EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computer Science

CASA-Report 11-27 April 2011

Modelling stretch blow moulding of polymer

containers using level set methods

by

J.A.W.M. Groot, C.G. Giannopapa, R.M.M. Mattheij

Centre for Analysis, Scientific computing and Applications

Department of Mathematics and Computer Science

Eindhoven University of Technology

P.O. Box 513

5600 MB Eindhoven, The Netherlands

ISSN: 0926-4507

PVP2010-25710

Modelling Stretch Blow Moulding of PolymerContainers Using Level Set Methods

J. A. W. M. Groot C. G. Giannopapa R. M. M. MattheijEmail: j.a.w.m.groot@tue.nl Email: c.g.giannopapa@tue.nl Email: r.m.m.mattheij@tue.nl

Dept. of Math. and Computer ScienceEindhoven University of TechnologyPO Box 513, 5600 MB Eindhoven

The Netherlands

ABSTRACT

Stretch blow moulding is a widely used technique e.g. for the production of PET bottles.In a stretch blow moulding process a hot preform of polymer is simultaneously stretched andblown into a mould shape. The process takes place at a fast rate and is characterised by largedeformations and temperature gradients. In this paper a computer simulation model for stretchblow moulding is presented. The model is based on finite element methods and uses a levelset method to track the interfaces between air, polymer and stretch rod. The PET behaviour ismodelled as a non-newtonian, isothermal fluid flow, based on a viscoplastic material model. Anapplication presented is the stretch blow moulding of a realistic PET water bottle. The model isvalidated by verifying volume conservation.

KEY WORDS: stretch blow moulding, process simulation, level set methods, Non-newtonian fluid flow,finite element methods

NomenclatureBr Brinkman numberdev deviatoric partFr Froude numberPe Péclet numberRe Reynolds numbercp specific heat [J kg−1K−1]g gravitational acceleration [m s−2]k modulus [Pa sm]Lc characteristic length [m]m reciprocal of power-law indexp pressure [Pa]t time [s]T temperature distribution [K]

Tp typical polymer temperature [K]Tm typical mould temperature [K]u flow velocity [m s−1]Vc characteristic velocity [m s−1]ε strain rate tensor [s−1]˙ε strain rate [s−1]λ thermal conductivity [W m−1K−1]µ (dynamic) viscosity [kg m−1 s−1]µc characteristic viscosity [kg m−1 s−1]ρ density [kg m−3]σ stress tensor [Pa]σ equivalent stress [Pa]φ level set function

INTRODUCTIONBlow moulding is a technique that was invented thousands of years ago for the production of glass

containers, e.g. bottles and jars. The automated blow moulding process as it is known today has seenrapid development during the last century, while at the same time the range of applications and diversityof materials has considerably increased. In spite of the fact that blow moulding has been used for manyyears, manufacturers still experience difficulties in optimising and controlling the process.

In polymer container manufacture three main blow moulding techniques can be distunguished: in-jection blow moulding, extrusion blow moulding and stretch blow moulding. In injection blow mouldingfirst a polymer preform is formed by injecting a polymer melt into a tube-shaped mould and subse-quently it is brought into another mould, where the final container is blown by inflating the preformwith air. Injection blow moulding is usually used to manufacture small or wide-mouthed containers.In extrusion blow moulding the polymer melt is extruded in a preform shape before blowing it into thecontainer shape. Extrusion blow moulding can be used for a wide variety of container shapes and has ingeneral a high production rate. Finally, stretch blow moulding is a popular manufacturing technique forthe production of so-called PET (polyethylene terephtalate) bottles and is subject of this paper.

Figure 1 shows a schematic drawing of the stretch blow process. Initially a preform is transferredon the stretch rod into a mould (Fig. 1(a)). The preform is typically produced by injection mouldinga polymer melt. Then the preform is simultaneously stretched with the stretch rod and inflated withpressurised air (Fig. 1(b)-1(c)). Finally, the resulting container is cooled down and ejected from themould.

