Mechanics of Materials 71 (2014) 74–84

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Mechanics of Materials

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Characterization and constitutive modeling of stress-relaxationbehavior of Poly(methyl methacrylate) (PMMA) across the glasstransition temperature

http://dx.doi.org/10.1016/j.mechmat.2014.01.0030167-6636/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +1 614 292 8404; fax: +1 614 292 3163.E-mail addresses: Mathiesen.2@osu.edu (D. Mathiesen), dupaix.1@

osu.edu (R.B. Dupaix).1 W185 Scott Laboratory, 201 West 19th Ave, Columbus, OH 43210, USA.2 E310 Scott Laboratory, 201 West 19th Ave, Columbus, OH 43210, USA.

Danielle Mathiesen 1, Dana Vogtmann 1, Rebecca B. Dupaix 2,⇑Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA

a r t i c l e i n f o

Article history:Received 26 July 2013Received in revised form 3 December 2013Available online 23 January 2014

Keywords:PMMAStress-relaxationConstitutive model

a b s t r a c t

Characterizing the large strain behavior of Poly(methyl methacrylate) (PMMA) across itsglass transition temperature is essential for modeling hot embossing. Its mechanical prop-erties vary significantly across the glass transition as well as with strain rate. Several pre-vious models have attempted to capture this behavior with limited success, and none haveconsidered stress relaxation. In this work, stress relaxation experiments are conducted onPMMA at various temperatures spanning the glass transition. The experimental data isthen used to develop a new constitutive model. As with earlier models for thermoplasticsaround the glass transition, the material model consists of two resistances: intermolecularand network. The key advantage of the new model is that the network resistance is repre-sented through an 8-chain hyperelastic model in parallel with a ROuse LInear EntangledPOLYmer (Rolie-Poly) element. The structure of the network interactions captures thestrain hardening at temperatures less than glass transition and the melt behavior at tem-peratures greater than glass transition to greatly improve stress relaxation predictions.Strain softening is introduced in the intermolecular branch to predict the stress at smallstrains. Glass transition is described through temperature dependent material properties.Results have been encouraging for the model’s ability to capture stress relaxation behaviorfrom temperatures 15� below to 25� above glass transition. In addition, it can capture therelaxation behavior at both large and small strains as well as with varying strain rates. Thisability to capture stress relaxation suggests it will greatly improve finite element modelpredictions for hot embossing with PMMA.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction is heated past its glass transition temperature. Then, a

Hot embossing or nano-imprint lithography is a lowcost processing technique that allows for micro- andnano-scale surface patterns, such as micro-channels, wells,and pyramids to be accurately applied to a polymer(Schneider et al., 2009). During hot embossing, a polymer

stamp with a pattern etched into it is applied to the poly-mer and compressed a finite amount. While held in thisposition for a specific hold time, the temperature is de-creased below glass transition. After, the stamp is removedas the polymer continues to cool as shown in Fig. 1. At thetemperatures hot embossing is performed, polymers arehighly sensitive to temperature fluctuations and strain rate(Dupaix and Cash, 2009). Creating a hot embossing proce-dure that is able to accurately utilize polymer glass transi-tion properties without simulations can be a costly andtimely endeavor. Therefore, a predictive constitutive model

Fig. 1. Schematic of hot embossing (A) polymer sample to be embossed and stamp with a raised pattern, (B) the polymer has been heated and pressureapplied to the stamp, and (C) cooling of the polymer sample with separation of the polymer from the stamp.

Fig. 2. Initial and deformed PMMA samples – PMMA samples before andafter a stress-relaxation test has been performed.

D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84 75

is necessary to model the process through finite elementanalysis.

One common material used in hot embossing isPoly(methyl methacrylate) or PMMA. PMMA has a glasstransition temperature, Tg, between 105 �C and 110 �C thatallows for easy processing (Lu et al., 2007). In addition, it ishighly biocompatible making it an ideal candidate forbiomedical applications, such as directed cell growth(Schneider et al., 2009). It has also been shown experimen-tally that high aspect ratios can be achieved using hotembossing (Lu et al., 2007; Narasimhan and Papautsky,2004).

