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An excellent example are semi-crystalline polymerswhere thetemperature andow history experiencedbythematerial during molding result in an inhomogeneous,anisotropic crystalline morphology. [1,2] As a result, samplestaken from an injection molded product of a semi-crystal-

line polymer like polyethylene, display different failurebehavior dependenton positionand orientation, e.g., toughparallel to ow and brittle in perpendicular direction. [3,4]

For amorphous polymers the effects of ow-inducedorientation during injection molding is much less pro-nounced. Frozen-in molecular orientation causes someanisotropy, [5,6] but its inuence on the yield stress canbe considered small with respect to itsabsolute value. [7] Onthe other hand, the thermal history experienced uponsolidication from the melt has a marked inuence on themechanical properties of molded polymer glasses. Deter-mining for the evolution of properties is the kinematicnature of the glass transition, T g, and the inherent non-equilibrium state below T g.[8,9] Differences in cooling ratehave large consequences for the yield stress of the product,and its time-to-failure under static or dynamic load. [1012]

Another important issue is that the change in yield stressmay lead to a change in failure mode from ductile tobrittle. [13,14]

With respect to mechanical analysis, considerable efforthas been directed toward the numerical simulation of deformationandfailureofglassypolymers.A numberof 3Dconstitutive models were developed and validated, e.g., inthe group of Mary Boyce at MIT, [1517] the group of PaulBuckley in Oxford, [1820] and in our Eindhoven group. [2123]

These developments enabled a quantitative analysis andprediction of short- and long-term failure (static anddynamic), [10,11] and demonstrated the intimate relationbetween the value of the yield stress and the lifetime inlong-term static loading. In our Eindhoven Glassy Polymermodel (the EGP-model), [23] changes in the value of the yieldstress, resulting from alterations in the thermodynamicstate, can be accommodated by adapting only a singleparameter Sa . Since the other parameters required areindependent of thermal history, only a single yield stressmeasurement will sufce to fully characterize the materialand enable numerical simulation of its mechanicalperformance. [10] Unfortunately, however, this might be atrivial exercise in the case of a standardized test piece withhomogeneous properties; it surelyis notin thecase of morecomplex product geometries, possessing a heterogeneousdistribution of yield stress throughout the product. More-over, fortrueproduct optimization onewouldliketo predictthe nal properties of a product in a virtual environment,without even the need of making a prototype.

In the present study, we aim to develop a predictive,quantitative methodology that will enable not onlyevaluationof thisprocess-inducedstructuraldevelopment,but also provide direct assessment of its inuence on the

resulting short- and long-term mechanical performance of the nal product. The basis is formed by a recentlydeveloped methodology that allows the evaluation of theevolution of the yield stress of glassy polymers duringprocessing. [2426] Themethodemploysstandardsimulation

tools to obtain the thermal history during cooling from themelt in each position of a molded product. If the agingkinetics are known, this information allows calculation of the yield stress distribution throughout the nal object. Bytranslating this yield stress information into the corre-sponding Sa -distributions, we open new possibilities fordirect numerical evaluation of the short- and long-termperformanceofproductsinaveryearlystageofdesign.Thiswill provide designers insight into consequencesof specicchoices of processing parameters on the time-to-failure of their products, and provide a means for true productoptimization.

Modeling

Multi-Mode EGP-Model

The basis of the EGP-model is the decomposition of theCauchystress into a hardeningstress s r anda driving stresss s , which is split up into a hydrostatic part (superscript h)and a deviatoric part (superscript d).[23] The deviatoric partis modeled as a combination of n parallel linked Maxwellelements, [22,27,28] which, for isothermalconditions,leads to:

s s r s hs

Xn

i1

s ds;i

Gr ~ B d k J 1 I Xn

i1Gi ~ B de;i: (1)

where Gr is the strain-hardening modulus, ~ B the isochoricleft Cauchy-Green strain tensor, k the bulk modulus, J thevolume change ratio, I the unity tensor, and G is the shearmodulus. The subscript e refers to the elastic part, thesubscript i refers to a specic mode, i 1; 2; 3; . . . ; n .

