Rheol Acta (2010) 49:23–44DOI 10.1007/s00397-009-0392-6

ORIGINAL CONTRIBUTION

Residual stresses in gas-assisted injection molding

Frederico J. M. F. Custódio · Patrick D. Anderson ·Gerrit W. M. Peters · António M. Cunha ·Han E. H. Meijer

Received: 2 June 2009 / Accepted: 19 September 2009 / Published online: 6 October 2009© The Author(s)2009. This article is published with open access at Springerlink.com

Abstract Residual stresses are a major issue in themechanical and optical behavior of injection-moldedparts. In this study, we analyze their development inthe case of gas-assisted injection molding (GAIM) ofamorphous polymers. Flow-induced residual stressesare computed within a decoupled approach, in whichelastic effects are neglected in the momentum balance,assuming a generalized Newtonian material behavior.In a staggered procedure, the computed viscous flowkinematics are used to calculate normal stresses em-ploying a compressible version of the Rolie-Poly model.For the computation of thermally and pressure-inducedresidual stresses, a linear thermo-viscoelastic model isused. A 3-D finite element model for GAIM is em-ployed, which is able to capture the kinematics of theflow front and whose capabilities to predict the thick-ness of the residual material layer have been validatedby Haagh and Van de Vosse (Int J Numer MethodsFluids 28:1355–1369, 1998). In order to establish a clearcomparison, the development of residual stresses isanalyzed using standard injection molding and GAIMfor a test geometry.

F. J. M. F. Custódio · P. D. Anderson (B) ·G. W. M. Peters · H. E. H. MeijerMaterials Technology, Eindhoven University of Technology,P.O. Box 513, 5600MB, Eindhoven, The Netherlandse-mail: p.d.anderson@tue.nl

A. M. CunhaUniversity of Minho, Braga, Portugal

Keywords Gas-assisted injection molding ·Viscoelasticity · Compressible Rolie-Poly model ·Flow-induced residual stresses · Thermallyand pressure-induced residual stresses ·Amorphous polymers · Numerical analysis

Introduction

Residual stresses in injection molding are responsiblefor the dimensional stability of molded parts and theanisotropy of their properties, i.e., mechanical and op-tical. There are mainly two sources of residual stresses(Baaijens 1991; Meijer 1997). The first, flow-inducedstresses, are viscoelastic in nature and originate fromthe resistance of molecules to attain a preferentialalignment with the flow direction (entropy driven). Thesecond, the so-called thermally and pressure-inducedstresses, originate from differential shrinkage inducedby the combined effect of inhomogeneous cooling andpressure.

Upon processing, polymer molecules in the meltbecome aligned within the flow direction, attaining adegree of orientation that is dependent on the strainrate and on the relaxation times of individual mole-cules. However, once molecules become orientated,stresses in the fluid start to develop, a phenomenausually depicted by an increase of the first normalstress difference. Such stress development is relatedto the decrease in entropy of the molecules. Frozen-in flow-induced stresses are known to dictate the long-term dimensional stability of injection-molded parts.Throughout the lifetime of a molded part, relaxationprocesses that yield shrinkage take place as oriented

24 Rheol Acta (2010) 49:23–44

molecules strive to reach for more favorable confor-mations. However, since relaxation times of moleculesbecome extremely high upon cooling through the glasstransition temperature, Tg, typically such processestake place over long times. As was shown by Caspers(1995), the resulting shrinkage depends on the mole-cular orientation. Furthermore, frozen-in orientationalso introduces anisotropy of physical properties, e.g.,mechanical, optical, and thermal. In addition to this, aswas pointed out by Struik (1978), flow-induced stressesalso affect the thermal expansivity of polymers belowTg, as they result in a negative contribution to thereversible coefficient of thermal expansion.

Differential shrinkage in injection molded parts orig-inates from the combined effect of inhomogeneouscooling and pressure. To understand how both mech-anisms contribute to the final residual stresses, it isworth to first consider the case of free quenching, inwhich only thermal stresses develop. Upon quenchinga flat sheet of an amorphous polymer, cooling becomesinhomogeneous since material near the surface coolsfirst and faster. Hence, each material point across thesheet thickness solidifies at a different time. As thematerial layers near the surface solidify and stiffen, theyshrink due to thermal contraction, imposing a compres-sive stress on the molten material in the core regionthat relaxes fast by viscous deformation. Later, whenthe core region starts to solidify, shrinkage thereof isprevented by layers previously solidified at the surface.As a consequence, the outer layers are put under com-pression while the core puts itself under tension uponshrinkage during cooling. Obviously, the residual stressprofile depends on the combined effect of inhomoge-neous cooling and changes in the elastic properties, inparticular the modulus, with temperature and time. Forinjection-molding parts, which is the case of interest inthe present study, the situation is more complex sincesolidification occurs in the presence of pressure that isonly released upon opening of the mold. Due to dif-ferences in compressibility between the solidified layersclose to the mold walls and the material which is under-going vitrification, a variation of the pressure level uponcooling leads to differences in residual stresses. Hence,the changes in density of each material point are locallydefined by the respective cooling and pressure history.Hastenberg et al. (1998) measured the residual stressprofile of amorphous polymers in injection-moldedflat plates using a modified removal layer technique.In contrast to free-quenching conditions, the authorsmeasured significant tensile stresses near the surface,followed by a compressive transition region and acore under tensile stresses. Such a profile, and remark-ably the presence of tensile stresses at the surface, is

believed to be responsible for mechanical failure phe-nomena referred to as environmental stress cracking,see Mandell et al. (1981). Following the conclusionsof Zoetelief et al. (1996), the residual stress profilein injection molding products is mostly determined bypressure, and not by differences in time of thermalcontraction across the part’s thickness. The situationfor semi-crystalline materials is somewhat different. Aswas shown by Kamal et al. (1988), the residual stressprofiles for high-density polyethylene (HDPE) are dif-ferent from those of polystyrene (PS) and polycarbon-ate (PC). More specifically, compressive stresses werefound to develop in the core region that increase totensile stresses towards the walls.

It has been reported (Baaijens 1991; Douven 1991;Zoetelief et al. 1996; Kamal et al. 1988) that the magni-tude of flow-induced stresses is about one to two orderslower than the magnitude of thermally and pressure-induced stresses. However, Wimberger-Friedl et al.(2003) showed, by comparing PS with PC, that the ratiobetween these two contributions to the final residualstresses depends on the polymer. Notably, the mag-nitude of flow-induced stresses is much higher for PSthan for PC. The lower molecular weight and relaxationtimes of PC chains, when compared with PS, make ofPC a difficult material to orient, thus enabling its usein optical applications that require low birefringencelike CDs and DVDs. However, the contribution of ther-mally and pressure-induced stresses to the total residualstresses is supposed to be higher for PC instead, sincethese stresses scale with the material modulus. Theauthors in Wimberger-Friedl et al. (2003) conclude thatboth flow-induced and thermally and pressure-inducedstresses induce, in equal magnitude, the birefringenceof PC injection-molded parts.

Even though some research was devoted to the com-putation of residual stresses in injection molding, therehas not yet been a study devoted to understanding howresidual stresses develop in gas-assisted injection mold-ing. Given the vast use of gas-assisted injection molding(GAIM) in polymer injection, the need for such type ofstudies motivates our investigation. The GAIM tech-nology is mostly used to produce parts with hollow sec-tions, structural ribs, bosses, or parts with regions withsignificant changes in thickness. Even though there aresome variants, typically the GAIM process evolves withthe following steps: first, polymer is injected until themold cavity becomes partly filled, next (or during thepolymer injection), the gas injection phase takes place,which is done in two phases: a gas penetration phase,during which the cavity walls are wet by the polymermelt, and a secondary penetration phase in which gas,under pressure control, holds the polymer part under

Rheol Acta (2010) 49:23–44 25

pressure while cooling takes place. In the end, the gasis vented and the part ejected. Some of the advan-tages of this technology are the decrease of the part’sweight, cycle time, injection pressure, clamping force,shrinkage, and warpage. Until recently, the numericalstudies reported on GAIM have mainly focused onthe prediction of the gas bubble shape and thicknessof the residual polymer layer as a function of processvariables, i.e., shot weight, gas pressure, etc. (Haaghand Van de Vosse 2001; Li et al. 2004; Parvez et al.2002; Polynkin et al. 2005). Given the vast industrialapplication and high potential of GAIM, it is essentialto also focus numerical studies on the prediction ofproperties. In this work, we propose a method for thecomputation of residual stresses in GAIM. We extendthe finite element 3-D model developed by Haagh andVan de Vosse (1998) for GAIM and follow a decou-pled approach in which the velocity–pressure problemis solved independently from the stress problem.Hence, at every computational time step, we computethe flow kinematics assuming a generalized Newtonianfluid behavior and plug them into a viscoelastic modelto compute residual stresses. Such a staggered-schemeapproach was validated by Baaijens (1991), where itwas shown that minor differences in terms of normalstresses result, compared to employing a viscoelasticconstitutive model in the momentum equation. Thedecoupled approach gathers the advantages of saving atremendous amount of computing effort and avoidingnumerical problems and flow instabilities that arise athigh Weissenberg numbers, which are typical for theinjection molding flow conditions. The study of suchinstabilities is beyond the scope of this work; see Hulsenet al. (2005).

