Material Characterization, Constitutive Modeling and Finite Element Simulation of Polymethyl Methacrylate (PMMA) for Applications in Hot Embossing

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Material Characterization, Constitutive Modeling and Finite Element Simulation of Polymethyl methacrylate (PMMA) for Applications in Hot Embossing

Doctoral Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Kamakshi Singh, B.E.

Graduate Program in Mechanical Engineering

*****

The Ohio State University 2011

Dissertation Committee:

Rebecca B. Dupaix, Advisor Jose M. Castro Amos Gilat Allen Yi

ABSTRACT

Polymethyl methacrylate (PMMA) is an amorphous thermoplastic used in various industrial applications. PMMA is compatible with human tissues and allows high resolution features to be embossed onto a surface, thus making it highly desirable for use in bio-medical, micro-optics, micro-fluidic devices, electronics, micro-electro-mechanical systems (MEMS), etc. The processes used to fabricate these devices capitalize on the fact that the mechanical behaviour of the polymers changes drastically around the glass transition temperature (Tg). The polymer is deformed at temperatures above the Tg where the material is more fluid-like and then cooled below the Tg where it behaves more like a solid. The changes in physical properties make this temperature regime highly favourable for these warm-temperature deformation processes. The same rationale also makes it more difficult to develop a continuum model which accurately predicts the polymer behaviour with temperature and strain rate dependence across the glass transition temperature. Most of the existing constitutive models do not achieve this task; they either work below or above glass transition, but not in both these regions. Hence, there is a greater need to develop a constitutive model for the polymer that can capture the material behaviour across the glass transition temperature (Tg - 20 to Tg + 60) relevant for hot

embossing applications.

The aim of this thesis is to develop such a material model for PMMA. First, material characterizations experiments were conducted on PMMA well across its glass transition temperature (Tg). This experimental data along with the existing data in the Dupaix lab was used in developing the material model. In order to develop the new material model for application in hot embossing that will work across the wide range of temperature and strain rates, two existing constitutive models on the polymer PMMA were studied: the

Dupaix-Boyce model and the Dooling-Buckley-Rostami-Zahlan model. From the aforementioned study, a new continuum model was developed to capture the mechanical behavior over a wider range of temperature across the glass transition. Experimental data was also collected from hot embossing experiments on the polymer PMMA across its glass transition temperature. This was done to better understand the process conditions of hot embossing and thus identify the vital parameters essential that the new developed material model must be able to capture. Finally, hot embossing simulations were performed on ABAQUS using the new material model. These results were used to validate the new material model. The new model worked extremely well for large strain deformations capturing the strain rate and temperature dependence, as well as stress relaxation of the material. The model was less accurate in capturing stress relaxation for small strain deformations. The strengths and weaknesses of the current model are discussed for future work improving the constitutive model.

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DEDICATION

To my parents, Prof. H. B. Singh and Mrs. Usha Singh

ACKNOWLEDGEMENTS

First of all, I would to thank my advisor Dr. Rebecca B. Dupaix for being the most wonderful advisor I could possibly imagine. Her guidance, encouragement and support throughout my graduate school has made my stay at The Ohio State University a very fulfilling experience. I genuinely appreciate the fact that she managed to find time to have our weekly meetings and discussions even with her new triplets.

I would also like to thank my committee members Prof. Amos Gilat, Prof. Jose M. Castro and Dr. Allen Yi for being part of my committee and giving their invaluable time and suggestions. In addition, I wish to thank Prof. Gary L. Kinzel, Assistant Chair of the

Department of Mechanical Engineering and Ms. Judith Ann Brown, Graduate Program

Coordinator of Department of Chemistry for giving me an opportunity to be a teaching assistant (TA) in their respective departments. Being a TA was a very enriching experience and helped me grow as both a student and a person.

I wish to thank my lab mates Arindam, Greg and Bill who’s work I followed; Guru for teaching me how to conduct experiments in our lab and for being a good friend; Parth,

Tom, Nimet, Srinath, Sarah, Ann, Jason, Sushma and all my friends and colleagues at

OSU. In particular, I am grateful to have known Venuka, Yashas and Shubham, they have been more like a family than friends.

I believe I am blessed indeed to have a family like mine, my parents have always supported me and have had faith and trust in my judgment; irrespective of their opinions on what they considered was the right thing for me. My sister, Dr. Namrata Singh, has been a pillar of strength in my life. She is my best friend, confidant, guardian and everything.

Last but not the least; I would like to thank my husband Dr. Navneet Singh for bearing with all my frustrations and complains regarding graduate school and otherwise , and for making me believe that I am a better person than I actually am. I am really, really looking forward to finishing graduate school and spending all our future times together.

And finally, no more travelling over the weekends!

VITA

January 1984…….……….……………………………………………BORN: Basti, India July 2005 – January 2006…………………………………………….….....Inplant Trainee Larsen and Toubro Limited Mumbai, India August 2002 - June 2006……….……………………………...…Bachelor of Engineering University of Mumbai Mumbai, India January 2007 – June 2007……………………….…………...Graduate Teaching Assistant Department of Chemistry The Ohio State University, Columbus September 2007 – June 2008………….……………………..Graduate Teaching Assistant Department of Mechanical Engineering The Ohio State University, Columbus July 2008 – March 2011…..…….…………………….……..Graduate Research Assistant Department of Mechanical Engineering The Ohio State University, Columbus

FIELDS OF STUDY Major Field: Mechanical Engineering

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TABLE OF CONTENTS

Page

Abstract ………………………………………………………………………………... ii

Dedication ……………………………………………...…………………………...... iv

Acknowledgements …………………………………...……………………………….. v

Vita …………………………………………………………………………………….. vii

List of Tables ……………………………………………………………………….…. xii

List of Figures …………………………………………………………………………. xiii

Chapter 1: Introduction ……………………………………...………………………… 1

1.1 Introduction ………………………………………...……………………… 1

1.2 Summary of Work …………………………………………………………. 3

Chapter 2: Literature Review ………………………………………………………….. 5

2.1 Experiments ……………………………………………………………..… 5

2.2 Constitutive Modeling ….…………………………………………………. 11

2.3 Hot Embossing Experiments ………………………………………………. 26

2.4 Hot Embossing Simulation ………………………………………………... 31

Chapter 3: Compression Experiments ………………………………………………… 34

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3.1 Introduction ……………………………………………………………….. 34

3.2 Material ……………………………………………………………………. 35

3.3 Experimental Set-up and Procedure ……………………………………….. 36

3.3.1 Experimental Set-up .……………………………………………. 36

3.3.2 Test Procedure …………………………………………………... 38

3.3.3 Limitations of plane strain compression fixture ………………… 40

3.3.4 Experimental Errors ……………………………………………... 41

3.4 Experimental Results ……………………………………………………… 45

3.5 Discussion …………………………………………………………………. 56

Chapter 4: Existing Constitutive Model ………………………………………………. 59

4.1 Introduction ……………………………………………………………….. 59

4.2 Dupaix-Boyce (DB) Model ……………………………………………..… 60

4.2.1 Details of Constitutive Model …………………………………… 60

4.2.2 Optimization of Material Constants …………………………….. 67

4.2.3 Comparison with Experimental Data ……………………………. 69

4.3 Dooling-Buckley-Rostami-Zahlan (DBRZ) Model ………………………. 73

4.3.1 Details of Constitutive Model …………………………………… 73

4.3.2 Replication of the DBRZ Model ………………………………… 79

4.3.3 Optimization of Material Constants …………………………….. 80

4.3.4 Comparison with Experimental Data ……………………………. 82

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4.4 Discussion …………………………….…………………………………… 87

Chapter 5: New Constitutive Model …………………………………………………... 90

5.1 Introduction …………………………………………………………….…. 90

5.2 Modifications to Dupaix-Boyce Model …………………………………… 91

5.2.1 Additional Molecular Relaxation Network ……………………… 91

5.2.2 Additional Transition Slope Constant …………………………… 95

5.2.3 New Model ………………………………………………………. 98

5.3 Results …………………………………………………...………………… 98

5.4 Discussion …………………………………………………………………. 106

Chapter 6: Hot Embossing Experiments ………………………………………….…… 110

6.1 Introduction ……………………………………………………………….. 110

6.2 Experimental Details – Material and Test Set-up …………………….…… 113

6.3 Experimental Results and Discussion ……………………………….…….. 115

6.4 Conclusion ……………………………………………………………….... 131

Chapter 7: Hot Embossing Simulations ………………………………….……………. 134

7.1 Introduction ……………………………………………………………….. 134

7.2 Finite Element Model ……………………………………………………... 136

7.3 Hot Embossing Simulations ……………………………………………….. 139

7.4 Analysis of New Model ...…………………………………………………. 148

7.5 Suggestions for New Model ……………………………………………..… 156

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7.6 Conclusion ………………………………………………………………… 161

Chapter 8: Conclusion & Future Work ………………………………….…………….. 163

8.1 Conclusion ……………………………………………………………….... 163

8.2 Future Work ……………………………………………………………….. 166

References ……………………………………………………………………………... 168

LIST OF TABLES

Page

Table 3.1 Previous uniaxial experiments conducted on PMMA by Palm (bullet) and

Ghatak (shaded area) ...... 46

Table 3.2 Plane strain experiments conducted by Palm (bullet) and also the new data

(shaded area) obtained in this work ...... 47

Table 4.1 Optimized material constants for the Dupaix-Boyce model using Nelder-

Mead Algorithm …………………………………………………………………… ...... 68

Table 4.2 Compression experimental data used for optimization of DB Model ……… 69

Table 4.3 Compression experimental data used for optimization of DBRZ Model …... 81

Table 4.4 Optimized material constants of the DBRZ model for the experimental data from work by Dupaix lab. Changed values are shown in bold ………………………... 83

Table 5.1 Compression experimental data used for optimization of new molecular relaxation constants in the New Model ………………………………………………... 94

Table 5.2 Material constants for the New Model ……………………………………... 99

Table 7.1 Logarithmic strain for the hot embossing simulation shown in figure 7.30 ... 162

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LIST OF FIGURES

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Figure 1.1 Temperature dependence of shear modulus ...... 2

Figure 2.1 Chemical Structure of the polymer PMMA ……………………………….. 5

Figure 2.2 Change in the values of modulus of the material across glassy-rubbery- melt region (Tg – Glass transition temperature, and Tm – Melting temperature) …….. 7

Figure 2.3 Generalized stress-strain curve for a polymer above its Tg .……………….. 9

Figure 2.4 Schematic of constitutive model by Haward and Thackray ………………. 13

Figure 2.5 Polymer deformation model by Argon, Boyce and co-workers …………… 14

Figure 2.6 Schematic of the eight chain model by Arruda and Boyce (a) undeformed,

(b) stretched in uniaxial compression, and (c) stretched in plane strain compression … 15

Figure 2.7 Schematic showing the stress-strain behavior of polymer with characteristic features ………………………………………………………………….. 16

Figure 2.8 Schematic of the constitutive model by Boyce, Socrate and Llana ……….. 17

Figure 2.9 Schematic of the constitutive model by Dupaix and Boyce ……………….. 18

Figure 2.10 Schematic representation of the modified Cooperation model by Ahzi and co-workers ……………………………………………………………………………... 19

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Figure 2.11 Schematic representation of the continuum model by Dooling, Buckley,

Rostami and Zahlan …………………………………………………………………… 21

Figure 2.12 A schematic representation of the originally proposed constitutive model by Anand and co-workers ……………………………………………………………... 23

Figure 2.13 A schematic showing the network contributions towards the total stress,

(a) below Tg and, (b) above Tg ……………………………………………………….... 24

Figure 2.14 A schematic representation of the current constitutive model by Anand and co-workers ………………………………………………………………………… 25

Figure 2.15 Schematic of hot embossing technique ...... 27

Figure 2.16 A schematic of the two-station hot embossing process ...... 29

Figure 2.17 Temperature and pressure profile with time for the mold ...... 30

Figure 2.18 Deformation profile of the polymer with time from onset of filling to almost fully filled cavity ...... 33

Figure 2.19 Schematic of bottom-up and lateral filling of cavities ...... 33

Figure 3.1 Polymerization of monomer MMA to form the polymer PMMA along with the chemical structures of the molecules ...... 35

Figure 3.2 Instron 5869 with High temperature environmental chamber ...... 37

Figure 3.3 Channel and die fixture for plane strain experiments ...... 38

Figure 3.4 Dessicator to store samples prior to testing ………………………………... 39

Figure 3.5 Stress-strain plot at 120 °C and strain rate of 4.0/min of a Sample 1 ...... 43

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Figure 3.6 Stress-strain plot at 120 °C and strain rate of 4.0/min of a Sample 2 and

Sample 3 ...... 44

Figure 3.7 Deformed and undeformed samples in plane strain and uniaxial compression ...... 48

Figure 3.8 A schematic of uniaxial and plane strain compression samples with load, flow and constrained direction ...... 49

Figure 3.9 Plane strain compression experimental data for PMMA at 120 °C ...... 50

Figure 3.10 Plane strain compression experimental data for PMMA at 130 °C ...... 50

Figure 3.11 Plane strain compression experimental data for PMMA at strain rate of

1/min ...... 51

Figure 3.12 Plane strain compression experimental data for PMMA at strain rate of

3/min ...... 51

Figure 3.13 Plane strain compression at strain rate of 1.0/min ………………………... 52

Figure 3.14 Plane strain compression at strain rate of 3.0/min ………………………... 53

Figure 3.15 Uniaxial compression of PMMA at 107 °C by Palm …………………….. 54

Figure 3.16 Uniaxial compression of PMMA at 107 °C by Ghatak …………………... 54

Figure 3.17 Uniaxial compression of PMMA with various temperatures at 3/min …… 55

Figure 3.18 Uniaxial compression of PMMA with various strain rates at 120 °C ……. 55

Figure 3.19 Stress-strain curve for compression of polymer below its Tg …………….. 56

Figure 4.1 Schematic of the Model by Dupaix and Boyce ...... 61

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Figure 4.2 Schematic showing the contributions of the two resistances to the total stress ...... 62

Figure 4.3 Individual contributions of the two resistances to the total stress, (a) intermolecular resistance, and (b) molecular network resistance ...... 63

Figure 4.4 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 102 °C with optimized set of constants ...... 70

Figure 4.5 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 110 °C with optimized set of constants ...... 70

Figure 4.6 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 115 °C with optimized set of constants ...... 71

Figure 4.7 Stress-strain data for PMMA in plane strain compression with experimental data and model simulation at 112 °C with optimized set of constants ..... 71

Figure 4.8 Stress-strain data for PMMA in plane strain compression with experimental data and model simulation at 120 °C with optimized set of constants ..... 72

Figure 4.9 Schematic of the Model by Dooling et al...... 74

Figure 4.10 Simulation of uniaxial tensile drawing of PMMA at 122 °C and nominal strain rate of 0.4/min, (a) From Dooling et al, and (b) Reproduction using the Dooling et al. …………………………………………………………………………...... 79

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Figure 4.11 Plots of true stress versus nominal strain rates (per min) for uniaxial drawing of PMMA at 120 °C. (a) Experimental data (Dooling et al.), and (b)

Simulation using the model by Dooling, Buckley et al...... 80

Figure 4.12 (a) Temperature-dependence of slippage viscosities normalized to their values at 128 °C, reproduced using Dooling et. al., (b) Modified temperature- dependence of slippage viscosities normalized to their values at 110 °C for obtaining the new set of constants for DBRZ model ...... 82

Figure 4.13 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 102 °C ...... 84

Figure 4.14 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 110 °C ...... 84

Figure 4.15 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 120 °C ...... 85

Figure 4.16 Stress-strain data for PMMA in plane strain compression with experimental data and model simulation at 112 °C ...... 85

Figure 4.17 Stress-strain data for PMMA in plane strain compression with experimental data and model simulation at 120 °C ...... 86

Figure 5.1 Schematic of Dupaix-Boyce Model ……………………………………….. 91

Figure 5.2 Schematic of the New Model ……………………………………………… 92

Figure 5.3 Plot showing value of ‘y’ versus temperature in Kelvin …………………... 93

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Figure 5.4 Temperature dependence of shear modulus as used in the Dupaix-Boyce

Model ………………………………………………………………………………….. 96

Figure 5.5 Experimental data collected by Palm showing temperature dependence of shear modulus …………………………………………………………………………. 97

Figure 5.6 Comparison between the experimental data and Model simulation using

New Model for uniaxial compression of PMMA at 82 °C ……………………………. 100

Figure 5.7 Comparison between the experimental data and Model simulation using

New Model for uniaxial compression of PMMA at 92 °C ……………………………. 100

Figure 5.8 Comparison between the experimental data and Model simulation using

New Model for uniaxial compression of PMMA at 102 °C …………………...... 101

Figure 5.9 Comparison between the experimental data and Model simulation using

New Model for uniaxial compression of PMMA at 110 °C …………………...... 101

Figure 5.10 Comparison between the experimental data and Model simulation using

New Model for uniaxial compression of PMMA at 115 °C …………………...... 102

Figure 5.11 Comparison between the experimental data and Model simulation using

New Model for uniaxial compression of PMMA at 120 °C …………………...... 102

Figure 5.12 Comparison between the experimental data and Model simulation using

New Model for uniaxial compression of PMMA at 130 °C …………………...... 103

Figure 5.13 Comparison between the experimental data and Model simulation using

New Model for plane strain compression of PMMA at 102 °C ………………………. 104

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Page Figure 5.14 Comparison between the experimental data and Model simulation using

New Model for plane strain compression of PMMA at 112 °C ………………………. 104

Figure 5.15 Comparison between the experimental data and Model simulation using

New Model for plane strain compression of PMMA at 120 °C ………………………. 105

Figure 5.16 Comparison between the experimental data and Model simulation using

New Model for plane strain compression of PMMA at 130 °C ………………………. 105

Figure 5.17 Model predictions for uniaxial and plane strain compression at 82 °C at the strain rate of 1/min ………………………………………………………………… 108

Figure 5.18 Model predictions for uniaxial and plane strain compression at 102 °C at the strain rate of 1/min ………………………………………………………………… 109

Figure 5.19 Model predictions for uniaxial and plane strain compression at 122 °C at the strain rate of 1/min ………………………………………………………………… 109

Figure 6.1 Temperature and pressure profile of the hot embossing process (a) more commonly used profile; (b) profile used in most experiments conducted in this chapter ...... 111

Figure 6.2 Channel and die fixture for hot embossing ...... 114

Figure 6.3 Die Profile ...... 115

Figure 6.4 Force versus Time data for different hold times at 92 °C …………………. 116

Figure 6.5 Force versus Time data for different hold times at 102 °C ………………... 117

Figure 6.6 Force versus Time data for different hold times at 112 °C ………………... 117

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Figure 6.7 Force versus Time data for different hold times at 122 °C ………………... 118

Figure 6.8 Force versus Time data for different hold times at 132 °C ……………….. 118

Figure 6.9 Force versus Time data for different hold times at 142 °C ………………... 119

Figure 6.10 Force versus time data for different temperatures at the hold time 180 s ... 119

Figure 6.11 Force versus time data for high temperatures at the hold time 180 s …….. 120

Figure 6.12 Representative displacements versus time plot for the given set of experiments ……………………………………………………………………………. 120

Figure 6.13 Force trend with time for various temperatures after the polymer samples were embossed to the desired depths ………………………………………………….. 122

Figure 6.14 Force trend with time for temperatures of 122, 132 and 142 °C after the polymer samples were embossed to the desired depths ……………………………….. 122

Figure 6.15 Percentage force in terms of peak force at the end of embossing with time after the polymer samples were embossed to the desired depths ……………………… 123

Figure 6.16 Percentage force in terms of peak force at the end of embossing with time after the polymer samples were embossed to the desired depths for temperatures of

92, 102 and 112 °C …………………………………………………………...……….. 123

Figure 6.17 Percentage force in terms of peak force at the end of embossing with time after the polymer samples were embossed to the desired depths for temperatures of

122, 132 and 142 °C ………………………………………………………………...… 124

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Figure 6.18 Samples embossed to a depth of 0.8 mm at 142 °C with different hold times showing dissimilar final depth of indentation. (a) Sample held for 180 seconds;

(b) sample held for 30 seconds ………………………………………………………... 125

Figure 6.19 Depth of the embossed sample at the end of the embossing cycle for various temperatures. The desired depth was 0.8 mm ………………………………… 126

Figure 6.20 Force versus Time data for different hold times at 142 °C ………………. 128

Figure 6.21 Depth of the embossed sample at the end of the embossing cycle for various temperatures. The desired depth was 0.8 mm ………………………………… 128

Figure 6.22 Embossing depths with variation in the hold times (30, 90,180, 240 and

300 seconds) at temperature of 142 °C ………………………………………………... 130

Figure 6.23 Force versus Time data for different hold times at 142 °C ………………. 130

Figure 6.24 Embossing depths with and without cooling at 142 °C ………………….. 131

Figure 7.1 Profile of the part and the section used to conduct hot embossing simulations …………………………………………………………………………….. 136

Figure 7.2 Dimension of the die used for hot embossing simulations ………………… 138

Figure 7.3 (a) Fine (1) , medium (2) and coarse (3) mesh region of the part; and (b)

Mesh of the part with the die in its initial position showing the dimensions of the fine, medium and coarse mesh ……………………………………………………………… 139

Figure 7.4 Force versus time with different hold times at 142 °C for hot embossing simulations …………………………………………………………………………….. 141

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Figure 7.5 Force versus Time experimental data for different hold times at 142 °C …. 142

Figure 7.6 Final displacement of the part after different hold times at 142 °C for hot embossing simulations ………………………………………………………………… 142

Figure 7.7 Hot embossing simulations conducted at 142 °C with 90 second hold without cooling (a) at end of deformation step, (b) at end of cooling step, and (c) at end of unloading step …………………………………………………………………. 143

Figure 7.8 Force versus time with different hold times at 132 °C for hot embossing simulations …………………………………………………………………………….. 144

Figure 7.9 Force versus Time experimental data for different hold times at 132 °C …. 144

Figure 7.10 Final displacement of the part after different hold times at 132 °C for hot embossing simulations ………………………………………………………………… 145

Figure 7.11 Force versus time with different hold times at 122 °C for hot embossing simulations …………………………………………………………………………….. 145

Figure 7.12 Force versus Time experimental data for different hold times at 122 °C ... 146

Figure 7.13 Final displacement of the part after different hold times at 122 °C for hot embossing simulations ………………………………………………………………… 146

Figure 7.14 Part embossed at 142 °C after a hold time of 90 seconds to different final temperatures …………………………………………………………………………… 147

Figure 7.15 Part embossed at 142 °C after a hold time of 90 seconds to different final temperatures for hot embossing simulations …………………………………………... 147

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Figure 7.16 True strain versus time for uniaxial compression at 130 °C with strain rate of 1/min and final strain of 1.4 using the New model 149

………………………………....

