PREDICTION OF AMPLITUDE AND WAVELENGTH
OF TROUGHS ON POLYETHYLENE WEBS
By
ADITYA GOTIMUKUL
Bachelor of Engineering in Mechanical Engineering
Jawaharlal Nehru Technological University
Hyderabad, Andhra Pradesh
2007
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May, 2010
ii
PREDICTION OF AMPLITUDE AND WAVELENGTH
OF TROUGHS ON POLYETHYLENE WEBS
Thesis Approved:
Dr. J.K.Good
Thesis Adviser
Dr. Raman P. Singh
Dr. Sandip P. Harimkar
Dr. A. Gordon Emslie
Dean of the Graduate College
iii
ACKNOWLEDGMENTS
First and foremost I would like to offer my sincere gratitude to Dr. J.K. Good for his
support, guidance and every other thing he has done for me. I appreciate him for patiently
guiding, helping, and supporting me in all situations through out my research work.
Secondly I would like to thank my committee members Dr. Raman P Singh and Dr.
Sandip P. Harimkar for willing to be my committee member. I would like to sincerely
extend my gratitude to Mr. Ron Markum for his invaluable help in developing the
experimental setups and conducting experiments. I would also like to thank WHRC
sponsors for funding my research and my friends at OSU for their help.
Last but not the least I would like to thank my parents Ashok Gotimukul, Usha
Gotimukul and my sister Ajitha Gotimukul for there continuous emotional and financial
support through out.
iv
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION......................................................................................................1
Troughing in Webs ..................................................................................................1
Reasons for Troughing of webs ...............................................................................2
II. LITERATURE REVIEW..........................................................................................5
Conference articles and Journal articles ..................................................................5
Research Objective ................................................................................................16
III. EXPERIMENTAL SETUP AND MATERIAL CHARECTERIZATION ...........17
Experimental Setup for Profile of Trough .............................................................17
Material Characterization.......................................................................................21
Modulus Testing ....................................................................................................21
Measurement of Poisson’s Ratio ...........................................................................23
IV. EXPERIMENTS AND MODELLING .................................................................27
Experiments to Determine Trough Profile.............................................................27
Modeling of Trough Formation Using ABAQUS .................................................28
V. RESULTS AND COMPARISSIONS……………………………………………32
Experimental Results……………………………………………………………..32
Simulation Results………………………………………………………………..37
Comparisons……………………………………………………………………..38
VI. CONCLUSION......................................................................................................47
Future Work...........................................................................................................48
REFERENCES ............................................................................................................49
APPENDICES .............................................................................................................50
v
LIST OF TABLES
Table Page
1.1 Tangent Modulus and Poisson’s Ratio of a LDPE Web……………………....26
5.1 Average Amplitude and Error of three different test specimens of 24” long ...36
5.2 Average Amplitude and Error of three different test specimens of 30” long…36
5.3 Average Wavelength and Error of three different 24 in long test specimen….36
5.4 Average Wavelength and Error of three different 30 in long test specimens...37
vi
LIST OF FIGURES
Figure Page
1.1 MD Troughs Formation………………………………………………………….2
1.2 Isotropic span of Web…………………………………………………………...6
2.2 Photograph of Sheet Depicting Troughs……………………………………….14
2.3 Dimensionless Wavelength Vs Strains………………………………………...15
3.1 Experimental Setup…………………………………………………………….19
3.2 Schematic Circuit Diagram…………………………………………………….20
3.3 50’ LDPE Web Stretched on the Floor to run Stretch Test……………………22
3.4 Stress Strain Curve of LDPE Material………………………………………...22
3.5 Elastic StressStrain Curve showing Young’s Modulus E…………………….22
3.6 Tangent Modulus at different Strains………………………………………….23
3.7 Photograph of two pair of Dots at a Strain of 0.0398…………………………25
3.8 Graph of Poisson’s ratio Vs Strain…………………………………………….26
4.1 Depicting Modeling of Web…………………………………………………..29
4.2 Showing the Two AmplitudeTime Curves…………………………………...30
4.3 Depicting the influence of ghost force on Outofplane Deformation………...31
5.1 Outofplane deformation of 24” test specimen at a strain of 0.016…………..32
5.2 Outofplane deformation of 24” test specimen at a strain of 0.132…………..32
5.3 Outofplane deformation of 27” test specimen at a strain of 0.1296…………32
5.4 Outofplane deformation of 30” test specimen at a strain of 0.033…………..34
5.5 Outofplane deformations of three 24” test specimens at strain of 0.049…….34
5.6 Depicting Measurement of Amplitude and Wavelength of Trough…………...35
5.7 The contour plot of outofplane deformation of a …………………………... 37
24” web at a strain of .049548.
5.8 Outofplane deformation of a 27” web from simulation at strain of .0555…....38
5.9 Wavelengths of the outofplane deformation in a 24 in test specimen………..39
5.10 Wavelengths of the outofplane deformation in a 27 in test specimen………..39
5.11 Wavelengths of the outofplane deformation in a 30 in test specimen………..40
5.12 Amplitudes of the outofplane deformation in a 24 in test specimen…………41
5.13 Amplitude of Outofplane Deformation in 27” Test Specimen……………….41
5.14 Amplitudes of the outofplane deformation in a 30 in test specimen…………42
5.15 Amplitudes of Outofplane deformation for 24” Specimen…………………..43
5.16 Amplitudes of Outofplane deformation for 27” Specimen…………………..44
5.17 Amplitudes of Outofplane deformation for 30” Specimen…………………..44
5.18 Amplitudes of 24” test specimen with inelastic material properties…………...45
1
CHAPTER I
INTRODUCTION
Throughing of Webs:
A Web is a continuous thin strip of material, made of paper, plastic films, textiles and
thin metals sheets. The webs often have to undergo several continuous processes prior to
forming a final product. The transportation of these webs during web processes is known
as web handling. Webs are often quite thin and such are subjected to instability. In the
process machinery webs are supported intermittently by rollers. The unsupported web
between the rollers is called free span.
During the transportation of webs, small out of plane deformations called troughs may
appear in the free span of the web. Formation of troughs in free span, hinder the
processes such as printing and coating etc due to non planar geometry in the web span.
Also these troughs may results in wrinkles on the rollers, which cause serious degradation
of material quality. The direction of travel of the web through a process machine is called
the machine direction (MD). The direction orthogonal to the machine direction, but still
in the plane of the web is called cross machine direction (CMD).
2
Figure 1.1MD Trough Formation
Reasons for Troughing of Webs:
The troughs in the free span occur due to compressive stresses in CMD. A free body
diagram of a web would show that there are no lateral forces at the edge of the web to
create compressive stresses. However CMD compressive stresses can arise that result in
trough formation for various reasons, some of which are:
a) Roller deflection: The deflection of the roller causes lateral compressive stress, as
the web seeks to align itself perpendicular to the axis of the deflected roller.
b) Variation of tension: Tensile stress ( x) due to tension in a web causes web strain
( x) in the MD and web strain ( y) equal to (– x) in CMD. Longitudinal strain
observed as the plastic films are processed in web form can be of the order of
0.001, although the strain increases and decreases during the process due to
changes in tension. Changes in width accompany these changes in longitudinal
tension due to Poisson’s ratio which is of the order of 0.3 and larger. Therefore,
an increase in width occurs when a web moves from high tension span to low
3
tension span. Changes in web tensions must occur at rollers where the change in
tension is balanced by frictional forces between the web and roller. As the web
tension decreases the web attempts to expand laterally on the roller which can
produce CMD compressive stress.
c) Increase in temperature or moisture: Plastic webs have a high coefficient of
thermal expansion, in some cases higher than 0.0001 per degree F and paper
usually expands significantly as it absorbs moisture. The processes such as drying
and corona or flame treatment involve heating of webs. In the process called
sizing, paper is made to absorb moisture. If lateral expansion of a web occurs near
a roller, frictional CMD forces can arise between the web and roller which
produce CMD compressive stresses, similar to the Poisson’s effect discussed in
case of variation of tension.
d) Viscoelastic memory: In draw or velocity controlled processes the web tension
can decrease inspan due to viscoelasticity. Decrease tension will result in CMD
expansion which can produce troughs.
e) Roller Imperfections: Both roller misalignment and roller diametrical taper are
capable of producing roughs in the web.
