1. IntroductionDue to their cost effectiveness and rapid response,numerical simulations have become an increasinglyimportant tool for the evaluation of part design,mould geometry and processing parameters in injec-tion moulding. Numerical investigations are able toestimate aspects of the physical model which other-wise would be difficult to quantify. They allow quickresponses on what will be the effects of processparameters changes on the final part. AlthoughComputer Aided Engineering (CAE) has been usedwith increasing success in the design and manufac-ture of polymer products and processes, the simula-tion of the injection moulding process involvingmicrostructures (!IM) presents many challenges.The flow behaviour of polymer melts in mouldmicro cavities is not fully understood. It is believedthat, due to the large surface-to-volume ratio, sur-face effects dominate the flow behaviour at themicroscale [1]. Kemann et al. [2] showed that stan-

dard injection moulding simulation packages arenot able to describe all of the effects in micro-moulding. The rheological data used in current pack-ages are obtained from macroscopic experimentsand they are not suitable for modelling microscaleflows. In fact, when flowing in micro channels apolymer has a strong tendency to slip [3]. If a clas-sic no-slip boundary condition is used in currentpackages, the consequences of wall slip cannot bepredicted. Furthermore, the microscale dimensionsof features and rapid filling rates typically occur-ring within micro cavities, ensure that the values ofshear rate, experienced by the polymer during themicro moulding process, are orders of magnitudehigher than those experienced in conventional injec-tion moulding. The extensional behaviour in con-traction flow or the pressure influence on the vis-cosity cannot be neglected as well. Also, the mould/melt heat transfer coefficient was found to be a crit-ical factor in determining the filling lengths [4]. A

417

On the performance of a viscoelastic constitutive model formicro injection moulding simulationsA. Gava, G. Lucchetta*

University of Padova, Department of Innovation in Mechanics and Management, Via Venezia, 1 - 35131 Padova, Italy

Received 16 September, 2011; accepted in revised form 5 December 2011

Abstract. The numerical simulation of the injection moulding process involving microstructures presents several chal-lenges, mainly due to the surface effects that dominate the flow behaviour at the microscale. In this paper a new approach,which employs weld lines as flow markers, is used to evaluate whether the numerical codes that are normally used to sim-ulate the conventional injection moulding process, are suitable to characterize the melt flow patterns in the filling of microfeatures. The Cross-WLF viscous model and the Giesekus viscoelastic model were evaluated using 3D models of a micropart implemented in two different numerical codes. A micro cavity was designed in order to compare the results of numer-ical simulations and experiments. While the viscous simulations were found to be inappropriate for multi-scale structures,the accuracy of micro filling predictions was significantly improved by implementing a viscoelastic material model.

Keywords: rheology, modelling and simulation, micro injection moulding

eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426Available online at www.expresspolymlett.comDOI: 10.3144/expresspolymlett.2012.44

*Corresponding author, e-mail: giovanni.lucchetta@unipd.it© BME-PT

significant decrease in the Nusselt number wasobserved concerning the laminar flow in microscalechannels. This means that a constant heat transfercoefficient may not be applicable to the heat trans-fer simulation involving flow through microscalechannels.The assumption of a generalized Newtonian fluid isgenerally used for traditional injection mouldingbecause the importance of elasticity compared toviscous effects is negligible. Because of high defor-mation rates during the injection phase of highspeed micro injection moulding, it is expected thatelastic effects will occur. Therefore there is a need totranslate the complex rheological behaviour of poly-meric fluids into suitable equations, and to use thesemodels to predict flow in micro cavities. During thelast decade substantial progress has been made inthe numerical simulation of viscoelastic flows.Extensions to non-isothermal and three-dimen-sional viscoelastic simulations are in progress [5].Several constitutive equations have been proposed,but none of them has been proven to be superior toothers [6].The main objective of this work is to evaluatewhether the present numerical commercial codesare suitable to characterize melt flow patterns in amicro-cavity, using both Autodesk MoldflowInsight®, i.e. a dedicated simulation software, and ageneral purpose fluid dynamic finite element (FEM)code such as Ansys Polyflow®. This paper alsoreports on the suitability of 3D general purpose com-putational fluid dynamic (CFD) software to be usedfor injection moulding simulations.

