On the effect of dimensionality and plasticity on the crumpling of a thin sheet

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On the effect of dimensionality and plasticity on

the crumpling of a thin sheet

A thesis presented for the degree of

Bachelor of Physics, 15 EC

Submitted by

Maurits Moeys 10213201

Under the guidance of Mehdi Habibi

Daniel Bonn Rudolf Sprik

Van der Waals-Zeeman Institute

Institute of Physics Amsterdam, the Netherlands

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Abstract

Crumpling is an everyday phenomenon, yet it’s physics is poorly understood. De-scribing a power-law relation between force and compaction, with an exponent β depending on the dimensionality of the folding, it is claimed that a theory for hi-erarchical folding captures the main features of the crumpling of a thin sheet. The validity of this model is examined by investigating the effect of plasticity and by elaborating on it’s dimensional arguments. Although β is found to deviate from the theoretical predictions, it is argued the theory needs not to be rejected. The effect of plasticity on β is quantified by analyzing the amount of layers of a crumpled structure, described by the layer exponent γ. This exponent is found to deviate from the models prediction, hence a redefinition of β is proposed.

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Populair wetenschappelijke samenvatting

Het verfrommelen van materiaal is een typisch verschijnsel van het dagelijks leven. De fysica hierachter beschrijft waarom je prullenbak zo snel vol raakt, hoe metaal van een auto zich gedraagt bij een botsing en hoe DNA effici¨ent verpakt kan worden in je cellen. Verfrommeling is natuurkundig echter moeilijk te beschrijven vanwege het grote aantal uiteenlopende processen dat hierbij betrokken is.

Er bestaat wel een theorie voor het georderd vouwen van dunne platen (bijvoor-beeld papier) waarmee getracht wordt verfrommeling te beschrijven. Het voorspelt dat de benodigde kracht voor het vouwen van papier toeneemt met een machtsfunc-tie. De macht waarmee de kracht toeneemt, β, hangt af de dimensionaliteit: het aantal richtingen waarop je vouwt.

De validiteit van deze theorie als een model voor verfrommeling wordt onderzocht door het effect te analyseren van dimensionaliteit en plasticiteit (de mate waarin materiaal onomkeerbaar deformeert). Met twee experimentele opstellingen worden materialen samengeperst in verschillende dimensionele regimes. Het blijkt dat di-mensionaliteit een vergelijkbare rol speelt bij verfrommelen als bij vouwen. Het effect van plasticiteit kan gemeten worden door het aantal lagen in de structuur van verfrommeld materiaal te analyseren. Het blijkt dat het effect van plasticiteit on-voldoende wordt voorspeld, waarop een aanpassing op het model wordt voorgesteld.

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Declaration

The work in this thesis is based on research carried out at the Van der Waals-Zeeman Institute, Institute of Physics, the Netherlands. No part of this thesis has been submitted elsewhere for any other degree or qualification and it is all my own work unless referenced to the contrary in the text.

Copyright c 2014 by Maurits Moeys.

“The copyright of this thesis rests with the author. No quotations from it should be published without the author’s prior written consent and information derived from it should be acknowledged”.

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Acknowledgements

I would like to extend firstly a huge vote of gratitude to Mehdi Habibi for being my first contact during my period of work in the lab and for his active participation in the interpretation of my data. Secondly, I would like to thank both Daniel Bonn and Mehdi Habibi for reviewing my thesis.

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List of Figures

1.1 Folding in one, two and three dimensions. . . 2

2.1 The three point flexural test . . . 4

3.1 Measurements on plasticity . . . 9

3.2 Cylinders used . . . 11

3.3 The 3d setup . . . 12

4.1 Measurements of β for Mylar . . . 13

4.2 Measurements of β for Paper and Aluminium . . . 14

4.3 Hierarchical structure after crumpling . . . 15

4.4 Linear results for 3d crumpling . . . 16

4.5 Data on γ . . . 17

4.6 Comparison of γ and β for different dimensionalities . . . 19

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List of Tables

2.1 Theoretical summary . . . 7

3.1 Properties of used materials . . . 10

4.1 Measurements of β . . . 14

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Chapter 1 Introduction

Crumpling of material is an occurrence that takes place typically at different scales of daily life. At large scales, collisions within our earth’s crust are the origin of earthquakes. Moving down to smaller scales, metal is crushed during an airplane crash (fortunately not that daily), material is crumpled into a ball in our hands and prote¨ıns are densely packed within our cells.

