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Traineeship Report of Kevin Soobbarayen

Master 2 Techniques Avancées en Calculs de Structures

École Normale Supérieure de Cachan

Cambridge University, Engineering Departement

Advanced Structures Group Laboratory

Supervizors : K. A. Seffen & S. D. Guest

Morphing multistable smart memory alloy shells :

Behavior of prestressed bimetallic strip

2

Abstract

This study deals with the behavior of prestressed bi-metal strips, which are capable to

change drastically their shape over a small range of temperature and under large deflexions.

We begin with an initially flat strip and a rolling-process, which applies a prestress, permit

to obtain a prestressed configuration where the tape has a plastic curvature over the chord,

and a null longitudinal curvature. The actuation of the strip is integrated and based

on both bi-metal action and an accurate control of temperature. The applied prestress

combined with this actuation give an interesting behavior for shape controlling. The study

we carried out is mainly an experimental work of this kind of strips. The aim of this part is

to characterize accuratly the behavior of a prestressed strip and to emphasize the relevant

parameters. We managed to obtain very good results because we were able to observe an

instability which occurs for a critical temperature and for a given initial curvature. We

also tried to make a modeling of this kind of strip and we investigated the model of a strip

of lenticular section and we can say that it is quiet efficient. It is able to reproduced the

flexural snap-through and two others phenomenons. The comparison between theory and

experiments is quiet good because we have a difference of less than 5%. This model shows

a hysteretic behavior which is very close to the cycle of smart memory alloy. Moreover it

seems that the strip is subject to elastic localized deformation near the configuration just

before the snap. The appearence of these two behavior is only an analytical result, and

our experiments needs to be enhanced.

Résumé

Cette étude porte sur le comportement de bandes de bi-metal précontraintes, capable de

changer drastiquement de forme sur un faible intervalle de température et avec prise en

compte des non linéarités géométriques. La précontrainte appliquée par un système de

rouleau, permet d’obtenir à partir d’une configuration initiale plate, une configuration

précontrainte où la bande possède une courbure plastique non nulle dans la largeur, et

nulle dans la longueur. L’actionnement de la bande est donc un actionnement intégré qui

s’appuie sur l’action du bi-métal et sur un contrôle précis de la température appliquée à

la bande. La précontrainte appliquée combinée à cet actionnement permet d’obtenir un

comportement intéressant du point de vue du contrôle de forme. L’étude que nous avons

menée comporte principalement une étude expérimentale de ce type de bandes. L’objectif

de cette partie expérimentale est de caractériser finement le comportement d’une bande

précontrainte et de mettre en avant les paramètres importants. Les résultats obtenus sont

plus que satisfaisants puisque l’on observe expérimentalement une instabilité qui génère

changement brutal de la courbure longitudinale, qui a lieu à une température critique et à

partir d’une courbure initiale donnée. Analytiquement, il s’avère que le modèle de plaque

à section lenticulaire permet de reproduire assez fidèlement cette instabilité et d’autre

comportement intéressant apparaissent. La comparaison entre théorie et expérience est

tout à fait satisfaisante puisque nous obtenons moins de 5% d’écart. Ce modèle met en

avant un comportement hystérétique et l’hystérésis qui apparaît est très proche du cycle

de transition entre les différentes microstructures des alliages à mémoire de forme. Enfin

il semblerait que la bande soit sujette à des déformations élastiques localisées, dans un

voisinage de la configurations avant l’instabilité. Ces deux phénomènes n’apparaissent

qu’analytiquement et une étude expérimentale plus poussée s’avère nécessaire.

3

Contents

1 Introduction

1.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2 Description of the global project . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Description of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Presentation of the problem : experimental work

2.1 Experimental procedure . . . . . . . . . . . . . . .

2.1.1 Production of bimetallic strips . . . . . . .

2.1.2 Description of an experiment . . . . . . . .

2.2 Observations . . . . . . . . . . . . . . . . . . . . .

2.2.1 Catching of the phenomenon . . . . . . . .

2.2.2 Post-processing . . . . . . . . . . . . . . . .

2.2.3 First results . . . . . . . . . . . . . . . . . .

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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3 Phenomenon modeling for two different shapes of the strip : lenticular

section and constant thickness

3.1 Large deflexions analysis : position of the problem with bi-metal action . . .

3.1.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . .

3.1.2 Problem position . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 A first approach : The strip of lenticular section . . . . . . . . . . . . . . . .

3.3 The strip of constant thickness . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 Resolution strategy with Mathematica . . . . . . . . . . . . . . . . . . . . .

3.4.1 Interests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.2 Description of the main functions . . . . . . . . . . . . . . . . . . . .

3.4.3 First results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Results

4.1 Analytical results for the strip of lenticular section . . . . . . . . . . . . . .

4.1.1 Stored strain energy analysis . . . . . . . . . . . . . . . . . . . . . .

4.1.2 Minimization of the strain stored energy for several increments of

temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1.3 Minimization of the strain stored energy for several κ

ˆ y0 . . . . . . .

4.1.4 Study of critical values of κ

ˆ x and κ

ˆT . . . . . . . . . . . . . . . . . .

4.1.5 Appearence of a hysteretic behavior . . . . . . . . . . . . . . . . . .

4.2 Experiments versus theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1 Analytical predictions versus experiments . . . . . . . . . . . . . . .

4.2.2 κ

ˆ x,crit : a constant of the problem . . . . . . . . . . . . . . . . . . .

4.2.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

6

6

7

8

10

10

10

12

12

12

14

15

16

18

18

18

20

21

22

23

23

23

24

25

27

27

27

29

30

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33

34

34

35

35

39

4.4

A

Aknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Mathematica algorithms

43

5

List of Figures

1.1

1.2

(a) Venus Flytrap (Y. Forterre, L. Mahadevan, J. Sktoheim, Nature 2005)

and the MIT plane [Pleaseinsertintopreamble] (b) Bistable screen (Cambridge Advanced Structures Group) [Pleaseinsertintopreamble] (c) Rolatube

(Cambridge Advanced Structures Group) [Pleaseinsertintopreamble] (d) Flickbracelet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Illustration of a cycle for a smart memory alloy . . . . . . . . . . . . . . . .

Pictures of a heated flat strip of bi-metal : creation of two main curvatures

κx ≥ 0 and κy ≥ 0 • If we apply a prestress which provides this kind of

curvatures, we will not have an interesting behavior . . . . . . . . . . . . . .

2.2 Pictures of a prestressed strip with opposite-sens bending • We will study

strips with this kind of curvatures κx0 = 0 and R = 1/κy0 ≤ 0 . . . . . . . .

2.3 Illustration of the rolling process • on the left : system of cylinders • on the

right : application of the prestress by the cylinders which gives a curvature

to the strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 We measure H and b with a micrometer and we calculate R • R = 1/κy0 =

b2

H

+

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8H

2

2.5 On the left : thermocouple • on the right thermal chamber . . . . . . . . . .

2.6 First sequence of picture for "small" initial curvature (R = 1/κy0 = 55mm)

• the variation of κx is continuous . . . . . . . . . . . . . . . . . . . . . . .

2.7 First sequence of picture for "high" initial curvature (R = 1/κy0 = 55mm)

• the variation of κx is not continuous, there is a critical temperature when

we can see a flexural snap-through . . . . . . . . . . . . . . . . . . . . . . .

2.8 Small deflexion analysis : correct for "small" values of temperature, i.e beL2

L4

fore snap-through • δ = Rx (1−cos(θ0 ), L = Rθ0 , so δ = Rx ( 2R

2 − 24R4 +...)

2

• If L Rx , Rx ≈ L2δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9 On the left : small deflexion analysis, we measure δ • on the right : large

deflexion analysis : we measure Rx . . . . . . . . . . . . . . . . . . . . . . .