Given the current understanding of the sources of the CMD forces which create troughs it
is still difficult to make troughs disappear by attempting to control these sources.
4
The focus of this research is that if it is given that web troughs will occur can their
amplitudes be predicted? The goal of this research is to quantify the wavelength and the
amplitude of these troughs when MD web strain is either in elastic or in the inelastic
region.
5
CHAPTER II
REVIEW OF LITERATURE
The web in a web line is subjected to tension in MD, but there is no evidence of CMD
forces that produce CMD compressive stresses. However for troughs to occur there must
be compressive stresses acting in a lateral CMD direction. The transverse cross section of
a troughed free span of thin web is similar to a buckled thin plate. Hence analysis of
troughed webs can be done similar to the buckling analysis of a rectangular plate which is
subjected to loads in both X and Y directions. Timoshenko and Gere [1] have analyzed
the buckling of a rectangular plate, subjected to loads in both the directions.
The differential equation for the deflection surface (w) in case of an isotropic plate, under
the action of membrane forces is:
¶ ¶
+ ¶
¶
+ ¶
¶
= ¶
¶
+ ¶
¶ ¶
+ ¶
¶
¶
x y
w
N
y
w
N
x
w
N
y D
w
x y
w
x
w
x y xy
2
2
2
2
2
4
4
2 2
4
4
4
2
1
2 {1}
Where Nx, Ny, and Nxy are the membrane forces which may serve to increase or decrease
the outofplane deformations.
In the case of a web in a web line where there are no shear stresses acting the deflection
equation can be rewritten as
{2}
Where x and y are the membrane stresses and t is the web thickness and
2 0 2
2
2
2
4
4
4 2 2
4
4
4
=
¶
 ¶
¶
 ¶
¶
+ ¶
¶ ¶
+ ¶
¶
¶
y
w
t
x
w
t
y
w
D
x y
w
D
x
w
D x y s s
12(1 2 )
3
n
= Et
D
6
Good and Biesel[2] has taken this further and derived an expression for the minimum
CMD compressive stress needed to buckle the web, known as the critical buckling stress.
For an isotropic web of width ‘b’ that spans the distance ‘a’ between two rollers the
governing differential that of equation {1}.
Figure 2.1Isotropic Span of Web
A solution is sought for the outofplane deformation ‘w’ of the form:
=
b
n y
Sin
a
m x
w A Sin mn
p p
{3}
where m and n are the half wave numbers in the x and y directions, respectively and Amn is
the maximum amplitude of outofplane deformation for a given buckled shape. By
choosing the displacement of the form {2}, the out of plane deformation is forced to vanish
at all four boundaries of the web span when m and n are positive integers. This condition
appears to be appropriate when web is in contact with rollers but no constraints exist on the
free web boundaries (y=0, b). During the experimental observation of troughs, the outofa
b
y
x
x
x N
s
N y ,s y
7
plane deformations near these free edges were minute compared to the outofplane
deformations associated with the troughs. This behavior could be due to the fact that,
compressive CMD stresses do not exist at the free boundaries. The combination of the
absence of troughs at edges and that web tension acts to restrict the outofplane
deformation ‘w’ supports the assignment of the simple support boundary condition to these
boundaries (y=0,b). The tension in the web restricts the half wave number in x direction
(m) to be unity. Substituting the expression {3} in to expression {2} and solving for y, a
relationship for buckling stress is produced of the form:
2 2 2
( 2 2 2 )2 4
a b n
b a n b e x
ycr
s s
s
+ +
=  {4}
Where
a h
D
e 2
p 2 s =
Observing expression {4} it can be determined that critical buckling stress ycr is a function
of half wave number (n) in y direction and tensile stress x in the x direction.
With the increase in magnitude of tensile stress and half wave number (n) in y direction,
stability of the web increases. To determine the correct value of n, for a given tension
requires consideration of minimum energy. Assuming n as being continuous for the
moment, the energy can be minimized by taking derivative of {4} with respect to n,
equating the result to zero, solving for n and substituting the result back into the {4}. The
resultant expression is
( ) ycr e e e x s = 2s + s 2 +s s {5}
From the expressions {4} and {5} it can be proved that very little ( y) CMD compressive
stress may induce instability in thin webs. If we select a=30”, E=600,000 psi, =0.3,
8
t=.001” , and x=1000 psi we will find that mere 1.55 psi y stress will induce troughs in
the web.
E. Cerda and L. Mahadevan [3] discuss about the wrinkling (they refer to trough as
wrinkles) in an elastic sheet under tension. The authors developed scaling laws for
amplitude and wavelength of trough, and assert these scaling laws are applicable to both
isotropic and anisotropic sheets that have been stretched either in the elastic or into the
inelastic range. All the authors’ developments consider isotropic materials stretched in
the elastic range. Extensions to anisotropic materials or to sheets stretched to the
inelastic range are not shown. They state that, when a thin elastic isotropic sheet of
thickness ’t’, width ‘W’ and length ‘L’ (where L>W>>t), composed of a material with
Poisson’s ratio ‘ ’ and young’s modulus ‘E’ is subjected to longitudinal strain ‘ ’, the
sheet remains flat until the applied strain do not exceeds the level strain c called the
critical stretching strain. Stretching the sheet further ( > c) causes the sheet to buckle
and form troughs.
In Cerda and Mahadevan’s case the troughs occur due to clamped boundaries. They do
not allow the sheet to contract laterally at the clamps which results in a biaxial stress state
at the clamps. The CMD stress is tensile near the clamps and compressive slightly further
from it. When sheet is stretched beyond the strain c, y becomes less than ycr, and the
web buckles.
The Authors developed the expressions for wavelength and amplitude by minimizing the
total energy. The total energy of a stretched sheet is U = UB + US , where UB is the
9
bending energy of the sheet and US is the energy due to stretching of the sheet, subject to
any geometric constraints.
The expression for strain energy in bending for the web stretched inbetween two clamps
is obtained by simplifying the total strain energy in bending given by Timeshenko [1]
( ) dxdy
x y
w
y
w
x
w
y
w
x
D w
¶ ¶
 ¶
¶
¶
¶
  ¶
¶
+ ¶
¶
¶ 2 2
2
2
2
2 2
2
2
2
2
2 1
2
u {6}
In the above expression the Authors assume the term
¶
¶
2
2
x
w
to be negligible; however
they do not give the reason for their assumption. The outofplane deformation of a
buckled web can be assumed to be of the form
=
b
n y
Sin
L
x
w ASin p p
where A is
the amplitude and = n/2b is the wavelength of the troughs. Substituting the out of plane
deformation in the expression {6} and solving gives the expression for bending energy.