A profound understanding of the physics of crumpling could therefore lead to nu-merous applications. From a better understanding of crumpling, there could arise models of earthquakes with a greater predictive performance, as well as it could open up possibilities for the way we handle our waste, in an attempt to dissolve the vast amounts of refuse dumps. It could help biochemists to understand the way DNA is wrapped around histones, paving the way for improved manipulation.

While it clearly plays a role in the above mentioned processes, the direct link be-tween an improved understanding of crumpling and these applications is not a priori evident. There is however a recent development in the field of graphene research in which crumpling is very relevant: crumpled graphene structures [1]. During the manufacturing process, van der Waals bonding causes the 2D graphene sheets to restack, resulting in loss of accessible surface area and other properties. One method to prevent aggregation from happening involves significantly reducing the size of the graphene sheets, which constitutes a great obstacle for large-scale production. It is shown that crumpled graphene structures are strongly aggregation-resistant, leading to higher accessible surface area compared to ordinary produced graphene.

Together with the electrical properties of graphene, this gives rise to possibilities

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Chapter 1. Introduction 2

of constructing a high-capacitance and high energy density capacitor as reported in [2], forming an superieur power source for various electronic devices. A better understanding of the underlying physics of crumpling of graphene could lead to an improved control of the structure’s properties, paving the way for large-scale pro-duction.

These applications invite us to explore the existing theoretical framework of the crumpling of thin sheets. Even though crumpling is a common action, the underly-ing physics is poorly understood because of the diversity of processes involved, such as non-linear strains, friction, plastic deformations, jamming into locking states and more [3]. However, an attempt was made by [4], describing ordinary, hierarchical folding of a thin sheet and comparing it to the process of crumpling. They claim their model captures the main features of crumpling. In my thesis, I aim at elaborat-ing on their efforts, and for this purpose I must first outline their work in more detail.

In their model of ordered folding, they predict a power-law relation between force and compaction with an exponent, β, depending on the dimensionality of the fold-ing. Three types of folding are considered: folding in one, two and three dimensions (figure 1a, 1b and 1c respectively). For the comparison with crumpling, their exper-imental setup consisted of a cylindrical shell (r = 15 cm) with a piston connected to a force transducer. Paper sheets of different sizes were placed inside the shell and compressed with an upper limit of F = 100 N

(a) (b) (c)

Figure 1.1: Folding in one, two and three dimensions.

They found a power-law relation between force and compaction. Their measurement of β resembled a value between what is predicted by their hierarchical model for 1d July 3, 2014

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Chapter 1. Introduction 3

and 2d folding. Validating their theory as a model of crumpling, they argue the piston setup to be a mixture of 1d and 2d compaction: the direction of the loading is one-dimensional, but the confinement by the cylindrical shell gives rise to a two-dimensional compaction effect.

In my thesis, I aim to further examine the validity of the hierarchical folding theory as a model for the crumpling of a thin sheet. This involves exploring the effect of plasticity and dimensionality. For the latter, I propose a setup to verify the validity of the argument aforementioned by elaborating on it. If the confinement by the shell is the cause of the 2d effect, one would expect this effect to increase as the radius of the cylinder shrinks. From here on, this argument will be referred to as the 1d−2d mixture argument. Also, an attempt will be made to experimentally explore the legitimacy of the model in the regime of 3d compaction.

As suggested by [5], it is thought that the plasticity (the susceptibility to irreversibly deform) of the material in question affects the force-compaction ratio. To explore this effect, experiments have been repeated for materials of different plasticity. The effect of plasticity on the amount of layers of a crumpled sheet will be investigated, and an attempt will be made to relate this result to the behavior of the crumpling exponent β.

For completeness of the argument, I shall first delve into the theoretical basis of the folding model.