Prior research on PMMA has included experimentalcharacterization in several deformation modes. Most rele-vant to this work is the large amount of uniaxial compres-sion data that has been gathered at small and large strains(Palm et al., 2006; Ghatak and Dupaix, 2010; Arruda et al.,1995). These experiments show that PMMA behaves as aviscoelastic solid below glass transition and a semi-viscousfluid above transition. While this data has been able tocharacterize thermal aging, strain rate, and temperaturedependence it does not describe the stress relaxation orcreep behavior that is present in all polymers (Palmet al., 2006; Ghatak and Dupaix, 2010; Boyce et al.,1988). The majority of stress relaxation experiments arebased on the works of McLoughlin and Tobolsky. By per-forming tensile stress relaxation experiments at smallstrains they obtained a master curve for PMMA (McLough-lin and Tobolsky, 1952). While this information is valuable,it does not give the relaxation profile for PMMA at largecompressive strains that are found locally during hotembossing. Others have performed small strain stressrelaxation in different stress states such as torsion (Takah-ashi et al., 1964) and multiaxial compression (Qvale andRavi-Chandar, 2004), again, with the results restricted tosmall strains.

Several constitutive models have been developed tocapture the behavior of PMMA with limited success. Manymodels are based on the Dupaix–Boyce model (Dupaix andCash, 2009; Palm et al., 2006; Ghatak and Dupaix, 2010;Ghatak, 2007; Palm, 2006). These models are able to pre-dict the stress in uniaxial compression at large strains attemperatures up to Tg + 10 �C but overestimate the stressat high temperatures (Palm et al., 2006; Ghatak and Dup-aix, 2010). The Doi–Edwards model was found to workwell above Tg + 15 �C as it is a fluid based model (Ghatakand Dupaix, 2010; Ghatak, 2007). The model developedby Anand et al. was successful at capturing the stress straincurve for temperatures across the glass transition and withmultiple strain rates, however it used over 40 constants

(Anand et al., 2009; Ames et al., 2009; Srivastava et al.,2010). Using this model they were able to predict the diefilling of a microchannel mold, however there was no pre-diction of the cooling or de-molding process (Srivastavaet al., 2010). While several models have worked for a spe-cific range of loading conditions, there is still no model thatis able to capture the behavior of PMMA at temperaturesacross different loading conditions.

Previous constitutive models have focused on modelingthe material in compression without considering the ef-fects of stress relaxation. Very little work has been doneon modeling the compressive response of PMMA duringstress relaxation at large held strains. Pfister performedsome experiments and developed a constitutive modelbut only in tension and at temperatures less than transi-tion (Pfister and Stachurski, 2002). The process of hotembossing involves large amounts of stress relaxation dur-ing the hold period, and often at finite deformation levels.Therefore, the goal of this paper is to characterize PMMA’sbehavior in stress relaxation across glass transition, evalu-ate the ability of current models to capture stress relaxa-tion, and develop a new constitutive model to describethe behavior, both in loading and stress relaxation.

2. Experiments

2.1. Experimental methods

The samples of PMMA used for testing were cylinderscut from commercial sheet stock supplied by Plaskolite,Inc. Before deformation, the dimensions were nominally8.8 mm in height and 10 mm in diameter. Fig. 2 showsan example of these specimens before and after testing.

An Instron 5869 screw driven (electromechanical)materials testing system was used with an Instron 3119-409 environmental chamber to heat the samples to the de-sired temperature. The system consisted of a static lower

Fig. 4. Stress-relaxation curves at 95 �C – stress vs. time for PMMA samplesat 95 �C strained at a rate of �1.0/min and held at different strains (�0.5,�1.0, and �1.5) for 180 s.

76 D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84

plate and a movable upper plate, both contained within theenvironmental chamber. An Instron 5800 controller run-ning Instron Bluehill software controlled the load frame.The software collected the displacement of the upper com-pression plate, and force was measured with a 50 kN loadcell. Force and displacement data were used to calculatetrue stress and true strain through an assumption of con-stant volume.

Samples were tested with a ramp-hold loading history,where the initial ramp was a constant true strain rate, fol-lowed by a hold period at a specified final strain. The testmatrix consisted of two different loading strain rates(�1.0/min and �3.0/min), three different final strains(�0.5, �1.0, and �1.5), and five different temperatures(ranging from 95 �C to 135 �C). Samples were placed in adesiccant chamber for at least 24 h prior to the test to con-trol the amount of moisture present. Teflon sheets wereplaced between the samples and compression platens toreduce friction and WD-40 was used to lubricate betweenthe plates and the Teflon film. Each sample was placed inthe environmental chamber for 30 min prior to testing toensure that it had uniformly reached the test temperature.Each test was run at least twice to ensure repeatability.

Fig. 5. Stress-relaxation curves at 105 �C – stress vs. time for PMMAsamples at 105 �C strained at a rate of �1.0/min or �3.0/min to differentstrains (�0.5, �1.0 or �1.5) and held for 180 s.

