INTERNATIONAL JOURNAL OF STRUCTURAL CHANGES IN SOLIDS – Mechanics and Applications Volume 2, Number 1, April 2010, pp. 53-63

_____________________________________________________ *Email: dupaix.1@osu.edu

53

Material Characterization and Continuum Modeling of Poly (Methyl Methacrylate) (PMMA) above the Glass Transition

Arindam Ghatak, Rebecca B. Dupaix *

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210

Abstract

Uniaxial compression tests were conducted on poly (methyl methacrylate) (PMMA) over a wide range of strain rates and temperatures in and above the glass transition (from 102°C to 130°C). PMMA exhibits different behavior close to the glass transition (below 115°C) as compared to temperatures farther above the glass transition. In the temperature range just above the glass transition, a clear yield point and strain hardening at higher strains is observed. At temperatures farther above the glass transition, PMMA shows more fluid-like behavior, with no clear yield point or strain hardening at high strains. This change in behavior with temperature poses difficulties in using some of the existing constitutive models, as illustrated by the use of two different models, namely, the Dupaix-Boyce model and the Doi-Edwards model. The data obtained is used to calibrate the two models in order to predict the behavior of PMMA across this industrially significant range of processing temperatures for hot embossing applications.

Keywords: uniaxial compression, constitutive model, PMMA, glass transition

1. Introduction

Poly (methyl methacrylate) (PMMA) is a thermoplastic used in applications ranging from microelectromechanical systems (MEMS), to micro-optics and medical devices. In these applications, it is often necessary to create micro-scale features on the polymer surface using techniques such as hot embossing. Hot embossing can produce micron-scale features and below in thermoplastics such as PMMA (Sotomayor Torres et al., 2003). The hot-emboss process involves localized surface deformation using a die at temperatures above the material’s glass transition. Before the die is withdrawn from the material, the polymeric material is cooled below the glass transition temperature in order to “freeze” the material into its final embossed shape. However, the material does experience some spring-back from its embossed position after die removal, even with substantial cooling. Therefore, it is important to understand the relationships between spring-back, rate of loading, and processing temperatures in order to predict and optimize embossing processes while retaining quality features.

In order to better understand the behavior of PMMA, uniaxial compression tests were conducted over a range of strain rates and temperatures. The stress-strain behavior obtained from the tests was then fit to the Dupaix-Boyce and Doi-Edwards models. A significant change in the behavior of PMMA was observed as it was heated to progressively higher temperatures. Beginning at temperatures around 115-120°C, PMMA begins to exhibit more fluid-like behavior. There is significant softening of the material at higher temperatures with no clear yield point and a loss of strain hardening. While the Dupaix-Boyce model successfully captures the behavior of PMMA between 102 and 115°C, the fluids-based Doi-Edwards model better captures its behavior between 120 and 130°C.

2. Background

Previous experimental work on PMMA near the glass transition has been conducted by Palm et al. (2006), G’Sell and Souahi (1997), and Dooling et al. (2002). Recent work by Palm et al. (2006) used the Dupaix-Boyce model (Dupaix and Boyce, 2007) to capture the behavior of PMMA at temperatures above the glass transition (θg), though the temperature range explored in that work was limited to temperatures up to θg+13°C (115°C). Another model by Dooling et al. (2002) also attempts to capture this temperature range with good results over the temperature range 114 to 190°C (all above θg), and includes rate dependence . A very recent model by Richeton et al. (2007) is able to capture the behavior of PMMA over a wide range of temperatures and rates, though it requires a large number of fitting constants and may have numerical problems in simulating cooling effects, as will be discussed later.

Fluids-based approaches to modeling polymer mechanical behavior have also been used at temperatures approaching the glass transition from above (Hirai et al., 2003; Juang et al., 2002a, 2002b; Rowland, 2005a, 2005b;

rata

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54

Scheer, 2005arelationship asand are especitransition, but and Edwards,

3. Experiment

Cylindricalcompression te140,000) and 5869 electromsoftware. Friccompression pwere heated totesting. Strain2.0.

