(Methyl Methacrylate) (PMMA)

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

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 Before the die is withdrawn from the material, the polymeric material is cooled below the glass transition temperature in order to “freeze” the material into its final embossed shape.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; _____________________________________________________ *Email: [email protected]

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Inte ernational Jourrnal of Structurral Changes In Solids, 2(1), 20 010 53-63

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Scheer, 2005aa, 2005b). Thhese models haave been usedd in hot embossing simulations, with a viiscosity-tempeerature relationship r ass the input matterial model. However, H thesee models oftenn fall short in ccapturing the behavior b closerr to θg and a are especially questionab ble below θg. We W seek a moddel that can caapture the full range of behav vior across thee glass transition, t but for illustrativee purposes show w the ability of one fluids-baased model by Doi and Edwaards (Doi, 19800; Doi and a Edwards, 1978, 1986) too capture the beehavior at tempperatures more than 15 degrees above θg. 3. 3 Experimenttal Details Cylindricall test specimeens of 10 mm m average diam meter and 7.7 78 mm averagge height weree used for unniaxial compression c teesting. The speecimens were machined from m sheet stock supplied s by Pllaskolite, Inc. (molecular ( weiight = 140,000) and were stored inn a dessicant chamber c prior to testing. The experiments were conductted using an Innstron 5869 electrom mechanical load d frame with a 50 kN load cell and an Instron I 5800 controller c run by Instron BluueHill software. Fricction was minimized by plaacing Teflon sheets s and WD D-40 lubricannt between thee specimen annd the compression c p platens, with caare taken to avooid contact bettween the lubriicant and the teest specimens. The test specimens were w heated too test temperatu ures ranging frrom 102°C to 130°C and weere allowed to equilibrate forr 20 minutes prrior to testing. t Strainn rates ranged from f -0.05/minn to -6.0/min annd specimens were w compresssed to a final trrue strain of -1.5 to 2.0. 2 4. 4 Experimenttal Results Figures 1 through t 6 show w how the streess-strain behavvior depends on o strain rate. Plots show truue stress versuus true strain at each temperature. t As A can be seen in these figurees, the materiall behavior stifffens with an inncrease in strainn rate. Furthermore, F the t material strrain hardens at temperatures nnear the glass transition t tempperature (Figurres 1-3). The am mount of o strain hardeening begins to decrease for teemperatures beetween 115°C and 120°C (Figures 3 and 4) and at temperratures above a 120°C, strain hardening is completeely absent (Figgures 5 and 6)). This behavioor is also noticceable at 115°C C and y point in aall of these casees is not readilly distinguishabble. The steep linear 120°C for testss at slower straain rates. The yield elastic e portion of the stress-sstrain curve eviident at lower temperatures gives g way to m much more com mpliant curves at and above a 115°C, coinciding c withh considerablee softening of thhe material at 115°C and aboove. Figures 7 and a 8 show tem mperature dependence through true stress-true strain plots over a range of temperatuures at strain rates of -1.0/min and -3 3.0/min respecctively. At highher temperaturres, the polymeer molecules caan more easily move past p one anoth her due to inccreased mobilitty. Therefore, the stress vallues are much lower at high her temperaturres, as illustrated i in th hese figures. The T stress-straiin curves at 1002°C, 107°C annd 110°C show w significantlyy higher stressees and more m strain-haardening comppared to the higher temperatuures. There is a distinct yielld followed byy a small amouunt of strain softenin ng evident at 102°C and 107°C at these higgher strain ratees. The reasonn for the soften ning at temperratures close c to θg (esppecially at high h strain rates) is the prior theermal history and a aging of thhe sample. Thee observations above make m it clear th hat different mechanisms m aree at work at tem mperatures belo ow and above 115°C. The reggion between 115°C 1 and a 120°C can n be seen as a transition t periood, wherein wee see “fluid-likke” stress-strain characteristics emerge, thoough a small amount of o strain harden ning still persists at higher sttrain levels. Overall, thhe experimentaal results are consistent c withh other results in the literatuure. In particuular, strong ratte and temperature t deependence is observed o and th his data clearlyy shows a transsition in behavvior from temperatures close to but above a θg to tem mperatures morre than 15 degrrees above θg.

Figure 1:U Uniaxial compression experimental data d at 102°C.

Figurre 2:Uniaxial comppression experimeental data at 110°C C.

Ghatakk and Dupaix / Chharacterization andd Modeling of PM MMA above Glass T Transition

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Figure 3: Uniaxial U compresssion experimental data d at 115°C.