Governing equations for the injectionmolding problem

The balance equations for mass, momentum, and en-ergy are now presented and simplified with respect tothe process requirements and modeling assumptions.The related constitutive equations and boundary con-ditions are given and justified. The general form of thebalance equations for mass, momentum, and internalenergy read:

ρ = −ρ∇ · u, (1)

ρ∂u∂t

+ ρu · ∇u = ∇ · σ + ρg, (2)

ρe = σ : D − ∇ · q + ρr + ρhr Rc, (3)

where ρ represents density and u the velocity field, σthe Cauchy stress tensor, g is the body force per unitmass, and e is the rate of change of internal energy.The terms on the right-hand side of the energy equa-tion, Eq. 3, represent the work done to deform thematerial, with D the rate of deformation tensor, theheat transferred by conduction, with q the heat flux,the heat transferred by radiation, r, and internal heatgeneration with Rc the reaction rate, and hr the reactionheat. To solve these equations, appropriate constitutiveequations have to be specified for the Cauchy stresstensor, the heat flux, and an equation of state for thedensity and internal energy, i.e., e = e(p, T ), where pand T denote pressure and temperature, respectively,introduced in the forthcoming section. Additionally,initial and boundary conditions have to be prescribed.We now state the basic assumptions to simplify theabove equations and to motivate the choice for consti-tutive relations.

Assumptions

The assumptions given below are quite standard forinjection molding. Justification of these can be found inliterature (Baaijens 1991; Douven 1991; Douven et al.1995; Haagh and Van de Vosse 1998) and are discussedin some detail further on when appropriate.

− Compressibility effects are negligible during thefilling phase.

− Flow kinematics are determined by kinematicboundary conditions.

− The melt behaves according to a generalizedNewtonian flow description.

− Inertial effects are negligible.− Thermal radiation is negligible.− No heat source is present.− Heat generated due to compression is negligible.− Isotropic heat conduction.

It is known that, upon processing, the thermal con-ductivity of polymers is increased in the flow directionand decreased in the direction normal to the flow.Such observations were reported by, e.g., Hansen andBernier (1972) and van den Brule and O’Brien (1990)and explained on the basis that the heat conductanceis much higher along covalent bonds than throughoutweak secondary bonds. Furthermore, the anisotropyresulting from the frozen-in molecular orientation candiffer substantially according to the molecular structureof the polymer, being favored by linear compact poly-mers, and for polymers with higher relaxation timesand molecular weight. All this is not taken into accountdue to the lack of experimental data. According to the

26 Rheol Acta (2010) 49:23–44

above assumptions, the governing equations are simpli-fied, yielding during the filling phase an incompressibleStokes flow problem. Thus, the continuity and momen-tum equation read:

∇ · u = 0, (4)

∇ p = ∇ · τ , (5)

where u is the velocity field and τ is the extra stresstensor. During the packing and holding phases, thecomplete form of the continuity equations is solved(Eq. 1). The simplified energy equation reads:

ρcpT = τ : D + ∇ · (λ∇T). (6)

In injection molding, the flow kinematics are mostlydetermined by kinematic boundary conditions, char-acterized by no-slip conditions at the walls, and by aprescribed flow rate at the gate. Therefore, the precisechoice of the constitutive equation for the stress tensorin the momentum equation has only a small effect onthe overall kinematics as long as the shear viscosity iscaptured correctly. Clearly, in regions with bifurcationsor close to the flow front, where significant elongationtakes place, this assumption is violated. Nevertheless,a viscous approach, vs a viscoelastic approach, has theadvantages of saving a tremendous amount of com-puting effort and avoiding numerical limitations andflow instabilities that arise at high Weisenberg numbers,which are typical for injection molding flow conditions;see Hulsen et al. (2005). The influence of viscoelasticinstabilities in injection molding was notably analyzedby Grillet et al. (2002) and Bogaerds et al. (2002);however, such analyses are beyond the scope of thiswork. According to the generalized Newtonian flowdescription, the extra stress tensor τ reads:

τ = 2η(T, Dd) D. (7)

Front-capturing

In order to track the polymer/air and polymer/gas inter-faces we use a front-capturing technique, also known aspseudo concentration method, which was proposed byThompson (1986). Each material point, or infinitesimalmaterial volume element, is labeled with a scalar c, andthe material labels for polymer, air, and gas core areconvected with the velocity u throughout the domain.Boundary conditions are made dependent on c. Themethod requires the addition of a pure (passive scalar)

convection equation that gives the evolution of thematerial label distribution:

∂c∂t

+ u · ∇c = 0. (8)

As initial condition, the material labels are set to zeroover the entire domain �, and at the inlet, the followingboundary conditions are assigned:

c (x, t = 0) = 0, x ∈ �, (9)

c (x, 0 < t < tfill) = 1, x ∈ �e. (10)

The interface is captured for c equal to 0.5. The materialproperties are made dependent on the local value ofthe concentration, c, and are discontinuous across theinterfaces polymer–air and polymer–gas. For the airor gas phase, c < 0.5, the fictitious-fluid properties areassigned, while for the case c ≥ 0.5, the polymer prop-erties are chosen. We also perform particle tracking,using Eq. 8, but instead of prescribing at the inletboundary a concentration value c, we prescribe a timelabel, t, convecting basically the flow history.

Flow-induced stresses

The total Cauchy stress tensor is the sum of an elas-tic and a viscous part: σ = σe + σv . The elastic partis split into a hydrostatic part and a deviatoric part:σ = −pI + τ + σv . For a proper description of thepolymer rheology, i.e., accurate residual stress predic-tions, a discrete number of relaxation times and theircorresponding moduli are required; see Bird et al.(1987), Macosko (1994). Using a multi-mode descrip-tion, the deviatoric part of the elastic extra stress tensoris given by:

τ d =m∑

j=1

Gj Bejd, (11)

in which Gj is the relaxation modulus of a specific mode,and Be the elastic Finger or conformation tensor.

Choice of the viscoelastic model: Although, in in-jection molding, the flow is shear-dominated, in theflow-front region, the polymer melt is stretched dueto the fountain flow. The flow generated is spatiallyin-homogeneous, combining shear and extensionalcomponents. In literature, results reported on the cal-culation of flow-induced stresses in injection moldingneglect fountain flow by adopting a lubrication ap-proximation that assumes negligible velocity gradients

Rheol Acta (2010) 49:23–44 27

parallel to the flow direction and negligible velocitiesacross the thickness direction. However, in our model,we capture the fountain flow in the overall kinematics,and thus, we need a model that is able to capturethe rheological behavior of polymers under shear andextensional flows. Likhtman and Graham (2003) pre-sented the Rolie-Poly model for the rheology of linearpolymers. Reported results show a good agreementwith rheological measurements in steady and transientregimes in both shear and elongation. The model isof the differential type, and in its single-mode form,it reads:

�Be + 1

τd(Be − I) + 2

(1 − √

(3/tr(Be)))

τR

×(

Be + β

(tr(Be)

3

)δ

(Be − I)

)

= 0. (12)