4. Experiment

Figures 1 tstrain at each tFurthermore, tof strain hardeabove 120°C, 120°C for testselastic portion above 115°C, c

Figures 7 astrain rates of -past one anothillustrated in thmore strain-hastrain softeninclose to θg (espmake it clear thand 120°C cansmall amount o

Overall, thtemperature deabove θg to tem

Figure 1:U

Inte

a, 2005b). Ths the input matally questionabfor illustrative1978, 1986) to

tal Details

l test specimeesting. The spewere stored in

mechanical loadction was minplatens, with cao test temperatun rates ranged f

tal Results

through 6 showtemperature. Athe material strening begins to

strain hardenis at slower straof the stress-s

coinciding withand 8 show tem-1.0/min and -3her due to inchese figures. Tardening compng evident at 1pecially at highhat different mn be seen as a tof strain harden

he experimentaependence is omperatures mor

Uniaxial compressi

ernational Jour

hese models haterial model. Hble below θg. W

e purposes showo capture the be

ens of 10 mmecimens were n a dessicant cd frame with imized by pla

are taken to avoures ranging frfrom -0.05/min

w how the streAs can be seen rain hardens at

decrease for teng is completeain rates. The ystrain curve evih considerablemperature dep3.0/min respec

creased mobilitThe stress-straipared to the hi02°C and 107h strain rates)

mechanisms aretransition perioning still persial results are cobserved and thre than 15 degr

on experimental d

rnal of Structur

ave been usedHowever, theseWe seek a modw the ability oehavior at temp

m average diammachined from

chamber prior a 50 kN load acing Teflon soid contact betrom 102°C to n to -6.0/min an

ess-strain behavin these figuretemperatures nemperatures beely absent (Figyield point in aident at lower

e softening of thendence throu

ctively. At highty. Therefore, in curves at 10gher temperatu°C at these higis the prior the

e at work at temod, wherein wests at higher st

consistent withhis data clearlyrees above θg.

data at 102°C.

ral Changes In

d in hot emboe models oftendel that can caf one fluids-baperatures more

meter and 7.7m sheet stock sto testing. Thcell and an I

sheets and WDtween the lubri130°C and wend specimens w

vior depends oes, the materialnear the glass tetween 115°C gures 5 and 6)all of these casetemperatures ghe material at gh true stress-her temperaturthe stress val

02°C, 107°C anures. There is gher strain rateermal history amperatures beloe see “fluid-liktrain levels. h other results y shows a trans

Figur

Solids, 2(1), 20

ssing simulatin fall short in capture the full ased model by than 15 degre

78 mm averagsupplied by Ple experiments

Instron 5800 cD-40 lubricanicant and the teere allowed to were compress

on strain rate. l behavior stifftransition tempand 120°C (Fi

). This behavioes is not readilgives way to m115°C and abotrue strain plo

res, the polymelues are much nd 110°C showa distinct yiel

es. The reasonand aging of thow and above ke” stress-strai

in the literatusition in behav

re 2:Uniaxial comp

010 53-63

ons, with a vicapturing the brange of behavDoi and Edwaes above θg.

ge height werelaskolite, Inc. (

were conductcontroller run nt between theest specimens. equilibrate for

sed to a final tr

Plots show trufens with an inperature (Figurgures 3 and 4)

or is also noticly distinguishab

much more comove. ts over a rang

er molecules calower at high

w significantlyld followed byn for the softenhe sample. The115°C. The regn characteristi

ure. In particuvior from temp

pression experime

iscosity-tempebehavior closervior across theards (Doi, 1980

e used for un(molecular weited using an Inby Instron Blu

e specimen an The test spec

r 20 minutes prrue strain of -1

ue stress versuncrease in strainres 1-3). The am and at temper

ceable at 115°Cble. The steep

mpliant curves

e of temperatuan more easily her temperatury higher stressey a small amouning at tempere observations gion between 1cs emerge, tho

ular, strong rateratures close

ental data at 110°C

erature r to θg e glass 0; Doi

niaxial ight = nstron ueHill nd the imens rior to .5 to -

us true n rate. mount ratures C and linear at and

ures at move

res, as es and unt of ratures above 115°C ough a

te and to but

C.

