but the fabric sheet is bent and not folded along curved lines.

Azimuthal and radial contraction: cones and anti-cones

The simplest examples of pneumatic Gaussian curvature (GC) generation are those of azimuthal and radial channels (radial and azimuthal directors respectively). Figure 6.4 shows realisations of these examples, with the director n superimposed, before and after inflation. Let us consider first a simple Archimedean spiral pattern (Fig. 6.4(a)). Upon inflation, the radial distances are contracted by λ, whereas azimuthal distances remain unchanged. The structure presents thus an angular surplus of 2π(λ -1 -1) and buckles into an anti-cone. This additional angular sector may be easily illustrated by cutting the circular structure along the radius and sealing the new edges (Fig. 6.4(c)). The structure remains flat and coils when inflated, the angular surplus matches the theoretical prediction. This coiling, although reminiscent of the overcurvature of curved path seen in the last chapter is however of different nature. Here, the structure coils because of the average metric in the inflated plate. All seam lines are under tension, being at the same time at the inner edge of one channel and at the outer edge of the next channel, in contrast with single inflated curved path, where the outer edge was under compression. This difference yields a larger overcurvature in the present case.

In the same manner, a purely radial pattern induces a perimeter contraction of amplitude λ, whereas radial distances remain constant. The structure has thus a missing angular sector of 2π(1 -λ) and buckles onto a cone of semi-angle at the apex ϕ = arcsin λ (Fig. 6.4(b)). Cutting again along the radius, this angular deficit may be directly observed and found to match the theoretical prediction.

From these simplest examples of topography change, we can already see mechanics emerging. The cones, with their radial channels, are easy to deform along their radii; their circular sections change under transverse forces since the welds between the channels can easily bend, without affecting the air volume inside the structure. Loading from the tip to flatten the cone is however resisted by circumferences resisting extension which would result in Gaussian curvature change and hence circumferential stretch (which in turn is resisted pneumatically). Conversely, the anti-cone does not present any soft mode since any attempt to bend the structure encounter an inflated beam (since the welds are curved), Figure 6.5: A logarithmic spiral director n forming a constant angle α with the radial vector e r . The corresponding seam is also a counterclockwise turning logarithmic spiral forming an angle π/2 -α with e r . which are stiff. It thus appears essential to have some curvature in the heat-sealed pattern in order to ensure a global structural stiffness, as we show in section 6.7.

Spiral patterns

Spiral channels offer great advantages since (i) for a circularly symmetric pattern, one can vary the angle that the director makes with the radial direction, α(r), and thereby create complex Gaussian curvature (GC) distributions, and (ii) with channels varying in direction, which avoids weak directions where bending can simply occur along seams. Here we revisit and directly apply some theories developed in the LCE community and more particularly by Mark Warner and his coworkers.

Logarithmic spirals

Logarithmic spirals are equiangular spirals which are self-similar curves (they don't have a typical length and look identical at every scale). Descartes first described these curves which have been later extensively investigated by Jacob Bernoulli, who called them Spira mirabilis, "the marvelous spirals". Their equation in polar coordinates is of the form r = exp[(cot α)θ] and they are a simple extension of the illustrative circular and radial director patterns (α = π/2, 0 respectively) in section 6.3. The angle α between the tangent and the local radial direction is constant (i.e does not depend on the radial distance r (see Fig. 6.5)). Hence the ratio of in-material (intrinsic) radius and circumference contraction is also constant, and they evolve into cones or anti-cones, depending on the value of α. When the director n is mainly azimuthal (α → π/2), the contraction will be also azimuthal and the inflated structure will be a cone (see Fig. 6.6). Conversely, when the director is 6.4. SPIRAL PATTERNS mainly radial (α → 0), the contraction will be mostly radial, inducing an angular surplus and the structure will buckle into an anticone (see Fig. 6.6).

The frontier between cone and anti-cone, i.e. the spiral pattern for which the structure remains flat on inflation, is thus expected to occur for an intermediate value of α close to 45 • (in the small deformation limit). The evolution is however more subtle in this large deformation configuration, since the material frame also evolves with the deformation and should make a 45 • angle in the inflated state in order to stay flat. In order to compute the radial contraction (that should be then compared with the azimuthal contraction to compute the angle deficit or surplus of the inflated structure), we need first to derive the equation of proto-radii, that is the material curves in the flat configuration that turn into geodesic radii in the inflated state (see Fig. 6.7).

In polar coordinates (in the frame of the radius e r , and azimuth, e θ in the deflated state), the metric tensor a has off-diagonal elements, according to equation 6.5 with a matrix rotation R of angle -α and reads:

a rr = λ 2 cos 2 α + sin 2 α, a rθ = a θr = - r 2 1 -λ 2 sin 2α,
a θθ = r 2 λ 2 sin 2 α + cos 2 α (6.6)
Proto-radii. Inflation induces a differential (with r) rotation of material points. As said before, radii in the inflated configuration evolve from proto-radii that are (nongeodesic) curves in flat space. Tangents t to curves in the flat reference state evolve as F • t (F is the deformation gradient tensor defined in equation 6.4), where the tangent to circles in polar coordinates (r, θ) are t c = e θ = (0, 1), and general curves have t = (dr(s)/ds, dθ(s)/ds), where t is a unit vector since we have taken a unit speed parametrisation in terms of arc length s. Since the director pattern is circularly symmetric, circles evolve to inflated/deflated circles. For a given curve to evolve to a radius in the inflated configuration, its evolved tangent must be orthogonal to that of circles, that is

(F • t c ) T • (F • t) ≡ t T c • a • t = 0.
The tangents t and t c are orthogonal under the metric a. The condition above for r(s) to be the ancestor of a radial geodesic is then a θr dr(s)/ds + a θθ dθ(s)/ds = 0

→ dr/dθ = -a θθ /a θr = 2r λ 2 sin 2 α + cos 2 α (1 -λ 2 ) sin 2α . ( 6.7)
The expression (6.7) is for a general α(r) but is particularly simple when the director integral curves are log spirals. Then α is constant and the director follows Then integrating equation. (6.7) gives the proto radius:

r(θ) = r(0)e cθ , with c = λ 2 + b 2 b(1 -λ 2 ) , (6.9)
which is also a logarithmic spiral, but with c = b, where c = cot β defines the angle β of the proto-radius log spiral. In Fig. 6.7(d), we plot the spiral angle β of the proto-radius as a function of the angle α of the director n.

Critical angle α c for a flat structure. Having captured the evolution of proto-radii into radii, we may now compute the semi-angle of the cone ϕ introduced in the previous section 6.3, given by the ratio of the new circumference l c , divided by 2π, to the length of the new (geodesic, inflated) radius u. The former reads naturally where t p is the tangent vector of the proto-radius, and we have taken out a dr/ds to change the intergral ds to dr. Putting in the first part of equation (6.7), dθ/dr = -a θr /a θθ , along the proto-radius, we obtain:

l c /2π = e θ • a • e θ = √
u = r 0 dr Det(a)/ √ a θθ = λ r 0 dr r √ a θθ . (6.12)
We have used an important invariant, Det(a) = λ 2 r 2 in polar coordinates that expresses the area change for this 2-D metric tensor. The cone semi-angle ϕ is then given by sin ϕ = l c /(2πu) = a θθ /(λr 2 ) = 1 λ (cos 2 α + λ 2 sin 2 α). (6.13)

Logarithmic spiral channels yield flat sheets when ϕ = π/2, which correspond for a logarithmic spiral director making an angle

α c = sin -1 1 √ 1 + λ (6.14)
with the radial direction. For an angle α > α c , the perimeters are more contracted than the radii, and the structure will buckle into a cone. Conversely, when α < α c , the radii are relatively shorter than the perimeters, and the structure yields an anti-cone. In Figure 6.6, a selection of channel/director angles are presented, and the cross-over between cones and anti-cones matches with good precision the theoretical prediction (α = 50 • for λ = 0.7). During inflation, the structure displays pronounced rotations at the same time, as the spiral channels evolve into modified logarithmic spiral channels. This rotation phenomenon makes contact with the evolution of the proto-radius logarithmic spiral into a geodesic radius, as predicted above. In Fig. 6.7, we printed in white the proto-radi corresponding for the values of α and λ associated with this structure. We observe that these curves evolve precisely to radial geodesics after inflation. Such log spiral directors have been imprinted in LC elastomers and glasses to produce cones and anti-cones, most notably by Broer and White groups (see the review [START_REF] White | Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers[END_REF]). They have been produced in arrays to give strong actuators that can lift several thousand times their own weight [START_REF] Guin | Layered liquid crystal elastomer actuators[END_REF] because stretch rather than bend predominates when evolution to a new metric is frustrated by a load.

The inverse problem for structures with constant Gaussian Curvature

The inverse problem is here more complex than in the case of baromorph, since we only have one degree of freedom, namely the director orientation, to program the metric.

SPIRAL PATTERNS

Curvature arises in this case from the spatial variation of the orientation of the metric tensor. We take here the metric to be in polar coordinates and the only variation is via the angle α(r). The Gaussian Curvature may be expressed, following the formula of Theorema Egregium and the work done by Mark Warner and coworkers [START_REF] Mostajeran | Encoding gaussian curvature in glassy and elastomeric liquid crystal solids[END_REF][START_REF] Mostajeran | Curvature generation in nematic surfaces[END_REF], as the spatial derivatives of the metric tensor. After some calculus and rearrangements (that may be found in the work by Mostajeran et al. [START_REF] Mostajeran | Curvature generation in nematic surfaces[END_REF]), the expression for the Gaussian curvature reduces to:

K = λ -2 -1 2 α + 3 r α sin(2α) + 2α 2 cos(2α) . (6.15)
For constant K, eqn. 6.15 is an ODE for α(r) with a simple solution (see [START_REF] Mostajeran | Encoding gaussian curvature in glassy and elastomeric liquid crystal solids[END_REF] for full details) :

α(r) = ± 1 2 arccos -C(K) r 2 2 + c 1 + c 2 r 2 , (6.16)
where C(K) = K/(λ -2 -1), and c 1 , c 2 are real constants of integration. These solutions bear a strong constraint:

∀r, -1 ≤ -C(K)r 2 + c 1 + c 2 r 2 ≤ 1 (6.17)
In the case of a compact disc, this inequality imposes c 2 = 0. In order to avoid any angular surplus or deficit at the center of the disc, we should have α(r = 0) = α c according to equation 6.14. This condition imposes the value of the other constant:

c 1 = - 1 -λ 1 + λ (6.18)
The maximal possible radius R max of the disc on which the smooth constant Gaussian curvature K structure may be programmed reads according to equation 6.17:

R max = 4(1 -λ) λK 1/2
for K > 0 (6. [START_REF] Bense | Buckling of elastomer sheets under non-uniform electro-actuation[END_REF])

R max = 4(1 -λ) -λ 2 K 1/2
for K < 0 (6.20)

On the boundary of this disc, the expression for α in (6.16) attains its extremal value of α = π/2 for K > 0 (i.e., purely azimuthal contraction) and α = 0 for K < 0 (i.e., purely radial contraction). Beyond this maximal radius, the contraction factor λ does not suffice. In order to grasp the idea of this limitation, one may for example think of a sphere. Starting from the north pole, the azimuthal contraction rate should tend to zero when approaching the south pole, which is not physically possible. Experimental realisations of structures of constant positive and constant negative Gaussian curvature may be found in Fig. 6.8. The spherical profile shown in Fig. 6.8(b) is in good agreement with the experimental one. Small deviations may be explained by the strongly anisotropic bending stiffness, that may cause deviations of the actual metric from the target metric, since bending energy may not be neglected in these structures (which are not that slender). Another explanation resides in the fact that the width w between two seams evolves with the radial distance r whereas the seam width e is constant: the contraction rate λ thus slightly varies with r and is not strictly constant as assumed in the theory (see Fig. 6.9).

The general, axisymmetric inverse problem.

Let us now consider the general axisymmetric inverse problem, that is programming any axisymmetric shape parametrized by its profile (see Fig. 6.10). Using the ideas of section 6.4.1 to calculate lengths of geodesic radii u in evolving structures, and the associated circumferential inflation 2πγ 1 , one can derive an explicit scheme for calculating the required director distribution to generate a general shape [START_REF] Warner | Nematic director fields and topographies of solid shells of revolution[END_REF], which generally results in a non-linear integral equation. For some shells, simple ODEs arise, as we shall see.