2.10 Dimensionless force-displacement diagrams • Red : 1/κy0 = 99mm • Black

: 1/κy0 = 55mm • Green : 1/κy0 = 23mm • Blue : 1/κy0 = 15mm . . . . .

7

7

2.1

3.1

3.2

3.3

4.1

11

11

11

12

12

13

14

15

15

16

Pictures of a heated flat strip of bi-metal : creation of two main curvatures

κx ≥ 0, κy ≥ 0 and R = 1/κy0 • If we apply a prestress which provides this

kind of curvatures, we will not have an interesting behavior . . . . . . . . . 20

Illustration of a lenticular section . . . . . . . . . . . . . . . . . . . . . . . . 22

Moment-curvature plot for a strip of lenticular section and subjects to oppositesense bending • left : influence of the initial plastic curvature κ

ˆ y0 without

heating (ˆ

κT = 0) • right : influence of the loading κ

ˆ T for a given κ

ˆ y0 . . . . 25

Plot of the stored strain energy as a function of κ

ˆ x and κ

ˆT • κ

ˆ y0 = {−1, −5, 10 − 15} 28

6

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

Plot of the strain stored energy as a function of κ

ˆ x for κ

ˆ y0 = −9 and κ

ˆT =

{0, 3, 4, 6} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Plot of the total strain stored energy as a function of κ

ˆ x for κ

ˆ y0 = −9 and

κ

ˆ T = {10, 15, 18, 22} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Plot of κ

ˆ T versus κ

ˆ x • Bifurcation diagram of the strip for κ

ˆ y0 = 0 and

κ

ˆ T ∈ [−4, 4] • blue : stable configurations • red : unstable configurations . .

Plot of κ

ˆ T versus κ

ˆ x • stable configurations of the strip for several κ

ˆ y0 •

blue : κ

ˆ y0 = −1 • purple : κ

ˆ y0 = −3 • black : κ

ˆ y0 = −3 • green : κ

ˆ y0 = −5

• red : κ

ˆ y0 = −7 • yellow : κ

ˆ y0 = −10 . . . . . . . . . . . . . . . . . . . . .

Plot of κ

ˆ T versus κ

ˆ y • stable configurations of the strip for several κ

ˆ y0 •

blue : κ

ˆ y0 = −1 • purple : κ

ˆ y0 = −3 • black : κ

ˆ y0 = −5 • green : κ

ˆ y0 = −7

• red : κ

ˆ y0 = −10 • yellow : κ

ˆ y0 = −15 . . . . . . . . . . . . . . . . . . . .

Evolution of κ

ˆ xcrit as a function of Abs(ˆ

κy0 ) • κ

ˆ y0 in[0, −40] . . . . . . . .

Evolution of κ

ˆ Tcrit as a function of Abs(ˆ

κy0 ) • (ˆ

κy0 ) in[0, −40]) . . . . . . .

Illustration of the hysteresis, plot of κ

ˆ T versus κ

ˆ x for different κ

ˆ y0 for

heating and cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Plot of κ

ˆ T against κ

ˆ x : theory against experiments • Blue dashed : theoretical prediction • green points : experiments • κ

ˆ y0 = {−2.1, −3.8, −9.1, −14.0}

Plot of κ

ˆ T against κ

ˆ x for different κ

ˆ y0 • (a) : theoretical prediction • (b) :

experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Plot of κ

ˆ T against κ

ˆ x for the constant thickness model• blue dashed : theoretical predictions • green point experiments . . . . . . . . . . . . . . . . . .

Plot of κ

ˆ T versus κ

ˆ x for the strip of constant thickness • Bifurcation diagram

of the strip for κ

ˆ y0 = 0 • blue : stable configurations • red : unstable

configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Plot of κ

ˆ T versus κ

ˆ y for the strip of constant thickness • stable configurations

of the strip for several κ

ˆ y0 • blue : κ

ˆ y0 = 30 • purple : κ

ˆ y0 = 40 • black :

κ

ˆ y0 = 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Superposition of κ

ˆ T versus κ

ˆ x (experiments) and contour plots of the energy

for several κ

ˆ y0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

29

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30

31

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32

33

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35

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36

37

38

Chapter 1

Introduction

1.1

Preamble

Smart structures are structures which are capable of morphing between very different

configurations : they adapt their shapes acording to their environment. In the nature,

this adaptation feature is the key for the survival of some plant species. Actually, the

famous carnivor plant, the Venus Flytrap, uses its multistability feature in order to eat.

In Y. Forterre’s work [16], a model of thin shallow shell is used in order to describe the

behavior of this plant, and it is shown that it exploits geometrical non linearity effects

combined with anisotropy and these features give access to a selection of different stable

configurations.

There are also several commercial products which have this morphing feature and it

goes of the simple flick-bracelet, to the deployable LCD screen. We can also cite the Rolatube, which is a composite, deployable post, used by militaries for communications. This

ability to change shape in function of needs is obtained by a tricky combination between

prestress on shell structures, anisotropy, and actuation. However, there are more complex

smart structures. Actually, since 2006, a group of researchers from Massachusetts Intitute

of Technology (MIT), works on a plane with smart wings. These wings are capable of

morphing according to the wind forces over the surface of the wings. The idea is to combined active materials with a piezo-electric actuation in the aim of increasing the plane

performances in term of fuel consumption. So, the point in designing smart objects is to

enhance their functionalities and their performances. This growing interest can be seen in

examples presented above and which are very differents (figure 1.1).

There is also an other good example of smart material : the smart memory alloy.

Memory alloy is an alloy which is capable to "remember" its original shape. This peculiar

feature is based on several changes of the microstructure with the temeprature. These

changes of microstructure define a cycle for the alloy which can be seen on figure 1.2.

But this kind of material is very expensive and their beahavior in shell structures is not

fully understood. But using cheaper, bi-metal structures, with a combination of initial

shape and prestress, we hope the expectation of hysteresis, so that it can be realised more

cheaply. The reason why hysteresis is important is that because many engineering morphing structures undergo non-linear and large displacements during the transition between

stable equilibria and that these give load paths with snaps, or jumps, which lead to different loading and unloading paths. But another important point concerns the amount

of energy needed to move between states: small amounts of actuation can trigger a large

change of shape, but it may be that a large amount of energy is needed to reverse the

change in shape. So, again, understanding the properties, as well as trying to find a way

of tuning them, in simple hysteretic systems, is key.

8

(a)

(b)

(c)

(d)

Figure 1.1: (a) Venus Flytrap (Y. Forterre, L. Mahadevan, J. Sktoheim, Nature 2005)

and the MIT plane • (b) Bistable screen (Cambridge Advanced Structures Group) • (c)

Rolatube (Cambridge Advanced Structures Group) • (d) Flick-bracelet

Figure 1.2: Illustration of a cycle for a smart memory alloy

1.2

Description of the global project

This traineeship is situated in the framework of smart structures designing and it is included in the project named Morphing shell structures : shape control and multiparameter

actuation. This project is a co-operation between S. D. Guest, K. A. Seffen from the Ingeneering Department of the University of Cambridge, and C. Maurini from the Institut Jean

Le Rond d’Alembert of the Université Pierre et Marie Curie, and financed by the RoyalSociety and the Centre National de la Recherche Scientifique (CNRS). The traineeship is

entitled Morphing multistable smart shells : behavior of prestressed bimetallic strips and

supervized by K. A Seffen. It took place in the Advanced Structures Group Laboraty of the

University of Cambridge. This Laboratory focuses on smart structures and its reseachers

9

are very experienced in this field.