U Et (A ) LW B
2
2
3
= l {7}
The expression for stretching energy for a web stretched inbetween the two clamps is
obtained by simplifying the stretching energy given by Timoshenko [1]
dxdy
y
w
x
w
N
y
w
N
x
w
N x y xy
¶
¶
¶
+ ¶
¶
+ ¶
¶
¶
2
2
1
2 2
{8}
Authors assume
2
¶
¶
y
w
Ny to be negligible: however they do not provide a reason for
their assumption, and solving the expression substituting the out of plane deformation of
the above mentioned form, gives the expression for stretching energy as
10
( ) LW
L
U Et A s
2
2 » g {9}
The Authors use a geometric constraint that they call “Geometric Transverse
Inextensibility”. Although not stated their constraint is a simplification of the large strain
expression:
¶
+ ¶
¶
¶ +
¶
+ ¶
¶
= ¶
2 2 2
2
1
y
w
y
v
x
u
y
v
yy e {10}
The strain yy is assumed to be – xx which is equal to – in the Authors variables. u,v,
and w are the deformation in x, y, and z dimensions respectively. Inextensibility would
imply that strain due to inplane deformation v would be negligible. Also the deformation
in the x direction (u) would be nearly constant for a given x location thus the 0
¶
¶
y
u
.
This leaves us with:
2
2
1
¶
=  » ¶
y
w
yy e ug {11}
After substitution of w and elimination of constants in the expression {11} leads us to the
Authors scaling law:
( l ) »ng 2 A {12}
Substituting expression {7} in expression {7} and {9}, total energy U can be expressed
as
LW
L
U Et Etgl ng
l
» + 2
2
2
3
{13}
Minimizing U with respect to gives a scaling law for the wavelength
11
( )
4
1
2
1
g
l » tL {14}
Substituting the {14} into the transverse inextensibility expression {11} gives a scaling
law for amplitude
( ) 4
1
2
1 A » ntL g {15}
E.Cerda and L.Mahadevan [4] deduced exact expressions with the prefactors for
amplitude and wavelength of troughs formed on thin stretched sheet.
To determine the criterion for selection of the wavelength and amplitude of wrinkles,
change in energies of bending and stretching must be accounted. Geometric constraints
are imposed using Lagrange multipliers (L). Let the outof plane displacement of the
initially flat sheet of area W.L be (x,y). x Î(0, l) as the coordinate along the sheet
measured from one end and y Î(0,W) , (W<<l) as the coordinate perpendicular to it
measured from its central axis. Then the total energy function can be written as
{16}
The bending energy UB due to the deformation which is predominantly in the y direction
is given by expression ¶
A
y B 2 2 dA ( )
2
1 z , where B is bending stiffness or flexural rigidity
of the sheet, given by [ ] 12(1 2 )
3
n
= Et
B and Us is the stretching energy in the presence of a
tension T(x) along x direction. The sheet satisfies the condition of transverse
inextensibility as it wrinkles under the action of a small compressive stress.
U U U L B s = + 
12
( ) 0
( )
2
1
0
2 =
¶  D dy
W
x b
yz {17}
where (x) ~ W is the imposed compressive transverse displacement.
Hence the term L in the expression {16} which accounts for the geometric constraints
can be expressed as
( ) dA
W
x
b x
A
y
¶  D ( )
( ) 2 z {18}
where b(x) is the Lagrange multiplier and (x) is the imposed compressive transverse
displacement. The EulerLagrange equation obtained from the condition of a vanishing
first variation of {16}, = 0
dz
dU
yields
¶4z  ( )¶2z + ( )¶2z = 0 y x y B T x b x {19}
For a stretched sheet T(x) is constant, and (x) ~ W is constant far from the boundaries
so that b(x) is constant. Away from the free edges in y direction the wrinkling pattern is
periodic so that (x,y) = (x,y+2 /kn), where kn =2 n/W, and n is the number of wrinkles.
At the clamped boundaries (0,y) = (l,y) = 0. Substituting a periodic solution of the form
e X (x) n
n
z = ikn y into the expression {19} yields a SturmLiouvillelike problem
2 0
2
2
+ = n n
n X
dx
d X w , Xn(0) = Xn(l) = 0 {20}
Where bk Bk T n n n w 2 = ( 2  4 ) / . b is the compressive stress and can be determined from the
nonlinear geometric constraint {18}. The solution to equation {20} when b is constant is
X A Sin x n n n = w l
m
n
w = p
13
For bending energy to be minimum there should be only on half sine wave along the
length, therefore m=1 hence
n= /L so that 2
2 n
n
2
2
n n Bk
l k
T
b (k ) = p + and the displacement
function is
l
A Cos(k y )Sin x n n n
z = + f p {21}
Plugging the obtained displacement function into the geometric constraint expression
{18} yields
= D
8
A2k 2W n n
{22}
After substituting in the expressions for bending energy UB and stretching energy Us,
the total energy can be written as
k l
T
U Bk l 2
n
2
2
n
= D + p D {23}
Minimizing the total energy {23} and using the geometric constraint {18} wavelength
=2 /k and amplitude A are obtained and are given as
2
4 1
1
l
T
B
2
l = p l
p
2
1
2
= D
W
A
Substituting the value of flexural rigidity B and tension T for a stretched sheet yields
Wavelength to be
{24}
( )
[ ( )]4
1
2
2
1
3. .1
2 lt
g  n
l = p
14
and Amplitude
{25}
To verify these expressions of wavelength E. Cerda and Mahadevan [4] had stretched
different lengths of polyethylene of thickness ~ 0.01cm and width of 12cm at the strain
levels of Î[0.01, 0.2]. The polyethylene sheet was clamped between the two aluminum
plates to enforce the boundary conditions. The sheet was first taped to one of the
aluminum plate using an adhesive tape so that slippage would not occur. A plot showing
4
1
1
g
on x axis and
2
1
(tL)
l on y axes is plotted with experimental values and
theoretical values, a quantitative agreement is obtained.
Figure 2.2 Photograph of the sheet depicting the troughs
( ) ( )
4
1
2 2
2
1
3 1
1
16 tl A
p  n
= n g
15
Figure 2.3Dimensionless wavelength Vs Strain
The expression obtained for amplitude and wavelength by using a double Sine function for
the outofplane displacement which Timoshenko and Gere yielded the same expressions
for amplitude and wavelength. Considering the expression {3} for the CMD compressive
stress, and minimizing it with respect to n and solving for n gives an expression for n
4
e
e x
L
b
n
s
s +s
= {26}
The wavelength can be expressed in terms of and width b. Consider there are n number
of half sine waves distributed uniformly throughout the width of the web. The distance
between the two same points on alternate half sine waves is the wavelength. Hence it can
be expressed as
b
n
2
l = .
Substituting the expression for e, expressing x in terms of strain and Young’s modulus
E and n=2b in expression {26} yields
4
3(1 2 )
1
2
n g
l p

= Lt
16
Similarly expression for amplitude can be obtained by considering the displacement
function for out of plane displacement to be
=
b
n y
Sin
L
x
ASin
z p p {27}
Substituting the expression {27} into the condition of transverse inextensibility {18} and
integrating it over x (0, L) and y (0, b) and using the expression for wavelength yields
4
1
3 2 (1 2 )
16 g
p n
n

A = Lt
Research Objective:
Cerda and Mahadevan have developed a condition of “Transverse Inextensibility”
to define the amplitude of troughs and the wavelength of troughs. The expression for
wavelength is equivalent to that which can be derived from Timoshenko expressions. The
expression for amplitude is novel. Cerda et al claim these expressions applicable to
isotropic and anisotropic materials in the elastic and inelastic domains of strain. The
Authors lend some proof in this context by wavelength measurements of troughs in
polyethylene web over a large range of strain. They provide no proof of their scaling laws
for amplitude and how they are impacted by inelastic strain.