The basic idea behind this programming scheme is to start at the center of the disc and see by which amount the perimeters should be contracted and thus which director angle α should be selected. If the summit is flat, then α = α c , which imposes the proto-radius contraction. An infinitesimal step dr on the flat non inflated structure corresponds thus to a step du(α) on the target shape. At this new location we repeat our reasoning: the azimuthal contraction γ 1 /r imposes the local angle α, which in turn yields a proto radius contraction du(α)/dr. In terms of mathematical expressions, equation (6.12) connects r and u, and we have du/dr = λr/ a θθ (r) (see Fig. 6.10). The new circumference divided by 2π was given by equation (6.10), that is γ 1 = √ a θθ . In a curve of revolution, we have the geometric relation (du) 2 = (dγ 1 ) 2 +(dγ 2 ) 2 . Differentiating with respect to r, and using → λr = 1 2 da θθ /dr 1 + (dγ 2 /dγ 1 ) 2 (6.21)

which can be useful for a wide range of curves for which one knows dγ 2 /dγ 1 .

Paraboloids. For paraboloids obtained from the revolution of γ 2 = 1 2 aγ 2 1 , we have (dγ 2 /dγ 1 ) 2 = a 2 γ 2 1 = a 2 a θθ , whereupon one can integrate Eqn. (6.21) with respect to r and a θθ :

λr 2 = 2 3a 2 (1 + a 2 a θθ ) 3/2 -1 . (6.22)
Recall that a θθ = r 2 (1 -(1 -λ 2 ) sin 2 α) (note that a θθ (r = 0) = 0, which is needed for the above integration). Equation (6.22) can then be solved for α(r) in the form:

sin 2 α λ (r) = 1 1 -λ 2 1 - 1 a 2 r 2 1 + 3λ 2 a 2 r 2 2/3 -1 . (6.23)
The subscript appearing in α λ (r) is to remind one that the form of α(r) only gives a paraboloid for this particular target λ.

NON-ISOMETRIC ORIGAMI

Catenoids. The catenoids that we consider are the revolution about the γ 2 axis of the catenary u = 1 a sinh(aγ 1 ) and where γ 2 = 1 a cosh(aγ 1 ). Returning to the relation du/dr = λr/ a θθ (r) = dγ 1 dr 1 a cosh(aγ 1 ), and remaining this time with the variable γ 1 ≡ √ a θθ , one obtains the set of equations:

λrdr = dγ 1 γ 1 cosh(aγ 1 ) → 1 2 λr 2 = γ 1 a sinh(aγ 1 ) - 1 a 2 cosh(aγ 1 ) + 1 a 2 . (6.24) Since √ a θθ = r 1 -(1 -λ 2
) sin 2 α(r), the above is an implicit equation for α λ (r). Taking the limit r → 0, one obtains the minimum for α as sin 2 [α λ (0)] = 1/(1 + λ). At the outer boundary of the disc that can be inflated to a catenoid, the director is tangential and sin 2 α λ (R max ) = 1, whence γ 1max = λR max which should be inserted into Eqn. (6.24) to give an equation for R max .

Although this pure geometrical approach is very satisfying, some deviations are observed between the target and actual shapes. Mechanics should be indeed taken into account, especially when the pattern is not very fine,for which the limit of slender structures does not remain valid. We will study in more details the mechanics of these inflated beams in section 6.7, but it is obvious that the stiffness of an array of straight inflated beams is strongly anisotropic. The structure may be indeed very easily bent along a seam, whereas the air beams are orders of magnitude stiffer. In the case of spirals, the structure starts to exhibit soft deformation modes when the director angle α is close to π/2 and does not vary much (that is, when the seam pattern is almost radial, as in the case of the cone (See Fig. 6.4). This strong anisotropy leads to surprising deformations in some cases and the structure may exhibit kinks (Fig. 6.12), especially when the surface metric imposes large curvatures along the air beams. However, we may harness this anisotropy to program localized bending of the structure at specific locations. The sharp bend will here not be smoothed out by elasticity as in the case of baromorphs, since the bending stiffness may differ by orders of magnitude depending on the orientation of the sollicitation. These localized folds are reminiscent of origami folding, as we shall now investigate in the next section.

Non-isometric origami

In this section, we aim at building morphing structures made of flat panels with localized folds induced by the metric distortion. In order to do so, we should have discrete changes of direction of the channels between regions of uniform, anisotropic in-plane distortion. This strategy is inspired by "non-isometric origami" [START_REF] Plucinsky | Programming complex shapes in thin nematic elastomer and glass sheets[END_REF][START_REF] Plucinsky | Patterning nonisometric origami in nematic elastomer sheets[END_REF][START_REF] Plucinsky | The deformations of thin nematic elastomer sheets[END_REF][START_REF] Modes | Materials with programmable shapes[END_REF][START_REF] Warner | Topographic mechanics and applications of liquid crystalline solids[END_REF], first investigated by the LCE community, that is of a completely different character from conventional, isometric folding of a sheet (typically paper). It is also different from modern, activated-fold origami [START_REF] Miskin | Graphene-based bimorphs for micron-sized, autonomous origami machines[END_REF][START_REF] Na | Programming reversibly self-folding origami with micropatterned photo-crosslinkable polymer trilayers[END_REF] which is based on bilayer effects and relies thus on isometric folding. In conventional origami, the 2D to 3D transformation relies on isometric Figure 6.12: Instead of the expected smooth anti-cone regularized at the apex, the inflated structure may exhibit kinks. The curvature imposed by the metric is too large for the air beam array, which prefers to localize the curvature at specific locations and remain mosty planar elsewhere.

folding, the sheet having by definition the same metric in both state. The structure is thus weak since it can be isometrically unfolded (the folds act as hinges). In non-isometric origami conversely, the faces change dimensions and resist with stretch at any attempt to unbend. The Gaussian Curvature is concentrated at vertices where there are angular deficits/surplus. Piecewise uniform channel/director fields offer (localised) Gaussian curvature at vertices where the fields meet [START_REF] Modes | Blueprinting nematic glass: Systematically constructing and combining active points of curvature for emergent morphology[END_REF][START_REF] Plucinsky | Programming complex shapes in thin nematic elastomer and glass sheets[END_REF]. Compatibility between the regions on distortion requires Rank-1 Connectedness (R-1C) at the boundaries where the uniform fields intersect. Physically, this R-1C condition is that deformations parallel to the boundary must be identical from one side (1) to the other (2) to prevent a length incompatibility along the boundary. It is expressed, in this nematic analogue, as n 1 • t = n 2 • t, where t is the vector along the line of separation: the directors must therefore make equal angles with the boundary. Simple units must be put together in a R-1C way. Three examples are shown in Fig. 6.13 where the change in angle is trivial to calculate [START_REF] Modes | Blueprinting nematic glass: Systematically constructing and combining active points of curvature for emergent morphology[END_REF] from the right triangles shown where opposite or adjacent sides change by a factor λ. Hence in Fig. 6.13(a) the summit angle χ in the flat deflated state evolves to the new angle χ after inflation which reads χ = 2 tan -1 1 λ tan(χ/2) and for (b) and (c) (which are essentially equivalent) one has χ = 2 tan -1 (λ tan(χ/2)). The structure favours local bending along radial seam lines starting from the vertex, which is possible when the director field is mainly azimuthal, leading to an angle deficit. This is the case for the bisector in Fig. 6.13(b) or along the edges in (c). When the director is mainly radial, yielding an angular surplus, the seams are thus rather azimuthal, and the weakest lines are the R-1C boundaries, along which the structure tend to kink. In a convex polyhedron, each apex exhibits an angular deficit (the structure has positive GC localized at the apex). A cube has three square faces meeting at each corner, so each sector has to develop a χ = π/2. The three protofaces meeting at a point while the sheet is still flat must have angles χ = 2π/3. Using the above transformation of the type (b), one requires a λ = tan(χ /2)/ tan(χ/2) = tan(π/4)/ tan(π/3) = 1/ √ 3 = .577 which is less than the available 2/π = 0.64 -cubes are inaccessible to our actuation strategy. Trying nevertheless to approach a cube, we may print a pattern inspired by Modes et al. [START_REF] Modes | Materials with programmable shapes[END_REF] (see Fig. 6.14). The inflated structures does show flat panels and sharp edges and apexes with angular deficit (Fig. 6.14(c)), and the structure folds, as predicted, along the weak seam lines starting from the vertices and not along the R-1C lines, as in the case of LCE. The pattern and its boundary must be therefore slightly updated (Fig. 6.15(a)). Each face is made of two thirds of a hexagon that should close into a square upon inflation. As explained before, the contraction rate λ is not sufficient to make the cube closed. However, the structure gets closer to a cube as it presents six faces (Fig. 6.15(b)). It should be also pointed out that each vertex may buckle one way or the other and may be snapped by pushing on it (Fig. 6.15(c)). Breaking the symmetry of the structure across the thickness may induce a preferential buckling direction (a kind of bilayer effect). Dodecahedra have pentagonal faces, three of which meet at a vertex. There being three, in the flat state χ = 2π/3. In the distorted state, the pentagonal angle is χ = 3π/5. The required λ = tan(3π/10)/ tan(π/3) = 0.79 is easily accessible by choosing the proper dimensionless seam width ξ in equation 6.2. In Fig. 6.16, each face is made in the flat state of 5/6 of a hexagon. The faces coil until self contact when inflated, creating pentagons. The structure thus closes to form a dodecahedron, as shown in Fig. 6.16.

Arrays of R-1C sectors can also form extended topographies rather than closed structures as above, as shown in Fig. 6.17. Inspecting the array of channels, or their orthogonal dual, i.e. the director field, one sees an array of ±1 topological defects in 2-D. In the inflated array, one sees these correspond to mountain peaks and troughs (+1) separated by saddles (-1). The structure here folds along the R-1C connections. From the saddle perspective, they are indeed the weakest (and thus preferred) lines of bending. The rigid- ity of these units is further enhanced by their being in an array. Attempts to deform the system induce stretch. Such systems are reminiscent of the LC solid arrays of White et al [START_REF] Guin | Layered liquid crystal elastomer actuators[END_REF] which can lift thousands of times their own weight when loaded and actuated. In addition to this, each saddle may be snapped to inverse the two upper and lower opposite corners and the structure may also fold along the seams connecting peaks and troughs enlarging remarkably the family of stable shapes that may be reached. In order to go from one shape to the other, at least one saddle should be snapped, wich means that each shape corresponds to a local minimum in the energy landscape. A few examples of shapes are shown in Fig. 6.17(c). There are some constraints in this system and the snapping of one saddle may impose the snapping of its neighbours. The count of all the possible stable states has not been carried out yet and will be the subject of future work.

Non-regular vertices and the rules they obey

Generally, vertices do not have to be formed by the meeting of identically-shaped regions of uniform director. But deviation from regularity can only be within strong conditions on the internal angles of sectors, χ i and on the angles α i that the director n i in the i th sector makes with the interface between the i -1 th and i th sectors, that is conditions on n i • t i ; see Fig.

α i+1 = π + χ i -α i (6.25) even i : α i+1 = χ i -χ i-1 + • • • + χ 2 -χ 1 + α 1 (6.26) odd i : α i+1 = π + χ i -χ i-1 + • • • -χ 2 + χ 1 -α 1 . (6.27)
The first expression arises from considering the triangle formed in the i th sector by the two boundaries and the director line, the latter two from iteration for i even or odd respectively.

For an n-sector vertex, α n+1 ≡ α 1 , whereupon the n even and odd cases yield:

even n : 0 = χ n -χ n-1 + • • • + χ 2 -χ 1 (6.28) odd n : α 1 = π/2 + 1 2 (χ n -χ n-1 + • • • -χ 2 + χ 1 ) , (6.29)
with of course the condition Σ n i=1 χ i = 2π. Thus for vertices with even sector number, α 1 can take any value, and the α i =1 then follow from eqns (6.26) and (6.27). For odd cases, the choice of α 1 is constrained by condition (6.29) and the subsequent α i =1 again follow from equations (6.26) and (6.27). Figure 6.18(a) shows an even, regular case where n = 6 and χ = π/3 in each sector. Choosing α 1 = 2π/3 gives all sectors the same α, or choosing α 1 > 2π/3 gives α 2 < 2π/3, with subsequent alternation. By contrast, (b) has χ 1 = χ 2 = χ 4 = π/3 as before, but χ 3 , χ 2 and χ 6 changed from π/3 such that condition (6.28) is respected, whence the director lines close and are R-1C as required.

