MECHANICAL AND VISCOELASTIC CHARACTERIZATION OF POLYVINYL ALCOHOL (PVA) HYDROGEL MEMBRANES USING

THE SHAFT-LOADED BLISTER TEST

A Thesis Presented

by

Edgar José Montiel - Rubio

to

The Department of Mechanical and Industrial Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in

Mechanical Engineering

in the field of

Mechanics and Design

Northeastern University

Boston, Massachusetts

June 2009

Abstract

The shaft loaded blister test was used as an alternative method to develop three different

tests to study the mechanical properties of polyvinyl alcohol (PVA) hydrogel on thin

circular membranes and see how the number of freeze/thaw cycles used in their

fabrication would affect the stiffness of the material. The first test consisted of a quasi-

static experiment; the membrane was clamped along its periphery and a spherical

indenter connected to the load cell of the instrument was then used to apply a

deformation. The second test was a creep test under the same geometric configuration,

but in this case the load consisted of the weight of a stainless steel ball, and a long focal

microscope was used to monitor the deformation. This allowed the study of the material

behavior over a long span of time. Finally, the third experiment was a cyclic loading test

that allowed the study of the dynamic properties of the material over a brief span of time,

as well as the energy storing capabilities of the same. By analyzing the results for

samples fabricated with 2, 3 and 4 freeze/thaw cycles in their processes, we demonstrated

that the increase of the number of these cycles increased the material stiffness, and also

that as the quantity of cycles were increased the material behaved more as an elastic solid

and less like a viscous fluid. The results of these tests were then applied to the design of a

sample fixture capable of holding a membrane for collagen cleavage and cell

differentiation studies. In both cases the goal is to study the sample biomechanical

behavior while a different set of stresses are applied to various regions of the membrane.

i  

Acknowledgments

This thesis could not be possible without the support of the people who have

contributed to my graduate studies in many ways. For this reason, I would like to take

this opportunity to express my gratitude to them.

First, I must thank my advisor, Dr. Kai-tak Wan, for giving me the opportunity to

form part of Northeastern University’s research community, and for his guidance and

dedication while we both settled in our new school. In addition, I would like to express

my appreciation to the other members of our group, to Scott Julien during the early days

of my work and to Jiayi Shi, Xin Wang, Guangxu Li and Zong Zong for their support and

company during long hours at 243A FR, the brand new lab we were lucky enough to

work in. I am thankful to Dr. Jeff Ruberti and his research group, for giving me access to

their lab and for their wise advice. Also, this thesis could not be possible without the help

given by Jon Doughty while fabricating our testing fixtures, and for making me feel at

home when I spent time in the machine shop.

Special thanks go to the National Science Foundation, for providing the financial

support of the present work through the CMMI # 0757140 and CMMI # 0757138 grants.

Finally, I would like to express my gratitude to my wife to be, Johana, for her

words of encouragement and sacrifice while we both worked towards our Masters

degrees; and to my family, my parents and grandparents, examples of hard work and

integrity, my sister Lisseth and my two brothers, Trino and Ender for their unconditional

love and companionship.

ii  

Table of contents

Abstract………………………………………………………………………………….................i

Acknowledgments……………………………………………………………………….………..ii

Table of Contents……………………………………………………………….…………….….iii

List of Figures……………………………………………………………………….…………....vi

List of Tables……………………………………………………….…………………………….ix

CHAPTER 1

INTRODUCTION………………………………………………………………………………....1

1.1 Material………………………………………………………………………………………...2

1.1.1 Hydrogels………………………………………………………………………………..…2

1.1.1.1 Polyvinyl Alcohol (PVA) Hydrogel…………………………………………………..3

1.2 Membrane production method………………………………………………………………....4

1.3 Significance of research and applications…………………………………………………...…5

1.4 Objective of research……………………………………………………………………….….6

1.5 Mechanical characterization methods for hydrogels…………………………………………..6

1.5.1 Extensiometry……………………………………………………………………………...6

1.5.2 Compression test…………………………………………………………………………...7

1.5.3 Bulge test…………………………………………………………………………………..7

1.5.4 Indentation test……………………………………………………………………………..8

1.5.5 Alternative tests…………………………………………………………………………....8

CHAPTER 2

BACKGROUND THEORY……………………………………………………………………...10

2.1 The shaft-loaded blister……………………………………………………………………….10

2.2 Time dependent behavior – viscoelasticity…………………………………………………...11

2.2.1 Creep……………………………………………………………………………………...12

2.2.2 Cyclic loading…………………………………………………………………………….13

iii  

2.2.3 The standard linear solid……………………………………………………………….…14

2.3 Large deformation in circular membranes…………………………………………………....15

2.3.1 Review of “Indentation of a circular membrane”………………………………………...15

2.3.1.1 The blister geometry……………………………………………………………….....16

2.3.1.2 Equations for the non-contact region…………………………………………………16

2.3.1.3 Equations for the contact region…………………………………………………...…19

2.3.1.4 Boundary conditions and solutions ………………………………………………..…20

CHAPTER 3

MATERIALS AND METHODS…………………………………………………………………22

3.1 The shaft-loaded blister test……………………………………………………………….….22

3.2 Viscoelastic tests………………………………………………………………………...……24

3.2.1 The creep test……………………………………………………………………………..24

3.2.2 The cyclic loading test…………………………………………………………………....26

3.3 A computational method for large deformation…………………………………………...….27

CHAPTER 4

RESULTS AND DISCUSSION………………………………………………………………….28

4.1 The shaft-loaded blister test………………………………………………………………..…28

4.2 Viscoelastic tests…………………………………………………………………………...…30

4.2.1 The creep test…………………………………………………………………………..…30

4.2.2 The cyclic loading test……………………………………………………………...…….33

4.3 A computational method for large deformation……………………………………….……..34

CHAPTER 5

APPLICATIONS OF THE BLISTER CONFIGURATION ON IN-VITRO EXPERIMENTS....35

5.1 On stem cell differentiation……………………………………………………………..........35

5.2 On collagen liquid crystals…………………………………………………………………...36

iv  

CONCLUSIONS…………………………………………………………………………..…….38

FUTURE WORK…………………………………………………………………………….….40

REFERENCES…………………………………………………………………………………..84

APPENDICES

Appendix 1: Mechanical drawings …………………………………………………...………….90

Appendix 2: Mathematica® code for the large deformation model …………………………....102

Appendix 3: Large deformation theory – Nomenclature…………...…………………...……....111

Author’s Curriculum Vitae……………………….……………………………………….…..112

v  

List of Figures

Figure 1.1. Temperature vs. time during the freeze/thaw cycling of the samples………………..42

Figure 1.2. Conventional techniques to mechanically characterize hydrogels…………………...43

Figure 2.1. The blister configuration……………………………………………………………..44

Figure 2.2. Transition between modes of deformation [46]……………………………………...45

Figure 2.3. The standard linear solid……………………………………………………………..46

Figure 2.4. (a) Step stress, (b) response from an elastic solid, (c) response from a linear

viscoelastic solid……………………………………………………………………………...…..47

Figure 2.5. Profile of a cyclic loading experiment………………………………………………..48

Figure 2.6. Blister diagram………………………………………………………………….……49

Figure 3.1. UTM Nano. Agilent Technologies – MTS……………………………….……….….50

Figure 3.2. Shaft-loaded test at different speeds………………………………………………….51

Figure 3.3. Shaft-loaded test with increasing steps of deformation……………………………....52

Figure 3.4. Fixture used in the shaft-loaded test……………………………………………….…53

Figure 3.5. Fixture used in the creep test………………………………………………………....54

Figure 3.6. Creep test setup showing the long-focal microscope………………………………...55

Figure 3.7. Sample image used to monitor the deformation used during a creep test on

a PVA hydrogel sample………………………………………………………………………..…56

Figure 3.8. Blister configuration in a creep test……………………………………………..……57

Figure 3.9. Free body diagram of the ball used in the creep test…………………………………58

Figure 3.9. Deformation pattern used in the cyclic loading tests (10 Hz)……………………..…59

vi  

Figure 3.10. TA.XT from Texture Technologies….………………………………………….…..60

Figure 3.11. Runge-Kutta algorithm applied in the Mathematica® code [52]…………………...61

Figure 4.1. Average elastic modulus obtained using shaft-loaded blister tests…………………..62

Figure 4.2. Load vs. deformation plot for a 2 freeze/thaw cycles sample………………………..63

Figure 4.3. Load vs. deformation plot for a 3 freeze/thaw cycles sample………………………..64

Figure 4.4. Load vs. deformation plot for a 4 freeze/thaw cycles sample………………………..65

Figure 4.5. Creep test on a 2 freeze/thaw cycles PVA hydrogel sample…………………………66

Figure 4.6. Creep test on a 3 freeze/thaw cycles PVA hydrogel sample…………………………67

Figure 4.7. Creep test on a 4 freeze/thaw cycles PVA hydrogel sample…………………………68

Figure 4.8. Cyclic loading test on a 2 freeze/thaw cycles PVA hydrogel sample………………..69

Figure 4.9. Cyclic loading test on a 3 freeze/thaw cycles PVA hydrogel sample………………..70

Figure 4.10. Cyclic loading test on a 4 freeze/thaw cycles PVA hydrogel sample………………71

Figure 4.11. Deformed membrane profiles for a/R=5 and stretch ratio at the outer edge λp=1….73

Figure 4.12. Load deflection curve for a/R=5 and stretch ratio at the outer edge, λp=1…………74

Figure 4.13. Stress resultant at pole for a/R=5 and stretch ratio at the outer edge, λp=1………...75

Figure 4.14. Radius of contact for a/R=5 and stretch ratio at the outer edge, λp=1……………...76

Figure 4.15. Deformed membrane profiles for a/R=5 and W/C1hR=1…………………………...77

Figure 4.16. Comparison between the deformation profile obtained with Wan’s formulation

for the transition between the bending and stretching modes and the large deformation theory

using the developed code…………………………………………………………………………78

Figure 5.1. Fixture built for cell differentiation and collagen cleavage experiments…………….79

Figure 5.2. One of the fixtures fabricated for the cell differentiation experiments………………80

Figure 5.3. Proposed stem cell differentiation experiment…………………………………….…81

vii  

Figure 5.4. Radial and tangential membrane stress variation along the radial direction [55]……82

Figure 5.5. Twisted plywood configuration of collagenous liquid crystals………………………83

Figure 5.6. Proposed collagen cleavage experiment…………………………………………...…84

viii  

ix  

List of Tables

Table 3.1.Nano UTM technical specifications…………………………………………………22

Table 3.2. Creep setup specifications…………………………………………………………….25

Table 3.3. TA.XT technical specifications…………………………………………………….…26

Table. 4.1. Elastic modulus of polyvinyl alcohol hydrogel obtained by shaft-loaded

blister tests………………………………………………………………………………………..28

Table. 4.2. Mechanical parameters of polyvinyl alcohol hydrogel obtained by tension

testing……………………………………………………………………………………………..30

Table. 4.3 Viscoelastic properties according to standard linear solid of polyvinyl

alcohol hydrogel obtained by creep tests………………………………………………………....31

Table. 4.4 Viscoelastic properties obtained from the cyclic loading test………………….……..33

CHAPTER 1

INTRODUCTION

Mechanical characterization of thin membranes is a current necessity in many

fields. Today different kinds of membranes are used in applications that range from

Micro Electro-mechanical Systems (MEMS), where a variety of thin films are applied on

the processes of deposition of different layers of the electronic circuitry [1]; to an

emerging field in the Biomedical industry where thin membranes are being used in drug

delivery systems [2], as scaffolding for cell cultivation in tissue engineering and many

more. Biomedical sciences have received special attention and will be the main focus of

this thesis since the experimental methods applied here were developed with the

particular interest of monitoring the mechanical properties in a nondestructive manner.

