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DRAFTProceedings of the ASME 2012 11 th Biennial Conference On Engineering Systems Design And Analysis

ESDA2012July 2-4, 2012, Nantes, France

ESDA2012-82725

MODELLING THE HEAT DURING THE INJECTION STRETCH BLOWINGMOULDING: INFRARED HEATING AND BLOWING MODELLING

Yun Mei Luo , Luc Chevalier and Franoise Utheza

Universit Paris-Est, Laboratoire Modlisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 bdDescartes, 77545 Marne-la-Valle, France

KEYWORDS

Initial heating conditions, thermal properties, thermo-visco-hyperelastic model

ABSTRACT

Effects of temperature: initial heating conditions or self

heating during the process, are very important during theinjection stretch blow moulding (ISBM) process of PET bottles.The mechanical characteristics of the final products, which aremainly controlled by the final thickness and the orientation ofthe molecular chains, depend strongly on the processtemperature. Modelling the heat transfer during the ISBMprocess is therefore necessary. In the first part of this paper, anexperimental study is presented in order to measure the initialtemperature distribution and to identify the thermal propertiesof the PET. An infrared camera has been used to determine thesurface temperature distribution of the PET sheets which areheated by infrared (IR) lamps. The Monte Carlo method is usedto identify the parameters best fit from the temperature

evolution. In the second part, a thermo-viscohyperelastic modelis used to predict the PET behaviour, taking into account thestrain rate and temperature dependence. A finite elementapproach implemented in matlab is used to achieve thenumerical simulation.

INTRODUCTION

The effects of temperature, initial heating conditions or selfheating during the process, are of fundamental importanceduring the injection stretch blow moulding (ISBM) process ofPET bottles. They control the preform temperature distribution

and strongly affect the blowing kinematics via the importantinfluence of temperature on mechanical behaviour. Thetemperature affects the orientation induced by biaxialstretching, which in turn, affects mechanical properties of PET.Besides, the final thickness distribution is controlled by thetemperature distribution inside the preform. Therefore, thermalproperties are one of the most important variables in the ISBM.

In order to achieve accurate simulation of the ISBM

process, it is necessary to: (i) measure the initial temperaturedistribution of the preform when the blowing operation begins;(ii) follow the history of the temperature field and consequentlyto identify the thermal properties of the PET; (iii) to model thebehaviour law of PET coupled to the thermal laws.

In this work, we modelled the PET behaviour in ISBMstrain rate and temperature conditions by a thermo-viscohyperelastic model which has been inspired from Figieland Buckley 2009 [1] and already presented by Luo et al. 2011[2].

A procedure is proposed for the identification of the thermalparameters from experimental results of a test where PET sheetsare heated using infrared (IR) lamps. The Monte Carlo method

is used to provide the parameters best fit from the temperatureevolution measured on the face in front of the lamps and fromthe rear face. The beginning of the temperature evolution givesinformation on the IR flux and the final stage of the heatinggives information on the convection exchange. Differencebetween the front and the rear faces gives information on thePET conductivity.

Using a finite element approach implemented in Matlab, thecoupled thermo-visco-hyperelastic model has been used tosimulate the ISBM process. We implement an axi-symmetricversion of the visco-hyperelastic model coupled to temperaturefor the numerical simulation. The self heating effect is notnegligible and has an important effect on the viscohyperelastic

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Author manuscript, published in "ASME 2012 11th Biennial Conference On Engineering Systems Design And Analysis, Nantes :France (2012)"
http://hal.archives-ouvertes.fr/http://hal-univ-mlv.archives-ouvertes.fr/hal-00777725


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model. Therefore we carried out non-isothermal visco-hyperelastic simulations in order to compare with theexperimental data [4].

EXPERIMENTAL PROCEDURE

According to Figure 1, the experimental apparatus whichconsisted in measuring, by thermal imaging, a PET sheetheating by infrared lamps. An FLIR B250 infrared camera withthe wavelength range 7.5 13 m is used to evaluate thesurface temperature distribution. Contrary to the thermocoupleswhich are easy to use but gives a local measure of thetemperature, the thermal imaging allows the determination ofthe surface temperature. The surface dimension of the sheet is60mm125mm with the thickness 1mm.

We tested three different distances between the lamps andthe PET sheet: 6.5 cm, 13 cm and 19.5 cm. For a constantheating IR flux, we can notice that the temperature decreaseswhile this distance increases. This a logical result because theintensity of the radiation decreases in proportion to the inverseof the square of the distance [5] which can be confirmed byplotting the radiation flux vs the distance. Increasing thedistance, the air convection process also takes more importanceand in regard of radiation.

FIGURE 1. Experimental heating set-up

(a) (b)

FIGURE 2. (a) IR lamps with PET sheet; (b) the dimensionof the PET sheet

IDENTIFICATION OF THE THERMAL PROPERTIES

The software FLIR quick report is used to measureaccurately the temperature. The thickness of the PET sheet is1mm. So the PET may be considered like an opaque mediumwhich means that only the radiation emitted from the sheetsurface is intercepted by the cameras sensor [6,7]. The blackpaint used is assumed to be opaque but because we are not ableto quantify its emissivity identification will be managed fromthermocouple measures. Nevertheless, this temperature fieldvisualization shows that some temperature heterogeneitiesappear on the edges of the sheet surface. According to theseresults, we can assume that the temperature is homogeneous inthe plane of the sheet and only varies in the thickness direction.Therefore, the identification can be done form a onedimensional model.