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Chapter 2 The hierarchical folding model

In this section, I shall discuss the theory of the hierarchical folding model as derived by [4]. The model arises from the consideration of the elastic bending energy involved in folding a sheet in two. For this, we exploit the energy stored in a fold as derived by [6]. Firstly, a more straightforward concept is introduced.

2.1 Flexural stiffness

When examining the folding of a sheet, the first important quantity one encounters is the flexural stiffness B, a measure for material’s tendency to resist bending. To find an expression for B of a plate, one can revert to complicated mathematical derivations as provided by [7], but an intuitive argument suffices for the scope of this thesis. Consider a compound placed in the setup of a three point flexural test (figure 2.1).

Figure 2.1: The three point flexural test

In this setup, material is fixed at two ends and a force is applied at the centre of mass. It is intuitively clear that increasing the thickness h will increase the resistance to bend, so B ∝ h. Another property is the Young’s modulus E, a measure of the stiff-ness of a material. Stiff material will be less likely to bend, so B ∝ E. The final

quan-tity we must consider is the moment of inertia, I, which is a measure for how far on average mass is distributed away from the centre of mass. When mass is distributed close to its centre (small I), a compound will flex more easily compared to when

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2.2. Establishing the power-law relation 5

mass is distributed more towards the ends. Thus, B ∝ I. For a plate, it is know that I ∝ h₁₂2 [8]. Combining the above arguments, we find

B ∝ EhI ∝ Eh

3

12 . (2.1.1)

This is in accordance with B = Eh₉3 for perfect incompressible material1, as derived in [4].

Knowing B is important because it plays a role in the characteristic force scale of the folding process. Thence, a scaling argument is needed to establish the power-law relations. The factor h3 _{is essential in this part of the derivation.}

2.2 Establishing the power-law relation

The next step towards deriving the power-law behaviour of folding a thin sheet is considering the elastic bending energy of a fold. For this, the approach by [6] is used. In their work, they describe a thin circular ring of paper folded along its center line, which then takes on a sadle-like configuration.

For a configuration like this, it is known that the energy per unit surface is pro-portional to the square of the mean root [9]. Then, energy of the folding line is shown to be Ef B = K B R Z 2π 0 [cos(θ 2) − cos( θ0 2)] 2_dξ_. _(2.2.2)

Here, the term in the large parentheses will result in a factor entirely dependent on the geometry of the sadle-configuration. For the scope of this thesis, only the the term K_B is important, where K is the stiffness of the fold, equivalent to B. It is shown by [4] that K_B = L_h, where L is the length of the side the sheet is folded along. Disregarding the non-essential part of the intergral,

Ef ∝

BL

h . (2.2.3)

1_{For a plate, the general expression is B =} Eh3

12(1−ν2) where ν = 1

2 is defined as Poisson’s ratio

for perfect incompressible material. This term follows from the detailed mathematical derivation and is thus beyond the scope of this thesis.

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From this, the energy needed for a single fold can be derived easily. Imagine a sheet of size L × W . The energy needed to fold the sheet along L is Ec= F L. The ratio

of Ef and Ec then leads to

F0 ∝

B

h, (2.2.4)

which is the fundamental force for one fold. With a scaling argument, the force for an arbitrary n amount of folds can be acquired. One should consider how the effective thickness hn increases with each fold, and one should relate the amount of

layers N to the compaction ratio ∆ ≡ Di

Di, the initial and final size of the sheet. For 1d folding, hn = nh in the area where layers are stacked.2 The thickness of

upcoming layers remain h, so that the flexural stifness remains unchanged. The amount of layers increases as N = n, and by folding a sheet, one observes N = ∆. Thus

1d : F ∝ F0n = F0N ∝ F0∆, (2.2.5)

For 2d and 3d folding, the amount of layers increase as N = 2n_{, so the effective}

thickness increases as hn = 2nh for the whole sheet. Consequently, the flexural

stiffness also changes: Bn = 23nB. For 2d and 3d, one can observe N = ∆ and

N = (∆)2 respectively. Thus

2d : F ∝ 22nF0 = F0N2 ∝ F0∆2, (2.2.6)

3d : F ∝ 22nF0 = F0N2 ∝ F0∆4. (2.2.7)

In conclusion, the above equations can be bundled more tidily. For this, one must observe the general expression for the relation between N and ∆, which reads

N ∝ ∆γ∗,

and the general relation between force and number of layers, which reads

F ∝ F0Nα

∗ .