Figuure 4: Uniaxial com mpression experim mental data at 120°°C.

Figure 5: Uniaxial U compression experimental data d at 125°C.

Figu ure 6: Uniaxial com mpression experim mental data at 130°°C.

Figure 7: Uniaxiaal compression expperimental data at a strain rate of -1..0/min

Figure 8: Uniaxiaal compression expperimental data at a sstrain rate of -3.0/m min

5. 5 Constitutivve Modeling In this secttion, we investiigate the abilityy of two modells to capture th he observed expperimental behhavior of PMM MA. 5.1. Dupaix-Booyce Model The Dupaix-Boyce moddel has previoously been ussed to capturre the mechannical behaviorr of PMMA up to temperatures t o θg +13°C (Paalm et al., 2006) and here wee attempt to ex of xtend its appliccation to even higher h temperaatures. The T details of the t model can be found in eaarlier work (Duupaix and Boycce, 2007), but a brief summarry is given heree.


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Figure 9: Schematic of the Dupaix-Boyce constitutive model

Figure 9, wherre the model is i interpreted with w two resisttances A schemattic of the consstitutive model is shown in F (intermolecula ( ar: I and netwoork: N). The paarallel nature oof the model im mplies that thee total stress is equal to the suum of the t Cauchy sttress in each branch: T = TI + TN and thee deformation gradient in each e branch iss equal to thee total deformation d g gradient: h deformationn gradient is decomposed into elastic and plastic parts FI = FN = F . Each multiplicativel m ly and the plasttic spin in the elastically loadded configuratiion is prescribeed to be equal to zero, to makke the formulation f unnique. A connstitutive modeel for each sprring is needed d to relate thee stress to the elastic deform mation gradient. g Con nstitutive modeels for each daamper relate thhe plastic strainn rate to the sshear stress, wh hich is then ussed to determine d the rate r of plastic straining. s The T constitutivve equations arre as follows. The T elastic parrt of (I) is takenn to be linearlyy elastic:

ΤΙ =

1 e C ⎡⎣ ΙnVΙ e ⎤⎦ JΙ

(1)

where w ume change, 1 ⎡ln V e ⎤ is the Hencky strain (Anand, 19799), C e is the fo ourth order tennsor of J I = dett FIe is the volu I ⎦ ⎣ J

elastic e constannts, and the supperscript e denootes the elastic part of the defformation graddient. The plasticc part of (I) is assumed a to folllow a thermallyy activated pro ocess: ⎡

γ& Ip = γ& 0 I e x p ⎢ − ⎣

Δ G I (1 − τ I / s I ) ⎤ ⎥ kθ ⎦

(2)

where w n energy of thhe material whhich must be overcome o befoore flow can begin, b Δ G I iss the activation γ&0I is a preexponential e facctor,

s I is the shear resistancce, taken to be 0.15 times thee shear moduluus, k is Boltzm mann’s constantt, θ is

the t absolute temperature of th he material, an nd

τ I is the maagnitude of the deviatoric streess (Dupaix annd Boyce, 20077).

The shear modulus m stronggly depends onn temperature aand is captured d through: μ =

5 1 1 ( μ g + μ r ) − ( μ g − μ r ) tannh( (θ − θ g )) + X g (θ − θ g ) 2 2 Δθ

(3)

where w μg repreesents the moddulus in the glaassy region, μ r represents thee modulus in thhe rubbery region, Δθ is the t temperaturre range across which the glasss transition occcurs, and X g is i the slope (off μ versus θ) ouutside the glass g transition n regime. θ g iss the glass trannsition temperaature of the maaterial. Since tthe modulus ( μ ) in the glass g transitionn region is alsoo strain-rate deependent, this iis taken into acccount by shift fting the glass transition t temperature t with strain rate as: a ⎧ θ g * : γ& Ip < γ& ref ⎪ ⎛ γ& Ip ⎞ θg = ⎨ (4) ⎜ ⎟ + θ g * : γ& Ip ≥ γ& ref log ξ 10 ⎪ ⎜ γ& ⎟ ⎝ ref ⎠ ⎩ * θ g is the referrence glass traansition temperrature taken too be 104 °C , ξ is a material cconstant, and γ&ref is the refeerence strain rate, equual to .00173//sec. The bullk modulus is also taken to be temperaturre dependent, following a similar fashion f to equaation (3):