The evolution of the elastic and volumetric strains isgiven by:

_ J Jtr D (2)

_~ B e;i ~ L D p;i ~ B e;i ~ B e;i ~ Lc D p;i : (3)The plastic deformation rate tensors D p;i are related to

the deviatoric stresses s ds;i by a non-Newtonian ow rule:

D p;i s ds;i

2h i t ; p; Sa ; (4)

T. A. P. Engels, L. C. A. van Breemen, L. E. Govaert, H. E. H. Meijer

830Macromol. Mater. Eng. 2009 , 294 , 829838

2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DOI: 10.1002/mame.200900227

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where t , the total equivalent stress, and p, the hydrostaticpressure, depend on the total stress, according to:

t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 s ds : s dsr ; p 13 tr s : (5)The viscosities are described by an Eyring ow rule,

which hasbeen modied [21,23] to takepressure dependenceand intrinsic strain softening into account:

h i h 0;it =t 0

sin h t =t 0 exp

m pt 0 exp S : (6)

where h 0;i is the zero viscosity of the ith mode, t 0 the

characteristic stress, and S is the state parameter thatcaptures the effect of thermal history (aging) and strainsoftening. Note that the viscosities are dened withrespect to the rejuvenated reference state. [17,23,29]

S is related to the equivalent plastic strain g p accord-ing to:S g p Sa R g p (7)

where Sa captures the initial thermodynamic state of thematerial, which is here assumed to be dened solely by itsthermal history, and R g p is the softening function:

R g p 1 r 0 exp g p

r 1

r 2 1=r 1

1 r r 10

r 2 1=r 1(8)

where r 0, r 1 , and r 2 are the tting parameters. Theevolution of the plastic strain, g p, is governed by themode with the longest relaxation time:

_g p t 1

h 1where t 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 s ds;1 : s ds;1r : (9)

Parameters used here are adopted from [27,28] and aregiven in the (Table 1 and 2). An illustration of the modelresponseformaterialswithdifferentthermalhistorieswith

respect to their rejuvenated reference state, i.e., quenchedandannealed, is given in Figure 1. Effectively an increase inSa shifts the relaxation time spectrum to longer relaxationtimes(Figure1left),leadingtoanincreaseintheyieldstress(Figure 1 right).

Evolution of the State Parameter

The model as presented above is capable of givingquantitative descriptions of various deformation modesonce the initial thermodynamic state of a material, Sa , isknown. [10,23,30] The value of the initial thermodynamicstate is obtained by a single mechanical test, but whenknown, the model even accounts for aging during theexperimental investigation. [10] This aging is capturedby introducing the evolution of the state parameterwith an effective time and was determined based onannealingexperiments below T g.[23] In Figure2 a schematic

Predicting the Long-Term Mechanical Performance of Polycarbonate from Thermal History

Table 3. Multi-mode Maxwell spectrum.

Mode h 0 G

Pa s 1 MPa

1 2.10 1017 3.50 102

2 3.48 1016 5.55 101

3 2.95 1014 4.48 101

4 2.84 1013 4.12 101

5 2.54 1012 3.50 1016 2.44 1011 3.20 101

7 2.20 1010 2.75 101

8 2.04 109 2.43 101

9 1.83 108 2.07 101

10 1.68 107 1.81 101

11 1.51 106 1.55 101

12 1.40 105 1.37 101

13 1.27 104 1.19 101

14 1.10 103 9.80 100

15 1.23 102 1.04 101

16 2.62 100 2.11 100

17 2.14 100 1.64 101

Table 1. Injection molded samples.

T m s y a) S

a

- C MPa

30 57.1 27.690 58.8 29.1

130 62.1 32.2

a)Measured at _ 1 10 3s 1 .

Table 2. Model parameters.