The asterisk indicates the variable is predicted by the hierarchical folding theory, and thus only depends on dimensionality. These equations can be combined to

2_{This relation holds for large n.}

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acquire the force as a function of compaction ratio:

N ∝ ∆γ∗ F ∝ F0Nα ∗    F (∆) ∝ F0∆β. (2.2.8)

where β ≡ α∗γ∗ and thus it depends only on dimensionality. Here, F0 is the

char-acteristic force scale determined by the flexural stiffness and the thickness, derived by considering the energy dissipated by and stored in a fold. Table 2.1 summarizes the theoretical predictions of the exponents. One can now experimentally explore the values for β for various crumpling setups, and make a comparison with the predictions of the hierarchical folding model.

Dimensionality Exponent

α∗ γ∗ β

1d 1 1 1

2d 1 2 2

3d 2 2 4

Table 2.1: An overview of the values of α∗, γ∗ and β as predicted by the hierarchical folding theory for different dimensionalities.

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Chapter 3 Methods

Three main experiments have been performed to further verify the validity of the hierarchical folding theory as a model for crumpling. Firstly, as an elaboration on the argument by [4], the 1d−2d mixture reasoning has been examined. All experi-ments have been repeated for twelve materials chosen for their varying properties. Secondly, a setup for 3d crumpling has been used to investigate β in this regime. Lastly, sheets have been crumpled by hand to find a relation between the plasticity and the amount of layers. Before illustrating the main experiments, the properties of these materials will be discussed.

3.1 Materials

Twelve materials, chosen for their varying thickness, plasticity and Young’s modu-lus, can be split into four categories: rubber (a white and a black rubber), Mylar (three different thicknesses), paper (bakery, waterproof and printing) and aluminium (two thin foils and a thicker one). The thickness was measured by using a digital caliper.

Rubber is considered to be purely elastic, which means the plasticity is zero: p = 0. The plasticity of paper, aluminium and Mylar remained to be measured. When a ribbon of Mylar is prepared into a cylindrical form with radius Ri (by wrapping it

around a cylinder), it will maintain some curvature κ = _R1

f after being released. One can see intuitively that Rf depends on Ri. The relation is a power-law: Rf ∝ (Ri)

1 p.

For each thickness of Mylar, approximately 10 ribbons were wrapped around cylin-ders with 0.45 mm < Ri < 3.75 mm. It is shown by [10] that mylar has a nonzero

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3.1. Materials 9

finite relaxation time. Therefore, the ribbons were held fixed and subsequently al-lowed to relax for both 15 minutes. Then, photographs from above were taken and Rf was measured by carefully fitting an animated circle onto the ribbon (figure

3.1a). Finally, the data was fitted (figure 3.1b) with a power-law from which p could be determined. One notable finding is that, within the accuracy of the experiment, it appears that p of Mylar does not depend on the thickness. Neglecting the vari-ation in h, this means that all Mylar sheets are intrinsically equal and thus can be considered as one material. For paper and aluminium foil, p was measured in the same way.

(a) (b)

Figure 3.1: (a) The measurement of κ of a ribbon of Mylar. (b) Plasticity measurements on different thicknesses of Mylar.

For most compounds, E was determined by performing an extension test. Here, a ribbon of material is placed between two clamps, from which one is connected to a force transducer. The ribbon is then stretched and the stress (force per unit cross section area) is measured versus the displacement ratio. Then, E is the gradient of the graph acquired. For aluminium foil, E was measured by a deflection experiment.