This project suggests innovating methods in designing and analysis for multistable

structures with integrated actuation and some shape controlling applications. The aim is

to develop new structures which are able to deform themselves continuously by a controled

way between several stable configurations, without the necessity of ponctual connections

or convential actuators. We focuse our analysis on thin structures in large deflexions by

taking account into geometrical non-linearities. Our main aim is to control all the phenomenons which govern shape control as anisotropy, prestress and actuation, in order to

obtain an interesting multistable behavior. In order to control the structure configuration

with efficiency and accuracy, we consider to use an actuation provided by active materials

: smart memory alloys. So we have to take into account two other effects : the influence of

temperature and its impact on the microstructure. All these aspects represent the global

project and the next section presents our study which is a part of this project.

1.3

Description of the study

This study deals with the behavior of prestressed bi-metal strip under thermal loading.

When a flat strip of bi-metal is heated with a uniform temperature, we can notice that

it curves, and the longitunal curvature depends linearly on the variation of temperature.

We tried to use this effect as an actuation to control the shape of a thin strip of bimetal. As explained above, if we start with an initially flat strip, we are able to control

continously the shape of the strip and the reponse is linear. This kind of control is not very

interesting because what is usefull in shape controlling is to be able to change drastically

the shape over a reduced range of temperature. To obtain this kind of feature, we study

the influence of prestress on the behavior of a bi-metal strip under thermal loading. As

explained above, this traineeship took place in the Advanced Structured Group Laboratory

which is very experienced in experiments on smart structures. So to carry out this study,

we first wanted to make accurate experiments to characterize the behavior of prestressed

bimetallic strips. We have a large quantity of bi-metal and all the equipments to work on

bi-metal as a thermal chamber, a thermocouple and all what we need to product strips.

We also want to make an analytical study of this problem to obtain a model of this kind

of structure.

In the first section, we present our experimental work. We show how to product bimetal strips and how to apply a prestress which provides a controlled curvature over the

chord. Then we explain how to make measures on a heated bi-metal strip in order to plot

force-displacement diagrams. In the second section, we present our analytical study. We

remind the theory for an "ordinary" strip and how we adapted it to bi-metal action. We

first used the "simple" model of a strip with a lenticular section and next the constant

thickness model both proposed by E. H. Mansfield in [1]. The equations provided by these

models are quiet complicated, so we chose to work with the soft Mathematica, and in this

section, we also describe our resolution strategy. In a fird part, we show our analytical

results and a comparison between theory and experiments.

10

11

Chapter 2

Presentation of the problem :

experimental work

In this chapter, we present the problem and all its parameters by presenting our experimental work. We first present the experimental procedure in order to explain how to

product prestressed bimetallic strips and the main features of the experiments. Then we

present our first observations by describing the behavior of a heated strip.

2.1

2.1.1

Experimental procedure

Production of bimetallic strips

We have a roll of bi-metal provides by the supplier Kanthal which is the brand for Sandvik’s heating technology products and services. The reference of the material is KANTHAL

155 TB1577A and the material properties are in Kanthal Thermostatic Bimetal Handbook,

for more informations see [9].

Material properties of KANTHAL 155TB1577A

Components

Specific curvature

Temperature Range

Modulus of elasticity

Heat treatment (guiding value)

Thickness

36 Ni/NiMn-steel

k = 28.5 ∗ 10−6 K −1

Normal- 20 to +350°C

E = 170 ∗ 103 N.mm−1

Ageing 2 hours at 350°C

0.2mm

Cutting bi-metal : to product bimetallic strip we have a machine which are able to cut

plates. So we use it to cut strips of chord a and length L, with a = 20mm and L = 150mm.

After this step, the strip is still flat.

Forming process : the aim is to obtain a curved strip and we have two cases : oppositesense or same-sense bending. When we heat a flat bimetallic strip, it produces positives

curvatures in x- and y-directions . In order to obtain an interesting behavior, we have to

apply a prestress which gives negatives curvatures. In reality, curvatures due to bi-metal

action has to be opposed to curvatures produced by the prestress. We chose to make

strips with a zero curvature in x-direction but non zero curvature in y-direction. All these

observations are shown in the following pictures.

12

Figure 2.1: Pictures of a heated flat strip of bi-metal : creation of two main curvatures

κx ≥ 0 and κy ≥ 0 • If we apply a prestress which provides this kind of curvatures, we will

not have an interesting behavior

Figure 2.2: Pictures of a prestressed strip with opposite-sens bending • We will study strips

with this kind of curvatures κx0 = 0 and R = 1/κy0 ≤ 0

To apply this prestress we need to use a rolling-process. This is a system of cylinders

in which we put our flat strip and it works as shown in the following pictures (figure 2.3).

The complete study which characterizes the residual stess due to the application of this

kind or forming process, is done by E. Kebdaze, S. D. Guest and S. Pellegrino in [12].

Figure 2.3: Illustration of the rolling process • on the left : system of cylinders • on the

right : application of the prestress by the cylinders which gives a curvature to the strip

Now we have to measure the curvature κy0 , and we do this as explained on the following

picture. We mainly study four kind of strip which correspond to four different initial

curvatures : R = 1/κy0 = {15, 23, 36.5, 55, 99} mm.

13

Figure 2.4: We measure H and b with a micrometer and we calculate R • R = 1/κy0 =

b2

H

+

8H

2

2.1.2

Description of an experiment

At the end of the forming process, we have a strip with a plastic curvature as shown

in figure 2.2. Now we introduce the strip into a thermal chamber, we fix it with a hook

and slowly increase the temperature and we measure it accuratly with a thermocouple.

Finally, we take pictures of the strip in order to measure the longitudinal curvature κx for

each increment of temperature.

Figure 2.5: On the left : thermocouple • on the right thermal chamber

2.2

2.2.1

Observations

Catching of the phenomenon

On this sequence of pictures we can see our first experiments, and we notice two cases.

The first case involves the strip with "small" initial curvature for which the behavior is

close to the initially flat strip : the changing of shape is continous. The second case

corresponds to strip with "high" initial curvature and for this kind of strip we observe a

flexural snap-through.

14

Temperature range : ∆T ∈ [0, 180]K

Temperature range : ∆T ∈ [200, 350]K

Figure 2.6: First sequence of picture for "small" initial curvature (R = 1/κy0 = 55mm) •

the variation of κx is continuous

15

Temperature range : ∆T ∈ [0, 90]K

Temperature range : ∆T ∈ [95, 300]K

Figure 2.7: First sequence of picture for "high" initial curvature (R = 1/κy0 = 55mm)

• the variation of κx is not continuous, there is a critical temperature when we can see a

flexural snap-through

2.2.2

Post-processing

In order to measure κx we chose to work with ImageJ. But we have to dinstinguish

two cases : the small deflexion analysis and the large deflexion analysis.

Small deflexion analysis : we can approximate κx = 1/Rx with Rx ≈

to measure δ (figure 2.8).

Large deflexion analysis :

shape with a circle.

L2

, so we have

2∗δ

the current curvature κx is high, so we are able to fit the

With ImageJ, we can measure δ or directly Rx as shown on the figure 2.9, but before

we have to set the scale to convert our measures in pixels into meters.

16

Figure 2.8: Small deflexion analysis : correct for "small" values of temperature, i.e before

L2

L4

snap-through • δ = Rx (1−cos(θ0 ), L = Rθ0 , so δ = Rx ( 2R

2 − 24R4 +...) • If L Rx , Rx ≈

L2

2δ

Figure 2.9: On the left : small deflexion analysis, we measure δ • on the right : large

deflexion analysis : we measure Rx

2.2.3

First results

By following the previous experimental procedure we were able to plot the evolution

of κx as a function of ∆T for several inital curvatures κy0 . However, it is more relevant to

plot ∆T versus κx in order to produce force-displacement diagrams. The following picture

(figure 2.10) shows diagrams for several κy0 . The presented diagrams use dimensionless

variables and this point will be explain further. This figure shows the appearence of the

17

flexural snap-through for a sufficient κy0 .