Over the last few decades computer simulation models have significantly contributed to a betterinsight in and understanding of blow moulding processes and nowadays they are widely used for pro-cess optimisation. Computer simulation models can help to minimise undesired variations in the wallthickness and reduce the weight while maintaining the strength. They offer a good alternative for trial-and-error methods with expensive blow moulding equipment or complicated measurements during amoulding process, which takes place in a closed construction at a fast rate. Moreover, simulations canbe used for comparison with measurements. The numerous papers on simulating the stretch blow mould-ing include references [1–8]. Various computational models have also been developed for modelling theforming of glass containers [9–14], thermoforming [15, 16] and extrusion blow moulding [6, 17–19].

The simulation model presented in this paper is based on finite element methods and uses a levelset method to track the polymer-air interfaces. Finite element methods have been widely used forthe modelling of blow moulding processes in literature. Finite element methods are usually coupledto interface tracking techniques, which attempt to find the interfaces implicitly and update the finite

preform mould

(a)

stretch rod pressurised air

(b)

container

(c)

Fig. 1. SCHEMATIC DRAWING OF A STRETCH-BLOW MOULDING PROCESS

element mesh as the interfaces evolve. The procedure of updating the mesh can become increasinglycomputationally expensive as the mesh size decreases or the mesh has to be updated more frequently. Amajor advantage of level set methods is that no re-meshing is required to distinguish between polymerand air as time progresses; the location of the interface can be marked by the so-called level set functioninstead [20–23]. Level set methods also have other attractive properties: they automatically deal withtopological changes, can in general be used with high order of accuracy [24] and extend easily to threedimensions [25]. The level set method in this paper is also used to describe the motion of the stretchrod.

The objective of this paper is to create a robust and computationally efficient model to be usedfor industrial purposes which can accurately capture the stretch blow moulding process of polymercontainers. The model takes as input information the preform shape, a uniform temperature distributionof the preform and a prescribed inlet air pressure. It computes the container shape, the products finalwall thickness as well as the stress and thermal deformations the polymer undergoes during the process.The model is applied to the stretch blow moulding of a 2D axial-symmetrical PET bottle with a relativelycomplex geometry. The model is validated by verifying conservation properties.

RHEOLOGICAL BEHAVIOURMuch research has been conducted on the rheological behaviour of polymers. Essentially, three

different kinds of models have been introduced in literature to describe the mechanics of a polymer meltsubject to inflation, namely hyperelastic, viscoelastic and viscoplastic models.

Regarding the short inflation time (∼ 0.5s), a hyperelastic constitutive relationship can be assumed[1, 26]. In particular, earlier papers propose isothermal, hyperelastic models, many of which were firstdeveloped for thermo-forming processes [1, 15, 27]. However, hyperelastic models have difficultieswith proper material characterisation and the prediction of the time-dependent behaviour of the polymerduring a blow moulding process [17, 26]. Moreover, in [28] it was observed that the behaviour of PETis highly rate dependent, which is an effect that hyperelastic models generally cannot deal with.

Viscoelastic and viscoplastic models take into account both strain hardening and strain rate effects.Viscoelastic models have been widely used for stretch blow moulding processes in literature. For ex-ample, in [29] a K-BKZ visco-elastic model was adopted to simulate the blow moulding process. In [3]a liquid-like visco-elastic constitutive equation of Johnson-Segalman type was used to describe themechanical behaviour of PET during a stretch blow moulding process. The simulation results were

in good agreement with the experimental data. In [30] the Christensen model was modified into avisco-hyperelastic model. Finally, in [31] the Buckley-Jones model was used to model the rheologicalbehaviour of PET during the stretch blow process. Viscoelastic models can also calculate the stressrelaxation during stretch blow moulding, which is a dominant deformation mode in relatively slow de-formation processes, such as extrusion. However, because of the short time duration of the stretch blowprocess, the loss of accuracy as a result of omitting stress relaxation is marginal, while the computa-tional time can be significantly decreased [32]. This can be a motivation to employ a viscoplastic model.Another advantage of viscoplastic models is that they are generally very stable, whereas difficulties withconvergence can be encountered when using hyperelastic or viscoelastic models [32]. Viscoplastic mod-els are mainly used for metal forming processes, but have not been used extensively for stretch blowmoulding. Viscoplastic models for stretch blow moulding were suggested in [5, 33–35].

