Mechanical Properties of Randomly

Crumpled Thin Sheets

Beni Snow, ‘19

Submitted to the

Department of Mechanical and Aerospace Engineering

Princeton University

in partial fulfillment of the requirements of

Undergraduate Independent Work.

Final Report

May 9, 2017

Andrej Kosmrlj

Mikko P. Haataja

MAE 340

35 pages

reader Copy

c© Copyright by Beni Snow, 2017.

All Rights Reserved

This document represents my own work in accordance with University regulations.

Abstract

Randomly crumpled thin sheets exhibit significant changes in their mechanical prop-

erties due to their geometry as compared to non-crumpled, or regular, materials.

For a sheet made of a given material at a given size and thickness, the geometrical

changes that result from crumpling lead to sheets that are stronger in certain ways,

but weaker in others.

A simple example of this is a sheet of paper. A non-crumpled piece of paper,

held at its edge, will flop over under its own weight. If the paper is crumpled and

then unfolded, it will become much more stiff and can hold itself up when supported

at one end. This crumpled and then flattened piece of paper is an example of a

planar crumpled sheet, meaning it is a flat sheet with the crumpling deformations,

as opposed to still being crumpled in a ball or formed into a cylinder or some other

shape.

Characterizing the changes in strength of these types of crumpled sheets is the

principal objective of this research. It was expected that the crumpled sheets would

become stronger against bending and compression forces, but weaker against tension

forces. This expectation matched the results.

In general, it was expected that the higher the level of crumple, the more pro-

nounced these changes would be. This generally true, although certain combinations

of material tested and level of crumple defied this expectation. It is possible that the

randomness of crumpling is responsible for such results. It is also possible that the

metric for defining level of crumple used was not a good assesment of a the changes

in a crumpled sheet.

It was also anticipated that, based on the data from one material, the change

in strength to a different material could be predicted. This was true for certain

geometries, but uncertainty and the inherent randomness and crumpling made it

difficult to verify.

The two materials tested were steel and aluminum sheets with a thickness of 0.002

inches (0.0508 mm). Three different quantitative levels of crumpling were examined,

qualitatively referred to as low crumple, medium crumple, and high crumple.

iii

Acknowledgements

First off, I would like to thank the entire MAE department for accommodating a

sophomore who for some reason thought it was a good idea to attempt independent

work a year early. You didn’t succeed in talking me out of it, and then I received

nothing but help and encouragement whenever I asked.

I would like to thank my adviser, Professor Andrej Kosmrlj for steering me towards

an area that was both interesting enough to put up with for a semester and within

my (rather limited) current abilities. He helped me work through all of the problems

I encountered, and spent a lot of time doing so. Thank you.

It would be a sin for me to write an acknowledgements without mentioning Mike

Voccaturo. He walked me through Whetstone bridges, strain gauges, translation

stages, and everything else I needed, and he provided me free materials and a space

to work. This project would not have been possible without his help.

Also thanks to Glen and Al in the MAE machine shop with helping out to fabricate

some of the parts for the testing apparatus.

I would feel guilty if I did not thank my parents, without whom I am sure that I

would not be at Princeton, and would not be able to work on this research.

Finally, it was a fantastic coincidence that Gorillaz and Mac Demarco released

new albums in the weeks leading up to the due date. Their music made writing this

up much more enjoyable.

iv

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Introduction 1

1.1 Mechanical testing of thin sheets . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Crumpled Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Experimental Setup 5

2.1 Translation stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Force gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Testing procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.2 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.3 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Data correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Results 13

3.1 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

v

4 Discussion and Conclusions 20

4.1 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

A Raw Data 24

vi

List of Figures

1.1 Bending setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Schematic of a Wheatstone bridge . . . . . . . . . . . . . . . . . . . . 6

2.3 Calibration curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Cylinders used to constrain the samples. . . . . . . . . . . . . . . . . 9