2.2. Experimental results

The results from the stress relaxation experimentsmatch what is currently known about polymers in com-pression as temperatures span the glass transition temper-ature, as shown in Fig. 3. At temperatures less than Tg,Figs. 4 and 5, the polymer exhibits a clear yield point, fol-lowed by a moderate amount of strain softening, then alarge region of strain hardening. Around Tg, Fig. 6, PMMAcombines solid and viscous effects. The yield point be-comes less pronounced and strain softening no longer oc-curs, however, a large amount of strain hardening is stillpresent. At temperatures greater than Tg, Figs. 7 and 8,the polymer behaves more like a fluid with no clear yieldpoint.

During relaxation, PMMA is dependent on temperature,final strain, and strain rate. To quantify the amount ofrelaxation present across different temperatures, the per-cent relaxed is defined as the maximum stress minus the

0 50 100 150 200 25005

10152025303540455055

Time (sec)

−Tru

e St

ress

(MPa

)

95◦C105◦C110◦C125◦C135◦C

Fig. 3. Stress-relaxation curves at multiple temperatures – stress vs. timecurves for PMMA samples at temperatures from Tg � 15 �C to Tg + 25 �Cstrained at a rate of �1.0/min to a final strain of �1.5, held for 180 s.

Fig. 6. Stress-relaxation curves at 110 �C – stress vs. time for PMMAsamples at 110 �C strained at a rate of �1.0/min or �3.0/min to differentstrains (�0.5, �1.0 or �1.5) and held for 180 s.

Fig. 7. Stress-relaxation curves at 125 �C – stress vs. time for PMMAsamples at 125 �C strained at a rate of �1.0/min or �3.0/min and held atdifferent strains (�0.5, �1.0 or �1.5) for 180 s.

Fig. 8. Stress-relaxation curves at 135 �C – stress vs. time for PMMAsamples at 135 �C strained at a rate of �1.0/min and held at differentstrains for 180 s.

Fig. 9. Percent relaxed vs. temperature – shows the percent relaxed fordifferent held strains (�1.5, �1.0, or �0.5) vs. temperature (Tg � 15 �C toTg + 25 �C) that were tested at a strain rate of �1.0/min.

D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84 77

steady relaxation stress divided by the maximum stress.This means that if all the stress were to be relaxed, the per-cent relaxed would equal 100%. If there were no relaxation,the percent relaxed would equal zero. As shown in Fig. 9, attemperatures less than transition the percent of relaxationdecreases with increasing final strain. However, at temper-atures greater than transition, the percent of relaxation in-creases slightly with increasing final strain. Near thetransition, the percent of relaxation does not have a strongdependence on the final strain. In addition, while observ-ing one final strain and increasing the temperature, thepercent relaxed will at first increase, then decrease.

Strain rate also has an effect on the relaxation profile ofPMMA. As strain rate increases the percent stress relaxedwith respect to the maximum stress increases, which canbe seen by looking at Figs. 5–7. Although the maximumstress also increases with increasing strain rate, theamount of stress relaxed is greater than the change in max-imum stress of the different strain rates at the same finalstrain. This indicates that the amount being relaxed away

does increase with strain rate and is not a side effect fromthe increased maximum stress.

These relaxation effects can be explained by under-standing the temperature and strain dependent polymermolecular movements. Below Tg, polymer molecules donot flow readily and require time to rearrange themselvesto reduce the stress. At small strains, there is no molecularalignment occurring like at large strains, allowing them torearrange easily into a lower stress state. Above Tg, poly-mer molecules flow and immediately arrange themselvesinto the lowest stress state, with the arrangement occur-ring even during loading. Small strains therefore are al-ready at a low stress state when the hold period begins,providing little remaining stress to be relaxed. More align-ment is allowed to accumulate with large strains that arerelaxed during the hold period. This effect also explainsthe large amount of relaxation that occurs with high strainrates. At high strain rates, the time necessary to rearrangethe molecules during loading is not available, so morestress is accumulated prior to the hold period. Once al-lowed to relax however, the polymers orient into a lowstress state similar to what was achieved for the lowerstrain rate.

3. Material model

3.1. Existing models

Previous models have been thought to fail at capturingthe spring back of PMMA during hot embossing becausethey were unable to capture the stress relaxation behavior.To verify this, models that have previously been successfulat capturing stress–strain behavior have been compared tothe experimentally obtained large compressive strainstress relaxation data. Dooling, Buckley, Rostami, and Zah-lan developed the first model tested (DBRZ). It consists ofthree branches as shown in Fig. 10, one governed by inter-molecular resistance and 2 branches governed by networkresistances, the details of which are described elsewhere(Dooling et al., 2002). K. Singh developed a second model

Fig. 10. Previous material models – (I) schematic as developed by Dooling, Buckley, Rostami, and Zahlan, divided into bond-stretching and conformationalcontributions (Dooling et al., 2002). (II) Schematic as developed by K. Singh, consisting of two resistances, intermolecular and network interactions, wherethe network interaction branches are activated with temperature (Singh et al., 2011).