All measurement of the properties are summarized in table 3.1

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3.2. Plasticity and layers 10

Property Rubber Mylar Paper Aluminium

Black White Average Waterproof Bakery Print A B Thick

h (µm) 460 600 N/A 25 60 105 15 15 35

p 0 0 0.52 0.52 - 0.67 0.67 0.67 - 0.88 0.88 0.88 0.86 - 1

E (GPa) 0.04 0.08 4 - - 3 61 53 10

Table 3.1: Relevant properties of the materials used in the experiments. No measurements of p of bakery and waterproof paper were done, but these are expected to be between the indicated values. Measurements of E for paper, Mylar and thick aluminium and measurements of p for thin aluminium and printing paper where done by M. Habibi (private communication). No measurements of E for Waterproof and Bakery paper were done.

3.2 Plasticity and layers

To find a relation between the plasticity and the amount of layers N , various sheets of white rubber, paper and Mylar have been isotropically crumpled into a ball by hand. For paper and aluminium, the structures were cut in two equal parts and N was averaged over both halfs. For Mylar and rubber, the structures were pierced two times by a thin needle in two perpendicular directions. The amount of layers was obtained by averaging over the amount of holes.

By varying the sizes of the balls, one can obtain a relation for the amount of layers as a function of compaction ratio ∆. The sizes of the balls were found by measuring the diameter in two perpendicular directions and averaging over them. It is known from [4] that the relation between N and ∆ is a power-law relation with an exponent γ called the layer exponent:

N ∝ ∆γ (3.2.1)

Since the plasticity of each of the materials is known, one can then obtain a relation for γ as a function of p. Ultimately, when crumpling measurements for each material have been done, one may try to relate γ and p to the crumpling exponent β.

3.3 Measuring β

Two experiments are proposed. Firstly, one to examine the validity of the 1d−2d mixture argument. All experiments have been performed three times for each of

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3.3. Measuring β 11

the twelve materials. Secondly, one to explore the value of β in the regime of 3d crumpling.

3.3.1 The 1d-2d setup

The original experiment by [4] was repeated for sheets of size 30 cm × 30 cm. They were compressed at constant speed by a piston connected to a force transducer, confined by a large cylindrical shell of rl = 15 cm (figure 3.3a). The force was

plotted as a function of displacement ratio, and was fitted by relation 2.2.8 so that β could be determined. During the fitting process, one had to realize that a sudden decrease in β at growing F can be explained by the system entering a locking state [11] in which no new crumples are made, and thus this range of data needed to be omitted.

(a) (b) (c)

Figure 3.2: (a) The large cylinder, (b) the middle cylinder and (c) the small cylinder.

The whole setup was then repeated for two cylinders of radii rm = 8 cm and rs = 5

cm (figure 3.2b and 3.2c). An attempt was made to prepare the experiments so that the initial conditions would be equal. This involved minimizing and, when inevitable due to the geometry, equalizing the proces of precrumpling.

3.3.2 The 3d setup

An amount of 22 strings was fixed at both ends, and hanged by one end on a force transducer. The bundle passed through a hole in a slab of Plexiglas, which itself July 3, 2014

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3.3. Measuring β 12

was constraint to a height x as seen in figure 3.3a. A sheet of size 30 cm × 30 cm was placed between the strings by subtly precrumpling it into a sphere, and is held fixed by pulling the rods upwards via manual control of the force transducer (figure 3.3b). In order to achieve isotropic compressing, the wires were then distributed equally over the surface of the sphere in such a way that no material came out in between.

Next, the material was further compressed by pulling the ropes upward at a constant speed using the force transducer, measuring force as a function of time. Photographs were taken every 20 seconds, from which the compaction ratio as a function of time could be determined (figure 3.3c). Combining the data, β could be determined by the same fitting method as in previous section. Again, an attempt was made to fix the initial conditions. This involved precrumpling the sheets in an equal manner, namely twisting the corners towards the centre of the sheet and grabbing it from the back.

(a) (b) (c)

Figure 3.3: (a) The setup hanging freely. The white arrow indicates the height x. (b) Aluminium in a precrumpled configuration. (c) Aluminium after the crumpling experiment.