The objective of this research is to determine if Cerda and Mahadevan’s claims are
credible or if not under what conditions they are credible.
17
CHAPTER III
EXPERIMENTAL SETUP AND MATERIAL CHARECTERIZATION
The equipment required for the Research was provided by the Web Handling Research
Center at Oklahoma State University. To conduct the experiments we needed a universal
testing machine, a sensor capable of capturing the troughs profile and load cell to
determine the load, and user interface to note the readings.
Experimental Setup for Profile of Trough:
The experimental setup consisted of equipment capable of holding, stretching the web
and measuring the wavelength and amplitude of troughs. An Instron Universal Testing
Machine was used to stretch the web; the maximum stroke of the machine was around 4
inches. The web was supported between the hydraulic ram and the load frame of the
Instron using two aluminum clamps. To have a good adherence, rough rubber strips were
used inbetween aluminum clamps and web, the rubber strips were adhered to the
aluminum clamps using a strong adhesive. As the web was stretched, inbetween the
clamps, a tensile load developed. An external Stype load cell was calibrated to measure
the low load levels applied to the web, as it was stretched at different strain levels. At low
strain levels, these troughs appear whose average amplitude is of the order of 102 inches.
A Laser sensor was used to capture the outofplane deformation associated with these
troughs. A Keyence model LC2100 laser sensor was used. The sensor is capable of
18
resolving a change in distance of 1/1000th of an inch, thus the outofplane deformation of
the troughs can be captured using this sensor. The sensor ejects a laser beam of light, this
beam after reaching the object gets reflected and return to the sensor. The distance
between the object and sensor is measured using the time taken by reflected beam to
reach the sensor. Using this sensor we can measure the outofplane deformation of a
point on the web. To get the profile of the trough across the width of the web, the
Keyence sensor is forced to move in the cross machine direction on a linear bearing. The
position of the Keyence sensor is measured using a YoYo pot. A YoYo pot transduces
linear motion to a change in resistance. The variable resistance becomes a part of a ballast
DC circuit where the voltage drop across the variable resistance is calibrated with respect
to the linear motion that requires measurement. A data acquisition system consisting of a
National Instruments SCB68 A/O board, a computer, and a LabView software program
were used to simultaneously record the output from the Keyence 2100 laser sensor and
the YoYo pot. In this way the trough amplitudes as a function of CMD location was
recorded.
19
Figure3.1Experiment Setup
20
Figure 3.2Schematic Circuit Diagram
NI SCB68
Power Supply
Red
Black
White
ACH2 ACH1
AIGN
D
32,67
29,64
ACH
O
ACH3
G +5 12 +12
G Out
G 12
+5 +12
G +5 12
+12
G Out
YoYo Pot
head
Lc 2100
Laser displacement meter
110V PS
21
Material Characterization:
The expression for wavelength and amplitude given by Cerda[] involve material
properties such as Young’s Modulus and Poisson’s ratio. Therefore to get correct values
of amplitude and wavelength it is required to have good knowledge of material
properties. To prove or disprove Cerda’s claim we must know how these web properties
change as the strain level enter the inelastic range.
There were two tests conducted to determine the Young’s Modulus, the Tangent Modulus
and Poisson’s ratio of a low density polyethylene web material, similar to that used by
Cerda.
Modulus Testing:
The stretch test was performed on a 50’ long and 10’’ wide test specimen of LDPE. A
load transducer was attached to the test specimen, and was elongated to a length of
approximately 65’. For every one unit change in the load applied recorded from the
transducer, the associated change in length or elongation of the specimen was noted.
Strain and stress can be calculated using the elongation and load respectively. Stress and
strain plotted on Y and X axes respectively gives stressstrain curve. The slope of stress
and strain curve in proportional range of stress and strain is the Young’s Modulus.
The tangent modulus at inelastic strain levels can be determined using the same test data.
22
Figure 3.3 60’’ web stretched on the floor to run the stretch test
0
200
400
600
800
1000
1200
1400
1600
1800
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Strain
Stress in Psi
Figure 3.4StressStrain Curve for a Polyethylene web material
y = 21511x
0
100
200
300
400
500
600
700
0 0.005 0.01 0.015 0.02 0.025 0.03
Strain
Stress in Psi
Figure 3.5Elastic Strain Vs Stress Showing Young’s Modulus (E) =21511
23
0
2000
4000
6000
8000
10000
12000
14000
16000
0 0.05 0.1 0.15 0.2 0.25 0.3
Strain
Tangent Modulus (Psi)
Figure 3.6Tangent Modulus along the strain
Measurement of Poisson’s Ratio:
The Poisson’s ratio varies from 0.3 to 0.5 for the polyethylene material. According to the
literature it seems that the Poisson’s ratio abruptly changes from 0.3 to 0.5 as soon as the
material reaches it plasticity. We were interested in determining the Poisson’s ratio at
each strain levels over the domain in strain where Young’s modulus and the Tangent
modulus were measured. Poisson’s ratio is defined as the minus ratio of lateral strain and
longitudinal strain. The measuring of longitudinal strain in the web was achieved by
tracking the movement of ram on the Instron machine. The change in the ram position is
the measure of change in the length of the web. Since the width of the web was only 6’’
inches it was hard to determine the change in width. Apart from this, the occurring of
troughs would hinder the measurement process.
24
A photographic method was used to determine Poisons ratio. The web was marked with
two pair of dots each apart by 1’’, one along MD and one along CMD, between the two
web clamps on the Instron machine and a photograph was captured using a high
resolution manual lens camera at various strain levels. A flat field macro lens was
selected as its focus remains constant through the field of view, as for a typical lens focus
varies, which would induce error in the Poisson’s ratio measurement. The distance
between the dots was measured using drawing tools within Microsoft Paint.
Since the distance between two pair of dots was known in terms of the number of pixels,
the scaling factors for determining the actual distance from the number of pixels between
the two points in the photographs could be defined as the ratio of the actual distance
between the points and the number of pixels. Photographs at each strain level were
captured and the change in distance between the two pair of dots in CMD and MD was
measured. The ratio of change in the distances and original length and width would give
the respective strains. Minus the ratio of the lateral strain and longitudinal strain would
give us Poisson’s ratio.
After the experiments were performed the Poisson’s ratio determined was greater than
0.5 even when the material was in elastic range, which indicated that there was an error in
the experiment. The reason for getting such values for Poisson’s ratio was due to
formation of troughs, even though the dots were marked at a place where the troughs
formation just started or the point where the CMD tensile stresses vanished. The trough
formation had a prominent effect on measurement of Poisson’s ratio.
25
Figure 3.7Photograph of the two pair of dots at a strain of 0.0398
In order to avoid this, a roller was placed such that it will touch the web right at the points
where the dots are marked to prevent outofplane trough deformations. As the stroke of
the Instron machine is 4’’ inches the roller cannot be in contact with the web at all the
strains. Therefore the roller was mounted such a way that it could be moved by hand in
the MD. Care was taken that the friction between the roller and the web will not hinder
the lateral movement of the dots. The Poisson’s ratio was measured over large range of
strain.
The test was conducted twice to check the repeatability and accuracy of the experiment.