The molecular dynamic mechanisms taken into accountin the derivation of this model, such as chain repta-tion, chain stretch, and convective constraint release,appear in the above equation, each associated with adifferent term, making this model particularly physi-cally intuitive. For higher strain rates, the Rolie-Polymodel produces a faster stress growth when comparedto the full theory. Following the approach taken inLikhtman and Graham (2003) for a multi-mode Rolie-Poly description of transient experimental data, we setβ to compensate for the over prediction of the steady-state stress at large rates. In the model, two time scalesare of importance: τd, the disengagement (reptation)time, and τR, the Rouse (stretching) time. The dis-engagement times are obtained from stress relaxationmeasurements, in which the relaxation modes are fit-ted. The Rouse time τR is estimated with the follow-ing relationship for mono-dispersed melts proposed byDoi and Edwards (1986):

τR = τd

3Z, (13)

in which Z , the number of entanglements per chain, iscalculated from the weight-averaged molecular weightMw and the averaged molar mass between entangle-ments 〈Me〉:

Z = Mw

Me. (14)

The value of τR estimated in this way is, accordingto Likhtman and Graham (2003), one to two timeslarger then those from a fit of transient experimentalshear data. However, since we do not have such data,we accept our estimated value as being reasonable forour approach, where the focus is on the application to

predicting residual stresses. Both choices, β = 0 and τR

obtained from Eq. 13, can be studied in more detailusing a set of fully rheologically characterized materi-als. However, this is outside the scope of our paper andwe restrict the injection molding analysis to the resultsbased on these assumptions.

Since compressibility is a key issue during packingand holding phases, we next introduce a compressibleversion of the Rolie-Poly equation, Eq. 12. For that,we adopt the approach used by Baaijens (1991), whoproposed a compressible version of the Leonov model.Assuming that the polymer cannot undergo a perma-nent plastic volume change, and that the deformationtensor F can be decomposed multiplicatively into anelastic part (Fe) and a plastic (Fp) part: F = Fe · Fp,the determinant of the deformation tensor J, is onlyattributed to elastic deformation:

Jp = det(Fp) = 1, Je = det(Fe) = det(F) = J. (15)

Volumetric changes embedded in Fe can be split fromthe deviatoric response according to Simo (1987) by thefollowing kinematic split:

Fe = J−1/3Fe, (16)

in which Fe is the volume preserving part of the elasticpart of the deformation tensor. The elastic Finger ten-sor (Be) relates to the elastic part of the deformationstensor Fe in the following manner:

Be = Fe · F ce ; Be = Fe · F c

e = J−2/3Be. (17)

Thermally and pressure-induced stresses

During packing and holding phases, compressibilitybecomes the key phenomenon that drives the devel-opment of residual stresses. The change from fillingto packing is also marked by a change in relevantmechanisms in the energy balance. Convection and vis-cous dissipation contributions become negligibly small,when compared with the filling phase, and heat conduc-tion becomes the dominant thermal transport process.For the computation of thermally and pressure-inducedstresses, we can simply employ a linear viscoelasticmodel as used by Baaijens (1991), since these originatefrom relatively small deformations. A viscous–elasticapproach has been adopted by others; however, suchmodels over predict stress and deformation values,since relaxation effects are neglected. In Zoetelief et al.(1996), Kamal et al. (1988), the authors have shown thata viscoelastic approach leads to a more correct positionof the compressive peak and more realistic stress valuesin the subsurface region and at the wall.

28 Rheol Acta (2010) 49:23–44

Only the final equations of the linear thermo-viscoelastic model are presented; for a full derivationof the model, the reader is referred to Douven et al.(1995). Again, the total Cauchy stress consists of avolumetric part, which accounts for the material elas-tic response to changes in volume, and a deviatoriccontribution, which accounts for conformation changes.It reads,

σ = −phI + τ de . (18)

When linearizing ρ = −ρtr(D), and assuming that norelevant pressure and temperature effects exist for t ≤ 0(before filling), an explicit relation for the hydrostaticpressure, ph, in the solid state is found:

ph =∫ t

0

(α

κT − 1

κtr(D)

)dt, (19)

where α is the thermal expansion coefficient and κ

the isothermal compressibility coefficient, which aredefined as:

α = − 1

ρ

(∂ρ

∂T

)

p, (20)

κ = 1

ρ

(∂ρ

∂p

)

T, (21)

respectively.The temperature dependence of the material’s

viscoelastic behavior and is described using the time–temperature superposition, i.e., assuming it to be ther-morheologically simple, implying that the relaxationtimes and viscosities are shifted from a reference tem-perature T0 to the current temperature T, by a shiftfactor aT :

η j = aT(T)η j0, τ j = aT(T)τ j0, (22)

with η j0 and τ j0 denoting the viscosity and relaxationtime at the reference temperature, T0. Only for T ≥ Tg

is the shift factor determined by the WLF equation; for

T < Tg, the relaxation times become so large that weset Tg as the lower bound limit for the time temperaturesuperposition. Hence, for T < Tg, aT = aT (Tg). TheWLF equation is given by:

logaT(T) = −C1(T − T0)

C2 + T − T0. (23)

When no deformation effects before t = 0 exist andassuming the material is thermorheologically simple,the deviatoric part τ d of the Cauchy stress tensor canbe written as:

τ d =m∑

j=1

τ dj , σ d

j =2∫ t

0Gj0e−(ξt j−ξsj)Dd(s)ds, Gj0 = η j0

τ j0,

(24)

where ξqj is the reduced relaxation modulus, which,according to the linear Maxwell model, is defined as:

ξqj =∫ q

o

1

aT(s)τ jds, q = t, s. (25)

The thermo-linear viscoelastic model is only solved forT ≤ Tg. Above the glass transition temperature, theresidual stresses are isotropic and equal to minus thepressure in the melt. The computation of thermally andpressure-induced stresses can be substantially simpli-fied by considering a series of assumptions commonlyemployed:

1. The material is assumed to stick to the mold foras long the pressure in the symmetry line remainspositive.

2. Continuity of stress and strain at the solid meltinterface.

3. The normal stress σ 22, see Fig. 1, is constant acrossthe part thickness and equals minus the pressure in

Fig. 1 Sketch of the cavitygeometry

80 mm

2 mm

50 mm

2

1O

Rheol Acta (2010) 49:23–44 29

the melt as long as the temperature at the symmetryline in the mold, T∗, is larger than the glass transi-tion temperature Tg.

4. In a coordinate system with the 22 direction perpen-dicular to the filling direction 11, the shear straincomponent ε12 is disregarded.

5. As long as pressure remains above zero, at thesymmetry line, the only non-zero strain componentis ε22.

6. Solidification takes place when the no-flow temper-ature (Tg) is reached.

7. Mold elasticity is disregarded.8. Frozen-in or flow-induced stresses can be

neglected.

A detailed discussion on these assumptions is givenby Baaijens (1991); we do not want to repeat all thisin this paper except for the last two assumptions. Bydisregarding the mold, elasticity errors are introducedin the pressure history inside the mold cavity. Thiswill, of course, have an effect on the development ofthe pressure-induced stresses. Baaijens (1991) showedthat introducing mold elasticity slows the decay of thepressure inside the mold cavity, causing some changesin the final profile of the thermally and pressure-induced stresses. Regarding assumption 8, it was shownin Baaijens (1991) and Zoetelief et al. (1996) that theorder of magnitude of flow-induced stresses is about 102

lower than the thermally and pressure-induced stresses.Moreover, the addition of flow-induced stresses to thetotal residual stresses is still a matter of some debate.Zoetelief et al. (1996) used the removal layer techniqueto measure residual stresses parallel and perpendicu-lar to flow direction. They found a difference of lessthan 20%, suggesting that the influence of flow-inducedstresses is small.