The approach proposed to compare the predictionsof numerical simulations to moulding results (Fig-ure 1) consists in the determination of the flow pat-tern by using weld lines as flow markers. This is analternative technique to the ‘short shots’ method,which predicts the shape of the free surfaces with alarge approximation due to stress relaxation andthermal contraction.

2. Experimental setup and data acquisitionThe mould micro cavity considered in this studywas designed to create an effective response vari-able to compare the results of numerical simula-tions and experiments. Obstacles as high as the totaldepth of the cavity were created. In this way, themelt flow was not allowed to climb over the fea-tures and when the separated melt fronts rejoined, atsome downstream location, weld lines were formed.Geometries and dimensions (Figure 2) were selectedaccording to existing industrial devices (blood sep-arators and micro pumps) and in order to exalt suchfactors that change their relevance when shiftingfrom conventional to micro injection moulding,such as:–"elongational flow,–"heat transfer in different thickness,–"wall slip,–"elastic behaviour of polymers.In the first part of the cavity, a convergent geometrywas created in order to pull out the extensional flowand elastic behaviour of the melt in convergent/divergent geometries. A step was created in the mid-dle part of the cavity in order to originate a three

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

418

Figure 1. Proposed approach for the filling validation

dimensional flow and to align the flow front beforethe micro channels entrance; furthermore, the direc-tion of the expected weld line, after the two holes,could give information regarding the shape of theflow front in the second part of the cavity.Channels as wide as 150, 200, 300, 450 and 600 !mwere created in the second part. Heat transfer in vari-able thickness and different rheological models wereconsidered. The gate (400 µm wide and 250 µmthick) was realised as small as possible, in order toapproximate it as an injection point.The experiments were performed using a high flu-idity class polystyrene, the PS 143 E produced byBASF (Ludwigshafen, Germany). Polystyrene isrelevant in micro injection moulding for its verylow viscosity, good biocompatibility, high opticalclarity, high transparency and high impact strengthcompared to silicon or glass. The polymer wasinjected into the mould cavity setting a constantspeed profile of 350 mm/s. The melt temperature inthe feeding zone was maintained at 230°C. Themould temperature was controlled by a heater andmaintained at 70°C.In order to obtain a closer control on the boundaryconditions and a good comparison between simula-tions and experiments, temperature and pressuretransducers were positioned near the cavity surface.

The following parameters were acquired during theinjection moulding process:–"hydraulic pressure,–"ram position,–"cavity pressure near gate,–"cavity surface temperature.The pressure and temperature transducers weremounted in the fixed mould part.In order to analyse the filling of the micro cavity,the position and direction of weld lines were usedas measurable outputs. To describe the shape of theweld lines, an accurate measurement of the weldline path on the sample was required. The investiga-tion strategy consisted in detecting the X and Ycoordinates of points on the weld lines with an opti-cal coordinate measurement machine (CMM)(Video Check IP 400, Werth Messtechnik, Giessen,Germany) and to plot them in a determined andrepeatable reference system.Measuring uncertainty was estimated taking intoaccount the optical CMM calibration on the meas-ured dimensional range, repeatability of the meas-urement on 5 repetitions, CMM resolution, definitionof the measurand and the influence of the tempera-ture (i.e. expansion of the plastic due to temperaturevariations). The combined expanded measuringuncertainty, calculated with a coverage factor k = 2

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

419

Figure 2. Micro part design (part volume 6 mm3) (a) and manufactured part (b)

(level of confidence 95%), resulted in 4 µm on theX direction and in 3 µm on the Y direction. Theinvestigated area and the injection moulded part arerepresented in Figure 3 [7].