Gr k t 0 Sa m r 0 r 1 r 2

MPa MPa MPa

26 3 750 0.7 0.08 0.965 50 3

Macromol. Mater. Eng. 2009 , 294 , 829838 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mme-journal.de 831

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representation is given of the evolution of the yield stresswith time. In this gure a division in two regions, i.e.,processing andservice life, canbe seen. Theinitial heightof the yield stress is directly determined by the processingconditions, but upon aging (service life) it starts to increaseand will, independently of its initial properties, follow thesame evolution with time. The rate of increase in the yieldstress is independent of the processing conditions, but isincreased by temperature. The initial region in which theyield stressdoes notincreasewith time, however, is relatedtothe processing conditions.The lengthof this regioncanbedened as the initial age, t a , of the material, i.e., the time atwhich the yield stress starts to increase.

The state parameter is directly proportional to the

uniaxial yield stress and captures the aging kinetics. Itisthe onlyparameterin themodel whichchangeswithtimeand is given by the following relation:

Sa t ; T c 0 c 1log t eff t ; T t a

t 0 (10)

where c 0 and c 1 are the two constants(whichcanbe derived from the intersect,i.e., intersection of the curve with thevertical axis at t 1 s, and slope of the linear relation of yield stress with

the logarithm of time, respectively)and their values equal 4.41 and 3.3,respectively, t a is the initial age of thematerial, t 01s and t eff is the effectivetime dened as:

t eff t ; T Z t

0a 1T T t

0 dt 0 (11)

The effective time is thus the timeaccelerated by temperature and capturesthe thermal history of the material. Theacceleration is governed by an Arrheniusdependency, given by:

aT T t exp DU a

R1

T t

1T ref (12)

where DU a is the activation energy of aging, R theuniversal gas constant, and T ref a reference temperature.

If the tensile tests are performed at the same strain rate,and it is assumed that the strain at yield is independent of applied strain rate, the yield stress can be given analogousto Equation 10 where the only variable is the effective timeof the material, and thus the thermodynamic state of the

material. The experimentally determined yield stress cannow be given by:

s y t ; T s y ;0 c log t eff t ; T t a

t 0 (13)

In this equation s y ;0 is the intersect of the linear relationof yield stress with the logarithm of effective time and c istheslope.Mind that s y ;0 will depend on the strain rate usedfor the determination of the master curve. Values for theparameters used in this study can be found in. [24]

Properties from ProcessingIn a previous study we showed that the yield stress of polycarbonate (PC) can even be predicted if the thermalhistory experienced during processing is taken intoaccount, [24] rendering a mechanical determination of thethermodynamicstateobsolete.Todoso,theevolutionoftheeffectivetime has to be followed starting from themomentthe glass starts to solidify during processing. In effect, theeffective timepreviously [23] wasusedasaparameterwhichcaptures the thermal history of the material from themoment we start evaluating the properties of the material,


108

104

100

104

108

1012

1016

1020

102410

0

101

102

103

104

time [s]

r e l a x a

t i o n m o d u l u s

[ M P a ]

Sa

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60

70

80

90

comp. true strain []

c o m p . t r u e s t r e s s

[ M P a ]

Sa

Figure 1. Left: Inuence of state parameter on relaxation modulus versus time. Right:inuence of state parameter on true stress versus true strain response. Dashed lines ():rejuvenated state; solid lines (-): increasing thermal histories, i.e., quenched andannealed, respectively.

log(time)

y i e l

d s t r e s s

processing related service

related

ta

Figure 2. Yield stress versus life-time: two regions; processingrelated and service related.



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i.e., the start of the experiment. The initial age weintroduced takes into account the thermal history thematerial already received prior to the experiment and ismostly unknown. However, if we start our numericalinvestigation already during processing, we can quantita-

tively predict the initial age of the material.[24]

ApplyingEquation 11 to the thermal history the material receivesduring processing gives:

T > T g : t eff ;c 0 and _t eff ;c 0

T T g : t eff ;c Z t c

0a 1T T c t

0 dt 0 (14)

here t eff ;c is the effective time which accumulates duringprocessing and will give the initial age, t a and T c t is thethermal history of the material during processing. From t a

thevalue of thestate parameter, Sa can be calculatedandusedas input for the investigation of the mechanical performance.T g is effectively used as an input parameter and

determined from themaximum of the loss angleof a DMTAmeasurement. Surely this is a very limited interpretation of the complex phenomenonknowas the glass transition. [8] Ina second study on processing inducedproperties [25] we usedthe framework of structural relaxation [3133] to predict theyield stress. This approach used a more physically accuratedescription of the glass transition, but did not improve onthe accuracy of the predictions, i.e., both methods givingquantitativeexcellentpredictions.Ofcourse theevolutionof the thermodynamic state also has a pronounced effect onother polymer properties, e.g., enthalpy, volume, creepcompliance, and stress-relaxation modulus, [9,3436] but it isgenerally accepted that the times scales on which agingaffects different properties are not the same and cannot benot correlated in a simple manner. [9,36]

The thermal history during processing is determined bymeans of numerical simulation of the injection moldingprocess. This is done using the commercial injectionmolding simulation package MoldFlow MPI. For the atrectangular plate, from which the tensile bars are taken, a2.5D approach is used in which the velocity and tempera-ture elds are computed 3D, but for the pressure a 2D

approximation, which is valid in the case of a thin product,is used. For the actual product full 3D computations areperformed. Of course any other nite element approach/package can be used.

Experimental PartMaterials

The material used in this study was an injection molding grade of PC: Lexan 141R, supplied as granules by Sabic Innovative Plastics(Bergen op Zoom, The Netherlands). The number-average mole-

cular weight and weight-average molecular weight were 9.2 and25.8 (kg mol 1), respectively.

Sample Preparation

Alltensileexperimentswereperformedonsamplesmachinedfrominjection-molded rectangular plates, see Figure 3 (left). The plateswithdimensions70 70 1 mm 3 were moldedon an Arburg320Sall-rounder 500150. The runner of the mold ensured uniformlling, as proven by several short shot experiments. The onlyvariable used during the injection molding is the mold tempera-ture. Melt temperature was kept constant at 285 8 C, as was thecooling time at 60s. From the plates, bars with dimensions70 10 1 mm 3 were cut parallel to the ow direction and ttedwith gauge sections of 33 5 1 mm 3, see Figure 3 (right).

To investigate the inuence of the temperature history on thedistributionof yieldstress over thethicknessof a product bymeansof micro-indentation, small bars with cross-sections 2 1 mm 2

were taken from the centers of the molded square plates by aprecisionmachiningoperation.Subsequently, increasingly thinnerlayers ( 1 mm) were removed from the surface of the 2 1 mm 2

cross-section by a microtoming operation under cryogenic (liquidnitrogen) conditions, this to minimize a possible inuence of machining on the thermodynamic state of the sample surface. ALeica RM 2165 rotation microtome was used; for each sample afresh glass knife was taken. To obtain different temperaturehistories, three mold temperatures were used 30, 90, and 130 8 C.

To verify the performance after injection molding and theinuence of thermal history thereupon,tensilesamples weretakenfrominjectionmolded platesprocessedat moldtemperatures of 30and 120 8 C.

Finally, as an example of a real product, a thick-walled cup-shaped sample, see Figure 4, was used. The bottom ring of the cuphas a diameter of 78mm. The cup itself starts with an outerdiameter of 65mm andhas a graduallydecreasing diameter up till60 mm at the top. We dene Figure 4 (left) as the upright position(the way in which the samples was loaded in the load-frame). Thethickness was around 3mm in all cross-sections. The cup-shapedsample was injection molded on an Arburg 320S all-rounder500150. The melt temperature was set to 285 8 C; the injectionrate to 50ccm s 1; and the packing pressure to 500bars. Moldtemperaturesweresetto30and130 8 C.Inbothcasesacoolingtimeof 120s was used.

Allsamples used in this study were stored at room temperatureafter injection molding. For PC no increase in yield stress was to beexpected at room temperature based on the time-temperature-superposition results of Klompen et al. [23] It was also experimen-


Figure 3. Injectionmolded samples and tensile barsmade thereof.