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

B =

57

1 1 5 ( B g + B r ) − ( B g − B r ) tanh( (θ − θ g )) 2 2 Δθ

(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: TN =

2 1 vkθ N −1 ⎛ λ N ⎞ ⎡ e ⎤ L ⎜ ⎟ ⎣⎢ B N − ( λ N ) I ⎦⎥ J N 3 λN ⎝ N ⎠

(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. L−1 is the inverse Langevin function defined as L( β ) = coth( β ) − (1/ β ) . The effective chain stretch λ N is given by the root mean square of the distortional applied stretch: λ N = ⎢ tr (B ⎣3 ⎡1

( )

B Ne = FNe FNe

T

e N

)⎤⎥

1/ 2

⎦

, FNe = ( J N )−1 / 3 FNe , J N = det FNe

(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 and causing a numerical singularity. To avoid this, we interpret the molecular network as being the current value of 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: ⎛ α /αc −1 ⎞⎛ α τ N ⎞ ⎟⎜ ⎟ ⎝ α 0 / α c − 1 ⎠ ⎝ α c vk θ ⎠

1/ n

(8)

γ& Np = C ⎜

where n is a power-law exponent, α is a measure of the orientation of the polymer chains with initial value α0 . a cutoff value, beyond which molecular relaxation ceases. α is calculated as:

α=

⎛ min(λ , λ , λ ) ⎞ 1 2 3 ⎟ − cos −1 ⎜ ⎜ λ2 + λ2 + λ2 ⎟ 2 2 3 ⎠ ⎝ 1

π

αc

is

(9)

where λi are the principal stretches. The parameter C is temperature dependent and is given by: ⎧ Q ⎫ C = D exp ⎨ − ⎬ ⎩ Rθ ⎭

(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.


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Table 1: 1 Material constannts for the Dupaix-Boyce model

Initial Elasttic Behavior

Flow Stress Resistancee Elasticity

Molecular Relaxation

Materiaal Property Glassy Moduluus Rubbery Moduulus Temperature Shift S Transition Sloppe

Symbol

Value

μg μr Δθ Xg

325 MPa 50 MPa 30 K -3 KPa/K

Rate Shift Factor

ξ

3K

Glassy Bulk M Modulus Rubbery Bulk Modulus Pre-exponentiaal Factor

Bg Br γ&0I

1.0 GPa 2.25 GPa

Activation Eneergy

ΔG

Rubbery Orienntation Modulu us Entanglement Density Temperature Coefficient C Second Tempeerature Parameeter Power-law Expponent Cutoff Orientaation

vkθ

N D Q/R

1/n

αc

1 7.5 × 1013 1/s − 19 2.12 × 10 J 8.0 MPa 500 1 .7 × 1 0 4 1//s 1 . 42 × 10

7

K

6.67 0.0012

Modeling M Resuults Figures 10 and 11 show the t experimenttal and simulateed stress-strainn curves for PM MMA at 102°C C and 110°C at strain rates r ranging from f -0.05/minn to -6.0/min. As can be seeen, the model predictions aree in good agreeement with thhe test results. r Howevver, as figure 12 1 shows, the model overpreedicts the stresss values at 115°C. The moddel overpredictss both the t initial moddulus and the amount a of straiin hardening. At even warm mer temperaturres, as the mateerial softens fuurther, this t model is completely c unaable to capture the stress-straain behavior. It I both overpreedicts the initiaal modulus andd yield stress, as well as predicts sttrain hardeningg that is not obbserved in exp periments at teemperatures more than 15 deegrees above a θg. In thhe next section n, we discuss th he Doi-Edwardds model in an attempt to cappture this behaavior at temperratures more m than 15 degrees d above θg. 5.2 5 Doi-Edwarrds Model The Doi-E Edwards modell was developeed to capture polymer p chain reptation and is a Non-New wtonian fluid model. m This T approach is used to illusstrate the poten ntial of fluids-bbased modeling g at temperaturres more than 15 1 degrees aboove θg. In I this approacch, the phenom menon of stresss relaxation is interpreted i as a reduction in material moduulus over time. This is i in contrast too the glass-rubbber model justt presented, whhich models sttress relaxationn as a transfer of o deformationn from elastic e to plastiic over time, with w unchangedd material propperties.