2.5 Samples. Low, medium, and high crumple from left to right. . . . . . 9

2.6 Bending of force sensor beam. . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Raw bending data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Corrected bending data. . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Combined and non-dimensional bending data. . . . . . . . . . . . . . 15

3.4 Raw compression data. . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Corrected compression data. . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Combined and non-dimensional compression data. . . . . . . . . . . . 17

3.7 Raw tension data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.8 Corrected tension data. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.9 Combined and non-dimensional tension data. . . . . . . . . . . . . . . 18

3.10 Combined tension and compression data. . . . . . . . . . . . . . . . . 19

4.1 Horizontal vs vertical folds. . . . . . . . . . . . . . . . . . . . . . . . 21

vii

Chapter 1

Introduction

1.1 Mechanical testing of thin sheets

Most sheets can be considered non-crumpled, or regular. They have regular geom-

etry and their behavior can accurately be predicted according to well-established

equations. [4]

Common regular sheets have had their mechanical properties extensively charac-

terized. In contrast, there is little work on sheets that have been crumpled before

testing begins. By applying the tests used on regular sheets to crumpled ones, the

differences to mechanical proprieties that result from the crumpled geometry may be

determined.

Three common tests for characterizing a sample of a regular sheet is the three

point bending test, the compression test, and the tension test.

1.1.1 Bending

In three point bending, a sample is rested on two holders a fixed distance apart.

These two points of contact between the holder and the sample are two of the three

points. The third point is a moving force gauge that presses down on the sample,

bending it between the two holding points. See figure 1.1

For metal samples, this is essentially creating a spring. A leaf spring is just this

setup–a thin piece of metal fixed at two points with a load applied to the middle.

As with all springs, the displacement of the center of the sample, where it comes

into contact with the force gauge, can be related to the applied force by means of a

spring constant k, using the equation

1

Figure 1.1: Bending setup

F = −ky, (1.1.1)

where F is the measured force, k is the spring constant, and y is the displacement.

The negative is to show that the force is in the opposite direction as the displacement.

The spring constant k in a regular sample is

k =48EI

L3, (1.1.2)

where k is the spring constant, E is the Youngs modulus of the material, I is the

moment of inertia of the sample, and L is the distance between the supports. [1] By

experimentally determining the spring coefficient, the effective Youngs Modulus may

be determined and compared to the true Youngs Modulus of the material.

1.1.2 Compression

A sample may also be fixed at one end, and have a force applied to the other to

compress the sample. With regular sheets, such a setup will result is a characteristic

buckling force. Before the buckling force is reached, the sample will hold rigid and

compress linearly. Afterwards, the sample will collapse. For a sample held in place

at one end and pinned at the other, the buckling force is given by the equation

F =2π2EI

L2, (1.1.3)

2

where F is the buckling force, E is the Young’s Modulus of the material, I is the

smallest area moment of inertia for the sample’s cross section, and L is the length of

the sample. [1] Although the setup used did not pin one end, due to the friction with

the force sensor it could no move, making it effectively pinned, so the equation still

holds.

Beyond the buckling strength, a sample will continue to compress and the change

in length will be related to the force, although not by a linear relationship as with

bending and the spring constant.

1.1.3 Tension

A third way of characterizing the properties of a sample is by applying a tension force.

Initially, a sample under tension will strain in a linear fashion. The ratio between the

stress applied and the strain is the Young’s Modulus of the material.

Eventually, a stressed sample will fail, but for metals the forces involved are quite

high (with stresses on the order of hundreds of megapascals) and outside the scope

of this work.

1.2 Crumpled Sheets

A simple example of a crumpled sheet is a piece of paper that has been crumpled into

a ball and then unfolded. Such an example has the two key attributes of a crumpled

sheet. One, the sheet has numerous random folds and creases. Two, it had the same

general shape as the regular form.

A rectangular sample that has been crumpled into a ball is a completely different

overall shape as compared to the non-crumpled rectangle, and is difficult to compare.

The sample crumpled ball, when spread out into the original rectangle shape, but

now with folds and creases, is a crumpled version of the original, which is useful for

comparison.