Injection molding case

To evaluate the development of flow-induced stressesin GAIM parts, we depart from an earlier study,Baaijens (1991), in which residual stresses were com-puted for injection-molded PC plates. Baaijens used aHele–Shaw formulation to predict the flow kinematicsand used the Leonov model to compute viscoelasticflow-induced stresses. In our study, we apply a fully3-D model to the same injection problem, i.e., geom-etry, material, and processing conditions, to study thedevelopment of flow-induced stresses when using con-ventional or GAIM. Small differences are obviouslyexpected when using a fully 3-D-based approach with-out the Hele–Shaw assumptions, which are: constant

pressure across the mold thickness, negligible velocityin the thickness direction, small velocity gradients, andnegligible thermal conduction parallel to the mid plane.The main difference, however, is that we capture thefountain flow, and the melt stretched at the flow frontregion will contribute to the final flow-induced stressprofiles. To compare with the results of Baaijens, wehave only to investigate a 2-D problem, of a cross sec-tion of the original 3-D geometry of 80 × 2 mm (length,height); see Fig. 1. The gate is located at the channelentrance. In Baaijens (1991), flow-induced stresseswere computed in an injection-molded PC plate. Next,we study the effect of GAIM on flow-induced stresses,using the same material, processing conditions, andgeometry. Again, we approximate the original 3-Dgeometry to a 2-D problem by taking a cross-sectionalong the channel’s length; see Fig. 1. The gate occupiesthe total height of the channel.

Constitutive relations

Viscoelastic model: The properties of the PC grade,taken from Baaijens (1991), are given in Table 1.To estimate the Rouse time, additional information isneeded, namely, the weight-averaged molecular weightMw and the averaged molar mass between entangle-ments 〈Me〉 of PC. Since, in Baaijens (1991), there isno specification of the material grade used, we took the

Table 1 Material parameters for PC

Parameters WLF equation:T0 = 200 [◦C]C1 = −4.217 [–]C2 = 94.95 [◦C]

Thermal properties:cp = 1500 [J kg−1]λ = 0.27 · 10−3 [J s−1mm−1K−1]

Viscoelastic properties:η′ = 3000η [Pa s]τ1 = 10−1 [s] η1 = 9.74 103 [Pa s]τ2 = 10−2 [s] η2 = 6.75 103 [Pa s]τ3 = 10−3 [s] η3 = 1.25 103 [Pa s]Mw = 28.5 [kg mol−1]Me = 1.790 [g mol−1]τR = 2.22 · 10−3 [s]

Tait parameters:s = 0.51 · 106 [◦C Pa−1]solid meltao = 868 · 10−6 868 10−6 [m3 kg−1]a1 = 0.22 · 10−6 0.577 10−6 [m3 kg−1 K−1]Bo = 395.4 · 106 316.1 106 [Pa]B1 = 2.609 · 10−3 4.078 10−3 [◦C−1]

30 Rheol Acta (2010) 49:23–44

value of Mw and Me from literature. In Souheng Wu(1989), the averaged molar mass between entangle-ments 〈Me〉 of PC was determined, and a value equalto 1.790 kg mol−1 was found. In Klompen (2005), theMw of a similar injection molding PC grade is given,and equals 25.8 kg mol−1. Inserting these values inEqs. 13 and 14, a value for the Rouse time τR equalto 2.22 · 10−3 s is found for the highest relaxation mode.For the other relaxation modes, the values of the cor-responding stretching relaxation times are found to betoo small to be relevant for the strain rates involvedin the process. Thus, with the exception of the highestrelaxation mode, stretching can be neglected and a non-stretch form of the Rolie-Poly equation can be adoptedinstead:

�Be + 1

τd

(Be − I

) + 2

3tr

(L.Be

) (Be + β

(Be − I

)) = 0.

(26)

Specific volume: To describe changes in specific vol-ume, we use the so-called Tait model, see Zoller (1982),which has been used vastly for amorphous polymers.The model equation reads:

ν (p, T)=(a0+a1

(T−Tg

))×(

1−0.0894 ln(

1+ pB

)),

(27)

in which Tg is the glass transition temperature, givenby Tg(p) = Tg(0) + sp, and the parameter B(T) byB(T) = B0exp(−B1T). The equation parameters a0, a1,B0, and B1 are different for the melt (T > Tg) andthe solid state (T < Tg). The total set of parameters isfitted on PVT experiments that are run at quasi equilib-rium conditions, under which ν is measured at varyingpressure and temperature. History effects are thereforenot taken into account. For a detailed explanation onthe model parameters, the reader is referred to Zoller(1982).

Viscosity model: The generalized Newtonian viscos-ity, Eq. 7, is computed from the steady Leonov modelat simple shear Baaijens (1991),

η(γ , T) = η0aT +n∑

k=1

2ηkaT

1 + xk, (28)

with xk defined as:

xk =√

1 + 4(θkaT γ

)2. (29)

The relaxation times θ j are shifted by aT , which iscomputed by the WLF equation, Eq. 23.

Boundary conditions

Assuming a computational domain �, Fig. 2, boundaryconditions are specified at �e, �w, and �v, designatingthe mold entrance, mold walls, and the air vents, re-spectively. We prescribe a volume flow rate while fillingthe mold cavity, by means of a fully developed velocityprofile, and an imposed pressure (i.e., normal stress)during the packing and holding phases. At the moldwalls, we use adjustable Robin boundary conditionsthat allow the change from slip to no-slip depending onthe material label c at the wall. If air touches the wall,c = 0, a slip boundary condition is assigned. For poly-mer, c ≥ 0.5, a no-slip condition is imposed by settinga traction force at the wall. Accordingly, the boundarycondition for the velocity and stress components ut andσt in tangential direction read:

aut + σt = 0 ∀ x ∈ (�w ∪ �v

), (30)

in which the dimensionless “Robin penalty parameter”a is defined as

a = a(c) ={ ≥ 106 if c ≥ 0.5: no slip or leakage

0 if c < 0.5: slip or leakage

Air is only allowed to exit the cavity at air vents, �v. Forthis, a Robin condition is assigned for the velocity andstress components un and σn in normal direction:

un = 0 ∀ x ∈ �w (31)

aun + σn = 0 ∀ x ∈ (�w ∪ �v

), (32)

in which a is again given by Eq. 31. However, in thiscase, the term “slip” should be replace by “leakage.”

An initial temperature field is prescribed over theentire domain corresponding to the air/fictitious fluidphase,

Ti = T0(x, t = 0) x ∈ �. (33)

At the injection gate �e, the injection temperature isprescribed,

T = Te(x, t) ∈ �e, t > 0. (34)

e

w

w

Γ

Γ

Γ

Γ

vΩ polymer air

Fig. 2 Computational domain

Rheol Acta (2010) 49:23–44 31

At the mold walls, we prescribe a Dirichlet bound-ary, via which a constant temperature is assigned,

T = Tw(x, t = 0, t) x ∈ �w ∪ �v, t ≥ 0. (35)

As a boundary condition for the computation of flow-induced stresses, we assume that the material injectedinside the mold cavity carries no deformation history.This assumption is not consistent with the fully devel-oped velocity profile prescribed at the inlet. However,given the high temperatures in the melt prior to injec-tion, the assumption that any deformation history inthe material is erased is a reasonable one. Thus, theboundary condition for the stress problem reads:

Be = I ∀ x ∈ �e, t ≥ 0. (36)

Thermally and pressure-induced stresses:post-ejection structural analysis

In our study, we only focus on the development ofresidual stresses inside the mold while the molded partis still kept under pressure. The situation when, due tothe absence of pressure, the part is allowed to moveand consequently shrink inside the mold is disregarded.The ejection phase, in which the part is ejected from themold cavity and allowed to shrink and deform accord-ingly to the residual stress field resulting in warpage,is dealt in a separate structural analysis. We followthe so-called residual-stress methods, in which com-puted residual stresses are used as loading conditionsin a structural analysis, to find the equilibrated resid-ual stress profile and corresponding displacement field.In a different approach, Baaijens (1991) and Douvenet al. (1995) computed displacements using shell typeof elements. More recently, Kennedy (2008) proposeda hybrid model in which shrinkage data measured oninjection-molded specimens are used to calibrate com-puted residual stresses.