3. Viscous and viscoelastic materialcharacterization

In order to have a closer control on the boundaryconditions setup and to improve numerical simula-tion results, the PS 143 E by BASF was character-ized both by means of a rotational and a capillaryrheometer.Steady shear measurements were performed on adual bore capillary rheometer (Rheo 2500, Ceast,Pianezza, Italy) at three different temperatures(190, 210 and 230°C) with a shear rate range of 100to 10000 s–1. The shear-dependent viscosity datawere fitted to the Cross model (Figure 4, Equa-tion (1)):

(1)

where !0 is the viscosity at zero shear and "! and nare data-fitted coefficients. The effect of tempera-ture on viscosity was accounted by means of theWilliams–Landel–Ferry (WLF) model (as definedby Equations (2)–(4)):

(2)

T! = D2 + D3·P (3)

A2 = A~2 + D3·P (4)

where D1, D2, D3, A1 and A~2 are constants to bedetermined while T! is a reference temperature [8].Model constants implemented in the numerical sim-ulations are summarized in Table 1. Pressuredependence of viscosity was measured by using athrottle apparatus arranged in series with the die ofthe capillary rheometer [8]. Data obtained fromexperiments in a pressure range of 40 MPa were fit-ted according to the Cross model and the D3 coeffi-cient was estimated as 2.1·10–7 K/Pa.Due to the high deformation rates, extensionaldeformations also play a significant role in theprocess. Thereby, for the viscous simulation theMoldflow® ‘unified’ viscosity model for mixed shearand extension deformations was used. In this modelthe apparent viscosity is modelled as a function ofthe extension rate and the shear viscosity using theextension viscosity model coefficients. These coef-ficients were determined using experimental pres-sure measurements in convergent flow. The appar-ent viscosity, !a, is modelled according to Equa-tion (5):

!a(T, P, #·, $·) = f($·)!s(T, P, #·) (5)

where !s is the shear viscosity calculated by theCross-WLF model and f is a transition functiondefined by Equation (6):

h0 5 D1 exp c 2 A11T 2 Tw 2A2 1 1T 2 Tw 2 d

hs 5h0

1 1 a h0 g?

twb 12n

hs 5h0

1 1 a h0 g?

twb 12n

h0 5 D1 exp c 2 A11T 2 Tw 2A2 1 1T 2 Tw 2 d

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

420

Figure 3. Moldflow® filling simulation results (a) and injec-tion moulded part (b)

Table 1. Cross-WLF model constants for PS 143ECross WLF

n 0.25433 D1 [Pa·s] 9.47833·1026 A1 56.342!0 [Pa·s] 1548.89 D2 [K] 428.15 A~2 [K] 2.192"! [Pa] 31210.18 D3 [K/Pa] 2.1·10–7

Figure 4. Experimental data and Cross-WLF fitting for thesteady shear viscosity of PS 143E

(6)

where the elongational rate is determined from theentrance pressure loss in a capillary according tothe Cogswell approach [9], while A and B are data-fitted coefficients respectively related to the magni-tude of the elongational effect and to the extensionrate of the transition to strong elongational stresses(Table 2).On the other hand, for the viscoelastic simulationdynamic experiments were performed on a rota-tional rheometer (ARES, TA Instruments, New Cas-tle, USA) at three different temperatures (190, 210and 230°C) in a frequency range % = 0.1–100 s–1. Aparallel disk geometry was chosen rather than acone plate apparatus for practical reasons. The sam-ples, prepared by compression moulding of the pel-lets, were maintained under nitrogen atmosphere toavoid polymer degradation by oxidation. Measure-ments were performed in frequency sweep modewith a 5% deformation amplitude. Three measure-ment points per decade were taken at increasing fre-quencies. The rheological functions were shifted toa master-curve at reference temperature of 230°Cusing the time temperature superposition [10]. Thehorizontal shift factor aT(T, Tref) follows from theloss angle (&) and the vertical shift factor bT(T, Tref)from the dynamic modulus (Gd) (see Equations (7)and (8)):