Figure 7.17 True stress versus time for uniaxial compression at 130 °C with strain rate of 1/min and final strain of 1.4 using the New model …………….……………… 150

Figure 7.18 True stress versus true strain for uniaxial compression at 130 °C with strain rate of 1/min and final strain of 1.4 using the New model ……………..……….. 150

Figure 7.19 True stress versus time for uniaxial compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 using the New model ……... 151

Figure 7.20 True stress versus true strain for uniaxial compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 using the New model . 151

Figure 7.21 True strain versus time for plane strain compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 from experimental data …… 152

Figure 7.22 True stress versus time for plane strain compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 from experimental data ...... 153

Figure 7.23 True stress versus true strain for plane strain compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 from experimental data …………………………………………………………………………………….. 153

Figure 7.24 True stress versus time for uniaxial compression at 130 °C with hold of

120 seconds, uniform die velocity of 0.01667 and final strain of 0.1 using New model 155

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Figure 7.25 True stress versus true strain for uniaxial compression at 130 °C with hold of 120 seconds, uniform die velocity of 0.01667 and final strain of 0.1 using

New model …………………………………………………………………………….. 156

Figure 7.26 Individual contributions of the two resistances to the total stress, (a) intermolecular resistance, and (b) molecular network resistance ...... 157

Figure 7.27 Schematic showing the contributions of the two resistances to the total stress ...... 158

Figure 7.28 Force versus time plot for hot embossing simulation at 130 °C with a hold time of 90 seconds with changed material constants of New model ……….…… 159

Figure 7.29 Displacement versus time plot for hot embossing simulation at 130 °C with a hold time of 90 seconds with changed material constants of New model ……... 159

Figure 7.30 Hot embossing simulations conducted at 130 °C with 90 second hold without cooling (a) at end of deformation step, (b) at end of cooling step, and (c) at end of unloading ………………………………………………………………………. 160

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CHAPTER 1

INTRODUCTION

1.1 Introduction

Polymethyl methacrylate (PMMA) is an amorphous thermoplastic used in various industrial applications. PMMA is compatible with human tissues and allows high resolution features to be embossed onto a surface, thus making it highly desirable for use in bio-medical, micro-optics, micro-fluidic devices, electronics, micro-electro-mechanical systems (MEMS), etc. One of the polymer processing techniques becoming popular for these applications is hot embossing. Hot embossing capitalizes on the fact that the mechanical behaviour of the polymers changes drastically around the glass transition temperature (Tg). The polymer is deformed at temperatures above the Tg where the material is more fluid-like and then cooled below the Tg where it behaves more like a solid. The changes in physical properties make this temperature regime highly favourable for these warm-temperature deformation processes. The same rationale also makes it difficult to develop a continuum model which accurately predicts how the polymer‘s mechanical behaviour changes with temperature and strain rate across the glass transition.

Most of the existing constitutive models do not achieve this task; they either work below

1 or above the glass transition, but not in both regimes. Hence, there is a great need to develop a constitutive model for the polymer that can capture the material behaviour across the glass transition temperature (Tg - 20 to Tg + 60) relevant for hot embossing applications. Material properties change with changes in temperature and strain rate during deformation, but this change is most dramatic across the glass transition temperature. For example, the change in shear modulus across the transition is shown in figure 1.1 [1]. The shear modulus changes by an order of magnitude from below to above glass transition temperature. Shear modulus in the figure is denoted by ‗µ‘ and the subscripts g and r stand for glassy and rubbery. Also, θg is the glass transition temperature.

Figure 1.1 Temperature dependence of shear modulus [1]

Several experiments have been conducted on PMMA to better understand the material behaviour. In addition, various material models have also been developed to capture the mechanical properties of the polymer. Many research groups have used these extensively developed constitutive models to run hot embossing simulations, while others have run hot embossing simulations with the use of simple continuum models and tabulated experimental data. A summary of this prior research will be covered in the following chapter.

1.2 Summary of Work

The goal of this thesis is to develop a material model for the amorphous polymer

Polymethyl methacrylate (PMMA) for simulating the hot embossing process. The hot embossing process can be made more economical by optimizing the operating parameters and conditions through simulations rather than trying to figure them out by trial and error.

Chapter 2 gives the background about the polymer PMMA and constitutive modeling of polymers. It discusses the uniaxial, plane strain and hot embossing experiments that have been performed on the polymer across its glass transition temperature. The chapter also discusses the existing and the new continuum models being developed to better predict various polymer processing techniques including hot embossing. Several hot embossing simulations conducted by different research groups are also studied.

Chapter 3 presents the material characterization experiments that were conducted on

PMMA. Uniaxial and plane strain compression experiments were carried out on the polymer well across its glass transition temperature (Tg). The data collected here was not

3 only used in developing the material model and obtaining the material constants but also in validating the constitutive model.

Two existing constitutive models: Dupaix-Boyce model and Dooling-Buckley-Rostami-

Zahlan model on the polymer PMMA are studied in Chapter 4. This is to better understand the material models and the findings were used to develop a new constitutive model that will work across the wide range of temperature and strain rates.

Chapter 5 discusses the new continuum model developed from the study of two existing constitutive models: Dupaix-Boyce and Dooling-Buckley-Rostami-Zahlan and various material models from the literature. An additional molecular network was added to the existing Dupiax-Boyce model to capture a wider range of temperature across the glass transition.

Chapter 6 presents the experimental data collected from conducting the hot embossing experiments on the polymer PMMA across its glass transition temperature. This was done to better understand the process conditions of hot embossing and thus identify the essential parameters that the new developed material model must be able to capture.

Hot embossing simulations are performed in Chapter 7 using the new material model developed in chapter 5 and are validated against the experimental data as shown in chapter 6. The strengths and weaknesses of the current model are discussed.

Chapter 8 discusses the conclusion of this work and the scope of future work in order to continue the goals of this thesis.

CHAPTER 2

LITERATURE REVIEW

2.1 Experiments

Researchers have been studying the polymer Polymethyl methacrylate (PMMA) from the late 1800‘s. The polymer is formed by polymerization of the monomer methyl methacrylate (MMA). A chemical structure of the polymer is shown in figure 2.1. The process to convert the monomer MMA into the polymer PMMA was discovered by the

German scientists Fittig and Paul in 1877 [2]. The polymer was first commercialized in

1936 and is now available in various brand names [2]. Due to the presence of different varieties of commercial PMMA, the mechanical and material properties they possess are also greatly varied.

Figure 2.1 Chemical Structure of the polymer PMMA 5

All polymers can be distinguished into two kinds based on their structure: amorphous and crystalline. Polymers whose molecular structure lack a definite repeating form, shape, or structure and have no definite shape are called amorphous polymers while polymers where a unit structure repeats itself are called crystalline in nature and has a definite shape, form and structure.

As mentioned before, PMMA is an amorphous polymer. These polymers show three distinct regions in mechanical behavior with variation in temperature: glassy, rubbery and melt. The glassy region is the temperature range where the polymer exhibits solid-like behavior up to around its glass transition temperature. The melt region is above the melting point of the polymer when it acts completely like a fluid. The rubbery region is between the glass transition and the melting temperature of the polymer where it behaves neither completely like a fluid nor a solid. The glassy and rubbery regions are of interest here in trying to predict material behavior around Tg. Figure 2.2 shows the change in the values of modulus of the material across these temperatures.

A reasonable amount of experimental data is available in the literature on PMMA and other polymers with industrial applications such as polyethylene terephthalate (PET), polycarbonate (PC) and many more in both the glassy and rubbery regions. The available data on all these polymers gives an insight to their behaviour with temperature and strain rate dependence at large strains.

Figure 2.2 Change in the values of modulus of the material across glassy-rubbery-melt region (Tg – Glass transition temperature, and Tm – Melting temperature)

A variety of experiments have been conducted on polymers below the glass transition.

Hydrostatic extrusion was performed on two different grades of PMMA by Hope et al.

[3], these grades had different glass transition temperature and molecular weight. The authors found that friction was an important criterion in determining process instabilities

(stick-slip) and steady state process behaviour. Kahar et al. [4] also performed hydrostatic extrusion on PMMA below its glass transition temperature and studied the development of orientation in the polymer using birefringence. They found that the mechanism of deformation depends both on the extrusion temperature and applied deformation, and the results were consistent with the existence of a molecular network as discussed in the article. Raha and Bowden [5] conducted plane strain compression on PMMA to study the

7 structural changes in glassy polymers with the help of birefringence measurements. They found that the model of dissociable cohension points gave a reasonable explanation to all the observations on the polymer of its optical and mechanical properties. Drotning and

Roth [6] studied the effect of moisture on the thermal expansion of PMMA. The results showed correlation between Tg and moisture content of the polymer, ―approximately -1

°C per 0.1% weight increase‖. Hasan and Boyce [7] studied stress-strain data and change in enthalpy with temperature by conducting experiments on PMMA, Polycarbonate (PC) and Polystyrene (PS). These experiments confirmed the presence of two separate deformation mechanisms as suggested by some research groups earlier [8-11]. A generalized stress-strain curve for polymers below the glass transition temperature (Tg) is shown in figure 2.3. Region (1) shows the elastic modulus of the material; region (2) shows the yield flow or strain softening; region (3) shows the gradual strain hardening; and region (4) shows the dramatic strain hardening at large strains. Hasan and Boyce [12] performed creep analysis by conducting compression tests on PMMA below its glass transition temperature. These observations were very helpful in understanding the mechanical behaviour of the polymers and giving insight for developing ‗a single unified model to predict nonlinear viscoelastic and viscoplastic deformation of amorphous polymers‘.

Arruda and Boyce [9] conducted both uniaxial and plane strain compression tests on

PMMA and PC at temperatures below the glass transition to study the effects of anisotropy on the deformation of the polymers. The stress-strain behaviour was found to strongly depend on the state of deformation. An eight-chain model for large strain

Figure 2.3 Generalized stress-strain curve for a polymer above its Tg

behaviour of amorphous polymers in the glassy range was developed to capture the dependence of strain hardening on the two states of deformation. This model was able to predict the two deformation states better than the earlier three-chain [13] and four-chain

[14] models. This work was further verified by Arruda et al. [15] by conducting more uniaxial and plane strain compression tests. In addition, birefringence measurements of the deformed samples were also taken to illustrate the anisotropic nature of these polymers under different states of deformation. A later paper by Arruda et al. [16] conducted uniaxial and plane strain compression experiments on PMMA at large strains below its glass transition temperature to study the effects of temperature, strain rate and thermo-mechanical coupling. Experiments were done at a strain rate of 0.001/sec to replicate isothermal test conditions and hence study the temperature dependence of yield, strain-softening and strain hardening. The yield, strain softening and the extent of strain 9 hardening of the polymer decreases with increase in temperature. All the experiments conducted here were used to verify the constitutive model developed by Arruda and

Boyce [17] following the work in the Boyce research group.

The last few paragraphs discussed the various experiments conducted on PMMA below its glass transition temperature, but the bulk of the experimental data on PMMA near and above Tg has focused on tension and compression tests. G‘Sell and Souahi [18] performed tension tests on three different grades of PMMA, one linear and two cross- linked, between 95 °C to 145 °C with glass transition temperatures of 108, 112 and 116

°C respectively. This study was done to observe the influence of the cross-linking on the plastic behaviour at large strains. The three grades of PMMA showed similar trends for the properties studied. It was observed that the ‗extent of strain hardening‘ of a polymer at large strains decreases with increase in temperature; the decrease is significant above the glass transition temperature but the strain hardening of the polymer continues to be noticeable. Tension tests on PMMA were also conducted by Dooling et al. [19] above its glass transition temperature (Tg). These experiments were conducted with variation in strain rate and temperature from around Tg to ~ Tg +75. Dooling et al. [20] also performed tension, shear and combined tension/shear creep experiments on aged PMMA below Tg to study the onset of nonlinear viscoelasticity in glassy polymers. Palm et al. [21] and

Ghatak and Dupaix [22] also have conducted uniaxial and plane strain compression experiments on PMMA between 82 °C to 132 °C (Tg ~ 102 °C). Richeton et al. [23] have conducted some uniaxial compression tests on PMMA up to ~ Tg +10 °C as well to study the polymer behaviour in order to develop a constitutive model for polymer under large

10 deformations. Ames et al. [24] have also conducted experiments on PMMA over a wide range of temperatures from 22 °C to 170 °C (Tg ~ 110 °C). All these experiments on

PMMA illustrated the changes in the mechanical behaviour of the polymer with variation in strain rate and temperature. The yield stress, strain softening and the extent of strain hardening of the material decreased with increasing temperature and decreasing strain rates.

Numerous experiments have been also been conducted on PMMA to study and understand several manufacturing processes in addition to develop a continuum model for the amorphous polymer. Juang et al. [25] performed some rheological experiments on

PMMA, polycarbonate (PC) and polyvinyl butyral (PVB) at temperatures around and much above Tg. The data was used to simulate hot embossing of these polymers.

2.2 Constitutive Modeling

Various material models exist in the literature for modeling amorphous polymers; one of the most prevalent applications of these constitutive models is to simulate different polymer processing techniques. Amorphous polymers have two solid states: glassy and rubbery states. The transition from the glassy to the rubbery state happens at the glass transition temperature. This transition makes it difficult to develop a constitutive model capable of capturing the different physical characteristics of the polymer over this transition.

A generalized stress-strain curve for polymers below the glass transition temperature (Tg) is shown in figure 2.3. Many features of this behavior change drastically with changes in

11 temperature, strain rate, strain state and even extent of deformation. At very small strains, the polymer behaves as a linear elastic material, however at large strains; the polymer behaves as both elastic and viscous. The elastic response at large strains is highly nonlinear. This elastic part of the behaviour is represented by a spring in a schematic of the continuum model. The viscous or fluid-like response is also nonlinear, but is occasionally modeled as simply Newtonian in the literature. The viscous elements are represented with dashpots in a schematic. The elastic and viscous responses of the polymers are also highly temperature dependent, sometimes requiring different equations to capture the physical mechanisms for temperatures below and above its Tg. Most of the constitutive models present in the literature started out as models which could predict polymer behavior using basic physical mechanisms using simple elastic and viscous elements. As the knowledge and understanding of polymers has evolved, these continuum models have become more and more complex and accurate.

Some of the early work on modeling of glassy polymers has been done by Robertson

[26], Argon [8], Haward and Thackray [11], Doi and Edwards [27], Doi [28], and

Edwards and Vilgis [29]. The model by Haward and Thackray [11] uses a Hookean spring in series with a parallel combination of an Eyring dashpot and a rubber elasticity spring. A schematic is shown in figure 2.4. A Hookean spring describes a strain or deformation that depends linearly on the applied force whereas a rubber elasticity spring is non-Hookean; i.e. it is non-linear elasticity, and the strain may or may not depend on temperature. An Eyring dashpot represents non-Newtonian fluid response, i.e. the

12 deformation depends either non-linearly on strain rate, or it depends linearly on strain- rate along with some other parameter.

Figure 2.4 Schematic of constitutive model by Haward and Thackray [11]

Work by Argon [8] on constitutive modeling of polymers was continued by Boyce, Parks and Argon [10]. This Argon model was later again modified by Arruda et al. [16] to study the effects of strain-rate, temperature and thermo-mechanical coupling on PMMA.

Temperature dependent strain hardening was incorporated in the model using previous work by Raha and Bowden [5] and Boyce [30]. A schematic of this polymer deformation model is shown in figure 2.5 [9]. The viscoplastic element is from the Argon model of rate and temperature dependent yield and the rubber elasticity Langevin spring is nonlinear in nature. The constitutive model developed in this thesis originates from the work by Boyce and co-workers. 13

Figure 2.5 Polymer deformation model by Argon, Boyce and co-workers [9]

Arruda and Boyce [9] developed an eight-chain model for rubber elasticity for large strain behaviour of amorphous polymers in the glassy range to capture the dependence of strain hardening on the two states of deformation: uniaxial and plane strain. This model was able to predict the two deformation states better than the three-chain and four-chain model. A schematic of the eight-chain model in its undeformed and deformed state is shown in figure 2.6. This work was further verified by Arruda et al. [15] to illustrate the anisotropic nature of these polymers under different states of deformation. More uniaxial and plane strain compression experiments were conducted on PMMA at large strains below its glass transition temperature by Arruda et al. [16] to study the effects of temperature, strain rate and thermo-mechanical coupling and hence test the accuracy of the eight-chain model. The yield, strain softening and the extent of strain hardening of the polymer decreases with increase in temperature. The eight-chain model continued to

14 perform better than the three-chain model by Wang and Guth [13] and the four-chain model by Flory and Rehner [14].

Figure 2.6 Schematic of the eight chain model by Arruda and Boyce (a) undeformed, (b) stretched in uniaxial compression, and (c) stretched in plane strain compression [9]

An elaborate review of constitutive models for rubbery materials was done by Boyce and

Arruda [31]. Various statistical and continuum models for incompressible rubber were discussed and compared against experimental data. The four major network models were also examined here: 3-chain network model [13], 4-chain network model [14], 8-chain network model and full network model. The eight-chain model preformed the best among all these models. ―The models differ in how the deformation of the chains is related to the deformation of the unit cell.‖ These network units are essential to accurately predict the polymer behavior at large deformations. The review article also highlighted the fact that although rubbery materials are generally considered incompressible, they are in fact a little compressible and the more accurate constitutive models include this detail.

This work was further continued by Llana and Boyce [32] in developing a constitutive model for PET above its glass transition. Uniaxial and plane strain compression experiments were performed on PET to study the polymer behavior over a wide range of temperature and strain rates. ‗The stress-strain behavior of PET above the glass transition temperature exhibits four characteristic features: a relatively stiff initial response (E), followed by a rollover to a yield or flow-like behavior (σ flow), followed by a gradual increase in stress with strain which is termed the initial strain hardening response (hi), followed by a dramatic increase in stress with strain at large strains‘ [32]. A schematic of the same is shown in figure 2.7 [32].