Hydrogels have been used as a substrate for many types of engineered tissue including

cartilage [3, 4], cornea [5, 6], skin [7] tendon [8] and vascular tissue [9].

One of the most challenging issues in tissue engineering could be to replicate the

mechanical characteristics of the natural counterpart. Many attempts to recreate different

prosthesis resulted in a final product with inferior mechanical properties [10, 11]. For this

reason, great attention needs to be dedicated to the characterization of these samples and

the variation of their fabrication process to achieve optimal properties that would be as

close as possible to the original tissue. But since in most of these cases we are dealing

with extremely delicate samples (less than a micron thick), standard methods such as the

ASTM standard tensile test [12], are not an option because the grips on this instrument

would very easily damage the frail films. This obstacle calls for a different method where

1

the sample fixture would minimize damage or not alter the membrane’s shape, and will

also allow the application of an external load, so a stress-strain relation can be obtained,

which is the basic principle of a mechanical characterization process.

Demonstration of various new testing methods on various model membranes and

the results will be shown in this thesis. We will also discuss the applications and

relevance to tissue engineering.

1.1 Material

1.1.1 Hydrogels

A hydrogel is a three-dimensional network of hydrophilic polymer chains that are

held together by covalent, ionic or physical bonds with high water content [13]. They can

be made from chains of natural polymers such as collagen or alginate or from synthetic

polymers such as polyvinyl alcohol (PVA) or polyacrylic acid (PAA). Lately, hydrogels

have received a lot of attention because of their high degree of biocompatibility and ease

of fabrication which has propelled a wide range of research for biomedical applications.

They are currently being used in areas like drug delivery, ophthalmology, prosthetic

tissues and orthopedics. One of the main reasons this kind of material is so versatile is its

ability of exhibiting a variety mechanical properties (i.e. elastic modulus, transparency),

by varying the parameters in its fabrication (i.e. polymer concentration) and the addition

of certain chemicals.

2

1.1.1.1 Polyvinyl Alcohol (PVA) Hydrogel

Currently the gross majority of hydrogels used in biomedical applications consist

of alginate or agarose types. A drawback of these hydrogels is their lack of integrity and

poor mechanical properties [10, 14] that fall short when compared to the natural

counterpart. A viable candidate currently being studied to be used in biomedical

applications is polyvinyl alcohol hydrogel since it has many of the key advantages of the

formers, such as insolubility at physiological temperature and the necessary porosity to

function as the extracellular matrix in multi-cellular organisms, with increased

mechanical properties that can also be tailored to fit a particular application. Some of

these recent applications include using PVA hydrogel as a mimetic substitute of articular

cartilage [15] and as a substrate for the cultivation of synthetic cornea [16].

Moreover, other beneficial characteristics of PVA hydrogel include its capability

to absorb large amounts of water and when carefully manufactured a good transparency

can be achieved. The hydrophilic behavior allows the use of PVA hydrogel as a drug

carrier and has been found as a carrier that can slowly and steadily deliver medication in

an individual [2] and transparency is essential for artificial cornea synthesis. Overall the

present study will focus its analysis on the application of PVA hydrogel as a scaffold for

tissue engineering with emphasis in self-arranged collagen structures such as the latter

synthetic corneas.

3

1.2 Membrane production method

For total dissolution PVA requires water temperatures of roughly 100 °C with a

hold time of at least 30 minutes. With this in mind the fabrication process used in this

thesis started with a ten percent solution (w/v) of polyvinyl alcohol, formed by

dissolving 10 g of polyvinyl alcohol (PVA, ACROS Organics, USA) into 100 ml of

deionized water. To achieve optimum solubility the mixture was stirred and heated at 98

°C (209 °F) for 90 minutes. This solution was then injected into molds to create 0.3 mm

(0.012”) thick membranes, by exposing the molds to a variable number of freeze/thaw

cycles to induce cross-linking [17]. Each freeze/thaw cycle consisted of a freezing phase

of 3 hours at -60 °C (-76 °F) and a thawing phase of 2 hours at 27 °C (80 °F) as shown in

Figure 1.1. Using two, three and four cycles three different types of membranes were

delivered. These were then cut into smaller size samples to accommodate in the designed

fixture.

There are other methods used to induce cross-linking in hydrogels, some include

irradiation and the addition of chemicals, the latter have been proven to decrease the

mechanical properties of the final product [18, 19].

As seen in the explanation above the fabrication procedure of this material adds

yet another advantage to the list, since it is extremely simple and involves only

commercially available products. Finally, it is worth mentioning in a way of reference

that hydrogels manufactured by freeze/thaw cycles are also referenced in the literature as

cryogels.

4

1.3 Significance of research and applications

The outcome of the present research will supply the knowledge necessary to link

the fabrication process with the resultant mechanical properties. Thus, allowing the

chemist to define the manufacturing process for a specific set of final properties.

There is currently work being done to determine the feasibility of application of

hydrogel membranes as drug delivery systems and as scaffolding in tissue engineering

just to name a couple. For the latter, as recent developments in cell differentiation

surface; such as how the environment in which cells live will determine the path they will

follow towards differentiation. A directly related parameter to this thesis, as is the

stiffness of the substrate, has been proven to determine the final type of cell that will

result from a pluripotent stem cell [20]. Greater detail on this subject will be given in

chapter 5. Therefore if there are certain mechanical properties known to be beneficial for

these applications, knowing the procedure to obtain them would definitely be

advantageous.

Moreover, as a comparison is done between actual experimental results and the

theoretical model, the validation of the latter will also be derived. As a consequence

insight of membranous bodies’ mechanics will be obtained, adding up to the pool of

knowledge to complex subjects such as cell adhesion and others.

5

1.4 Objective of research

The main objectives of the present research are to review the relevant methods

related to the deformation of thin membranes and to use this methods to characterize the

mechanical properties of PVA hydrogel samples. This will also serve, at the same time,

as a comparison between theoretical and experimental results.

1.5 Mechanical characterization methods for hydrogels

1.5.1 Extensiometry

This is one of the most used methods at the present time to determine the

mechanical properties of hydrogels. It has been applied on previous mechanical

characterization studies on different kind of hydrogels [21, 22]. The procedure resembles

the standard tensile test, where the sample is fixed between two grips and then a uniaxial

strain is applied to monitor the stress and strain relation. As previously mentioned this

method brings up the issue of concentrated stresses on the grips, an alternative created for

this regard has been the use of a ring sample shown in Figure 1.2 (b). Other shortcomings

that could be considered would be the limited geometry of the samples (only strips or

rings), the fact that the material is only being deformed in one direction, and its

destructive nature that only allows for one test per sample. With this set of data different

mechanical parameters can be determined such as the elastic modulus, yield strength and

ultimate stress. Moreover, the viscoelastic properties of the material could be extracted if

the sample is deformed until a certain level and the deformation is kept constant while

6

monitoring the load on the sample, or a sinusoidal deformation is applied over a short

time interval.

1.5.2 Compression test

The compression test is another technique that has been extensively applied in

previous characterization studies to determine the mechanical properties of different

types of hydrogels [14, 24]. It basically consists of a sample pressed between two plates,

thus by monitoring the applied force and the resultant deformation one would be able to

determine the mechanical properties of a hydrogel sample. One advantage of this method

is that the only geometrical requirement for the sample is that it has two opposite flat

surfaces. When the latter geometrical characteristic is not complied the processing of the

data gets overcomplicated because the contact area in which the pressure is being applied

is hard to estimate.

1.5.3 Pressurized blister test

Another commonly used method to examine the mechanical properties of

hydrogels is the pressurized blister test also known as the inflation test or bulge test [25,

26, 27, 28]. Intuitively, this test consists of a gradual inflation through a hole in the

substrate or making use of hydrostatic pressure while keeping track of the bulge profile,

either using a laser or a CCD camera to take pictures by a determined interval. Then by

using a theoretical model such as [29, 30] or a finite element package, the mechanical

characteristics of the materials can be determined. Some of the drawbacks of this

7

technique include the possibility of leakage and difficulty to control the applied pressure,

but still some have managed to perform this test even on scaffolds embedded with live

cells [31, 32].

1.5.4 Indentation test

Indentation has recently gained great popularity in material characterization,

including soft materials like hydrogels. This technique works by indenting the sample at a

single point to a predetermined depth and measuring the required force to create the

indentation [33, 34, 35, 36]. Using the obtained force-displacement chart, the elastic

modulus can be determined. A critic element in the process is the indenter tip geometry,

since it will determine the approximated contact area in the calculations, thus affecting

the final results [37]. Recent advances in the field have improved the instrument used

allowing the analysis of viscoelastic properties by maintaining a fixed displacement and

observing the stress relaxation phase. These improvements also include the current better

resolutions, which have gotten to the point where good results are possible in the nano

scale [38] allowing the characterization of extremely thin films and coatings even in

multiple points, allowing the determination of localized material properties. A diagram

for this test is shown in Figure 1.2 (e).

8

1.5.5 Alternative tests

Several other techniques have been applied in the study of the mechanical

behavior of hydrogels. Lin et al [39] used a method called spherical ball inclusion, where

a magnetic sphere is embedded within the hydrogel. A magnetic force is then applied to

the sphere, causing it to move inside the material. From this deformation and the known

applied force the mechanical properties can be determined, with the major disadvantage

that can only be applied to transparent materials.

Another method is the micropipette aspiration [40], used accurately in biological

cells only. In this method a suction pressure is applied to the sample and by measuring

the amount of material that gets inside the pipette the mechanical characteristics of the

hydrogel can be determined, even though a fair measurement resolution can be obtained,

the scale of the experiments are usually in the cellular scale and not in the tissue level. A

final method worth mentioning is the ultrasound elastography [41]. In this method a sonic

pulse is transmitted from one end of the sample to the other and the speed of the sound

wave is measured by direct observation, serves as a parameter to estimate the elastic

modulus of the hydrogel sample. All these methods are explained by a simple diagram in

Figure 1.2.