78 D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84

successful in capturing the uniaxial behavior as shown inFig. 10 (Singh et al., 2011). Again it is broken into two resis-tances with the intermolecular interactions describedthrough a linear elastic spring with a thermally activateddashpot. The network resistance is described throughtwo thermally activated branches consisting of a highlynonlinear spring and dashpot.

3.2. New material model

The new material model consists of two resistances:intermolecular and network, as shown in Fig. 11. The inter-molecular interactions are represented with a linear elasticspring and thermally activated dashpot, denoted as resis-tance A. Network interactions consist of an 8-chain hyper-elastic spring, resistance B, in parallel with a ROuse LInearEntangled POLYmer element with finite extensibility, resis-tance C. This arrangement allows each section to have thesame deformation gradient, F as in Eq. (1). The total stress,T, is equal to the sum of the stress from each section shownin Eq. (2).

F ¼ FA ¼ FB ¼ FC ð1Þ

T ¼ TA þ TB þ TC ð2Þ

This new material model is based off of the originalDupaix–Boyce model with a few modifications (Dupaixand Boyce, 2007). Previously it was shown that the Dup-aix–Boyce model cannot capture PMMA’s behavior atT > Tg + 10 �C (Ghatak and Dupaix, 2010). To improve high

Fig. 11. New material model – schematic of new material model consist-ing of two resistances caused by intermolecular or network reactions.

temperature predictions, the network interactions mustbe altered. One way to do this is to use the Rolie-Poly mod-el (Likhtman and Graham, 2003). Likhtman and Grahamderived a simple constitutive model for ROuse LInearEntangled POLYmer melts, or Rolie-Poly, from the full ver-sion presented by Likhtman and Graham (2003), Grahamet al. (2003). This model is a refined version of Doi–Ed-wards, incorporating the effects of reptation, chain stretch,convective constraint release, and contour length fluctua-tions (Graham et al., 2003). Recently, De Focatiis et al.has shown the predictive capability of this model at largedeformations in pre-oriented polystyrene (De Focatiiset al., 2010). There, they combined the multi-mode bond-stretching component of the Oxford Glass-Rubber modelwith a multi-mode Rolie-Poly model to make a glass-meltmodel (Buckley and Jones, 1995; De Focatiis et al., 2010).Since they were using the Rolie-Poly model in semi-solidstates, finite extensibility was incorporated into the Ro-lie-Poly model. Using this model they were able to predictthe melt stage behavior very well but were under predict-ing the strain hardening at low temperatures because themodel is unable to capture the subentanglement chain ori-entation (De Focatiis et al., 2010).

By combining the melt stage behavior of Rolie-Polywith the 8-chain hyperelastic spring of Arruda–Boyce, net-work interaction predictions at temperatures near andabove transition will improve (Arruda and Boyce,1993a,b). The 8-chain hyperelastic spring without molecu-lar relaxation will provide the strain hardening caused bychain orientation at temperatures near and below Tg,where the De Focatiis et al. model fails (De Focatiis et al.,2010; Arruda and Boyce, 1993a,b). At these temperatures,the stress in the network interactions is dominated bythe hyperelastic spring and Rolie-Poly has little effect indi-cating little relaxation. As temperatures increase to themelt stage, the 8-chain hyperelastic spring will decreaseits contribution by increasing the number of rigid links be-tween entanglements and the Rolie-Poly model will thenbecome dominant to describe the melt and relaxation ef-fects in the network. Similar to De Focatiis et al., finiteextensibility as proposed by Kabanemi and Hetu will beincorporated into Rolie-Poly since it will have a small effectat semi-solid states near Tg (Kabanemi and Hétu, 2009).

Finally, to improve small strain predictions strain soft-ening must be incorporated. To do this, the intermolecularinteractions branch of the original Dupaix–Boyce model

D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84 79

was modified to include strain softening. At temperaturesless than transition, a moderate to large region of strainsoftening is observed. To capture strain softening, theathermal shear resistance was made plastic strain ratedependent (Boyce et al., 1988).

The following is a summary of the equations requiredfor the new model:

3.2.1. Intermolecular interactions: resistance AThe deformation gradient of resistance A is first multi-

plicatively broken down into its elastic and plasticcomponents.