Figure 2.7 Schematic showing the stress-strain behavior of polymer with characteristic features [32]

This constitutive model from the Boyce research group was further developed to capture these four characteristic features of the polymer PET by Boyce et al. [33]. A schematic representation of this constitutive model by Boyce, Socrate and Llana is shown in figure

2.8. The Cauchy stress of the polymer is divided into two resistances to deformation: intermolecular resistance and molecular network resistance. The intermolecular resistance response is seen as initial stiffness and flow (i.e. yield stress), whereas the molecular network response is divided into network orientation and molecular relaxation.

Figure 2.8 Schematic of the constitutive model by Boyce, Socrate and Llana [33]

This model was further developed by Dupaix and Boyce [1, 34] for the amorphous polymer poly(ethylene terephthalate)-glycol (PETG), and material constants were obtained using the new model to fit the experimental data for PET [35]. This constitutive model was able to predict material behaviour for both PET and PETG, even though PET undergoes strain-induced crystallization near Tg unlike PETG. Work by Palm et al. [21,

36] used this Dupaix-Boyce continuum model for another amorphous polymer PMMA.

This material model was found to correspond well with uniaxial compression data from room temperature to about Tg +10 °C. These experiments were done at various strain

17 rates to large strains and the model was able to predict the strain rate dependence as well.

The following work by Ghatak and Dupaix [22] showed that the same model could not work at much higher temperatures (~ Tg +15 and higher). The Dupaix-Boyce model

(figure 2.9) will be modified in this thesis to capture experimental data of PMMA over a wider range of temperature above Tg as well.

Figure 2.9 Schematic of the constitutive model by Dupaix and Boyce

Richeton et al. [37] developed a constitutive model for amorphous polymers which was derived from the cooperative model developed by Fotheringham, Cherry and Bauwens-

Crowet [38-40]. Here, the cooperative model was used for the first time to predict compression at impact loading rates. All the model predictions were validated by experiments conducted from room temperature to temperatures above Tg for the polymers

PMMA and PC. This cooperative model was soon after modified by Richeton et al. [41] into a three-dimensional constitutive model and again validated against compression experiments conducted on PMMA and PC. This was a thermomechanical model for large

18 deformations of the polymer which was able to capture temperature and strain rate dependence as well as the effects of hydrostatic pressure. A schematic of the same is shown in figure 2.10. However, this model was shown to work mostly in the glassy region of the polymer PMMA up to ~ Tg +10 °C and not at higher temperatures.

Figure 2.10 Schematic representation of the modified Cooperation model by Ahzi and co-workers [41]

Gerlach et al. [42] developed a glass-rubber continuum model using previous work done by Buckley and co-workers [43, 44] to simulate the modeling of biaxial oriented PET films. The Cauchy stress in this model has two contributing components; bond-stretching and conformational. ‗The bond-stretching component dominating deviatoric response at short times and low temperatures and the hydrostatic response at all times and the conformational component dominating deviatoric response at long times and high temperatures‘ [45]. Adams et al. [45] conducted experiments on PET to study the 19 dependence of yield stress on strain—rate ratio between the glass transition temperature and the rubber-like plateau region of the polymer. These results were used to verify the earlier proposed model developed by the Buckley group. The constitutive model results were in good agreement with the experimental data. Continuing the work, Adams et al.

[46] added more features on the model to successfully replicate hot-drawing of PET.

This model included the elements for entanglement slippage and crystallization and thus better explained the strain-stiffening of PET through around its Tg and well above the glass transition temperature. Work by Dooling et al. [20] extended this glass-rubber model for glassy polymers at and below their glass transition temperature. The stress- induced crystallization was removed from the model to be used for the amorphous polymer PMMA. The material model was able to predict creep for PMMA, however the model was not able to predict strain recovery after unloading. The same model was developed further by incorporating an additional conformational contribution for total stress by Dooling et al. [19]. This model was able to predict the polymer behaviour accurately at temperatures above the glass transition of PMMA only (~Tg to Tg+75) relevant for thermoforming. The authors did not try to use the model for temperatures below Tg. This continuum model (figure 2.11) in its current state can be successfully used for simulating the hot-drawing of PMMA at moderate strain rates. ‗This is a fully defined three-dimensional constitutive model that has consistency with physical theories but is structured in simpler form for greater computational efficiency in process modeling‘ [19].

Figure 2.11 Schematic representation of the continuum model by Dooling, Buckley,

Rostami and Zahlan

Figiel and Buckley [47] studied different kinematic structures of constitutive models for amorphous polymers. One was based on additive decomposition of velocity gradient tensor and another on multiplicative decomposition of deformation gradient tensor. They found the difference between the approaches only for problems involving finite rotation.

It was observed that for consistency with experimental data, constitutive models for highly elastic flows of polymers should be based on either (i) additively of elastic and viscous velocity gradients – with symmetry of viscous velocity gradient or (ii) multiplicability of elastic and viscous deformation gradients - with symmetry of convected viscous velocity gradient. The approach used in this thesis work and the Boyce group is based on the latter.

Ames et al. [24] first proposed a constitutive model for amorphous polymers based on their original work and work by Ahzi [23, 37], Buckley [19, 43] and Boyce [1, 33] group.

A schematic of this originally proposed continuum model is shown in figure 2.12. Ames,

Anand, Chester and Srivastava [48, 49] developed the constitutive model further for 21 amorphous polymers which worked from room temperature to around the glass transition temperature of the polymer. These results were validated by conducting simple compression experiments on polymethyl methacrylate (PMMA), polycarbonate (PC) and cyclo-olefin polymer (Zeonex-690R). ―The constitutive model performs well in reproducing the following major intrinsic features of the macroscopic stress–strain response: (a) the strain rate and temperature dependent yield strength; (b) the transient yield-peak and strain-softening which occurs due to deformation-induced disordering; (c) the subsequent rapid strain-hardening due to alignment of the polymer chains at large strains; (d) the unloading response at large strains; and (e) the temperature rise due to plastic dissipation.‖ Srivastava et al. [50] have recently modified the earlier constitutive model [48, 49] developed by the Anand research group for amorphous polymers which works well across the glass transition temperature. The results have been verified by comparing them to experiments conducted on PMMA, PC and Zeonex-690R. This current material model works from room temperature to around Tg +60. This model being first of its kind is very useful and brings a better understanding of the polymer properties.

However, it does use a large number of constants to accurately predict the polymer behavior (~45). This material model uses two kinds of network to capture the experimental data, intermolecular (α = 1) and molecular network (α = 2 and 3). The most unique feature of this model was to use two different and independent molecular networks to capture the material behavior well across the glass transition temperature.

One molecular network contributes to stresses below Tg (α = 2) while another to stresses above Tg (α = 3). A schematic showing the network contributions towards the total stress

22 is shown in figure 2.13. Figure 2.13 (a) is for temperatures below Tg and figure 2.13 (b) is for temperatures above Tg. A schematic representation of the constitutive model is also shown in figure 2.14.

Figure 2.12 A schematic representation of the originally proposed constitutive model by

Anand and co-workers [24]

(a)

(b)

Figure 2.13 A schematic showing the network contributions towards the total stress, (a) below Tg and, (b) above Tg [50]

Figure 2.14 A schematic representation of the current constitutive model by Anand and co-workers [50]

Many other material models have been developed for polymers using fluid mechanics fundamentals rather than solid mechanics. Such models are not discussed in detail here since the approach used in this thesis is based on solid mechanics fundamentals. The advantage of using a solid mechanics based model is that it is capable of predicting the polymer behavior below the glass transition temperature unlike a continuum model based on fluid mechanics. In addition, such a model can predict mechanical properties such as spring-back of the material, which is essential for accurately simulating hot embossing.

However, a few of such material models are discussed here for completeness. Juang et al. 25

[25] used fluid-based models (Power-law Model and Ellis Model or Carreau Model) to run finite element simulation of hot embossing of polymethyl methacrylate (PMMA), polycarbonate (PC) and polyvinyl butyral (PVB). They conducted some rheological experiments on PMMA, PC and PVB to study the dynamic shear viscosity and the transient extensional viscosity. Yao et al. [51] too used a viscous model for hot embossing simulations of acrylonitrile-butadiene-styrene copolymer (ABS). The simulations produced pressure, strain rate and velocity vector during the cavity filling process but did not predict the features such as spring back and residual stresses. All these fluid mechanics based continuum models are not capable of capturing the polymer behaviour at temperatures below Tg. Even most of the solid mechanics based material models are not capable of predicting the material behaviour across a wide range of temperatures crucial for these hot embossing manufacturing processes.

2.3 Hot Embossing Experiments

Hot embossing is a manufacturing technique used for a wide number of applications from macro to nanometer scales. Hot embossing is a technique of imprinting microstructures on a substrate (here, polymer) using a master mold (tool or a die). It mainly consists of the following steps. First, the polymer heated above its glass transition temperature.

Secondly, the polymer is stamped with the desired imprint at this temperature (above Tg) where it behaves more fluid-like. Also, the force required to deform the polymer at this temperature is much smaller than that required at temperatures below the glass transition

(Tg) of the polymer. Thirdly, the polymer is cooled below its Tg and the mold is removed.

At these cooler temperatures, the polymer becomes solid-like and the deformation can be arrested. A schematic of the hot embossing technique is shown in figure 2.15 [52].

Figure 2.15 Schematic of hot embossing technique [52]

Datta and Goettert [53] devised a number of steps to find the optimum hot embossing process condition without use of any material model. The method was able to determine

27 optimum hot embossing conditions for PMMA, PC and polypropylene (PP). The most significant process parameters were found to be molding temperature, molding force, molding rate, hold time, demolding temperature and demolding rate. Optimal force and temperature during molding resulted in complete filling of mold, hence producing high quality parts and longer mold insert lifetime.

Guo et al. [54] conducted hot embossing experiments on PMMA to study the demolding forces. The two main causes of demolding forces are: frictional force which causes surface adhesion and thermal stress which causes shrinkage between the mold and polymer. The authors concluded that the friction force was the main cause of damage to the polymer part and reducing the friction would improve the part quality.

Yao et al. [51] came up with a new scheme to make the hot embossing process faster, instead of heating and cooling the same stamp, they used two different stages. A schematic of the same is shown in figure 2.16. They used a viscous model and time- dependent thermal boundary conditions to simulate the hot embossing process for high- density polyethylene (HDPE) and acrylonitrile-butadiene-styrene (ABS) copolymer. The simulations produced pressure, strain rate and velocity vector during the cavity filling process but did not predict the features such as spring back and residual stresses.

Figure 2.16 A schematic of the two-station hot embossing process [51]

Juang et al. [55] conducted hot embossing experiments on PMMA, PC and PVB and used fluid-based models (Power-law Model and Ellis Model or Carreau Model) to run finite element simulation for hot embossing of PMMA. The simulations were run using experimental data collected by the same authors [25]. For more optimum embossing of polymer parts on micrometer scale, they found that the thermal cycle should be 25-40 °C to minimize the thermal stresses and the de-molding process should be automated. The temperature and pressure profile of the hot embossing process is shown in figure 2.17.

The temperature profile represents the mold insert and the polymer for isothermal processes and only the mold insert for non-isothermal processes.

Hot embossing process is mainly used for imprinting surface features but the process has been tried for embossing of discrete microparts by Kuduva-Raman-Thanumoorthy and

Yao [56]. The polymer parts are attached to a thin residual film after embossing which mechanically detached during the rubber-assisted ejection action. High embossing pressure and long embossing time are required for obtaining thin residual films for easy removal. These experiments were done on high-density polyethylene (HDPE) and acrylonitrile-butadiene-styrene (ABS) copolymer.

The traditional hot embossing process has also been adapted to fabricate 3D multi-metal layers structure on PMMA [57].

Figure 2.17 Temperature and pressure profile with time for the mold [55]

2.4 Hot Embossing Simulation

Embossing of various different polymers including PMMA was done by Lu et al. [58, 59] using finite element simulations on DEFORM. A tabular input data form was used with

30 values of stress, strain, strain rate and temperature rather than a constitutive model for the polymer. Lu et al. [59] simulated laser/IR-assisted microembossing of PMMA. The authors conducted rheological experiments on PMMA for experimental data above Tg and obtained data from literature [18] for temperatures below Tg. The simulations worked but could not predict the displacement values correctly. Lu et al. [58] developed a method to emboss high-aspect-ratio-features. This was done by sacrificing the template and hence avoiding the de-molding step where the micro-parts typically get damaged. However, lack of materials available for templates make embossing of high Tg polymers such as

PMMA difficult.

Computational work by Rowland et al. [60] captures the polymer flow behaviour very accurately, It studies the impact of polymer film thickness and cavity size on the flow of polymer during embossing using nanoimprint lithography (NIL) and also shows how total time to fill the cavity is substantially larger than an almost filled cavity. A diagram showing the same is depicted in figure 2.18. Rowland et al. [61] also studied NIL using embossing tools with different shapes and sizes. Differences in cavity size can lead to non-uniform filling and polymer flow. The results suggest that ―non-uniform embossing tool design should be tailored based on the principles of squeeze flow‖.

Scheer et al. [62] investigated imprinting polystyrene (PS) at lower temperatures (around

Tg). The glass transition temperature of PS used was 95 °C. The temperature up to 130 °C was called the glass transition region, the temperature from 130 to 210 °C was called the viscous-elastic plateau region and above 210 °C was called the viscous flow region. The lower limit of the viscous-elastic plateau was also found to be the practical lower

31 temperature limit for these imprinting experiments. It was observed that the time required to fill the cavities increases substantially by lowering the temperatures any further.

Scheer et al. [63] also studied the effects of temperature and molecular weights with variation of imprint pressure on PS. Two different filling behaviours were observed: bottom-up and lateral filling. The same is shown in figure 2.19. This behaviour depended on the polymer height ‗h‘ below an elevated stamp and the lateral size ‗s‘ of the stamp feature. For h > s, a bottom-up filling was observed while for h < s, the lateral filling was observed. Scheer et al. [64] studied the effects of molecular weight and viscosity at temperatures around the glass transition of PMMA, PS and polyvinyl chloride (PVC).

They observed that the optimum process temperature depends on the size of the cavity and pattern of the imprint.

Yao et al. [51] too used a viscous model for hot embossing simulations of HDPE and

ABS. The simulations produced pressure, strain rate and velocity vector during the cavity filling process but did not predict the features such as spring back and residual stresses.

Guo et al. [54] simulated hot embossing of PMMA without using a material model and found that friction force was the main cause of damage to the polymer part and reducing the friction would improve the part quality.

Figure 2.18 Deformation profile of the polymer with time from onset of filling to almost fully filled cavity [60].

Figure 2.19 Schematic of bottom-up and lateral filling of cavities [63].

CHAPTER 3

COMPRESSION EXPERIMENTS

3.1 Introduction

Compression experiments were conducted on Polymethyl methacrylate (PMMA) with variation in temperatures, strain rates and strain states to gather data about material behaviour. The experimental data existing in our lab [36, 65] for uniaxial and plane strain compression cover most of the temperature range from 82 to 132 °C and strain rates from 0.2/min to 4.0/min to a strain level of 1.4 or 140%. The experiments performed here are to complete some gaps in the data, all these together should provide sufficient data base for material characterization for Dupaix-Boyce model [1, 34]. There is some overlap of experiments conducted to ensure consistency within the data.

The data obtained from uniaxial compression tests are very consistent with the existing data on PMMA [16, 18, 19]. Plane strain compression data obtained by Palm [36] in the lab compares well with the experimental data available in the literature for PMMA [9] at the room temperature (~25 ºC). There is no data available for plane strain compression of

PMMA with variation in temperature and strain rates at large strains around and above Tg in the literature aside from the work done by Palm [21].

3.2 Material

The material used for all the experiments was Polymethyl methacrylate (PMMA). This is a commercially available grade: Plaskolite OPTIX Acrylic Plastic provided by Plaskolite,

Inc. The components are Polymethyl methacylate (PMMA) 99.5 % (min.) and Methyl methacylate (MMA) 0.5 % (max.) with the molecular weight of about 140,000 g/mol.

The Material Safety Data Sheet (MSDS), technical data sheet and physical, mechanical, thermal and chemical properties can be found on the Plaskolite, Inc. website. The polymer supplied by Plaskolite as Acrylic sheets were of 48‖ by 96‖ (4‘ by 8‘) with thickness of approximately 0.375‖.

The monomer MMA generally undergoes free radical vinyl polymerization to form the polymer PMMA. This is shown in figure 3.1 along with the chemical structures of the molecules.

Figure 3.1 Polymerization of monomer MMA to form the polymer PMMA along with the chemical structures of the molecules.

3.3 Experimental Set-up and Procedure

3.3.1 Experimental Set-up

The experiments were conducted on an Instron: Universal Testing Machine (UTM). This was a 5800 series dual column tabletop system for mid-range testing up to 50kN. The particular machine used was an Instron 5869 screw driven load frame run with Instron

5800 controller and Bluehill 2.0 software. Instron UTM had a load cell of 50 kN and speed range of 0.001 – 500 mm/min. Instron model 3119-409 high temperature environmental chamber was used to achieve the desired high temperatures. The range of use for this chamber is from -70 to 250 °C. Liquid N2 or CO2 needs to be used to achieve the lower temperatures. The equipment uses only electricity to reach higher temperatures.

Figure 3.2 shows the Instron 5869 with the high temperature environmental chamber.

The samples used for the uniaxial compression tests were 8.8 mm in height and 10 mm in diameter. The samples used for plane strain compression were 8.8 mm in height and 9.6 mm in width and breadth. Plane strain experiments were carried out using a channel and die fixture as shown in Figure 3.3. The samples were stored in a desiccator for at least 24 hours before the tests to dehumidify the polymers. The dessicant chamber is as shown in figure 3.4. The glass transition temperature of the PMMA was found to be ~ 102 °C (375

K) using Differential Scanning Calorimetry (DSC) [36].

Figure 3.2 Instron 5869 with High temperature environmental chamber.

Figure 3.3 Channel and die fixture for plane strain experiments

3.3.2 Test Procedure

The oven is switched on at the desired temperature for at least 2 hours to ensure that the chamber is heated uniformly and to avoid any inconsistencies. This time is much longer than the time required to achieve any desired temperature. The specification of the environmental chamber 3119-409 states that the time required to reach the maximum temperature is around 30 minutes. Also, the recovery time to attain the desired temperature again after the door has been opened for 1 or 2 minutes is 4 or 5 minutes respectively.

The polymer samples were taken out of the desiccant chamber and the dimensions of the part were measured using a vernier calliper.

Teflon sheets were used to reduce the friction in the set-up. Lubricant WD-40 was applied between the Teflon sheets and the compression anvils. The sample was placed

Figure 3.4 Desiccators to store samples prior to testing

between the Teflon sheets which in turn were placed between the compression anvils; the polymer sample was not exposed to the lubricant. This set-up was sufficient for uniaxial

39 compression. For plane strain compression, additional Teflon sheets were used between the sample and the channel and die fixture to reduce the friction.

Once the sample was placed between the compression anvils, the load cell was calibrated and the displacement was set to zero before the start of the test.

All samples were placed in the chamber for 20 minutes before the start of the test to attain the test temperature. The time for the sample at room temperature to reach the desired temperature was estimated using Newton‘s law of cooling. The value of thermal conductivity used for the same was 0.209 W.K-1.m-1. These 20 minutes at high temperatures also relieve process-induced residual stresses.

After 20 minutes, the test was run according the test profile entered by the user. At the end of the test, the raw data (time, displacement and force) was stored in the computer which was used to calculate the stress-strain curves.

The test was repeated for the same test conditions until repeatable data was obtained.

3.3.3 Limitation of plane strain compression fixture

Plane strain experimental data at higher true strain than ~ 1.45 cannot be used due to the limitations and inaccuracies of the fixture. At this compression, the polymer samples are very close to the walls of the ‗die and channel‘ fixture and hence the data obtained may not represent the true stress values.

Assuming that the volume of the polymer sample does not change, the maximum final dimension of the compressed sample along the length is constrained by the length of the fixture. The length and breadth of the fixture is 41 mm and 9.6 mm respectively whereas the length and breadth of the sample is 9.6mm and its height is 8.8 mm.

Since, Initial Volume of the polymer sample is equal to the Final volume of the sample, the maximum deformed height of the polymer sample can be calculated by equating the initial and final volumes of the sample.

(9.6)*(9.6)*(8.8) = (9.6)*(41)*(hmaximum)

hmaximum = 2.0605 mm.

The maximum true strain that can be accurately obtained in the fixture can be calculated using equation 3.1 with the initial and the maximum deformable height.