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tally shown that PC stored at room temperature did not show anincrease in yield stress for at least a period of 3 years. [37]

Methods

Micro-indentation experiments were performed on a nano-indenter XP (MTS NanoInstruments, Oak Ridge, Tennessee) under

displacement control. The indenter has a at tip, effectively a at-ended cone witha top angle of70 8 and a circular contact area withdiameter 10 mm. Correction for tip-sample misalignment wasperformed using a specially designed alignmenttool.For details onthe experimental technique see; [30,38] and for the alignment toolsee.[38]

Tensile tests were performed on a Zwick Z010 universal tensiletesterataroomtemperatureof23 8 C. Experimentswere performedby applying constant linear strain rates _" _x=l0 or engineeringstresses s F = A0 . Unless indicated otherwise, a standardconstant linear strain rate of 10 3 s 1 was used. All tensile yieldstresses listed in the results section are engineering yield stresses,and taken as the mean value of ve experiments.

The cup-shaped samples were tested on a Zwick 1475 tensiletesting machine. The cups were placed in the machine with thetaperedsection to the top (see Figure 4 left). The bottomplate usedhasa atcircularrecessto t theouterdiameter of thebottom ringof thecup. Samples were loadedwith constant displacementrates,or constant forces. Experiments were performed at a roomtemperature of 23 8 C and corrected for the nite stiffness of thesetup.

Results

Inuence of Thermal History on the Yield-StressDistribution

Previously [24,25] distributions of yield stresses throughoutapolymer product were predictedbasedon thedifferencesintemperature history experienced during molding. Fastcooling, e.g., at the mold surface, limits physical agingleading to a lowyield stress, while duringslow cooling, e.g.,inthecoreoftheproduct,agingismorepronouncedasistheincrease in yieldstress. Verication wasdone using a meanyield stress calculated based on area averaging, andexcellent agreement was found for different mold tem-peratures and mold thicknesses. Here we will attempt toalso validate the distributions, rst by using micro-

indentation. Figure 5 (left) shows results of indentationwith the use of a at-tip and indeed a measurabledistinction is found in force-displacement results for aquenched ( s y 62MPa)a and an annealed ( s y 70MPa)a

material. [30]

The yield stress distribution was studied on the cross-section of samples produced with different mold tempera-tures (resulting in different properties, see Table 1). Toachieve a smooth cross-sectional surface with little plasticdeformation of thesurface, thesamples were rst precisionmachined, followed by cryogenic microtoming of thesurface. Indents were made over the total width of thesamples, see Figure 5 (right), and the resulting indentationforces (at an indentation depth of 2 mm) are presented inFigure 6 (left). Although the calculated yield stressdistributions (Figure 6 (right)) and the yield stressmeasurements (Table 1), both suggest differences in

indentation forces, they are not measured (The decreasein indentation force at theedgesis most likelythe inuenceof a local decrease in stiffness due to the presence of thesurface.) The conclusion could be that, despite our precau-tions, sample preparation still inuences the thermody-namic state of the surface, preventing to measure proper-ties as they result from processing.

We therefore try to obtain samples that are preparedwithouttheneedofapost-processingmachiningoperation,by stacking twelve (1 mm thick, vacuum dried) sheets of extruded PC (200 200mm 2), separated by thin sheets of aluminium ( < 0.1 mm thick) to prevent sticking. Twothermocouples are added, one at a surface sheet, onebetween thetwomiddle sheets. Thestack ofsheetsis placedin a hot press at 200 for 10 min, to erase any previousthermal history and orientation from the sheets, andsubsequently the stack is cooled in a cold press at 18 8 C. Thestack is removed from the press once the center tempera-ture is 18 8 C, and the sheets are separated. Subsequentlytensilebarsaremachinedfromthecenterofthesheetswithgauge sections of 33 5 1 mm 3.


Figure 5. Left: force versus indentation depth for a quenched andan annealed material. Right:residual indents on a sampleof moldtemperature 130 8 C.