In addition, some different models have been proposed for stretch blow moulding. For example, anelasto-visco-plastic model was suggested in [2] and a neo-Hookean hyperplastic consitutive relationshipwas employed in [36]. For the latter model it was concluded that the model is in reasonable qualitativeagreement with the experimental results, but the model failed to accurately predict the container shape.

In this paper the viscoplastic model presented in [5, 37, 38] is used to describe the rheological be-haviour of the polymer. The model is based on the Levy-Mises flow rule:

dev(σ) =23

σ

˙εε, (1)

where the strain rate ˙ε and the equivalent stress σ are given by

˙ε =

√23

ε: ε, (2)

σ =

√32

dev(σ):dev(σ). (3)

where : is the double dot inner product, defined by

T:T= tr(TT), (4)

for any tensor T. In [5, 33] the material behaviour was determined by performing an uniaxial tensiletest. From the test the following relation was deduced:

σ = k(T ) ˙εm. (5)

In [35, 39] the G’Sell-WLF material model was proposed for the rheological behaviour of PET:

σ = k0 exp(

C1(T −Tref)

C2 +T +Tref

)(1− exp(−Aγ))exp(Bγ

C) ˙εm, (6)

where γ is the cumulated strain rate. If constants A and B are disregarded, the uniaxial tensile test in (5)is obtained with

k(T ) = k0 exp(

C1(T −Tref)

C2 +T +Tref

). (7)

In (7) the WLF equation for the temperature dependence of the viscosity can be recognised [40]. Finally,substitution of (5) in (1) results in the isotropic, viscous polymer flow,

dev(σ) = 2µε, (8)

with non-newtonian viscosity

µ( ˙ε,T ) =k(T )

3˙εm−1. (9)

GOVERNING EQUATIONSThis section presents the governing equations for the stretch blow moulding process. These involve

conservation of mass, momentum and energy of the polymer melt and air and the evolution of thepolymer-air interfaces. A two-phase fluid flow formulation is considered, i.e. the governing equationsdescribe the flow of both air and molten polymer.

MechanicsThe mechanics of the system can be described by the momentum and mass equations for an incom-

pressible fluid:

ρdudt

= ∇·σ+ρg, (10)

∇·u = 0, (11)

where ddt := ∂

∂t +u·∇ denotes the material time derivative. For isotropic, viscous fluids the constitutiveequation for the stress tensor is given by

σ = 2µε− pI, (12)

where the strain rate tensor relates to the flow velocity as

ε =12(∇u+∇uT) . (13)

System of equations (10-11) is written in dimensionless form in order to perform a quantitative analysisof the problem. Consider a typical: velocity Vc, length scale Lc and viscosity µc. Then introduce thedimensionless variables

t∗ :=VctLc

, x∗ :=xLc

, u∗ :=uVc

, µ∗ =µµc, p∗ :=

Lc pµcVc

. (14)

Superscript ∗ is used to indicate that variables and operators are in terms of the dimensionless variables.The gravity force can be written as −ρgez, where ez is the unit vector in axial direction. Substitution of

the dimensionless variables (14) into the flow equations (10-11) and division by the order of magnitudeof the viscous term, µcVcL−2

c , leads to the dimensionless flow equations

Redu∗

dt∗= ∇

∗· (2µ∗ε∗)−∇∗p∗− Re

Frez, (15)

∇∗·u∗ = 0, (16)

where

Re =ρVcLc

µc, (17)

Fr =V 2

cgLc

(18)

are the Reynolds number and the Froude number, respectively. The dimensionless form is useful forassessing the order of magnitude of the different terms in (15) and is the basis for simplifications in thesimulation model.