Figure 10: Expeerimental and simu ulation results for uniaxial u compresssion tests at the glass transittion temperature, 102°C. 1

Fig gure 11: Experimeental and simulatioon results for uniaxxial compressiion tests at 110°C


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Figuree 12: Experimentall and simulation reesults for uniaxial compression tests at 115°C. . The origin nal model development, as contained c in (D Doi, 1980) is cast c in a form tto predict stresss-strain behavvior in uniaxial u compression. Detailled derivationss of these equaations can be found f in Dupaaix (2003). Thee components of the stress tensor ass a function of time are takenn to be t

σ αβ ( t ) = G 0

∫ dt

'

μ ' ( t −t ' ) Q αβ [ F ( t , t ' )]

−∞

where w the relaxxation modulus

μ'

( (11)

is given by b

∑

μ ' (t ) =

p odd

⎛ −t exp ⎜ ⎜τ p p π τp ⎝ 8

2

2

⎞ ⎟⎟ ⎠

( (12)

τ p is a time constant, c and p is a set of inntegers. We usse only p = 1 since it is shoown in (Dupaixx, 2003) that aadding further f terms does d not signifiicantly change the results whhile increasing computation c tim me. The other variables are the deformation gradient F, F the componeents of the Caauchy stress teensor σ αβ , thee unit tangent t vector to the polymeer chain segmeent u, the curreent time t, a refference time inn the past t ' , Boltzmann’s constant k, k and the temp perature of the material θ . Qαβ is given by: Qαβ α (F ) =

(Fu )α (Fu )β Fu

2

1 − δ αβ 3

=∫ 0

s

⎫⎪ u )α (Fu )β 1 d 2 u ⎧⎪ (Fu − δ αβ ⎬ ⎨ 2 4π ⎪ 3 Fu ⎪⎭ ⎩

(13)

where w indiicates a volume average com mputed as an inntegral over the surface of a unit sphere. This T is effectiveely an ο orientation o ten nsor which accoounts for the deeformation graadient operatingg on the unit vector along thee chain backbone. Uniaxial Deforrmation Assumingg uniaxial defo ormation in thee z-direction aand incompresssibility of the m material, the deformation d graadient becomes b ⎡ ⎢ ⎢ F=⎢ ⎢ ⎢ ⎢ ⎣

1

λ 0 0

0 1

λ

0

⎤ 0⎥ ⎥ 0⎥ ⎥ λ⎥ ⎥ ⎦

(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:

λ 2 u z2 − λ − 1u x2 λ u z2 + λ − 1 ( u x2 + u y2 )

Q zz − Q xx = F3 ( λ ) ≡

(15)

2

ο

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

, (16)

8

,

For loading at a constant strain rate starting at t = 0, ⎧⎪ e x p ( ε& t ) : ' & − e x p ε ( ) : t t ⎪⎩

λ (t , t ' ) = ⎨

(

t' < 0

)

(17)

t' ≥ 0

Taking only the p=1 term and simplifying, equation (16) becomes: ⎛ −t ⎜τ p ⎝

σ zz − σ xx = Z F3 (ex p ( ε& t ))τ p ex p ⎜

where Z = G 0

8

π 2τ p

⎞ ⎟⎟ + Z ⎠

∫

t 0

⎛ −s d s ex p ⎜ ⎜τp ⎝

⎞ ⎟⎟ F3 ( ex p ( ε& s ) ) ⎠

(18)

.

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:

Z = 1.1× 10 −3θ 2 − 0.8859θ + 178.4393

(19)

τ p = 2 × 10 −12 θ 2 − 15.14θ + 2899.34

(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.


Figure F 13: Experim mental and simulaation results for uniaxial compressionn tests at 115°C using the Doi-Edwards D modeel.

Figure 15: Experiimental and simulaation results for unniaxial compressioon tests at 125°C using the Doi-Edwards D modeel.

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Figure 14: Experimental and simulation resultss for uniaxial comppression teests at 120°C usingg the Doi-Edwardss model

Figure 16: Experimental and siimulation results for f uniaxial comprression tests at 130°C usingg the Doi-Edwardss model.