Figure 4. Injection molded cup.

a Note: listed yield stresses are engineering values measured at_" 1 10 3s 1



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Figure 7 (left) shows the calculated and measuredtemperatures (heat capacities used as input for thenumerical results were taken from the ATHAS DataBank [39,40] ), and Figure 7 (right) shows the calculated, [24]and measured yield stress distributions. The calculateddistribution overestimates the experimental one by

2 MPa, i.e., a deviation of 3%. When the predicteddistribution is lowered by 2 MPa (dashed line), an excellentagreement between experimental and numerical distribu-tions is obtained. In conclusion it proves to be difcult tomeasure yield stress distributions caused by inhomoge-neous cooling via direct indentation measurements, due toerrorsinducedby thesample preparationmethod. But by using samples that didnot require any post-fabrication machin-

ing, the calculated distribution indeedcan be measured, and the results arecorrect with an error of 3% only.

Inuence of Processing on the FinalProperties of a Product

The 3D constitutive framework of thepredictive tool developed [23] allows topredict the performance for an arbitrarysample geometry under any loadingcondition. [10] To validate this, tensileexperiments are performed at differentstrain rates and stresses for samplesmade using mold temperatures of 30and 120 8 C. Next a more complex geo-metry is analyzed, using a cup-shapedproduct.

Figure 8 (left) shows the coolingproles over the thickness in the centerofsquaresamples,seeinsetFigure8(left),as obtained from a numerical simulationof the injectionmolding process for moldtemperatures of 30 and 120 8 C. Cooling is

fasternear thesurface than at thecenter,andcooling rates aremuchhigherfor thelowmoldtemperature.Subsequentlywecompute point wise the resulting localvalue of Sa and the related value of the

yield stress, see Figure 8 right. Near thesurfacewendamuchloweryieldstresscompared to that in the center of theplate, related to the higher cooling ratesoccurring there.Toenablecomparison of the numerical predictions with experi-mentally determined yield stresses, athickness-weighted average of the com-puted Sa distribution was determined.This yielded a value of Sa 27.4 for

samples with a mold temperature of 30 8 C, and Sa 31.8for those with a mold temperature of 120 8 C.

Experimental and numerical results on tensile tests at astrain rate of 10 3 s 1 are compared in Figure 9. Using themulti-mode EGP-model the stress responses up to yield areaccurately predicted, while the postyield localization isstronger in the experiments than in the simulations. Thiscanbeattributedtosamplepreparation.Thetensilebarsaremade by a machining operation and are not polishedafterwards giving them a rather rough surface, resulting instrong localization, which is not captured by simulation.


0 200 400 600 800 10000

50

100

150

200 measured

calculated

surface

center

time [s]

t e m p e r a

t u r e

[ C ]

center surface

1 2 3 4 5 6 7 8 9 1 0 11 1250

55

60

65

70

sheet #

y i e l

d s t r e s s

[ M P a ]

modelmodel output minus 2 MPa

Figure 7. Left: cooling histories as measured (-) and as calculated (). Right: measured(* ) and predicted (solid drawn line) yield stress distributions.

0 5 10 15 20 25 300

50

100

150

200

250

300

Tmold

=30 C

Tmold

=120 C

time [s]

t e m p e r a

t u r e

[ C ]

surface to center

surface to center

0 0.25 0.5 0.75 1

50

60

70

center surface

normalized thickness []

y i e l

d s t r e s s

[ M P a ]

Tmold

=120 C

Tmold

= 30 C

Figure 8. Left: temperature versus time during the cooling of the injection moldedsamples. Right: corresponding predicted yield stress distributions.

1 0.5 0 0.5 120

21

22

23

24

25


i n d . f o r c e

@ 2

m

[ m N ]

Tm

= 130 C

Tm

= 90 C

Tm

= 30 C

1 0.5 0 0.5 1

45

50

55

60

65


y i e l d s t r e s s

[ M P a ]

Tm

=130 C

Tm

= 90 C

Tm

= 30 C

Figure 6. Left: measured indentation force distributions over the thickness of sampleswith different thermal histories. Right: calculated yield stress distributions over thethickness of samples with different thermal histories.


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However,whileitinuencesthewidthoftheyieldpeak,theheight is unaffected.