The Poisson’s ratio at different strain levels from both the experiments were plotted on
the graph shown below and a curve was fit so that a specific value could be determined
26
for Poisson’s ratio at all strain levels, using the expression from the curve fit. The curve
fit equation was ‘ n(g) = 80.62g3  31.81g 2 + 4.008g + 0.2994 ’. {28}
0
0.1
0.2
0.3
0.4
0.5
0.6
0.00E+00 2.00E02 4.00E02 6.00E02 8.00E02 1.00E01 1.20E01 1.40E01 1.60E01 1.80E01 2.00E01
Strain
Poisson's Ratio
Figure 3.8Graph of Poisson’s ratio VS strain
Strain Tangent Modulus ET Poisson’s ratio
0.045 13745 psi .422
0.054 9482 psi .436
0.064 7500 psi .447
0.079 5674 psi .457
0.097 4705 psi .462
0.135 2204 psi .463
Table 3.1 Tangent Modulus and Poisson’s Ratio of LDPE Web
27
CHAPTER IV
EXPERIMENTS AND MODELLING
Experiment to Determine the Trough Profile:
A 6’’ wide 100 gauge polyethylene web was used in the study. The experiments were
conducted on specimens with different aspect ratios (length/width) ranging from 4 to 5 in
their undeformed state. A web of fixed length was installed in aluminum clamps. The
clamps were setup with high friction surfaces to prevent slippage of the web in the clamp
in the MD and CMD directions. Care was exercised such that the two aluminum clamp
surfaces lie in the same plane. This was done to prevent bending and torsional loads from
influencing the result. Also the web was fixed such that it was perfectly orthogonal to
both the clamps. A servo hydraulic material testing system (Instron Model 8502) was
used to precisely stretch the web. Two black lines were drawn on the web, where it enters
the clamps at beginning of the experiment. If the lines remain straight after the
experiment it was an indication that slippage did not occur during the experiment.
The Finite Element Method was used to analyze the internal stresses in the web, fixed at
both the ends, but subject to MD tension. CMD tension resulted in at the near vicinity of
clamps, and then CMD compressive stresses developed away from the clamps before the
stresses died out. These regions of pockets of compressive CMD stresses were located
about 10’’ away from the clamps irrespective of the test span length or aspect ratio of test
specimen. The results of these analyses were that all test specimens were chosen with
28
lengths exceeding 20” such that CMD compressive stresses and hence instability would
occur.
Three sets of experiments were conducted and each set had different specimen length and
an aspect ratio. To test the repeatability each specimen length was tested thrice. In the
first set of experiments the specimen was chosen to be 24’’ long. This specimen was
stretched to a strain level of 0.132. The troughs started appearing at a very low strain
level of 0.005. The outofplane deformation of the trough was captured using a Keyence
(Model 2100) laser sensor. The second and third set of experiments, were conducted on
test span lengths of 27’’ and 30’’. The troughs were formed at almost same strain level as
formed on 24’’ test specimen.
In order to confirm that these troughs were not formed due to the pocket of compressive
stress, a 12’’ test specimen was tested, stretching it to a strain level of 0.254. There were
no prominent troughs formed on the web even at highest strain level achieved.
Modeling of Troughs Formation Using ABAQUS:
Finite Element Modeling was used to model the trough formation witnessed in the
laboratory. The FEA package ABAQUS Explicit was used to model the laboratory
procedure. The web was modeled in a 3D modeling space as single section with shell
elements. A structured mesh with quad dominated element shape was used. A ‘S4R’
element which is a 4 node shell element with reduced integration. The clamping of the
web at the ends was modeled using the boundary condition. At one end, the web
movement was constrained in 6 DOF (Ux, Uy, Uz, Rxy, Ryz and Rzx) and on the other end
29
the movement was constrained in 5 DOF(Uy, Uz, Rxy, Ryz and Rzx) leaving the web free to
move in the direction of the applied displacement Ux. To model the stretching of the web
displacements were enforced to the Ux DOF. For every strain level an enforced
displacement was given as a boundary condition. The displacement was applied in steps
using an amplitudetime curve.
Figure 4.1 Depicting the modeling of the web and positions of ghost force
The global seed of the mesh was chosen to be 0.05”; that is each element has an edge of
length 0.05”. The accuracy of the trough wavelength depended on this mesh size. The
finer the mesh more provided greater the accuracy of the wavelength, but also increased
the run time of the simulation. Thus this mesh size was chosen to optimize the run time of
the simulation without jeopardizing the accuracy of the wavelength.
Mathematically instability will not occur when the structure is subjected to tension.
Commercial finite elements codes which are in use as of today cannot automatically
simulate the behavior of the thin structures buckling in tension. To simulate the buckling
behavior of the web in ABAQUS an outofplane load must be applied, called ghost
force, in order to induce some instability in the structure. In order to minimize the
influence of the ghost load on amplitude and wavelength on troughs formed, it was
applied in such a way that it vanishes after instability was induced in the structure. In
30
order to accomplish this, a second amplitudetime curve was used to vary the amplitude
of the ghost force. Pairs of equally spaced positive and negative concentrated ghost forces
were applied to web. The stresses induced due to these forces were negligible when
compared to tensile stresses that resulted from stretching of the web. The nonlinear
analysis occurred over 10 time solution steps. The enforced displacements which induced
the strain desired in the simulation became maximum in time step 2. The ghost load
became maximum in the time step 5 and vanishes in the time step 7. The outofplane
deformations were examined in the time step 8 through 10 to determine if instability had
occurred.
Figure 4.2 Showing the two Amplitudetime curves
The simulations were run at several strain levels for the three different lengths web
specimens tested (23”, 27” and 30”). After completion of the simulation, the outofplane
deformation (Uz) were examined and used to calculate the amplitude and wavelength of
the troughs formed.
31
The simulation was run initially using ABAQUS Standard, but the amplitude of the
troughs appeared to be dependent on the magnitude of ghost force. Then ABAQUS
Explicit was used to run the simulation. The dependency of the amplitude of outofplane
deformations due to the magnitude of ghost forces in case of ABAQUS Explicit is shown
in the figure. Since there is no dependency it is assumed that the amplitudes of the outofplane
deformations computed are realistic.
Figure 4.3Depicting the influence of ghost force on Outofplane Deformation with time.
32
CHAPTER V
RESULTS and COMPARISIONS
Experimental Results:
The outofplane deformations for the three test span lengths at different strain levels
were obtained. The outofplane deformations for a particular strain level of 0.0165 on a
24” long web is shown below. The results for other strain levels and web lengths are
shown in the appendix.
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation (in) x
Figure 5.1Outofplane deformation of 24” test specimen at a strain of 0.016
33
0.12
0.1
0.08
0.06
0.04
0.02
0
0 1 2 3 4 5 6
CMD Location (in)
Outofplane deformation (in) x
Figure 5.2Outofplane deformation of 24” test specimen at a strain of 0.132
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation (in) x
Figure 5.3Outofplane deformation of 27” test specimen at a strain of 0.1296
34
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0 1 2 3 4 5 6
CMD Location (in)
OutofPlane Deformation (in) x
Figure 5.4Outofplane deformation of 30” test specimen at a strain of 0.033
To verify the repeatability of the experiment all the test specimens were tested for three
times and the results where compared.
0.12
0.1
0.08
0.06
0.04
0.02
0
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation (in) x
I Sample
II Sample
III Sample
Figure 5.5Outofplane deformations of three 24” test specimens at strain of 0.049
35
Observing the above outofplane deformation graphs, there is a general left to right
decrease in the deformation, this could be due to the fact that the line of travel of the
keyance sensor may not be perfectly parallel to the web when mounted on to the Instron
machine. This may not affect the accuracy of the amplitude as the width of a typical
trough is less than one half of an inch and the method of measuring the amplitude of a
trough is independent of the positions of other trough. The actual deformation can be
obtained by deducing the angle between the web plane and line of travel of the sensor
using the linear regions on the either side of the buckled web.