FA ¼ FeAFp

A ð3Þ

The superscript e represents the elastic and p, the plas-tic contribution to the total deformation gradient.

Using polar decomposition, the elastic and plastic por-tions of the deformation gradient from Eq. (3) are furtherbroken down into stretch and rotational components inEq. (4).

FeA ¼ Ve

AReA

FpA ¼ Vp

ARpA

ð4Þ

The velocity gradient is found through substituting theelastic and plastic portions of the deformation gradientinto the definition of the velocity gradient in Eq. (5).

LA ¼ _FAF�1A

LA ¼ _FeAFe�1

A þ FeA

_FpAFp�1

A Fe�1A ¼ Le

A þ ~LpA

ð5Þ

The plastic portion of the velocity gradient is defined inEq. (6)as the sum of the symmetric rate of plastic straining~Dp

A and anti-symmetric spin ~WpA.

~LpA ¼ ~Dp

A þ ~WpA ð6Þ

The anti-symmetric spin is prescribed to be zero( ~Wp

A ¼ 0). The rate of plastic straining is then defined as

~DpA ¼ _cANA ð7Þ

where NA ¼ 1ffiffi2p

sAT0A is the normalized deviatoric stress, T0A is

the Cauchy stress, with sA ¼ ½12 trðT0AT0AÞ�1=2 and

T0A ¼ TA � 13 trðTAÞI.

The elastic portion is represented through a linearlyelastic spring in Eq. (8).

TA ¼1JA

Le ln VeA

� �ð8Þ

where JA ¼ det FeA; L

e is a fourth order tensor of elastic con-stants, and Ve

A is the stretch component of the elastic defor-mation gradient. The fourth order tensor of elasticconstants depends on the bulk and temperature dependentshear modulus, l through Eq. (9).

Le ¼ 2lUþ 3K � 2l3

1� 1 ð9Þ

where U is the fourth order and 1 is the second order iden-tity matrix.

The plastic strain rate in A, _cpA, follows a thermally acti-

vated process according to Eq. (10).

_cpA ¼ _cOA exp

�DGð1� sA=sÞkh

� �ð10Þ

_cOA is a pre-exponential factor, DG is the activation energy,s is the shear resistance, k is Boltzmann’s constant, and h isabsolute temperature.

The shear resistance s, is dependent on the temperatureand the plastic strain rate to include the effect of strainsoftening found in PMMA at temperatures below Tg.

_s ¼ h 1� ssss

� �_cp

A ð11Þ

h is the slope of the yield drop with respect to the plasticstrain, s is the current athermal deformation resistance ofthe material, and sss is the steady state deformation resis-tance, which depends on temperature.

3.2.2. Network interactions: resistance B and C3.2.2.1. Resistance B. An 8-chain hyperelastic model pro-vides the strain hardening at temperatures near and belowTg, represented as resistance B, through Eq. (12).

TB ¼1JB

mkh3

ffiffiffiffiNp

�kL�1

�kffiffiffiffiNp� �

�BB � ð�kÞ2I

h ið12Þ

where �k is the effective chain stretch, �k ¼ ½13 trð�BBÞ�1=2

with�BB ¼ �FB

�FTB , �FB ¼ ðJBÞ

1=3FB, and JB ¼ det FB. L�1 is the inverseLangevin function, LðbÞ ¼ cothðbÞ � ð1=bÞ. The number ofrigid links between entanglements is N, chain density ism, and mkh is rubbery modulus. This section decreases itscontribution at temperatures greater than Tg by having Nincrease with increasing temperature.

3.2.2.2. Resistance C. Resistance C is represented through aRolie-Poly element with finite extensibility. Finite extensi-bility is incorporated through the nonlinear spring coeffi-cient, ksðkÞ. The Rolie-Poly model shown in Eqs. (13) and(14)describes how the conformation tensor, r, changesdue to LC , the velocity gradient described by LC ¼ _FCF�1

C

_r ¼ LC � rþ r � LC þ fðrÞ ð13Þ

fðrÞ ¼�1sdðr� IÞ�2ksðkÞ

sR1�

ffiffiffiffiffiffiffi3

trr

r !rþb

trr3

� �d

ðr� IÞ" #

ð14Þ

The nonlinear spring coefficient in Eq. (15)is approxi-mated through a normalized Padé inverse langevin func-tion; in the original Rolie-Poly model it equals 1 (Pfisterand Stachurski, 2002). sd is the reptation time, sR is theRouse time, b is the convection constraint release coeffi-cient, and d is a negative power.

ksðkÞ ¼3� k2

k2max

1� 1

k2max

1� k2

k2max

3� 1

k2max

ð15Þ

The stress Tc relates to the conformation tensor r by

TC ¼g0

sdksðkÞðr� IÞ ð16Þ

80 D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84

g0 represents the temperature dependent zero shear ratepolymer viscosity and sd remains the reptation time.