Figure 10 (left) shows the yield stress versus the strainrate for samples made with mold temperature 30 and120 8 C, and Figure 10 (right) the applied stress versus thetime-to-failure in a creep experiment. Solid lines arepredictions based on our modeling approach and theyare in excellent agreement with experi-mental results. The difference in moldtemperature of 90 8 C between the twosamples, results in an increase in failuretime of about a factor 100 if the samplesare loadedwith thesame stress. A similareffect was already found and predictedfor a quenched versus an annealedmaterial, [23] but mind that here thiseffect is predicted based on the tempera-ture history the material experiencedduring its fabrication in the injectionmolding process.

The model thus far proves to predictthe performance of simple tensile barsvery well. Of course the question rises

whether it will also perform well on a more complexproduct. To investigate this a cup-shaped sample is chosen,as discussed in the Experimental Part, and shown inFigure11(middle).Forthe analysis of theinjection moldingprocess, a full 3Dmesh isbuild, seeFigure 11(left), while for

the structural analysis a 2.5D axi-sym-metrical mesh provedto besufcient, seeFigure 11 (right). The dimensions of themesh for the mold lling analysis arebased on the dimensions of the mold,whereas the dimensions of the mesh forstructural analysis are based on dimen-sions of actual samples, which howeverdiffer only slightly from the dimensionsof the mold.

Analysis of the injection moldingprocess is again performed with thecommercial injection molding simula-tion package (MoldFlow MPI). A full 3D

analysis is used with a mesh which haseight elements over the thickness and a total of 800000elements. The large amount of elements is necessary toobtainasufcientlydetailedtemperatureinformationoverthe thickness of the cup. In Figure 12 the distribution of thestateparameter Sa is given fora quarter cross-section takenfromthecenterheightofthecup.Asexpected,thepredictedvaluesof Sa aremuch higher for the 130

8 C samples than for


0 0.02 0.04 0.06 0.08 0.10

10

20

30

40

50

60

70

strain []

e n g . s t r e s s

[ M P a ]

Tm

= 120 C

Tm

= 30 C

model prediction

Figure 9. Experimental and numerical results of tensile tests at astrain rate of 10 3 s 1 for samples with mold temperatures30 and1208 C.

45

50

55

60

65

70

strain rate [s 1 ]

e n g . y i e l

d s t r e s s

[ M P a ]

10 5 10 4 10 3 10 2 10 1

Tm

= 120 CTm = 30

Cmodel prediction

101

102

103

104

10540

45

50

55

60

65

70

timetofailure [s]

a p p l

i e d s t r e s s

[ M P a ]

Tm

= 120 CTm = 30

Cmodel prediction

Figure 10. Left: yield stress versus applied strain rate. Right: applied stress versus time-to-failure. Both for samples with mold temperatures 30 and 120 8 C.

Figure 11. Cup-shaped sample (middle) and injection moldinganalysis mesh (left) and structural analysis mesh (right) madethereof.

0 10 20 300

10

20

30 Tm = 30 C

x [mm]

y [ m m

]

0 10 20 300

10

20

30

Sa

Tm

= 130 C

x [mm]

y [ m m

]

40

35

30

25

20

15

10

5

0

Figure 12. Distribution of Sa over a quarter cross-section for T m 308 C (left), and for

T m 1308 C (right).



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the 30 8 C samples. The distributions are, however, morepronounced forthe last one. These ndings correspond wellwith the results observed for the yield stress of the tensilebars molded at different mold temperatures, see Figure 8(right).

To facilitate the numerical analysis, again a thickness-averaged value of Sa is used, similar to what was done forthe tensile bars. The calculated area mean values areSa 25.0 for the 30

8 C mold temperature samples andSa 34.9forthe130

8 C samples.Theresultsof thestructuralanalysis are shown in Figure 13 and 14, where markers

indicate experimental results and solid drawn lines are thepredictions. For the 130 8 C samples the loading path asshown in Figure 13 is predicted accurately. This also holdsfortheratedependenceofthemaximumforce,seeFigure14(left), and the stress dependence of the time-to-failureunder staticload,see Figure14 (right). Forthe 30 8 C samplesthe overall loading path is described well, although weobserve an underestimationof themaximum force of 6%.This deviationappearsto besystematicas itis also observedin therate dependence of themaximum force,see Figure14

(left), and the stress dependence of the time-to-failureunder static load, see Figure 14 (left). Please note that theslopes of the predictions in Figure 14 in all cases correlatewell with experimental data and all predictions are within10% accuracy.