The amplitude and wavelengths from all the three test spans were obtained and the error
was calculated.
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 1 2 3 4 5 6 7
CMD Location (in)
O utofPlane D eformation (in) x
2Apeak
2A2
2A1
4
A A A A
A peak 1 2 3
avg
+ + +
=
3 2A
1 l
2 l
2
1 2 l = l +l
Figure 5.6 Depicting measurement of Amplitude and Wavelength
36
The table depicts the error in the amplitude and wavelength from three different test
specimens of same length.
Amplitudes of 24" Test span at different strain levels (PP Values)
Strain I II III Avg (in) Error(in)
0.0165 0.0222 0.02075 0.02167 0.02154 0.000734
0.033 0.0294 0.028 0.0278 0.0284 0.000872
0.0495 0.0325 0.0331 0.0368 0.034133 0.002329
0.066 0.0376 0.031 0.0391 0.0359 0.004309
0.0825 0.0368 0.0281 0.0347 0.0332 0.00454
0.099 0.04025 0.0268 0.0337 0.033583 0.006726
0.1155 0.03712 0.031 0.0347 0.034273 0.003082
0.132 0.0323 0.031 0.0304 0.031233 0.000971
0.1485 0.0328 0.0295 0.0225 0.028267 0.00526
Table 5.1Average Amplitude and Error of three different test specimens of 24” long
Amplitudes of 30" Test span at different strain levels (PP Values)
Strain I II III Avg (in) Error(in)
0.0165 0.0164 0.0158 0.018 0.0167 0.0011
0.033 0.0192 0.02 0.021 0.02 0.00090
0.0495 0.0228 0.0208 0.0213 0.0216 0.0010
0.066 0.0234 0.02 0.0236 0.022 0.002
0.0826 0.021 0.017 0.019 0.0028
0.099 0.0213 0.0148 0.0180 0.004
0.1157 0.02 0.0126 0.0163 0.005
0.132 0.019 0.012 0.0155 0.004
Table 5.2Average Amplitude and Error of three different test specimens of 30” long
24 in Test Specimen
Strain I II III Average(in) Error(in)
0.0165 0.744 0.776 0.759 0.759 0.016
0.033 0.67 0.7003 0.7 0.69 0.017
0.0495 0.618 0.639 0.574 0.61 0.033
0.066 0.6005 0.5946 0.543 0.579 0.031
0.0825 0.556 0.581 0.5257 0.554 0.027
0.099 0.56525 0.5685 0.484 0.539 0.047
0.1155 0.55825 0.519 0.517 0.531 0.023
0.132 0.519 0.511 0.489 0.506 0.015
0.1485 0.5485 0.4866 0.4447 0.493 0.052
Table 5.3Average Wavelength and Error of three different 24 in long test specimen
37
30 in Test Specimen
Strain I II III Average(in) Error(in)
0.0165 0.892 0.964 0.999 0.951 0.0545
0.033 0.822 0.8035 0.8145 0.8133 0.0093
0.0495 0.7265 0.701 0.759 0.728 0.029
0.066 0.6795 0.64625 0.6966 0.674 0.0256
0.0826 0.669 0.673 0.6035 0.648 0.039
0.099 0.623 0.6645 0.553 0.613 0.056
0.1157 0.6105 0.6265 0.559 0.598 0.035
0.132 0.61 0.624 0.5536 0.595 0.037
Table 5.4Average Wavelength and Error of three different 30 in long test specimens
Simulation Results:
The Simulation was run for all the three test specimens at different strain levels and the
amplitude and wavelength were measured from the outofplane deformations obtained in
the simulation.
Figure 5.7The contour plot of outofplane deformation of a 24” web at a strain of .0495
38
The outofplane deformation of each node was obtained using a probe value function
available in Abaqus, these vales when plotted against the width gives us the trough
profile.
0.03
0.02
0.01
0
0.01
0.02
0.03
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
Figure 5.8Outofplane deformation of a 27” web from simulation at strain of .0555
Comparisons:
The values of amplitude and wavelength from experiments and ABAQUS simulation
were compared against the closed form solution {24, 25} given by Cerda.
The Poisson’s ratio needed to determine the amplitude and wavelengths of troughs from
the closed form solution is obtained from the expression for Poisson’s ratio interms of
strainn (g ) = 80.62g 3  31.81g 2 + 4.008g + 0.2994 . The web was 6” wide with a thickness
of 0.0012”.
39
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.05 0.1 0.15 0.2
CMD Location (in)
Wavelength (in) x
Experiment
Simulation
Cerda {24}
Figure5.9Wavelengths of the outofplane deformation in a 24 in test specimen
0
0.2
0.4
0.6
0.8
1
1.2
0 0.05 0.1 0.15 0.2
Strain
Wavelength (in) x
Experiment
Simulation
Cerda {24}
Figure 5.10Wavelengths of the outofplane deformation in a 27 in test specimen
40
0
0.2
0.4
0.6
0.8
1
1.2
0 0.05 0.1 0.15
Strain
Wavelength (in) x
Experiment
Simulation
Cerda {24}
Figure 5.11Wavelengths of the outofplane deformation in a 30 in test specimen
Finally from the figure 4.9, 4.10, 4.11 the results for the wavelengths from experiments,
simulations and closed form solution are agreeing. The closed form expression {24} for
wavelength developed using energy theory, involves the material properties in CMD. The
web under tension in MD has very small stresses in CMD, with no change in CMD
material properties, thereby making the wavelength expression valid even in the inelastic
region.
41
The graphs below are representing amplitudes for the three test spans from experiments,
simulation and closed form expression.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 0.05 0.1 0.15 0.2
Strain
Amplitude (in) x
experiment
Simulation
Cerda {25}
Figure 5.12Amplitudes of the outofplane deformation in a 24 in test specimen
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.05 0.1 0.15 0.2
Strain
Amplitude (in) x
Experiment
Simulation
Cerda {25}
Figure 5.13Amplitude of Outofplane Deformation in 27” Test Specimen
42
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.05 0.1 0.15
Strain
Amplitude (in) x
Experiment
Simulation
Cerda {25}
Figure 5.14Amplitudes of the outofplane deformation in a 30 in test specimen
Since the expression for amplitude given by Cerda was agreeing neither with the
Simulation results nor with the experiments, validation of Cerda’s expression can be
questioned. It appeared that his expression for amplitude was an developed by integrating
the transverse inextensibility constraint along both length and width. The results from
experiments as well as simulations can only be obtained and compared at a particular
position along the length. Therefore in order to obtain an expression for average
amplitude along the width I developed an expression by integrating inextensibility
condition along the width at X=L/2, where X is the variable representing along the length
and L is the total length of the test specimen.