&(%, T) = &(%aT, Tref) (7)

Gd(%, T) = bTGd(%aT, Tref) (8)

The shift factors obtained experimentally (Table 3)were plotted as a function of temperature and fittedaccording to the WLF equation.Dynamic experiments allowed to determine therelaxation spectrum, i.e. the set of relaxation times,

by fitting with experimental data the expressionspredicted by the viscoelastic Giesekus model in thecase of small-amplitude oscillatory shear flow. TheGiesekus model was selected because it gives satis-factory predictions in the standard rheometricaltests that appear most relevant for the flow underconsideration [11]. The Giesekus model, belongingto the class of Maxwell-type differential constitu-tive equations, is capable of describing the complexrheological behaviour of a material at various defor-mation histories. By introducing several uncoupledor coupled discrete relaxation modes it can beshown that the area of concordance between predic-tions of the model and experimental results for poly-mer solutions can be considerably extended [12].The model can be written as shown by Equation (9):

(9)

where ' is the relaxation time, "=v is the viscoelasticpart of the extra-tensor, !v is the viscoelastic contri-bution to viscosity, ( is a dimensionless mobilityparameter and D

=is the rate of strain tensor [11].

Multiple relaxation times have been used in order tobetter fit the viscoelastic behaviour at different shearrates and improve accuracy of simulation results.Complex viscosity, loss and storage moduli werefitted according to a 4-mode Giesekus model (Fig-ure 5).In the model, nonlinear effects are introduced bytaking into account an average anisotropy of themolecular conformation during flow. The strengthof influence to the conformation and the retroaction

lt,5 1 a I

51al

hvt5v b t5v 5 2hv D

5 1u5 2

f1e. 2 5 1 1A·e.

B 1 e.

lt,5 1 a I

51al

hvt5v b t5v 5 2hv D

5 1u5 2

f1e. 2 5 1 1A·e.

B 1 e.

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

421

Table 2. Extension viscosity coefficients for PS 143E

Table 3. Horizontal and vertical shift factors obtained fromexperimental data

Temperature [°C] aT bT

190 8.62998 1.25456210 2.70511 1.21066230 1 1

Figure 5. Comparison of complex viscosity, loss and stor-age moduli predicted by a 4-mode Giesekus modelwith experimental data for PS at reference T =230°C

Coefficients A B0.309 297.26

to the flow is determined by an anisotropic mobilityparameter (. At ( = 0 the isotropic Maxwell modelis recovered. When ( is set to unity the model pre-dicts similar behaviour in elongation flows as thecorotational Maxwell model. At intermediate val-ues of (, the Giesekus model fits steady and tran-sient shear flows better than any other differentialconstitutive equation [5]. A best fit for the ( param-eter (Figure 6) was obtained by minimizing thedeviation from the steady shear experiments con-ducted on the capillary rheometer with the functiondefined by Equation (10):

(10)

where j indicates the individual data points.The knowledge of both the flow field and the fluidproperties determines the character of the flow. Inparticular, the Deborah number is used to character-ize the fluid elasticity, which is defined as the ratioof the material characteristic relaxation time ' to thecharacteristic flow time t (Equation (11)):

(11)

The weighted relaxation time was calculated asshown by Equation (12):

(12)

The estimated Deborah number is 11.6. This valueconfirms the hypothesis that the filling behaviour isinfluenced by viscoelastic effects. In this case thevalue calculated for the Deborah number is notcomparable to values typical of the conventionalinjection moulding process. This establishes thatthe elastic behaviour of the polymer melt has to beconsidered.