Fig. 12. Stress vs. time as predicted by DBRZ model – stress vs. time data forsamples loaded at a rate of �1.0/min to a final strain of �1.5, held for180 s using the model proposed by Dooling–Buckley–Rostami–Zahlan.

3.2.3. Temperature dependenceTo incorporate the temperature dependent nature of

PMMA, some material constants depend on temperature, h.The shear modulus of resistance A, a part of the fourth

order tensor of elastic constants, is dependent on temper-ature through

l ¼ 12ðlg þ lrÞ �

12ðlg � lrÞ tanh

5Dhðh� hgÞ

� �ð17Þ

where lg is the glassy shear modulus, lr is the rubberyshear modulus, hg is the glass transition temperature, andDh is the temperature range over which the glass temper-ature range occurs. Shear rate dependence in resistance Ais incorporated by defining the glass transition tempera-ture as

hg ¼h�g þ n log

_cpA

_cref

: _cp

A P _cref

h�g : _cpA < _cref

8<:

9=; ð18Þ

where h�g is a reference temperature of 374 K, n is a mate-rial constant and _cref is 0.00173 s�1.

The steady state deformation resistance in A is temper-ature dependent through Eq. (18).

Sss ¼ Sf0l½S

fss þ ð1� Sf

ssÞ tanhðqAðh� hAÞÞ� ð19Þ

where Sf0 is an initial constant and l is the shear modulus

at the given temperature. Sfss is a fraction greater than zero

such that Sf0Sf

ssl represents the mean steady state deforma-tion resistance. qA is a ratio that controls the slope near thetransition temperature for the shear resistance, hA.

In resistance B the number of rigid links between entan-glements, N, is temperature dependent as in Eq. (20). Astemperature increases past transition, the hyperelasticspring’s contribution decreases as the behavior becomesmore melt like by having N increase with temperature.

N ¼ Nmax � N0ð1þ expðqBðh� hBÞÞÞ�1 ð20Þ

In Eq. (20), Nmax is the maximum number of rigid linksbetween entanglements, which occurs at high tempera-tures. Nmax � N0 is the number of rigid links at low temper-atures. qB is a ratio that controls the slope near the melttransition temperature, hB. To ensure conservation of mass,vðhÞNðhÞ ¼ const so entanglement density, vðhÞwill vary asNðhÞ given the constant is defined at a temperature lessthan Tg, where NðhÞ is at its minimum (Arruda et al., 1995).

In resistance C, similar temperature dependence isfound with the zero-shear rate viscosity, g.

g ¼ g0½1þ expðqcðh� hgÞÞ��1 þ g0I ð21Þ

In Eq. (21), g0 describes the difference between the maxi-mum and minimum viscosity and g0I is a mean viscositywith the qc ratio controlling the slope near transition.

The total list of 25 material constants fit for this PMMAis provided in Appendix A.

4. Fitting of material models

4.1. Existing model fits

Two models were examined to see whether they wouldfit the new large strain compressive stress-relaxation datagathered here. K. Singh previously fit the constants usedfor both models to uniaxial, large compressive strain dataof the same PMMA used for the stress relaxation experi-ments (Singh et al., 2011). The DBRZ model is shown inFig. 12, at 110 �C and 135 �C. Clearly this model does notfit the large strain compressive relaxation data well. It isable to capture the uniaxial compressive behavior for theloading period only at 110 �C and does a poor job at135 �C. This uniaxial behavior is consistent with what hasbeen seen previously with this model where it had beenshown to work for a narrow range of temperatures (Singhet al., 2011). Since the model was fit to data around theglass transition temperature, it fits that data better thanthe higher temperature of 135 �C. During the hold perioda very small amount of stress relaxes and it quickly reachesa large steady relaxation stress. At temperatures aroundtransition it relaxes a greater amount than at higher tem-peratures, which does match the trend shown in Fig. 9,however the amount it relaxes for both temperatures isinsufficient.

Shown in Fig. 13 is the model developed by K. Singh.This model clearly provides a better fit to the large com-pressive strain stress relaxation data than the DBRZ model.While it does an adequate job capturing the uniaxialbehavior prior to the hold period, it does a poor job captur-ing the relaxation profile of both temperatures. At 110 �C itover predicts the steady relaxation stress, while at 135 �C itunder predicts the steady relaxation stress. Both modelsfail to capture the relaxation profiles of PMMA at largecompressive strains, which limit their effectiveness at pre-dicting the behavior during hot embossing.