5

tT

Figure 3: U

Figure 5: U

Figure 7: Uniaxia 5. Constitutiv

In this sect

5.1. Dupaix-Bo

The Dupatemperatures oThe details of t

Ghatak

Uniaxial compress

Uniaxial compress

al compression exp

ve Modeling

tion, we investi

oyce Model

ix-Boyce modof θg +13°C (Pathe model can

k and Dupaix / Ch

sion experimental d

ion experimental d

perimental data at

igate the ability

del has previoalm et al., 200be found in ea

haracterization and

data at 115°C.

data at 125°C.

a strain rate of -1.

y of two model

ously been us6) and here we

arlier work (Du

d Modeling of PM

Figu

Figu

.0/min

ls to capture th

sed to capture attempt to exupaix and Boyc

MMA above Glass T

ure 4: Uniaxial com

ure 6: Uniaxial com

Figure 8: Uniaxia s

he observed exp

re the mechanxtend its applicce, 2007), but a

Transition

mpression experim

mpression experim

al compression expstrain rate of -3.0/m

perimental beh

nical behaviorcation to even ha brief summar

5

mental data at 120°

mental data at 130°

perimental data at amin

havior of PMM

r of PMMA higher temperary is given here

55

°C.

°C.

a

MA.

up to atures. e.

(tdmfgd

T

w

e

w

e

t

wtggt

f

56

A schemat

(intermoleculathe Cauchy stdeformation gmultiplicativelformulation ungradient. Condetermine the r

The constitutiv

where IJ det=

elastic constanThe plastic

where IGΔ is

exponential fac

the absolute teThe shear m

where μg reprethe temperaturglass transitionglass transitiontemperature w

*gθ is the refer

strain rate, equfashion to equa

Inte

tic of the consar: I and netwotress in each gradient: F =I

ly and the plastnique. A connstitutive moderate of plastic s

ve equations ar

eIFt is the volu

nts, and the supc part of (I) is a

s the activation

ctor, Is is the

mperature of thmodulus strong

21μ =

esents the modre range acrossn regime.

gθ isn region is alsoith strain rate a

rence glass traual to .00173/ation (3):

ernational Jour

Figure 9:

stitutive modelork: N). The pabranch: TT =

FF == N. Each

tic spin in the nstitutive modeels for each dastraining.

re as follows. T

ume change, 1J

perscript e denoassumed to foll

pIγ =&

n energy of th

shear resistanc

he material, angly depends on

21)(

21

rg μμ −+

dulus in the gla which the glass the glass trano strain-rate deas:

⎪⎩

⎪⎨

⎧

=g ξθ

ansition temper/sec. The bul

rnal of Structur

Schematic of the

is shown in Farallel nature o

NI TT + and theh deformationelastically loadel for each spramper relate th

The elastic par1 eCJΙΙ

⎡Τ = ⎣

ln eIJ

⎡ ⎤⎣ ⎦V is the

otes the elastic low a thermally

0 e x pIγ⎡ Δ−⎢

⎣&

he material wh

ce, taken to be

nd Iτ is the man temperature a

tan)(21

rg μμ −

assy region, rμ

ss transition ocnsition temperaependent, this i

⎠

⎞⎜⎜⎝

⎛

ref

pI

g

γγ

θ

&

&log

:

10

*

rature taken tolk modulus is

ral Changes In

Dupaix-Boyce co

Figure 9, wherof the model ime deformation

n gradient is ded configuratiring is needed

he plastic strain

rt of (I) is takenenVΙ⎡ ⎤Ι⎣ ⎦

Hencky strain

part of the defy activated pro

(1 /I I IG sk

τθ−

hich must be o

0.15 times the

agnitude of theand is captured

(5nh( gθθθ

−Δ

represents theccurs, and

gX iature of the mais taken into ac

+⎟⎟⎠

⎞<

pIg

refp

I

γθ

γγ

&

&&

:*

o be 104 C° , ξalso taken to

Solids, 2(1), 20

nstitutive model

re the model implies that the

gradient in edecomposed

ion is prescribed to relate then rate to the s

n to be linearly

(Anand, 1979

formation gradocess:

) ⎤⎥⎦

overcome befo

e shear modulu

deviatoric stred through:

()) gg X θθ −+

e modulus in this the slope (ofaterial. Since tccount by shift

≥ refγ&

is a material cbe temperatur

010 53-63

is interpreted we total stress is each branch is

into elastic ed to be equal

e stress to the shear stress, wh

y elastic:

9), eC is the fo

dient.

ore flow can b

us, k is Boltzm

ess (Dupaix an

)gθ

he rubbery regif μ versus θ) outhe modulus ( μfting the glass t

constant, and re dependent,

with two resistequal to the su

s equal to theand plastic

to zero, to makelastic deform

hich is then us

ourth order ten

begin, 0Iγ& is a

mann’s constant

nd Boyce, 2007

on, θΔ is utside the

μ ) in the transition

refγ& is the refefollowing a s

tances um of

e total parts

ke the mation sed to

(1)

nsor of

(2)

a pre-

t, θ is

7).

(3)

(4)

erence imilar

Ghatak and Dupaix / Characterization and Modeling of PMMA above Glass Transition

57

))(5tanh()(21)(

21

grgrg BBBBB θθθ

−Δ

−−+= (5)

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:

( )2113

eNN N N

N N

vk N LJ N

λθ λλ

− ⎛ ⎞ ⎡ ⎤= −⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠T B I (6)

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. 1L− is the inverse Langevin function defined as ( )( ) coth( ) 1/L β β β= − . The effective

chain stretch Nλ is given by the root mean square of the distortional applied stretch:

( )2/1

31

⎥⎦⎤

⎢⎣⎡= e

NN tr Bλ

( )TeN

eN

eN FFB = , ( ) e

NNe

N J FF 3/1−= , eNNJ Fdet=

(7)

This model has previously been given temperature dependence, through the constants ν and N (Arruda et al. 1995). However, this can cause numerical problems in simulations involving cooling. The argument inside the inverse Langevin function (

√) must always be less than one. If N is prescribed to increase with temperature, as is done in the

Richeton et al. (2007) model, and a simulation is performed at an elevated temperature, then a relatively large value of may be achieved. Now, as the material cools, N will decrease, possibly to the point that √ becomes smaller than

the current value of and causing a numerical singularity. To avoid this, we interpret the molecular network as being independent of temperature. Instead, temperature is viewed as facilitating reptation, through the molecular relaxation dashpot. The rate of molecular relaxation for resistance N is given by:

1 /

0

/ 1/ 1

np c N

Nc c

Cvk

α α ταγα α α θ

⎛ ⎞ ⎛ ⎞−= ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

& (8)

where n is a power-law exponent, α is a measure of the orientation of the polymer chains with initial value 0α . cα is a cutoff value, beyond which molecular relaxation ceases. α is calculated as:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

++−= −

23

22

21

3211 ),,min(cos

2 λλλ

λλλπα (9)

where λi are the principal stretches. The parameter C is temperature dependent and is given by:

exp QC DRθ

⎧ ⎫= −⎨ ⎬⎩ ⎭

(10)

where D and Q/R are material parameters.

There are a total of 15 parameters used in this model, given in Table 1. The two constants that are perhaps the most physically intuitive are the glassy and rubbery modulus. These are intended to represent the initial elastic stiffness of the material below and above the glass transition. The constants were obtained from data fit close to the glass transition temperature, so the “glassy modulus” used here is somewhat lower than would be expected for PMMA at room temperature, because it was obtained just a few degrees below the glass transition. The rubbery modulus may seem a bit high for the very compliant stress-strain curves seen above the glass transition, however, this constant is determined from the very early part of the stress-strain curve before any yielding or flow occurs. Since flow occurs at very low stress levels above the glass transition, the initial elastic behavior has a fairly small effect on the overall stress-strain curve above the glass transition.