The crumpled piece of paper is a planar crumpled sheet, which is the type discussed

in this report. Other crumpled geometries, such as crumpled cylinders and crumpled

spheres, have been the subject of limited research, [2] although in general, crumpled

materials have been largely overlooked.

Kosmrlj & Nelson, 2013, examined warped membranes, which are similar to crum-

pled planar geometries. [3] Although the published research was a theoretical model,

they also examined 3D printed plastic membranes that was approximately planar,

3

the research is the most similar to the research conducted for this report, and served

as the inspiration. Kosmrlj and Nelson examined the change in mechanical properties

with the change in height profile of the material. A higher range of height variation in

the material is a sign of a more crumpled sample. They only examined the change in

the linear regime, before buckling occurs. In that regime, they generally found that

as the height profile, and therefore the level of crumple, increased, the resistance to

compression increased, while the resistance to tension forces decreased. [3]

Such a finding makes sense with an intuitive understanding of these sheets. Re-

turning to the example of the crumpled paper, it is understood that crumpled paper

bends less easily than non-crumpled paper. A flat sheet of paper will flop over under

its own weight. A crumpled one will not. A non-crumpled piece of paper is quite

resistant to tension. Pulling on such a sheet does not result in much strain, whereas

a crumpled sheet can strain significantly.

Other crumpled geometries, such as spheres and cylinders, have received more

attention. This may be because such shapes are used to build pressure vessels, rocket

bodies, and other objects that are subject to significant forces that may cause localized

crumpling.

John Hutchinson in 2010 described how localized deformation which he called

crumpling reduce the buckling strength of spheres and cylinders. [2]. This decrease in

strength for compressive forces is opposite that increase in strength found by Kosmrlj

& Nelson. This discrepancy may be because the structural integrity of a round shape,

such as a cylinder of sphere, is dependent on the nearly perfect roundness of the shape.

Any change in that reduces the structural integrity. Planar geometries do not depend

on the perfection of their geometry.

1.3 Objectives

The primary objective for this research was to characterize how the strength and

rigidity of a crumpled sample change compared to a non-crumpled, regular, sample. It

was anticipated that the geometry of the sample, meaning the level of crumpledness,

would dictate the properties of the material. This implies that there should be a

relationship between samples with the sample level of crumple, even if the samples are

comprised of different materials. This is analogous how the same equations describe

bending, buckling, and other deformations for non-crumpled materials, with only a

parameter for some property of the material, such as its Young’s Modulus.

4

Chapter 2

Experimental Setup

2.1 Translation stage

In order to generate stress-strain curves, the displacement of the sample must be

measured. This was accomplished by means of a vertically mounted translation stage.

The stage was mounted on a threaded shaft so that each turn of the shaft moved the

stage up or down by 1/40 of an inch (0.635 mm).

The translation stage is the the vertical structure in figure 2.1. The moving part

of the translation stage is the aluminum slider in the middle with the blue L-bracket

and the thin brass beam mounted on it.

For the bending and compression tests, a data measurement was taken every half

turn (0.3125 mm) and every quarter turn (0.15875 mm) for the tension tests.

2.2 Force gauge

The other half of the data measured was the force exerted on the sample. This was

captured by use of a full bridge strain gauge setup. A strain gauge is a resistor

that varies its resistance when strained. By placing a strain gauge on a beam, and

then by bending the beam by exerting force on the end of that beam, the strain

gauge lengthens or shortens (depending on which side of the beam it is on) and the

resistance changes. With a constant input current, the output voltage will change

based on the beam’s curvature. By calibrating the force exerted on the beam to the

output voltage, the strain gauge and beam setup can be used as a force sensor. [5]

To increase the accuracy of the strain gauges, they are arranged in a full bridge

5

Figure 2.1: Experimental setup

setup. In this setup, 2 strain gauges are placed on the bottom of the beam, and 2

are placed on top. They are wired into a Wheatstone bridge, which allows a small

change in resistance in the strain gauges to result in a detectable voltage change. [5]

A schematic of a Wheatstone bridge may be found in figure 2.2.