From the known values of maximum deformable height, hfinal of 2.0605 mm and the original height of the sample, horiginal of 8.8 mm, the maximum true strain achievable on this ‗die and channel‘ fixture is, εtrue = 1.452.

3.3.4 Experimental Errors

Several sources of experimental error were identified during the testing. The steps taken to reduce or eliminate the effects of these errors are discussed below.

(i) Manual errors

Some stress-strain plots observed were very different from one another for the

same test conditions. Such data was discarded and the tests were repeated

until repeatable data was obtained.

For example, figure 3.5 shows a stress-strain plot at 120 °C and strain rate of

4.0/min. The sample was probably not in contact with the compression anvil

when the test started; this resulted in the test stopping well before the desired

true strain or displacement was obtained. The compression anvil stopped 41

moving after a certain desired displacement which resulted in an incomplete

and inaccurate test. Figure 3.6 shows the stress-strain data for the same test

conditions with different samples. These results were used unlike that of

sample 1.

Though this is an extreme case of the polymer sample being significantly off

rather than being just in contact with the compression anvil at the start of the

test, the same scenario could have happened at other times in much smaller

magnitudes. Care was taken to reliably start all tests with the sample

appropriately positioned between the anvils.

(ii) Non-uniform air-flow around the sample

The sample temperature was measured for a few experiments using a

thermocouple to ensure that the desired or set temperature was obtained. The

thermocouple was stuck to the polymer sample using a cellophane tape. The

end of the thermocouple was not covered with the tape. The temperatures

measured were within 2-3 °C of the set point in uniaxial compression but were

off by as much as 10 °C in some cases of plane strain compression.

This was most likely due to the fact that the channel and die fixture did not

allow the hot air to freely circulate around the polymer sample. A more

accurate way to conduct the experiments needs to be determined to conduct

the high temperature plane strain compressions experiments in the future.

Also, since the thermocouple on polymer sample could not be seen from

inside the channel and die fixture, it cannot be conclusively said that the

temperature measured was of the polymer sample not the air around it. The

polymer could have been at a higher temperature since it was also in contact

with the metallic fixture.

(iii) Equipment errors or inaccuracies

The extended range test frame of the Instron adds some compliance to the

system, leading to some inaccuracies in force and/or displacement

measurements.

The accuracy range of environmental chamber was + 2 °C, so the temperature

could easily have been + 2 of the set temperature or even more at some

instances.

The die and channel fixture used for plane strain compression were not rigid,

hence the samples did not undergo absolutely true plane strain compression.

4.5 Sample 1 4

3.5

2.5

1.5 -TrueStress (MPa) 1

0.5

0 0 0.5 1 1.5 -True Strain

Figure 3.5 Stress-strain plot at 120 °C and strain rate of 4.0/min of a Sample 1. 43

30 Sample 2 Sample 3 25

10 -TrueStress (MPa)

0 0 0.5 1 1.5 -True Strain

Figure 3.6 Stress-strain plot at 120 °C and strain rate of 4.0/min of a Sample 2 and

Sample 3.

(iv) Sample variability

There could be numerous differences amongst the polymer samples. For

example, each sample could not have been a perfect cube with all perfectly

parallel and perpendicular surfaces.

(v) Environmental factors

The environmental conditions could be very diverse from one test to another.

There could have been differences in temperature or humidity

(vi) Miscellaneous

There could have been other unsuspecting causes for errors in the

experiments.

Of all the listed errors, the most significant was the non-uniform air-flow around the sample for plane strain compression. As mentioned before, the temperatures measured were off by as much as 10 °C in some cases of plane strain compression. The stress-strain data would look very different if the polymer samples were at 10 °C lower than assumed, the significance of this error and how it may affect the outcome will be discussed in chapter 4 with the experimental data. All the other error were less significant and would not change the basic trends observed.

3.4 Experimental Results

As stated earlier, these experiments are filling the gaps in the existing data on PMMA in the lab. Previous work by Palm and Ghatak [21, 22, 36, 65] had covered most of the experimental work on PMMA. Table 3.1 shows the previous uniaxial experiments conducted on PMMA by Palm (bullet) and Ghatak (shaded area). Table 3.2 shows the plane strain experiments conducted by Palm (bullet) and also the new data obtained in this work (shaded area).

Plane strain compression experiments were conducted at temperatures of 120 and 130 °C and strain rates of 0.2/min, 0.4/min, 1.0/min, 3.0/min and 4.0/min to a strain level of 1.4 or 140%. Temperature and strain rate dependence was observed in the true stress versus true strain curves. Figure 3.7 shows the original/undeformed and deformed shapes of uniaxial and plane strain samples and figure 3.8 shows a schematic of uniaxial and plane strain compression samples with load, flow and constrained direction [32]. The original

45 shape of the samples is shown by the dashed lines and the deformed shape of the samples is shown by solid lines.

Compressive Temperature in °C ▼ strain rate per minute

▼ 25 82 92 102 107 110 112 115 120 122 125 130 132

0.05

0.06 ●

0.2 ● ● ● ● ● ● ● ●

0.4 ● ● ● ● ● ● ● ●

0.6 ●

0.8

1.0 ● ● ● ● ● ● ● ●

2.0

3.0 ● ● ● ● ● ● ● ●

4.0 ● ● ● ● ● ● ● ●

5.0

6.0 ●

Table 3.1 Previous uniaxial experiments conducted on PMMA by Palm (bullet) and

Ghatak (shaded area).

Compressive Temperature in °C ▼ strain rate per

minute ▼ 25 82 92 102 107 112 120 122 130 132

0.06 ●

0.2 ●

0.4 ●

0.6 ●

1.0 ● ● ● ● ● ● ● ●

3.0 ● ● ● ● ● ● ●

4.0 ●

6.0 ●

Table 3.2 Plane strain experiments conducted by Palm (bullet) and also the new data

(shaded area) obtained in this work

More plane strain experiments could have been conducted at lower temperatures but the primary goal is to have adequate experimental data to be used to develop a material model. A considerable number of uniaxial experiments have been conducted on the polymer PMMA which will provide the required database for the constitutive model, in addition, a sufficient amount of experimental data on plane strain is available to ensure that the material model captures the strain state dependence.

Figure 3.7 Deformed and undeformed samples in plane strain and uniaxial compression

Figures 3.9 – 3.12 show experimental data for plane strain compression. As observed in figures 3.9 and 3.10, at a given temperature, stress decreases with decreasing strain rate.

Also, the stresses decrease with increasing temperature as observed in figures 3.11 and

3.12. Figure 3.11 – 3.12 have been plotted using new data generated here and previous data by Palm [36]. The initial stiffness and strain softening are higher/more pronounced at lower temperatures and at higher strain rates, consistent with other data in the literature for amorphous polymers.

Figure 3.8 A schematic of uniaxial and plane strain compression samples with load, flow and constrained direction [32].

50 -0.2/min -0.4/min 40 -1/min -3/min -4/min 30

20 True Stress (MPa) Stress True

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 True Strain

Figure 3.9 Plane strain compression experimental data for PMMA at 120 °C

50 -0.2/min -0.4/min 40 -1/min -3/min -4/min 30

20 True Stress (MPa) Stress True

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 True Strain

Figure 3.10 Plane strain compression experimental data for PMMA at 130 °C 50

100 82 C 92 C 80 102 C 107 C 112 C 60 120 C 130 C

40 True Stress (MPa) Stress True

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 True Strain

Figure 3.11 Plane strain compression experimental data for PMMA at strain rate of

1/min

100 82 C 92 C 80 102 C 107 C 112 C 60 120 C 130 C

40 True Stress (MPa) Stress True

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 True Strain

Figure 3.12 Plane strain compression experimental data for PMMA at strain rate of

3/min 51

Additional experimental data is shown to illustrate the consistency of the all data used for further development of the material model.

Figure 3.13 shows plane strain compression data at strain rate of 1.0/min by Palm [36], the plots at 122 and 132 °C is comparable to the plots at 120 and 130 °C in figure 3.11.

Similarly, plane strain compression data at strain rate of 3.0/min at 122 and 132 °C by

Palm [36] in figure 3.14 is comparable to the plots at 120 and 130 °C in figure 3.12. Any data collected past a strain of 1.4 should be disregarded for reasons discussed in section

3.3.3.

Figure 3.13 Plane strain compression at strain rate of 1.0/min [36].

Figure 3.14 Plane strain compression at strain rate of 3.0/min [36].

Although uniaxial compression experiments on PMMA were not conducted here, some data from previous work [36, 65] has been shown to overlap with the stress-strain data.

Figure 3.15 shows uniaxial compression of PMMA at 107 °C by Palm and figure 3.16 shows the same by Ghatak. Some data plots are also shown to illustrate the trends amongst stress-strain data. Figure 3.17 shows uniaxial compression of PMMA with various temperatures at 3/min while figure 3.18 shows uniaxial compression of PMMA with various strain rates at 120 °C.

Figure 3.15 Uniaxial compression of PMMA at 107 °C by Palm [36].

40 -0.2/min 35 -0.4/min 30 -1.0/min -3.0/min 25

-TrueStress (MPa) 10

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -True Strain Figure 3.16 Uniaxial compression of PMMA at 107 °C by Ghatak [65].

80 82 C 70 92 C 102 C 60 107 C 112 C 50 120 C 40 130 C

TrueStress (MPa) 20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 True Strain

Figure 3.17 Uniaxial compression of PMMA with various temperatures at 3/min

25 -0.2/min -0.4/min 20 -1/min -3/min -4/min 15

10 TrueStress (MPa) 5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 True Strain

Figure 3.18 Uniaxial compression of PMMA with various strain rates at 120 °C.

3.5 Discussion

The plane strain experiments conducted, along with the data existing in our lab [36, 65] complete the desired data base for constitutive modeling of the polymer PMMA. The plots show four major regions, these are easier to observe at low temperature stress-strain data (below Tg) (1) a straight line depicting the initial elastic region and yield stress, (2) followed a region of strain softening or flow where the stress values decrease, (3) initial gradual strain hardening where the stress value again begin to increase, and (4) finally dramatic strain hardening at large strains. These major four regions of the stress-strain curve can be seen in figure 3.19. All the four features show temperature and strain rate dependence.

Figure 3.19 Stress-strain curve for compression of polymer below its Tg.

The following observations were made from the experimental data. At a particular strain rate, the temperature dependence can be easily seen as shown in figure 3.11, 3.12, 3.13,

3.14 and 3.17. (i) The stresses are lower at higher temperatures since the polymer chains can easily move past each other which in turn lower the force required to deform the material. (ii) The initial elastic region is represented by a straight line, this line is steeper at lower temperatures. This elastic region almost disappears once we move about 10-15

°C above Tg. (iii) The following region of yield or strain softening is again more prominent at temperatures below the glass transition. This region also decreases with increase in temperature and is not observed at higher temperatures. (iv) Strain softening does not occur at higher temperatures. This is because the polymer sits for at least 20 minutes before the start of the tests and hence the residual stresses/thermal history is removed at these warm temperatures. (v) Both the initial gradual strain hardening and the dramatic strain hardening decrease with increase in temperature. The stress-strain lines become less steep at higher temperatures. Also, the onset of dramatic hardening happens at larger strains with increase in temperature. (vi) Strain hardening virtually disappears at around 15 °C above Tg. The polymer begins to behave more fluid-like at these temperatures.

Similarly, at a particular temperature, the effects of strain rate can be easily observed as seen in figure 3.9, 3.10, 3.15, 3.16 and 3.18. (vii) Stresses decrease as the strain rate decreases. (viii) The changes in initial elastic region, yield and strain hardening with increasing strain rates were same as with decreasing temperatures. At lower rates, the heat generated by the plastic deformation of the polymer has time to dissipate unlike at

57 higher strain rates. At higher rates, the heat is not able to dissipate as quickly as it is being generated leading to increase in the temperature of the polymer sample.

The stresses were higher in plane strain than in uniaxial compression for the polymer.

However, all the observations showed same trends amongst the stress-strain data for the polymer in both the modes of deformation, uniaxial and plane strain compression.

The initial modulus, yield and strain hardening, all these decrease with increase in temperature and decrease in strain rate. All the results show both temperature and strain rate dependence. For a constitutive model to be able to capture all of this observed behavior, it must incorporate the following features: (i) initial modulus, (ii) followed by yield or flow, (iii) initial gradual strain hardening, and (iv) dramatic strain hardening [1].

Only a material model which is able to include all these features with temperature, strain rate and strain state (mode of deformation) dependence will accurately predict the material behavior. The existing constitutive model for the polymer will be tested against the experimental data in the following chapter and the observations will be used to improvise and hence develop a new material model.

CHAPTER 4

EXISTING CONSTITUTIVE MODELS

4.1 Introduction

Very few material models have been developed to try and capture the material behaviour of polymers across the glass transition temperature. In this chapter, two existing constitutive models for PMMA with experimental data are studied: the model by Dupaix and Boyce [1, 21, 34, 65] and the model by Dooling and Buckley [19]. The material models are assessed in their ability to capture the flow behaviour of the material over a very wide range of strain rates and temperatures of significance in industrial applications, in particular, hot embossing. The continuum model for PMMA by Dooling et al. [19] was investigated to incorporate some of its components into the Dupaix-Boyce model. Both models are then fit to experimental data for PMMA obtained in our lab over the desired ranges of temperature and strain rate. To quantify which of these two models fits better to the experimental data, the error was calculated with the optimized set of constants for both the continuum models. The optimized set of constants were obtained using the

Nelder-Mead Algorithm [66, 67]. Results from this study will be used to develop a new model for PMMA.

The model by Dooling, Buckley, et al. [19] has been considered for this purpose since the make-up of the model is very similar to the Dupaix-Boyce model. Dooling-Buckley model is differential in nature unlike most of the models in literature for polymers above the glass transition temperature. In addition, this model uses springs and dampers to capture material behaviour similar to the Dupaix-Boyce model. Both models have temperature dependence and although the authors of Dooling et al. [19] do mention that the model would be inaccurate at high rates of deformation, they do not analyse its strain rate dependence. Nevertheless, our exploration of the model found that it does contain relatively accurate strain rate dependence at moderate rates. Also, the experiments conducted for the Dupaix-Boyce model were compression tests and those for Dooling et al. [19] model were tension tests. A detailed description of the models is presented here.

4.2 Dupaix-Boyce (DB) Model

4.2.1 Details of Constitutive Model

A schematic of the constitutive model is shown in Figure 4.1. The constitutive model consists of two resistances: intermolecular, I and network, N. Resistance I denotes the intermolecular interaction between the polymer chains and resistance N denotes the network interactions of the polymer chains. Each spring relates the elastic deformation gradient to stress and each damper relates the plastic strain rate to the shear stress. The spring on the intermolecular resistance is linear and the dashpot follows a thermally activated process. The bulk modulus (B) and shear modulus (μ) change with temperature making the intermolecular component temperature dependent. The spring on the

60 molecular network resistance is highly non-linear and captures strain stiffening. The damper on the molecular part follows an empirical power law and includes both strain and temperature dependence.

Figure 4.1 Schematic of the Model by Dupaix and Boyce

The two resistances are in parallel, thus the total stress (T) is equal to the sum of the

Cauchy stresses and the total deformation gradient (F) is equal to deformation gradient in each component.

The total stress (T) is given by

T = TI + T N (4.1) and the deformation gradient for each branch is given by

F  F  F I N (4.2) where subscript I and N denote the contribution from intermolecular interactions and the molecular network, respectively.

Figure 4.2 shows the combined contribution of the two resistances: intermolecular interaction (I) and the molecular network interactions (N) to the total stress. Their individual contributions are shown in figure 4.3. The intermolecular interaction accounts for the initial yield of the material whereas molecular network interaction accounts for the subsequent strain hardening of the material.

Figure 4.2 Schematic showing the contributions of the two resistances to the total stress

The deformation gradient is decomposed into elastic and plastic parts multiplicatively.

These are further divided into rotation (R) and stretch (V) components.

ep FFFi i i (4.3)

FVRe e e (4.4) i i i

p p p FVRi i i (4.5) where the subscript i denotes the resisitance I or N, and the superscripts e and p denote the elastic and plastic contributions, respectively. In all of these equations, repeated subscripts does not imply summation. The velocity gradient (L) is defined as follows:

1 LFFi i i (4.6)

From (4.3), (4.4), (4.5) and (4.6), the velocity gradient becomes

LFFFFFFLLe e1  e p p  1 e  1  e  p (4.7) i i i i i i i i i

(a) (b)

Figure 4.3 Individual contributions of the two resistances to the total stress, (a) intermolecular resistance, and (b) molecular network resistance.

p The plastic velocity gradient in the current configuration ( Li ) is defined as a sum of the

p p p antisymmetric plastic spin and the symmetric plastic rate of stretching LDWi i i  .

p The antisymmetric plastic spin is assumed to be equal to zero Wi  0. The plastic rate of stretching is

pp DNi  i i (4.8)

1/2 1 where NT1/ 2 ' ,   (1/ 2)tr ( TT'' ) , and TTT' tr  . T is the Cauchy i i i i i i i i3 i i

p stress tensor in each spring element, while  i is the plastic strain rate in each dashpot.

Resistance I: Intermolecular Interactions

The elastic part of resistance I can be taken as follows:

1 ee TVIII L ln (4.9) JI

1 where J  det Fe is the volume change, ln Ve is the Hencky strain [68], is the I I J I fourth order tensor of elastic constants, and the superscript e denotes the elastic part of the deformation gradient.

The plastic part of resistance I is assumed to follow a thermally activated process:

p GsIII1/  II0 exp  (4.10) k

64 where GI is the activation energy of the material which must be overcome before flow

can begin,  0I is a pre-exponential factor, sI is the shear resistance, taken to be 0.15 times the shear modulus, k is Boltzmann‘s constant,  is the absolute temperature of the

material, and  I is the magnitude of the deviatoric stress [1].

The shear modulus strongly depends on temperature and is captured through:

1 1 5   (   )  (   )tanh( (  ))  X (  ) (4.11) 2 g r 2 g r  g g g

where μg represents the modulus in the glassy region, r represents the modulus in the rubbery region,  is the temperature range across which the glass transition occurs, and

X g is the slope (of μ versus θ) outside the glass transition regime.  g is the glass transition temperature of the material. Since the modulus (  ) in the glass transition region is also strain-rate dependent, it is taken into account by modifying the glass transition temperature as:

* p   g : I  ref  p  g      *  log  I   :  p   (4.12)  10   g I ref   ref 

 * where g is the reference glass transition temperature taken to be 102 °C,  is a material

 constant, and ref is the reference strain rate, equal to .00173/sec.

The bulk modulus is also taken to be temperature dependent, following a similar fashion to equation (4.12):

1 1 5 B  (B  B )  (B  B ) tanh( (  )) (4.13) 2 g r 2 g r  g

Resistance N: Molecular Network Interactions

Resistance N consists of a highly non-linear spring and a dashpot. While the spring captures the strain-stiffening effects in the polymer, the dashpot represents the molecular relaxation at higher temperatures or lower strain rates. The elastic spring in (N) makes use of the Arruda-Boyce 8-chain model:

1 vk N  2 TBIL1 N e  (4.14) NNN  J NN3  N where N is the number of rigid links between entanglements, and v is the chain density.

These are the only two material constants in this equation. L1 is the inverse Langevin

1 function defined as L( ) coth( ) . The effective chain stretch  N is given by the  root mean square of the distortional applied stretch:

1/ 2 1 e  N   trBN  (4.15) 3 

e e e T e 1/ 3 e e BN  FN FN  , FN  J N  FN , J N  det FN (4.16)

The rate of molecular relaxation for resistance N is given by:

1/n p /1 cN     N  C    (4.17) 0 /1 cc  vk   where n is a power-law exponent,  is a measure of the orientation of the polymer chains

with initial value0 . c is a cutoff value, beyond which molecular relaxation ceases. is calculated as:

  min( , , )     cos 1  1 2 3  (4.18) 2  2 2 2   1  2  3 

where λi are the principal stretches.

The parameter C is temperature dependent and is given by:

Q CDexp (4.19) R where D and Q/R are the material parameters.

4.2.2 Optimization of Material Constants

The optimized constants of the Dupaix-Boyce model are as shown in Table 4.1. These constants were optimized using the Nelder-Mead Algorithm [66, 67]. The error used in the optimization was found by comparing the model predictions with four representative experimental stress-strain plots. The compression experimental data on PMMA that was used for optimization is shown in Table 4.2.