Figure 2.10: Dimensionless force-displacement diagrams • Red : 1/κy0 = 99mm • Black :

1/κy0 = 55mm • Green : 1/κy0 = 23mm • Blue : 1/κy0 = 15mm

2.3

Conclusion

This experimental work shows the appearence of an instability which is linked to the

applied prestress. This behavior is the one we were expecting for because it permits to

obtain two very different shapes for a very small range of temperature. We also hoped

to observe a hysteretic behavior, that is to say a different behavior between heating and

cooling. With this hysteresis we could have a cycle which could be similar to the cycle of

smart memory alloy, but this point will be developped further.

In order to conclude this chapter, we can say that our work provides very accurates

results but the forming process is quiet limited. Actually, the highest initial curvature

which can be produced by the rolling process is around 1/15mm−1 , and this limitation

may hide a hysteretic behavior.

18

19

Chapter 3

Phenomenon modeling for two

different shapes of the strip :

lenticular section and constant

thickness

In this chapter, we present our analytical work. The main idea of this part is to adapt

existing models of strip by including bi-metal effects. We first explain how to make this

adaptation then we try to study a lenticular section model and a constant thickness model.

Finally, we explain our resolution strategy with Mathematica and our first results.

3.1

Large deflexions analysis : position of the problem with

bi-metal action

In large deflexion theory, we take into account of the middle-surface stresses arising from

the straining of the middle-surface. This kind of straining occurs when a plate tends to be

a non-developpable surface. In large deflexion, the strain e is of the form :

1

1

e(u, w) = (∇u + ∇uT ) + ∇w ⊗ ∇w

2

2

with u the in-plane displacement and w the deflexion, this is the non-linear theory of

Föppl-Von Kàrman. The coupling between the middle-surface strain and the out of plane

displacement is given by the Gauss Egregium theorem :

L(e) =

∂ 2 exy

∂ 2 ex ∂ 2 ey

+

−

2

= κx κy − κ2xy = G

∂x2

∂y 2

∂x∂y

G is called the gaussian curvature and when it is not zero, we have a non-developpable

surface with two main curvatures. So we place our study in the frame of the Föppl-Von

Karmàn theory.

3.1.1

Description of the model

The geometry of the strip is presented in the chapter two. We keep the geometry of

the strip with a chord a, a length L, and initial curvature due to prestress κy0 ≤ 0 and

initial curvature κx0 = 0. We assume that these two curvatures in x- and y-directions are

uniforms and we can define an initial deflexion w0

1

w0 = − κy0 y 2

2

20

(3.1)

Then, the applied temperature is uniform, so it does not depend on the spatial coordinates.

Involving the boundary conditions, we can say that the edges of the strip are free. When

the strip is subjected to a total moment M , the current longitudinal curvature is κx and

the current deflexion w is of the form :

1

w = − κx x2 + w(y)

¯

2

(3.2)

With this form of the current deflexion w, we can make two assumptions, in one hand

the longitudinal curvature remains uniform and in other hand, the current curvature in

y-direction κy is not uniform but depends on the spatial coordinate y. That is why we

¯ and this assumption is the key in our study. Now we have to take into

introduce w(y),

account the bi-metal effect. For an ordinary strip with an initial deflexion w0 , the generalised Hooke’s law in bending for plate are :

mxordinary = D ((κx − κx0 ) + ν(κy − κy0 ))

(3.3)

myordinary = D ((κy − κy0 ) + ν(κx − κx0 ))

(3.4)

With

κx = −

∂2w

∂x2

κy = −

∂2w

∂y 2

(3.5)

But the bi-metal action produces the same curvatures in both x- and y-directions, so

we can introduce the curvature κT which is the curvature that would be adopted by the

plate in both x- and y-directions during heating. This parameter depends on both the

temperature variation and the specific curvature k : κT = k∆T . The specific curvature

k characterizes the difference between the thermal expansion coefficients for each layer of

the strip, and it is a material property given by the supplier Kanthal. So, we can rewrite

the generalised Hooke’s law for a bimetallic plate as following :

mx = D ((κx − κx0 ) + ν(κy − κy0 ) − κT (1 + ν))

(3.6)

my = D ((κy − κy0 ) + ν(κx − κx0 ) − κT (1 + ν))

(3.7)

3

Et

With D = 12(1−ν

2 ) the bending rigidity. If we consider the following strip, the twisting

moment is such that mxy is zero. Morever, the mid-surface forces per unit length are such

that Ny and Nxy are zero too.

21

Figure 3.1: Pictures of a heated flat strip of bi-metal : creation of two main curvatures

κx ≥ 0, κy ≥ 0 and R = 1/κy0 • If we apply a prestress which provides this kind of

curvatures, we will not have an interesting behavior

Now we know how to include bi-metal action in the existing model of plates. The next

step is to define the problem to solve.

3.1.2

Problem position

• Determination of w

¯ : In E. H. Mansfield book [1], the condition of compatibility for

plates with variable rigidity D, variable thickness t, in the framework of large deflexion

analysis is of the following form, where Φ is the force function :

1

1

2 1 2

4

∇

∇ Φ − (1 + ν)

, Φ + E 4 (w, w) − 4 (w0 , w0 ) = 0

(3.8)

t

t

2

4 (f, g) =

∂2f ∂2g

∂2f ∂2g

∂2f ∂2g

−

2

+

∂x2 ∂y 2

∂x∂y ∂x∂y

∂y 2 ∂x2

(3.9)

Subsitution of equations 3.2 and 3.1 into the conditon of compatibility 3.8 now yields to :

d2 Nx

d2 w

¯

− E κx 2 = 0 ⇒ Nx = Etκx w

¯

(3.10)

dy 2

t

dy

The equation of compatibility is not different from the one of an ordinary strip, that is

why we do not show the complete calculation of Nx , but it is quiet easy. For the complete

calculation of Nx see [1]. Now we have to write the equilibrium and it is determined by

d2 my

= κx Nx , and substitution of equation 3.7 into the condition of equilibrium, which is

dy 2

simply the "beam" equation, now yields to :

d2 my

d2 d2 w

¯

= κx Nx ⇒ 2

+ κy0 − νκx − (1 + ν)κT = −κx Nx

(3.11)

dy 2

dy

dy 2

If we introduced the expression of Nx given by equation 3.10 into the equation of equilibrium 3.11, we obtain the differential equation which provides w

¯:

d2

d2 w

¯

D(

+

κ

−

νκ

−

(1

+

ν)κ

)

+ Etκ2x w

¯=0

(3.12)

y0

x

T

dy 2

dy 2

Now, we have to write the boundary conditions to complete the problem and these conditions express the fact that the longitudinal edges are free. That is to say, the moment my

22

is zero along the edges. Moreover there is not shear force along the edges and these two

conditions are of the form :

2

d w

¯

+

κ

−

νκ

−

(1

+

ν)κ

=

0, no moment

D

y0

x

T

dy 2

y=±a/2

(3.13)

2

d

d w

¯

D

+ κy0 − νκx − (1 + ν)κT

= 0, no shear force

dy

dy 2

y=±a/2

Finally, we can calculate w

¯ by solving equation 3.12 and the conditions 3.13 provides the

complete expression for w.

¯

• Determination of the total applied moment : the applied moment is of the form

R a/2

¯

So we first compute w

¯ as previously explained, and we can calculate

−a/2 (mx +Nx ∗ w)dy.

mx and Nx . The problem is to determine all the equilibrium configurations, so it is a

minimization problem which reduces itself to find the roots of M in terms of κT and κx .