The methods applied in this thesis will be explained further in the text with

greater detail.

9

CHAPTER 2

BACKGROUND THEORY

2.1 The shaft-loaded blister

When looking at how a thin, free standing membrane is deformed, a different set

of stages are observed throughout the process; the geometrical configuration will be the

determining factor of the stage that will be observed.

The first stage is dominated by the indentation produced by a spherical probe

pressing down the elastic surface. This deformation can be estimated using the Hertz

contact model [42], as long as the indentation is much smaller than the membrane

thickness and the sphere radius. Thus, the equation for the deformation produce by

indentation is

916

/ 1 //

34

(2.1)

The following stage is dominated by a bending process which is ruled by the same

expression of the bending of plates [43] and will be considered valid as long as ,

and .

1 (2.2)

Finally there is a transition stage as the deflection is produced mainly by the

stretching of the membrane, until the deformation is considered as pure stretching [44].

10

4 / 1 //

,

behaves in an almost completely elastic fashion. In principle, however, all real materials

(2.3)

Where R is the sphere radius, a is the membrane radius, h the membrane

thickness, and wo is the initial or instant deformation at the pole (see Figure 2.1).

Figure 2.2 (Normalized F vs w) illustrates the transition between these stages. The

needed force to get a certain deformation will then be given by

(2.4)

Where and are constants and shown above in equations 2.1, 2.2 and 2.3

[45].

2.2 Time dependent behavior - viscoelasticity

When a body is submitted to stress or strain, rearrangements take place inside the

material as a response to the excitation. In any real material these arrangements

necessarily require a finite time. The time required, however, may be very short or very

long. When the changes take place so rapidly that the time is negligible compared to the

time scale of the experiment, the material is considered as purely viscous. In a purely

viscous material, all the energy required to produce the deformation is dissipated as heat.

When the material rearrangements take virtually infinite time, we speak of a purely

elastic material. In a purely elastic material the energy of deformation is stored and may

be recovered completely upon release of the forces acting on it. Water comes close to

being a purely viscous material; and steel, if deformed to no more than a percent or two,

11

are viscoelastic. Some energy may always be stored during the deformation under

appropriate conditions, and energy storage is always accompanied by dissipation of some

energy [47, 48].

2.2.1 Creep

Creep is the time-dependent change in strain following a step change in stress.

Unlike

nd

another

creep in most metals, polymer’s creep is usually recoverable at low strains of less

than one percent. Figure 2.4 shows the typical response of an elastic material and a linear

elastic material. For the case of the elastic material it is shown how the strain response

follows a similar proportional pattern, whereas in the viscoelastic material a strain with a

decreasing rate follows the instantaneous one. For the most general case of a viscoelastic

solid the total strain is the sum of three essentially different parts: immediate elastic

deformation, the delayed elastic deformation and the Newtonian flow which is

identical with the deformation of a viscous fluid obeying Newtown’s law of viscosity.

Note also that once the stress is removed there is an instant strain recovery, a

part that will slowly decrease its recovery rate. An analogous to this behavior is

the stress relaxation; in this case a step strain is applied to the material and held constant.

It is then observed how the initial stress starts to decrease demonstrating a “relaxation”

phenomenon attributed to the material’s viscoelasticity.

12

2.2.2 Cyclic loading

t the transient experiments and provide information corresponding

to very sho

strain at an angle (See Figure 2.5).

y the following equation

(2.5)

Where is the maximum amplitude of the strain, thus using the constitutive

equation we obtain (for details on this derivation see [48])

cos (2.6)

and two frequency dependent functions, the storage modulus , and the loss modulus

.

⁄ cos (2.7)

⁄ sin

⁄ tan

To supplemen

rt times, the strain may be varied periodically, usually with a sinusoidal

alternation at a frequency in radians/sec. If the viscoelastic behavior is linear, it is

found that the stress will also alternate sinusoidally but will be out of phase with the

The strain will be determined b

sin

sin cos sin sin

(2.8)

(2.9)

13

2.2.3 The standard linear solid

(shown in Figure 2.3) is known to be the simplest

mathem

The additional spring element to what is known as the Maxwell Unit (spring and

dashpot in

eing the infinite

or equilibrium strain, oduli,

viscosity, the relaxation time and the strain at a certain time .

0∞ 0

The standard linear solid

atical representation of a viscoelastic material, such as a polymer, that allows a

very slow creep deformation but not an unlimited one such as the one exhibited by warm

tar.

series), gives the model the required equilibrium to offer a decent

approximation to a polymer’s behavior. Though not as acurate as it would be needed to

obtain a close fit over a wide time range, it gives some good results without over

complicating the model into an imense network of dashpots and springs.

The equations below rule the behavior of the model with ∞ b

0 the instant strain, and the elastic m the effective

1 / (2.10)

∞0 1 (2.11)

1

(2.12)

14

2.3 Large deformation in circular membranes

or membrane is much greater than its

thickness the equations used in general stop be

circular m

.3.1 Review of “Indentation of a circular membrane”

onsisting of a set of ordinary

differential equations with the purpose of

ang and Feng’s work, where

the approached the problem where a load is applied to a circular membrane by a smooth

sphere.

When the deformation applied to a plate

ing valid and a new set of equations that

correspond to the large deformation theory are a better approximation of the real results.

We will begin with an overview of the work published in “Indentation of a

embrane” by Yang and Hsu [49], who worked on the task of reducing the

complex set of differential equations in the general theory of large deformation to a more

practical one that allows a less mathematically demanding approach to the specific case

of axisymmetric indentation of a circular membrane.

2

Green and Adkins [50] first developed a model c

understanding the large deformation process of

circular membranes. In their work they assumed that the circular membrane was initially

flat, limiting the application for this particular case. Yang and Feng [29] then proposed a

simpler method, with only 3 ODEs instead of 22 as the previously referenced; and though

it was limited to axisymmetric deformation, it was still valid for the case when the

membrane had some initial deformation or pre-stretching.

Yang and Hsu then developed a model based on Y

15

2.3.1.1

addition of some new terms and variables that are only relevant for this specific case. The

the rigid sphere, that contacts the membrane at its center,

has a radiu

2.3.1.2 Equations for the non-contact region

Conveniently plane polar coordinates were used to describe the position of any

med membrane. A point P’ on the deformed membrane is located by

int to the axis of symmetry and , the

meridian arc length b

umferential stretch ratios respectively.

The same subscripts will be used throughout this review to define the corresponding

directions.

The blister geometry

We will be using the same nomenclature as the previous chapters, with the

membrane has a radius “a” and

s “R”. As the sphere proceeds it creates an axisymmetric surface with the area

of contact at centered with the membrane.

point in the undefor

two coordinates: , the distance from the po

etween the center of the membrane and the given point. Figure 2.6

shows the segment PQ and it’s deformed state at P’Q’. The deformation is then

described by a solution with the following form:

(2.13a)

(2.13b)

The following are the meridian and circ

(2.14a)

16

(2.14b)

Without the presence of external forces in the non-contact region, the

homogeneous equations of equilibrium in the meridian and normal directions are,

respectively,

10

directions, respectively; and are the principal curvatures of the arcs

in the corresponding directions.

The following introduced variables ease the symbolic manipulation

(2.15)

0 (2.16)

where and are the stress resultants per unit edge length in the meridian and

circumferential

(2.17)

(2.18)

(2.19)

with the latter variables and can be expressed in the following way

(2.20a)

(2.20b)

where the prime denotes differentiation with respect to r.

17

The material is assumed to be elastic, isotropic and incompressible. Its

mechanical property can be described with the known strain-energy function , ,

wher and are strain invariants expressed by the following equations

(2.21a)

(2.21b)

using this function a stress-resultant stretch-ratio relation was derived in term of the

newly introduced variables

2

and

(2.22a)

2

Yang and Feng demonstrated that the system of equations governing the unknowns

, , , , , , , and can be reduced to three first-order differential equation

Using Mooney’s strain energy function

(2.23)

where and are material constants and ⁄ . Then the governing differential

(2.22b)

where is the undeformed thickness of the membrane, assumed to be constant.

s.

3 3 3 3

equations in the noncontact region are

, , , (2.24a)

, , , , , , (2.24b)

18

1 (2.24c)

where

1 3 3 1 13 1 (2.25)

1 1 11 1 (2.26)

and ⁄ is a dimensionless radius. The right hand side of equation 2.24 are all

s of , , and .

2.3.1.3 Equations for the contact region

In this region the membrane deforms following the shape of the rigid sphere, a

function

known surface. Hence and are related by equation of the spherical indenter

(2.27)

here ⁄ and ⁄ . With this relation the number of equations is reduced by w

one. Leaving

(2.28)

sin / (2.29)

The friction between the sphere and the membrane is neglected. Then,

substituting the latter equations in the first equilibrium equation, the following expression

is obtained

19

3 3 1 13 1

1(2.30)

and using the defined stretch ratios

(2.31)

Equations 2.30 and 2.31 constitute the governing equations over the contact

region.

needed in order to assure they comply with the boundary conditions.

The second governing equation, though not needed to obtain a solution, gives the

pressur

Proper integration of both the equations for the contact and noncontact regions is

e distribution between the sphere and the membrane as

2 22

1 1 (2.32)

2.3.1.4 Boundary conditions and solutions

f integration, we have the

condition of symmetry that gives

In the contact region at 0, the beginning point o

0 0 (2.33)

and assuming a value for the stretch ratio at the pole

0 (2.34)

20

The total indentation distance as shown in Figure 2.6 is , and is the contact

ane and the sphere. It is important to

remember that this radius should never exceed the semi-sphere.

radius between the undeformed portion of the membr

The weight of the sphere or load exerted by a spherical probe can be determined

by integrating the vertical component of at the membrane’s edge .

2 1 / (2.35)

With given values for and , equation (2.24) can be integrated with following

initial conditions

(2.36)

(2.37)

1 /2 , (2.38)

where is the stretch ratio that takes a range of values to create a family of solutions

the noncontact region.

Regardless of the solution method, it is required that the following continuity

conditions are met

in

(2.39)

(2.40)

which

ained in the noncontact region.

⁄

means that the values obtained from the expressions for the contact region are

equal to the ones obt

(2.41)

21

CHAPTER 3

3.1 The shaft-loaded blister test

The quasistatic tests were performed in a MTS® Universal testing machine

Figure 3.1. The technical details of this instrument are given in

ength of the experiments, some extending to almost

half an hour (more than enough to cause dehydration), care was taken to always keep the

samples immersed in water, for this reason the fixture designed for this experiment

consist

Table 3.1.Nano UTM technical specifications

MATERIALS AND METHODS

(UTM) nano shown in

Table 3.1 below. Due to the time l

ed of a liquid cell capable of holding the membrane in place and maintain its entire

volume under water.