Material Property Symbol Value

Glassy Modulus μg 293 MPa  Rubbery Modulus r 55 MPa Temperature Shift  48 K X Transition Slope g -4.2 KPa/K Initial Elastic Rate Shift Factor  3.6 K Behavior Glassy Bulk Bg 0.64 GPa Modulus Rubbery Bulk Br 2.73 GPa Modulus Pre-exponential  13 0I 7.08 10 1/s Flow Stress Factor Activation Energy G 2.14 1019 J Rubbery Orientation vk 8.0 MPa Modulus Resistance Elasticity Entanglement N 500 Density Temperature D 1.34 107 K Coefficient Second Temperature Molecular QR/ 1.75 104 1/s Parameter Relaxation Power-law 1/n 7.4 Exponent  Cutoff Orientation c 0.0017

Table 4.1 Optimized material constants for the Dupaix-Boyce model using Nelder-Mead

Algorithm.

Stress State Temperature Strain rate

1 Uniaxial compression 102 °C 1.0/min

2 Uniaxial compression 112 °C 1.0/min

3 Uniaxial compression 112 °C 3.0/min

4 Plane strain compression 112 °C 1.0/min

Table 4.2 Compression experimental data used for optimization of DB Model.

4.2.3 Comparison with Experimental Data

The new set of optimized constants was subsequently used to predict the remaining experimental data. This was done to ensure that constants fit all the stress-strain data and not just the representative stress-strain plots used for optimization. The stress-strain plots for the optimized set of constants are shown in following figures. Figures 4.4-4.6 show the experimental data versus the model simulation for uniaxial compression at temperatures 102 °C, 110 °C and 115 °C respectively. As was seen in previous work with this model, the optimized set of constants is also not able to capture the experimental data at 115 °C as shown in figure 4.6. Figures 4.7-4.8 show the experimental data versus the model simulation for plane strain compressions at temperatures 112 °C and 120 °C respectively.

25 -0.05/min exp -1.0/min exp 20 -2.0/min exp -3.0/min exp -0.05/min sim 15 -1.0/min sim -2.0/min sim

10 -3.0/min sim -True Stress (MPa)-TrueStress 5

0 0 0.5 1 1.5 -True Strain

Figure 4.4 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 102 °C with optimized set of constants

25 -0.05/min exp -2.0/min exp 20 -3.0/min exp -6.0/min exp -0.05/min sim 15 -2.0/min sim -3.0/min sim

10 -6.0/min sim -True Stress (MPa)-TrueStress 5

0 0 0.5 1 1.5 -True Strain

Figure 4.5 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 110 °C with optimized set of constants

10 -0.05/min exp -1.0/min exp 8 -3.0/min exp -0.05/min sim -1.0/min sim 6 -3.0/min sim

4 -True Stress (MPa)-TrueStress 2

0 0 0.5 1 1.5 2 -True Strain

Figure 4.6 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 115 °C with optimized set of constants

50 -0.2/min exp -0.4/min exp 40 -1.0/min exp -3.0/min exp -4.0/min exp 30 -0.2/min sim -0.4/min sim 20 -1.0/min sim -3.0/min sim

-True Stress (MPa)-TrueStress -4.0/min sim 10

0 0 0.5 1 1.5 -True Strain

Figure 4.7 Stress-strain data for PMMA in plane strain compression with experimental data and model simulation at 112 °C with optimized set of constants

30 -0.2/min exp -0.4/min exp 25 -1.0/min exp -3.0/min exp 20 -0.2/min sim -0.4/min sim 15 -1.0/min sim -3.0/min sim

10 -True Stress (MPa)-TrueStress

0 0 0.5 1 1.5 -True Strain

Figure 4.8 Stress-strain data for PMMA in plane strain compression with experimental data and model simulation at 120 °C with optimized set of constants

In uniaxial compression, the model works well for temperatures about 112 °C (Tg + 10) and for true strain rates of 0.2 – 4.0 /min. Nonetheless, the model fails to work at higher temperatures (above Tg + 15) where the behaviour of the polymer is more fluid-like. The model predicts much higher initial stresses and also over predicts the strain hardening at higher strains. For plane strain compression, the model is better able to predict stress- strain data at higher temperatures than in uniaxial compression. The model predicts the stress value more accurately at lower temperatures (115 °C or lower) in uniaxial compression but is unable to capture the extent of strain hardening. At 112 °C, the model over predicts the initial yield stress and is also unable to predict the strain hardening.

Also, at 120 °C the model over predicts the initial stress; however, it does perform better at higher strains.

Although the DB model does differentiate between uniaxial and plane strain compression, it is not able to capture either of the stress states accurately at temperatures much higher than the glass transition temperature (Tg + 10 °C). The constitutive model over predicts the initial yield stress in both the cases, however it over predicts the strain hardening in uniaxial compression and under predicts the same in plane strain compression. The model does predict higher stresses for plane strain compression than uniaxial compression which is both as expected and desired but not as high as observed.

4.3 Dooling-Buckley-Rostami-Zahlan (DBRZ) Model

4.3.1 Details of Constitutive Model

A schematic of the model is shown in Figure 4.9. The total stress predicted by the model

b consists of two main components: a bond-stretching stress (  ) and a conformational stress ( c ). The bond stretching part is dominant in the initial glassy region whereas the conformational stress is dominant later in the rubbery regime.

The total stress ( ) is given by

 b   c**ww   c 1 1 2 2 (4.20)

ww1 12 (4.21)

Figure 4.9 Schematic of the Model by Dooling et al. [19]

where the subscripts 1 and 2 represent the two conformational stresses with ‗w‘ being the weighting factor found experimentally. The spring in the bond-stretching part is linear and in the conformational part is non-linear. The bond-stretching component has a linear viscoelastic relaxation time which is temperature dependent and the conformational component has temperature dependence through the viscosity term.

The stress in the model consists of two components: a bond-stretching part (b) and a conformational part (c). The bond-stretching stress is due to stretching of primary and secondary bonds and the conformational stress is due to conformational entropy. These components act independently and in parallel. Thus, the deformation gradient (F) and the

Cauchy stress tensor (σ) are as follows:

The rate of deformation tensor (D) (symmetric part of the velocity gradient) is:

The model consists of one ‗bond stretching‘ (b) and two ‗conformational‘ (c) contributions in parallel. The weighing factors (w) were found experimentally. The total

Cauchy stress is given by

B: Bond-stretching Stress

The bond-stretching stress (σb) is further decomposed in elastic and viscoplastic components. The elastic part (e) is due to elastic distortion of inter- and intra-molecular potentials and the viscoplastic part (v) is due to thermally activated short-range diffusion of molecular segments. It follows that the rate of deformation Db is given by:

The hydrostatic part of the bond-stretching response is taken to be linear elastic with bulk

b b modulus K . The mean stress σ m and the volume ratio J are defined by

and they are related as follows:

b where σ m0 is the initial, built-in, bond-stretching stress resisting collapse of the entropic network. The deviatoric rate of deformation of the elastic and viscous parts are De΄ and

Dv΄. Sb is the deviatoric part of σb and a hat denotes the time derivative with respect to a reference frame rotating with the material:

Here, Gb is the bond-stretching contribution to the shear modulus, and the relaxation time

(τ).

The relaxation time is given in terms of the linear viscoelastic relaxation time ( τ0 ) and a stress-dependent factor ‗a’ derived from a three-dimensional generalization of the Eyring flow model.

From (4.26) and (4.29), the deviatoric part of the rate of deformation D gives

where (identified with the Jaumann rate of stress) and are given by

where, W is the spin tensor, is the octahedral shear stress and σ is the strain-

induced mean stress σ σ , while and are the shear and pressure activation volumes, respectively.

Dependence of the viscoelastic relaxation time on temperature T and structure (via fictive temperature ) is as follows:

* where is the linear viscoelastic relaxation time for a reference temperature T and

structural state (fictive temperature) [69]. Above the glass transition, structural equilibrium obtains Tf = T.

C: Conformational Stress

The conformational part is also decomposed into network stretch and slippage. The network stretch (n) associated with stretch as in the bond-stretching part and the slippage

(s) is associated with relaxation of molecular orientation or slippage of the entanglements:

The conformational stress is directly obtained by differentiation of the conformational free energy density Ac. Theories of rubber elasticity yield expressions for Ac in terms of

n the principal network stretches λ i (i = 1 …. 3). Thus, the principal components of the

Cauchy stress are obtained from

The Edwards and Vilgis [29] expression for Ac was employed. The network behaviour of uncrosslinked PMMA is modelled as a form of Edwards–Vilgis rubber, where there are no chemical crosslinks:

77 where Ns is the number density of slip-links, η is a parameter specifying the looseness of the entanglements (η = 0 for a crosslink), and α is a measure of the inextensibility of the entanglement network. It is related to the number n of freely orienting (Kuhn) segments per network chain: .

An isotropic, isochoric Newtonian flow rule is invoked and the slippage rate of deformation is:

where γ is a slippage viscosity that depends only on temperature. The temperature dependence of the slippage viscosity is modelled in terms of the Fulcher equation

where, Tf = T above the glass transition, and the limiting or Vogel temperature for the slippage process may not be identical to that for the segmental flow process.

Using (4.25), eq. (4.34) and (4.37) becomes:

Eq. (4.35) becomes:

4.3.2 Replication of the DBRZ Model

The glass transition temperature of the PMMA used in their paper was ~ 110 °C. This model was shown to work very well in uniaxial tension above the glass transition temperature but the parameters were limited to temperatures at and above 120 °C. The results from the Dooling et al. model were first successfully replicated to understand the working of model and write the accurate codes.

The simulation of uniaxial tensile drawing of PMMA showing all the components of the stress from the paper [19] and their reproduction is shown in Figure 4.10. Figure 4.11 (a) shows the experimental data for uniaxial drawing of PMMA at 120 °C [19] and 4.11 (b) shows simulation results obtained using the model.

(a) (b)

Figure 4.10 Simulation of uniaxial tensile drawing of PMMA at 122 °C and nominal strain rate of 0.4 /min, (a) From Dooling et al, [19], and (b) Reproduction using the

Dooling et al.

(a) (b)

Figure 4.11 Plots of true stress versus nominal strain rates (per min) for uniaxial drawing of PMMA at 120 °C. (a) Experimental data (Dooling et al, [19]) (b) Simulation using the model by Dooling, Buckley et al. [19]

4.3.3 Optimization of Material Constants

The constants for the Dooling-Buckley-Rostami-Zahlan (DBRZ) model were optimized using the Nelder-Mead Algorithm [66, 67]. These material constants were optimized in two sets one at a time. The two groups were constants of i) Bond-stretching Component, and ii) Conformational Component. The error calculated to optimize was found using four representative data sets of stress-strain plots. The compression experimental data on

PMMA that was used for optimization is shown in Table 4.3

The first three of the four mentioned conditions are the same as used for the optimization of Dupaix-Boyce (DB) model. The fourth set of stress-strain data used in optimizing DB model was from a plane strain compression experiment. This particular set of data was

80 not used in the DBRZ model since the model was not able to capture the difference between uniaxial and plane strain compression.

Stress State Temperature Strain rate

1 Uniaxial compression 102 °C 1.0/min

2 Uniaxial compression 112 °C 1.0/min

3 Uniaxial compression 112 °C 3.0/min

4 Uniaxial compression 120 °C 1.0/min

Table 4.3 Compression experimental data used for optimization of DBRZ Model.

Given that the glass transition temperature of the PMMA used for fitting the model was different from the PMMA used by Dooling et. al. [19], the slippage viscosity would also change. The temperature dependence of the slippage viscosity normalized at 128 °C from the Dooling et. al [19] article is as shown in figure 4.12 (a). Since the DBRZ model was developed for temperatures above glass transition (Tg), the same slippage viscosity temperature dependence could not be used for temperatures at and below Tg without modification. First, the curve was shifted along the x-axis with the peak being moved from 120 °C to the Tg of the PMMA used in the experiments (102 °C). In addition, a constant slippage viscosity was used for temperatures at and below Tg. The new and modified temperature dependence of the slippage viscosity is as shown in figure 4.12 (b).

Slippage viscosities normalized at 110 C Slippage viscosities normalized at 128 C 16 10 4 10 X: 102 15 3 Y: 5.653e+015 10 10

2 14 10 10

1 10 13 10 0 10 12

Normalizeds) (TPaviscosity 10

Normalizeds) (TPaviscosity -1 10

-2 11 10 10 110 120 130 140 150 160 170 180 60 80 100 120 140 160 180 Temperature (C) Temperature (C)

(a) (b)

Figure 4.12 (a) Temperature-dependence of slippage viscosities normalized to their values at 128 °C, reproduced using Dooling et. al. [19], (b) Modified temperature- dependence of slippage viscosities normalized to their values at 110 °C for obtaining the new set of constants for DBRZ model.

4.3.4 Comparison with Experimental Data

The new set of optimized constants for the experimental data from the work in our lab, plus from previous work by our group [36, 65] are as shown in Table 4.4. Some of the material constants remain unchanged from the initial work by Dooling et. al. [19]. The new optimized constants are shown in bold. The experimental data (Dupaix Lab) and model simulations (DBRZ Model) for the stress-strain data for uniaxial compression are shown in figures 4.13-4.15. Figures 4.16-4.17 show the experimental data (Dupaix Lab) and model simulations (DBRZ Model) for the stress-strain data in plane strain compression at 112 °C and 120 °C respectively.

Material Property Symbol Value

Unrelaxed Shear Modulus G 0.83 GPa

Bulk Modulus K 2.12 GPa

* Glass viscosity μ o 3.5 GPa.s

Bond- Cohen/Turnbull Constants C 217 K

Stretching Limiting Temperature T∞ 302.3 K

Component Activation Enthalpy ΔHo 289 kJ/mol

3 3 Shear Activation Volume Vs 4.423 10 m /mol

3 3 Pressure Activation Volume Vp 0.399 10 m /mol

* Reference State Temperature T 373 K

26 -3 Slip-link Density Ns 3.01 10 m

Inextensibility Factor α 0.15

Slip-link Mobility Factor η 0.0 Conformational Slippage Cohen/Turnbull s Component C 21.6 K Constant

s Limiting Slippage Temperature T ∞ 373 K

* Structural State Temperature T f 380 K

Slippage Network Fractions (k = 1, 2) wk 0.677 0.323

* Spectrum Viscosities (k = 1, 2) γ k 3.92 576 TPa.s

Others Glass Transition Temperature Tg 375 K (~ 102 °C)

Table 4.4 Optimized material constants of the DBRZ model for the experimental data from work by Dupaix lab [36, 65]. Changed values are shown in bold.

30 -0.05/min exp -1.0/min exp 25 -2.0/min exp -3.0/min exp 20 -0.05/min sim -1.0/min sim 15 -2.0/min sim -3.0/min sim

10 True Stress ( MPa ) ( StressMPa True 5

0 0 0.5 1 1.5 Nominal strain

Figure 4.13 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 102 °C

30 -0.05/min exp -2.0/min exp 25 -3.0/min exp -6.0/min exp 20 -0.05/min sim -2.0/min sim 15 -3.0/min sim -6.0/min sim

10 True Stress ( MPa ) ( StressMPa True 5

0 0 0.5 1 1.5 Nominal strain

Figure 4.14 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 110 °C 84

Figure 4.15 Stress-strain data for PMMA in uniaxial compression with experimental data and model simulation at 120 °C

30 -0.4/min exp -1.0/min exp 25 -3.0/min exp -4.0/min exp 20 -0.4/min sim -1.0/min sim 15 -3.0/min sim -4.0/min sim

10 True Stress ( MPa ) ( StressMPa True 5

0 0 0.5 1 1.5 Nominal strain

Figure 4.16 Stress-strain data for PMMA in plane strain compression with experimental data and model simulation at 112 °C 85

30 -0.4/min exp -1.0/min exp 25 -3.0/min exp -4.0/min exp 20 -0.4/min sim -1.0/min sim 15 -3.0/min sim -4.0/min sim

10 True Stress ( MPa ) ( StressMPa True 5

0 0 0.5 1 1.5 Nominal strain

Figure 4.17 Stress-strain data for PMMA in plane strain compression with experimental data and model simulation at 120 °C

For uniaxial compression, the constants do a very good job in capturing the experimental data at 102 °C and 110 °C, as well as the strain rate dependence. The DBRZ model predicts the stress well but the experimental data shows more strain hardening than the model predicts. This set of constants also does not work very well at 120 °C, though they are better able to predict the initial yield stress than the DB model as shown in figure

4.15. In addition, the model is unable to capture any difference between the two strain states: uniaxial and plane strain compression. At 120 °C the model appears to capture the plane strain compression data well, but in reality the model predicts the same stresses in both uniaxial and plane strain compression as seen in figure 4.15 and 4.17.

4.4 Discussion

Uniaxial and plane strain experiments were conducted on PMMA over a wide range of temperatures and strain rates to gather material behaviour. The data obtained from uniaxial compression tests are very consistent with the existing data on PMMA [16, 18,

19]. The PMMA used in the aforementioned experiments had different glass transition temperatures than the material used in the literature by other groups, so the stress values are scaled slightly differently. Plane strain compression data obtained by Palm [36] in the

Dupaix lab compares well with the experimental data available in the literature for

PMMA [9] at room temperature (~23 °C). There is no data available for plane strain compression of PMMA with variation in temperature and strain rates at large strains around and above Tg in the literature aside from the work done by Palm [36].

The Dupaix-Boyce model [21, 22, 34] works for temperatures up to Tg + 10 °C but not for temperatures higher than Tg + 15 °C [22]. As a consequence, there is a need to improve the current version of the model to try and capture the experimental data for higher temperatures as well. An existing constitutive model for PMMA by Dooling et al.

[19] has been studied to understand the pros and cons of material modelling better. This model was selected since it works well for PMMA above the glass transition.

The experimental data was fit to these two existing models i) Dupaix-Boyce (DB) model, and ii) Dooling-Buckley-Rostami-Zahlan (DBRZ) model, and the material parameters were optimized using the Nelder-Mead algorithm [66, 67]. The following observations were made from the above mentioned study (i) The DB model works well for temperatures at and slightly above glass transition temperature (Tg). (ii) As we move

87 above Tg, the DB model over-predicts the stress for uniaxial compression and under- predicts the stress in plane strain as seen in figures 4.6 and 4.7 respectively. (iii) The

DBRZ model also works well at and above Tg as shown in figures 4.13-15. In addition, it does a better job of capturing the experimental data at Tg +18 (120 °C) than the DB model. (iv) The DBRZ model is unable to capture the behaviour in plane strain compression, as it does not differentiate between the two strain states (Figures 4.15 and

4.17 predict the same stresses). The reason why the DB model is able to distinguish between uniaxial and plane strain compression is the presence of the Arruda-Boyce spring in the molecular network resistance [17]. (v) The DBRZ model is better able to predict the change in the initial yield with temperatures across the glass transition temperature. (vi) The stresses predicted at higher strains (30 - 140 %) are not very accurate for either of the models.

The DBRZ material constants were fit to experimental data close to the Tg of the polymer. Hence, the model predicts the data better at temperatures close to the Tg than it does at higher temperatures (> Tg +20 °C) for the given set of material constants. But, we have already seen the DBRZ model to work well at temperatures well above Tg (Tg +18 to

Tg +50 °C) in work by Dooling et al. [19]. From our study and previous work, it appears that this model is able to capture the polymer behaviour over a somewhat narrow range of temperatures, but in order to span the full temperature range from below, through, to above Tg, at least two different sets of material constants would be required.

One way to achieve this would be to incorporate an additional parallel network on the molecular relaxation side, thereby capturing the experimental data better. However, as

88 seen from studying the DBRZ model, an additional network with the same temperature dependence only helps in fine tuning the curve and not in capturing the dramatic change in material behaviour both below and above Tg. A more successful solution will likely require two networks on the molecular relaxation side, but with different set of constants, similar to the approach followed in Srivastava et al. [50]. The two networks could be interpreted as different mechanisms that are active in the two different temperature regimes. Based on the results of this work, the next chapter is devoted to developing a simple model with two strain hardening/molecular relaxation networks to better capture the full range of temperatures relevant for processing of PMMA.