4. Numerical simulationsAs a first approach a three-dimensional finite ele-ment Moldflow® analysis was performed in orderto simulate the micro injection moulding process.The main material functions considered are the‘unified’ viscosity model and the two-domain Taitmodel for the pvT data. Non-Newtonian, non-isother-mal flow solutions were obtained by solving themomentum, mass and energy governing equations.No-slip boundary conditions were imposed on thecavity walls filled by the polymer, while on theunfilled part, a free boundary condition allowed forthe formation of the typical fountain flow. The melttemperature is a potential problematic parameter interms of modelling. In practice, the melt tempera-ture is only indirectly controlled through the barreltemperature zones. It was therefore decided to con-sider it as equal to the barrel temperature. The mouldtemperature was defined as the mean value of themould surface acquired by the temperature trans-ducers. The simulations were performed imple-menting the ram speed profile as set in the machine.The numerical simulation was carried out using the3D mesh shown in Figure 7. A sensitivity analysisof software simulation to the mesh dimension wasconducted. It was decided, as a result, to mesh the

l 5a

4

i51hi li

a4

i51hi

De 5l

t

R5aN

j51c aG9 1vj 2

G9exp,j 2 1b 2

1 aG0 1vj 2G0exp.j

2 1b 2 dR5aN

j51c aG9 1vj 2

G9exp,j 2 1b 2

1 aG0 1vj 2G0exp.j

2 1b 2 d

De 5l

t

l 5a

4

i51hi li

a4

i51hi

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

422

Figure 6. Fitting of non-linear parameter. Comparison ofsteady shear viscosity (as a function of shear rate)with linear viscosity (as a function of frequency).

Figure 7. Mesh of the 3D model considered in Moldflow®

simulation

part model together with the feeding system and todecrease the mesh size in the cavity zones wherethe weld lines were experimentally detected.As a second approach a three dimensional viscoelas-tic simulation was performed in the Ansys Polyflow®

environment. Polyflow® is a finite-element programprimarily designed for the analysis of industrialflow processes dominated by non-linear viscousphenomena and viscoelastic effects. The theoreticalfoundation is provided by the general principles ofcontinuum mechanics, together with phenomeno-logical and kinetic theoretical models for describingthe rheological behaviour of the fluid.The model was created and meshed in Gambit® andsimulations were performed in Polydata® imple-menting the same previous boundary conditions.Stress (one stress field for each mode, or relaxationtime), velocity and pressure are computed simulta-neously. Furthermore, the problem involves flow,heat transfer by conduction and convection and heatgeneration by viscous dissipation. Energy, momen-tum and mass governing equations were solved inthe fluid domain implementing the viscoelasticproperties of the material.In the viscoelastic simulation, the total extra-stresstensor is divided into a purely viscous part (New-tonian) and a viscoelastic part (Equation (13)):

"= = "=n + "=v (13)

where the subscripts n and v stand for Newtonianand viscoelastic contributions, respectively. TheNewtonian part, which can be seen as the stressresponse associated with fast relaxation modes, iswritten as shown by Equation (14):

"=v = 2!n D=(u=) (14)

"=v is computed according to the Giesekus model[11].When a multi-mode viscoelastic model is used, thetotal extra-stress tensor is decomposed into a sumof individual viscoelastic components and anypurely-viscous component. To prevent ambiguousdefinition of purely-viscous component, the corre-sponding viscosity factor is defined together withthe first mode. Consequently, the remaining modeswill not contain any purely-viscous components.In this work a numerical analysis was performeddefining three solid and one fluid sub-domains(Figure 8):

–"a fluid domain (SD1)–"a plunger (SD2)–"the inferior part of the mould (SD3)–"the superior part of the mould (SD4).The finite element meshes are represented in Fig-ure 9. Three dimensional finite elements are definedfor fluid gob, mould and plunger. In particular thefluid domain and the superior part of the mould aremeshed by triangles while the inferior part of themould and the plunger are meshed by quadrilater-als. The 3D mesh model is represented in the initialconfiguration, before pressing the fluid. At a pre-scribed time the plunger moves downwards andpresses this fluid domain in the mould cavity.