Maximum load 500 mN

Load resolution 50 nN

Maximum extension 150 mm

Displacement resolution 35 nm

Ex 0.5 µm/s to 5 /s tension rate mm

To determine d successive faster speeds, starting at 0.02 mm/min

up to 2 mm/min, we Fig order to determine

the fastest speed possible to m of a quasistatic test. As long as the slopes

in the line traced in the force vs. displacement plot were maintained it is safe to assume

that no significant dynamic were experienced. Therefore we chose a speed of 2 mm/min

the optimal spee

re applied to a sample as shown in ure 3.2 in

aintain the basis

22

to cond

loading and unloading phases. Starting from 0.5 mm and increasing the same value until

we rea

thesis work.

uct our tests. This test was done on a four freeze/thaw cycles sample following the

assumption of it being the stiffest kind within the group, and this allows us to establish it

as a reference for the other two kinds.

Then the maximum deformation to be applied needed to be determined. For this

purpose a different plot, shown in Figure 3.3, was used. At the determined speed obtained

from the previous test subsequent deformation steps of deformation were applied, each

time a slightly larger deformation was applied while monitoring the force through the

ched 2.5mm. A maximum deformation of 2.5 mm was selected since the ratio

between elastic energy and dissipated energy for this case gave a sense of sufficiently

little plastic deformation (from the hysteresis loop formed by the loading/unloading

curve), a requirement since we are assuming the deformation in this experiment is solely

elastic; and also because this amount of deformation made us certain that the transition

between a bending and stretching regime would be observed.

Two different geometries (but same fixture design shown in Figure 3.4) were

used; the first one consisted of a sample with a 10 mm diameter. The larger one with a 20

mm diameter (2a) gave better repeatability therefore it was maintained throughout this

23

3.2 Viscoelastic tests

.2.1 The creep test

For this test a different fixture was needed, the requirements were to be small

could be immersed in water and most importantly one

should be able to have a complete view of the membrane profile without any visual

a similar design [23] we were left with the fixture shown in Figure

3.5, a s

Using the elastic modulus obtained in the previous section – quasistatic test – we

were a

a point load force.

e tested

virtuall

3

enough so that the whole setup

obstruction. Following

imple design that compresses the edge of the sample between two clear acrylic

cylinders allowing for a clear view of the membrane profile while it is being deformed.

As shown in Figure 3.5 by fixing the sample in the setup we obtained the 20 mm

circular membrane. Then by laying a stainless steel sphere in the freestanding membrane,

we had a stable load to be applied for the duration of the test.

ble to select a stainless steel sphere of 4.5mm diameter was selected since its

weight of 433 mg (4.25 mN) was expected to deform the membrane sufficiently without

being so massive that it would had ruled out the assumption of

As for the experiment itself the temperature was monitored though not controlled;

observing a very modest change in temperature of only a few degrees Celsius throughout

the duration of the test. The average temperature was 24 degrees Celsius, with a

maximum and minimum of 26 and 22 degrees respectively. All three samples wer

y simultaneously and in the same tank to assure equal conditions that would allow

an objective comparison.

24

The deformation was obtained through digital pictures taken by a home

assembled “long focal microscope”, the instrument itself is shown in Figure 3.6 and its

specifications are in Table 3.2. This kind of setup was first used by Liu to determine the

mechanical properties of thin elastomeric membranes [51]. Pictures were taken each

minute for the first ten minutes, each ten minutes for the following 50 minutes and then at

ramdom intervals within the one month period. A sample picture is shown in Figure 3.7,

using a geometric reference the deformation was calculated by measuring the pixels in an

image-manipulation software (i.e. Adobe Photoshop) and comparing to the

aforementioned reference.

Table 3.2. Creep setup specifications

Focal distance 90 mm

Displacement resolution 0.03 mm

The actual force applied to the sample was determined the following way. First

the stainless steel robalance (OHAU P214CN). Then

e effects of buoyancy were taken into account (as shown in Figure 3.9), leaving a

resultant force of 3.78 mN.

4.25 0.47 3.78 (3.3)

Where is the buoyancy force, obtained by multiplying gravity ( ), the sp

volume ( ) and the density of water ( ); is the weight of the sphere and is

the reaction of the membrane which is equal in magnitude to the resultant force ( ).

ball was weighed in a mic S, model E

th

0 (3.1)

(3.2)

| |

here

25

3.2.2 The cyclic loading test

y effe s th

e rate

while the tests were performed. The experiment consisted of a rapid pre-travel, where an

initial deformation of 1.5 mm was applied to the sample in order to increase the

magnit

Texture Technologies’® TA.XT Texture Analyzer (Figure 3.10) was used to

perform this test; its specifications are shown in Table 3.3. The reason why the test was

performed at the highest possible frequency is because any lower frequency, as close as 8

Hz, resulted in a large noise-to-signal ratio.

Table 3.3. TA.XT technical specifications

For the cyclic loading we used the same fixture used in previous experiments but

this time to avoid any complex buoyanc ct e sample was not immersed in water.

Because the length of the test was considerably short, the samples did not d hyd

ude of the forces measured in the following step, while performing the

oscillations. The latter were a series of oscillations of 0.5 mm at 10 Hz or 62.83 rad/s, as

shown in Figure 3.9, selected since it gave a sensed force range that was in concordance

with the instrument’s force resolution and the frequency was the highest possible with the

former.

Maximum load 1 Kg

Load resolution ~0.1mN

Distance capacity 0.1 - 295 mm

Displacement resolution 1 μm

Speed 40 mm/s to m/s capacity 0.01m

26

3.3 A computational method for large deformation

Mathematica® cod

In order to solve the ordinary differential equations a Mathematica® code was

developed by Scott Julien, a member of our research group, to apply the Runge – Kutta

iterative approximation method. It basically contains a section of integration for the

tact with the spherical indenter;

rest of the membrane not in contact with the

indenter; and a final section where the applied load is calculated integrating the vertical

component of the stress at the edge, as explained before. The code was created following

the algorithm shown in Figure 3.11.

ious experiments. Thus, demonstrating which is the

best fit

e using Runge-Kutta

contact region, the section of the membrane in con

another one for the noncontact region, the

The procedure starts by holding passive the stretch ratio at the pole, and

varying the contact radius, until the value at the outer edge for the variable equals the

preset . For more details see Appendix 2, which includes the complete script.

The results of this section will allow us to compare the hyperelastic model to the

linear elastic model used in the prev

for the blister setup, and therefore emit a recommendation as to which model

should be used to predict the material behavior under stress or strain.

27

CHAPTER 4

RESULTS AND DISCUSSION

4.1 The shaft-loaded blister test

Three samples of each kind were tested and for each one of these three

onsecutive tests were performed (n=9). Overall, given the results shown in Table 4.1 and

it has been proven that the stiffness of polyvinyl alcohol hydrogel

does indeed increase by the number of freeze/thaw cycles involved in the manufacture

eze/thaw cycles used in the fabrication improve the

roduct by a creating a stiffer membrane. That said, the

number or freeze/thaw cycles that can be used has a limit. It was seeing that as the

number of cycles increased the capability of the hydrogel of baring water decreased, an

observation made by Fromageau et al [41] as well, who said that this could be a

consequence of the crystals formed inside the hydrogel structure during the freezing

phase that tended to expulse water. According to the former, samples fabricated with

more than 10 freeze/thaw cycles tended to be extremely dehydrated and could not be

considered hydrogels anymore.

c

shown in Figure 4.1

process. Thus, the number of fre

mechanical properties of the final p

Table. 4.1. Elastic modulus of polyvinyl alcohol hydrogel obtained by shaft-loaded blister tests

Average elastic modulus (KPa)

Maximum elastic modulus (KPa)

Minimum elastic modulus (KPa)

2 cycles (n=9) 180 ± 8 197 169 3 cycles (n=9) 230 ± 10 261 204 4 cycles (n=9) 630 ± 18 680 589

28

It is known that the physical properties are also dependent on possible

dehydra on carried ing a preparation, the speed of

in atures, the minim emperature and the volume of the

order xclusively on t fects caused by the ation of

freeze/thaw cycles, care was given to reproduce all the other conditions on each case.

Assuming that the material is incompressible a Poisson ratio of 0.5 was assumed

in all applicable cases. The elastic modulus was calculated by using the plot of applied

force vs. central deformation and performing a curve fitting using the equations for the

theoretical models explained in chapter two.

By looking at the sample deformation graphs (Figures 4.2 -4.4) it is noticeable

how the theoretical models fail to accurately comply with the experimental results

throughout the complete range of deformation. For this reason based on Wan’s statement

[46] that indicates that the required force to obtain a certain deformation is dependent on

the latter deformation elevated to the power ( ), where is a number between 1

and 3,

.

As shown in Table 4.1 the elastic modulus from the tested samples range from

ti during the heat t the first step of

increasing and decreas g temper um t

sample [41]. In to judge e he ef vari

we fixed the value of this power to two ( 2), as we noticed this was the slope

shown for the stretching region in our data, and varied the value of the elastic modulus E,

until the best possible fit was obtained. We show that when is equal to two the best fit

is obtained, for this reason this value was used for all the cases, thus obtaining the elastic

moduli that are reported on the present thesis

169 KPa (close to the 160 KPa average elastic modulus reported by Förster et al for the

central bovine cornea [53]) to 680 KPa (close to the 630 KPa average reported by Selzer

29

et al for a healthy common carotid artery [54]. This confirms the feasibility of using PVA

hydrogel as scaffold for different kinds of cell cultures.

s additional proof of the results

discussed above.

These results agree with a similar work done by J. Fromageau et al [41] where the

elastic modulus was calculated by ultrasound elastography in samples that differed only

by the number of freeze/thaw cycles used in their fabrication. As additional comparison,

we performed a series of tensile tests on strips of the same membranes used for the blister

tests. The results are shown in Table 4.2, and serve a

Table. 4.2. Mechanical parameters of polyvinyl alcohol hydrogel obtained by tension testing

Average Elastic modulus (KPa)

Average ultimate stress (KPa)

Average ultimate strain (mm/mm)

2 cycles 210 41 .864 3 cycles 320 141 1.643 4 cycles 575 Machine’s limits reached

4.2 Viscoelastic tests

4.2.1 The creep test

e 3.7 shows a portion of the deformation profile of a PVA hydrogel during

e creep test taken with the system microscope. The results shown herein are for one

sample of each kind. The time length of the experiment made it impossible to run more

parison. It can be seeing how it is possible to obtain a

the applied setup. Due to the opacity of the samples it was not

possible to confirm the thickness of the samples visually, for this reason we relied

Figur

th

experiments to have statics com

decent resolution with

30

comple

r models, such as the Maxwell and Voigt,

available for describing the creeping deformation. The Maxwell model predicts that the

deformation

initially finite and then increased with

time to

ol hydrogel

tely on the measurement taken by the micrometer, approximated to some degree

(with about 10% variation) because the samples were not rigid enough to stop the

micrometer knob without some compression.