To calculate the Gaussian curvature associated with a vertex, one must calculate each χ i and hence the total angular deficit/surplus Σ n i=1 χ i -2π. Not all the sectors' n fields are of the symmetric form of Fig. 6.13, and one requires the metric tensor in its nondiagonal form. Considering the triangle of Fig. 6.18(c), the side c is parallel to n and hence simply changes its length as c → λc. The other sides change as, for instance,

b 2 → b • a • b = f 2 b b 2 with f 2 b = λ 2 cos 2 β + sin 2 β if
one considers the factors of b • n required in contracting with the metric tensor, eqn. (6.4) (with β = α -χ). Use of the cosine rule in the triangle before and after deformation then gives:

cos χ = λ 2 f a f b cos χ + (f 2 a -λ 2 )a 2 + (f 2 b -λ 2 )b 2 2f a f b ab . ( 6.30)
The second term depends on factors a/b = sin β/ sin α and the inverse, so of course the size of the sector is irrelevant.

Experimental realizations of irregular vertices have not been conducted yet and will be the subject of future work. Irregular vertices enrich indeed strongly the possibilities of non-isometric origami.

6.6 Adding a second degree of freedom in the metric distortion 6.6.1 Width of the seam It might be convenient to have a second degree of freedom in the distortion of the flat metrics to increase the possibilities of shape programming: one obvious strategy, as discussed above , is to play with the relative seam width ξ, to change the homogenized in-plane contraction perpendicular to the seam direction (see equation 6.2). Of course welding the two sheets on a larger area decreases the structural stiffness of the structure since the welded portions have a very low bending stiffness. However, as we shall see in the latter, the structure as a whole may be stiff even though a large part of the structure is weak.

Hemisphere Let us consider purely radial airways, and thus azimuthal contraction. Along the curvilinear coordinate u, there is no length change (u = r in this case). In order to program a portion of a sphere of radius R, the evolution of the azimuthal contraction λ ⊥ as a function of the radial in-material distance r reads (see Section 1.3 for the details):

λ ⊥ (r) = R r sin r R = 1 - r 2 6R 2 + o((r/R) 2 ) (6.31)
Together with equation 6.2, the relative seam width ξ(r), may be easily computed, yielding the pattern shown in Fig. 6.19(a). The maximum contraction is 2/π which enables us to program a hemisphere. At the equator, we have indeed r/R = π/2. The structure does buckle, when inflated, into a bowl shape that is close from a hemisphere (Fig. 6.19(a) top). However, the structure remains very floppy, because the center part is nothing more than two passive layers of fabrics and because the structure may be easily folded along any radial seam. Saddle In order to program a saddle of constant negative Gaussian curvature, we know that the perimeters should be relatively longer than the radii : P (u) = 2πu(1 -Ku 2 /6 + o(u 2 )), where u is the radial coordinate along the inflated surface. With purely azimuthal seams, and thus radial director, the perimeters are unchanged upon activation, but the radii contract. The contraction should thus reads at the second order:

λ(r) = 1 + Kr 2 6 (6.32)
Helicoid Let us consider parallel seams of varying width. The contraction should be maximal at the center line of the ribbon and can be chosen as π/2; at the outer edge, no contraction is needed. Thus the maximum admissible ratio l/p, where l is the width of the ribbon and p the pitch of the helicoid, reads:

l p = 1 2 λ -2 max -1 4π 2 (6.33)
where λ max = 2/π (Note that λ max is actually the minimal value of λ, but correspond to the maximal contraction). The targeted contraction rate λ ⊥ as a function of the coordinate along the seam x (the origin being chosen at the center of the ribbon) can be expressed

CHAPTER 6. INFLATABLE MAZES as: λ ⊥ (x ) = λ max 1 + 4πx 2 /p 2 (6.34)
Using equation 6.2, the proportion of the seam width ξ reads:

ξ(x ) = ( 1 + 4πx 2 /p 2 -1)/(λ -1 max -1) (6.35)
Additionnally, two parallel stripes are sealed along the outer edges of the ribbon to ensure bending stiffness along the helicoid direction.

The dome is very floppy, whereas the helicoid and the saddle are relatively stiff : they show indeed no soft modes, that is bending possibilities along a seam without encountering any air beam. They indeed both present a boundary that is not contracted along its tangent t: the contraction is perpendicular to the boundary, which is thus stiffened by the presence of the curved air beam. This is possible only for negative curvatures, since the plate may only contract (and not extend) within a fixed boundary length. Another requirement in order to obtain a stiff well defined structure is that the welded portions are under tension. As they are very slender, they would indeed buckle at the onset of small compressive forces.

Structural Stiffness

In this section, we tackle the question of the stretching and bending stiffness and maximum admissible moment of straight inflated textile beams and arrays of zigzag beams, in order to better understand the mechanics of the structures we construct.

Stiffness of straight lines

Consider two textile rectangles sealed on their edges of length L, width w, and thickness t, with t w L. Upon inflation, if p Et 3 /w 3 , that is, if the pressure energy is large compared to the bending energy of the textile sheet -which is the case in our experiments -the structure deforms into an air beam of circular cross section with radius r = w/π (in the case where p Et/w, that is, if the pressure energy is small compared to the stretching energy of the textile sheet, which is also the case experimentally).

Stretching. Along the beam, the stretching modulus Y of the inflated structure is given by the sheets property and reads Y = πEt. There is indeed no soft mode, and stretching along the air beam direction means stretching the sheets (the potential contribution from pressure scales as pw which is order of magnitudes smaller than stretching the sheet). Perpendicular to the seams, stretching the structure means elongating the cross-section towards a non volume-maximizing solution, and eventually stretching the sheets. From Laplace law, we know that no matter the state of extension of the structure, the cross sections of the beams will remain portion of circles of varying spanned angle β and corresponding radius of curvature R = w(1 + ε)/β (see Fig. 6.20(a)), where ε is the (uniform) strain of the fabric. The tension T in the sheet reads:

T = Etε = pw(1 + ε)/β (6.36)
A simple force balance (Fig. 6.20(a)) gives the following expression for the force F (per unit length): F = 2T cos(β/2) (6.37)

The corresponding total displacement ∆ reads:

∆ = L -L 0 = 2w (1 + ε) sin(β/2) β - 1 π (6.38)
Combining equations 6.36-6.38, one may eliminate T and ε, and express both the force and the displacement as a function of β

F = 2pw cos(β/2) β -Σ (6.39) ∆ = 2w sin(β/2) β -Σ - 1 π -Σ (6.40)
where Σ = pw/(Et) is a small dimensionless number as discussed above. These expressions are plotted in Fig. 6.20(b) (dashed lines) and match precisely the force-displacement curves measured experimentally (solid lines) for various pressures. The seam-line width e (and thus the actual airways width w), difficult to measure, has been used as a fitting parameter and yields reasonable values, close to the estimated measurements.

Linearization of the ratio F L 0 /(l∆) around the unloaded inflated state (β = π) gives For small deformations, the fabric can be assumed as nextensible. As a consequence, the stretching modulus does not depend on the material and is only set by the pressure and the geometry of the air beams: dividing the force by the pressure yields the collapse of the curves (Fig. 6.20(b) inset). However, for large deformations (β → 0), the radius of curvature increases, the tension T = pR = pw/β in the fabric increases, up to a regime in which the assumed inextensibility of the fabric sheet is not valid anymore (since T = Etε).

The maximal slope is given by the stiffness of the sheet and reads 2Et as shown in Fig. 6.20(b). Experimental measurements are well described by this simple argument (Fig. 6.20(b)).

Bending. For small bending deformations, that is, in the linear regime, the pressure does not play any role, since bending is at the first order a volume-conserving transformation [START_REF] Main | Beam-type bending of spacebased inflated membrane structures[END_REF][START_REF] Pontavice | Propulsion par cerf-volant: envol et pérégrinations[END_REF]. The bending stiffness B ⊥ per unit width of the beam is thus the bending stiffness of the envelope in the inflated geometry, and reads:

B ⊥ = Ew 2 t/(4π) (6.42)
In order to probe this prediction experimentally, slender air beams are manufactured and subjected to a three-points bending test for various pressures, as shown in Fig. 6.21(a) and (b). Measuring the applied force as a function of the vertical displacement ∆z (Fig. 6.21(c)), we observe for small displacements a linear response, as predicted by classical Euler-Bernoulli beam theory, which classically reads ∆z = F L 3 48EI (6.43)

We may get a hint that Euler-Bernoulli beam theory may be applied for our thin-wall air beams by looking at the profile of the deflection when subjected to a three-point test.

The theoretical prediction (shown in dashed red line in Fig. 6.21(b)) matches indeed very well the experimental pictures as long as the force-displacement response remains linear. The resulting slope appears in our experiments to be mostly independent on the applied pressure, as predicted by the theory (equation 6.42). Extracting from the slope the bending stiffness B = πEI/2w, we obtain a reasonable agreement (at least in terms of scaling law) between the experimental bending stiffness and the theoretical one, as shown in Fig. 6.21(d). The experimental bending stiffness appears nonetheless to be always smaller than the theoretical value. Moreover, we do observe a small pressure dependence. Several reasons may explain this mismatch. First, the perfect circular geometry of the cross section assumed in the theory is not exactly valid, since it would mean an infinite curvature at the seam. Increasing the pressure tends to approach this idealized scenario (the cross section slightly evolves with the applied pressure) and may explain why a small pressure dependence is observed. Moreover, the indentation of the beam may locally change the geometry of the cross-section, impacting once again the bending stiffness. Finally, the mechanical properties of impregnated fabric sheets are more complex than in the case of an ideal homogeneous material. The presence of woven fibers induces some anisotropy and the effective Young modulus we should use in our estimates may differ from the value measured in tensile tests. However, despite these numerous sources of inaccuracies, the bending stiffness B of an inflated air beams does scale as Ew 2 t with a pre-factor not that far from 1/4π.

Although the linear response is mostly pressure independent, such an air beam dramatically collapses upon a critical load F c (see Fig. 6.21(c)) that is strongly pressure dependent, as shown in Fig. 6.21(c). The critical moment per unit width M c may be CHAPTER 6. INFLATABLE MAZES easily computed from the critical load F c (M c = πF c L/8w) and plotted as a function of the pressure for various air beams in Fig. 6.21(e), varying both the material and the width of the cross-section. The critical moment when the pressure vanishes seem to depend only on the sheet properties, whereas the dependence on pressure appears to be affine, the slope varying with the cross-section size (Fig. 6.21(e)).

Following Calladine [START_REF] Calladine | Theory of shell structures[END_REF] and Seide and Weingarten [START_REF] Seide | On the buckling of circular cylindrical shells under pure bending[END_REF], we know that the critical torque M c for which the air beam collapses is obtained when the extreme fibre reaches the critical buckling stress σ c in the case of uniaxial compression of a cylindrical shell. The critical stress reads σ c = Eπt/( √ 3w). In order to reach this critical compressive stress, the imposed moment must additionally overcome the pressure-induced longitudinal tension in the beam, which classically reads: σ p = pw/2πt.

Hence the maximum bending moment per unit width that the air beam may sustain reads:

M c = πwt/2[σ c + σ p ]M c = πEt 2 2 √ 3 + pw 2 4π (6.44)
The ovalization of the cross section upon curvature, the well known Brazier effect, may be overseen in these examples, since the typical applied pressure prevents the onset of cross-section shape change. The theoretical critical momentum with (dashed lines) and without (solid lines) the ovalization of the cross section as a function of the pressure is shown in Fig. 6.21(f).

A few interesting features are worth being noticed. First, the maximum moment has two completely different contributions: one term in Et 2 , which depends only on the sheet properties, comes from the classical critical buckling stress at the "extreme fibre". The second term scales as pw 2 and is conversely independent of the sheet, but rather depends on the size of the channel and the applied pressure. As said before, our objects are in the regime of vanishing flexural stiffness of the sheet (p Et 3 /w 3 ) and sheet inextensibility (p Et/w). Hence, the contribution of both terms are in principle of the same order of magnitude, and should be taken into account. Maybe one should add that there is a maximum admissible pressure in the structures: indeed, above a given tension T max in the sheet, delamination between the two heat-sealed layers may occur. From Laplace law, we know that T max = p max w/2π. The maximum admissible pressure thus scales as ∼ 1/w such that the maximum moment scales with w and not w 2 as suggested by equation 6.44.