Test results above 115°C C show significcantly differennt behavior, inccluding signifiicant softening g of the materiial, no distinguishable d e yield point, and a reductio on in and ultim mate eliminatiion of strain hardening. h At higher h temperratures around a 125°C and 130°C, PMMA P behavees like a pseuudo-fluid with no strain harddening. The Dupaix-Boyce D m model works w well for capturing low wer temperatu ure behavior, but b a fluids-baased model, suuch as the Doii-Edwards moddel, is better b for captuuring the behavvior at temperaatures more thaan 15 degrees above a the glasss transition tem mperature. Thee DoiEdwards E modeel, in particularr, was also sho own to have soome limitationss even in the w warmest temperrature regime. While W the t model repllicated the test results well att higher tempeeratures of 125°C and 130°C, it could not capture c the obsserved strain hardeninng at temperattures closer too the glass trannsition (115°C C and 120°C). Furthermore, although the m model successfully prredicts stress values v at relativvely high strainn rates, it was not n able to effeectively capturee the behavior of the material m at the slowest strain rate. s does not incorporate strrain hardening at higher straiins. Strain harddening The Doi-Eddwards model in its present state models m exploreed in the fluidss literature (succh as Giesekuss, 1982; Ianniruuberto and Maarrucci, 2001; Wiest, W 1989) arre one possible p directtion for improovement. This would be esppecially imporrtant in the traansition zone between b 115°C C and 120°C, which is characterizeed by material softening duee to high temperature but alsso strain harden ning at high sttrains. More M critically y, since PMM MA behaves diffferently in thee two differennt temperature ranges, in this work it had to be modeled m using g two different models. In ord der to perform 3-D simulatioons, the two moodels would haave to be seam mlessly integrated, i a non-trivial task since one mod del is differenttial and the othher integral in nature. n Futuree work in this area a is certainly c warraanted to try too develop a coombined modeel that can cappture this transsition from straain hardening--based network-like n b behavior to moolecular relaxaation-dominateed, fluid-like behavior. b Thee recent modell of Richeton, et al. (2007) ( is cleaar progress in modeling thee large strain behavior overr the wide tem mperature range across the glass transition. t Ho owever, there are a some conceerns with how that model woould be able too handle non-iisothermal situuations where w the material is cooled while highly stretched. s We anticipate thatt many of the iideas in their paper p will be helpful h in i future modeeling developm ment. Another reecent model off Ames, et al. (2009) is alsoo a large step forward in m modeling behavvior in this com mplex temperature t reegime. Their model m essentiallly adds a thirdd branch to the schematic useed in the Dupaiix-Boyce modeel and

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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 Mech.,46, 78-82. Ames, N.M., Srivastava, V., Chester, S.A., and Anand, L., 2009, A thermo-mechanically-coupled theory for largedeformations 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 above the glass transition, Mechanics of Materials, 39, 38-52. G’Sell, C. and Souahi, A., 1997, Influence of Crosslinking on the Plastic Behavior of Amorphous Polymers at Large 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 tensorial mobility, Journal of Non-Newtonian Fluid Mechanics, 11, 69-109. Hirai, Y., Yosikawa, T., Takagi, N., Yoshida, S., and Yamamoto, K., 2003, Mechanical Properties of Poly-methyl methacrylate (PMMA) for Nano Imprint Lithography, Journal of Photopolymer Science and Technology, 16, 615620. 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. Palm, G., Dupaix, R.B., Castro, J., 2006, Large strain mechanical behavior of poly (methyl methacrylate) (PMMA) near the glass transition temperature, Journal of Engineering Materials and Technology-Transactions of the ASME, 128, 559-563. Rowland, H.D., Sun, A.C., Schunk, P.R., and King, W.P., 2005a, Impact of polymer film thickness and cavity size on polymer flow during embossing: Toward process design rules for nanoimprint lithography, Journal of Micromechanics and Microengineering, 15, 2414-2425. Rowland, H.D., King, W.P., Sun, A.C., and Schunk, P.R., 2005b, Simulations of nonuniform embossing: the effect of asymmetric neighbor cavities on polymer flow during nanoimprint lithography, Journal of Vacuum Science and Technology, 6, 2958-2962.

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Richeton, J., Ahzi, S., Vecchio, K.S., Jiang, F.C., and Makradi, A., 2007, Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates, International Journal of Solids and Structures, 44, 7938-7954. Scheer, H.-C., Bogdanski, N., Wissen, M., Konishi, T., and Hirai, Y., 2005a, Polymer time constants during low temperature nanoimprint lithography, Journal of Vacuum Science and Technology, 23, 2963-2966. Scheer, H.-C., Bogdanski, N., and Wissen, M., 2005, Issues in nanoimprint processes: the imprint pressure, Japanese Journal of Applied Physics, 44, 5609-5616. Sotomayor Torres, C.M., Zankovych, S., Seekamp, J., Kam, A.P., Clavijo Cedeño, C., Hoffmann, T., Ahopelto, J., Reuther, F., Pfeiffer, K., Bleidiessel, G., Gruetzner, G., Maximov, M.V., Heidari, B., 2003, Nanoimprint lithography: an alternative nanofabrication approach. Materials Science and Engineering, 23, 23-31. Wiest, J. M., 1989, A differential constitutive equation for polymer melts, Rheologica Acta, 28, 4-12.