The equation describing the output voltage of the bridge is

VoutVin

=R3

R3 +R4

− R2

R1 +R2

(2.2.1)

Figure 2.2: Schematic of a Wheatstone bridge

6

This equation implies that if all the resistors have the same resistance, then the

output voltage will be zero.

When the beam is bent, the resistance on the strain gauges that are stretched

increase, and the resistance on the strain gauges that are compressed decrease. Re-

sistors 1 and 3 are placed on one side of the beam so they are lengthened and have

their resistance increase, and resistors 2 and 4 are placed on the other side of the

beam so they are compressed and have their resistance decrease. Equation 2.2.1 also

implies change that if resistors 1 and 3 increase their resistance by x%, and resistors

2 and 4 decrease their resistance by x%, then the output voltage will be x% of the

input voltage, for small values of x.

All the force sensors used in this researched used 350 ohm resistors, meaning their

non-strained resistance was 350 ohms. Typical percent changes when strained are

around 0.1% or less. A input voltage of 5 volts was used, resulting in an output

voltage in the millivolt range.

To make such small voltages easier to accurately read, the output voltage was sent

through a AD623 amplifier, which amplified the signal by roughly a factor of 1000.

This amplified voltage was read by a Photon microcontroller, built by Particle, which

measured the voltage and then sent the reading to a MATLAB script which logged

the voltage.

Four separate beams were used for the testing of this research, with thicknesses

ranging from 0.007 inches to 0.063 inches. (0.1778 to 1.6002 mm). These beams were

made from either brass or aluminum. In order to ensure that at zero force the gauge

would read zero voltage, it was essential that the beams had no curvature. Cutting

metal so thin without bending it is impossible, so a stack of thin metal sheets were

bolted together and milled to shape. Each gauge was 2.5 inches long and 1 inch wide

(63.5 mm long and 25.4 mm wide). To maximize the range of forces that could be

measured, each gauge had two positions at which the force could be exerted. Position

one was 1.25 inches (31.75 mm) from the attachment point to the translation stage.

Position two was 2 inches (50.8 mm) away.

2.2.1 Calibration

Once a force gauge was created by attaching four strain gauges to a beam, the sensor

had to be calibrated. This was accomplished by measuring the voltage with a given

mass hung from the end of the gauge. Since the force exerted on the beam is simply

the mass times the acceleration due to gravity, this allows for a series of measured

7

Figure 2.3: Calibration curve

voltages to be associated with known forces. This data was then used to create a first

degree polynomial fit (linear relationship) which allowed for any measured voltage to

be correlated with the force that would create such a voltage. An example calibration

curve with the fit line is shown in figure

2.3 Sample preparation

Preparing crumpled samples that had a reproducible degree of crumpledness was a

key component of this research. This was accomplished by taking a 3 inch square

(76.2 mm) piece of the material to be tested.

The materials used were AISI 1008 steel and 1145 aluminum.

For a crumpled sample, the square had to first be volumetrically constrained. The

size of the constraint volume is what determines the level of crumpledness.

The constraining was done in a right cylinder, meaning a cylinder where the height

is equal to the diameter. This allows for a single length dimension to reflect the size

of the cylinder. The smaller the cylinder, the more crumpled the sample.

Three cylinder diameters were used: 2, 3 and 4 cm, corresponding to the high,

medium, and low levels of crumple. It is worth noting that the aluminum samples

were produced in all three levels of crumpledness, but the steel samples were only

produced in low and medium crumple. This was because the steel was too stiff to be

constrained to the smallest cylinder.

The cylinders were 3D printed out of ABS plastic on a Stratasys 3D printer. A

8

Figure 2.4: Cylinders used to constrain the samples.