In a structure which is an assembly of straight beams, the bending stiffness B along the seams is very week because it may only involve bending of the seams, with the inflated region only rotating around these hinges. The stiffness of such hinges scales classically as Et 3 which is orders of magnitude smaller than the stiffness in the transverse direction (Ew 2 t with w t). This strong stiffness anisotropy is something that should be avoided for the overall stability and stiffness of the object. Soft structural modes may indeed cause the failure of the structure. Note that this soft mode is possible because the sealed lines are rectilinear, providing the rotation axis. Hence, some direction change is needed in the network of heat-sealed lines, such that no matter the direction and position of bending 6.8. GENERAL INVERSE PROBLEM solicitation, a stiff mode will be encountered.

General inverse problem

Let us now turn to the general inverse problem, that is finding the seam pattern that makes the structure morph into an arbitrary target shape. With this metrics distortion strategy (unidirectional contraction of amplitude λ), we were not able to find a simple intuitive recipe as in the case of baromorphs. In order to solve numerically this highly complex problem, we started a collaboration with Mark Pauly and Julian Panetta from the Laboratory Graphics and Geometry (EPFL). We present here very briefly the different steps of such a strategy and what we managed to achieve so far.

The procedure they envision consists in the following steps:

• a parametrization: flatten the target 3D surface to the plane with a mapping that stretches everywhere the surface by 1/λ along one direction and leaves the orthogonal direction unstretched. This is so that the inverse mapping from the plane to the target surface contracts lengths by λ 2/π just like the air channel mechanism. In practice, λ can be slightly varied by playing with the seam width, giving more freedom in the parametrization, as we shall see later.

• Trace air channel walls along the unstretched direction field with thickness determined by the value of 1/λ at each point.

• Simulate the inflation of these air channels and build atop this simulation an optimization of the wall curves to better fit the inflated structure to the target surface.

First parametrization

The question is the following: we seek a mapping f : M → R 2 of a target arbitrary surface M ⊂ R 3 to the plane that is "compatible" with our inflation mechanism. Said differently, we want to flatten an arbitrary shell by stretching the structure in a way that is compatible with the anisotropic contraction of airways.

The most obvious requirement on f can be expressed in terms of the singular value decomposition (SVD) (σ 0 and σ 1 ) of its Jacobian at each point on the surface:

∇f = U σ 0 0 0 0 σ 1 0 V T , (6.45)
where U ∈ R 2×2 is a matrix whose two columns give the directions in the plane along which lengths are stretched or preserved respectively, σ 0 and σ 1 are the stretching amounts along these directions, and V ∈ R 3×3 indicates the directions in 3D space that are stretched by the mapping. To be compatible with the air channel mechanism, f must have σ 1 = 1 and σ 0 in some admissible stretching range 1/λ min , π 2 . Setting λ min = 2 π would mean all air channels are of "full width," while choosing λ min ∈ 2 π , 1 means that the fused line has CHAPTER 6. INFLATABLE MAZES a finite width (see equation 6.2). It appears that allowing λ min > 2 π (i.e. a variation in the width of the seam line) enables significantly more flexibility in the space of surfaces that can be flattened (Note that λ min is actually the maximal allowed value of λ, but correspond to the minimal contraction). Mappings of this form were already constructed by Aharoni et al. [START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF], which ran an iterative "local-global" optimization to fit ∇f to the desired form by minimizing the energy:

E[f ] := min U,V,λ 1 2 M ∇f -U λ -1 0 0 0 1 0 V T 2 F
dA. (6.46) Their approach alternates between (i) computing a closed-form solution to the minimization over U , V and α (found by computing the SVD of the current Jacobian ∇f for each triangle and setting α to σ 0 clamped to the admissible range); and (ii) solving for the optimal mapping f while holding the newly computed U , V and α fixed. Provided such a mapping exists and as long as this method is run from a reasonable initialization, it quickly reaches a parametrization f with singular values close to the admissible range. As an illustration, we first aim at inflating a flat inflatable that would shape into the roof of the Lilium tower on inflation (see Fig. 6.22). The Lilium tower is a planned skyscraper to be built in Warsaw, designed by Zaha Hadid. This shape has both positive and negative Gaussian curvature and no specific symmetry. Unfortunately, this first approach results here in noisy "stretch factors" λ -1 and orientations that would be difficult to trace air channels through, as shown in Fig. 6.23(a).

To encourage more structure in the stretching factor and orientation fields, some additional regularization terms are needed in equation (6.46). After adding these terms, the local-global optimization approach can no longer be used. We instead must directly tackle where the sums in the regularization terms are over the mesh edges (indexed by adjacent triangles i and j). These regularization terms are expressed in terms of functions of the singular value decomposition (SVD) of the Jacobian ∇f on triangles i and j: we have defined λ -1 i to be triangle i's largest singular value (σ 0 ) and φ i to be the angle between the corresponding left singular vector and the x axis.

Weights w φ and w λ control the trade-off between preferring smoothness in the corresponding fields and fitting to the admissible singular values. Parameters p φ and p λ control the type of smoothness/structure we ask for. For instance, setting p φ = 2 requests that stretching orientations vary smoothly across the surface, while p φ = 1 requests more of a "zig-zagging" map with patches of nearly straight air channels that abruptly change direction across a sparse network of seams.

The results presented in Fig. 6.23(a) and (b) use 1/λ min = 1.4 (corresponding to a CHAPTER 6. INFLATABLE MAZES minimum channel width of 70% of the channel spacing). The result from the regularized minimization shows indeed more structure and smoother variations (Fig. 6.23(b)), enabling Julian to construct the airways pattern.

Construction of the stripe pattern

We use the algorithm from Knöppel et al. [START_REF] Knöppel | Stripe patterns on surfaces[END_REF] to construct a stripe pattern aligned with u 1 (where u 1 is the direction of the second singular value σ 1 , that is the direction which does not stretch when flattening the surface). A spatial frequency is specified, which controls the scale of the air channels (increasing this parameter packs more air channels into the sheet).

The "stripe field" is effectively a distance field to the channel wall centers in dimensionless units: the field is 0 at the wall center and increases at an approximately constant rate as one moves perpendicularly to the wall. The field reaches a peak value of π at the center of the channel before dropping back to 0 at the next wall center (the spacing between wall centers is always 2π in these scaled units). Recall that the channel wall width adjusts the amount of contraction induced by channel inflation. Consequently, for a given parametrization f , we must choose the wall widths according to the stretch σ 0 that f applies when flattening the target surface to the plane. For instance, if σ 0 is the maximum value of π 2 , then the walls must have zero width so that the air channels occupy the full span of 2π units between their walls. For a general, spatially varying σ 0 , the formula for seam widths is given by equation 6.2 and the seam pattern may be drawn accordingly.

Numerical inflation simulation

On the simulation side, Julian has implemented the approach described by Skouras et al. [START_REF] Skouras | Designing inflatable structures[END_REF], which models the sheets as incompressible neo-Hookean membranes with no bending stiffness. His implementation supports enabling and disabling the "Tensionfield Theory" approximation (assuming that the compressive stiffness of the membrane is zero to represent xrinkling and buckling). The numerical inflation gives, for large enough pressures, results very close to the actual experiments: a pattern of azimuthal and radial seams are shown in Fig. 6.25 leading to an anticone and a wrinkled cone, as in the experiments (see Fig. 6.4).

We can now simulate the inflation of the pattern constructed from the regularized minimization to see how well the design in Fig. 6.23(b) works. Unfortunately, the sheet immediately deploys into a very different shape, shown in Fig. 6.26 (even if we "cheat" by initializing the sheet onto the target surface). It can be checked that this behavior is not due to a failure of the parametrization f to properly encode the target shape. The reason for the failure of reproducing the target shape resides indeed in the mechanics of the structure, that we completely neglected so far. The structure is indeed, as detailed in section 6.7, much stiffer along the channels (that turn into air beams on inflation) than perpendicular to them. In the example of interest, the long air channels spanning across the shape must bend significantly to fit to the surface, but the strong stiffness anisotropy favours bending perpendicular to the channel and the channels remain mostly straight. The parametrization algorithm should be adapted to consider more than just the geometry and the smoothness of the channels; it must ensure that the air channels bend as little as possible in their deployed configuration to mitigate the effects of stiffness anisotropy.

Adding a bending regularization term

To this end, Julian developed a "homogenized" anisotropic bending energy model to capture the reluctance of the inflated tubes to bend. Standard beam bending energy may be used (see section 6.7):

1 2 (κ -κ 0 ) 2 ds, (6.49) where κ is the curvature of the beam's centerline and κ 0 its natural curvature Here, we choose to penalize the normal curvature of the air beams (and neglect the change in geodesic curvature). Imagine the area element dA at a point x on our surface M is filled with many parallel inflated tubes all oriented in direction v 1 (i.e., the channel directions implied by the parametrization f ). Summing the bending energy over all the segments of tubes fitting inside dA gives us an energy proportional to κ(v 1 ) 2 dA, where κ(v 1 ) is the normal curvature of the surface in the v 1 direction at this area element. This normal curvature can be written as v T 1 S(x)v 1 where S(x) = κ 1 d 1 ⊗ d 1 + κ 2 d 2 ⊗ d 2 is the shape operator at x expressed in terms of the surfaces' principal curvatures and principal curvature directions. The total bending energy in the homogenized inflated sheet for a given direction field v 1 on M is thus proportional to: The parametrization and its derived stripe patterns/air channel walls change substantially with the new bending regularization term (see Fig. 6.27(b) in comparison with Fig. 6.24(b)). This design provides a much more convincing reproduction of the qualitative CHAPTER 6. INFLATABLE MAZES features of the Lilium tower model (see Fig. 6.27(c)). The result is of course better if we constrain the boundary of the inflated structure (see Fig. 6.27(d)). We show the inflated design here with the target surface superimposed in translucent red. Remarkably, inflating the experimental structure leads to a shape very close to the target shape (the features of this complex surface are qualitatively reproduced).

The second optimization of the wall curves built atop the inflation simulation to better fit the inflated structure to the target surface is currently in progress. However, the results are already very impressive: we have tested experimentally the pattern and the inflated structure is remarkably close to the numerical simulation (and to the target surface), as shown in Fig. 6.27(e). At this point, the limitations of such a strategy have not been investigated yet. Many designs could now be imagined, with additional openings or cuts in the initially flat surface.

Conclusion and perspectives

In this chapter, we introduced a very easy-to-make and stiff pneumatic shape-morphing strategy. Two superimposed flat fabric sheets impregnated with heat-sealable layers are bonded together along a specific pattern of locally almost parallel seam lines, defining a channel direction. Upon inflation, the sheet locally bends perpendicular to the seam lines, leading to an average unidirectional contraction of the surface. In that sense, this strategy may be seen as self-actuated origami tessellations, since the sheet does not stretch but bends (folds in the case of origami) to distort the average metric, the local metric being preserved. The geometrical formalism appears to be equivalent to deformations in liquid crystal elastomers (with a Poisson ratio equal to zero) and we used many concepts and theories developed by the LCE community to efficiently program, in collaboration with Mark Warner, cones, anticones and any axisymmetric shape (with some constraints on the amplitude due to the finite contraction rate λ). We also harnessed the strong stiffness anisotropy to program local folding along specific seam lines, leading to nonisometric origami surfaces (also first investigated by the LCE community), such as a dodecahedron.

We introduced a way to gain one degree of freedom in the average metric distortion by making "zigzag" patterns of locally controlled angle. This strategy proved itself to make the stiffness of the inflated structure more isotropic, leading to more stability. Finally, in collaboration with Mark Pauly and Julian Panetta, we started developing an optimizationbased solution to the inverse problem, with very promising preliminary results.

Many questions remain however unanswered and will be the subject of future work: the influence of slits or holes, changing the topology of the surface, have not been investigated yet [START_REF] Modes | Angular deficits in flat space: remotely controllable apertures in nematic solid sheets[END_REF]. First preliminary results, shown in Fig. 6.28, demonstrate that we can benefit from openings in the structure to program truncated cones and anticones of very large angle deficit/surplus.

The "zigzag" strategy could be deepened with asymmetry between the "zig" and the "zag", leading to non-zero shear in the metric distortion. This question is currently inves-6.9. CONCLUSION AND PERSPECTIVES tigated by Tian Gao, in collaboration with Antonio DeSimone. The reference surface of zero Gaussian curvature could be a cylinder and not a flat compact disc as investigated so far. The fabrication process would still be two-dimensional, the two superimposed flat sheets being closed into a cylinder with an additional welded line after the printing of the pattern.