For the flow problem the following boundary conditions are imposed:

• no-slip conditions for the polymer melt and no-stress conditions for air on the mould wall,• no-slip conditions for the polymer melt and an inlet pressure for air at the mould entrance,• axial symmetry conditions on the axis.

ThermodynamicsThe heat transfer in polymer, air and equipment is described by the heat equation for incompressible

continua:

ρcpdTdt

=−∇·q+2µε:∇u. (19)

If the heat transfer due to radiation is neglected, the heat flux q [W m−2] is the result of thermal con-duction,

q = −λ∇T, (20)

In order to write (19) in dimensionless form, a dimensionless temperature distribution is introduced,

T ∗ :=T −Tm

Tp−Tm, (21)

where Tm is the typical temperature of the mould and Tp is the typical polymer temperature. Substitutionof the dimensionless variables (14) and (21) in the heat equation (19) leads to the dimensionless form,

PedT∗

dt∗= ∇

∗2T∗+2µ∗Br ε∗:∇

∗u∗, (22)

where

Pe =ρcpVcLc

λ, (23)

Br =µcV 2

c

λ(Tp−Tm

) (24)

are the Péclet number and the Brinkman number, respectively. The heat transfer is discussed into moredetail further on for the PET bottle simulation.

Evolution of the interfaces: a level set methodA level set method is used to capture the interfaces in the two-phase fluid flow problem. Level set

methods are based on an implicit formulation of the interfaces by means of so-called level set functions,which enables the two-phase fluid flow problem to be solved on a fixed mesh in the flow domain. Whilethe governing equations are formulated in the whole flow domain, level set methods can be used todifferentiate between material properties.

Level set methods seem to be attractive for this problem because the interfaces are accurately cap-tured, topological changes are naturally dealt with and complicated re-meshing algorithms are avoided.The latter is particularly an issue for blow moulding processes, since a moving mesh can become rapidlydistorted during a simulation because of the large deformations. As a result the mesh would have to beupdated frequently, which can become increasingly computationally expensive. Other advantages oflevel set methods are a relatively easy generalisation to three dimensions and straightforward calcula-tion of properties of the interfaces, such as the normal and curvature. A drawback of level set methodsis that they are in general not conservative. However, in [13] a change in mass of less than 1% duringthe glass blow moulding process simulations is reported, which can even be further improved by usinghigher order time integration schemes or by taking smaller time steps.

The basic idea of level set methods is to embed each moving interfaces between the polymer, airand stretch rod as the zero level set of a level set function φ. The equation of motion of the interfacesfollows from the fact that the material derivative of the level set function is equal to zero:

dφ

dt=

∂φ

∂t+u·∇φ = 0. (25)

Initially, the level set function is defined as the signed Euclidean distance function to the interfaces.The flow velocity u is found by solving flow problem (15-16) and boundary conditions. Each polymersurface corresponds to the zero level set of one level set function, which means two level set problemshave to be solved to capture both interfaces. Furthermore, to completely avoid moving mesh algorithms,also the surface of the stretch rod is described by a level set function. However, the level set problemfor the stretch rod moving with constant velocity ur =Vrez is trivial, as

φr(x, t) = φr,0(x+ur min{t− t0, tstretch− t0}), (26)

where φr(x, t) is the level set function corresponding to the stretch rod at position x and time t, tstretch isthe stretch time, t0 is the time at which the stretch rod starts moving and

φr,0(x) = φr(x,0). (27)

In conclusion, three level set functions are used to describe the evolution of the moving interfaces:φi for the inner polymer surface, φo for the polymer surface and φr for the stretch rod surface. Level setfunctions φi and φo are found by solving (25) and φr is given by (26). Figure 2 gives an overview of thecontinuum domains and the corresponding signs of the level set functions. The continuum domains aredefined as follows:

air : max{min{φi,φo},φr}< 0,polymer : min{φi,φo}> 0,

stretch rod : φr > 0.