3.2

A first approach : The strip of lenticular section

As shown previously, the calculation of w

¯ is quiet difficult, so we tried to make it easier.

Our idea was to study a strip with a lenticular section, it seems to be difficult but in reality

it is not. This kind of strip is studied in [1], and in his book, the author describes the

integration of equation 3.12 for variable rigidity D and thickness t. For such a strip the

thickness t, and hence the rigidity D, varies as follows:

t = t0 1 − (2y/a)2

3

D = D0 1 − (2y/a)2

Et30

D0 =

12(1 − ν 2 )

(3.14)

By introducing equation 3.14 into 3.12 we find

1

w(y)

¯

= − Ky ∗ y 2

2

κy0 − νκx − (1 + ν)κT

with Ky =

1 + κ2x

(3.15)

At this step, it is convenient to introduce the following non-dimensional terms :

1/2

a2 1 − ν 2

(κx , κy , κy0 , κT )

(3.16)

(ˆ

κx , κˆy , κ

ˆ y0 , κ

ˆT ) =

4t0

5

R a/2

We can now calculate M = −a/2 (mx + Nx ∗ w)dy,

¯

we define η = κT (1 + ν) and we use

dimensionless variables, we find after manipulations :

M=

β

κ

ˆ 2x )2 (−1

+ ν 2)

( − ηˆ2 κ

ˆx + κ

ˆ x (−(1 + κ

ˆ 2x )2 − κ

ˆ 2y0 + κ

ˆ x (3 + κ

ˆ 2x )ˆ

κy0 ν + ν 2 )

(3.17)

+ ηˆ(1 + κ

ˆ 4x − 2ˆ

κx κ

ˆ y0 − ν + κ

ˆ 2x (2 + ν))

16Et40

ˆ = βM . Now

with β =

, and we use the non-dimensional moment M

21a(5(1 − ν 2 ))1/2

ˆ in terms of ηˆ and κ

we have to find the roots of M

ˆx.

(1 +

23

Figure 3.2: Illustration of a lenticular section

3.3

The strip of constant thickness

In this section we want to write the total moment applied M for a strip with a constant

thickness. For this kind of strip, the equation 3.12 which gives w

¯ is of the form :

d4 w

¯ Etκ2x

+

w

¯ = 0,

4

dy

D

(3.18)

and the boundary conditions are the same. The general solution for this equation is given

by :

w

¯ = C1 cosh βy cos βy + C2 sinh βy sin βy,

(3.19)

and the two constants C1 , C2 are modified to include κT (1 + ν). To carry out this point,

we use a result from K. A. Seffen and S. Pellegrino in [4]. Actually, the end moment for

an opposite-sense bending and for an ordinary tape spring are :

F2

2

M = Da κx + νκy0 − νF1 (+κy0 + νκx ) +

(+κy0 + νκx )

(3.20)

κx

2

F1 =

λ

1

F2 =

2λ

cosh λ − cos λ

sin λ + sinh λ

cosh λ − cos λ

sin λ + sinh λ

−

sinh λ sin λ

(sin λ + sinh λ)2

(3.21)

1/2

a(3(1 − ν))1/4 κx

λ=

t

For a bimetallic tape, each of the terms in equation 3.20 needs to be replaced by taking

into account the terms of the generalised Hook’s law for bimetallic strip (equation 3.7) :

κx + νκy0 needs to be replaced with : κx + νκy0 − κT (1 + ν)

(3.22)

κy0 + νκx needs to be replaced with : κy0 + νκx − κT (1 + ν)

Thus, equation 3.20 for a bimetallic tape is :

F2

(κy0 + νκx − κT (1 + ν))2

M = Da κx + νκy0 − κT (1 + ν) − νF1 (κy0 + νκx − κT (1 + ν)) +

κx

(3.23)

24

At this step, it is convenient to introduce the following non-dimensional terms :

a2 (3(1 − ν 2 ))1/2

(κx , κy , κy0 , κT )

t

If we define η = κT (1 + ν), and after manipulation we find :

F2

2

(ˆ

κy0 + ν κ

ˆ x − η)

M =α (ˆ

κx + ν κ

ˆ y0 − η) − νF1 (ˆ

κy0 + ν κ

ˆ x − η) +

κ

ˆx

(ˆ

κx , κˆy , κ

ˆ y0 , κ

ˆT ) =

(3.24)

(3.25)

Et4

ˆ . Now we have to

, and we use the non-dimensional moment M

3a(3(1 − ν 2 ))1/2

ˆ in terms of ηˆ and κ

find the roots of M

ˆx.

with α =

3.4

3.4.1

Resolution strategy with Mathematica

Interests

During the previous modelling part, we were able to reduce the problem to a minimizaˆ for both lenticular

tion problem. Thus we only have to find the roots of the moment M

section and constant thickness models. If we remember what we explained in the chapter

two when we described our experiments, we can notice that we apply a uniforme temperature in the thermal chamber, then we wait until the longitudinal curvature stops to

change, and finally we take a picture. So the equilibrium configurations correspond to the

longitudinal curvature which minimizes the total stored strain energy for a given value of

ˆ , that is

the temperature. In order to minimize the energy, we have to find the roots of M

ˆ for a given value of κ

to say find κ

ˆ x which cancels M

ˆ T . The moment is seen as a function

of the only variable κ

ˆ x and κ

ˆT , κ

ˆ y0 and ν are parameters.

One significant advantage Mathematica provides is that it can handle symbolical expressions. The functions for which we want to find the roots are quiet complicated, so it is

more convenient to use a soft like Mathematica. For example, if we look at the total applied

moment for the constant thickness case, if we see κ

ˆ T as a parameter, it is not possible to

isolate κ

ˆ x , so we have to use a numerical method applied on the symbolical expression of

ˆ.

M

3.4.2

Description of the main functions

The two main functions which permit to find the roots of the moment are ResolutionAlea

and Resolution, they are in appendix (see algorithms 2 and 1). Both are based on the

function FindRoot which is implemented in Mathematica. This function uses a classical

Newton-Raphson method in order to find the zeros of a given function. But it has to be

initialized with a starting point, and the main difference between the functions ResolutionAlea and Resolution is the choice of this starting point. The structure of these functions

are quiet similar and their arguments are mainly the symbolical function which represents

the moment and all the parameters κ

ˆT , κ

ˆ y0 , ν.

• The loading κ

ˆ T : this parameter is a vector which describes the evolution of the loading. We choose a maximum value and an increment to define it. So we have to find the

roots of the moment as many times as the length of κ

ˆ T . We will mainly work with loading

of constant increments.

ˆ for each increment of load• ResolutionAlea : this function calculates all the roots of M

ing. So for each κ

ˆ T , we choose a random starting point κ

ˆ xstart in a given range. This is

25

determined graphically by plotting the moment. This function was written because when

we plot the moment as a function of the longitudinal curvature, we see that the moment

has at the most three roots.

• Resolution : this function calculates one of the roots of the moment for each increment

of loading. This kind of resolution is quiet different from the above function and it permits

to follow a equilbrium branch, either stable or unstable. So we have to precise a first

starting point for the first increment of loading. Then we calculate the corresponding root

and we initialize the next resolution with the previous root, multiplied by a coefficient in

order to perturb it. This manipulation of the starting point prevents the Newton-Raphson

to converge on the previous root. We also have to separate two cases : when we heat the

strip the loading increases, when we cool the strip the loading decreases. If we are heating,

we perturb the starting point by increasing it and vice versa.