Figures 4.5 through 4.7 show the strain variation over time for each respective

case. A typical viscoelastic behavior is observed in all three cases, with little variation in

the amount of creep deformation between kinds.

The standard linear solid model seemed to be a suitable viscoelastic model for the

material. When comparing to the other simple

increases linearly against time but never reaches a plateau, while the Voigt

model depict the deformation initially as zero and then gradually increasing with time.

However the deformation of our samples were

finally reach a plateau. Such behavior cannot be well described either by Maxwell

or Voigt, but the Standard Linear Solid generally provides a good description.

The calculated parameters used to obtain the best fit in each case according to the

standard linear solid are shown in Table 4.3 below. The decrease in the relaxation time as

the number freeze/thaw cycles are increased, suggests that as the number of cycles are

increased the material behaves more like an elastic solid and less like a viscous fluid.

Table. 4.3 Viscoelastic properties according to standard linear solid of polyvinyl alcohobtained by creep tests

Relaxation time - τ (min) E1 (KPa) E2 (KPa) η (MPa s )

2 cycles 22 185.32 8.48 2.675 3 cycles 15 290 32.64 0.725 4 cycles 8 706 70 7.64

31

The initial elastic deformation also confirmed the results of the quasistatic tests

with a minor offset; in the greatest case, seen in the three freeze/thaw cycles sample, the

static tests was 230 KPa, about 17% smaller.

relevan firmed with this experiment is the existence of a constant strain

al stage o formation, or plateau. This is significant because it allow us to

assure

It is believed that the use of an incubator or humidity chamber would benefit the

output

ailable.

elastic modulus (E1) was found to be 290 KPa, whereas the average elastic modulus

calculated in the quasi

A t fact con

in the fin f de

to a certain extent that when PVA hydrogel is used as substitute for cartilages or as

part of an artery stent, it will not incessantly deform under a constant load, but it will be

subject to some initial deformation that will stabilize at some point.

of the experiment as the setup would not need to be submerged in water in order to

keep the sample from dehydrating. This will allow us to get rid of the effects of

buoyancy, and thus a greater applied force would be obtained, while maintaining the

same ideal geometry. Moreover, the quality of the pictures used to obtain the deformation

would also increase and a greater displacement resolution would be av

Suspected thickness variations made it difficult for the stainless steel sphere to lie

in the exact membrane center. This may also carry a minor error in our calculations as the

ideal axisymmetric case is not strictly being applied, a fact that could very likely be

overridden with the use of a slightly greater force.

32

4.2.2 T

zero, as almost all the energy invested in the deformation is stored and restored once the

s all the energy applied in the deformation will be dissipated as heat. It is possible to

the results obtained in the creep experiments, which means that as the number of

freeze/thaw cycles involved in the fabrication are increased the material starts behaving

less like a viscous fluid. Since while using the same 10 Hz

frequen

Loss tangent

he cyclic loading test

Based on the notion that an elastic solid will show a phase angle that approaches

deformation is removed; and a viscous liquid will have a phase angle close to 90 degrees,

a

confirm

more like an elastic solid and

cy for the sinusoidal deformation waves the phase angle, which indicates the ratio

between stored and dissipated energy decreases.

Table. 4.4 Viscoelastic properties obtained from the cyclic loading test

Phase angle (degrees)

Storage modulus - E’- (KPa)

Loss modulus - E’’- (KPa)

2 cycles 75.80 ± 1 3.95 ± 0.310 28.98 ± 2.28 114.54 ± 0.50 3 cycles 71.34 ± 1 2.96 ± 0.180 48.3 ± 2.93 143.02 ± 0.82 4 cycles 60.74 ± 1 1.78 ± 0.075 218.14 ± 9.23 389.36 ± 3.75

As shown in Table 4.4, for the case of the two cycles sample we started with a

phase angle of 75.8°, and started decreasing to 71.34° for the three freeze/thaw cycles

sample, to f uted to the

increasing recoverable elastic ach , pr e

free les us rial fa

inally 60.94° for the four cycles samples. These results are attrib

force during e deformation cycle oportional to th

number of ze/thaw cyc ed in the mate brication.

33

4.3 A c

nally to observe how the

theoretical model matches with the experimental results.

Using a value of 0.1 for , in the Mooney strain-energy function, because we are

ing the material to be a linear elastic solid, the graphs obtained by Yang and Hsu

inal ones. Thus, the developed

s validated. These are shown in Figures 4.11 through 4.15.

an et al [55] for the transition

between bending and stretching for a point loaded blister. It can be seen that both models

match

uld be

attribut

omputational method for large deformation

Numerical Solutions

The results pertaining to this section will be presented as graphical plots only.

First as a comparison to validate the created code, and fi

assum

were reproduced in order to compare to the orig

Mathematica® code wa

An additional graph shown in Figure 4.16, illustrates the results obtained with this

code making use of the large deformation theory, and comparing them with the

theoretical results given by the expression used by W

closely, except on the early stage of deformation where the large deformation

model fails to comply with the behavior shown when bending predominates. Also, when

deformation is relatively large, close to thirty times the membrane thickness, there is a

“turn-around” in the results given by the large deformation code. This effect co

ed to the fact that, differing from the bending/stretching transition where the load

is applied at a single point, in large deformation this load is distributed along the contact

area of the spherical indenter and as the deformation becomes larger the membrane could

begin to wrap around the sphere; thus, giving this abnormal output.

34

CHAPTER 5

APPLICATIONS OF THE BLISTER CONFIGURATION ON IN-VITRO

5.1 On stem cell differentiation

Since the molecular chemistry revolution little attention had been given to other

factors in cell behavior and natural processes. Recent

EXPERIMENTS

advances in how

echanotransduction affects the differentiation process of stem cells have been made,

opening the perspective that was almost universally accepted, where the expression of

mical markers or indicators that would trace

that mechanical cues are sensed even faster than the time

be recognized by a cell [56]. Furthermore, on the

breakthrough experim

5.2, is composed of three parts; two upper plates that constrain the substrate membrane,

m

genes within a cell depended solely on che

the path that cells would follow into differentiation.

Now, it has been proven

it takes for a chemical signal to

entation carried out by Engler [20], three different specialized cells

(neurons, muscle cells and bone cells) were originated from a single kind of

mesenchymal stem cells only by varying the stiffness of the culture substrate matrix.

According to these experiments a soft substrate (0.1 – 1 KPa) will give rise to neurons, a

medium one (8 – 17 KPa) to muscle cells, and a stiff (25 – 40 KPa) substrate will give

rise to bone cells.

In an effort to prove the effects of the substrate stiffness, and see if in fact cells

can sense strains applied to the environment they reside in, a special fixture that makes

use of the blister configuration was designed. The fixture, illustrated in Figure 5.1 and

35

and a third plate that can have cone or a ball at the center, responsible for the membrane

deformation.

sample, and based on the referenced results different specialized cells should be expected

on a single culture or m

The fixture allows us to exert a stress that varies in the radial direction through the

embrane sample. Therefore, by knowing that the stress in the

fixture

iments are carried out.

will be reduced towards the edge from its maximum at the center, an optimistic

result could be to obtain segmented cultures of distinct differentiated cells that vary in the

radial direction. Bone cells could be expected to develop at the center, followed by

muscle cells and finally neurons at the very edge, as shown in Figure 5.3. Whether, cell

interaction tends to produce a single kind of cells is uncertain, and will remain that way

until the exper

The stress variation proportion can be seen in Figure 5.4.

5.2 On collagen liquid crystals

Collagen liquid crystals are a type of cholesteric liquid crystals, this means they

are a form of organized structure in which individual elements (fibrils) are arranged

parallel to each other in planes, and each consecutive plane is rotated a constant angle

from the previous one. Giraud-Guille explains this structure applying a twisted plywood

model shown in Figure 5.5 [57].

36

A great number of research projects are now focusing on ways of obtaining these

organized structures in the laboratory and in

same fixture, explained in the previous section, and the

latter p

ent of the remaining fibers.

d in

Figure 5.6 caused by the lack of sufficient stiffness of the digested area to resist cleavage.

using them to recreate other complex natural

materials such as the cornea and tendons [58].

In collagen the mechanical stimulus produced by a minor strain (1-10%)

according to Wyatt et al [59] causes a stiffening effect that increases the cleavage time of

the fibers. Therefore by using the

rinciple. By loading a liquid crystal membrane in the fixture as we add the

collagenases enzyme to the sample we should be left with only fibers that were submitted

to enough stress, predicting some sort of radial arrangem

There is no certain knowledge of what the results of the experiment will be, but

two possible options are that a set of arranged fibers are left, because they were the ones

submitted to sufficient stress to be subject of the stiffening needed to outlast the rest of

the fibers during enzymatic cleavage. The second option that we are able to presume,

since the stress has its maximum at the center and decreases from that point towards the

edge, is that we are left with a round portion of the original membrane as illustrate

37

CONCLUSIONS

he results of this thesis allow the statement of the following conclusions:

It has been confirmed that changes can be made to the fabrication process of

set of specific mechanical properties

within the limits of the material. Particularly we established the effects of varying the

umber of freeze/thaw cycles that are required to induce cross-linking and strengthen the

fabrication process. Finding that from the softest material tested, with

two freeze/thaw cycles, the stiffness - measured by means of the elastic modulus - was

increased over three times when compared to the stiffest sample fabricated with four

freeze/t

Both of the transient tests, creep and cyclic loading, served to demonstrate that the

variation in the number of freeze/thaw cycles also had an effect on the viscoelastic

properties of the material. It was observed that as the number of cycles was increased the

material behaved more like an ideal elastic solid, showing less creep deformation and

more efficient elastic energy storage.

is demonstrated that neither the linear elastic model nor the

hyperelastic model gave a fair fit to the experiments results. A fact that will be discussed

in the future work section.

ifferentiation and collagen cleavage experiments.

Based on the idea of simultaneously applying a range of stresses on a membrane sample,

T

polyvinyl alcohol (PVA) hydrogel, to pursue a

n

material during the

haw cycles.

Data analys

The design and fabrication of a fixture based on the shaft-loaded blister geometry

used in this thesis resulted in a setup that will give our research group and collaborators

the possibility to perform live cell d

38

of a cell culture scaffold and collagen membrane respectively, to evaluate the

physiological effects and understand the behavior of these biomaterials.