M

rrtt

am

5

TIie

58

Modeling Resu

Figures 10 rates ranging fresults. Howevthe initial modthis model is cstress, as well above θg. In thmore than 15 d

5.2 Doi-Edwar

The Doi-EThis approach In this approacis in contrast toelastic to plasti

Figure 10: Expe tests

Inte

Initial Elast

Flow

Resistance

Molecular

ults

and 11 show tfrom -0.05/minver, as figure 1dulus and the acompletely una

as predicts sthe next sectiondegrees above

rds Model

Edwards modelis used to illus

ch, the phenomo the glass-rubic over time, w

erimental and simuat the glass transit

ernational Jour

Table 1

tic Behavior

Stress

e Elasticity

Relaxation

the experimentn to -6.0/min. 12 shows, the amount of straiable to capture train hardeningn, we discuss thθg.

l was developestrate the poten

menon of stressbber model justwith unchanged

ulation results for ution temperature, 1

rnal of Structur

1: Material constan

MateriaGlassy ModuluRubbery ModuTemperature STransition SlopRate Shift Fac

Glassy Bulk MRubbery Bulk Pre-exponentiaActivation EneRubbery OrienEntanglement Temperature CSecond TempePower-law ExpCutoff Orienta

tal and simulateAs can be seemodel overprein hardening. the stress-stra

g that is not obhe Doi-Edward

ed to capture pntial of fluids-bs relaxation is it presented, wh

d material prop

uniaxial compress102°C.

ral Changes In

nts for the Dupaix

al Property us ulus Shift pe tor

Modulus Modulus

al Factor ergy ntation ModuluDensity

Coefficient erature Parameponent

ation

ed stress-strainen, the model edicts the stres

At even warmain behavior. Ibserved in expds model in an

polymer chain based modelinginterpreted as hich models st

perties.

sion Fig

Solids, 2(1), 20

-Boyce model

Symbolμg

Bg Br

us

eter 1/n

n curves for PMpredictions are

ss values at 11mer temperaturIt both overpreperiments at te

attempt to cap

reptation and g at temperatura reduction in tress relaxation

gure 11: Experime compressi

rμθΔgX

ξ

0Iγ&GΔ

vkθND

/Q R

cα

010 53-63

Value 325 MPa 50 MPa 30 K -3 KPa/K 3 K

1.0 GPa 2.25 GPa

137.5 10× 12.12 10−×8.0 MPa 500

41 .7 1 0× 1/71042.1 × K

6.67 0.0012

MMA at 102°Ce in good agre5°C. The mod

res, as the mateedicts the initiaemperatures mpture this beha

is a Non-Newres more than 1material modu

n as a transfer o

ental and simulatioion tests at 110°C

1/s 19 J

/s K

C and 110°C at eement with thdel overpredictserial softens fu

al modulus andore than 15 de

avior at temper

wtonian fluid m15 degrees aboulus over time.of deformation

on results for uniax

strain he test s both urther, d yield egrees ratures

model. ove θg. This

n from

xial

u

w

f

tk

wo

b

. The originuniaxial compstress tensor as

where the relax

pτ is a time cfurther terms d

The other

tangent vector k, and the temp

Qα

where οindi

orientation ten

Uniaxial Defor

Assumingbecomes

Ghatak

Figure

nal model devression. Details a function of

xation modulu

constant, and pdoes not signifivariables are

to the polymeperature of the

( ) ( )= α

αβF

FuF

icates a volumnsor which acco

rmation

g uniaxial defo

k and Dupaix / Ch

e 12: Experimental

elopment, as cled derivationstime are taken

(αβσ

s 'μ is given b

'μ

p is a set of inicantly change the deformati

er chain segmematerialθ . Q

( )− αβ

βα δ31

2u

Fu

e average comounts for the de

ormation in the

haracterization and

l and simulation re

contained in (Ds of these equan to be

)( '0 dtGt

t

μ∫∞−

=

by

2 2

8( )p odd

tp π

= ∑

ntegers. We usthe results whon gradient F

ent u, the curreQαβ

is given by:

(∫

⎪⎩

⎪⎨⎧

=s

dπ4

2

0

Fuu

mputed as an ineformation gra

e z-direction a

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

λ

0

0

1

F

d Modeling of PM

esults for uniaxial

Doi, 1980) is cations can be f

([)( '' tQtt Fαβ−

2 expp p

tτ τ

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

se only p = 1 hile increasing cF, the compone

ent time t, a ref

) ( )−βα δ

31

2Fu

Fuu

ntegral over thadient operating

and incompress

⎥⎥⎥⎥⎥⎥

⎦

⎤

λλ0

01

00

MMA above Glass T

compression tests

cast in a form tfound in Dupa

)], 't

since it is shocomputation timents of the Ca

ference time in

⎪⎭

⎪⎬⎫

αβδ

e surface of a g on the unit v

sibility of the m

Transition

at 115°C.

to predict stresaix (2003). The

own in (Dupaixme. auchy stress te

n the past 't , B

unit sphere. Tector along the

material, the d

5

ss-strain behave components

(

(

x, 2003) that a

ensor αβσ , the

oltzmann’s co

This is effectivee chain backbo

deformation gra

59

vior in of the

(11)

(12)

adding

e unit

nstant

(13)

ely an ne.

adient

(14)

International Journal of Structural Changes In Solids, 2(1), 2010 53-63

60

where λ is the applied z-direction stretch and the nonzero components of can be expressed as:

2 2 1 2

3 2 2 1 2 2( )( )

z xzz xx

z x y

u uQ Q Fu u u

ο

λ λλλ λ

−

−

−− = ≡

+ + (15)

This can be evaluated analytically using spherical coordinates for u and the stress is given by:

, ,

8

, (16)

For loading at a constant strain rate starting at t = 0,

( )( )( )

'

'' '

exp : 0,

exp ( ) : 0

t tt t

t t t

ελ

ε

⎧ <⎪= ⎨− ≥⎪⎩

&

&

(17)

Taking only the p=1 term and simplifying, equation (16) becomes:

( )( )3 30(exp( )) exp exp exp

t

zz xx pp p

t sZ F t Z ds F sσ σ ε τ ετ τ

⎛ ⎞ ⎛ ⎞− −− = +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫& &

(18)

wherep

GZτπ 208

= .

The two constants to be determined are Z and τp. Temperature dependence is introduced by making Z and τp

functions of temperature through the following curve-fit expressions:

3 21.1 10 0.8859 178.4393Z θ θ−= × − + (19)

34.289914.15102 212 +−×= − θθτ p (20)

where temperature θ is in Kelvin. Modeling Results

Figures 13-16 show the experimental and simulated (using the Doi-Edwards model) true stress-strain curves from 115 to 130°C at strain rates of -0.05/min, -1.0/min and -3.0/min. The Doi-Edwards model is able to capture the initial slope of the stress-strain curves successfully. However, the fit at higher strains is not as good, especially at temperatures between 115°C and 120°C. The reason for this is that at temperatures around 115°C, there is a small amount of strain hardening taking place, and strain hardening effects are not included in the Doi-Edwards model. At higher temperatures, where strain hardening is absent from the data, the Doi-Edwards model does a much better job. At all temperatures, the Doi-Edwards model can partially capture rate dependence, but its predictive abilities are very poor for the lowest strain rate. 6. Disucssion and Conclusions

Uniaxial compression tests were conducted over a wide range of temperatures and strain rates on PMMA to verify and improve upon the models described in (Doi, 1980; Doi and Edwards, 1978, 1986; Palm et al., 2006). The experimental results were consistent with previous experimental data in this temperature range (Dooling et al., 2002; G’Sell and Souahi, 1997; Palm et al., 2006). Test results up to about 115°C show material characteristics of typical polymer materials which undergo elastic deformation followed by plastic deformation accompanied by strain hardening, especially at higher strain rates and lower temperatures.