• Determination of the stability : when we found a root of the moment, we now have

to check if it is stable or not. This step is quiet simple because we juste have to calculate the second derivative of the total stored strain energy, which corresponds to the first

derivative of the moment, for the couple {ˆ

κx,root , κ

ˆ T,root }. If it is positive, the configuration

is stable and if it is not, the configuration is unstable. The function called Trilist is the

one which makes this analysis.

3.4.3

First results

With our code written in Mathematica, we are able to plot the applied moment and

to show the influence of the parameters κ

ˆ y0 and κ

ˆ T . Figure 3.3 shows the evolution of the

applied moment as a function of the longitudinal curvature and for several values of κ

ˆ y0

and κ

ˆT .

If we have a flat strip, we can say that the evolution of the applied moment is linear,

and we find the behavior predicted by the supplier Kanthal. Moreover, the initial curvature

introduces a high non-linearity in the moment-curvature relationship but only for small

values of κ

ˆ x , and we still have a linear evolution for large values of κ

ˆ x . So, by introducing

an initial plastic curvature, we rigidify the strip when κ

ˆ x is small and this behavior is very

intuitive.

Now if we heat the strip for a given value of κ

ˆ y0 , we can notice that the non-linear area

dicreases and the moment-curvature evolution tends to be linear. Moreover, we can say

that the applied moment has several roots : only one for small and large κ

ˆ T , three in a

certain range of κ

ˆ T . Here we can see the significance of the starting point which initialized

the resolution. For exemple the point (ˆ

κx = 0, κ

ˆ T = 0) is always a root of the moment, so

if we start the resolution near this point, the algorithm will converge on this point.

Finally, we also observe a peak in the moment-curvature relationship and we it is

explained in seffen pellegrino, that this kind of behavior leads to the formation of localized

deformations. So, when the moment has an up-down-up profile and when we reached the

limit point, the strip is subjected to localization.

26

Figure 3.3: Moment-curvature plot for a strip of lenticular section and subjects to oppositesense bending • left : influence of the initial plastic curvature κ

ˆ y0 without heating (ˆ

κT = 0)

• right : influence of the loading κ

ˆ T for a given κ

ˆ y0

3.5

Conclusion

In this part we managed to adapt the existing theory of Mansfield with bi-metal

action, in a large deflexion analysis. We investigate two kinds of sections : lenticular and

constant thickness. We were able to write analytically the applied moment for both, and we

wrote and efficient code to compute the equilibrium configurations of the heated strip. We

also gave few features of the behavior of the strip of lenticular section, and our comments

seem to be in line with the articles of the bibliography.

However, the moment-curvature relationship found for the constant thickness does not

work very well. Actually, this relationship has the same features as the one of the lenticular

section, that is to say, the same up-down-up profile and the same behavior towards all the

parameters. But we have a scale problem, the values of the loading and the initial curvature

seem to be too high. This is the reason why we do not compare this to any experiments

yet, or directly to the results of the lenticular section. However, the analytical predictions

do not compare as well, but this will be validated later.

Why is it so complicated for the strip of constant thickness? The answer to this question

is given by E. H. Mansfield in [1], he managed to emphasize a boundary-layer phenomenon

which occurs for this model. Actually, Mansfield explains that near the free edges, the

function w

¯ is an oscillating but rapidly decaying function of distance from the free edges.

This phenomenon is classical in the large deflexions analysis of plates. For the strip of

lenticular section, the problem is simpler because the bending rigidity tapers smoothly

to zero at the boundaries and this feature avoids the appearence of this boundary-layer

phenomenon.

27

28

Chapter 4

Results

4.1

4.1.1

Analytical results for the strip of lenticular section

Stored strain energy analysis

The stored strain energy is a way to explore the nature of the load-free shapes, and

hence, the moment-free shapes. In order to calculate the energy, the only thing we need

is the expression of the transverse displacement w (equation 3.2). Then we can write the

energy as follows :

Z

0

0

V = Ubending

+ Ustretching

dA

2

1

0

(4.1)

= D (∇2 (w − w0 ) − (1 − ν) 4 (w − w0 , w − w0 )

with Ubending

2

1

0

N2

and Ustretching

=

2Et x

In the chapter two, we derived the equilibrium governing equation which is a result of

an energy minimization. This energy minimization leads to find the roots of the applied

dV

moment because M =

. So we explore the energy landscape by plotting the logarithm

dκx

of the non-dimensional strain stored energy Vˆ as a function of κ

ˆ x and κ

ˆ T for several values

of κ

ˆ y0 (figure 4.1). It seems that for small values of κ

ˆ y0 , all the κ

ˆ x which minimize Vˆ are in

a kind of "valley". But when we increase κ

ˆ y0 , we notice that these minimums are localized

in two main areas : one near κ

ˆ x = 0, and the other for κ

ˆ x 0. These two regions are

separated by a potential barrier which could be jumped over as soon as κ

ˆ T is enough large.

29

Figure 4.1: Plot of the stored strain energy as a function of κ

ˆ x and κ

ˆT • κ

ˆ y0 =

{−1, −5, 10 − 15}

We plot the total strain stored energy as a function of κ

ˆ x for given a value of κ

ˆ y0 and

for different values of κ

ˆ T (see figure 4.2 and 4.3. We notice that for small values of κ

ˆT ,

the energy has its only one minimum for κ

ˆ x ≈ 0 (ˆ

κT ≤ 3). But when we increase κ

ˆT

(ˆ

κT ≥ 3), an other minimum appears and it corresponds to κ

ˆ x 0. So, the strip can have

two different configurations and this confirms the previous observations. We will illustrate

further these two configurations.

30

Figure 4.2: Plot of the strain stored energy as a function of κ

ˆ x for κ

ˆ y0 = −9 and κ

ˆT =

{0, 3, 4, 6}

Figure 4.3: Plot of the total strain stored energy as a function of κ

ˆ x for κ

ˆ y0 = −9 and

κ

ˆ T = {10, 15, 18, 22}

4.1.2

Minimization of the strain stored energy for several increments of

temperature

To find all the configurations of the strip for all values of κ

ˆ T , we have three steps :

ˆ

ˆ by using the

we first compute the applied moment M , then we find the roots κ

ˆ xcrit of M

31

function ResolutionAlea with M athematica, and finally we evaluate the sign of the second

derivative of Vˆ in order to determine if the configuration is stable or not.

By this way we are able to minimize Vˆ for each values of κ

ˆ T . On the following picture

(figure 4.4) we plot κ

ˆ T as a function of κ

ˆ x , we can see that for small values of κ

ˆ T we have a

non-linear regime which is only due to both bi-metal action and the size of the strip. Then

after a κ

ˆ Tcrit we have either a linear regime in which κ

ˆ x increases as κ

ˆ T , or a non linear

regime where κ

ˆ x tends to zero as κ

ˆ T increases. We have also an unstable path in red.

The first non-linear regime appears because of the competition between bi-metal action

in the two directions of the plane of the strip. After κ

ˆ Tcrit , the strip has to choose one of

the blue path and it chooses the easiest. Actually, the strip has a chord a and a length L :

if a ≥ L, κ

ˆ x tends to zero whereas κ

ˆ y increases with κ

ˆ T , if L ≥ a, κ

ˆ y tends to zero whereas

κ

ˆ x increases with κ

ˆ T . The last case is our case. These obsevations emphasize a first kind

of instability which is linked to the appearence of a bifurcation.