CHAPTER 5

NEW CONSTITUTIVE MODEL

5.1 Introduction

As discussed in the previous chapter, there is a need to develop a new model in order to predict the material behaviour across the entire temperature range from below, through, to above glass transition (Tg or θg). Most material models for PMMA existing in the literature are in one of the two temperature regions, (i) from below Tg and/or up to ~ Tg

+20, [21, 22, 41, 48, 70] and (ii) around Tg +20 and higher [19, 25, 51, 60]. From what we observed in Chapter 4, it seems that the same set of material constants cannot capture the material behavior across the complete temperature regime. Anand and co-workers

[50] have recently developed a material model which spans the whole temperature region. This model is first of its kind and gives new insight into modeling polymer behavior across Tg. The material model has two different molecular resistances, one each for below and above Tg; this reiterates the point that a single network is not capable of this task and confirms the fact that two networks for the different temperature regions are essential. The two networks could be interpreted as different mechanisms that are active in the two different temperature regimes. A model following Anand‘s approach will be

90 developed here with the goal of reducing the complexity and the number of material constants that are required. To incorporate this revision in the existing DB model, an additional parallel network on the molecular relaxation side is added, thereby capturing the experimental data better.

5.2 Modifications to Dupaix-Boyce Model

5.2.1 Additional Molecular Relaxation Network

The existing Dupaix-Boyce (DB) model has been discussed in Chapter 4. A schematic of the same is shown in figure 5.1.

Figure 5.1 Schematic of Dupaix-Boyce Model

An additional molecular relaxation network was added to capture the molecular relaxation at temperatures higher than glass transition to the Dupaix-Boyce model. Figure

5.2 shows the schematic of the new model. As before, the resistances I and N are in parallel, thus the total stress (T) is equal to the sum of the Cauchy stresses in the two

91 networks. The stress in the network interactions resistance (N) is calculated by taking the weighted average of the branches N1 and N2. The total deformation gradient (F) is equal to deformation gradient in each component.

Figure 5.2 Schematic of the New Model

The total stress (T) and total deformation gradient are:

T = TI + x*T N1 + y*T N2 (5.1)

F = FI = F N1 = F N2 (5.2) where x + y = 1. Through the total stress is sum of three different stresses, all the three componenets are not active at all temperatures. The value of ‗x‘ and ‗y‘ change with temperature. Between Tg and Tg +20 °C, the value of ‗x‘ moves from 1 to 0 and the value of ‗y‘ from 0 to 1. This simplifies eq. (5.1) to:

(5.3)

The values of ‗x‘ and ‗y‘ change from 0 to 1 or vice versa over a range of 20 °C. The range Tg to Tg +20 (in this case: 102-122 °C or 375-395 K) was selected and not Tg -10 to

Tg +10 since the stress values begin to drop drastically only after Tg. The expression for

‗y‘ is given below.

(5.4) where T in the temperature in Kelvin. The plot of ‗y‘ with temperature is shown in figure

5.3.

0.8

0.6

0.4 Valuey of

0.2

0 375 380 385 390 395 Temperature in Kelvin

Figure 5.3 Plot showing value of ‗y‘ versus temperature in Kelvin.

All the calculations for finding the stress in molecular network II are same as in molecular network I. The only difference is in the values of four material constants used 93 for molecular relaxation. The original model had 15 material constants: 7 for the initial stiffness; 2 for flow; 2 for network stretching and orientation; and 4 for molecular relaxation. The four new material constants were added for the molecular relaxation (N2) for temperatures higher than the Tg. These were obtained by fitting the data above Tg using the Nelder-Mead Alogorithm [66]. The modified model now has 19 constants. The first fifteen are the same as before and material constants 16 through 19 are the new optimized values of temperature coefficient (D), second temperature parameter (Q/R), power-law exponent (1/n) and cut-off orientation (αc) for temperatures above Tg. The new constant were found using experimental data at 120 and 130 °C. These are shown in table

5.1.

Stress State Temperature Strain rate

1 Uniaxial compression 120 °C 1.0/min

2 Uniaxial compression 130 °C 1.0/min

3 Uniaxial compression 130 °C 3.0/min

4 Plane strain compression 130 °C 1.0/min

Table 5.1 Compression experimental data used for optimization of new molecular relaxation constants in the New Model.

5.2.2 Additional Transition Slope Constant

The additional molecular network significantly improved the model predications but was not performing very well below the glass transition temperature of the polymer.

The model was predicting lower yield stress values at temperatures below the glass transition. The yield stress depends on the initial modulus of the material and the initial modulus is calculated using the shear and bulk modulus in the DB model. The relationship amongst these variables is shown in equation 5.5-7.

1 1 5   (g  r )  (g  r )tanh( ( g ))  X g ( g ) 2 2  (5.5)

1 1 5 B  (B  B )  (B  B ) tanh( (  )) (5.6) 2 g r 2 g r  g

3EB   (5.7) 9BE where µ is the shear modulus, µg is the glassy shear modulus, µr is the rubbery shear modulus, Δθ is the temperature shift, θ is the absolute temperature, θg is the glass transition temperature, Xg is glass transition slope constant, B is the bulk modulus, Bg is glassy bulk modulus, Br is the rubbery bulk modulus, and E is the modulus of elasticity.

Thus, the yield stress prediction was improved by improvising the shear modulus calculation below the glass transition temperature. The model uses a single glass transition slope constant, Xg; the same value of slope is used to calculate the shear

95 modulus on either side of the glass transition temperature (Tg or θg). Figure 5.4 shows the temperature dependence of shear modulus as used in the Dupaix-Boyce Model [34].

Figure 5.4 Temperature dependence of shear modulus as used in the Dupaix-Boyce

Model [34]

However, the experimental data obtained by Palm [36] as shown in figure 5.5 suggests otherwise. The figure shows two distinct slopes on either side of the glass transition temperature. Therefore, there is a need to use two different slopes below and above the glass transition to accurately predict the shear modulus and hence the yield stress.

Figure 5.5 Experimental data collected by Palm [36] showing temperature dependence of shear modulus.

Hence, a second glass transition slope constant was added to the model. As discussed in

Chapter 4, equation 4.11, the shear modulus is calculated using the following equation.

(5.8)

Here, Xg was replaced by Xr and Xg representing the slope above and below Tg respectively. Xr stands for the slope in the rubbery region, i.e. above Tg and Xg stands for the slope in the glassy region, i.e. below Tg (θg).

(5.9)

where

The magnitude of X will be smaller in the rubbery range since a large value would cause the shear modulus to have negative values at higher temperatures. However, this restriction does not apply at lower temperatures. The constitutive model now has 20 constants.

5.2.3 New Model

The New Model now has 20 material constants and is able to capture the experimental data quite well. The comparison between the experimental data and the model simulations will be made in the next section. The final values of material constants are shown in table 5.2.

5.3 Results

The new model has been fit to the experimental data available in the Dupaix Lab. The results are shown in the following figures. Figure 5.6 and 5.7 are for uniaxial compression below the glass transition temperature at 82 and 92 °C respectively. Figures

5.8 – 5.12 show comparison between the data and simulation results for uniaxial compression of PMMA at Tg and temperatures higher than Tg, at 102, 110, 115, 120 and

130 °C respectively.

Material Property Symbol Value

Glassy Modulus μg 350 MPa  Rubbery Modulus r 30 MPa

Temperature Shift  30 K X Initial Elastic Glassy Transition Slope g - 7.0 MPa/K

Behavior Rubbery Transition Slope - 4.2 KPa/K Rate Shift Factor  3.6 K

Glassy Bulk Modulus Bg 0.64 GPa

Rubbery Bulk Modulus Br 2.73 GPa

 13 Pre-exponential Factor 0I 7.08 10 1/s Flow Stress G 19 Activation Energy 2.14 10 J Rubbery Orientation Modulus vk 8.0 MPa Resistance Elasticity Entanglement Density N 500

7 Temperature Coefficient D 1.34 10 K

QR/ 4 Molecular Second Temperature Parameter 1.75 10 1/s Relaxation (1) Power-law Exponent 1/n 7.4  Cutoff Orientation c 0.0017

Temperature Coefficient K

Molecular Second Temperature Parameter 1/s Relaxation (2) Power-law Exponent 1/n 6.52

Cutoff Orientation 0.00045

Table 5.2 Material constants for the New Model

60 -1.0/min exp -3.0/min exp 50 -4.0/min exp -1.0/min sim 40 -3.0/min sim -4.0/min sim 30

20 -TrueStress (MPa)

0 0 0.5 1 1.5 -True Strain

Figure 5.6 Comparison between the experimental data and Model simulation using New

Model for uniaxial compression of PMMA at 82 °C.

60 -1.0/min exp -3.0/min exp 50 -4.0/min exp -1.0/min sim 40 -3.0/min sim -4.0/min sim 30

20 -TrueStress (MPa)

0 0 0.5 1 1.5 2 -True Strain

Figure 5.7 Comparison between the experimental data and Model simulation using New

Model for uniaxial compression of PMMA at 92 °C. 100

25 -0.05/min exp -1.0/min exp 20 -2.0/min exp -3.0/min exp -0.05/min sim 15 -1.0/min sim -2.0/min sim -3.0/min sim

10 -TrueStress (MPa) 5

0 0 0.5 1 1.5 -True Strain

Figure 5.8 Comparison between the experimental data and Model simulation using New

Model for uniaxial compression of PMMA at 102 °C

25 -0.05/min exp -2.0/min exp 20 -3.0/min exp -6.0/min exp -0.05/min sim 15 -2.0/min sim -3.0/min sim -6.0/min sim

10 -TrueStress (MPa) 5

0 0 0.5 1 1.5 -True Strain

Figure 5.9 Comparison between the experimental data and Model simulation using New

Model for uniaxial compression of PMMA at 110 °C 101

10 -0.05/min exp -1.0/min exp 8 -3.0/min exp -0.05/min sim -1.0/min sim 6 -3.0/min sim

4 -TrueStress (MPa) 2

0 0 0.5 1 1.5 2 -True Strain

Figure 5.10 Comparison between the experimental data and Model simulation using New

Model for uniaxial compression of PMMA at 115 °C

4 -0.05/min exp -1.0/min exp 3.5 -3.0/min exp -0.05/min sim 3 -1.0/min sim -3.0/min sim 2.5

1.5 -TrueStress (MPa) 1

0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -True Strain

Figure 5.11 Comparison between the experimental data and Model simulation using New

Model for uniaxial compression of PMMA at 120 °C 102

2 -0.05/min exp 1.8 -1.0/min exp -2.0/min exp 1.6 -3.0/min exp -0.05/min sim 1.4 -1.0/min sim -2.0/min sim 1.2 -3.0/min sim

0.8

0.6 -TrueStress (MPa)

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -True Strain

Figure 5.12 Comparison between the experimental data and Model simulation using New

Model for uniaxial compression of PMMA at 130 °C

Plane strain compression data was also compared with the New model simulations. The results are shown in figures 5.13 – 16. The model does not capture the strain hardening at large strains in plane strain compression very well.

103

80 -1.0/min exp 70 -3.0/min exp -1.0/min sim 60 -3.0/min sim

-TrueStress (MPa) 20

0 0 0.5 1 -True Strain

Figure 5.13 Comparison between the experimental data and Model simulation using New

Model for plane strain compression of PMMA at 102 °C

50 -0.2/min exp -0.4/min exp 40 -1.0/min exp -3.0/min exp -4.0/min exp 30 -0.2/min sim -0.4/min sim -1.0/min sim 20 -3.0/min sim

-4.0/min sim -TrueStress (MPa) 10

0 0 0.5 1 -True Strain

Figure 5.14 Comparison between the experimental data and Model simulation using New

Model for plane strain compression of PMMA at 112 °C 104

30 -0.2/min exp -0.4/min exp 25 -1.0/min exp -3.0/min exp 20 -0.2/min sim -0.4/min sim -1.0/min sim 15 -3.0/min sim

10 -TrueStress (MPa)

0 0 0.5 1 -True Strain

Figure 5.15 Comparison between the experimental data and Model simulation using New

Model for plane strain compression of PMMA at 120 °C

15 -0.2/min exp -0.4/min exp -1.0/min exp -3.0/min exp 10 -0.2/min sim -0.4/min sim -1.0/min sim -3.0/min sim

5 -TrueStress (MPa)

0 0 0.5 1 -True Strain

Figure 5.16 Comparison between the experimental data and Model simulation using New

Model for plane strain compression of PMMA at 130 °C 105

5.4 Discussion

The new model does a good job of capturing the experimental data in uniaxial compression. The results are not quite equally good for plane strain compression. The uniaxial compression results are discussed first.

At temperatures below the glass transition of the polymer, the model under predicts the initial yield peak (figure 5.6 – 7). Also, at these temperatures of 82 and 92 °C, the model is not able to capture dramatic strain hardening at large strains. The model predicts more flat stress-strain data than the experimental data observed over the strain of ~ 0.6.

However, the model does show very clear strain rate dependence. A better fit of the constants would probably improve the results at these lower temperatures ( < Tg ).

Nevertheless, the results are significantly better at higher temperatures, i.e. from glass transition temperature of the polymer (Tg) to much higher temperatures (i.e. from 102 °C to 130 °C). At Tg, as seen in figure 5.8, the model prediction for the yield peak is close to the experimental value. The model does not try to capture the strain softening followed by the initial peak but does seem to come closer to the actual stress values at large strains as strain hardening begins. At higher temperatures, the material does not exhibit strain softening hence the model seems to be working more accurately at these higher temperatures.

At 110 °C, the model does an excellent job of predicting the stress-strain plots. The initial modulus is predicted fairly accurately followed by strain hardening at larger strains.

Figure 5.9 shows the comparison between the model prediction and experimental data with good strain rate dependence. At 115 °C, the model over predicts the initial elastic

106 modulus but does match the data closely at higher strains (figure 5.10). Model predictions at these two temperatures are even more relevant since the material constants were not fit to experimental data at these temperatures, and yet the results look quite promising.

At even higher temperatures of 120 and 130 °C, the model predicts higher yield point but is fairly accurate in the capturing the general stress-strain data plot (figure 5.11 – 12).

Also, at 120 °C, the model seems to be able to capture the strain hardening at higher strains. In figure 5.12, the initial yield peak predicted by the model is closer to the actual data at lower strain rate. However, here the model continues to predict strain hardening at large strains but the experimental data does not seem to follow the same trend. At higher temperatures, the material begins to behave more like a fluid and the strain hardening decreases.

For plane strain compression comparisons between the new model and the experimental data, the model predicts the values of the initial yield but is unable to capture the dramatic strain hardening at large strain. At 102 °C, the initial stress values calculated by the model are close to the actual values (figure 5.13). However, the model is unable to capture the strain hardening at larger strains ( < 0.6 ). The same trend is observed at higher temperatures of 112, 120 and 130 °C as well. The model values are close to the experimental values for strain of 0.6 - 0.7, but the experimental data experiences more dramatic strain hardening than the material model predicts (figure 5.13 – 16).

It is also important to note that the temperatures measured for plane strain compression experiments may not be accurate as discussed in the experimental errors section of chapter 3. The actual temperatures of the polymer samples were less than assumed. Plane

107 strain compression simulations were run again at temperatures 8 °C below the assumed test temperatures. The new model did predict higher yield stress values for these new simulations but again failed to capture the strain hardening of the polymer.

The model does have strain state dependence as seen in figure 5.17 - 19. The plot shows the model predictions at 82 °C, 102 °C and 122 °C at the strain rate of 1/min. The stress predictions for uniaxial compression have lower values than in plane strain compression, yet the difference between the stress data is not as apart as observed from the actual experimental data.

A more accurate prediction of the experimental data can be obtained by using the new model discussed above with a better fit of material parameters.

50 Uniaxial Plane Strain 40

20 -TrueStress (MPa) 10

0 0 0.5 1 1.5 -True Strain

Figure 5.17 Model predictions for uniaxial and plane strain compression at 82 °C at the strain rate of 1/min 108

35 Uniaxial 30 Plane Strain

-TrueStress (MPa) 10

0 0 0.5 1 1.5 -True Strain

Figure 5.18 Model predictions for uniaxial and plane strain compression at 102 °C at the strain rate of 1/min

4 Uniaxial 3.5 Plane Strain

2.5

1.5

-TrueStress (MPa) 1

0.5

0 0 0.5 1 1.5 -True Strain

Figure 5.19 Model predictions for uniaxial and plane strain compression at 122 °C at the strain rate of 1/min 109

CHAPTER 6

HOT EMBOSSING EXPERIMENTS

6.1 Introduction

Hot embossing (HE) is a simple operation with high accuracy in replicating micro- features. The relative cost of the embossing tools is also quite low [55, 71], making it a popular manufacturing process. In this chapter, hot embossing experiments are conducted on polymethyl methacrylate (PMMA) across the glass transition temperature of the polymer with variations in temperature, and hold and cooling times. The results will be compared with hot embossing simulations, later in the next chapter, to investigate the effectiveness of the new model in predicting hot embossing of PMMA. These experiments will not only give data for comparison with the simulation results, but will also help to better understand the hot embossing process of the polymer.

During the hot embossing process, the polymer is heated above the Tg where the force required to emboss is much smaller, and then the polymer is cooled down below the glass transition to retain the feature. Figure 6.1 shows the temperature and pressure profiles of a typical hot embossing process, figure 6.1(a) shows the more commonly used temperature and pressure profile while figure 6.1(b) shows the profile of the hot

110 embossing experiments conducted in this chapter. The main difference is that in these experiments the polymer is cooled after the pressure/force is released. A few experiments were also conducted using the typical hot embossing profile wherein the cooling of the material is started before the pressure/force is released.

(a)

(b)

Figure 6.1 Temperature and pressure profile of the hot embossing process (a) more commonly used profile; (b) profile used in most experiments conducted in this chapter. 111

Various experiments have already been done on PMMA over a large range of temperatures. G‘Sell and Souahi [18] performed tension tests on three different grades of

PMMA between 95 °C to 145 °C with glass transition temperatures of 108, 112 and 116

°C. Dooling et al. [19] also performed tension tests on PMMA above its glass transition.

These experiments are less relevant here since compression tests are more similar to the hot embossing process. Simple uniaxial and plane strain compression experiments were conducted by Arruda and Boyce [9] and Arruda et al. [16] below the glass transition temperature. Hasan and Boyce [12] also performed creep analysis by conducting compression tests on PMMA below its glass transition temperature. Palm et al. [21] and

Ghatak and Dupaix [22] conducted uniaxial and plane strain compression experiments between 82 °C to 132 °C (Tg ~ 102 °C). Richeton et al. [23] conducted some uniaxial compression tests on PMMA up to ~ Tg +10 °C as well. Ames et al. [24] have conducted experiments on PMMA over a wide range of temperatures from 22 ˚C to 170 °C (Tg ~

110 °C). These uniform compression tests provide valuable data for developing a constitutive model, but as they do not involve embossing any features of the polymer nor measuring the final dimension of the polymer sample they have limited use in validating

FEM simulations. Anand and Ames [72] did conduct indentation experiments on PMMA at room temperature to study the load versus depth curves.

Very few indentation experiments have been performed on PMMA with the intention of developing a constitutive model for hot embossing. In this work, our goal was to fill this gap in the literature by conducting experiments from 92 °C (Tg -10) through 142 °C (Tg

+40) in increments of 10 °C. All of the samples were first embossed to a depth of 0.8 mm

112

(800 µm), then constant strain was maintained for different hold times of 30, 90 and 180 seconds. Afterward, the load was removed and the samples were cooled to room temperature (23 °C). The hot embossing experiments performed here in this transitional temperature regime (from below Tg through temperatures well above Tg) are to gather data to help us better understand the behavior of PMMA when subjected to non-uniform loading, holding under strain, and cooling, as a step to improve hot embossing simulations, in addition to validating the new model.

6.2 Experimental Details - Material and Test Set-up

The material used for all the experiments was Polymethyl methacrylate (PMMA). This is a commercially available grade: Plaskolite OPTIX Acrylic Plastic provided by Plaskolite,

Inc with molecular weight of about 140,000 g/mol. The components are Polymethyl methacylate (PMMA) 99.5 % (min.) and Methyl methacylate (MMA) 0.5 % (max.).

The samples used had a square cross-section (9.6 mm in width and breadth) and were 8.8 mm in height and the samples were kept in a dessicator for at least 24 hours before the tests. Hot embossing experiments were carried out using a channel and die fixture as shown in Figure 6.2. The glass transition temperature of the PMMA was found to be ~

102 °C (375 K) using Differential Scanning Calorimetry (DSC) [21]. The compression experiments were conducted on an Instron 5869 screw driven load frame with a 50 kN load cell and an Instron 5800 controller run by Instron Bluehill software.

