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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
32
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
29
30
31
31
32
33
34
35
35
36
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


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)

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 =


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

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

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

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
Cambridge 2007.
[16] Y. Forterre, J. M. Skotheim, J. Dumais, L. Mahadevan, How the Venus flytrap snaps,
Nature 2005.
43

[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
of Cambridge.
[19] T. Tadaki, K. Otsuka, K. Shimizu, Shape memory alloys, Annu. Rev. Matter Sci.
1988.
[20] K. Otsuka, C. M. Wayman, Shape memory materials, Cambridge University Press
1998.
[21] W. Zaki, Comportement thermo-mécanique des matériaux à mémoire de forme : modélisation macroscopique, chargement cyclique et fatigue, Thèse de l’École Polytechnique
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

}




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