4.2. New model results

The proposed model predicts stress relaxation for tem-peratures from Tg � 15 �C to Tg + 25 �C for a held strain of

Fig. 13. Stress vs. time as predicted by K. Singh’s model – stress vs. time datafor samples loaded at a rate of �1.0/min to a final strain of �1.5, held for180 s using the model proposed by K. Singh.

Fig. 15. Stress relaxation prediction at 95 �C – stress vs. time of new modelat 95 �C for samples strained at a rate of �1.0/min to different final strains(�1.5, �1.0, �0.5). Experimental data is shown with bold lines andsimulation of the new model with fine lines.

D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84 81

�1.0 as shown in Fig. 14. The model captures the uniaxialcompression behavior prior to relaxation accurately for alltemperatures tested. During relaxation it captures the ini-tial relaxation rate for all temperatures tested and the stea-dy state stress for all temperatures. Figs. 15–17 show themodel’s ability to predict stress relaxation for differentheld strains at temperatures below, at, and above Tg.

Below glass transition temperature, Fig. 15, held straindependence has a severe effect on the relaxation profile.At small held strains, the model predicts the large amountof relaxation with respect to the maximum stress. While atlarge held strains, the model predicts the small amount ofrelaxation with respect to the maximum stress. Addition-ally at all strains, it does a great job capturing the initialrelaxation rate and long-term steady state stress. Prior torelaxation, it predicts the uniaxial compression behaviorincluding the large amount of strain softening present atsmall strains and strain hardening at large strains that pre-vious models have failed to capture.

0 50 1000

5

10

15

20

25

30

35

−Tru

e St

ress

(MPa

)

Fig. 14. Stress vs. time for new model at multiple temperatures – stress vs. time simfinal strain of �1.0 at multiple temperatures (95 �C, 105 �C, 110 �C, 125 �C, and

Near the glass transition temperature, Fig. 16, the mod-el predicts the stress relaxation behavior for the small tomedium held strains and slightly under predicts at largestrains. Prior to relaxation, it predicts the initial yield andsubsequent strain hardening present at small to mediumstrains. However at large strains, it under predicts theamount of strain hardening occurring. During relaxation,at small to medium held strains, the model exhibits thelarge initial relaxation rate while quickly settling to thecorrect steady state stress. At large held strains of �1.5, ithas a large initial relaxation rate greater than found exper-imentally but reaches the correct steady state stress. Nearthe glass transition temperature the model captures theuniaxial compression behavior through medium strainsand the held strain dependence of stress relaxation.

Above glass transition temperature, Fig. 17, held straindoes not affect the relaxation profile as dramatically ascompared to lower temperatures, however, the percent of

150 200Time (s)

95°C105°C110°C125°C135°CSim

ulation results (solid line) for samples strained at a rate of �1.0/min to a135 �C). Experimental data denoted by symbols.

Fig. 16. Stress relaxation prediction at 110 �C – stress vs. time of newmodel at 110 �C for samples strained at a rate of �1.0/min to differentfinal strains (�1.5, �1.0, �0.5). Experimental data is shown with boldlines and simulation of the new model with fine lines.

Fig. 17. Stress relaxation prediction at 135 �C – stress vs. time of newmodel at 135 �C for samples strained at a rate of �1.0/min to differentfinal strains (�1.5, �1.0, �0.5). Experimental data is shown with boldlines and simulation of the new model with fine lines.

Fig. 18. Stress relaxation prediction at 105 �C, 110 �C, and 125 �C at strainrate of �3.0/min – stress vs. time for the new model at differenttemperatures for samples strained at a rate of �3.0/min, to a strain of�1.0, held for 180 s. Experimental data is shown with bold lines and thenew model shown with fine lines.

82 D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84

relaxation that occurs is much less. Prior to relaxation, themodel predicts the lack of strain softening at small strainsand decrease in strain hardening at large strains. At smallheld strains, the model captures the moderate relaxationrate and the steady state stress seen experimentally. Atlarge held strains, the large initial relaxation rate is pre-dicted followed by fast leveling off to the steady statestress. The new material model captures the stress relaxa-tion profile at temperatures greater than glass transitionfor small to large deformations.