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Chapter 4 Results and discussion

4.1 The 1d-2d setup

For each material, a series of three experiments per cylinder (radii rl, rm and rs)

was performed. Measurements of β were obtained for each cylinder in every series. As an example, figure 4.1a shows the data of an experiment on Mylar, h = 19 µm, compressed in the smallest cylinder. There is a strong power-law relation, and when plotted on a log-log scale, the legitimacy for a power-law fit is evident. For this particular example, it was found that βs≈ 1.32.

(a) (b)

Figure 4.1: (a) Force versus compaction ratio on two scales. (b) Measurements of β for the large, middle and small cylinder

All 27 experimental realizations on Mylar showed a similar power-law behaviour, and as discussed on page 10, regardless of the thickness, the data of Mylar could be considered ensemble. It was then found that βl ≈ 1.13, βm ≈ 1.22 and βs ≈ 1.37.

As seen in 4.1b, β increases with decreasing cylinder size. This in accordance with the 1d−2d mixture argument. One may therefore interpret this dataset as a

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4.1. The 1d-2d setup 14

tion for hierarchical folding theory as a model for crumpling of a thin sheet.

The values measured for β for all other materials are summarized in table 4.1. For clarity, similar plots of β as a function of cylinder size for paper and aluminium are shown in figure 4.2a and 4.2b. Contrary to the results of Mylar, it can be seen that β shows a decrease for smaller cylinders. This phenomenon is not supported by the 1d − 2d mixture argument. It is however not a priori clear if this is due to invalidity of the mixture reasoning or due to the effect of another mechanism at work. Possibly, the process of precrumpling has an effect on the measurements of β. Initial conditions might have varied due to the arbitrary crumpling by hand and due to the varying geometrical constraints of the cylinders.

(a) (b)

Figure 4.2: (a) β for paper for different cylinder sizes. (b) The same, but for aluminium.

Property Rubber Mylar Paper Aluminium

Black White Average Waterproof Bakery Print A B Thick

βl 2.00 1.75 1.13 1.14 1.22 1.12 1.75 1.61 1.90

βm 1.96 2.09 1.22 1.41 1.56 1.56 1.68 1.87 1.74

βs - - 1.37 1.33 1.24 1.52 1.44 53 1.43

Table 4.1: All values measured for β for the different materials using different the three different sizes of cylinders. Measurements for rubber with the large cylinder were done by M. Habibi (private communication).

Even though it was endeavoured, it was geometrically impossible to accomplish iden-tical precrumpled structures because of the varying sizes of the cylinders. Sheets for July 3, 2014

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4.1. The 1d-2d setup 15

the large cylinder were nearly unharmed, whereas for the small cylinder they were strongly deformed. It was unsuccessfully attempted to demonstrate the impact of deviating initial conditions by performing an ad hoc experiment for the small cylin-der: before compressing and comparing β of two sheets, one was rolled up without making any crumples, another was randomly precrumpled. No significant difference was measured after three trials. This suggests that arbitrary differences as a result of the precrumpling by hand is not responsible for the decrease in β. The effect of the varying geometrical constraints by the cylinders is however still open to discus-sion.

A possible explanation for the decrease of β is suggested by observing the struc-tures compressed in the smallest cylinder. Figure 4.3 shows a strong hierarchy which has a lot in common with 1d hierarchical folding, as shown in figure 1.1a. This reveals a link between ordered folding and the process of crumpling. Possibly, an increased confinement could somehow mimic the environment of 1d compres-sion, for which the folding theory predicts β = 1. Although the decrease in β is not predicted by the mixture argument, it thus appears to support the same principle. Therefore, one can argue it is likely that another unidentified process is at work simultaneously, and can refrain from rejecting the mixture reasoning.

Figure 4.3: Mylar crumpled in the smallest cylinder. A strong hierarchical is observable.

Considered on its own, the hierarchy of the crumpled structure argues in favour of the fold-ing theory as a model of crumplfold-ing. It is however unclear why Mylar did not show a decrease in β for smaller cylinders. Possibly, β has a smaller tendency to decrease for materials with a low plasticity. However, for black and white rubber

with p = 0, β was measured to be around 2, which can not be identified as a 1d process. Still, the thickness of the used rubbers were much larger compared to the other materials, and the data was excessively noisy, making it incompatible for good comparison. It is possible the large thickness increases the effect of friction. Even

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4.2. The 3d setup 16

more, the effect of friction as a whole is disregarded, while it is arguable it can affect β.