Several layers could be stacked on top of each other with different seam patterns connecting two successive layers, encoding different shapes. This process would require the careful calibration of the hot head speed in order to bond together only the two upper sheets, leaving the underneath layers untouched. The simultaneous inflation of several layers remains an open question.

This project being more recent than the baromorphs, the potential applications have not been extensively studied yet. We are currently discussing with the french company Decathlon in order to develop new outdoor gears, such as tents, shape-changing dishware, workout equipment, etc.

Conclusion

When observing everyday's life around us, inflation seems limited to boring shapes. Indeed, due to Laplace law that governs the tension in inflated membranes, the resulting structures are generally spherical (as observed for soap bubbles or rubber balloons) and sharp corners are rounded. Although playing with the initial shape of the structure, tubes or rings may be obtained, the overall shape is still round and does not present flat portions.

Controlling the shape and the deformation of inflated structures has been the focus of many studies (playing with the architecture and the material used). Inflatable paddle boards and windsurf (that are essentially flat when inflated) are based on the drop stitch technology (Fig. 6.29(a)). Two superimposed flat flexible panels are connected by a mesh of quasi-inextensible threads (typically made of nylon) of uniform length, creating a stiff yet very light structure: similarly to an I-beam, the two flat panels are far from the neutral line, enhancing considerably the bending stiffness. Such structures can be easily deflated and rolled to fit inside a small car trunk. In 1956, the Goodyear company even started developing the Goodyear Inflatoplane, an inflatable experimental aircraft, that has been flying and tested until 1972, when the project was finally cancelled (Fig. 6.29(b)). The blooming field of soft robotics investigates the movements that can be achieved upon inflation: for instance, using a tilted mesh of fibers around a conventional cylindrical balloon leads to a net contraction upon inflation, similarly to muscles. Joseph Laws McKibben, a nuclear physicist, invented this device in 1957 for prosthetic applications, and more specifically for his crippled daughter. Twisting, bending and elongation may CONCLUSION also be triggered by disposing adequately the fibers [START_REF] Gorissen | Elastic inflatable actuators for soft robotic applications[END_REF].

In this manuscript, we presented a new way to envision inflatable structures based on a metric approach. Inserting a hollow mesostructure inside a rubber plate enables us to orient the stresses and thus the strains in particular directions when subjected to an internal pressure. These deformations lead to a new metric in the slender structure, which can be incompatible with the initially flat geometry. The structure thus buckles out of plane to adopt a shape minimizing its elastic energy (given the pre-strains imposed by the internal inflation). This idea was investigated in chapter 3 for elongated cavities with rectangular cross section, leading to a strong anisotropy in the local strain. We showed that circular (respectively radial) channels and subsequent radial (respectively azimuthal) expansions lead to cones (respectively anti-cones) on inflation. The critical buckling pressure was also derived for cones using the theory of non-Euclidean plates. Adding a gradient in the channel density, the stretch perpendicular to the channel's direction may also be controlled and domes, helicoids or saddles can be easily programmed. We also presented a geometrically exact inverse recipe to design the channel layout coding for a target arbitrary shape when inflated at a specific pressure p. In chapter 4, we presented preliminary results on controlled isotropic in-plane deformation with cylindrical cavities or pillars. We observed and rationalized a superelastic instability, with the apparition of highly-stretched bulges that propagate through the plate at constant pressure, leading to the out-of-plane buckling of the initially regular and homogeneous structure.

In the second part of this thesis, we turned to quasi-inextensible thin sheets to promote more structural stiffness. In chapter 5, we started with simple ribbons of constant width: straight ribbons classically become cylindrical air beams with circular cross-section when inflated. Rings, however, buckle out of plane and present a rich wrinkle pattern. We showed that this instability is the consequence of an incompatible curvature of the inflated ring. Open curved paths indeed overcurve when pressurized and we rationalized the crosssection's profiles, the overcurvature and the main features of the wrinkling pattern with the optimization of the enclosed volume under inextensibility constraint, neglecting the bending stiffness of the sheet.

We finally studied a related system consisting in two flat superimposed fabric sheets sealed together along a specific pattern of lines. Upon inflation, the structure bends in order to maximize its volume, given the complex constraints imposed by the seams. At the local scale, the sheets bend between two locally parallel seam lines to form a quasicircular cross section, yielding an in-plane contraction of magnitude 2/π. Changing the orientation of the seam lines, the local contraction direction changes and the metric of the structure may be distorted in a non-Euclidean way. The direction of the seam lines appears to be analogous to the orthogonal dual of the director field in liquid crystal elastomers. Printing logarithmic spirals patterns, cones and anti-cones may be programmed, coupled with material rotation in the structure. Spiral patterns may be extended to program any axisymmetric shape. We then proposed two strategies to gain one degree of freedom in the metric distortion, namely the varying width of the seam line and the angle of zigzag patterns. We used this additional lever to program helicoids and axisymmetric shapes. We also investigated the local folding of flat panels connected by sharp edges, revisiting the concept of "non-isometric origami" developed in the liquid crystal elastomers community. We now aim at solving numerically the general inverse problem and presented finally the first results towards this goal. This PhD is a preliminary experimental study of pneumatic shape morphing structures and many questions remain unanswered. The mechanical theory should be deepened and generalized. Specifically, shaping dynamics has been barely investigated and entails fundamental questions, since the metrics distortion and the subsequent shaping impacts the frequency modes of the structure. For complex shapes, the actuated structure could be stuck in some undesired minima and the selection of one or the other embedding of the target metric remains an open question.

In terms of designs, it would be very interesting to break the up-down symmetry in our structure to induce a bilayer effect, as in the very recent work of Zhou and coworkers [START_REF] Zhou | Biasing buckling direction in shape-programmable hydrogel sheets with through-thickness gradients[END_REF] in the context of hydrogel swelling: this bias induces a preferred direction of bending. More interestingly, the components of the target shape tensor may potentially be programmed (the degrees of freedom and limitations of such target curvatures remain mostly unknown). The stacking of several independent layers or even the embedding of several independent networks in one layer could open the way to interesting dynamics, the final shape possibly depending on the order of the inflation. The zigzag pattern strategy, efficiently used in the context of inflatable mazes to gain one degree of freedom, could be applied to the baromorphs.

Although both strategies (baromorphs and inflatable mazes) rely on completely different mechanisms, they both lead to similar homogenized metric distortions: the lengths along the channels remain unchanged and expand (baromorphs) or contract (inflatable mazes) in the perpendicular direction, by an amount that can be adjusted (channel density, seam line width). It is also interesting to point out the surprising fact that we could find a simple analytic recipe to solve the inverse problem in the case of baromorphs and not in the case of inflatable mazes. Baromorphs are relatively more demanding to manufacture. They are inherently soft (and thus cannot sustain their weight at the meter scale) and present the advantage to adopt a continuous family of shapes that are smooth at the channel scale. The amplitude of the shape is thus tunable, which might open the way for medical applications.

In contrast, inflatable mazes are very easy to manufacture. They become much stiffer when inflated and can consequently be used to make large scale structures. The surface is not smooth, since it is made of a collection of small air beams. The shape is also mostly independent of the applied pressure (when the pressure is sufficiently large to reach the infinitely bendable regime).

To conclude, both techniques are interesting from a fundamental/academic point of view, since they offer a way to circumvent (and benefit from) the geometric limitations imposed by Gauss' Theorema Egregium to produce for the first time relatively stiff and

Introduction 1 2 . 1 3 Baromorphs 3 . 1

 121331 Elasticity, Geometry and Morphing 1.1 First strategy to change the shape: bilayer effect . . . . . . . . . . . . . . . 1.2 Stretching vs. Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Geometry in plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Differential geometry of surfaces . . . . . . . . . . . . . . . . . . . . 1.3.2 Gauss Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Non-Euclidean plate theory . . . . . . . . . . . . . . . . . . . . . . . 1.4 Examples from Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Synthetic shape morphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . At small deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 At finite deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction and idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Tutorial -Manufacturing baromorphs . . . . . . . . . . . . . . . . . . . . . 3.2.1 Design of the mould . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Printing of the mould . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Manufacturing protocol . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Possible failures and mitigation strategy . . . . . . . . . . . . . . . . 3.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Experimental target strain measurement . . . . . . . . . . . . . . . . . . . . 3.4 Theoretical target strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Naive analytical approach . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS

Figure 1 :

 1 Figure 1: Inflating to shape. (a) Baromorphs: network of airways embedded in a rubber plate coding for the family of shapes. (b) Programming of the outline of curved mylar balloons. (c) Inverse problem for shape morphing of welded fabric structures.

Figure 1 . 1 :Figure 1 . 2 :

 1112 Figure 1.1: Same pine cone closed and opened, depending on the humidity: wet (left) and dry (right) respectively. Credit : https://www.warrenphotographic.co.uk/08748pine-cone-open-and-closed

1. 1 .

 1 FIRST STRATEGY TO CHANGE THE SHAPE: BILAYER EFFECT

Figure 1 . 3 :

 13 Figure 1.3: Multigait soft robot based on the bilayer effect from Shepherd et al.[START_REF] Shepherd | Multigait soft robot[END_REF]

Figure 1 . 7 :

 17 Figure 1.7: Geometry on surfaces. No matter how a surface is isometrically folded in 3D (extrinsic curvature), the Gaussian curvature (or intrinsic curvature, the product of the two local main curvatures) is conserved at any point. It is thus impossible to fold a flat sheet of zero Gaussian curvature (a) onto a sphere (positive Gaussian curvature (b)) or a saddle (negative Gaussian curvature (c)) without modifying the distances along the surface.

Figure 1 . 8 :

 18 Figure 1.8: The distance D of a neighbouring point Q to the tangent plane at P of a surface is half the second fundamental form.

Figure 1 . 11 :

 111 Figure 1.11: Paper cones and anticones, reproduced from Dervaux et al. [56]. Closing a disc with a missing sector (a) leads to a cone (b). Adding a surplus sector into a full disc (c), we get an anti-cone or e-cone (excess angle cone).

Figure 1 . 12 :

 112 Figure 1.12: Swelling hydrogel shape morphing. (a) Radially controlled isotropic swelling rate of NIPAM hydrogels leading to the buckling of initially flat discs by Klein et al. [13]. (b) Halftonetechnique for the 2D coarse-grained control of the local swelling rate in hydrogels. A sphere may be programmed with this technique.[START_REF] Kim | Designing responsive buckled surfaces by halftone gel lithography[END_REF]. (c) Direct printing of hydrogels swelling anisotropically due to flow induced fibre orientations.[START_REF] Gladman | Biomimetic 4d printing[END_REF]

Figure 1 . 13 :

 113 Figure 1.13: Liquid crystal elastomer shape morphing. (a) Orientation of the director field controls the anistropic contraction direction in the nematic phase.A structure with spikes is programmed upon heating[START_REF] Ware | Voxelated liquid crystal elastomers[END_REF]. (b) Inverse problem in order to program an arbitrary shape (here a face), finding the optimal director field leading to the target shape on heating.[START_REF] Aharoni | Geometry of thin nematic elastomer sheets[END_REF][START_REF] Aharoni | Universal inverse design of surfaces with thin nematic elastomer sheets[END_REF]

1. 5 .

 5 Figure 1.14: Shape-morphing with dielectric elastomers. (a) Secondary buckling pattern in a growing circular patch surrounded by a passive sheet [19]. (b) Dome programmed with spatially varying electric fields [20] (c) Cone programmed using the anisotropy of composite dielectric elastomers [21].

Figure 1 . 16 :

 116 Figure 1.16: Inflatable structures. (a)-(d) Finite stretch regime with rubber-based materials p ∼ Et/L: some examples of inflatable structures in which the material is significantly stretched by the pressure forces: (a) Dog sculptures made of elongated rubber balloons, (b) inner tubes of tyres, (c) a soft robotic gripper. In this regime, we present in the second chapter of this thesis a inflatable shape morphing strategy, referred to as baromorphs (d). (e)-(i) Inextensible highly bendable regime Et 3 /L 3p Et/L: in this doubly asymptotic regime, the sheet may be considered as quasi-inextensible but with vanishing bending stiffness. Mylar balloons (e) and numbers (f), scientific ballooning (g), or large inflatable architectures (h) (source: buildair.com) are typical examples from day life. In Chapter 6, we will investigate a shaping strategy in this regime (i).