Teflon sheets were placed (i) between the sample and the channel (bottom part of the fixture), and (ii) between the compression anvils and the die and the channel fixture.

113

Lubricant WD-40 was applied between the Teflon sheets and the compression anvils; the sample was not exposed to the lubricant. Teflon was not applied between the sample and the die (top part of the fixture) since the accuracy of the die profile (Figure 6.3) would be compromised. The die used was made of aluminium since it is a good conductor of heat.

An Instron model 3119 high temperature environmental chamber was used to achieve the desired temperature. The chamber is switched on for at least 2 hours prior to the testing to ensure that the chamber is heated uniformly. All samples are placed in the chamber for 20 minutes before the start of the tests. All tests were performed at least twice and until repeatable data is obtained.

Figure 6.2 Channel and die fixture for hot embossing

114

Figure 6.3 Die Profile

6.3 Experimental Results and Discussion

The hot embossing experiments were conducted at temperatures ranging from 92 °C (Tg -

10) to 142 °C (Tg +40). The samples were compressed for 6 seconds at a displacement rate of 0.13333 mm/s resulting in a total displacement of 0.8 mm (800 µm). The compression anvil was then held in position for an additional 30, 90 or 180 seconds resulting in a total time of 36, 96 and 186 seconds, respectively. Additional experiments were conducted for longer hold times of 240 and 300 seconds at 142 °C. The load was then removed and the samples taken out of the temperature chamber directly to room temperature (25 °C). These experiments follow the temperature and force profile as shown in figure 6.1(b). The force and displacement data during the compression and holding stages was recorded.

A few cooling experiments were also conducted at 142 °C while the polymer samples were being held. In these cases, the environmental chamber door was opened after some time to allow the room temperature air to cool the sample while the polymer sample was

115 still held in the deformed state. These experiments follow the temperature and force profile as shown in figure 6.1(a). The force and displacement data during the compression and holding stages was recorded here as well.

Figures 6.4-6.9 show the force versus time data for temperatures of 92 °C to 142 °C in 10

°C increments. All the experiments were conducted at least twice or until repeatable data was obtained. There is more noise at higher temperatures (132 and 142 °C) as seen in figure 6.8 and 6.9. This is due to the fact that the data is in the range of 5-40 N which is close to the lower end of the range for the 50 KN load cell used. The force versus time data for all temperatures with a hold time of 180 seconds is shown in figures 6.10 and

6.11 for easier comparison. In all of the plots at a particular temperature, the force-time curves at different hold times correlate well, which illustrates the high repeatability of these experiments. A representative displacement versus time plot is shown in figure

6.12.

3000 30 s Hold 90 s Hold 2500 180 s Hold

2000

1500 Force (N) Force 1000

500

0 0 50 100 150 200 Time (s)

Figure 6.4 Force versus Time data for different hold times at 92 °C 116

2000 30 s Hold 90 s Hold 180 s Hold 1500

1000 Force (N) Force

500

0 0 50 100 150 200 Time (s)

Figure 6.5 Force versus Time data for different hold times at 102 °C

500 30 s Hold 90 s Hold 400 180 s Hold

300

Force (N) Force 200

100

0 0 50 100 150 200 Time (s)

Figure 6.6 Force versus Time data for different hold times at 112 °C

117

120 30 s Hold 90 s Hold 100 180 s Hold

60 Force (N) Force 40

0 0 50 100 150 200 Time (s)

Figure 6.7 Force versus Time data for different hold times at 122 °C

35 30 s Hold 30 90 s Hold 180 s Hold 25

Force (N) Force 10

-5 0 50 100 150 200 Time (s)

Figure 6.8 Force versus Time data for different hold times at 132 °C

118

30 30 s Hold 90 s Hold 25 180 s Hold

15 Force (N) Force 10

0 0 50 100 150 200 Time (s)

Figure 6.9 Force versus Time data for different hold times at 142 °C

3000 92 C 102 C 2500 112 C 122 C 132 C 2000 142 C

1500 Force (N) Force

1000

500

0 0 50 100 150 200 Time (s)

Figure 6.10 Force versus time data for different temperatures at the hold time 180 s 119

90 122 C 80 132 C 142 C 70

40 Force (N) Force

0 0 50 100 150 200 Time (s)

Figure 6.11 Force versus time data for high temperatures at the hold time 180 s

30 s Hold 0.8 90 s Hold 180 s Hold 0.7

0.6

0.5

0.4

0.3 Displacement (mm) Displacement 0.2

0.1

0 0 50 100 150 200 Time (s)

Figure 6.12 Representative displacements versus time plot for the given set of experiments 120

Forces at the three hold times of 30, 90 and 180 seconds for all six temperatures are shown in figure 6.13. Force at time zero is the peak force attained at the end of the embossing of the samples. As the embossing die is held at constant height, the force drops due to stress relaxation. The scale of the force changes drastically above the glass transition temperature; the force required to compress the polymer is much lower at temperatures of 122 °C (Tg +20) and higher. Figure 6.14 shows the force versus hold time plots for temperatures of 122 °C and higher to more clearly illustrate the force at higher temperatures.

The force during the hold period is also plotted as a percentage of the peak force after the polymer samples were embossed to the desired depths. These are shown in figures 6.15-

6.17. Figure 6.15 shows the percentage of peak force with hold times for temperatures from 92 to 142 °C. Two different trends are observed (i) between 92 and 112 °C as shown in figure 6.16, and (ii) between 122 and 142 °C as shown in figure 6.17.

121

3000 92 C 102 C 2500 112 C 122 C 2000 132 C 142 C

1500 Force(N) 1000

500

0 0 50 100 150 200 Hold Times (s)

Figure 6.13 Force trend with time for various temperatures after the polymer samples were embossed to the desired depths.

100 122 C 132 C 80 142 C

Force(N) 40

0 0 50 100 150 200 Hold Times (s)

Figure 6.14 Force trend with time for temperatures of 122, 132 and 142 °C after the polymer samples were embossed to the desired depths. 122

100 92 C 102 C 80 112 C 122 C 132 C 60 142 C

40 % Peak% Force

0 0 20 40 60 80 100 120 140 160 180 Time (s)

Figure 6.15 Percentage force in terms of peak force at the end of embossing with time after the polymer samples were embossed to the desired depths.

100 92 C 102 C 80 112 C

40 % Peak% Force

0 0 20 40 60 80 100 120 140 160 180 Time (s)

Figure 6.16 Percentage force in terms of peak force at the end of embossing with time after the polymer samples were embossed to the desired depths for temperatures of 92,

102 and 112 °C. 123

100 122 C 90 132 C 142 C 80

50 % Peak% Force 40

20 0 20 40 60 80 100 120 140 160 180 Time (s)

Figure 6.17 Percentage force in terms of peak force at the end of embossing with time after the polymer samples were embossed to the desired depths for temperatures of 122,

132 and 142 °C.

From 92 to 112 °C, the force (in percentage) drops off more quickly with increasing temperature whereas from 122 to 142 °C, the force drops off less quickly with increasing temperature. It is essential to note that it is the percentage of peak force that follows this trend and not the values of actual force. At higher temperatures, since the peak force at the end of embossing is not very high, the change over time is much less drastic unlike the change of force at temperatures much closer to the glass transition temperature.

After the force analysis, the depth dimensions of the samples were also measured. This was done at the end of the complete embossing cycle, i.e. heating of the sample to the desired temperature, embossing of the profile using the die, hold the die at the final depth

124 and taking the sample out of the temperature chamber back into the room temperature (23

°C). The final dimensions of the deformed samples were measured using vernier calipers.

Two samples of polymers embossed to a depth of 0.8 mm at 142 °C with different hold times are shown in figure 6.18. Figure 6.18(a) shows the sample held for 180 seconds and figure 6.18(b) shows the sample held for 30 seconds. The dissimilar final depth can even be observed with bare eyes.

Figure 6.18 Samples embossed to a depth of 0.8 mm at 142 °C with different hold times showing dissimilar final depth of indentation. (a) Sample held for 180 seconds; (b) sample held for 30 seconds.

Figure 6.19 shows the depth of the embossed sample at the end of the embossing cycle for various temperatures. The desired depth was 0.8 mm. As expected, the depth of the polymer samples was closer to the desired depth for longer hold times, i.e. for hold times of 180 seconds as compared to 30 or 90 seconds. However, the more interesting observation is the trend of the final depths measured at various temperatures. This trend could be attributed to two opposing factors contributing to the dimensions of the polymer sample. First, as the temperature of the sample rises, the material begins to flow more 125 freely resulting in higher spring back once the die is released. This factor is more prominent in temperature between 92 and 122 °C, resulting in less than the desired final dimension of the polymer samples. Secondly, as the temperature of the polymer material rises, stress relaxation happens much faster leading to a quicker reduction in elastic stress, lower force and lower spring back. This cause is more prominent from 122 to 142

°C. The transition from ‗the increasing spring back with increasing temperatures‘ being the larger contributing factor to ‗the faster stress relaxation at higher temperatures‘ being a more prominent cause seems to happening around 122 °C for the given grade of

PMMA.

0.8 30 s 0.7 90 s 180 s 0.6

0.5

0.4

0.3 Depthmmin

0.2

0.1

0 80 100 120 140 160 Embossing Temperature in C

Figure 6.19 Depth of the embossed sample at the end of the embossing cycle for various temperatures. The desired depth was 0.8 mm.

126

Force required for the embossing decreases with increase in temperature, and a significant drop is observed around 122 °C (Tg +20) as seen in figures 6.3-6.8 and 6.10.

Also, from figures 6.3-6.8, force values change significantly with different hold times for the temperature of 92 °C but the change is much less at higher temperatures. The desired embossed depth was 0.8 mm. The trend observed in the depth of the samples also changes at 122 °C as shown in figure 6.19. The depth dimension is closer to 0.8 mm for temperatures close to or below the glass transition and again at higher temperatures.

As seen in figures 6.9 and 6.19, even though the force values did not change substantially at 142 °C with longer hold times, the embossing depth continues to change with increasing hold times. To analyze this trend, more experiments were conducted at 142 °C with longer hold times of 240 and 300 seconds. Figure 6.20 shows the force versus time plot for the same. Figure 6.21 shows the depth of the embossed sample at the end of the embossing cycle for various temperatures as seen in figure 6.19 along with these additional hold times. The embossing depth with variation in the hold times at this higher temperature of 142 °C is shown in figure 6.22.

127

30 180 s Hold 240 s Hold 25 300 s Hold

15 Force(N) 10

0 0 100 200 300 Time (s)

Figure 6.20 Force versus Time data for different hold times at 142 °C

0.8 30 s 0.7 90 s 180 s 0.6 240 s 300 s 0.5

0.4

0.3 Depthmmin

0.2

0.1

0 80 100 120 140 160 Embossing Temperature in C

Figure 6.21 Depth of the embossed sample at the end of the embossing cycle for various temperatures. The desired depth was 0.8 mm. 128

Although the embossing depths continue to increase with increasing hold times, figure

6.20 shows that the embossing depths do begin to stabilize and do not change as significantly with longer hold times as before. However, all these experiments were done without any cooling during the embossing of the sample. Cooling the sample while applying the force on the polymer would also significantly affect the embossing depths.

A few preliminary experiments were done to study this scenario as well, which is illustrated in figure 6.1(a).

The polymer sample was held under the compression anvil for 240 seconds, while the environmental chamber door was opened after 180 seconds to allow the room temperature air to cool the sample. The temperature of the environmental chamber dropped from 142 °C at 180 seconds to 70 °C at end of 240 seconds. In another test, the sample was held for 300 seconds and the environmental chamber door was opened after

180 seconds. The temperature of the environmental chamber dropped from 142 °C at 180 seconds to 56 °C at end of 300 seconds. In this testing, the temperature of the polymer sample was not able to be directly measured during this time. The force versus time plots for these are shown in figure 6.23. Also, the embossing depths for hold times of 240 and

300 seconds with and without cooling are compared in figure 6.24. As expected, the depths increase with increase in hold times and cooling times.

129

0.8

0.7

0.6

0.5

0.4

0.3 Depthmmin

0.2

0.1

0 0 50 100 150 200 250 300 350 Hold Times in seconds

Figure 6.22 Embossing depths with variation in the hold times (30, 90,180, 240 and 300 seconds) at temperature of 142 °C

25 240 s Hold with cooling after 180 s 300 s Hold with cooling after 180 s 20

Force(N) 10

0 0 50 100 150 200 250 300 350 Time (s) Figure 6.23 Force versus Time data for different hold times at 142 °C 130

0.8 Hold without cooling Hold with cooling 0.75

0.7

0.65

Depthmmin 0.6

0.55

0.5 200 250 300 350 Hold Times in seconds

Figure 6.24 Embossing depths with and without cooling at 142 °C

6.4 Conclusion

Simple features were embossed on PMMA over a wide range of temperatures from below the glass transition temperature to well above the Tg. Force and depth analysis were made with variation in temperature and hold times. The polymer behavior changes drastically from below Tg to around 10-20 °C above Tg making it difficult to be consolidated in a single material model. Experiments were performed in this vital temperature regime to better understand the polymer behaviour.

In the discussion above, we observed that longer hold times result in depths closer to the desired dimension. At lower temperatures (below and around Tg), the final dimension of the samples is better but the force values are very high, whereas as the temperature is

131 increased, the force values are much lower but the depth dimension is less accurate. But as we move over 122 °C, the depth dimensions begin to improve. If we continue to increase the embossing temperature, the depth dimensions get better although there is not a very significant advantage over the force applied (as seen in figure 6.11). In addition, it was observed that depth dimension improved with increasing the hold and cooling times.

Samples held for the same duration but with longer cooling times also gave better final dimensions. The difference in the final dimension of the polymer samples with or without cooling should be less significant at temperatures closer to Tg. This is because at these relatively lower temperatures the polymer is more solid – like and shows less spring back or flow as compared to temperatures 30 – 40 °C above Tg. The best temperatures to emboss the polymer would be at least 30 °C above Tg where the final dimensions of the sample start to improve and the force required to emboss is low as well. After the polymer is embossed with the feature, it needs to be cooled below Tg; this final temperature can be decided based on the following factors: time and equipment required to cool the sample, accuracy of the final dimension desired, and the cost of the entire process.

From the results obtained in this chapter, it can be deduced that the two most crucial aspects for a material model for amorphous polymers that is to be used for hot embossing simulations are the following. First, the temperature range of the constitutive model needs to be from well below Tg through at least 30 °C above the Tg of the polymer. It can be deduced that a material model that works accurately from Tg-10 through Tg+30 °C would be able to capture the material behavior across the most important part of the temperature

132 range essential to simulate hot embossing process for PMMA. Secondly, the model must be able to capture stress relaxation over time in order to be used for hot embossing simulations. The polymer sample may be deformed with any feature but in order to precisely determine the final dimension of the sample, the stress relaxation of the material has to be taken into account. The continuum model will be inadequate in predicting the hot embossing experiments if it cannot describe the stress relaxation or the extent of spring back of the material.

Most of the hot embossing experiments performed here were isothermal; the samples were heated to a temperature, embossed, held at that temperature and then directly taken out to room temperature (23 °C). A few experiments were conducted with cooling the sample while it was held under pressure at 142 °C. The results obtained from these preliminary hot embossing experiments will now be compared with hot embossing simulations in the next chapter to validate the new model.

133

CHAPTER 7

HOT EMBOSSING SIMULATIONS

7.1 Introduction

In this chapter, hot embossing simulations were run to validate the new model. The simulation results were compared with the hot embossing experimental data obtained in chapter 6. As discussed in chapter 6, during hot embossing the polymer is heated above its glass transition temperature. A force/pressure is then applied to press the contoured features into the polymer surface. This force is held over the polymer for some time; this can either be with or without cooling of the sample. Finally, the force is released and the desired feature is embossed on the polymer.

To find the most optimum process conditions for hot embossing by trial and error can be a very expensive affair. One needs to (i) determine the die dimensions after taking into consideration the extent of spring back of the material, (ii) optimize the hold and cooling time so that the final dimensions of the embossed product are accurate in addition to the process taking the least time possible, (iii) determine the starting temperature at which the embossing of the polymer will occur and whether both the polymer and the die will be heated to that initial temperature, and (iv) determine the final temperature at which the

134 die will be released from the polymer. All these conditions need to be considered with regard to the ease of production, time required, cost of all the processes and the final value, use and accuracy of the product desired. Hence, the most economical way to choose the process condition parameters is to run various hot embossing simulations and as a result, find the most optimum parameters to be used to conduct the hot embossing experiments.

Hot embossing simulations are run either using tabulated data or using constitutive material models. As discussed in chapter 5, both methods have advantages and disadvantages; collecting tabular data requires a large amount of experimental data, a much smaller set of data can be used for the constitutive modelling. At the same time, using tabular data to interpolate or extrapolate the results is much simpler than developing complex constitutive equations to predict the material behaviour. Either of the techniques could go wrong if the experimental data is not collected over a sufficiently wide range of conditions; a dramatic change in a trend of a material behaviour or some physical mechanism can be overlooked.

Here, preliminary hot embossing simulation will be conducted on PMMA to test the new model and find if the model can capture the important and crucial aspects of hot embossing. These simulations will be carried out in ABAQUS using the user defined

UMAT subroutine to code the new constitutive model.

135

7.2 Finite Element Model

The new model has been used for all of the hot embossing simulations. The details of the hot embossing finite element simulations conducted are discussed in this section. The embossing simulations were conducted on a representative repeating section of the complete profile. The profile and its section used for simulations are shown in figure 7.1.

Figure 7.1 Profile of the part and the section used to conduct hot embossing simulations.

136

The die used for all the hot embossing simulations is shown in figure 7.2. The die was considered as a rigid body for all the simulations. This die has rounded corners, a sharp corner gives very high values of forces and stresses in the part and also leads to convergence issues of the finite element simulations. The die was placed 5 units above the polymer part before start of the analysis and moved down 9 units from the polymer surface with the total displacement of the die being 14 units. The part dimensions were 36 units in the horizontal direction and 60 units in the vertical direction. The part had 3 regions with fine, medium and coarse meshes. The finer mesh was used for the region of the part closer to the die. Further away the mesh was changed to medium and then coarse.

The fine elements were present from 0 to 24 units in both horizontal and vertical direction with 32 elements in horizontal and 12 elements in vertical direction. The medium elements were present from 24 to 32 units in both horizontal and vertical direction with 8 elements in both horizontal and vertical direction. The coarse elements were present from 32 to 36 units in horizontal direction and 32 to 60 units in vertical direction with 2 elements in horizontal and 8 elements in vertical direction. There are a total of 594 elements with 384 fine, 104 medium and 106 coarse elements. The mesh of the part is shown in figure 7.3, figure 7.3 (a) shows the fine (1) , medium (2) and coarse

(3) mesh region of the part whereas figure 7.3 (b) shows the mesh with respective dimensions/units for each kind of mesh. Element type used for the part was a 2D, 4-node bilinear, plane strain element - CPE4.

For boundary conditions, the die used was taken to be a rigid material. The motion of the die was restricted in 1- (motion along the x or horizontal axis) and 6- (rotation about the x

137 or horizontal axis) conditions. The part used was defined to behave as described in the

UMAT (new model). The left and right sides of the part were not allowed to move in the

1- (motion along the x or horizontal axis) direction and the bottom was constrained in the

2- (motion along the y or vertical axis) direction.

Steps used in the input file for hot embossing simulations were as follows. (i) Lowering

Step: The die starts to move vertically down from 4.9 units above the part and just touches the part. (ii) Contact Step: The die comes in contact with the part and move down by 0.2 units. (iii) Deformation Step: The die continues to move down into the polymer by

8.9 units for a total displacement of 14 units from the starting position. (iv) Hold and

Cooling Step: The die is held in contact with the polymer for the duration of the step with or without cooling. (v) Unloading Step: The die moves back up to its original position of

5 units above the undeformed part. (vi) Pause Step: There is no further movement of the die or part in this step. This is the final step to ensure that the simulation has completed.

Figure 7.2 Dimension of the die used for hot embossing simulations

138

Figure 7.3 (a) Fine (1) , medium (2) and coarse (3) mesh region of the part; and (b) Mesh of the part with the die in its initial position showing the dimensions of the fine, medium and coarse mesh.

7.3 Hot Embossing Simulations

Preliminary hot embossing simulations were conducted using the new model. Each sample was embossed to a depth of 9 units. These simulations were conducted at 122,

132 and 142 °C with variation in hold and cooling times.