In conclusion, the data does not support the 1d − 2d mixing argument. A possi-ble explanation for the decrease of β could be the effect of precrumpling. Observing the crumpled structures, it is suggested that strong confinement gives rise to a 1d crumpling effect, associating crumpling with ordered folding and thus showing a parallel with the mixture reasoning. Rather than rejecting the mixture reasoning, it is more likely to consider a simultaneous process that grows in dominance for smaller cylinders.

4.2 The 3d setup

In figure 4.4 the data of an experiment on aluminium is shown. It shows a clear linear behavior. After 25 realizations for different materials, it was not succeeded in achiev-ing a dataset that shows a strong power-law behavior as with the previous setup,

therefore no appropiate measurements on β were made.

Figure 4.4: Linear results

It is argued that the failure of the measurements is due to the precrumpling of the sheet. In order to be able to equally spread the wires over the surface of the material in a manner that no mate-rial comes out in between, it was often necessary to precrumple the sheet to a large extent. Pos-sibly, within the force sensor’s range of F = 100

N, not enough work was available for the sheet to be significantly crumpled any further, holding off the power-law regime. Instead, a linear regime of elastic com-pression was dominant. One should try to repeat the experiment with a higher up-per force limit in order to obtain a power-law relation between force and compaction

In the next section, investigation on the amount of layers of a 3d crumpled structure

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4.3. The effect of plasticity 17

will be done. Since no measurements for β were obtained for 3d crumpling, data of previous research on exactly the same materials have been used for comparison. This data was acquired by M. Habibi (private communication).

4.3 The effect of plasticity

By comparing horizontally the deviating measurements for β in table 4.1, one can already safely claim that p has an effect on the crumpling exponent. By exploring the behaviour of γ, an attempt will be made to associate p with β more explicitly.

Figure 4.4a shows data on the number of layers N of 35 crumpled Mylar balls with radius 1,4 cm ≤ r ≤ 4,8 cm. There was a clear power-law relation, and the layer exponent (as defined by equation 3.2.1) was measured to be γ ≈ 1, 76. For

Figure 4.5: (a) The amount of layers for Mylar balls of different sizes. The slope of this fit is γ (b) Measurements of γ for materials of different plasticities. (c) A comparison of β and 2γ. Here, α∗ = 2 such that β = 2γ (d) A comparison of αγ and β, where α ≈ 3.5.

paper, aluminum and white rubber, γ was obtained in an equivalent way. In figure July 3, 2014

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4.4b the dependence of γ on the plasticity is shown. Therefore, to associate p with β, one may start with a comparison of the layer γ and crumpling exponent β. Firstly, by viewing 4.4c, one can immediately observe for β and γ a similar trend. Since γ depends on the plasticity, this suggests β also depends on p.

Secondly, a more theoretical approach is attempted. From equation 2.2.8, recall that F ∝ F0Nα

∗

where the exponent is predicted1 _{to depend on the dimensionality:}

α∗ = 2 for 3d folding. From the scaling argument followed that N ∝ ∆γ∗ _where

γ∗ = 2 for 3d compaction. Combining these relations, equation 2.2.8 was found, which read

F ∝ F0∆β.

However, for 3d crumpling, it was found that γ deviates from the theoretical γ∗ = 2 (figure 4.4b). One can therefore adjust this equation:

N ∝ ∆γ F ∝ F0Nα ∗    F ∝ F0∆β. (4.3.1)

Note that β ≡ α∗γ∗, whereas β ≡ α∗γ is now modified according to experimental findings. Physically, one can interpret this substitution as taking into consideration the effect of plasticity: where α∗ and γ∗ depend only on dimensionality, γ depends also on p. This results in an crumpling exponent β that depends both on plasticity and dimensionality.

Figure 4.4c actually shows a comparison between β3d and β, because β ≡ α∗γ = 2γ.