  2.1).

Figure 2 . 1 :

 21 Figure 2.1: Pressurized elastic tube. (a) Schematic of the tube of interest: inner radius R, thickness e, applied pressure p which induces stresses σ θθ and σ xx in the membrane. (b) Balancing the forces and using Hooke's law, the strains may be computed as a function of the dimensionless pressure pR/(Ee). Interestingly, no deformation occurs along the tube direction e x .

Figure 2 . 2 :

 22 Figure 2.2: Pressurized hyperelastic tube. (a) Picture of an inflated cylindrical balloon. We observe the coexistence of two phases: one highly stretched and the other slightly stretched. (b)Deformations as a function of the non dimensional pressure pR/(µe) using a Gent model for the elastomer for various values of the first invariant limit J m (respectively 10, 20 and 30 from lighter to brighter colors). For long enough polymer chains, we obtain a S-shape for the strain-pressure curve. It implies that for one given pressure, two very different states are possible, as observed in the experiments.

Figure 3 . 1 :

 31 Figure 3.1: Principle of pressure actuated baromorph plate. a, Schematic of actuation: the pressure inside the airways induces anisotropic inflation of the plate (higher strain normal to the airways than along the channels). b, 3D printed mould used to cast the baromorph illustrated in panel c. c, Actuation of the plate: suction (left) tends to contract the plate in the azimuthal direction, leading to a bowl (positive Gaussian curvature), while inflation (right) leads to an excess angle and a transformation into a saddle shape (negative Gaussian curvature). d, Evolution of the cap of an acetabularia alga from a bowl to a saddle shape due to a preferential growth in the azimuthal direction (adapted from Serikawa et al. [71])

Figure 3 . 2 :

 32 Figure 3.2: (a) Picture of a 3D printed mould. (b) Dimension and parameter definition of the airways.

Figure 3 . 3 :

 33 Figure 3.3: Manufacturing steps of a baromorph. (a)The degassed elastomer mixture is poured into the clean mould. In parallel, the rest of the mixture is poured on a clean flat surface. (b) A flat transparent plate (typically made of plexiglas is placed on top of the mould in order to ensure a controlled constant thickness of the top membrane. The mixture on the flat surface is spread with a spreader letting a controlled film thickness e. Note the mismatch between the gap set by the spreader and the thickness of the obtained film. After the mixture has cured (c), we may remove the elastomer (d). In parallel, a thin layer of the same elastomer, newly mixed, is spread on the flat membrane and will serve as a glue. (e) The moulded part is placed on the flat membrane and gently pressed in order to make sure that there is contact everywhere. (f) Once the thin layer is cured, the exceeding area of the membrane may be cut.

Figure 3 . 4 :

 34 Figure 3.4: Top view pictures of the speckled baromorph ring before (a) and after (b) inflation.The deformation is tracked using DIC program CorreliQ4 on Matlab[START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF]. The outer diameter of the ring is 8 cm.

  .

  4.

Figure 3 . 5 :

 35 Figure3.5: Extraction of the radial and azimuthal strain (respectively perpendicular and parallel to the channels) from the displacement measurements. (a) The contrast between the ring and the background is enhanced through brightness threshold. The polar coordinates origin is located at the center of the ring. (b) For each small angular sector and various pressures, the radial displacement u(r) is computed and plotted as a function of the radius, both expressed in pixels. (c) The radial strain ε ⊥ = du/dr computed from the slope of the linear fit of the previous plot, is then plotted for each angle value θ. (d) The azimuthal strain ε is computed for each orientation angle θ by averaging the ratio displacement over radial distance u/r.

Figure 3 . 6 :

 36 Figure 3.6: Average radial and azimuthal strain as a function of the pressure. Errorbars correspond to the standard deviation. The object is made of polyvinylsiloxane (Elite Double 8), of Young's modulus E = 250 ± 15 kPa,

Figure 3 . 7 :

 37 Figure 3.7: Schematic cross section of a baromorph plate.

Figure 3 . 8 :Figure 3 . 9 :

 3839 Figure 3.8: Target strains in the baromorph plate parallel (ε ) and perpendicular (ε ⊥ ) to the channels direction as a function of pressure. (a) Target strain for different values of Φ with Ψ = 0.69±0.05. (b) Target strain for different values of Ψ with Φ = 0.5±0.02. Solid lines correspond to the model without any fit parameter (in our simplified model ε vanishes). The baromorph rings are made of polyvinyl siloxane (Elite Double 8 from Zhermack) of Young's modulus E = 250 ± 15 kPa.

Figure 3 .

 3 Figure 3.10: (a) Shape programming of a cone. Stretching the radii of a flat disk by an amount (1 + ε t r ) = 1/ cos α leads to a cone of slope tan α. (b) For a baromorph, this can be obtained by a configuration made of concentric and regularly spaced circular air channels.

Figure 3 .

 3 Figure 3.11: (a)-(f) Actuation of a regular concentric circular airways pattern for growing values of the pressure ((a) corresponds to a suction and (b)-(f) correspond to the data points shown in (h)). (g) Concentric circular channel pattern. (h) Corresponding slope angle α as a function of the pressure. The solid line corresponds to the geometric prediction cos α = 1/(1 + ε r (p/E, Ψ, Φ)).

Figure 3 . 12 :

 312 Figure 3.12: Collapse of the instability threshold for various plates when pressure is rescaled by the typical critical pressure p c from equation 3.27. Red diamonds (Ψ = 0.78 ± 0.05, Φ = 0.5, R = 50 mm, H = 43.8 ± 0.2 mm); Blue triangles (Ψ = 0.74, Φ = 0.5, R = 40 mm, H = 5.4 mm); Green flags (Ψ = 0.67, Φ = 0.5, R = 40 mm, H = 6.0 mm); Orange squares(Ψ = 0.60, Φ = 0.5, R = 40 mm, H = 6.7 mm); Grey circles (Ψ = 0.47, Φ = 0.5, R = 50 mm, H = 6.4 mm); Purple flags (Ψ = 0.68, Φ = 0.2, R = 50 mm, H = 4.4 mm).

Figure 3 . 13 :

 313 Figure 3.13: Effect of the pressure on both bending and stretching stiffness of a baromorph cantilever beam for a fixed geometry (Φ = 0.5, Ψ = 0.69, H = 4.4 mm, w = 30 mm). (a) Picture of a baromorph cantilever bending under its own weight. The channels are in the longitudinal direction. (b) Shape of a sagging beam for different clamping lengths and applied pressures. Experimental shapes (continuous lines on the left side) are compared with theoretical predictions (dashed lines on the right side). (c) True stress as a function of the strain for different applied pressures. The theoretical prediction is plotted as a dashed line.

Figure 3 . 14 :

 314 Figure 3.14: Evolution of the angle α as a function of the pressure. Symbols correspond to Red diamonds (Ψ = 0.78 ± 0.05, Φ = 0.5, R = 50 mm, H = 43.8 ± 0.2 mm); Blue triangles (Ψ = 0.74, Φ = 0.5, R = 40 mm, H = 5.4 mm); Green flags (Ψ = 0.67, Φ = 0.5, R = 40 mm, H = 6.0 mm); Orange squares(Ψ = 0.60, Φ = 0.5, R = 40 mm, H = 6.7 mm); Grey circles (Ψ = 0.47, Φ = 0.5, R = 50 mm, H = 6.4 mm); Purple flags (Ψ = 0.68, Φ = 0.2, R = 50 mm, H = 4.4 mm).

Figure 3 . 15 :

 315 Figure 3.15: Experimental (continuous lines) versus computed (dashed lines) cone profiles for various values of the applied pressure, with Φ = 0.5, Ψ = 0.75, R = 50 mm, H = 4 mm.

Figure 3 . 16 :

 316 Figure 3.16: Axisymmetric baromorphs. (a) Portion of a sphere: corresponding channel pattern with an increasing density of concentric channels along the radius (left), pictures of the actual structure for various pressures (top) and the prediction of the axisymmetric model from equations 3.37 and 3.38(bottom). (b) Gaussian-like shape: corresponding channel pattern with a decreasing density of concentric channels along the radius (left), pictures of the actual structure for various pressures (top) and prediction of the axisymmetric model (bottom). The theory and the experiments do not exactly correspond to the same parameters.

Figure 3 .

 3 Figure 3.17: Negative Gaussian Curvature baromorphs. (a) Anti-cone. Pattern made of radial regularly spaced channels (left), picture of the inflated structure (center) and its corresponding Gaussian curvature, concentrated around the apex (right). (b) Saddle shape. Pattern made of radial channels with a gradient in channel density (left), picture of the inflated structure (center) and its corresponding negative Gaussian curvature distributed on the surface (right).

Figure 3 . 18 :

 318 Figure 3.18: 3D scanning technique. (a) Picture of one of the four reference patterns projected on a white planar surface. (b) Distorted pattern on the 3D surface of the baromorph. (c) Reconstruction of the 3D surface with our program. (d) Gaussian curvature computed with a local quadratic fit of the surface.

Figure 3 . 19 :

 319 Figure 3.19: Large angle truncated cone and helicoid. (a) Pattern for the truncated cone. (b) Pictures of the structure before and after inflation. (c) Schematic of the geometry of a truncated cone. (d) Pattern for the helicoid. (e) Picture of the baromorph before (flat) and after (twisted) inflation. (f) Schematic of the geometry of an helicoid:

Figure 3 . 20 :

 320 Figure 3.20: Shape programming of a face. The target shape (a) and the corresponding contour lines (b), along which the network of channels are displayed (c) with a local width computed to give rise to the target metrics upon inflation at a given pressure p. (d) and (e): two pictures of the deformed baromorph, made of Dragonskin 10 Medium from Smooth-On.

Figure 3 . 21 :

 321 Figure 3.21: Mask programmed with a cut : (a) Network of airways embedded in the plate, the red line highlighting the cut; (b) Experimental shape transformation.

Figure 3 . 22 :

 322 Figure 3.22: Pictures at various pressure of a baromorph made of concentric channels with a gradient in channels density (more channels towards the center than in the outer region). Upon suction, channels collapse radially and the structure buckles onto a saddle shape, whereas the structures shifts to a spiky Gaussian-like structure upon inflation, its amplitude varying continuously with the pressure.

Figure 3 . 23 :

 323 Figure 3.23: Fast actuation of a baromorph plate at approximately 3 Hz. Left: amplitude A of the deformation as a function of time. Right: two pictures of the object in the fully inflated and deflated states.

Figure 3 . 24 :

 324 Figure 3.24: Scalability of the baromorphs: (a) Shape of structures (made of Elite Double 16 Fast from Zhermack) with the same design at different scales and three different pressures: R = 15 mm, h = 1 mm, e = 0.4 mm, d = d w = 0.4 mm for the smallest sample, all dimensions multiplied by 2 and 4 respectively for the larger ones.(b) Deflection of the same baromorphs under gravity when clamped at one edge.

Figure 3 . 25 :

 325 Figure 3.25: Two pictures of the Airwood technology developed by Arca Ebenisterie. (Left) playing with the boundary, different bas-reliefs may be obtained. (Right) a football emerging from the flat wooden board.

Figure 3 . 26 :

 326 Figure 3.26: Making the Ludum table. (a) Casting the silicon mould on the backing plexiglas mould. (b) Moulding the actual structure in polyurethane elastomer. (c) Detail of the object, the rendering is very fine and the path of the CNC machine may be seen. (d) Polyurethane structure buckles into a truncated cone upon inflation. (e) The same structure covered by a finely cut wood sheet. (f) Jonas and Baptiste making the finishing touches.

Figure 3 . 27 :Figure 3 . 28 :

 327328 Figure 3.27: Ludum table at the "Révélations" exhibition.

  Another potential application of the baromorphs technique is the development of new medical tools. We met the Doctor Jean Bergounioux, head of the paediatric intensive care and reanimation unit in Garches hospital Raymond Poincaré. He shared with us the complications met during long-lasting intubation of patients. Intubation is the insertion of a flexible plastic tube into the trachea to maintain an open airway (see Fig 3.29(a)).

Figure 3 .

 3 Figure 3.29: (a) Intubation of a patient: an endotracheal tube is inserted in the trachea and a cuff is inflated in order to prevent leakage. (b) Different types of inflating cuffs: they are made of thin elastic membrane, with varying designs and stiffnesses. (c) The cross section of the trachea has a horse shoe shape and is not symmetric.