On second thought, the predicted Sa 25 for the 3 mmthick cupsappears ratherlow, especiallywhen compared tothe Sa 27.4 for 1 mm thick tensile samples at the samemold temperature of 30 8 C. Re-examination of the Sadistributions learned that at the surface Sa values are aslow as 69, which is unrealistic. Such values correspond toyield stresses normally only encountered after mechanicalrejuvenation. The explanation for this anomaly is found inthe fact that the cooling proles for the surface nodescontaininsufcientinformation,i.e.,thetemperatureattherst time increment is already below T g, thereby giving anunderestimation of the actual thermodynamic state. [Seealso the rough Sa values at surfaces in the inset of Figure 12(left)].The useof the commercial injectionmoldingpackagehere limits the detail of our investigation, since the timeresolved information which can be extracted is limited to100datapoints, apparently posinga problem in the3D casebut not for the 2.5D case. To demonstrate the impact of thisshortcoming we recalculated the area mean value of Sawhile excluding the surface nodes, resulting in a value of Sa 30.2 for the 30

8 C sample and a small but insignicantdifferenceforthe130 8 Csample.Predictionsfor Sa 30.2areadded as dashed lines in Figure 13 and 14 and give a slightoverestimation of the data. A better calculation of thecooling proles therefore is expected to result in

25 < Sa < 30.2.Asa nal remarkwe note that thenumerically calculated

and experimentally observed failure modes were, thus far,all ductile. The cup-shaped samples which were injectionmolded with a high mold temperature, however, display atransition from ductile failure [open symbols, Figure 14(right)] to brittle failure [closed symbols, Figure 14 (right)].The model still predicts the failure times accurately, sincedespite the change in failure mode, the kinetics of plasticow determine the onset to failure. This is in agreement

with experiments on loaded poly(vinylchloride) pipes [41] and on tensile bars of PC with different molecular weights. [10]

Conclusion

A previously developed modelingapproach which allows the predictionof the yield stress distribution as itfollows directly from the temperaturehistory experiencedduringprocessing [24]

is validated more extensively. Predictedyield stress distributions are experimen-


0 2 4 6 80

5

10

15

20

25

30

35

40

45

displacement [mm]

f o r c e [

k N ]

Tm

= 130 C

Tm

= 30 C

model prediction

Figure 13. Experimental and numerical results of compressiontests of a cup at a loading rate of 0.045mm s 1 for samples withmold temperatures 30 and 130 8 C.

104

103

102

101

100

1010

10

20

30

40

50

loading rate [mm/s]

m a x . l

o a d [ k N ]

Tm = 130 C

Tm = 30 C

model prediction

100

101

102

103

104

105

106

1070

10

20

30

40

50

timetofailure [s]

a p p l

i e d l o a d

[ k N ]

Tm = 130 C

Tm = 30 C

model prediction

(open markers) (closed markers)

Figure 14. Left: maximum load versus loading rate. Right: applied load versus time-to-failure. Both for samples with mold temperatures 30 and 130 8 C.


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tally validated and found to be in good agreement. Forsimpletensilebarstheperformanceundervariousconstantstrain rates and stresses was also found to be in excellentagreement with experimental results. Finally, the predic-tion of the performance of a more complex product in the

form of an injection molded cup was investigated fordifferent loading rates and applied loads and againexcellent agreement was observed, including the tremen-dous inuence of different mold temperatures, that canchange the lifetime of polymer products with more thantwoorders of magnitude. Giventhe good overall agreementbetween predictions and experimental results the frame-work presented in this study facilitates the mechanicaloptimization of polymer products without the need forperforming any mechanical tests.

Acknowledgements: Authors are grateful to the Dutch Polymer

Institute (DPI) for nancial support (grant 578).

Received: February 13, 2009; Revised: July 30, 2009; Publishedonline: October 28, 2009; DOI: 10.1002/mame.200900227

Keywords: mechanical properties; modeling; performance;processing

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