The scale difference spurs to consider also replacing α∗ by a parameter α, leading to e

β ≡ αγ. Physically, it is not a priori clear how this can be interpreted, and although no direct measurements were made for this exponent, the data suggests α ≈ 3, 5 for an acceptable fit (figure 4.4d).

In conclusion, the experimental findings suggests that both plasticity and

dimension-1_{The parameters that are described by the folding theory are indicated by an asterisk. Thus,}

these parameters depend only on dimensionality of the configuration.

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ality are important for the crumpling of a thin sheet. Quantitatively, this invokes replacing β with eβ so that the force is now associated with both dimensionality and plasticity:

F (∆) ∝ F0∆βe. (4.3.2)

When comparing γ with the measurements of β for the cylinders of the 1d−2d setup, one would immediately expect a difference in scale, but might argue to observe a similar trend in β as in figure 4.4a. Remarkably, contrary to 3d crumpling, figure 4.6 does not show decrease in β for increasing p, but a slight increase. It is not a priori clear what establishes this difference in behavior. To investigate whether equation 4.3.2 holds beyond 3d compaction, γ must be measured for the 1d−2d regime.

Figure 4.6: Comparison of γ for 3d crumpling with β for the 1d−2d setup.

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Chapter 5 Conclusion

An attempt was made to explore the validity of the hierarchical folding theory as a model for the crumpling of a thin sheet, in which folding of thin sheets is described by a power-law with an exponent β depending on the geometry of the folding. For this purpose, the effect of plasticity and dimensionality was investigated experimentally. The latter was examined by considering two experiments in different dimensional regimes

Firstly, it was unsuccesfully attempted to measure β for a 3d setup. Merely lin-ear results were obtained. It is argued that the linlin-ear regime is due to the large extent of precrumpling: the upper force limit of F = 100 N might have been too low to significantly crumple the sheet any further. Perhaps increasing the force limit will result in a power-law relation.

Secondly, cylinders of decreasing size were used to investigate the 1d − 2d mixture argument, which states that increasing the confinement will result in an increase of β. For Mylar, it was found that β behaves accordingly. On the contrary, a decrease in β was observed for paper and aluminium sheets. It is demonstrated that varying initial conditions due to the arbitarty precrumpling by hand is unlikely to be the source of this behaviour, although the effect of diverging geometrical constraints is still open to discussion. The hierarchy of the crumpled structures bears resem-blance with 1d folded sheets, which suggest a mechanism at work that promotes, rather than 2d, a stronger 1d compaction effect. Even though this explanation is incompatible with the 1d−2d mixture argument, the underlying principle of dimen-sional dependency seems to be mutually supported, suggesting the mixture argument might not need to be rejected. It is however unclear why Mylar behaved differently

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Chapter 5. Conclusion 21

from paper and aluminium. Perhaps plasticity determines the material’s predilec-tion for 1d or 2d compacpredilec-tion at high confinement. This could be examined further by properly repeating this experimental setup for purely elastic material like rubber.

To investigate the effect of plasticity, the amount of layers N of 3d crumpled struc-tures was analysed. Different materials were crumpled into balls of different sizes, from which the layer exponent γ could be measured. It was found firstly that γ behaves similarly to β3d, obtained in previous research. Secondly, it was found that

γ depends strongly on the plasticity, which means the experimental γ deviates from the theoretical prediction based on dimensionality, γ∗ = 2. It is then suggested to redefine β ≡ α∗γ∗ → β ≡ α∗_{γ such that γ}∗ _{→ γ, which implies the crumpling}

exponent is now dependent on both dimensionality and plasticity. Lastly, β can be rescaled to nearly perfectly fit the experimental data of β3d, by changing β → eβ

such that α∗ = 2 → α ≈ 3.5. It is however unclear how this substitution should be interpretated physically. A suggestion for future research would therefore be to investigate the behaviour of α.

Also, when comparing γ found for 3d crumpled balls with β from the 1d−2d piston setup, no similarity was found. To see whether γ can predict the behavior of β in this dimensional regime, the behavior of γ for structures crumpled in the 1d − 2d experiment should be investigated.

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On the effect of dimensionality and plasticity on the crumpling of a thin sheet