Boundary conditions employed inside the mold, dur-ing filling, packing, and holding phases, and those afterthe ejection phase employed in the structural analysis(addressed in a separate section) have to be considered.For the injection phases, filling, packing, and holding,two situations are distinguished for which adequateboundary conditions have to be prescribed:

– Constrained quench with molten core—The pressureinside the cavity is positive and the temperature inthe nodes located at the symmetry line of the part,T∗, is still above the glass transition temperature Tg.

p ≥ 0, T∗ ≥ Tg(p). (37)

All strain components are zero except for ε22, whichmust obey

∫ h/2h/2 ε22dx2 = 0. With respect to the local

coordinate system, the Cauchy stress componentsread:

σ22 = −p, σ11 = ˜σ11 + ba

( − p − ˜σ22). (38)

– Constrained quench with solid core—The pressureinside the cavity is positive but the temperature inthe nodes located at the symmetry line of the part,T∗, is below the glass transition temperature Tg:

p ≥ 0, T∗ < Tg(p). (39)

The material still contacts the wall; the only differ-ence with the previous situation is that σ22 is pre-scribed according to Eq. 40, assuming

∫ h/2h/2 ε22dx2 =

0. The Cauchy stress components read:

σ22 =

h/2∫

h/2

˜σ22/a dx2

h/2∫

h/2

1/a dx2

. (40)

σ11 = ˜σ11 + ba

(σ22 − ˜σ22

). (41)

In order to find the equilibrated residual stress pro-file after ejection, we perform a geometrically non-linear analysis using the commercial finite elementpackage MSC Marc. Residual stress values obtainedduring the filling, packing, and holding phases are in-terpolated to Gaussian points, imposing on the part anon-equilibrium stress state. The domain is spatiallydiscretized with four-noded bilinear quadrilateral el-ements, and the material behavior is described by alinear-elastic constitutive equation. The Poisson ratiofor PC ν is taken to be 0.37; see van Krevelen (1990).For the gas-assisted injection case, since after the gasinjection no polymer is injected to seal the gate, wecannot simply run a structural analysis on the entire

O

1

2

Fig. 3 Boundary conditions employed in the non-linear geomet-rical analysis for the GAIM case

32 Rheol Acta (2010) 49:23–44

O

1

2

Fig. 4 Boundary conditions employed in the non-linear geomet-rical analysis for the conventional injection case

domain. However, given the homogeneity of the stressfield throughout the part’s length, see Fig. 15, we canassume it to be symmetric over the length. In Fig. 3,we illustrate the computational domain with the properboundary conditions to exclude rigid body motions. Forthe conventional injection case, no symmetry assump-tion can be used over the length, since the residualstress profile changes throughout the part’s length. Wetherefore carry out a dynamic analysis, in which con-straints of rigid body motions are not required. Sincethe acceleration values are very small, inertia forcesnegligibly affect the computed stress field. The part ismechanically supported, as shown in Fig. 4. As outputof the analysis, Cauchy stresses and logarithmic strainsare obtained. The magnitude of the computed strainsis of O ∼ 10−3, thus validating the use of a linear-elastic approach. A viscoelastic approach would havebeen more consistent within the modeling frameworkpresented here. However, it was the scope of this studyto analyze the development of residual stresses insidethe mold during the filling, packing, and holding injec-tion phases. The structural analysis carried out uponejection is only done so that an equilibrium stress stateis obtained.

Processing conditions

Conventional injection molding

The polymer is injected at an average velocity of120 mm s−1 at 320◦C. After filling, tf = 0.67 s, a shortpacking phase, tp = 0.71 s, follows until an injectionpressure of 50 MPa is reached. This pressure is thenmaintained during a holding phase of 4 s. After thistime, the gate is assumed to freeze off instantaneously.The temperature at the mold walls is set to 80◦C. InTable 2, the processing conditions are summarized.

Table 2 Characteristic values of injection molding process vari-ables for thermoplastics

Packing pressure Holding time Avg velocity[MPa] [s] [mm s−1]

Polymer GasConventional 50.0 4 120 –GAIM 0.145 2 120 130

Gas-assisted injection molding

The conditions used for GAIM, in terms of injectiontemperature and speed, are equal to those used in theconventional case apart from the injection time that isset to 0.44 s. The gas is injected over a limited heightof the inlet (1 mm) and its injection speed is set to avalue slightly higher than that of the polymer. The gas isinjected immediately after the polymer injection; thus,there is no delay time between the polymer and thegas injections. The remaining processing conditions arelisted in Table 2.

Computational aspects

We use a finite element solution algorithm to solvethe flow and heat transfer problems in 3-D, developedearlier in our group by Haagh and Van de Vosse (1998).The Stokes and energy equation are coupled but solvedwithin each time step in a staggered manner. TheStokes equations, Eqs. 1 and 5, that compose the flowproblem are solved by a velocity–pressure formulationthat is discretized by a standard Galerkin finite elementmethod (GFEM). Since, during the filling phase, theflow is incompressible, and in the subsequent phases(packing and holding) compressible, two different weakforms are found after performing the Galerkin finite el-ement discretization. The system of equations is solvedin an integrated manner; both velocity and pressure aretreated as unknowns. In case of 2-D computations, thediscretized set of algebraic equations is solved using adirect method based on a sparse multi-frontal variantof Gaussian elimination (HSL/MA41)—direct solver(HSL); for details, the reader is referred to Amestoyand Duff (1989a, b) and Amestoy and Puglisi (2002).In 3-D computations, the resulting system of linearequations consists of generally large sparse matrices,and often, iterative solvers are employed, which usesuccessive approximations to obtain a convergent solu-tion. Furthermore, they avoid excessive CPU time andmemory usage. In our 3-D computations, we use a gen-eralized minimal residual solver (GMRES), see Saadand Schultz (1992), in conjunction with an incompleteLU decomposition preconditioner. The computationaldomain is discretized with elements with discontin-uous pressure of the type Crouzeix–Raviart—Q2 Pd

1 ,2-D quadrilateral or brick 3-D finite elements, in whichthe velocity is approximated by a continuous piece-wise polynomial of the second degree, and the pres-sure by a discontinuous complete piecewise polynomialof the first degree. The degrees of freedom at thenodal points correspond to the velocity components,

Rheol Acta (2010) 49:23–44 33

while, at the central node, the pressure and pressuregradients are computed. The integration on the ele-ment is performed using a nine-point (2-D) or 27-point(3-D) Gauss rule. Special care has to be given to solvethe front-capturing convection equation. Convection-dominated problems give rise to unstable solutionswith spurious node-to-node oscillations, referred to aswiggles. To overcome this problem, the streamline-upwind Petrov–Galerkin (SUPG) method, proposed byBrooks and Huges (1982), is the most employed and,thus, is adopted in our model. Flow-induced stressesare computed by adopting an explicit scheme, whichis presented in Appendix A, to numerically integratethe Rolie-Poly equation. For the numerical solutionof thermally and pressure-induced stress problem, thelinear thermo-viscoelastic constitutive model is writtenin an incremental form; see Baaijens (1991); Douvenet al. (1995). In this way, the stress state at time tn+1 canbe evaluated using the completely determined stressstate at tn, together with the pressure and temperaturehistory at tn+1. Details of the incremental form are givenin Appendix B.

Results & discussion

Flow-induced residual stresses

Conventional injection molding

In Fig. 5, we show the first normal stress difference N1,defined as N1 = τ 11 − τ 22, computed at the end of the

filling phase. The results obtained by Baaijens (1991)are also shown. The most important difference is thehigh value of N1 close to the mold wall, which resultsfrom the contribution of the fountain flow. Baaijens(1991) used a Hele–Shaw-based injection model thatonly takes into account shear contributions, and conse-quently, the N1 profile evidences a single peak halfwayin between the part mid-plane and the mold wall, whereshear rates attain their maximum. Such a profile isnot in agreement with experimental observations, seeKamal and Tan (1979), in terms of measured flow-induced birefringence, since it fails to predict the highlyoriented skin layer, which is induced by the steadyelongational flow in the advancing flow front; see, e.g.,Tadmor (1974). The material that is deposited at theskin layers travels through the center region of the flow,where shear rates are minimal, before it is stretched atthe flow front and subsequently quenched at the walls.Our model fully captures the spatial inhomogeneity ofthe flow field, and thus, both shear and extensionalcomponents contribute to the final N1 profile. Theinfluence of the fountain flow on the flow kinematicsis illustrated in Fig. 6, by plotting time labels duringthe polymer injection. Figure 7 shows the first normalstress difference profiles at increasing upstream dis-tances from the flow front. The effect of fountain flowon flow-induced stresses is depicted by a negative N1value in the center line, resulting from the compressionthe material undergoes as it approaches the flow front,and large and positive N1 values close to the wall,denoting the extension of the fluid elements that are

(a)

020

4060

80

00.511.52

0

0.02

0.04

0.06

0.08

0.1

0.12

height [mm]length [mm]

N1

[MP

a]

(b)

Fig. 5 First normal stress difference (N1) results at the end of the filling phase: a results computed by Baaijens (1991) and b from oursimulation

34 Rheol Acta (2010) 49:23–44

P3 P2 P1

Fig. 6 Time labels during injection, denoting the contribution offountain flow to the flow kinematics

deposited in the skin layer. In regions where there is noinfluence of the fountain flow, here designated as fullydeveloped flow regions, N1 values are zero at the centerline. The large stress values at the walls result alsofrom the contribution of the singularity at the contactpoint between the polymer interface and the wall. Theeffect of this singularity on the stress values increasesupon mesh refinement, resulting in unrealistically largestress values at the walls. Similar observations werereported by Mavridis et al. (1988). A way to exclude theeffect of the contact-point singularity from the overallstress computations would be to set Be = I ∀ c < 1,implying that stresses could only develop behind theflow front. However, such an approach would excludethe local phenomena occurring at the flow front, 0.5 <

c < 1.0, and lead to zero stress values at the walls,which are not meaningful. The stress values at the wallsreported here are thus only indicative, but since theyqualitatively agree with the experimentally observedtrends, we include them in our results.