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

423

Figure 8. Sub-domains definition defined on Polyflow®

environment

Figure 9. Finite element mesh of the model in Polyflow®

environment

A major difficulty to be overcome in the simulationarises from the fact that the filling domain evolvesconsiderably over time. The position of the front isan unknown, which means that the limit of thedomain under investigation (i.e. filled with polymermelt) is an unknown. This is called a free surfaceproblem and has given rise to a large number ofmethods, which can be classified into two cate-gories. The first approach consists in using controlvolumes defined on a fixed mesh, covering theentire domain to be filled, with the use of an addi-tional variable representing the volume fraction ofthe injected polymer within the control volumes (andwithout front discretization). The second approachis based on accurately tracking the flow front andadapting the mesh, at each time step, in order tocover only the filled domain. This latter front tack-ing-remeshing approach was selected to solve themicro injection moulding problem, since an accu-rate representation of the front at an affordablecomputational cost was required in this small scaleproblem. Indeed, the second class of methodsrequires refined finite element meshes in the frontvicinity. Besides this, the exact position of frontmeetings was intrinsically interesting as related tothe weld line position. The fluid domain was, there-fore, considered as a free surface and a Lagrangianremeshing technique was applied. In the Polyflow®

environment, remeshing techniques are based onlyon the positions and displacements of the boundarynodes, and not on kinematic considerations, unlessa Lagrangian or streamwise method is used forremeshing. Tangential remeshing preserves the origi-nal node distribution along a surface for three dimen-

sional moving domains. In transient iterative param-eters a maximum value of 10–6 s was set as the timestep in order to contain deformation of the elementsbefore remeshing.

5. ResultsAn extensive measurement campaign of the experi-mental and simulated weld lines positions was car-ried out covering weld lines located in several areasof the micro component. Two different outputswere considered: the weld line number 1 and theline 1 of the weld line number 3. These two weldlines were selected because they originate in zonesof the micro cavity where the elastic behaviour ofthe fluid should not be neglected. In correspon-dence with the first part of the micro cavity, a con-traction flow is opposed to an expansion flow. Twodifferent streamlines were acquired at 1.2 ms bothin Moldflow® (yellow) and in Polyflow® (red) sim-ulations (Figure 10). In the expansion flow, thestreamline in the viscoelastic simulation reveals amore elastic behaviour if compared to the viscoussimulation. On the other hand, the streamline at thecontraction exit is delayed due to normal stressesand resistance to elongational deformation duringthe contraction flow. As a consequence, the vis-coelastic numerical weld line moves towards theexperimental one. As a second case, the horizontalline in the 150 µm wide micro channel was consid-ered. Figure 11 shows how the viscous numericalsimulation overpredicts the ease at which the poly-mer would flow through this channel; on the otherhand, the viscoelastic one underpredicts (with alower absolute error) the weld line position. The dif-

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

424

Figure 10. Streamlines at 1.2 ms (a) and subsequent weld line formation (b)

ference may be justified considering geometricalconstraints and the viscoelastic nature of the poly-mer itself. Normal stresses and resistance to elonga-tional deformation reduce the filling length in themicro channel. This effect is not as clear as in theother channels with higher dimensions, because asthe channel dimensions decrease the materialappears to be more rigid due to the constraints at thewall.Differences between numerical and experimentalresults may be related to inadequate boundary con-ditions. No-slip conditions were imposed on thecavity walls filled by the polymer whereas wall slipis expected to occur due to the increased shearstress. A complete validation of the approach willbe possible when more reliable models about localviscosity and heat transfer will be available.