3. 9 .Figure 3 .

 93 Figure 3.30: (a) 3D schematic and cross section of a tubular baromorph in order to fit the shape of the trachea. (b) First prototypes of cylindrical baromorphs before (left) and after (right) inflation.

Figure 3 . 31 :

 331 Figure 3.31: Two layers baromorph. (a) Two identical channels patterns are stacked in a single structure. When a higher pressure is exerted in the upper or lower layer, the structure tends to curve downwards and upwards respectilvely. (b) Snapping instabilities may be triggered by out of phase actuation of the two layers.

3. 1 3 . 3

 133 Introduction and idea . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Tutorial -Manufacturing baromorphs . . . . . . . . . . . . . . . . 3.2.1 Design of the mould . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Printing of the mould . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Manufacturing protocol . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Possible failures and mitigation strategy . . . . . . . . . . . . . . 3.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental target strain measurement . . . . . . . . . . . . . . 3.4 Theoretical target strain . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Naive analytical approach . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Linear homogeneization . . . . . . . . . . . . . . . . . . . . . . . 3.5 Shape changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Starting with a cone . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Scaling law for buckling threshold. . . . . . . . . . . . . . . . . . 3.5.3 Axisymmetric plate equations for circular channels . . . . . . . . 3.5.4 Geometric programming of axisymmetric shapes . . . . . . . . . 3.5.5 Programming negative Gaussian curvature shapes . . . . . . . . 3.5.6 3D scanning and computation of Gaussian curvature. . . . . . . 3.5.7 Other examples of lateral expansion: truncated cones and helicoids.

3. 6 Figure 4 . 1 :Figure 4 . 2 :

 64142 Figure 4.1: Concept of isotropic baromorphs : (a) 3D printed mould; (b) Schematic top view: a regular hexagonal lattice of cylinders of different radii; (c) Schematic radial cut of the isotropic baromorph: a small channel interconnects the cavities. (d) Pictures of the baromorph at rest and inflated

Figure 4 . 4 :

 44 Figure 4.4: Upon inflation, the baromorph deforms, changing thus the actual ψ and φ parameters, leading to geometric non-linearities. In the model proposed, the deformed pillars are approximated by cylinders, neglecting the boundary effect on the actual shape of the pillar, and accepting the deformation mismatch at the boundary pillar-membrane (the pillars shrink in the xy-plane, whereas the membrane is stretched).

Figure 4 . 5 :

 45 Figure 4.5: Fit of a tensile test on a ribbon made of Elite Double 8 from Zhermack with the incompressible Gent model: the two parameters, namely the shear modulus µ and the limited value of the first invariant J m are found to best fit the experimental curve for the values of 7.2.10 4 Pa and 14.2 respectively.

Figure 4 . 6 :

 46 Figure 4.6: (a) Deformation response of the plate as a function of the dimensionless applied pressure p/µ for various values of Ψ, Φ and J m . Triangles correspond to experimental measurements, solid lines to the theoretical response. Snapshots of the plate are shown in insets, corresponding to some data points on the blue curve. The deformation may be easily measured by following the evolution of the average distance between pillar centres. (b) Minimum value of J m (color scale) in order to have the superelastic instability as a function of the geometrical parameters Ψ and Φ.

Figure 4 . 7 :

 47 Figure 4.7: Series of pictures of a large plate embedding a regular hexagonal lattice of pillars upon inflation. The boundary effect imposed by the outer wall is thus reduced. Nevertheless, bulges appear in the plate and then propagate through the whole structure, at a nearly constant pressure.

Figure 4 . 8 :

 48 Figure 4.8: Baromorphs with a deployable stiffening structure. (a) pattern of the kirigami embedded at the top and bottom of the plate. The structure may be stretched along the zigzag direction. (b) and (c) First experimental tests with the soft swelling material from Miriyev et al.[START_REF] Miriyev | Soft material for soft actuators[END_REF]. Upon heating, the initially flat structure buckles onto a conical shape prescribed by te kirigami pattern.

Contents 4 . 1 81 4. 2 83 4. 3 86 4. 4

 41812833864 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model and assumptions . . . . . . . . . . . . . . . . . . . . . . . . Experimental realisation and quantitative comparison with the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

CHAPTER 5 .Figure 5 . 1 :

 551 Figure 5.1: Flat inflatable (a) The number "9" before and after inflation: the initially open outline over-curves and self contact appears closing the nine. (b) Details of the surprising wrinkling pattern on the envelope: hierarchical radial wrinkles appear except on two parts: at the inner line and along a band between the mid line and the outer line, the latter being compressed.

Figure 5 . 2 :

 52 Figure 5.2: Schematics of the inflation of an infinitely long rectangle balloon of width w.

Figure 5 . 3 :

 53 Figure 5.3: Dido building Carthage, William Turner, 1815

Figure 5 . 4 :Figure 5 . 5 :

 5455 Figure 5.4: Schematic of a ring of inner radius R and width w before and after inflation. Axes and coordinates are defined.

Figure 5 . 6 :

 56 Figure 5.6: (a) Actual shape of military parachutes. (b) Pumpkin balloon made by Sir G.I. Taylor.

Figure 5 . 8 :

 58 Figure 5.8: Cross-section of a closed inflated annulus. (a) Profile picture of a closed inflated annulus, the axisymmetry of the structure being constrained by two plates to prevent out-of plane buckling. The ring is made of a 16 µm thick polypropylene sheet of inner radius R = 25 mm and outer radius R + w = 130 mm. Dashed lines correspond to the theoretical cross sections. Note that the wrinkles extend through the whole torus. (b) Theoretical (solid lines) and experimental (triangles) rescaled cross sections of inflated closed rings for various aspect ratios S = R * /(1 + R * ) with R * = R/w. r * = r/w and z * = z/w correspond the rescaled radial and vertical coordinates, respectively.

Figure 5 . 9 :Figure 5 .

 595 Figure 5.9: Photographs of two experimental realizations of inflating an annulus: the structure surprisingly buckles out of plane. For textile, allowing some shear deformation, the structure is smooth (a), whereas two opposed folds appears for thin polypropylene sheets (b).

Figure 5 . 11 :

 511 Figure 5.11: An opened ring over curves upon inflation by an angle ∆α.

  )

Figure 5 . 12 :

 512 Figure 5.12: Overcurvature of an open torus. (a) Picture of a flat curved path, before and after inflation: the inflated structures coils, inducing an overcurvature factor λ = α/α 0 . (b) Dimensionless perimeter difference u * as a function of the curvilinear coordinate s * for various overcurvature strains λ, in the case R w (circular section upon inflation). u * = 0 in the flat state; green line, upon inflation with λ = 1; red line, for ∆α tens 137 • (the red cross indicates the abscissa s * tens under tension); dashed blue line, for λ = 1.5 (this solution is not physically relevant since it implies azimuthal extension). (c) Experimental and theoretical (solid black line) target curvature change as a function of the slenderness S. Triangles: experiments with 16 µm thin sheets of polypropylene; diamonds: experiments with 4 µm thin sheets of polyethylene. (d) Wrinkles are absent along a band of finite width, highlighted in blue. The red line corresponds to the theoretically calculated profile (within the limit of inextensibility) and the red crosses mark the positions of the tensions line for R/w = 3.

Figure 5 .

 5 Figure 5.13: (a) Rescaled theoretical profiles for overcurvature factors λ varying between 1 and 1.5 for an inner outer ratio R * /(R * + 1) = 0.1. The black line is the profile for λ max = 1.29 that maximises the total volume. (b) Rescaled perimeter contraction u * as a function of the curvilinear coordinate s * . When overcurvature is increased, a region which is not under compression appears and then expands. (c) Evolution of the volume as a function of the overcurvature factor λ.As long as no tension line appears, the profile is almost the same (the only difference comes from the fact that the inner radius is not R anymore but R/λ, the profile is thus selected in the family of curves of Fig.5.8). When compression vanishes at one point, the condition u * ≤ 0 plays its role, and the shape strongly deviates from the family of curves in the closed ring condition. The portion under vanishing compression extends until a maximal volume is encompassed by the inextensible envelope.

Figure 5 . 14 :

 514 Figure 5.14: Experimental and theoretical cross-section profiles of open annuli. The theoretical results from the Lagrangian describe accurately the experimental cross sections. Contrary to close ring profiles, open rings ones do not significantly depend on the slenderness S of the ring.

Figure 5 . 15 :

 515 Figure 5.15: Comparison of opened and closed ring cross sections for a slenderness S = 0.1. (a) Theoretical predictions: the shapes significantly deviate in closed (solid line) and open (dashed line) condition. (b) Picture of an inflated open ring. In order to avoid boundary effects, flat rigid panels of the shape of the predicted cross section are sealed on both ends of the open ring. c-d: superposition of the experiments (red line on the picture) and the theoretical curves (blue lines) for open (c) and closed (d) conditions

Figure 5 . 16 :

 516 Figure 5.16: Evolution of the overcurvature upon inflation. Coiling factor λ as a function of the slenderness: triangles and circles represent actual experiments with curved paths made of 4 µm thick polyethylene and 16 µm thick polypropylene sheets respectively. The black solid line is the theoretical prediction based on our model.

Figure 5 . 17 :

 517 Figure 5.17: Location of the wrinkles. (a) Upon inflation a "C" mylar balloon coils to form the letter "O" and exhibits a regular wrinkling pattern. (b) Superposition of the predicted profile and a picture of the balloon, at a point where the slenderness S = 0.26. The location of the wrinkles can be interpreted in our model as the regions where the perimeter compression u * < -0.01. A flat, uncompressed region is indeed observed on the balloon.

Figure 5 . 18 :

 518 Figure 5.18: Programming of a lemniscate. (a) Target outline, (b) normalized curvature of the lemniscate κ tar and of the fabrication state 1/R * , for a given path width w, as a function of the curvilinear coordinate v * = v/w. The flat state curvature 1/R * is computed using equation [5.21]. The corresponding pattern on top of a picture of the structure in the flat state is shown in (c). (d) Upon inflation, the structure coils into the target lemniscate outline.

Figure 5 . 19 :

 519 Figure 5.19: Inverse problem for getting any curved shape. (a) and (e), target path; (b) and (f), normalized target and rest curvature κ tar and 1/R * for a given path width w, as a function of the curvilinear coordinate v * = v/w of a portion of the target path, highlighted in (a) by a dashed box. The curvature of the flat path is computed using the prediction for curvature change plotted in Fig. 5.16, that is according to equation 5.21. (c) and (g), flat path computed by the inverse model on top of a photograph of the experimental realization. (d) and (h), same path under pressure, fitting closely the target curve.

Figure 5 .Figure 5 . 21 :

 5521 Figure 5.20: a-b. Two configurations of sharp angles of π/2 at rest. c-d. Corresponding shapes obtained upon inflation. A change in angle is observed, independent of the outer path geometry.

Figure 5 . 22 :

 522 Figure 5.22: From left to right: flat state before inflation, buckled structure for intermediate pressures and a planar swim ring for high pressure. Note the amount of stretching of the inner seam, the red dashed lines corresponding to the initial inner perimeter. Wrinkles are not present in the inner part of the swim ring, indicating tension in the sheet.

Contents 5 . 1 99 5. 2

 51992 Straight ribbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curved paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.1 Closed rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.2 Inflating open rings . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2.3 Inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 6 . 2 :

 62 Figure 6.2: Strategies for welding. (a) Printing the welding pattern using a soldering iron mounted on a 2-axes robot. (b) Heat press of a "sandwich" made of the two fabric sheets and a baking paper sheet with specific cuts where the two superimposed sheets fuse together.

Figure 6 . 3 :

 63 Figure 6.3: Schematic principle of the deformation upon inflation. (a) Upon inflation, the cross section between two locally parallel seams of width e distant by w deforms onto a nearly circular cross section, causing an in-plane contraction of magnitude λ 2/π. (b)Taking into account the finite width e of the seam-line, the actual contraction may be easily computed (see (c)). Green triangles are experimental measurements with a TPU-coated nylon fabric 20 den from Extremtextil, the solid lines correspond to the theoretical prediction for e = 1.3 mm. (c) The unit repeat length has a fraction ξ of weld and 1 -ξ of free sheet. A section before and after inflation. The direction of contraction, n is equivalent to a nematic director in a liquid crystal solid. The contraction of the 1 -ξ fraction is by a factor of 2/π in the ideal case, or by a factor > 2/π if there is length taken up by the bend near the seam.