Strain rate dependence across the glass transition tem-perature is shown in Fig. 18. Here, it is clear that the modelpredicts the long-term relaxation behavior of PMMA for all

of the temperatures at a faster strain rate. While it capturesthe long-term relaxation behavior, it struggles to capturethe uniaxial compression behavior completely at tempera-tures less than transition. It slightly over estimates theamount of strain hardening present at fast strain rates.One reason this may be occurring is heating of the polymerdue to the deformation is not included in the model. If itwere, as the polymer is deformed it would increase in tem-perature, especially at high strain rates, which would thendecrease the strain hardening effects. However, at temper-atures greater than transition the model is able to captureboth the uniaxial compression and stress-relaxationbehavior. Even though the model is unable to completelycapture strain rate dependence at temperatures less thantransition, it does a good job at predicting the long-termrelaxation behavior important in hot embossing at alltemperatures.

4.3. Model limitations and future work

While this model captures the stress-relaxation behav-ior of PMMA from Tg � 15 �C to Tg + 25 �C it has not beenverified for temperatures outside this region to experimen-tal data. Also, it struggles to capture the uniaxial compres-sion strain rate dependence at temperatures less thantransition. In the future, if strain rate dependence at tem-peratures less than transition was to be improved, someof the new material constants may need to be made strainrate dependent. However, to capture the current behavior25 material constants were required, 8 of which to capturetemperature dependence alone. Therefore, it will be neces-sary to determine whether additional constants will bebeneficial to improve a small subset of simulations.

Table A1Material constants for new material model.

Resistance A constants (linear elastic with temperature activateflow)

lg 400 MPalr 2 MPaDh 40 KB 2.25 GPan 1 K_cOA 7.0 * 1013 1/sDG 2.14 * 10�19 Jh 500 MPa

Sf0

0.40

qA 1/10

Sfss

0.7

hA 383 K

Resistance B constants (8-chain hyperelastic spring)Nmax 107.7N0 106.1qB .12hB 399 KðvkhÞN 10 MPa

Resistance C constants (Rolie-Poly)g0 220 MPa ssD 100 ssR0 125 sb 0.1d �0.5k 1.40g0I 21 MPa sqc 1/7

D. Mathiesen et al. / Mechanics of Materials 71 (2014) 74–84 83

Additionally, cooling has not yet been considered. Toincorporate cooling, the temperature dependence of thematerial constants will need to be addressed. As the sam-ple is cooled, the experimentally measured stress will notincrease during the hold period. However, because someof the material constants, specifically the number of rigidlinks between entanglements, depend strongly on temper-ature it may need to be made a function of the maximumtemperature of the sample prior to cooling. This wouldbe appropriate because if the sample were deformed attemperatures greater than Tg, it would already have in-creased the number of links between entanglements.Therefore when it is cooled, the structure would be main-tained. Finally, thermal expansion of the polymer has notbeen included. In the future this can easily be introducedwith one additional material constant.

Future work includes implementing this material mod-el in finite element simulations and the incorporation ofcooling. Cooling experiments will be performed to quantifythe behavior of PMMA during the cooling process experi-enced during hot embossing to further verify the model.The model will be implemented in a finite element simula-tion to capture the die-filling, cooling behavior, and springback after the mold is removed. Simulation results will becompared against hot embossing experiments to verify themodel’s accuracy.

5. Conclusions

Stress relaxation behavior in PMMA has been docu-mented for Tg � 15 �C to Tg + 25 �C in compression at vary-

ing strain rates. The behavior matches current knowledgeof PMMA behavior in uniaxial compression and sheds lighton its behavior in stress relaxation. Using this data, the per-cent relaxed was calculated to quantify the amount ofrelaxation at different temperatures and strains. This datawas then used to develop a new model for PMMA to cap-ture the stress relaxation which previous models havebeen unable to capture. To do this, the new model utilizeda unique arrangement of three resistances in parallel.Resistance A captured intermolecular interactions by com-bining a linear spring and thermally activated dashpot tocapture the initial elastic region, yield, and subsequentstrain softening. It provides a large initial relaxation andis the dominant resistance at temperatures less than tran-sition. Network interactions were captured through an 8-chain hyperelastic spring, resistance B, in parallel with aRolie-Poly element, resistance C. The 8-chain hyperelasticspring captured the chain orientation at temperatures nearand below Tg while melt behavior was captured by the Ro-lie-Poly element. The combination of these three differentresistances is what captures the varying behavior of PMMAacross different temperatures. This model has been shownto capture the small and large strain behavior of PMMA instress relaxation for temperatures from Tg � 15 �C toTg + 25 �C.

Acknowledgements

The authors thank National Science Foundation forfunding this work under the NSF CMMI Grant No.0747252 and Plaskolite, Inc. for supplying the material.

Appendix A

See Table A1.

Appendix B. Supplementary data

Supplementary data associated with this article can befound, in the online version, at http://dx.doi.org/10.1016/j.mechmat.2014.01.003.

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