6. ConclusionsIn this paper a new approach, which employs weldlines as flow markers, is used to evaluate whetherthe commercially available numerical codes aresuitable to characterize the melt flow patterns in themicro moulding process. A micro cavity wasdesigned and manufactured in order to create aneffective response variable to compare the results ofnumerical simulations and experiments. Conven-tional three dimensional simulations were testedand found to be inappropriate for multi-scale struc-tures, typically in micro-injection moulded parts. Itwas expected that differences between experimentsand numerical investigations would be due to theassumption of a generalized Newtonian fluid, gen-erally used for traditional injection moulding, where

the importance of the material elasticity comparedto viscous effects appears to be negligible. Becauseof high deformation rates during the injection phase,it was expected that viscoelastic effects mightoccur.Careful material characterization was conducted bymeans of both capillary and rotational rheometryand data obtained were fitted according to a non lin-ear viscoelastic model (Giesekus model). Threedimensional viscoelastic numerical simulations werethen performed to evaluate whether the implemen-tation of a viscoelastic material model could improvethe accuracy of micro filling simulations. Improve-ments in the viscoelastic simulation results wereobserved in predicting the weld lines position. Fur-ther differences between experiments and numeri-cal simulations are to be related to the absence of arobust local heat transfer model.

References [1] Ho C-M., Tay Y-C.: Micro-electro-mechanical-sys-

tems (MEMS) and fluid flows. Annual Review ofFluid Mechanics, 30, 579–612 (1998).DOI: 10.1146/annurev.fluid.30.1.579

[2] Kemmann O., Weber L., Jeggy C., Magotte O., DupretF.: Simulation of the micro injection molding process.in ‘Proceedings of SPE Annual Technical Conference– ANTEC 2000. Orlando, Florida, USA’ 576-580(2000).

[3] Chien R-D., Jong W-R., Chen S-C.: Study on rheolog-ical behavior of polymer melt flowing through micro-channels considering the wall-slip effect. Journal ofMicromechanics and Microengineering, 15, 1389–1396 (2005).DOI: 10.1088/0960-1317/15/8/003

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

425

Figure 11. Line 1 of the weld line number 3

[4] Yu L., Koh C. G., Lee L. J., Koelling K. W., Madou M.J.: Experimental investigation and numerical simula-tion of injection molding with micro-features. PolymerEngineering and Science, 42, 871–888 (2002).DOI: 10.1002/pen.10998

[5] Peters G. W. M., Schoonen J. F. M., Baaijens F. P. T.,Meijer H. E. H.: On the performance of enhanced con-stitutive models for polymer melts in a cross-slot flow.Journal of Non-Newtonian Fluid Mechanics, 82, 387–427 (1999).DOI: 10.1016/S0377-0257(98)00173-6

[6] Larson R. G.: A critical comparison of constitutiveequations for polymer melts. Journal of Non-Newton-ian Fluid Mechanics, 23, 249–269 (1987).DOI: 10.1016/0377-0257(87)80021-6

[7] Gava A., Tosello G., Hansen H. N., Salvador M., Luc-chetta G.: A new approach for the validation of fillingsimulations in micro injection moulding. in ‘Proceed-ings of the 9th International Conference on NumericalMethods in Industrial Forming Processes – NUMI-FORM, Porto, Portugal’ 307–312 (2007).DOI: 10.1063/1.2740829

[8] Cardinaels R., Van Puyvelde P., Moldenaers P.: Evalu-ation and comparison of routes to obtain pressure coef-ficients from high-pressure capillary rheometry data.Rheologica Acta, 46, 495–505 (2007).DOI: 10.1007/s00397-006-0148-5

[9] Macosko C. W.: Rheology: Principles, measurementsand applications. Wiley-VCH, New York (1994).

[10] Ferry J. D.: Viscoelastic properties of polymers. Wiley,New York (1980).

[11] Giesekus H.: A simple constitutive equation for poly-mer fluids based on the concept of deformation-depen-dent tensorial mobility. Journal of Non-NewtonianFluid Mechanics, 11, 69–109 (1982).

[12] Armstrong R. C., Brown R. A., Quinzani L. M.,McKinley G. H., Byars J. A.: Measurement of velocityand stress fields in complex polymer flows. in ‘Proceed-ings of the XI International Congress on Rheology,Brussels, Belgium’ 16–23 (1992).

Gava and Lucchetta – eXPRESS Polymer Letters Vol.6, No.5 (2012) 417–426

426