39

FUTURE WORK

This thesis focused mainly in the mechanical aspects of polyvinyl alcohol

aterial. Further work needs to be done to understand other facets of its

fabrication process, including chemical parameters such as polymer concentration,

iochemical and biological parameters as biocompatibility and required porosity; the

ope or parameters critical to successfully suit live cells and embedded collagen

networks.

As to possible improvements in the applied methodology, it would be

advantageous to develop a more extensive Mathematica® code, that requires less human

interact

ider range of frequencies.

hydrogel as a m

b

sc

ion to perform the required iterations and with focus on real data fitting. With this

in mind, it might be possible to obtain a closer fit between both theoretical and real data.

Also, the development of a method or code for Agilent’s® UTM is required so that the

cyclic loading test, carried out in Texture Technologies’ ® TA.XT, can be performed on

this instrument to take advantage of its better force resolution and ability to perform this

test on a w

Moreover, a varied number of experimental setups are being developed by our

research group; some of them have been recently fabricated by the students in

Northeastern’s Capstone project. The idea is to provide experimental results to all the

theoretical models that have been developed in the recent past, such as the pressurized

blister test, the shaft-loaded pre-stressed blister test, and contact experiments with

membranous bodies; all of which will be benefited with the outcome of this thesis and

will increase the general understanding of membrane mechanics.

40

Finally, the designed fixture needs to be put to use so the results of a first

experimental phase can be observed. Experimentation with stem cells is planned to study

the differentiation path they follow in relation to the localized strain of their substrate.

Cultivation of cartilage (i.e. articular cartilage) might also be feasible by the application

of stress in cell-seeded scaffolds, obtaining prosthesis that would self-heal and therefore

would not need to be replaced because of normal wear and tear.

The results of this first in-vitro experimentation would then be used to determine

if modi

fications to the rig are needed in response to specific problems. Some of the issues

that might arise are: material incompatibilities, appearance of bacteria, and very likely a

fine tuning of the maximum deformation will be required to guarantee the existence of

the required stress ranges that would guarantee for example the presence of more than a

single kind of differentiated cells.

41

Figure 1.1. Temperature vs. time during the freeze/thaw cycling of the samples.

42

Figure 1.2. Conventional techniques to mechanically characterize hydrogels: (a) strip extensiometry; (b) ring extensiometry; (c) compression test; (d) pressurized blister test; (e) indentation, (f) spherical ball inclusion, (g) micropipette aspiration, (h) elastography (F = Force, P = Pressure).

43

Figure 2.1. The blister configuration.

44

Figure 2.2. Transition between modes of deformation [46]. Where w: central deformation, h: membrane thickness, F: applied force, κ: flexural rigidity and a: membrane diameter.

45

Figure 2.3. The standard linear solid.

FF

η E2 

E1 

46

h is identical with the deformation of a viscous fluid obeying Newtown’s law of viscosity.

Figure 2.4. (a) Step stress, (b) response from an elastic solid, (c) response from a linear viscoelastic solid. ε1 is the immediate elastic deformation, ε2 the delayed elastic deformation and ε3 the Newtonian flow whic

47

Figure 2.5. Profile of a cyclic loading experiment, showing the phase angle, δ.

Mem

bran

e D

efle

ctio

n, w

Time x frequency, ωt

Load

,F

δ

0 ωt = π/2 ωt = π ωt = 3π/2 ωt = 2π

wo

Fo

48

Figure 2.6. Blister diagram. See Appendix 3 for nomenclature.

49

figuration the sensor is pointing upward. In order to perform the tests the spherical nt downward in for the liquid cell to be able to work without dripping the

water out.

Figure 3.1. UTM Nano. Agilent Technologies – MTS. Note the instrument is upside down, in its normal conindenter needed to poi

50

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250 300 350 400 450 500

App

lied

Forc

e (m

N)

Central Deformation (μm)

0.02 mm/min

0.1 mm/min

0.2 mm/min

1 mm/min

2 mm/min

Figure 3.2. Shaft-loaded test at different speeds, obtained with Agilent UTM.

51

Figure 3.3. Shaft-loaded test with increasing steps of deformation(0.5, 1, 1.5, 2, and 2.5 mm), obtained with Agilent UTM.

52

 

6mm

Figure 3.4. Fixture used in the shaft-loaded test. For more details see drawings in Appendix 1.

53

20 mm

12 mm

Two transparent cylinders clamped the hydrogel membrane (Ø 20mm)

while allowing visual access to monitor the deformation

Figure 3.5. Fixture used in the creep test. For more details see drawings in Appendix 1.

54

Figure 3.6.a) Creep test setup showing the long-focal microscope.

Figure 3.6.b) Creep test fixture

55

Stainless steel sphere (Ø 4.5mm)

PVA Membrane (Ø 20mm –

thickness 250μm)

Figure 3.7. Sample image used to monitor the deformation used during a creep test on a PVA hydrogel sample.

56

Figure 3.8. Blister configuration in a creep test.

57

Figure 3.9. Free body diagram of the ball used in the creep test.

Stainless steel ball

) Membrane’s reactionBuoyancy ) 

Ball weight ) 

58

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Central  de

form

ation (m

m)

time (sec)

Figure 3.9. Deformation pattern used in the cyclic loading tests (10 Hz).

59

a)

b)

Figure 3.10. a) TA.XT from Texture Technologies, shown here with long focal microscope and workstation; b) a fixture showing the blister formation, loaded in the TA.XT.

60

ALGORITHM RUNGE – KUTTA , , , ,

This algorithm computes the solution of the initial value problem at ′ , ,equidistant points

, , … , ; 2

here is such that this problem has a unique solution on the interval . ,

INPUT: Initial values , , step size , number of steps

OUTPUT: Approximation to the solution

1 , where 0, 1, … , 1

For n=0, 1, …, N-1 do:

,

,

,

2 2

End Stop End RUNGE-KUTTA

Figure 3.11. Runge-Kutta algorithm applied in the Mathematica® code [52].

61

Figure 4.1. Average elastic modulus obtained using shaft-loaded blister tests.

0

100

200

300

400

500

600

700

Ela

stic

Mod

ulus

(KPa

)

2 cycles 3 cycles 4 cycles

62

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5

App

lied

forc

e (m

N)

Central deformation (mm)

Figure 4.2. Load vs. deformation plot for a 2 freeze/thaw cycles sample. (Triangles: experimental data; Theoretical lines are the gray lines as follows, dash-dotted: large deformation; dashed: pure stretching; dotted: pure bending; solid: pure stretching variation with n=2).

63

0.01

0.1

1

0.1 1

App

lied

forc

e (m

N)

Central deformation (mm)

 

 

 

 

-0.5

0.5

1.5

2.5

3.5

4.5

5.5

0 0.5 1 1.5 2 2.5

App

lied

forc

e (m

N)

Central deformation (mm)

Figure 4.3. Load vs. deformation plot for a 3 freeze/thaw cycles sample (Triangles: experimental data; Theoretical lines are the gray lines as follows, dash-dotted: large deformation; dashed: pure stretching; dotted: pure bending; solid: pure stretching variation with n=2).

0.01

0.1

1

0.1 1

App

lied

forc

e (m

N)

Central deformation (mm)

64

 

 

-0.5

1.5

3.5

5.5

7.5

9.5

11.5

13.5

0 0.5 1 1.5 2 2.5

App

lied

forc

e (m

N)

Central deformation (mm)

Figure 4.4. Load vs. deformation plot for a 4 freeze/thaw cycles sample (Triangles: experimental data; Theoretical lines are the gray lines as follows, dash-dotted: large deformation; dashed: pure stretching; dotted: pure bending; solid: pure stretching variation with n=2).

0.01

0.1

1

10

0.1 1

App

lied

forc

e (m

N)

Central deformation (mm)

65

3.38

3.39

3.4

3.41

3.42

3.43

3.44

3.45

3.46

3.47

Central deformation (m

m)

Time (min)

PVA 2CMembrane radius (a) = 10mmMembrane thickness (h)= 100umBall radius (R) = 2.25mmBall weight = 4.25 mNActual force = 3.78 mNRelaxation time (τ) = 22 min

1 10 102 103 104 105

τ = 22 min 

Figure 4.5. Creep test on a 2 freeze/thaw cycles PVA hydrogel sample.

66

2.9

2.92

2.94

2.96

2.98

3

3.02

3.04

3.06

3.08

3.1

Central deformation (m

m)

Time (min)

PVA 3CMembrane radius (a) = 10mmMembrane thickness (h)= 100umBall radius (R) = 2.25mmBall weight = 4.25 mNActual force = 3.78 mNRelaxation time (τ) = 15 min

1 10 102 103 104 105

τ = 15 min 

Figure 4.6. Creep test on a 3 freeze/thaw cycles PVA hydrogel sample.

67

2.16

2.18

2.2

2.22

2.24

2.26

2.28

2.3

Central deformation (m

m)

Time (min)

PVA 4CMembrane radius (a) = 10mmMembrane thickness (h)= 100umBall radius (R) = 2.25mmBall weight = 4.25 mNActual force = 3.78 mNRelaxation time (τ) = 8 min

1 10 102 103 104 105

τ = 8 min 

Figure 4.7. Creep test on a 4 freeze/thaw cycles PVA hydrogel sample.

68

Figure 4.8. Cyclic loading test on a 2 freeze/thaw cycles PVA hydrogel sample.

69

Figure 4.9. Cyclic loading test on a 3 freeze/thaw cycles PVA hydrogel sample.

70

Figure 4.10. Cyclic loading test on a 4 freeze/thaw cycles PVA hydrogel sample.

71

0

1

2

3

4

5

0 1 2 3 4 5

δ/R

ρ/R

λo=1.05λo=1.10λo=1.20λo=1.40λo=1.60λo=2.00

Figure 4.11. Deformed membrane profiles for a/R=5 and stretch ratio at the outer edge λp=1, where λo is the stretch ratio at the pole.

72

0

5

10

15

0 2 4

w__  C1 h R

δ/R

Figure 4.12. Load deflection curve for a/R=5 and stretch ratio at the outer edge, λp=1.

73

0

1

2

3

0 5 10 15

T1(0)_  C1 h 

w__  C1 h R

Figure 4.13. Stress resultant at pole for a/R=5 and stretch ratio at the outer edge, λp=1.

74

0.0

0.3

0.6

0.9

1.2

1.0 1.5 2.0

ρ(rc)

λo

1.0

Figure 4.14. Radius of contact for a/R=5 and stretch ratio at the outer edge, λp=1. The plateau represents the maximum contact radius, which is that of the spherical indenter being used, and logically may not exceed this value.

75

0.0

0.2

0.4

0.6

0 2 4 6 8

δ/R

ρ/R

10

λp=1.10

λp=1.20

λp=1.60

λp=2.00

Figure 4.15. Deformed membrane profiles for a/R=5 and W/C1hR=1, where λp is the stretch ratio at the outer edge.