( ) 0
2
1
0
2
=
 D
dy
w
x
dy
dw b
{29}
43
Substituting
=
b
n y
Sin
L
x
w ASin p p
=
¶
¶
= b
n y
Cos
b
A n
y
w
x L
p p 2
2
2 2 2 2
2
and integrating the equation {29} will yields
[ ] 4
1
2 2 4 2 2
2
b n L
L b
A
+
=
p
ng
{30}
Substituting the value of
( ) 4
1
2 2
2 2
t
12L 1
1
2L
A
 +
=
p
p g n
gn
{31}
The graphs below represents the amplitudes from experiments, simulation and from the
new expression {31} developed for the three different test spans.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0 0.05 0.1 0.15 0.2
Strain
Amplitude (in) x
Experiments
Simulation
Gotimukul {31}
Figure 5.15Amplitudes of 24” Test Specimen
4
3(1 2 )
1
2
n g
l p

= Lt
44
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0 0.05 0.1 0.15 0.2
Strain
Amplitude (in) x
Experiment
Simulation
Gotimukul {31}
Figure 5.16Amplitudes of 27” Test Specimen
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.05 0.1 0.15
Strain
Amplitude (in) x
Experiment
Simulation
Gotimukul {31}
Figure 5.17Amplitudes of 30” Test Specimen
From the above graphs we can concur that the new expression developed by averaging
the outofplane deformation along the width alone is not in good agreement with the
45
results from experiments and simulation but it can be proclaimed that it’s better than that
of Cerda’s expression.
In order to see the effect of the inelastic material properties on the closed form expression
for average amplitude along the width, it was expressed in stress and tangent modulus.
The expression {31} for amplitude was modified by expressing strain in terms of
Modulus and stress.
( ) 4
1
2 2
2
T
2
t
12L E 1
1
2L
A

+
=
p
s n
p
gn
{32}
The tangent modulus (ET) used in developing the results was obtained from the Table 3.1
Stress ( ) from the load obtained from load cell and Poisson’s ratio from the expression
{28}.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 0.05 0.1 0.15 0.2
Strain
Amplitude (in) x
Experiments
Simulation
Gotimukul {31}
Cerda {25}
Gotimukul TM {32}
Figure 5.18Amplitudes of 24” test specimen with inelastic material properties
46
It can be inferred from the graph that using the expression in terms of inelastic material
properties would still predict the amplitudes to be of the same order which is
contradicting the experimental and simulation results.
47
CHAPTER V
CONCLUSIONS
Experiments were conducted on a polyethylene web to study the behavior of the troughs,
at different strains both in linear and non linear range. The conclusions drawn from this
research includes
1. From the figures 5.8, 5.9, 5.10 it can be concluded that the closed form expression
{24} for wavelength of the troughs claimed by Cerda is capable of predicting the
wavelength for strain range of 0.0165 to 0.166. These strains proceed well into the
plastic range. However the CMD modulus enters expression {24} and the stresses
in this direction are small.
2. From the figures 5.12  5.17 it appears that the two closed form expressions {24}
and {31} for average amplitude of troughs developed by Cerda and Gotimukul
are not in agreement with the experiment and simulation results. They do however
help over estimate the amplitude of the troughs. Gotimukul overestimates by a
factor of ~ 2.5 and Cerda overestimates by a factor of ~ 3.6. This conclusion is
applicable only in the linear elastic range.
48
3. Neither of the two closed form solutions is accurate in predicting the amplitudes
of the troughs in both linear as well as non linear region.
Future Work:
It has been shown in this research that wavelength of the troughs on a web can be
predicted, but prediction of amplitudes in both elastic and inelastic region is still in the
ambiguity. Further research can be done in developing a closed form solutions, not based
on linear energy theory, which can predict the amplitudes of the trough.
49
REFERENCES
[1] S.Timoshenko, S.P., Gere, J.M., Theory of Elastic Stability, 2nd edition McGrawHill,
1961, p.324332.
[2] Good, J.K, and Biesel, J.A, “Instability of Webs: The Prediction of Troughs and
Wrinkles” Advances in Pulp and Paper Research Oxford 2009 Vol 1 pp 517
[3] Cerda, E, RaviChandar, E and Mahadevan, L “Wrinkling of an Elastic Sheet Under
Tension” Nature Publishing Group 2002 and pp 419579.
[4] Cerda, E, Mahadevan, L “Geometry and Physics of Wrinkling” The American
Physical Society 2003 Vol 90, No. 7 and pp 0743021 – 0743024
50
APPENDICES
The experiments were conducted on different test spans at different strain levels, in the
results chapter, the out of plane deformation at two strain levels were shown. The out of
plane deformation of the test specimens at other strain levels are shown below.
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation(in)
I Sample
II Sample
III Sample
Outofplane deformation of three different 24” test specimen at a strain of 0.0165
51
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
I Sample
II Sample
III Sample
Outofplane deformation of three different 24” test specimen at a strain of 0.033
0.12
0.1
0.08
0.06
0.04
0.02
0
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation (in)
I Sample
II Sample
III Sample
Outofplane deformation of three different 24” test specimen at a strain of 0.0495
52
0.12
0.1
0.08
0.06
0.04
0.02
0
0 2 4 6
CMD Location (in)
Outofplane Deformation(in)
I Sample
II Sample
III Sample
Outofplane deformation of three different 24” test specimen at a strain of 0.066
0.12
0.1
0.08
0.06
0.04
0.02
0
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
I Sample
II Sample
III Sample
Outofplane deformation of three different 24” test specimen at a strain of 0.0825
53
0.12
0.1
0.08
0.06
0.04
0.02
0
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
I Sample
II Sample
III Sample
Outofplane deformation of three different 24” test specimen at a strain of 0.099
0.12
0.1
0.08
0.06
0.04
0.02
0
0 1 2 3 4 5 6
I Sample
II Sample
III Sample
Outofplane deformation of three different 24” test specimen at a strain of 0.1155
54
0.12
0.1
0.08
0.06
0.04
0.02
0
0 2 4 6 8
CMD Location (in)
Outofplane Deformation (in)
I Sample
II Sample
III Sample
Outofplane deformation of three different 24” test specimen at a strain of 0.132
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0 1 2 3 4 5 6
CMD Location (in) Outofplane Deformation(in)
Outofplane deformation of 27” test specimen at a strain of 0.0185
55
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation (in)
Outofplane deformation of 27” test specimen at a strain of 0.037
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation(in)
Outofplane deformation of 27” test specimen at a strain of 0.0555
56
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
Outofplane deformation of 27” test specimen at a strain of 0.074
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0 1 2 3 4 5 6
Outofplane deformation of 27” test specimen at a strain of 0.0925
57
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation (in)
Outofplane deformation of 27” test specimen at a strain of 0.111
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0 1 2 3 4 5 6 7
CMD Location (in)
OutofPlane Deformation(in)
Outofplane deformation of 27” test specimen at a strain of 0.1296
58
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
I Sample
II Sample
III Sample
Outofplane deformation of three 30” test specimen at a strain of 0.0165
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
I Sample
II Sample
III Sample
Outofplane deformation of three 30” test specimen at a strain of 0.033
59
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0 1 2 3 4 5 6
CMD Location (in)
OutofPlane Deformation(in)
I Sample
II Sample
III Sample
Outofplane deformation of three 30” test specimen at a strain of 0.0495
0.03
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0 1 2 3 4 5 6
CMD Location(in)
Outofplane Deformation(in)
I Sample
II Sample
Outofplane deformation of two 30” test specimen at a strain of 0.066
60
0.05
0.04
0.03
0.02
0.01
0
0.01
0.02
0 1 2 3 4 5 6
CMD Location (in)
OutofPlane Deformation(in)
I Sample
II Sample
Outofplane deformation of two 30” test specimen at a strain of 0.0826
0.04
0.03
0.02
0.01
0
0.01
0.02
0 1 2 3 4 5 6
CMD Location (in)
OutofPlane Deformation(in)
I Sample
II Sample
Outofplane deformation of two 30” test specimen at a strain of 0.099
61
0.05
0.04
0.03
0.02
0.01
0
0.01
0.02
0 2 4 6
CMD Location (in)
OutofPlane Deformation(in)
I Sample
II Sample
Outofplane deformation of two 30” test specimen at a strain of 0.1157
The Outofplane deformations of the test specimen of different lengths from ABAQUS
simulations at different strains are shown below.