Figure 4.4: Plot of κ

ˆ T versus κ

ˆ x • Bifurcation diagram of the strip for κ

ˆ y0 = 0 and

κ

ˆ T ∈ [−4, 4] • blue : stable configurations • red : unstable configurations

4.1.3

Minimization of the strain stored energy for several κ

ˆ y0

In our study the chord of the strip is smaller than its length, so we will study the

stable path along which κ

ˆ x increases with κ

ˆ T . On the following picture (figure 4.5), we

can see the behavior of the strip for different κ

ˆ y0 . We always have a non-linear regime for

small κ

ˆ T and a linear one for large value of κ

ˆ T . Moreover, in the non linear regime, the

gradient increases with κ

ˆ y0 and this phenomenon characterizes a change of rigidity : the

more the initial curvature due to prestress is important , the more the bending rigidity

will be important and this is quiet intuitive. We also notice that for a certain value of

κ

ˆ y0 , there is a flexural snap-through which traduces an instability and the gap between

the configurations before and after the snap becomes bigger when κ

ˆ y0 increases. Finally

the curvature κ

ˆ x before the snap seems to be a constant of the problem and its value is

around 0.8, but we will study this feature further.

32

Figure 4.5: Plot of κ

ˆ T versus κ

ˆ x • stable configurations of the strip for several κ

ˆ y0 • blue

: κ

ˆ y0 = −1 • purple : κ

ˆ y0 = −3 • black : κ

ˆ y0 = −3 • green : κ

ˆ y0 = −5 • red : κ

ˆ y0 = −7

• yellow : κ

ˆ y0 = −10

Now, if we look at the evolution of κ

ˆ y (figure 4.6), we notice that it tends to zero as κ

ˆx

increases : the strip tends to be a developpable surface. In the chapter three, we found :

κ

ˆy =

κ

ˆ y0 − ν κ

ˆ x − (1 + ν)ˆ

κT

2

1+κ

ˆx

κ

ˆ x 7→ ∞ ⇒ κ

ˆ y 7→ 0

(4.2)

(4.3)

Figure 4.6: Plot of κ

ˆ T versus κ

ˆ y • stable configurations of the strip for several κ

ˆ y0 • blue

: κ

ˆ y0 = −1 • purple : κ

ˆ y0 = −3 • black : κ

ˆ y0 = −5 • green : κ

ˆ y0 = −7 • red : κ

ˆ y0 = −10

• yellow : κ

ˆ y0 = −15

If we look more accuratly, we can see that the flexural snap-through occurs when κ

ˆ y is

near zero. Actually, when κ

ˆ y is near zero, the level of energy of the strip is such that it is

33

able to jump over the potentiel barier and to go to the other "valley" of minimums. Moreover, as explained previously, the non-linear regime takes place when κ

ˆ y is not zero, and

this regime is emphasize by the competition between bi-metal action in x- and y-direction

and also the prestress. At the end of this regime, the snap occurs and then, we find a

linear regime which is the same as if the strip was initially flat : during this regime, the

bending stifness does not depend on κ

ˆ y0 .

4.1.4

Study of critical values of κ

ˆ x and κ

ˆT

In this section we try to see if κ

ˆ xcrit , the longitudinal curvature just before the snap, is

a constant of this problem or not. To observe this feature, we make the same computation

as in the previous section and we find κ

ˆ xcrit , by testing the gap between two successive

configurations : if this gap is greater than 1, we keep this configuration. We do this analysis for several κ

ˆ y0 and we plot κ

ˆ xcrit versus κ

ˆ y0 . The following picture (figure 4.7) shows

that κ

ˆ xcrit does not depens on κ

ˆ y0 .

Figure 4.7: Evolution of κ

ˆ xcrit as a function of Abs(ˆ

κy0 ) • κ

ˆ y0 in[0, −40]

Moreover, we made the same calculation for κ

ˆ T and we plot its evolution as a function

of κ

ˆ y0 (see figure 4.8). We observe on the following pictures that κ

ˆ T versus κ

ˆ y0 is a linear

function and its gradient is ±0.6. So if we have a strip with a given value of κ

ˆ y0, we can

predict that κ

ˆ Tcrit will be ±0.6 ∗ κ

ˆ y0.

34

Figure 4.8: Evolution of κ

ˆ Tcrit as a function of Abs(ˆ

κy0 ) • (ˆ

κy0 ) in[0, −40])

Finally, the two last pictures show that the flexural snap-through occurs only for

Abs(ˆ

κy0) ≥ 6 (see figure 4.7 and 4.8). If Abs(ˆ

κy0) ≤ 6, linear and non-linear regimes

are joined continuously without snap.

4.1.5

Appearence of a hysteretic behavior

The aim of this study is to find a way to reproduced the structural effect of the

smart memory alloys. Actually, the aim of this work is to find a simple way, with classical

materials, which permits to be close to the behavior of smart memory alloys.

By using the same resolution strategy, we were able to emphasize a hysteretic behavior

as we can see on the following picture (figure 4.9). The hysteretic behavior appears as soon

as κ

ˆ y0 is enough large to observe a snap, so for Abs(ˆ

κy0 ) ≥ 6. Moreover, the size of the

hysteresis increases with κ

ˆ y0 . This peculiar behavior is very close to the behavior of SMA,

it permits to choose between very different configurations of the strip, over a small range

of temperature : that is why it is interesting for shape controlling. In order to observe this

hysteresis, we use the function Resolution.

35

Figure 4.9: Illustration of the hysteresis, plot of κ

ˆ T versus κ

ˆ x for different κ

ˆ y0 for heating

and cooling

4.2

4.2.1

Experiments versus theory

Analytical predictions versus experiments

Now we want to compare our analytical prediction with experiments and we managed to

obtain very good results : we have a difference of less than 5%. So we can conclude that

our model of a strip of lenticular section is quiet efficient. The figure 4.10 shows this good

result.

36

(a)

(b)

(c)

(d)

Figure 4.10: Plot of κ

ˆ T against κ

ˆ x : theory against experiments • Blue dashed : theoretical

prediction • green points : experiments • κ

ˆ y0 = {−2.1, −3.8, −9.1, −14.0}

4.2.2

κ

ˆ x,crit : a constant of the problem

Experimentally, we find that κ

ˆ xcrit is also a constant and its value is 0.8 too : the analytical

prediction were right (see figure 4.11).

(a)

(b)

Figure 4.11: Plot of κ

ˆ T against κ

ˆ x for different κ

ˆ y0 • (a) : theoretical prediction • (b) :

experiments

4.2.3

Comments

If we try to see how bad are the predictions provided by the constant thickness model, we

can also plot the temperature-curvature, and for κy0 = 1/15mm−1 we have the following

37

results (figure 4.12) and as explained previously, we have very bad result.

Figure 4.12: Plot of κ

ˆ T against κ

ˆ x for the constant thickness model• blue dashed : theoretical predictions • green point experiments

However, we can observe an instability which is a bifurcation, which occurs when κy0

is zero. This is similar to the result of the strip of lenticular section but in this case, the

temperature range is more important as we can see on the following picture (figure 4.13)

Figure 4.13: Plot of κ

ˆ T versus κ

ˆ x for the strip of constant thickness • Bifurcation diagram

of the strip for κ

ˆ y0 = 0 • blue : stable configurations • red : unstable configurations

Finally, we can also observe an other kind of instability which a flexural snap-through,

but it appears for κ

ˆ y0 ≥ 30 and it is higher from the lenticular section and also the

experiments (figure 4.14).

38

Figure 4.14: Plot of κ

ˆ T versus κ

ˆ y for the strip of constant thickness • stable configurations

of the strip for several κ

ˆ y0 • blue : κ

ˆ y0 = 30 • purple : κ

ˆ y0 = 40 • black : κ

ˆ y0 = 50

So, the three last comments permit to say that our moment-curvature relationship

has the good qualitative features because it presents the two kind of instability, but the

temperature range, the value of the initial curvature, and the value of the longitudinal

curvature are too high. That is why we can say that we have a scale problem and this part

is of ongoing work

Involving the strip of lenticular section, if we superpose plots of κ

ˆ T vs κ

ˆ x obtained

experimentally and contour plots of the energy, we can notice that the real measures are

confined in the two "valleys" as explained above. The figure 4.15 shows this results.