F

dawbEt

m

mp

Mmicn(twi

t

Figure 13: Experim tests at

Figure 15: Experi tests at

Test resultdistinguishablearound 125°C works well fobetter for captuEdwards modethe model replstrain hardeninsuccessfully prmaterial at the

The Doi-Edmodels explorepossible direct120°C, which More criticallymodeled usingintegrated, a ncertainly warranetwork-like b(2007) is cleatransition. Howhere the matin future mode

Another retemperature re

Ghatak

mental and simula115°C using the D

imental and simula125°C using the D

s above 115°Ce yield point,

and 130°C, Pr capturing lowuring the behavel, in particularlicated the test ng at temperatredicts stress vslowest strain dwards model ed in the fluidstion for improis characterize

y, since PMMg two different on-trivial task anted to try tobehavior to moar progress in owever, there aerial is cooled

eling developmecent model ofegime. Their m

k and Dupaix / Ch

ation results for unDoi-Edwards mode

ation results for unDoi-Edwards mode

C show significand a reductio

PMMA behavewer temperatuvior at temperar, was also shoresults well at

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International Journal of Structural Changes In Solids, 2(1), 2010 53-63

62

depending on whether the material is below or above the glass transition, either branch 2 or branch 3 becomes active in the model. The main drawback to using this model in simulations is the need to fit close to 50 material constants, so it seems there is still room for improvement in developing a practical model for hot embossing simulations.

In spite of the limitations shown here for the Dupaix-Boyce model at temperatures more than 15 degrees above the glass transition, even a limited model may be fairly successful in predicting hot embossing outcomes (Cash and Dupaix, 2008). It largely depends on the precise temperature range of interest, the relevant time scales (strain rates), as well as the level of deformation expected in the embossing operation. Regardless of the choice of model, since all of these models were developed from data collected on isothermal experiments, care must be taken when modeling processes involving potentially large temperature changes so as to avoid introducing artifacts into the simulations. References Anand, L., 1979, On H.Hencky’s Approximate Strain Energy Function for Moderate Deformations, J. Applied

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deformations of amorphous polymers. Part II: Applications. International Journal of Plasticity, 25, 1495–1539. Arruda, E.M., Boyce, M.C., and Jayachandran, R., 1995, Effects of strain rate, temperature and thermomechanical

coupling on the finite strain deformation of glassy polymers, Mechanics of Materials, 19, 193-212. Dupaix, R.B. and Cash, W., 2009, Finite Element Modeling of Polymer Hot Embossing: Effects of Temperature on

Springback and Residual Stress, Polymer Engineering and Science, 39, 531-543. Doi, M., 1980, A constitutive equation derived from the model of Doi and Edwards for concentrated polymer solutions

and polymer melts, Journal of Polymer Science: Polymer Physics Edition, 18, 2055-2067. Doi, M. and Edwards, S.F., 1978, Dynamics of concentrated polymer systems. Part 3 – the constitutive equation.

Journal of the Chemical Society – Faraday Transactions II, 74, 1818-1832. Doi, M. and Edwards, S.F., 1986, The theory of polymer dynamics, Oxford University Press.***** Dooling, P.J., Buckley, C.P., Rostami, S., and Zahlan, N., 2002, Hot-drawing of poly(methyl methacrylate) and

simulation using a glass-rubber constitutive model, Polymer, 43, 2451-2465. Dupaix, R.B., 2003, Temperature and rate dependent finite strain behavior of poly (ethylene terephthalate) and poly

(ethylene terephthalate)-glycol above the glass transition temperature, PhD Thesis, MIT, Boston, MA.***** Dupaix, R.B., Boyce, M.C., 2007, Constitutive modeling of the finite strain behavior of amorphous polymers in and

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Strains, Journal of Engineering Materials and Technology, 119, 223-227. Giesekus, H., 1982, A simple constitutive equation for polymer fluids based on the concept of deformation-dependent

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Ianniruberto, G. and Marrucci, G., 2001, A simple constitutive equation for entangled polymers with chain stretch, Journal of Rheology, 45, 1305-1318.

Juang, Y.J., Lee, L.J., and Koelling, K.W., 2002a, Hot embossing in microfabrication. Part I: Experimental, Polymer Engineering and Science, 42, 159-165.

Juang, Y.J., Lee, L.J., and Koelling, K.W., 2002b, Hot embossing in microfabrication. Part II: Rheological characterization and process analysis, Polymer Engineering and Science, 42, 551-566.

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