φi < 0

airφi > 0

φo > 0φo < 0

air

stretchrod φ

r=0

polymer

φr>0 φ r < 0

φi = 0

φo = 0

Fig. 2. SIGN OF THE LEVEL SET FUNCTIONS

One of the difficulties encountered in level set methods is maintaining the desired shape of thelevel set function. The flow velocity does not preserve the signed distance property, but may insteadconsiderably distort and stretch the shape of the function, which eventually leads to additional numericaldifficulties [20, 41]. To avoid this the evolution of the level set function is stopped at a certain point intime to rebuild the signed distance function. This process is referred to as re-initialisation. In this papera triangulated fast marching method is used [20, 42]. Fast marching methods build the signed distancefunction outward starting from a narrow band around the zero level set and subsequently marching alongthe nodes. Further details about the fast marching algorithm used can be found in [13, 43, 44].

DISCRETISATION METHODA Galerkin Finite Element Method is employed for the discretisation of the governing equations.

The 2D axial-symmetrical finite element model has been implemented in COMSOL 3.5 with MATLAB.Each time step the system of equations is solved using COMSOL 3.5, while the re-initialisation of thelevel set functions is done in MATLAB between successive time steps. The level set problems are solvedusing the convection-diffusion application mode in COMSOL 3.5. First order elements have been usedfor the spatial discretisation: Mini-elements for the flow problem and Lagrange elements for the otherproblems [45–47]. For the temporal discretisation an IDA scheme has been used [48], which employs a

variable-order variable-step-size BDF method. The discretised system of equations is solved using theBiCGstab method with a geometric multigrid method as a preconditioner [49].

PET BOTTLE SIMULATIONIn this section the stretch blow moulding process of a PET (polyethylene terephtalate) bottle is

considered. First the governing equations are simplified based on a quantitative analysis of the process.Then the simplified model is applied for the simulation of the stretch blow moulding process. Finally,the results of the simulations are discussed. Table 1 depicts typical values for the simulation, which arealso used for the analysis.

Quantity Symbol [Unit] Value

PET Air

Characteristic length Lc [m] 1.00 ·10−2

Characteristic velocity Vc [m s−1] 1.00 ·10−1

Viscosity µ [kg m−1s−1] 1.00 ·105 1.00 ·10−5

Density ρ [kg m−3] 1.20 ·103 1.00

Thermal conductivity λ [W m−1K−1] 0.24 1.00 ·10−2

Specific heat cp [J kg K−1] 1.05 ·103 1.00 ·103

Inlet pressure pin [Pa] 1.0 ·106

Polymer temperature Tp [K] 3.73 ·102 2.98 ·102

Mould temperature Tm [K] 2.83 ·102

Rod temperature Tr [K] 3.13 ·102

Reynolds number Re [-] 1.20 ·10−5 1.00 ·102

Froude number Fr [-] 1.02 ·10−1

Péclet number Pe [-] 5.25 ·104 1.00 ·102

Brinkman number Br [-] 4.63 ·102 6.67 ·10−7

Table 1. TYPICAL VALUES FOR STRETCH BLOW MOULDING A PET BOTTLE

Quantitative analysis of stretch blow modelFirst a quantitative analysis of the rheological behaviour of the PET melt is performed. The Deborah

number De measures the ratio of elastic forces to viscous forces. For stretch blow moulding of PET theDeborah number is given by

De = τpin

µ∼ 1, (28)

where τ = 0.1 is the typical relaxation time of PET [3]. Thus, the elastic forces are of the same orderof magnitude as the viscous forces during the stretch blow process. The typical yield strength of PET at

T = 373K is σy = 10 MPa [50, 51]. For comparison the typical stress during stretch blow moulding is

σc = µcVc

Lc= 1 MPa. (29)

Thus, the isotropic yield criterium σc ≤ σy is barely satisfied, that is, the typical stress for stretch blowmoulding is close to the elastic limit. It can be concluded that the rheological behaviour is elasto-visco-plastic. On the other hand, the choice of a viscoplastic material model is not unreasonable, particularlyin view of modelling simplifications by disregarding the elastic effects.