76

App

lied

load

φ=(

Fa^2

)/(2π

Dh)

Normalized shaft displacement Wo=wo/h

107

106

105

104

103

102

10

1

10-1

10-2

10-1 1 10 2

a)

10

Figure 4.16. a) Comparison between the deformation profile obtained with Wan’s formulation for the transition between the bending and stretching modes and the large deformation theory using the developed code. b) close up at final stage. a/R: ratio between the membrane and indenter ratio. Solid line (green): a/R=5; dash-dotted (red): a/R=20; dashed (blue): a/R=50.

App

lied

load

φ=(

Fa^2

)/(2π

Dh)

Normalized shaft displacement Wo=wo/h

106

105

104

30 40 50 60 70 80 90 100

a/R=5

b)

77

58 mm

12 mm

9 mm30 mm

a)

b)

c)

12 mm9 mm

Figure 5.1. Fixture built for cell differentiation and collagen cleavage experiments. Either ball (a) or a cone is used to induce the deformation [b) exploded, c) assembled].

78

Figure 5.2. One of the fixtures fabricated for the cell differentiation experiments.

79

Figure 5.3. Proposed stem cell differentiation experiment. A possible outcome is shown, where there are three segmented areas of specializes cells: a) Bone cells in the area of highest stress; b) muscle cells in the area of medium stress; and c) neuron in the area with the lowest stress.

80

Where /

; /

⁄;

Figure 5.4. Radial and tangential membrane stress variation along the radial direction. Extracted from Wan et al. [55]

81

Figure 5.5. Twisted plywood configuration of collagenous liquid crystals, as proposed by Giraud-Guille [57].

82

Figure 5.6. Proposed collagen cleavage experiment. Two of the possible results are shown: a) the membrane diameter remains, but only the fibers submitted to sufficient strain survive the enzymatic cleavage (thus exhibiting an organized arrangement); b) all fibers remain but all the collagen at the low stress are gets digested by the enzyme.

83

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APPENDIX 1

1.40

1.05 .125

.165

.63

.45

.95

.25

.63

2.36

1.56

.12

.15

.03

.25

R [USE .8 DRILL BIT].40

SHEET 1 OF 1UNITS INCHESSCALE 1.000A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME CELL_20

DATEENG APPR01

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L LIQUID CELLDATECUST APPR

19-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY OR THE AUTHOR, IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA NONE NONE

1

2

A

B

C

D

1

2

D

C

B

A

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

1.40

.32

.125

.63

.63

1.56

1.56

.24

.13

SHEET 1 OF 1UNITS INCHESSCALE 1.500A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME CAP_20

DATEENG APPR02

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L LIQUID CELL CAPDATECUST APPR

19-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA NONE NONE

1

2

A

B

C

D

1

2

D

C

B

A

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

SHEET 1 OF 1UNITS INCHESSCALE 1.500A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME ASSY_DOWN

DATEENG APPR03

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L

CREEP FIXTURELOWER ASSEMBLYDATECUST APPR

19-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

RING

LOW OUTER CYLINDER

LOW INNER CYLINDER

LOWER PLATE

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

.75

.875

.33

SHEET 1 OF 1UNITS INCHESSCALE 2.000A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME BOTTOM_IN_TUBE

DATEENG APPR04

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L LOW INNER CYLINDERDATECUST APPR

19-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

1.125

1.25

.43

SHEET 1 OF 1UNITS INCHESSCALE 1.000A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME BOTTOM_OUT_TUBE

DATEENG APPR05

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L LOW OUTER CYLINDERDATECUST APPR

19-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

.975

1.125

.13

SHEET 1 OF 1UNITS INCHESSCALE 2.000A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME RING

DATEENG APPR06

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L RINGDATECUST APPR

19-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

1.26

.885

.75

1.50

.17 .02

.04

1.50

.57

SHEET 1 OF 1UNITS INCHESSCALE 1.000A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME BOTTOM

DATEENG APPR07

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L LOWER PLATEDATECUST APPR

19-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

6-32 UNC - 2b TAP THRU #36 DRILL ( 0.110 ) THRU -( 4 ) HOLE

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

1.50 1.50

.27

.15

.57

.14

.75

.875

SHEET 1 OF 1UNITS INCHESSCALE 1.500A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME ASSY_UP

DATEENG APPR08

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L

CREEP FIXTUREUPPER ASSEMBLYDATECUST APPR

19-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

R15

75 75

1603

58.47

24.04

SHEET 1 OF 1UNITS MILIMTSSCALE 0.750A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME CONE_BASE

DATEENG APPR09

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L CONE PLATEDATECUST APPR

22-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

4-40 UNC - 2b TAP THRU #43 DRILL ( 2.260 ) THRU -( 4 ) HOLE

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

3

45

R15

110

0.5

75

3

34

36.5

39.5

31

SHEET 1 OF 1UNITS MILIMTSSCALE 0.750A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME BOTTOM

DATEENG APPR010

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L MIDDLE PLATEDATECUST APPR

22-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

4-40 UNC - 2b CLEAR #32 DRILL ( 2.950 ) THRU -( 4 ) HOLE

4-40 UNC - 2b TAP THRU #43 DRILL ( 2.260 ) THRU -( 4 ) HOLE

SIZE

DRAWN BY DATE TITLE

DWG NO. REV

38.5

R15

36.48

2.95

34

31

20

21

75

SHEET 1 OF 1UNITS MILIMTSSCALE 0.750A

DATEMFG APPR

Non Machined Rolled,Cast,Forged

Angle = 0.55-30 = 0.430-120 = 0.5120-400 = 1.0400-1000 = 2.0M

ETR

IC PART FILE NAME TOP

DATEENG APPR011

Machined Ra Ra = 63 uin 1.6 um

Angle = 0.5Fraction = 1/320.X = 0.10.XX = 0.010.XXX = 0.005IM

PE

RIA

L UPPER PLATEDATECUST APPR

23-Jun-09EDGAR MONTIEL

SURFACE FINISHUnless otherwise

specified

TOLERANCESUnless otherwise

specified

DEPARTMENT OF MECHANICALAND INDUSTRIAL ENGINEERING

NORTHEASTERN UNIVERSITY

THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING. ANY REPRODUCTION IN PART OR IN WHOLE WITHOUT THE WRITTEN PERMISSION OF THE DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NORTHEASTERN UNVERSITY, OR THE AUTHOR IS PROHIBITED.

Third Angle Projection

REVISION HISTORY

APPRDATEBYDESCRIPTIONREV

MATERIAL CONDITION FINISH

PMMA

1

2

A

B

C

D

1

2

D

C

B

A

APPENDIX 2

H∗GOVERNING EQUATIONS∗LClear@F, x, y, t, r, αD;

F@x_, y_, t_, r_D =

1

r y

t Ix Iy2 t4 − 3M − t Iy4 t2 − 3M + α It3 Iy4 t2 + 1M − x y2 Iy2 t4 + 1MMM ë IIy4 t2 + 3M I1 + α t2MM;

G@x_, y_, t_, r_D =1

r×Iy2 − x2M Iy2 t4 − 1M I1 + α y2M

t Iy4 t2 − 1M I1 + α t2M ;

H@x_, y_, t_, r_D =x

y F@x, y, t, rD + G@x, y, t, rD;

J@x_, y_, t_, r_D =1

r Hx − tL;

H∗MATERIAL PROPERTIES HRatio of Mooney−Rivlin parameters: α = C2êC1L∗Lα = 0.1;H∗GEOMETRIC PARAMETERS∗LH∗ Normalized quantities, normalization constant = R, radius of sphere ∗LH∗ rc = contact radius in the undeformed plane, a = undeformed film radius∗LH∗ Keep changing rc until the last t@100D = λp ∗LH∗ Here,

λp = 1 Hi.e. last entry of t columnL. Calculation possible for λp = other numbers ∗LH∗ Note that there is no input for λp in this code ∗LH∗ All input variables: α, rc, a, λ0, and λp ∗LH∗ Can calculate weight of sphere at the end ∗Lrc = @% required input %D;

a = 5 @% diameter ratio %D;H∗INITIAL CONDITIONS∗LH∗ Stretch ratio at apex, stress should be equi−biaxial at this point∗Lλ0 = @% required input %D;

y@0D = λ0;

ξ@0D = 0;

r@0D = 0;

x@0D = λ0;

t@0D = λ0;

irxyttbl = 880, r@0D, x@0D, y@0D, t@0D<<;H∗INTEGRATION OVER CONTACT REGION FOR x, y, and t∗Ln = 10; H∗number of steps∗Lhc =

rc

nêê N; H∗step size∗L

h = hc;H∗∗Integration over first step∗∗LH∗ ki are the Runge−Kutta constants∗Li = 0;

yk1@iD = y@iD;

ξk1@iD = ξ@iD;

rk1@iD = r@iD;

xk1@iD = x@iD;

tk1@iD = t@iD;

k1y@iD = 0;

k1ξ@iD = yk1@iD;

yk2@iD = y@iD +h

2 k1y@iD;

ξk2@iD = ξ@iD +h

2 k1ξ@iD;

rk2@iD = r@iD +h

2;

xk2@iD = yk2@iD Cos@ξk2@iDD;

tk2@iD =Sin@ξk2@iDD

rk2@iD ;

k2y@iD = F@xk2@iD, yk2@iD, tk2@iD, rk2@iDD;

k2ξ@iD = yk2@iD;

yk3@iD = y@iD +h

2 k2y@iD;

ξk3@iD = ξ@iD +h

2 k2ξ@iD;

rk3@iD = r@iD +h

2;

xk3@iD = yk3@iD Cos@ξk3@iDD;

tk3@iD =Sin@ξk3@iDD

rk3@iD ;

k3y@iD = F@xk3@iD, yk3@iD, tk3@iD, rk3@iDD;

k3ξ@iD = yk3@iD;

yk4@iD = y@iD + h k3y@iD;

ξk4@iD = ξ@iD + h k3ξ@iD;

rk4@iD = r@iD + h;

xk4@iD = yk4@iD Cos@ξk4@iDD;

tk4@iD =Sin@ξk4@iDD

rk4@iD ;

k4y@iD = F@xk4@iD, yk4@iD, tk4@iD, rk4@iDD;

k4ξ@iD = yk4@iD;

y@i + 1D = y@iD +h

6 Hk1y@iD + 2 k2y@iD + 2 k3y@iD + k4y@iDL;

ξ@i + 1D = ξ@iD +h

6 Hk1ξ@iD + 2 k2ξ@iD + 2 k3ξ@iD + k4ξ@iDL;

r@i + 1D = r@iD + h;

x@i + 1D = y@i + 1D Cos@ξ@i + 1DD;

t@i + 1D =Sin@ξ@i + 1DD

r@i + 1D ;

irxyttbl = Append@irxyttbl, 8i + 1, r@i + 1D, x@i + 1D, y@i + 1D, t@i + 1D<D;