Figure 2.5: Samples. Low, medium, and high crumple from left to right.

picture of the cylinder may be found in figure 2.3. Note that 2, 3 and 4 cm are

the inside dimensions of the cylinders. The cylinder walls are 0.4 mm thick, so the

cylinders appear larger than they actually are on the inside.

After being constrained to the appropriate cylinder, the square was removed and

flatten once to a flat plane by spreading the corners to their original distance apart,

and then allowing the sample to return to whatever geometry was natural for it. This

ensured a reliable and repeatable flattening procedure. The now crumpled sheet was

then cut in half into two 3 by 1.5 inch samples in order to fit in the testing apparatus.

See figure 2.4 for a picture of such samples, ordered left to right as low, medium, and

high.

At this point the sample preparation diverged depending on which type of test

was required.

9

For the bending tests, no further work was required.

For the compression tests, a line was drawn on the sample 0.5 inches (12.7 mm)

from one edge. This line marked the amount of sample that was to be clamped

in place underneath the strain gauge during testing. This means that although the

entire sample measured 3 inches (76.2 mm) long, the effective length tested was 2.5

inches (63.5 mm).

The samples used for the tension tests had the sample line drawn at 0.5 inches for

clamping, but also had a small hole made 0.5 inches from the other end. This was to

allow a piece of copper wire to be tied through the sample and onto the beam of the

force sensor to let the sensor pull on the sample. A wire was used instead of a clamp

since a heavy clamp mounted on the beam would make measuring the small forces

produced very difficult. The line and the hole resulted in an effective sample length

of 2 inches (50.8 mm).

It was essential that each sample was tested only once. The samples plastically

deformed after each test, so a totally new sample was used every time.

2.4 Testing procedure

2.4.1 Bending

For bending tests, a sample was placed horizontally on ’U’ shaped bracket to allow

for a three point bending test. In such a test, the sample lies on two points, and the

force sensor acts as the third point in between the two supports and pushes down on

the sample, bending it. The two supports were 2.375 inches (60.325 mm) apart.

For the bending tests, a data point was captured every half turn, meaning every

0.3175 mm. The strain was done for all tests statically, meaning the translation stage

did not move while a measurement was being taken.

2.4.2 Compression

The compression process was similar to the bending process, except instead of the

sample resting on a support, it was placed vertically with the bottom end clamped

into place (using the 0.5 inch tab that was marked for this purpose.

The force sensor was then placed on the top end and moved downwards, com-

pressing the sample. The data acquisition frequency was identical to bending.

10

2.4.3 Tension

For tension testing, the sample was placed in the same clamp as with compression

testing. The force sensor had to be rotated 180 degrees relative to compression and

bending testing, since it would pull on the sample rather than push.

The sample was connected to the force sensor by a short length of copper wire

tied through the hole near the top of the sample and then tied through a set of two

holes in the force sensor.

The data points were taken twice as frequently for tension as compared to ten-

sion/bending. This was to ensure adequate sampling over the course of the short

strains produced. A data point was taken every 0.15875 mm.

2.5 Data correction

In order to have enough sensitivity in the force sensor to measure forces that were

as low as the 100 millinewton range, very thin beams were required. These beams

would bend, making the displacement at the base of the beam different from that at

the end. See figure 2.5 for a schematic of this displacement. The figure is the force

gauge for either compression or bending testing, where the beam is pushing down on

the sample, meaning the sample pushed up on the beam.

To correct for this, the displacement delta of the end of the beam had to be

subtracted from the measured strain of the translation stage. As this was typically

on the order of a millimeter or less, actually measuring the strain would be very

error-prone.

By assuming that the strain gauge was a uniform beam (neglecting the holes used

to connect to the tension samples) and that the bending under it’s own weight was

negligible, the beam’s displacement for small displacements could be calculated as a

bent Euler beam [1] using the equation

δ =FL3

3EI, (2.5.1)

where F is the applied force, L is the length of the beam from the point of applied

force to the connection point to the translation stage, E is the Young’s Modulus of

the beam material (either aluminum or brass) and I is the moment of inertia of the

beam.