139

The force over time plots for simulations conducted at 142, 132 and 122 °C are shown in figures 7.4, 7.8 and 7.11 respectively. Hot embossing (with hold) experiments were also conducted at these temperatures in chapter 6. The experimental data for the same is shown in figures 7.5, 7.9 and 7.12 respectively. A simulation result for hot embossing at

142 °C at end of deformation step, cooling step, and at end of unloading step is shown in figure 7.7. As seen from the comparison of the experimental data with the hot embossing simulations, the model fails to capture the stress relaxation of the material with time. The model predicts the stress relaxation to happen immediately (within a second) whereas the actual data shows a much slower relaxation of the stresses in the material.

Figure 7.6, 7.10 and 7.13 shows the final depth of the polymer parts after different hold times at 142, 132 and 122 °C respectively. Looking at the plots from the model predictions, one can say that the spring back reduces with longer hold times, but since all the values are very close to each other, the influence of hold times can be neglected. At the same time, it is known and seen in the experimental data that hold times do significantly affect the final dimensions of the part. This discrepancy can again be explained by the fact that the model predicts a much faster rate of stress relaxations than what the material actually experiences. Since most of the stresses relax within a second of hold, the amount of spring back does not change with longer hold times for simulated data. However, as discussed above, the model predictions do not match the experimental data.

Simulations with cooling of 90 seconds were also done with the embossing temperature of 142 °C. The parts were cooled to 132 and 122 °C. These results were compared with

140 the 90 seconds hold simulations at 142 °C without any cooling. The final depth of the part for all the three scenarios is shown in figure 7.14. The force versus time plots for the same are in figure 7.15. Since the model fails to capture the rate of stress relaxation; it is not possible to deduce a lot of trends or information from these results.

The current problems with the new model and the ways to improve it will be discussed in the next section.

45 1 sec Hold 40 30 sec Hold 35 90 sec Hold 180 sec Hold 30 240 sec Hold 25 300 sec Hold

20 Force(N) 15

0 0 50 100 150 200 250 300 350 Time (seconds)

Figure 7.4 Force versus time with different hold times at 142 °C for hot embossing simulations

141

30 30 s Hold 90 s Hold 25 180 s Hold

15 Force (N) Force 10

0 0 50 100 150 200 Time (s)

Figure 7.5 Force versus Time experimental data for different hold times at 142 °C

8.97

8.96

8.95

8.94

8.93

8.92

8.91

Displacement(mm) 8.9

8.89

8.88

0 50 100 150 200 250 300 Time (seconds)

Figure 7.6 Final displacement of the part after different hold times at 142 °C for hot embossing simulations

142

(a) (b) (c)

Figure 7.7 Hot embossing simulations conducted at 142 °C with 90 second hold without cooling (a) at end of deformation step, (b) at end of cooling step, and (c) at end of unloading step

143

60 1 sec Hold 30 sec Hold 50 90 sec Hold 180 sec Hold 40

30 Force(N) 20

0 0 50 100 150 200 250 Time (seconds)

Figure 7.8 Force versus time with different hold times at 132 °C for hot embossing simulations

35 30 s Hold 30 90 s Hold 180 s Hold 25

Force (N) Force 10

-5 0 50 100 150 200 Time (s)

Figure 7.9 Force versus Time experimental data for different hold times at 132 °C

144

8.96

8.94

8.92

8.9

8.88 Displacement(mm)

8.86

8.84 0 20 40 60 80 100 120 140 160 180 Time (seconds)

Figure 7.10 Final displacement of the part after different hold times at 132 °C for hot embossing simulations

80 1 sec Hold 70 30 sec Hold 90 sec Hold 60 180 sec Hold 50

Force(N) 30

0 0 50 100 150 200 250 Time (seconds) Figure 7.11 Force versus time with different hold times at 122 °C for hot embossing simulations

145

120 30 s Hold 90 s Hold 100 180 s Hold

60 Force (N) Force 40

0 0 50 100 150 200 Time (s)

Figure 7.12 Force versus Time experimental data for different hold times at 122 °C

8.95

8.9

8.85 Displacement(mm)

8.8 0 20 40 60 80 100 120 140 160 180 Time (seconds)

Figure 7.13 Final displacement of the part after different hold times at 122 °C for hot embossing simulations

146

8.952

8.95

8.948

8.946

8.944

Displacement(mm) 8.942

8.94

8.938 125 130 135 140 Tempearture (C)

Figure 7.14 Part embossed at 142 °C after a hold time of 90 seconds to different final temperatures.

45 142 - 142 C 40 142 - 132 C 35 142 - 122 C

20 Force(N) 15

0 0 20 40 60 80 100 120 140 Time (seconds) Figure 7.15 Part embossed at 142 °C after a hold time of 90 seconds to different final temperatures for hot embossing simulations 147

7.4 Analysis of New Model

From the hot embossing simulations conducted in the last section, the model predicts the material behavior to be highly plastic, much more than that of the actual material. This could be due to the fact that the optimized constants of the new model were obtained solely from the loading part of simple compression experiments conducted on the polymer PMMA.

Simple uniaxial simulations were conducted for further analysis of the new model in predicting load-unload and load-hold-unload data, and in capturing stress relaxation of the polymer PMMA to see if it predicts the material to be highly plastic. A single element model was used for conducting these simulations. The part was a simple cube of side 1 unit. The base of the cube was restricted to move in 2-direction. Element C3D8H (a 3-D quadratic brick element) was used to run the finite element simulation.

(i) First, uniaxial compression simulation was conducted at 130 °C without hold at the strain rate of 1/min (0.01667/sec) with the final strain of 1.4 at the end of compression to see if the new model predicts the polymer to be highly plastic. The plots for true strain and true stress versus time are shown in figure 7.16 and 7.17. Figure 7.18 shows the true stress versus true strain for the uniaxial compression at 130 °C with strain rate of 1/min and final strain of 1.4 using the new model.

(ii) Another uniaxial compression simulation using new model was also conducted at 130

°C with the strain rate of 1/min (0.01667/sec) with the final strain of 1.4 at the end of compression, but with a hold of 120 seconds. The plot for true stress versus time is shown

148 in figure 7.19 while figure 7.20 shows the true stress versus true strain for the uniaxial compression.

1.5

1 TrueStrain 0.5

0 0 20 40 60 80 100 120 140 160 Time (seconds)

Figure 7.16 True strain versus time for uniaxial compression at 130 °C with strain rate of

1/min and final strain of 1.4 using the new model

149

0 TrueStress (MPa)

-2

-4 0 20 40 60 80 100 120 140 160 180 Time (seconds)

Figure 7.17 True stress versus time for uniaxial compression at 130 °C with strain rate of

1/min and final strain of 1.4 using the new model

3 TrueStress (MPa) 2

0 0 0.5 1 1.5 True Strain

Figure 7.18 True stress versus true strain for uniaxial compression at 130 °C with strain rate of 1/min and final strain of 1.4 using the new model 150

3 TrueStress (MPa) 2

0 0 50 100 150 200 250 Time (seconds)

Figure 7.19 True stress versus time for uniaxial compression at 130 °C with strain rate of

1/min, hold of 120 seconds and final strain of 1.4 using the new model

3 TrueStress (MPa) 2

0 0 0.5 1 1.5 True Strain

Figure 7.20 True stress versus true strain for uniaxial compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 using the new model 151

(iii) Experimental data from a simple plane strain compression experiment conducted at

130 °C with a strain rate of 1/min (0.01667/sec), hold time of 120 seconds with a final true strain of 1.4 is also illustrated here for comparison with the new model results. The plots for true strain and true stress versus time are shown in figure 7.21 and 7.22. Figure

7.23 shows the true stress versus true strain for the uniaxial compression at 130 °C with strain rate of 1/min and final strain of 1.4.

1.5

1 TrueStrain 0.5

0 0 50 100 150 200 250 Time (seconds)

Figure 7.21 True strain versus time for plane strain compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 from experimental data

152

3 TrueStress (MPa) 2

0 0 50 100 150 200 250 Time (seconds)

Figure 7.22 True stress versus time for plane strain compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 from experimental data

3 TrueStress (MPa) 2

0 0 0.5 1 1.5 True Strain

Figure 7.23 True stress versus true strain for plane strain compression at 130 °C with strain rate of 1/min, hold of 120 seconds and final strain of 1.4 from experimental data 153

Although the above two cases (case ii and iii) were under exactly same conditions of temperature, strain rate, final true strain and hold duration, one to one comparison cannot be made since the new model prediction is for uniaxial compression and the experimental data is for plane strain compression. However, the true stress versus time plots (figure

7.19 and 7.22) and the true stress versus true strain plots (figure 7.20 and 7.23) look very alike and promising. The model seems to show a very realistic rate of stress relaxation for these experiments and matches well with the experimental data. The fact that the model does not predict these slower rates of stress relaxation in the hot embossing simulation has to be due to the differences between the hot embossing simulation and the simple compression simulations. The major difference between the two simulations conducted using the new model are is the much smaller final strains for the embossed parts compared with the simple compression experiments.

More simple compression simulations on the single element were done to investigate how the new model works at these smaller strains (~ 0.1) and constant velocity compression

(unlike constant strain rate compression) as these are more similar to the hot embossing experiments performed in chapter 6. The simulation discussed next is again at 130 °C with a hold time of 120 seconds. However, the load and unload steps were conducted with uniform velocity of 0.01667 units/sec and a final true strain of ~ 0.1. The plots for true stress versus time and true stress versus true strain are shown in figures 7.24 and 7.25 respectively. This simulation shows a much faster stress relaxation (figure 7.24) than seen in the experimental data (figure 7.8) at a similar condition. This explains the results seen in the hot embossing simulations by new model.

154

The observations from this section were used to understand the drawbacks of the model and hence suggest changes for future improvement.

1.5

0.5 TrueStress (MPa)

0 0 20 40 60 80 100 120 140 Time (seconds)

Figure 7.24 True stress versus time for uniaxial compression at 130 °C with hold of 120 seconds, uniform die velocity of 0.01667 and final strain of 0.1 using new model

155

1.5

0.5 TrueStress (MPa)

0 0 0.05 0.1 0.15 True Strain

Figure 7.25 True stress versus true strain for uniaxial compression at 130 °C with hold of

120 seconds, uniform die velocity of 0.01667 and final strain of 0.1 using new model

7.5 Suggestions for New Model

From the results obtained using the new model for hot embossing simulations and single element simulations, we observe that even though the model captures the stress relaxation correctly for large strains, it fails to capture the material behavior at small strains. At these small strains, the model predicts the bulk of contribution to the total stress from the intermolecular resistance (I) with only a small contribution from the molecular network resistance (N). Figure 7.26 show the individual contributions of the two resistances and figure 7.27 shows the contributions of the two resistances to the total stress. This leads to a highly plastic response of the material with almost no spring back at these strains.

However, as seen from the experiments, the material has some elasticity even at these

156 small strains (~ 0.1) and the PMMA samples do show some amount of spring back at these small strain, high temperature experiments. At large strains, as the contribution of the molecular network resistance (N) increases, the strain hardening of the material increases and the model predictions show a higher elastic response. Thus, the new model is able to capture the stress relaxation of the material at large strains unlike at small strains.

Figure 7.26 Individual contributions of the two resistances to the total stress, (a) intermolecular resistance, and (b) molecular network resistance.

Material constants of the new model were changed to establish if the model can capture the hot embossing experimental data. Hot embossing simulations were conducted after changing some material constants. Results from a representative simulation at 130 °C

(403 K) with a hold time of 90 seconds without any cooling are shown here. In this

157 particular case, the rubbery bulk modulus of the material was changed from 2.73e9 to

2.73e6 Pa (the shear modulus of the material was also changed accordingly).

Figure 7.27 Schematic showing the contributions of the two resistances to the total stress

However, it is important to note that the bulk modulus may not be directly affecting the polymer behavior at small strains. As discussed before in chapter 5, equations 5.5-7, the bulk modulus and shear modulus together determine the Young‘s modulus of the material, which in turn affects the elastic-plastic transition at small strains. Figure 7.28 shows the force versus time plots while figure 7.29 shows the displacement versus time plot for this simulation. Figure 7.28 looks more similar to the experimental data at 132 °C

(figure 7.8) than the previous predictions of the new model (figure 7.7). The simulation results for the same at end of deformation step, cooling step, and at end of unloading step is shown in figure 7.30. However, none of these set of constants were able to capture the

158 experimental data conducted at large strains. The conclusion of all these results is discussed in the final section of the chapter.

15 Force(N) 10

0 0 20 40 60 80 100 Time (seconds)

Figure 7.28 Force versus time plot for hot embossing simulation at 130 °C with a hold time of 90 seconds with changed material constants of new model

3 Displacement(mm)

0 0 20 40 60 80 100 120 Time (seconds)

Figure 7.29 Displacement versus time plot for hot embossing simulation at 130 °C with a hold time of 90 seconds with changed material constants of new model 159

(a) (b) (c)

Figure 7.30 Hot embossing simulations conducted at 130 °C with 90 second hold without cooling (a) at end of deformation step, (b) at end of cooling step, and (c) at end of unloading

160

7.6 Conclusion

The hot embossing simulations conducted here were compared with the experimental hot embossing data in chapter 5 to validate the new model. The simulations indicated that the model failed to predict the experimental data and demonstrated a very fast rate of stress relaxation. The first impulse was to conclude that the model is not capable of capturing the stress relaxation data well, but as the analysis of the new model showed, it does extremely well in predicting the stress relaxation of the material at large strains. This was very reassuring since the material constants were obtained from simple large strain compression experiments. When the material constants were changed to provide a more realistic stress relaxation response at small strains, the hot embossing simulations improved and correlated better with the hot embossing experiments. However, the model with the newly modified constants no longer captures experimental data at large strains.

To be able to accurately simulate hot embossing, the model must be capable of predicting the material behavior for both small and large strains. Even though the strain of the part along the direction of deformation is small, there will be various elements with large strains. Table 7.1 shows the logarithmic strains (LE) for the hot embossing simulation conducted with the modifed rubbery bulk modulus (figure 7.30). The values of the true strains go as high as ~ 0.64, this reinforces the importance of capturing material behavior at both small and large strains.

161

Max-in-plane Min-in-plane LE 11 LE 22

Minimum -6.19116E-03 - 638.29E -03 -570.671E-03 - 333.272E -03

Maximum 604.443E-03 - 9.00616E-03 295.355E -03 551.66E -03

Table 7.1 Logarithmic strains for the hot embossing simulation shown in figure 7.30.

From these observations, we can conclude that even if the model is capable of capturing the stress relaxation, temperature and strain rate dependence of experimental data at large strains, it may still not work for small strains (~ 0.1). This constitutive model can be used to simulate processes at large strains (> 1) but the material constants will need to be refit for simulating processes at smaller strains.

For future work, the continuum model material constants have to be refit to be able to capture the material data for both large and small strains. A working model can be studied for the following effects on the polymer part; (i) the embossing and final temperature, (ii) die geometry, (iii) feature spacing of the die/part, (iv) variation in hold and cooling time, and (v) the cooling rate.

162

CHAPTER 8

CONCLUSION AND FUTURE WORK

8.1 Conclusion

The goal of this thesis was to develop a constitutive model to simulate hot embossing for the amorphous polymer polymethyl methacrylate (PMMA). During hot embossing the polymer is heated above its glass transition temperature (Tg). A force/pressure is then applied to press the contoured features into the polymer surface. This force is held over the polymer for some time; this can either be with or without cooling of the sample, finally, the force is released and the desired feature is embossed on the polymer. The part is generally cooled below its glass transition before the end of the process. A constitutive model must be able to incorporate the following features: (i) initial modulus, (ii) followed by yield or flow, (iii) initial gradual strain hardening, and (iv) dramatic strain hardening

[1] with temperature, strain rate and strain state dependence in order to predict the polymer behavior accurately for hot embossing process.

Simple uniaxial and plane strain experiments were conducted on the polymer PMMA in our lab with variation in temperature and strain rate to gather material characterization data. This data was used to determine the material constants for the Dupaix-Boyce (DB)

163 constitutive model. The DB model was able to capture material behaviour from well below Tg to around Tg + 10 °C, but not at any higher temperatures. The next step was to modify the DB model so that it could predict the material behaviour even at temperatures well above the glass transition of the polymer. To better understand the material behaviour at these higher temperatures, another constitutive model by Dooling-Buckley-

Rostami-Zahlan (DBRZ) on PMMA was studied. This model was shown to work well exclusively above the Tg of the material. Both the continuum models were fit to the data in Dupaix lab and the material parameters were optimized using the Nelder-Mead algorithm [66, 67]. With the new set of optimized constants fit to the experimental data of

PMMA over a wide range of temperatures, the DBRZ model also failed to capture the material behavior well above Tg. It was deduced that a single set of material constants is not capable of predicting the polymer behavior over a wide range of temperature relevant for hot embossing of the polymer. To resolve this issue, two networks on the molecular relaxation side of the DB model were used; one for temperatures up to the glass transition and another for temperatures from Tg to well above Tg of PMMA. Similar approach was effectively used to capture material behavior of polymer by Srivasatava et al. [50] in

Anand‘s group. The New model with the additional molecular relaxation network for higher temperatures works very well in capturing the experimental data in uniaxial compression but not as well for plane strain compression.

To evaluate the new model for hot embossing simulation, hot embossing experiments were conducted to be compared with the model predications. Simple features were embossed on PMMA over a wide range of temperatures from below the glass transition

164 temperature to well above the Tg. Force and depth analysis was made with variation in temperature and hold times. As expected, the following trends were observed; (i) the force required to emboss the part reduced with increasing temperatures, (ii) depth dimension improved with increasing the hold and cooling times, and (iii) same hold times with longer cooling times also gave better final dimensions. Various factors such as time and equipment required to heat and cool the sample, accuracy of the final dimension desired, and the cost of the entire process are essential before deciding on the final process conditions. From the hot embossing experimental data obtained and studied, the most vital parameters which a constitutive model needs to incorporate in predicting the hot embossing accurately were found to be the following; (i) a material model that works accurately from Tg - 10 through Tg + 30 °C, and (ii) the model must be able to capture stress relaxation over time. Finally, hot embossing simulations were conducted to validate the New model using the hot embossing experimental data gathered in chapter 6.

The model failed to predict the experimental data and demonstrated a very fast rate of stress relaxation.

The new model was able to capture the stress relaxation of PMMA for simple compression experiments at large strain but not for hot embossing experiments at small strains. The model may be successfully used to simulate processes at large strains (~ 1) but the material constants will need to be refit for simulating processes at smaller strains

(~ 0.1) as well. It is essential to note that even though most of the material would undergo small strain deformation during hot embossing, there would be regions near the corners and edges that would be undergo large strain deformation. Thus, in order to accurately

165 replicate the hot embossing experiments, the constitutive model must be able to capture all the material behavior such as strain hardening, strain softening, stress relaxation, etc. with temperature and strain rate dependence at both large and small strains.

8.2 Future Work

Sufficient experimental data is available in the Dupaix lab for material characterization of the amorphous polymer polymethyl methacrylate (PMMA) at large strains. Most of the future work would be in (i) conducting compression experiments with stress relaxation at small strains, (ii) gathering more hot embossing experimental data, and (iii) in changing or refitting the material constants so that the new model is able to capture the material behavior at both large and small strains.

Compression experiments need to be conducted at very small strains with stress relaxation in addition to variations in temperature and strain rate. These results can be used to improve and validate the new model against small strain deformation. The model must be capable of capturing small strain deformation in addition to the large strain deformation in order to predict micro-hot embossing accurately.

More hot embossing tests need to be carried out with cooling of the samples during the hold step. Also, a more elaborate study of all these tests needs to be conducted with analysis (i) of the final dimensions of the features along with the depth, (ii) on a smaller scale, and (iii) monitoring the polymer temperature during all these experiments, especially during cooling. The hot embossing experiments conducted here were with a

166 embossed depth of 800 µm of the polymer samples. Additional experiments will also be required at much smaller scales to be relevant for micro-hot embossing.

The continuum model material constants have to be refit to predict the material behavior at both large and small strains more accurately. Since the material model works well at large strains, the refitting of constants for the intermolecular network (I) of the model should be adequate. This is because substantial contribution to the total stress comes from the intermolecular network at small strains. A working model can be studied for the following effects on the polymer part; (i) the embossing and final temperature, (ii) die geometry, (iii) feature spacing of the die/part, (iv) variation in hold and cooling time, and

(v) the cooling rate.

167

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