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.
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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