Consider the dimensionless form of the flow equations (15-16). Obviously, the orders of magnitudeof the inertia forces (Re ∼ 10−5) and the gravitational forces (Re

Fr ∼ 10−4) are small for PET. Thisindicates that the flow problem for PET is dominated by viscous forces only. However, this is clearlynot the case for air. In order to simplify the governing equations in the whole domain, air is replaced bya fictitious fluid with the same physical properties as air, but with a much higher viscosity, e.g. µair = 1.Then the Reynolds number of the fictitious fluid (Re∼ 10−3) is small enough to reasonably neglect theinfluence of the inertia and gravitational forces. On the other hand, the viscosity of the fictitious fluidis still much smaller than the viscosity of glass, so that the pressure drop in the air domain is negligiblecompared to the pressure drop in the glass domain [13, 52, 53]. As a result the momentum equation forboth PET and fictitious fluid can be described by the Stokes flow equations.

Next consider the dimensionless form of the heat equation (22). The Péclet number is large for bothPET and air, which indicates that the contribution of conduction to the heat transfer is much smaller thanthe contribution of convection. The Brinkman number is moderately large for PET, but still a few ordersof magnitude smaller than the Péclet number. The Brinkman number of the fictitious fluid (Br∼ 10−2)is considerably larger than the Brinkman number of air, but still small compared to the Péclet number.It can be concluded that also the influence of dissipation on the heat transfer is marginal compared toconvection. Moreover, the Graetz number Gz is large (Gz = 1

4πρVcLccp/λ ∼ 8.6 · 103), so that alsothermal conduction towards the wall is negligible. Thus the heat transfer is convection dominated,hence the temperature is preserved along streamlines. This means that if the initial temperature of thefictitious fluid and PET is uniform, it will remain uniform. Based on the assumption that the temperaturedistribution of PET is uniform, the heat equation is omitted in the simulation and the coefficient k in (5)is treated as a constant.

In conclusion, the system of governing equations for the stretch blow moulding process of the PETbottle are (∗ is omitted):

∇·(2µε)= ∇p

∇·u = 0

dφi

dt= 0

dφo

dt= 0,

(30)

The governing equations hold for the fictitious fluid, PET and the stretch rod. For the stretch rod anartificial viscosity µr is imposed, which is much larger than the viscosity of PET, so that the strain ratein the rod is negligible. For the simulations the value µr = 1012 has been used.

ResultsThe stretch blow moulding of a 2D axial-symmetrical PET bottle is simulated. The height of the

bottle is 21.6 cm, the radius of the bottle is 1.5 cm at the neck to 3.15 cm at the widest part. Figure 3shows a typical mesh for the stretch blow moulding simulation.

21

.6cm

1.5cm

3.15cm

21

.6cm

Fig. 3. TYPICAL MESH FOR PET BOTTLE

Figure 4 shows the preform and initial position of the rod. The stretch rod is initially positioned at aheight h0 of 9.72 cm from the bottle mouth. The radius r0 of the stretch rod is 0.375 cm. At t = t0 therod starts moving from this position with constant speed Vr. The level set function for the stretch rod interms of 2D axial-symmetrical coordinates (r, z) is

φr(r, z, t) = r0−{

r, if z(z, t)≥ 0,√r2 + z2(z, t), if z(z, t)< 0,

(31)

where

z(z, t) = z+Vr min{t− t0, tstretch− t0}−h0. (32)

A disadvantage of modelling the motion of the stretch rod by a level set function is that automatically ano-slip condition is imposed on the stretch rod boundary, which is not always realistic or desirable. Oneway to allow some slip is to define a gradual transition of the viscosity over the stretch rod boundary.The stretch time tstretch can either be given manually or is determined by the simulation model. In thelatter case the stretching is stopped when the polymer hits the bottom of the mould. This method is usedfor the simulations.