11

Figure 2.6: Bending of force sensor beam.

12

Chapter 3

Results

There were seven different types of samples tested: three levels of crumpled aluminum,

two levels of crumpled steel, plus a non-crumpled control for both materials. Each

sample type underwent the bending, compression, and tension tests, for a total of

21 types of testing. In an attempt to minimize the effects of randomness, each test

consisted off five individual trials, for a total of 105 trials. The plots of the complete

data set may be found in the appendix. All plots in this section are an average of the

five trials.

3.1 Bending

The uncorrected bending data for aluminum and steel may be found in figure 3.1.

Uncorrected means without modifying the strain data to account for beam bending.

The error bars show half of a standard deviation over the range of the five trials

measured. While a full standard deviation is more standard, such a range would

make the graphs very difficult to read.

The data was then corrected to account for the bending of the force sensor beam.

Note how the data points are no longer at identical intervals for all of the materials

tested. The corrected bending data may be found in figure 3.2. The error bars are

the same half of a standard deviation.

The data may be made non dimensional by taking strain as a percentage of the

sample length, and by creating a force coefficient according to the equation

FL2

EI, (3.1.1)

13

Figure 3.1: Raw bending data.

Figure 3.2: Corrected bending data.

14

Figure 3.3: Combined and non-dimensional bending data.

where F is the force, L is the sample length, E is the Young’s Modulus of the mate-

rial, and I is the moment of inertia of the sample’s cross section. The coefficient is

independent of the material, which allows the aluminum and steel data to be more

easily compared. This non dimensional combined data is found in figure 3.3. The

error bars are not shown to ease in the readability of the data.

3.2 Compression

Similar the bending, the uncorrected data is shown in figure 3.4 for aluminum and

steel, respectively. The corrected data is shown in figure 3.5, and the non dimensional

combined data is shown in figure 3.6. The error bars (when shown) are still half of a

standard deviation.

The absolute forces for steel were much larger than for compression, and larger for

compression than for bending. These two factors combine to require thicker beams

used in the force sensors for compression of steel. Since the displacement of the force

sensor beam is inversely proportional to the moment of inertia of the beam, and the

moment of inertia is proportional to the cube of the beams thickness, the experiments

run with thicker beams have much smaller corrections. This is particularly notable in

15

Figure 3.4: Raw compression data.

Figure 3.5: Corrected compression data.

the right sides of figures 3.4 and 3.5, which are nearly identical despite the correction.

3.3 Tension

Again, the first figure (3.7) is the uncorrected tension data, the next (figure 3.8) is the

corrected data, and the third figure (3.9) is the combined non dimensional data. For

tension, the equation for the force coefficient was different as compared to bending

and compression. It was

F

EA, (3.3.1)

where F is the force, E is the Young’s Modulus, and A is the cross sectional area.

The reason for the different coefficient is the strain of a sample under tension is not

a function of the moment of inertia, but rather the area.

16

Figure 3.6: Combined and non-dimensional compression data.

Figure 3.7: Raw tension data.

17

Figure 3.8: Corrected tension data.

Figure 3.9: Combined and non-dimensional tension data.

18

Figure 3.10: Combined tension and compression data.

Finally, the compression and the tension plots may be combined to form a full

plot of forces axial the the sample. This is shown in figure 3.10. This plot also shows

minor artifacts from the data correction, such as the non-crumpled aluminum briefly

experiencing a backwards displacement in the lower right quadrant. This is when

the force on the sample increases so rapidly that the correction equation returns a

negative value for δ.

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Chapter 4

Discussion and Conclusions

The data confirms the expectation from previous research that the compression and

bending strengths for crumpled materials would increase, while the tension strength

would decrease.

4.1 Bending

For the bending experiments, the sample can be thought of as a small spring, requiring

a force to bend it a certain distance. Under this model, the slopes of the lines of the

force/displacement charts are the spring constants of the samples. Viewed this way,

there is a 10 fold increase in the spring constant between the highest crumpled samples

and the uncrumpled controls. The less crumpled samples fell somewhere in between

in no particular order.