6. 3 .Figure 6 . 4 :

 364 Figure 6.4: Radial and azimuthal directors: (a) An Archimedean spiral approximating azimuthal channels (radial n), results in an anticone ("ruff") on inflation, (b) Radial channels (azimuthal n) induce a cone of half summit angle ϕ = sin -1 λ. (c) Angular surplus of 147 • or (d) angular deficit of 108 • , made apparent when a disc has instead a radial cut and does not change its topography.

  r(θ) = r(0)e bθ , with b = cot(α). (6.8) CHAPTER 6. INFLATABLE MAZES

Figure 6 . 6 :Figure 6 . 7 :

 6667 Figure 6.6: Anti-cones and cones arising from channel systems with director angles α to the radii shown. Note that the channels at are the complement of α. The critical angle where the inflated structure remains flat is an α c = sin -1 (1/ √ 1 + λ) ≈ 50 • , as predicted by the theory.

Figure 6 . 8 :

 68 Figure 6.8: Programming of structures with constant Gaussian Curvature. (a) Pattern and (b) photographs of the inflated structure for a portion of a sphere (constant positive GC). The dashed line represents the target profile. (c) Pattern and (d) photographs of the inflated structure for a saddle shape (constant negative GC).

Figure 6 . 9 :

 69 Figure 6.9: Subdivision of inflated channels. The contraction rate λ is locally impacted by the width change of the airways, more particularly by the resulting higher proportion ξ of seams.

Figure 6 .Figure 6 . 11 : 1 2

 66111 Figure 6.10: A reference disc where a point at radius r maps on inflation to a point on a shell given by (γ 1 , γ 2 ) and geodesic radius u.

Figure 6 . 13 :

 613 Figure 6.13: Sectors of uniform director, simply arranged parallel (a) or perpendicular (b) to the bisector, or parallel to the sides (c), necessarily having an internal line of R-1C along the bisector. The director fields just outside the sectors are shown, indicating how these regions connect to their neighbours across the boundary vectors t. The right triangles with sides a and b give simple rules for the change of apex semi angle χ/2 to χ /2 after distortion. Below each is a pneumatic realisation, with triangular sections opening out (a), or closing in (b) and (c), with the last example having the upper side of the triangle developing an angle.

Figure 6 . 14 :Figure 6 . 15 :

 614615 Figure 6.14: Attempting a non isometric cube. In contrast with LCE, the inflated structure tends to bend along the weak lines of the seam and not at the R-1C boundary. (a) Pattern for a non isometric cube in LCE [167] for a contraction λ = 0.58, the red lines being the target edges of the cube. The green lines are the actual edges of the inflated structures. (b) Printed flat structure. (c) On actuation, the structure buckles onto a structure made of flat panels connected by sharp edges (highlighted by the green lines)

Figure 6 .

 6 Figure 6.16: (a) A pattern of channels schematically in flat space (left), that will inflate to a dodecahedron (right). The green lines show the position of the edges. (b) Upon inflation, the structure gradually closes to form the target dodecahedron.

Figure 6 . 17 :Figure 6 .

 6176 Figure 6.17: An array of alternating positive (+1) and negative (-1) topological defects. (a) Pattern and picture of the flat state. (b) Upon inflation, the structure exhibits mountain peaks and troughs (+1) separated by saddles (-1). (c) Each saddle may be snapped to inverse the two upper and lower corners and the structure may fold along the seams connecting peaks and troughs enlarging remarkably the family of stable shapes that may be reached.

6. 6 .Figure 6 . 19 :

 6619 Figure 6.19: Simple shape programming varying the relative width of the seam ξ. (a) A hemisphere with purely radial seam lines and its corresponding pattern. The black portions correspond to the welded non inflated part of the structure. Although the obtained shape is close to the target one (red profile), the structure is very floppy along the meridians. (b) A saddle with purely azimuthal seam lines of varying width.(c) A helicoid made out of a pattern of parallel seam lines of varying width.

Figure 6 . 20 :

 620 Figure 6.20: Stretching of the structure perpendicular to the seam. (a) Schematic cross section of one inflated beam at rest and under traction, with the spanned angle β, the sheet strain ε, the net displacement ∆ and a zoom on the force balance at the seam. (b) Force-displacement curve for various pressures, with experimental measurements in solid lines, and theoretical prediction in dashed lines, the seam width of the line e, difficult to measure, being used as a fitting parameter. Inset: rescaled force-displacement curve in the quasi-inextensible regime: the force does not depend on the sheet's properties. (w = 5.5 mm, e = 1.7 mm, l = 75 mm, E = 4.5.10 8 Pa, and t = 0.1 mm)

CHAPTER 6 .

 6 INFLATABLE MAZESus an analytic expression for the stretching modulus perpendicular to the seam direction Y ⊥ for infinitesimal deformation.

1 Figure 6 . 21 :

 1621 Figure 6.21: Three-point bending test of a straight inflated beam. (a) Experimental setup. (b) Pictures for various imposed deflections with the superimposed theoretical profile in red dashed line. The four pictures, labeled by a,b,c,d, correspond to different stages of a bending test, the force-displacement curves being shown in (c) for a air beam of deflated width w = 9 mm made of a polypropylen sheet of thickness t = 50 µm and Young's modulus E = 2.2 GPa. (d) Experimental bending stiffness per unit width B versus theoretical bending stiffness of air beams. Experimental (e) and theoretical (f) critical moment per unit width M c as a function of pressure.

Figure 6 . 22 :

 622 Figure 6.22: Target shape (the roof of the Lilium tower) for testing the general inverse problem framework.

Figure 6 . 23 :

 623 Figure 6.23: Effect of regularization on the orientation and stretch factor. (a) Unregularized and (b) regularized patterns.

Figure 6 .

 6 Figure 6.24: (a) Stripe patterns following Knöppel et al. [173]. (b) Corresponding airways pattern with the local relative seam width in accordance with the stretch factor map in Fig. 6.23(b).

Figure 6 . 25 :

 625 Figure 6.25: Inflated simulations of flat inflatables. Mesh and sealed vertices with azimuthal (a) and radial (c) seam pattern and their corresponding shape on inflation: an anticone (b) and a wrinkled cone (d). The bending stiffness of the sheet is neglected.

Figure 6 . 26 :

 626 Figure 6.26: Inflation simulation: the obtained shape does not match at all the target shape although the metric prescription is satisfied.

1 Figure 6 . 27 : 1 -

 16271 Figure 6.27: General inverse problem. (a) Target shape, which correspond to the top of the Lilium tower. (b) Seam pattern found by the regularized optimization. (c) Numerical inflation of the corresponding structure, with its boundary fixed. (d) Differences between the target (red) and the numerically inflated shape. (e) Actual shape of a physical structure upon inflation.

Figure 6 . 28 :

 628 Figure 6.28: Truncated cones and anticones with large angle deficit/surplus, benefiting from the hollow center part to have more freedom in the shape-programming.

Figure 6 .

 6 Figure 6.29: (a) Drop-stitch fabric from Haining Gebo Textile Ltd., available at 15 $ per running meter (minimum purchase quantity: 1000 m) (b) Goodyear Inflatoplane in flight. (c) McKibben artificial muscles deflated (left) and inflated (right) (photo credit: Okoyama University).

Figure

  Figure B.1: (a) Pattern generated by the Matlab code. (b) Corresponding flat inflatable morphing into a shape close to the targeted catenoid (code line 26).

Part I Baromorphs Chapter 2 Inflation of a cylindrical rubber balloon Contents 1.1 First strategy to change the shape: bilayer effect . . . . . . . . . 9 1.2 Stretching vs. Bending . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Geometry in plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

		1.3.1 Differential geometry of surfaces . . . . . . . . . . . . . . . . . . 16
		1.3.2 Gauss Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
		1.3.3 Non-Euclidean plate theory . . . . . . . . . . . . . . . . . . . . . 18
	1	.16(d) and (i) but have similar geometrical features.

.4 Examples from Nature . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Synthetic shape morphing . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 3 Baromorphs Contents 2.1 At small deformations . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 At finite deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Table 4 .

 4 .4). 1: Material properties deduced from a simple traction test by fitting the curve with the expected response of a Gent hyperelastic model.

	CHAPTER 4. ISOTROPIC BAROMORPHS AND PROPAGATIVE INSTABILITY
	Material	shear modulus µ First invariant limit J m
	Ecoflex 0050	3, 0.10 4 Pa	23
	Elite Double 8	7, 2.10 4 Pa	14.2
	DragonSkin 10	15.10 4 Pa	18
	DragonSkin 20	21.10 4 Pa	7.6

CONCLUSION reversible self-morphing structures. The question of their potential utility naturally arises. This question that has been marginally apprehended during this PhD goes beyond the strict scope of mechanics and will be very interesting to investigate.

Appendix A

Code for generating a mold with concentric circular channels

Here, we propose an example of a code to generate a mold with concentric circular channels and a gradient in wall thickness (i.e. a gradient in the channel density Φ) using the free software OpenScad. The generated mold is shown in

Appendix B

Code for generating a log spiral seam pattern

An example of a code for generating the seam pattern to program a catenoid upon inflation (with the zigzag strategy) is presented. The target profile should be simply given on line 26 of the code. However, one should ensure that this target shape is reachable given the limited contraction factor λ. The seam pattern generated with the code is presented in lambda =0.72; % effective contraction rate R= linspace (0 ,1 ,300); % radial coordinate in the flat state . d=0.1; % maximal width of the channels % initialization of the parameters r= zeros (size(R)); % radial in -space coordinate in the deformed state z= zeros (size(R)); % vertical coordinate in the deformed state . phi= zeros (size(R));% angle between the radial direction and the seam. S= zeros (size(R)); % curvilinear coordinate along the profile lambda_theta = zeros (size(R)); % contraction in the azimuthal direction lambda_r = zeros (size(R)); % contraction in the radial direction % initializing the parameters such that there is no angle deficit at the center . phi (1)=acos (1./ sqrt (1+ lambda )); phi ( 2

Published articles

Two articles, entitled "Bio-inspired pneumatic shape-morphing elastomers" [START_REF] Siéfert | Bio-inspired pneumatic shapemorphing elastomers[END_REF] and "Programming curvilinear paths of flat inflatables" [START_REF] Siéfert | Programming curvilinear paths of flat inflatables[END_REF] are condensed versions of chapters 3 and 5 respectively. Preprints are available on open access on Benoît's website (https://blog.espci.fr/benoitroman/) and on the platform HAL.

Bio-inspired pneumatic shape-morphing elastomers

Emmanuel Siéfert, Etienne Reyssat, José Bico & Benoît Roman 2019, Nature Materials, vol. 18, no 1, p. 24 Shape-morphing structures are at the core of future applications in aeronautics, minimally invasive surgery, tissue engineering and smart materials. However, current engineering technologies, based on inhomogeneous actuation across the thickness of slender structures, are intrinsically limited to one-directional bending. Here, we describe a strategy where mesostructured elastomer plates undergo fast, controllable and complex shape transformations under applied pressure. Similar to pioneering techniques based on soft hydrogel swelling, these pneumatic shape-morphing elastomers, termed here as 'baromorphs', are inspired by the morphogenesis of biological structures. Geometric restrictions are overcome by controlling precisely the local growth rate and direction through a specific network of airways embedded inside the rubber plate. We show how arbitrary three-dimensional shapes can be programmed using an analytic theoretical model, propose a direct geometric solution to the inverse problem, and illustrate the versatility of the technique with a collection of configurations. Inflatable structures offer a path for light deployable structures in medicine, architecture, and aerospace. In this study, we address the challenge of programming the shape of thin sheets of high-stretching modulus cut and sealed along their edges. Internal pressure APPENDIX C. PUBLISHED ARTICLES induces the inflation of the structure into a deployed shape that maximizes its volume. We focus on the shape and nonlinear mechanics of inflated rings and more generally, of any sealed curvilinear path. We rationalize the stress state of the sheet and infer the counterintuitive increase of curvature observed on inflation. In addition to the change of curvature, wrinkles patterns are observed in the region under compression in agreement with our minimal model. We finally develop a simple numerical tool to solve the inverse problem of programming any 2-dimensional (2D) curve on inflation and illustrate the application potential by moving an object along an intricate target path with a simple pressure input.

Programming curvilinear paths of flat inflatables