In [5, 54] the material parameters m = 0.54, k = 292529 are given for the uniaxial tensile stress test(5). Figure 5 shows the viscosity-strain rate relation given by (9).

0.25cm

9.7

2cm

PET

stretch rod

PET

air

Fig. 4. PREFORM AND INITIAL POSITION OF THE ROD FOR THE

SIMULATION

10−2

10−1

100

101

10210

4

105

106

µ[kgm

−1s−

1]

˙ε [s−1]

Fig. 5. NON-NEWTONIAN VISCOSITY FOR UNIAXIAL TENSILE STRESS TEST

Figure 6 shows results of the stretch blow moulding simulations for a PET bottle at different timesfor inlet pressure pin = 1 MPa. First only air is blown inside without stretching. Then at t = 0.1 sthe stretch rod starts moving with speed Vr = 0.45m s−1. The white arrows depict the flow velocityvectors. As expectable in stretch blow moulding, the polymer form first bulges at the top, while thebottom is stretched, then the bulging gradually extends to the bottom. The bottle in Fig. (6(f)) has analmost uniform thickness, with a minimum of approximately 1 mm at the bottom part and a maximumof slightly more than 3 mm at the neck part. This matches with the thickness profile of a realistic waterbottle, which has an almost uniform thickness of 1.5 mm.

In order to assess the accuracy of the simulation model presented in this paper, the glass volumeconservation is verified. Figure 7 shows the percentage volume change in time. The volume change hasa maximum of 2.5%. The volume conservation can be further improved using smaller time steps andhigher mesh quality. This phenomenon was also observed in [13, 14, 43, 55].

(a) t = 0.1S (b) t = 0.2S (c) t = 0.3S

(d) t = 0.4S (e) t = 0.5S (f) t = 0.62S

Fig. 6. STRETCH BLOW MOULDING PROCESS FOR PET BOTTLE

WITH pin = 1 MPa AND Vr = 0.45m s−1 AT DIFFERENT TIMES.

CONCLUSIONSThis paper presents the development of a computer simulation model for the stretch blow moulding

process of polymer containers. The model is based on finite element methods and uses a level set methodto track the interfaces between polymer and air. The rheological behaviour of the polymer is describedby a viscoplastic model.

The model was used to simulate the stretch blow moulding of a 2D axial-symmetrical PET bottle. Itwas verified that the polymer flow can be described by the non-newtonian, isothermal Stokes equations.The results showed that the model is robust and gives a good representation of the stretch blow mouldingprocess. The model was validated by verifying the volume conservation. The method gave a volume

0 0.1 0.2 0.3 0.4 0.5 0.6

−2

−1.5

−1

−0.5

0

0.5

1

% g

lass

vol

ume

chan

ge

time [s]

Fig. 7. Volume conservation of the stretch blow moulding process for the PET bottle using the IDA time discretisation scheme with time step

5 ·10−4.

change of less than 2.5% for the simulations of the PET bottle stretch blow moulding. This givesconfidence that the model can be used for industrial practice. Optimisation of the bottle thickness will bea next step in the development. Comparison between the simulations and experiments or a viscoelasticmodel would also be a future consideration.

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PREVIOUS PUBLICATIONS IN THIS SERIES:

Number Author(s) Title Month

11-23 11-24 11-25 11-26 11-27

B.V. Matioc G. Prokert L.M.J. Florack H.C. van Assen F. Lippoth G. Prokert M. Rudnaya R.M.M. Mattheij J.M.L. Maubach H.G. ter Morsche J.A.W.M. Groot C.G. Giannopapa R.M.M. Mattheij

Hele-Shaw flow in thin threads: A rigorous limit result Multiplicative calculus in biomedical image analysis Classical solutions for a one phase osmosis model Gradient-based sharpness function Modelling stretch blow moulding of polymer containers using level set methods

Apr. ‘11 Apr. ‘11 Apr. ‘11 Apr. ‘11 Apr. ‘11

Ontwerp: de Tantes,

Tobias Baanders, CWI