The lowest level of crumple did have a tendency to experience sudden drops in

the force applied, before beginning to rise with the strain again. The plot of all

the individual bending runs for the low crumple steel and aluminum found in the

appendix show this clearly, but can even be seen once averaged out over five trials.

This is thought to be because on the lowest crumple level had a tendency to form

a few number of large folds rather than many smaller ones. See figure 2.4 and 4.1.

These larger crumples would bend and since each crumple was responsible for a

large percentage of the sample’s strength, a significant drop in strength would occur.

With more, smaller crumples, when one crumple bends it does not make a large

difference.

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Figure 4.1: Horizontal vs vertical folds.

4.2 Compression

For compression, the plots did not shown a roughly linear trend, but all the crumpled

samples did shown increases in strength. The lack of linear trend is expected, as even

regular sheets only have a small linear regime before they buckle. There was less of a

clear stratification by crumple level as with bending, but that is because of the nature

of the crumples.

Some of the folds in the crumpled material run vertically, while others run hor-

izontally. See figure 4.1. The right sample in that image has mostly folds oriented

in the vertical direction when the sample is being compressed. The left sample has

mostly a single large horizontal fold.

The horizontally folded sample will have much, much less resistance to compression

forces as compared to the sample with the vertical folds. This makes the differences

between crumple levels less noticeable. Note the large spread of the error bars in all

of the data.

Although this is consistent with the research conducted on planar material, the

research on cylinders and spherical shells showed a decrease in compression strength.

This is thought to be because those structures rely on not exceeding a certain buckling

strength, and that buckling strength is smaller in crumpled materials.

This is not a factor for the planar materials, since their buckling strengths are

so low compared to the forces in this experiment that essentially the entire trial was

conducted on a buckled structure.

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Equation 1.1.2 shows that for uncrumpled aluminum and steel, the critical buck-

ling force is and 94 and 270 millinewtons, respectively. Such forces were often exceeded

before the first data point was taken.

4.3 Tension

The tension data, while it showed a decrease resistance to pulling forces in the axial

direction, did not reveal much differentiation in the levels of crumple, similar to the

compression data. This may also be due to the orientation of the folds. A vertical

fold is much more difficult to ’pull out’ of the material than a horizontal one.

4.4 Conclusion

This research demonstrated that crumpled sheets are easier to stretch (tension) than

regular sheets, but harder to compress or bend. This fits with the little prior research

on planar crumpled geometries. It does not match with the research on circular

shapes, but that is too be expected, as those geometries derive their strength from

their perfect form.

At low levels of crumple, the sheets become very random. This is because a small

number of individual folds and creases make the sheets anisotropic so the direction of

the force is very relevant to the behavior of the sheet. The higher levels of crumple

seem to be less susceptible to this issue.

Therefore, further research would benefit on examining higher levels of crumple,

or determining a method for classifying the level of crumple that eliminates this issue

of directionality.

For example, perhaps the volumetric constraint is not the best method of classi-

fying crumple. Perhaps the mean height and/or length of the folds is a better metric.

22

Bibliography

[1] R.C. Hibbeler. Mechanics of Materials. Prentice Hall, 2014.

[2] John W Hutchinson. Knockdown factors for buckling of cylindrical and spherical

shells subject to reduced biaxial membrane stress. International Journal of Solids

and Structures, 47(10):1443–1448, 2010.

[3] Andrej Kosmrlj and David R. Nelson. Mechanical properties of warped mem-

branes. Physical Review E, 88(1):012136, 2013.

[4] Stephen P. Timoshenko. Theory of Elasticity. Mcgraw-Hill College, 2010.

[5] Michael Vocaturo. Personal communication.

23

Appendix A

Raw Data

Below are all of the plots for the corrected data. Each of the 21 plots has the five

runs for each combination of test. material, and crumple level. They are ordered first

bending, then compression, then tension.

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25

26

27

28