Regarding the shear-induced peaks in Fig. 5, it canbe seen that we predict higher values than Baaijens’ re-sults, although the magnitude of the stresses is similar.The fact that we solve the 3-D problem without using

0 0.5 1 1.5 2

–0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

height [mm]

N1

[MP

a]

fully developed region P3close to the flow front P2behind flow front P1

Fig. 7 N1 profiles across the channel height in the vicinity ofthe flow front and at increasing upstream distances from the flowfront, positions 1, 2, and 3 according to Fig. 6

γ [s– 1]

ψ 1

[Pa·

s2]

.

Fig. 8 First normal stress coefficient ψ1 predicted by the Leonovand Rolie-Poly model

the Hele–Shaw assumptions will, to a certain degree,contribute to this effect. Though the Hele–Shaw modelaccurately predicts pressure fields for thin-walled parts,as is the case of the simulation here, the predictedtemperature field can be erroneous, due to the absenceof the fountain flow, causing the heat convected fromthe core to the walls in the flow front to not be takeninto account. In the case of our simulations, this couldcause a faster decrease in the cross-sectional height,caused by a thicker solidified layer, which might beresponsible for higher shear rates, therefore inducinghigher values of N1. Additionally, one might expect thechoice of the viscoelastic model to influence the stressresults since we use a compressible version of the Rolie-

0 0.5 1 1.5 2

0

0.02

0.04

0.06

0.08

0.1

0.12

height [mm]

N1

[MP

a]

EntranceL = 30.0 [m]L = 60.0 [m]End

Fig. 9 N1 profiles across the channel height for different posi-tions along the cavity length

Rheol Acta (2010) 49:23–44 35

(end of filling) (end of packing)

t = 2.0 s t = 2.5 s

t = 3.0 s t = 4.0 s

Fig. 10 Time evolution of N1 in the convention injection molding case

36 Rheol Acta (2010) 49:23–44

Poly model instead of the compressible Leonov modelemployed in the study of Baaijens (1991). To investi-gate the influence of the viscoelastic model, we plotin Fig. 8 the first normal stress coefficient, defined asψ1 = N1/γ 2, see Macosko (1994), vs shear rate, forboth the Rolie-Poly and Leonov models, using theviscoelastic data in Table 1. It can be seen that bothmodels predict almost identical results; thus, minor dif-ferences might be expected, in terms of the magnitudeof the shear-induced peaks, when using the Leonov orRolie-Poly model.

Figure 9 shows the values of N1 at different locationsalong the mold cavity at the end of the filling phase. Itclearly shows that N1 is maximum close to the middleof the cavity and decreases to zero towards the endof the cavity, due to the small deformation history the

material experiences there. Notice the double peak inthe curve at he entrance of the cavity. We speculate thatthis peak is related to re-melting of the material duringfilling. Since we did not anticipate on this transienteffect, we did not analyze it, i.e., store the full 3Dthermal history. This effect is not new; a similar resultwas found by Douven (1991), see Fig. 6.11 in his thesis.

Figure 10 shows the evolution of N1 during thetotal injection molding cycle. After filling, the resid-ual stresses are drastically reduced by fast relaxationat the characteristic high melt temperature, and theirvalue remains only at the walls, where stress is frozenduring filling. Upon packing, a third peak in betweenthe shear-induced peak, developed during filling, andthe center line develops. Although the strain rates arelow during packing and holding, the relaxation times

(end of polymer filling) (during gas filling)

t = 2.0 s

Fig. 11 Time evolution of N1 in the GAIM case

Rheol Acta (2010) 49:23–44 37

0 0.5 1 1.5 2

0

0.02

0.04

0.06

0.08

0.1

0.12

height [mm]

N1

[MP

a]t = 0.42 [s]t = 0.46 [s]t = 0.52 [s]t = 0.58 [s]

(a)

0 0.5 1 1.5 2

0

0.02

0.04

0.06

0.08

0.1

0.12

height [mm]

N1

[MP

a]

EntranceL = 30.0 [m]L = 60.0 [m]End

(b)

Fig. 12 a N1 profiles at x = 0.02 m, depicting the transition from polymer filling to gas filling. b N1 profiles along the cavity length afterthe completion of gas filling

increase due to cooling. They are clearly responsiblefor the development of significant orientation in thesephases of the process, since a prerequisite for molecularorientation is sufficient strain rate: ε > 1/(τ(T, Mw),with τ(T, Mw) the temperature and molecular weight-dependent relaxation times and sufficient strainε = εt > 1.

Gas-assisted injection molding

Figure 11 shows the evolution of N1 during the GAIMcycle. The transition from polymer filling to gas fillingis marked by a sudden decrease in N1. The transitionis illustrated in Fig. 12, where normal stress differencesare plotted over the height of the channel for differenttimes at a position close to the gate corresponding tox1 = 0.02 m. At t = 0.42 s, gas is injected and betweent = 0.52 s and t = 0.58 s, the gas front passes throughx1 = 0.02 m. Since the gas is only injected through thecore of the part, significant relaxation occurs in theregion of the shear-induced peaks. Only the moltenmaterial located in the core region is displaced. Thus,the deformation rates are drastically decreased, andsince temperatures are still well above the glass transi-tion temperature, the flow-induced stresses are allowedto relax until a new N1 profile is achieved; see Fig. 12afor t = 0.58 s. Nevertheless, at the flow front, deforma-tion of the material takes place as the molten polymeris pushed forward by the gas pressure. This is shownin Fig. 12b, in which normal stress differences, plottedat the end of the polymer filling phase for differentpositions along the cavity length close to the mold

wall remain unchanged throughout the channel. Whencompared with the N1 results obtained with conven-tional injection molding, Fig. 9, there is an impressivereduction in flow-induced stresses. It can be seen thatthe height and width of the shear-induced peaks issignificantly decreased by gas injection. Moreover, thepacking and holding phases do not induce further orien-tation. This is due to the lower pressure level inside thecavity (and the fact that no further material is pushedinto the cavity). Since the gas is able to exert pressureeverywhere inside the cavity, the pressure level neededis lower (in this example, two orders of magnitude) thanin the conventional case. As a consequence, in GAIM,the resulting frozen-in orientation and related flow-

0 1 2 3 4 50

5

10

15

20

25

30

35

40

45

50

55

time [s]

Pre

ssur

e [M

Pa]

L = 8.0 [mm]L = 40.0 [mm]L = 75.0 [mm]

Fig. 13 Pressure history inside the cavity at different locationsfor the standard injection molding case

38 Rheol Acta (2010) 49:23–44

(end of filling) t = 1.3 s

t = 2.4 s t = 3.3 s

t = 4.4 s (upon ejection)

Fig. 14 Time evolution of thermally and pressured-induced stresses in the convention injection molding case

Rheol Acta (2010) 49:23–44 39

(end of filling) t = 1.4 s

t = 1.8 s t = 2.0 s

t = 3.5 s (upon ejection)

Fig. 15 Time evolution of thermally and pressured-induced stresses in the GAIM case

40 Rheol Acta (2010) 49:23–44

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

time [s]

Pre

ssur

e [M

Pa]

L = 8.0 [mm]L = 40.0 [mm]L = 75.0 [mm]

Fig. 16 Pressure history inside the cavity at different locationsfor the GAIM case

induced stresses are solely determined during the fillingphase.