39

Figure 4.15: Superposition of κ

ˆ T versus κ

ˆ x (experiments) and contour plots of the energy

for several κ

ˆ y0

40

4.3

Discussion

In this study, we show how to introduce high non-linearties in the moment-curvature

relationship for plates. The two keys are the introduction of prestress combined with bimetal action. The bi-metal action creates a competition between curvatures created in xand y-directions and the prestress leads to an instability. Thus, we obtain an interesting

behavior for bimetallic strips, and we can now change drastically the shape of a prestressed

strip over a small range of temperature.

In our experimental work, we show the appearence of this instability phenomenon. We

were able to characterize this instability and the conclusion is that the initial curvature

has to be more than 1/55mm−1 to observe the snap. Moreover, the non-linear regime can

be shown on our experimental results. It is shown that the bending rigidity is very high

for small values of the longitudinal curvature for prestressed strip. But our experiments

are quiet limitated, because of the forming process. Actually, the highest initial curvature

due to prestress which can be produced is around 1/15mm−1 and this point has to be

enhanced.

Then, we were able to adapt Mansfield’s theory by including bi-metal action. This point

was quiet difficult but we obtained good results for the strip of lenticular section. Actually,

we adapt this model and we observe a high non-linearity and a flexural snap-through in

the temperature-curvature relationship. Moreover, we also see that the moment-curvature

plot shows an up-down-up profile and this means that the strip is subjected to elastic

localized deformations. This kind of localization is called propagating instabilities and the

complete study can be read in the paper of Kyriades [3]. However, when we tried to make

the complete analysis of a strip of constant thickness, we encountered a scale problem.

Actually, when we study the moment-curvature relatioship it seems to be right, because

qualitatively it presents the same features as the moment-curvature relationship for the

strip of lenticular section. That is to say, it presents high non-linearity and an up-down-up

profile. Morever, the temperature-curvature plot shows the same flexural snap-through

ˆ, κ

but the values of M

ˆ T and κ

ˆ x are very high and so very different from the lenticular

section model. We are still investigating this point and we hope to solve this mathematical

problem.

So we compared the lenticular section results with our experiments, and we were very

pleased to see that this model is very efficient. Actually, when we compare the experimental

and analytical temperature-curvature plots, we find a difference of less than 5% and this

is a very good result. Morever, the fact that the critical value of κ

ˆ x is a constant is also

verified experimentally. Then, this model shows the appearene of a hysteretic behavior and

this is a very good point in order to be close to SMA’s behavior. With this hysteresis, we

can define a cycle during which the strip can change considerably its shape and according

to me, it is the most important result of this study. Unfortunately, we were not able to

study this behavior experimentally because of the limitation of the forming process. The

highest initial curvature which can be produced by this process is not enough large to

observe a clear hysteresis : the configurations adopted by the strip during heating and

cooling are too close. Involving the phenomenon of localization, it did not appear clearly

during our experiments.

To conclude this work, we can say that we find a good way to obtain an insteresting

behavior for shape controlling. We managed to build an efficient model which is very close

to experiments. But we have to enhance the mentionned points above. The next step of

this study, is to try to extend this work to shape memory alloy.

41

4.4

Aknowledgements

The work presented in this paper was supervized by Dr Seffen, University Senior

Lecturer in Engineering. I gratefully thank Dr Seffen for his precious help during my work

and for his kindness during all my stay at Cambridge. I also want to thank Mr Maurini

for having proposed me this traineeship.

42

Bibliography

[1] E. H. Mansfield, The bending and stretching of plates (second edition), Cambridge

University Press 1989.

[2] K. A. Seffen, Z. You, S. Pellegrino Folding and deployment of curved tape spring, International Journal of Mechanical Sciences, 1999.

[3] S. Kyriades Propagating instabilities in structures, Advanced in applied mechanics.

Boston Academic Press, 1994.

[4] K. A. Seffen, S. Pellegrino Deployment dynamics of tape spring, A Proceedings of the

Royal Society 1998.

[5] K. A. Seffen, ’Morphing’ bistable orthotropic elliptical shallow, A Proceedings of the

Royal Society 2007.

[6] K. A. Seffen, S. D. Guest, Prestressed morphing bistable and neutrally stable shells,

Journal of Applied Mechanics 2011.

[7] K. A. Seffen, Mechanical memory metal : a novel material for developping morphing

engineering structures, Elsevier 2006.

[8] K. A. Seffen, Hierarchical multi-stable shapes in mechanical memory metal, Elsevier

2006.

[9] KANTHAL thermostatic bimetal handbook, available on http://www.kanthal.com.

[10] A. Fernandes, C. Maurini, S. Vidoli, Multiparameter actuation for shape control of

bistable composite plates, Elsevier 2010.

[11] S. Vidoli, C. Maurini, Tristability of thin orthotropic shells with uniform initial curvature, A Proceedings of the Royal Society 2008.

[12] E. Kebadze, S. D. Guest, S. Pellegrino, Bistable prestressed shell structures, Elsevier

2004.

[13] M. A. C. Stuart, W. T. S. Huck, J. Genzer, M. Müller, C. Ober, M. Stamm, G. B.

Sukhorukov, I. Szleifer, V. V. Tsukruk, M. Urban, F. Winnik, S. Zauscher, I. Luzinov,

S. Minko, Emerging applications of stimuli-responsive polymer materials, Nature 2010.

[14] P. Schmid, F. J. Hernandez-Guillen, E. Kohn, Diamond switch using new thermal

actuation principle, Elsevier 2006.

[15] Eduard Toews von Riesen, Active Hyperhelical Structures, Thèse de l’université de

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[16] Y. Forterre, J. M. Skotheim, J. Dumais, L. Mahadevan, How the Venus flytrap snaps,

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[17] D. A. Galletly, S. D. Guest , Bistable composite slit tubes : shell model, Elsevier 2004.

[18] M. Schenk, Textured shell structures, First year PhD report, Clare College, University

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2006.

44

Appendix A

Mathematica algorithms

Algorithm 1 ResolutionAlea

ˆ , {ˆ

Require: M

κy0 , κ

ˆ T , ν} , {a, b} , p

1: for i = 1 to size(ˆ

κT ) do

2:

for j = 1 to p do

3:

κ

ˆxinit,j ←RandomReal{a, b}

4:

const ← κ

ˆxinit,j , κ

ˆ y0 , κ

ˆ ,ν

Ti

ˆ (ˆ

5:

κ

ˆx

= FindRoot M

κx , const), κ

ˆx, κ

ˆx

sol,i,j

init,j

end for

7: end for

8: return κ

ˆ xsol

6:

Algorithm 2 Resolution

ˆ , {ˆ

Require: M

κy0 , κ

ˆ T , ν} , κ

ˆ xinit,0 , sense,

1: κ

ˆxinit ← κ

ˆ xinit,0

2: for i = 1 to size(ˆ

κT ) do

3:

const ← {ˆ

κxinit , κ

ˆ y0, κ

ˆ Ti , ν}

ˆ (ˆ

4:

κ

ˆx

= FindRoot M

κx , const), {ˆ

κx , κ

ˆx

sol,i

5:

6:

7:

8:

9:

10:

if sense = 1 then

κ

ˆ xinit = κ

ˆ xsol,i (1 + ) (heating)

else

κ

ˆ xinit = κ

ˆ xsol,i (1 − ) (cooling)

end if

end for

return κ

ˆ xsol

45

init

}