0.0025
0.002
0.0015
0.001
0.0005
0
0.0005
0.001
0.0015
0.002
0.0025
0 1 2 3 4 5 6
CMD Location
Outofplane Deformation(in)
Outplanedeformation of a 24” test specimen from simulation at a strain of 0.00467
62
0.006
0.004
0.002
0
0.002
0.004
0.006
0 1 2 3 4 5 6 7
CMD Locarion (in)
OutofPlane Deformation(in)
Outplanedeformation of a 24” test specimen from simulation at a strain of 0.00834
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
0 1 2 3 4 5 6
CMD Loaction (in)
Outofplane Deformation(in)
Outplanedeformation of a 24” test specimen from simulation at a strain of 0.0165
63
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
0.025
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 24” test specimen from simulation at a strain of 0.033
0.025
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
0.025
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 24” test specimen from simulation at a strain of 0.0495
64
3.00E02
2.00E02
1.00E02
0.00E+00
1.00E02
2.00E02
3.00E02
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 24” test specimen from simulation at a strain of 0.066
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
0.025
0 1 2 3 4 5 6 7
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 24” test specimen from simulation at a strain of 0.0825
65
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
0.025
0 1 2 3 4 5 6 7
CMD Location (in)
OutofPlane Deformation(in)
Outplanedeformation of a 27” test specimen from simulation at a strain of 0.0185
0.03
0.02
0.01
0
0.01
0.02
0.03
0 1 2 3 4 5 6 7
CMD Location (in) OutofPlane Deformation(in)
Outplanedeformation of a 27” test specimen from simulation at a strain of 0.037
66
0.03
0.02
0.01
0
0.01
0.02
0.03
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 27” test specimen from simulation at a strain of 0.055
0.03
0.02
0.01
0
0.01
0.02
0.03
0 1 2 3 4 5 6 7
CMD Location (in)
OutofPlane Deformation(in)
Outplanedeformation of a 27” test specimen from simulation at a strain of 0.0747
67
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0 1 2 3 4 5 6
CMD Location (in)
OutofPlane Deformation(in)
Outplanedeformation of a 27” test specimen from simulation at a strain of 0.0925
0.01
0.008
0.006
0.004
0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 30” test specimen from simulation at a strain of 0.033
68
2.50E02
2.00E02
1.50E02
1.00E02
5.00E03
0.00E+00
5.00E03
1.00E02
1.50E02
2.00E02
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 30” test specimen from simulation at a strain of 0.0495
0.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 30” test specimen from simulation at a strain of 0.066
69
0.015
0.01
0.005
0
0.005
0.01
0.015
0 1 2 3 4 5 6
CMD Location (in)
Outofplane Deformation(in)
Outplanedeformation of a 30” test specimen from simulation at a strain of 0.0825
Stretch Test Data:
Thickness(in) Width(in) Area(sqin) Length(in)
Tangent
Modulus (psi)
0.0012 10 0.012 600
Load(pound) Delta L L Strain Area Stress
1 2.375 600 0.00395833 0.012 83.33333
2 4.5 600 0.0075 0.012 166.6667
3 6.5625 600 0.0109375 0.012 250
4 8.875 600 0.01479167 0.012 333.3333
5 11.25 600 0.01875 0.012 416.6667
6 13.9375 600 0.02322917 0.012 500
7 16.9375 600 0.02822917 0.012 583.3333
8.3 18.31 600 0.03051667 0.012 691.6667
9 23.4375 600 0.0390625 0.012 750
10 27.075 600 0.045125 0.012 833.3333 9482.7586
11.1 32.875 600 0.05479167 0.012 925 7500
12 38.875 600 0.06479167 0.012 1000 5673.7589
13 47.6875 600 0.07947917 0.012 1083.333 4705.8824
14 58.3125 600 0.0971875 0.012 1166.667 2203.8567
15 81 600 0.135 0.012 1250 1967.2131
16.2 111.5 600 0.18583333 0.012 1350 1259.8425
17 143.25 600 0.23875 0.012 1416.667 2040.8163
18 167.75 600 0.27958333 0.012 1500 4297.5207
19.3 182.875 600 0.30479167 0.012 1608.333 5276.8284
70
Poisson’s Ratio Data:
Strain Poisson's Ratio Strain
Poisson's
Ratio
4.35E03 0.33 6.52E02 0.483
4.39E03 0.329 6.90E02 0.4734
8.69E03 0.33 7.41E02 0.457
8.75E03 0.329 7.40E02 0.4638
1.30E02 0.346 8.26E02 0.445
1.30E02 0.33 9.00E02 0.479
1.74E02 0.395 9.89E02 0.472
2.17E02 0.39 1.02E01 0.451
2.61E02 0.385 1.10E01 0.449
2.70E02 0.329 1.12E01 0.454
3.04E02 0.378 1.20E01 0.452
3.48E02 0.38 1.21E01 0.441
3.98E02 0.395 1.33E01 0.448
4.38E02 0.39 1.34E01 0.452
4.82E02 0.4372 1.47E01 0.467
5.20E02 0.467 1.47E01 0.466
5.22E02 0.443 1.59E01 0.4779
5.72E02 0.424 1.60E01 0.46
6.09E02 0.4574 1.76E01 0.4683
6.51E02 0.4521 1.77E01 0.44
VITA
Aditya Gotimukul
Candidate for the Degree of
Master of Science
Thesis: Prediction of Amplitude and Wavelengths of Troughs on Polyethylene Webs.
Major Field: Mechanical Engineering
Biographical:
Education:
Completed the requirements for the Master of Science in Mechanical
Engineering at Oklahoma State University, Stillwater, Oklahoma in May 2010.
Received Bachelors of Technology in Mechanical Engineering from Jawaharlal
Nehru Technological University, Hyderabad, AP, India in May 2007
.
ADVISER’S APPROVAL: Dr. James. K. Good
Name: Aditya Gotimukul Date of Degree: May 2010
Institution: Oklahoma State University Location: OKC or Stillwater, Oklahoma
Title of Study: PREDICTION OF AMPLITUDE AND WAVELENGTH OF TROUGHS
ON POLYETHYLENE WEBS
Pages in Study: 70 Candidate for the Degree of Master of Science
Major Field: Mechanical Engineering
Scope and Method of Study: Outofplane deformations occur in free spans of webs,
which may be composed from polymers, during their transportation through
process machinery. These troughs may become wrinkles when they transgress a
roller. As the amplitudes of these outofplane deformations increase the
propensity for web wrinkles increase. The goal of this research is to determine if
the amplitude and wavelengths of the troughs can be estimated.
Findings and Conclusions: Closed form expressions for amplitude and wavelength of
troughs in case of a stretched web were obtained from the literature. Experiments
were conducted to stretch the web and measure the outofplane deformation.
Simulation of troughs formation was done using ABAQUS Explicit. The outofplane
deformations from both experiments and simulations were obtained at
different strain levels for different specimen lengths. Amplitude and wavelength
were inferred from the outofplane deformations. These amplitudes and
wavelengths were compared with the closed form expressions. From the
comparisons it was concluded that the wavelength expression was accurate in
predicting the wavelengths of the troughs in the elastic and in the inelastic range
of the material. The amplitude expressions given by Cerda or developed by
Gotimukul are not accurate in predicting the amplitude of the troughs. However
these expressions are aids for overestimating the amplitudes of the troughs.