Thermally and pressure-induced residual stresses

Conventional injection molding

Figure 13 shows the calculated pressure history insidethe cavity in three different positions, x1 = 0.008 m,x2 = 0.04 m, and x3 = 0.075 m. The pressure decay ismostly determined by α and κ , and its evolution hasa direct impact on the pressure-induced stresses. Afterthe filling phase, t f = 0.67 s, the change to packing ismarked by a steep rise in pressure. Figure 14 showsthe evolution of the thermally and pressure-induced

stress components. At the end of filling, only tensilestresses develop in the regions near to the mold walls,induced by the hampered thermal contraction whilethe pressure is maximum at the injection gate L = 0and zero at the end L = 0.08 m. In the packing phase,the pressure increases to become uniform inside themold cavity. Shrinkage upon cooling makes the pres-sure lower during the holding phase, and upon furthermaterial solidification, the pressure keeps on decaying.The pressure history is reflected in the stresses in thesolidified layers compared to the bulk. Residual stressesbecome compressive in layers that solidify under a highpressure, and tensile in those solidified under a lowpressure. At the end of the holding phase, the stressesin the solidified layers increase proportionally with thepressure relief.

Gas-assisted injection molding

Figure 16 shows the pressure history at three differentpositions along the cavity’s length, x1 = 0.008 m, x2 =0.04 m, and x3 = 0.075 m for the GAIM case. Thetransition from polymer to gas injection can be seenby a decay and a subsequent growth of the pressure att f = 0.3 s at positions x1 and x2. The difference betweenthe computed pressure histories for the conventionaland GAIM cases immediately suggests that the residualstresses in GAIM should differ substantially. In GAIM,the end of filling occurs at a decaying pressure, sinceonly the molten-liquid material in the core region isdisplaced, and the packing, and subsequently, holding,pressures are much lower. Also for the GAIM case,during the holding phase, the pressure inside the cavity

–1 –0.5 0 0.5 1

100

150

200

250

300

350

z/h [–]

T [

o C ]

t = 0.654 [s]t = 2.0 [s]t = 4.5 [s]

(a)

–1 –0.5 0 0.5 1

100

150

200

250

300

350

z/h [–]

T [

o C ]

t = 0.9 [s]t = 1.4 [s]t = 3.5 [s]

(b)

Fig. 17 Temperature history at x2 = 0.04 m for: a conventional injection molding; b GAIM

Rheol Acta (2010) 49:23–44 41

–1 –0.5 0 0.5 1

–50

–40

–30

–20

–10

0

10

20

z/h [–]

σ 11 [M

Pa]

t = 0.65 [s]t = 0.7 [s]t = 1.7 [s]t = 2.7 [s]t = 4.4 [s]

(a)

–1 –0.5 0 0.5 1

–50

–40

–30

–20

–10

0

10

20

z/h [–]

σ 11 [M

Pa]

t = 0.9 [s]t = 1.4 [s]t = 2.0 [s]t = 3.5 [s]

(b)

Fig. 18 Thermally and pressured-induced stresses at x2 = 0.04: a conventional injection molding; b GAIM

remains constant, and therefore, the polymer materialsolidifies at a constant pressure. Figure 15 shows theevolution of the thermally and pressure-induced σ11

stress component. The profiles at the end of filling showa slightly higher tensile stress than in the conventionalcase. This can be related to the increased cooling inGAIM, which results from the absence of a moltencore and a longer filling time (Fig. 16). In Fig. 17,temperature profiles are plotted in the middle of thepart at the end of the respective filling phases forboth conventional injection molding and GAIM. Theincreased cooling in the case of GAIM is clear. As aresult of a shorter cooling time and a higher coolingrate, the thermal contraction prevented during fillingis expected to be higher, thus inducing higher tensile

–1 –0.5 0 0.5 1

–50

–40

–30

–20

–10

0

10

20

z/h [–]

σ 11 [M

Pa]

conventionalGAIM

Fig. 19 Thermally and pressured-induced stresses at x2 = 0.04 mupon ejection

stresses. As cooling proceeds, the tensile stresses prop-agate to reach the inner surface, resulting in a arc-typeprofile. Given the significant lower pressures, especiallyin the packing phase, the development of a compressivepeak does not occur inside the mold. Close to the endof the slit, the stress profile shows some instabilities,which were found to be dependent on changes of thethickness of the solidified layer, which could not bemade smoother via mesh refinement. Thus, the stressfield is very sensitive to changes in the thickness of thepreviously solidified layer. Figure 18a and b give theevolution of the residual stress profiles at position x2

in time, for conventional injection molding and GAIM.The final residual stress profiles obtained from thestructural analysis are given in Fig. 19. It can be seenthat, for GAIM, the tensile stresses at the wall are muchlower compared to the conventional case.

Conclusions

A numerical study to assess the effect of GAIM onthe development of residual stresses was conducted.Firstly, the evolution of flow-induced stresses in bothconventional and GAIM was assessed. A compressibleversion of the Rolie-Poly model was proposed andapplied for the computation of flow-induced stresses.Computational results show a significant decrease inthe magnitude of stresses during the gas-filling phase.Moreover, it was found that the stress level did notevolve further during the packing and holding phases.Hence, the magnitude of flow-induced orientation andrelated stresses in GAIM is much lower than in the con-ventional case and is set during the filling phase only.

42 Rheol Acta (2010) 49:23–44

Secondly, the development of thermally and pressure-induced stresses in GAIM was investigated. A thermo-viscoelastic model was used to compute thermally andpressure-induced stresses. Computed stress profiles forGAIM during filling and holding phases exhibit slightlyhigher tensile stresses at the surface; however, themost noticeable difference is the absence of a compres-sive region, which is typical for conventional injectionmolded parts. This can be explained by the fact thatthe pressures for GAIM are much lower than thoseof conventional injection molding during the packingand holding phases. Knowing that, in injection molding,pressure-induced stresses overrule the contribution forthe total residual stresses, it has been shown that GAIMcan drastically change the residual stresses in injectionmolded parts. The final equilibrated stress profile showsthat GAIM can significantly reduce tensile stressesat the walls. This is obviously of importance, sincetensile stresses can initiate crazing and, thereafter, sur-face cracking. Our study supports that GAIM has astrong effect on the behavior of injection molding partsand their properties. It reduces the level of frozen-in orientation and can therefore minimize the part’slong-term dimensional changes and the anisotropy inphysical properties. The simulations can be improvedwith constitutive models for density and heat conduc-tion with better characterized material data. Analysesto assess the influence of processing conditions and ofthe part’s geometry on computed stresses, which werebeyond the scope of our study, are of interest to furtherexploit the GAIM characteristics to arrive at superiorproducts with enhanced dimensional stability.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

Appendix A

Numerical integration of the Rolie-Poly model

To perform the numerical integration of the Rolie-Poly equation, Eq. 12, we use a second-order Adams–Bashford explicit scheme. For the first two time steps,the numerical integration follows a first-order forwardEuler method. Accordingly, the time marching schemereads:

for time step ≤ 2,

Ben+1 ≈ Bn

e + f(u, Be, ∇u, (∇u)T) |n�t, (42)

for time step > 2,

Ben+1 ≈ Bn

e +(

3

2f(u, Be, ∇u, (∇u)T) |n

−1

2f(u, Be, ∇u, (∇u)T) |n−1

)�t. (43)

Appendix B

Incremental formulation of the thermo-viscoelasticmodel

We will now discretize the linear thermo-viscoelasticmodel, expressing it in an incremental formulation. Attime t = tn, the model, given by Eqs. 19 and 24, reads:

σ n = −phnI +

m∑

j=1

τ djn , (44)

phn =

∫ tn

0

(α

κT − 1

κtr(D)

)ds, (45)

τ djn = 2

∫ tn

0Gje−(ξn−ξ(s))/τi0 εdds. (46)

The subscript n indicates the evaluation at time tn, atwhich the model variables are fully determined. Forthe evaluation of the stress at the next step, tn+1, itis required to know the temperature and strain fields.We introduce the following incremental variables forT and ξ :

