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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 Before the die is withdrawn from the material, the polymeric material is cooled below the glass transition temperature in order to “freeze” the material into its final embossed shape.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;
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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
tures closer tovalues at relativ
rate. in its present s
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MA behaves difmodels. In ordsince one mod
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modeling theare some concewhile highly s
ment. f Ames, et al.
model essential
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cantly differenon in and ultimes like a pseu
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(2009) is alsolly adds a third
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n Figure 14: te
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udo-fluid with but a fluids-baan 15 degrees a
ome limitationseratures of 125nsition (115°Cn rates, it was n
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tial and the othel that can caped, fluid-like bbehavior overthat model woanticipate that
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MMA above Glass T
Experimental and ests at 120°C using
xperimental and sists at 130°C using
cluding signifiion of strain hno strain hard
ased model, suabove the glasss even in the w°C and 130°C
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nt temperature ons, the two moher integral in npture this transbehavior. Ther the wide temould be able tot many of the i
forward in m schematic use
Transition
simulation resultsg the Doi-Edwards
imulation results fg the Doi-Edwards
icant softeninghardening. At hdening. The Duch as the Dois transition tem
warmest temper, it could not cFurthermore,
ectively capture
at higher straiarrucci, 2001; Wansition zone bso strain harden
ranges, in thiodels would hanature. Futuresition from strae recent modelmperature rano handle non-iideas in their p
modeling behaved in the Dupai
6
s for uniaxial comps model
for uniaxial comprs model.
g of the materihigher temper
Dupaix-Boyce mi-Edwards modmperature. Therature regime. Wcapture the obsalthough the me the behavior
ins. Strain hardWiest, 1989) arbetween 115°Cning at high sts work it had ave to be seame work in this aain hardening-l of Richeton, ge across the isothermal situpaper will be h
vior in this comix-Boyce mode
61
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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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