H∗∗Integration over remaining steps∗∗LClear@iD;

DoByk1@iD = y@iD;

ξk1@iD = ξ@iD;

rk1@iD = r@iD;

xk1@iD = x@iD;

tk1@iD = t@iD;

k1y@iD = F@xk1@iD, yk1@iD, tk1@iD, rk1@iDD;

k1ξ@iD = yk1@iD;

yk2@iD = y@iD +h

2 k1y@iD ê. λ0 → λ0v;

ξk2@iD = ξ@iD +h

2 k1ξ@iD;

rk2@iD = r@iD +h

2ê. r0 → 0;

xk2@iD = yk2@iD Cos@ξk2@iDD;

tk2@iD =Sin@ξk2@iDD

rk2@iD ;

k2y@iD = F@xk2@iD, yk2@iD, tk2@iD, rk2@iDD;

k2ξ@iD = yk2@iD;

yk3@iD = y@iD +h

2 k2y@iD;

ξk3@iD = ξ@iD +h

2 k2ξ@iD;

rk3@iD = r@iD +h

2ê. r0 → 0;

xk3@iD = yk3@iD Cos@ξk3@iDD;

tk3@iD =Sin@ξk3@iDD

rk3@iD ;

k3y@iD = F@xk3@iD, yk3@iD, tk3@iD, rk3@iDD;

k3ξ@iD = yk3@iD;

yk4@iD = y@iD + h k3y@iD;

ξk4@iD = ξ@iD + h k3ξ@iD;

rk4@iD = r@iD + h ê. r0 → 0;

xk4@iD = yk4@iD Cos@ξk4@iDD;

tk4@iD =Sin@ξk4@iDD

rk4@iD ;

k4y@iD = F@xk4@iD, yk4@iD, tk4@iD, rk4@iDD;

k4ξ@iD = yk4@iD;

y@i + 1D = y@iD +h

6 Hk1y@iD + 2 k2y@iD + 2 k3y@iD + k4y@iDL;

ξ@i + 1D = ξ@iD +h

6 Hk1ξ@iD + 2 k2ξ@iD + 2 k3ξ@iD + k4ξ@iDL;

r@i + 1D = r@iD + h ê. r0 → 0;

x@i + 1D = y@i + 1D Cos@ξ@i + 1DD;

t@i + 1D =Sin@ξ@i + 1DD

r@i + 1D ;

H∗ "Append": add to end of table∗Lirxyttbl = Append@irxyttbl, 8i + 1, r@i + 1D, x@i + 1D, y@i + 1D, t@i + 1D<D,8i, 1, n − 1<F;H∗ End of "Do" loop∗L

H∗INTEGRATION OVER NON−CONTACT REGION∗Lm = 90; H∗number of steps∗Lhnc =

a − rc

mêê N; H∗step size∗L

h = hnc;H∗∗Integration over all steps∗∗LClear@iD;

DoBxk1@iD = x@iD;

yk1@iD = y@iD;

tk1@iD = t@iD;

rk1@iD = r@iD;

k1x@iD = H@xk1@iD, yk1@iD, tk1@iD, rk1@iDD;

k1y@iD = F@xk1@iD, yk1@iD, tk1@iD, rk1@iDD;

k1t@iD = J@xk1@iD, yk1@iD, tk1@iD, rk1@iDD;

xk2@iD = x@iD +h

2 k1x@iD;

yk2@iD = y@iD +h

2 k1y@iD;

tk2@iD = t@iD +h

2 k1t@iD;

rk2@iD = r@iD +h

2;

k2x@iD = H@xk2@iD, yk2@iD, tk2@iD, rk2@iDD;

k2y@iD = F@xk2@iD, yk2@iD, tk2@iD, rk2@iDD;

k2t@iD = J@xk2@iD, yk2@iD, tk2@iD, rk2@iDD;

xk3@iD = x@iD +h

2 k2x@iD;

yk3@iD = y@iD +h

2 k2y@iD;

tk3@iD = t@iD +h

2 k2t@iD;

rk3@iD = r@iD +h

2;

k3x@iD = H@xk3@iD, yk3@iD, tk3@iD, rk3@iDD;

k3y@iD = F@xk3@iD, yk3@iD, tk3@iD, rk3@iDD;

k3t@iD = J@xk3@iD, yk3@iD, tk3@iD, rk3@iDD;

xk4@iD = x@iD + h k3x@iD;

yk4@iD = y@iD + h k3y@iD;

tk4@iD = t@iD + h k3t@iD;

rk4@iD = r@iD + h;

k4x@iD = H@xk4@iD, yk4@iD, tk4@iD, rk4@iDD;

k4y@iD = F@xk4@iD, yk4@iD, tk4@iD, rk4@iDD;

k4t@iD = J@xk4@iD, yk4@iD, tk4@iD, rk4@iDD;

x@i + 1D = x@iD +h

6 Hk1x@iD + 2 k2x@iD + 2 k3x@iD + k4x@iDL;

y@i + 1D = y@iD +h

6 Hk1y@iD + 2 k2y@iD + 2 k3y@iD + k4y@iDL;

t@i + 1D = t@iD +h

6 Hk1t@iD + 2 k2t@iD + 2 k3t@iD + k4t@iDL;

r@i + 1D = r@iD + h;

irxyttbl = Append@irxyttbl, 8i + 1, r@i + 1D, x@i + 1D, y@i + 1D, t@i + 1D<D;

,8i, n, m + n − 1<F;

H∗COMPUTATION OF PROFILE POINTS∗LH∗∗POINT−TO−POINT CHANGES IN TRANSVERSE DISPLACEMENT Hout of planeL, ∆δ ∗∗LH∗ c − contact region, and nc − non−contact region ∗L∆δc@i_D =

hc

2 Iy@iD2 − x@iD2M 1

2 + Iy@i + 1D2 − x@i + 1D2M 1

2 ;

∆δnc@i_D =hnc

2 Iy@iD2 − x@iD2M 1

2 + Iy@i + 1D2 − x@i + 1D2M 1

2 ;

Clear@iD;H∗∗∗Within contact region∗∗∗LDo@

irxyttbl@@i + 1DD = Append@irxyttbl@@i + 1DD, ∆δc@iDD,8i, 0, n − 1<D;

Clear@iD;H∗∗∗Within non−contact region∗∗∗LDo@

irxyttbl@@i + 1DD = Append@irxyttbl@@i + 1DD, ∆δnc@iDD,8i, n, n + m − 1<D;H∗∗∗At r=a∗∗∗L

i = n + m;

irxyttbl@@i + 1DD = Append@irxyttbl@@i + 1DD, 0D;

irxyttbl = Prepend@irxyttbl, 8"i", "r", "x", "y", "t", "∆δ"<D;

irxyttbl êê MatrixFormH∗∗TOTAL TRANSVERSE DISPLACEMENT AT EACH POINT, δ ∗∗LDo@

irxyttbl@@kDD = Append@irxyttbl@@kDD, Sum@irxyttbl@@j, 6DD, 8j, k, Length@irxyttblD<DD;

,8k, 2, Length@irxyttblD<D;

irxyttbl@@1DD = Append@irxyttbl@@1DD, "δ"D;

irxyttbl êê MatrixFormH∗∗RADIAL DISPLACEMENT AT EACH POINT, ρ ∗LDo@

irxyttbl@@kDD = Append@irxyttbl@@kDD, irxyttbl@@k, 2DD irxyttbl@@k, 5DDD;

,8k, 2, Length@irxyttblD<D;

irxyttbl@@1DD = Append@irxyttbl@@1DD, "ρ"D;

irxyttbl êê MatrixForm

H∗ Last entry of the above table should be equal to a = 5 ∗LH∗PLOTTING PROFILE∗LH∗∗Building a table of profile points∗∗Lδtbl = irxyttbl@@All, 7DD;

ρtbl = irxyttbl@@All, 8DD;

ρδtbl = 8Prepend@8δtbl@@1DD<, ρtbl@@1DDD<;

Clear@iD;

Do@ρδtbl = Append@ρδtbl, Prepend@8δtbl@@iDD<, ρtbl@@iDDDD;

,8i, 2, Length@δtblD<D;

ρδtbl êê MatrixFormH∗∗Plotting points∗∗LListPlot@Take@ρδtbl, 82, Length@ρδtblD<, 81, 2<DD;H∗w = WEIGHT OF BALL−BEARING∗Lxa = irxyttbl@@Length@irxyttblD, 3DD;

ya = irxyttbl@@Length@irxyttblD, 4DD;

ta = irxyttbl@@Length@irxyttblD, 5DD;

ρa = irxyttbl@@Length@irxyttblD, 8DD;

w == 4 π ρa I1 + α ta2M ya

ta−

1

ya3 ta3 1 −

xa2

ya2

1

2

APPENDIX 3

  

Appendix 3: Large deformation theory – Nomenclature

Distance from the point to the axis of symmetry

Meridian arc length between the center of the membrane and the point

Meridian stretch ratio

Circumferential stretch ratio

Meridian stress resultant per unit edge

Circumferential stress resultant per unit edge

  Principal curvature in the meridian direction

Pri n   ncipal curvature in the circumferential directio

, , Variables for the symbolic manipulation

, Strain invariants

Un ss -deformed thickne

Mooney strain energy

, Material constants

Material c ⁄ onstants ratio

dius Dimensionless ra

Dimensionless distance from the point to the axis of symmetry

Dimensionless meridian arc length betw embrane and een the center of the m

the point

, Symbolic manipulation functions

Pressure

Stretch ratio at the pole

Weight or load applied to the membrane

Stretch ratio for a range of values

Axial deformation

111  

CURRICULUM VITAE

EDGAR JOSÉ MONTIEL-RUBIO

EDUCATION

Master of Science in Mechanical Engineering Northeastern University, June 2009 Boston, MA. United States of America Bachelor of Science in Mechanical Engineering Universidad del Zulia, March 2007 Maracaibo, Estado Zulia. Venezuela

PROFESSIONAL EXPERIENCE

Research Assistant. January 2008 to June 2009 Department of Mechanical and Industrial Engineering. Northeastern University Boston, MA. United States of America Applications Engineer. August 2006 to August 2007 Engineering Department. Petrotex Maracaibo, Estado Zulia. Venezuela

PUBLICATION

“Mechanical Characterization of a Freestanding Polyvinyl Alcohol Hydrogel Membrane” IMECE2008-67446 Proceedings of the ASME International Mechanical Engineering Congress and Exposition 2008 Boston, Massachusetts, United States of America

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