�ξn+1 = ξn+1 − ξn, �Tn+1 = Tn+1 − Tn. (47)

It is assumed that ξ and T vary linearly between twodiscrete time steps, implying that ε and T are constantover each time increment. It yields

T = �Tn+1

�tn+1, ε = D�t, for t ∈ [tn, tn+1], (48)

Next, we discretize Eq. 19, and evaluate the integralsusing the trapezium rule:

pn+1 = pn + β�Tn+1 − Ktr(D), (49)

β = 1

�tn+1

∫ tn+1

tn

α

κds, (50)

K = 1

�tn+1

∫ tn+1

tn

1

κds. (51)

Rheol Acta (2010) 49:23–44 43

The discretization of Eq. 24, in combination with thetrapezium rule, leads to:

m∑

j=1

τ djn+1

=m∑

j=1

ξ jn+1τdjn + 2G�εd

n+1, (52)

G =m∑

j=1

Gj0

2(1 + ξ jn+1), (53)

ξ jn+1 = exp[−�tn+1

2τ j0

(1

aTn+1

+ 1

aTn

)]. (54)

After extensive rewriting of Eq. 44 for t = tn+1, thefollowing discretized form for σ is found:

σ n+1 = σ + Ktr(�εn+1)I + 2G�εnn+1, (55)

where

σ = −(pn + β�Tn+1

)I +

m∑

j=1

ξ jn+1τdjn . (56)

The quantities σ , K, and β are evaluated when thestate at tn is determined and the temperature history isknown up until tn+1. The stress components of the linearthermo-viscoelastic model are written with respect tothe local base O1, defined in Fig. 1. Due to assumption5 and the knowledge of σ22, the incremental straincomponent �ε22 can be eliminated from Eq. 55:

�ε22 = σ22 − ˜σ22

a, a = 3K + 4G

3. (57)

The final form of the discretized model reads:

σ n+1 = a�ε11 + g, (58)

where

a = a − b 2

aand g = σ11 + b/a

(σ22 − σ22

), (59)

with

a = 4G + 3K3

, b = −2G + 3K3

. (60)

References

Amestoy P, Duff I (1989a) Memory management issues in sparsemultifrontal methods on multiprocessors. Int J SupercomputAppl 7:64

Amestoy P, Duff I (1989b) Vectorization of a multiprocessormultifrontal code. Int J Supercomput Appl 3:41

Amestoy P, Puglisi C (2002) An unsymmetrical multifrontal lufactorization. SIAM J Matrix Anal Appl 24:553

Baaijens F (1991) Calculation of residual stresses in injectionmolded products. Rheol Acta 30:284–299

Bird R, Armstrong R, Hassager O (1987) Dynamics of polymerliquids, vol 1. Wiley-Interscience, New York

Bogaerds A, Hulsen M, Peters G, Baaijens F (2002) Stabilityanalysis of injection molding flows. J Rheol 48:765–785

Brooks A, Huges T (1982) Streamline upwind/petrov-galerkinformulation for convection dominated flows with particu-lar emphasis on the incompressible navier-stokes equations.Comput Methods Appl Mech Eng 32:199–259

Caspers L (1995) Vip, an integral approach to the simulationof injection moulding. PhD thesis, Eindhoven University ofTechnology, Eindhoven

Doi M, Edwards S (1986) The theory of polymer dynamics.Claredon, Oxford

Douven L (1991) Towards the computation of properties of injec-tion moulded products. PhD thesis, Eindhoven University ofTechnology, Eindhoven

Douven L, Baaijens F, Meijer H (1995) The computation ofproperties of injection-moulded products. Prog Polym Sci20:403–457

Grillet A, Bogaerds A, Peters G, Baaijens F, Bulters M (2002)Numerical analysis of flow mark surface defects in injectionmolding flow. J Rheol 46:651–669

Haagh G, Van de Vosse F (1998) Simulation of three-dimensional polymer mould filling processes using a pseudo-concentration method. Int J Numer Methods Fluids 28:1355–1369

Haagh G, Van de Vosse F (2001) A 3-d finite element model forgas-assisted injection molding: simulations and experiments.Polym Eng Sci 41:449–465

Hansen D, Bernier G (1972) Thermal conductivity of poly-ethylene:the effects of crystal size, density and orientationon the thermal conductivity. Polym Eng Sci 12:204

Hastenberg C, Wildervank P, Leeden A (1998) The measure-ment of thermal stresses distributions along the flow path ininjection-molded flat plates. Polym Eng Sci 32:506–515

Hulsen A, Fattal R, Kupferman R (2005) Flow of viscoelasticfluids past a cylinder at high weissenberg number: stabilizedsimulations using matrix logatithms. J Non-Newton FluidMech 127:27–39

Kamal M, Tan V (1979) Orientation in injection moldedpolystyrene. Polym Eng Sci 19:558–563

Kamal M, Fook RA, H AJR (1988) Residual thermal stressesin injection moldings of thermoplastics: a theoretical andexperimental approach. Polym Eng Sci 42:1098–1114

Kennedy PK (2008) Pratical and scientific aspects of injectionmolding simulation. PhD thesis, Eindhoven University ofTechnology, Eindhoven

Klompen E (2005) Mechanical properties of solid polymers: con-stitutive modelling of long and short term behaviour. PhDthesis, Eindhoven University of Technology, Eindhoven

Li C, Shin J, AI I (2004) Primary and secondary gas penetrationduring gas-assisted injection molding. Part 2: simulation andexperiment. Polym Eng Sci 44:992–1002

Likhtman A, Graham R (2003) Simple constitutive equationfor linear polymer melts derived from molecular the-ory: Rolie-poly equation. J Non-Newton Fluid Mech 114:1–12

Macosko C (1994) Rheology: principles, measurements, andapplications. Wiley-VCH, New York

Mandell J, Smith K, Huang D (1981) Effects of residual stress andorientation on the fatigue of injection molded polysulfone.Polym Eng Sci 21:1173–1180

44 Rheol Acta (2010) 49:23–44

Mavridis H, Hrymak A, Vlachopoulos J (1988) The effect offountain flow on molecular orientation in injection molding.J Rheol 32:639–663

Meijer H (1997) Processing of polymers. In: Meijer H (ed)Material science and technology, vol 18, chap 1. VCH,Verlagsgesellschaft mbH, Weinheim

Parvez M, Ong N, Lam Y, Tor S (2002) Gas-assisted injectionmolding: the effects of process variables and gas channelgeometry. J Mater Process Technol 121:27–35

Polynkin A, Pittman J, Sienz J (2005) 3d simulation of gas as-sisted injection molding analysis of primary and secondarygas penetration and comparison with experimental results.Int Polym Process 20:191–201

Saad Y, Schultz M (1992) Gmres: a generalized minimal resid-ual algorithm for solving nonsymetric linear systems. SIAMJ Matrix Anal 13:121–137

Simo J (1987) On a fully three-dimensional finite-strain viscoelas-tic damage model: formulation and computational aspects.Comput Methods Appl Mech Engrg 60:153–173

Souheng Wu EI (1989) Chain structure and entanglement.J Polym Sci, Part B: Polym Phys 27:723–741

Struik L (1978) Orietation effects and cooling stresses in amor-phous polymers. Polym Eng Sci 18:799–811

Tadmor Z (1974) Molecular orientation in injection molding.J Appl Polym Sci 18:1753–1772

Thompson E (1986) Use of pseudo-concentrations to followcreeping viscous flows during transient analysis. Int J NumerMethods Fluids 6:749–761

van den Brule B, O’Brien S (1990) Anisotropic conductionof heat in a flowing polymeric material. Rheol Acta 29:580–587

van Krevelen D (1990) Properties of polymers. Elsevier ScienceB.V., Amsterdam

Wimberger-Friedl R, Bruin J, Schoo F (2003) Residual bir-refringence in modified polycarbonates. Polym Eng Sci 43:62–70

Zoetelief W, Douven L, Housz J (1996) Residual thermal stressesin injection molded products. Polym Eng Sci 36:1886–1896

Zoller P (1982) A study of the pressure-volume-temperaturerelationships of four related amorphous polymers: polycar-bonate, polyarylate, phenoxy and polysulfone. Polym Sci20:1453–1464