The temperature balance equation in the 1D case can bewritten:

2

2 0 pT

C T k z

=

& (1)

where: the mass density, C p the specific heat capacity and kis the material's conductivity.

With the boundary conditions:

( ). s f k T n h T T = on the face in front of thelamps;

( ). r k T n h T T = on the rear face.where s is the heat flux resulting from IR lamps. Theconvective heat transfer coefficient h is different as the face infront of the lamps or the face in the back. T is the surroundingbulk temperature.

While the IR flux s estimation can be obtained at thebeginning of the temperature evolution, the convectionexchange h estimation can be rather got from the final stage ofthe heating. The PET conductivity k can be calculated from thedifference between the front and the rear faces.

The density is measured directly and the Monte Carlomethod is used to identify the other parameters to best fit theexperimental results. So the thermal proprieties are listed in thetable1. This identification shows that h f is almost useless inregard of the rear coefficient hr .

TABLE 1. THE VALUE OF THERMALPROPERTIES

Parameters Values (kg/m 3) 1345

C p (J /kg.K) 1880k (W/m.K) 0.102

h f (W/m2.K) 0

hr (W/m2.K) 28.59

s (W/m 2) 1460

Infrared camera

PET sheet

Infraredlamps

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Figure 3 represents the experimental temperature evolutionon the surface in front of the lamps T f and the one on the rearface T r . With the identified parameters, the curves obtained fit

the experimental results (points).

FIGURE 3. The experimental results (points) and thenumerical results (curves) with optimal thermal properties

THE VISCOHYPERELASTIC BEHAVIOUR OF PET

We proposed an incompressible visco-hyperelastic modelwith nonlinear forms for both elastic and viscous parts [2].

I q I p D N += 2

( )

( )==

vvv

ee

DT

G

,,2

2

& (2)

where is the Cauchy stress tensor, D v is the symmetric part ofthe viscous velocity gradient, the subscript ^ denotes thedeviatoric part of the tensor, N is the small value of theviscosity of the Newtonian branch of the Zener like model usedin order to solve the ill-conditioned problem. e is the elasticpart of the Eulerian strain measure defined by:

( ) I Bee = 21 (3)where Be is the elastic left Cauchy Green tensor. p is a Lagrangemultiplier associated to the global incompressibility condition,and q is the multiplier associated to the incompressibility of theelastic part.

==

==

=

0

,0

,1det

vv

e

DtraceV div

DtraceV div

B

v

v (4)

The assumption of an additive decomposition of elastic andviscous velocity gradient is adopted to describe the kinematicstructure of this model:

ve D D D += (5)

Combining equations, 1, 2 and 4 in the Oldroyd derivationof the elastic left Cauchy-Green tensor leads to:

0. =+ eee

B BG

t

B

(6)

G is the elastic shear modulus and is the viscosity. In order torepresent the strain hardening and strain rate effect andtemperature dependency, we choose two rheological functions

for elastic and viscous parts: G( e) and ( v, v & ,T). e is the

equivalent elastic strain andv

& is the equivalent viscous strainrate.

We consider a Hart-Smith like model to represent the elasticpart G( e):

( )( )210 3exp = I GG , ( )e Btrace I =1 (7)For the non-linear viscous part of the model, we follow the

same method as in Cosson and Chevalier [3] to representmacroscopically the strain hardening effect:

( ) ( ) ( )vv f T hT &.,0= (8)

with ( ) ( )( )( )am

a

ref vv f 1

1

+= &&&

where ref & is a reference strain rate that can be taken equal to 1s-1. The strain hardening effect is related to the h function whichincreases continuously with v . We detailed the identificationprocedure for the h function in [2]. The strain hardening effectis influenced by the temperature. Therefore, h is a function of T also:

( ) ( ) ( ) ( )( )( )( ) N vv

vv

T

K T T hT

lim

00

1

exp1.,

= (9)

We choose the Williams-Landel-Ferry (WLF) model for theevolution of 0(T):

( )ref

ref T T T C

T T C a+=

2

1ln (10)

where C 1 and C 2 are the WLF parameters, T ref =90 oC. We

propose the evolution of vlim(T) in the following way:

( )( )

+=

2

1lim_lim 1 BT

T T B ref ref vv (11)

whereC vref v o90lim_lim_

= .During the stretching, the self-heating effect can be

important in the specimen, especially for the high strain rate

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(32s -1 for example). The adiabatic heating is due to the viscousdissipation which generates an increase of temperature in thePET sheet. We can estimate this temperature rise, for the case

32s-1

at 90o

C, if assuming that the stretching is fast enough sothat all dissipation is converted in self-heating (no exchangewith ambient air). Consequently, the global temperature balanceequation, which is integrated over the volume and over the time,writes:

: pt t C Td dt Dd dt = & (12)The conduction term is neglected in this analytical solution

because the biaxial stretching process is considered fast enoughin regard of the time needed to propagate the temperaturethrough the thickness. The characteristic time for diffusion is

sk ect pd 572 == and the characteristic time for capacity is

shcet c 132== . The time for the process is about 2s. IfEq.12 is used to estimate temperature as a post processing ofthe isothermal stretching simulation,, we can estimate thetemperature has increased about 4 oC when the stretch ratioreaches 2.8,. If we take this temperature evolution into accountin our model of viscosity, the final value of the viscosityreduces of 25% (3.6 10 5 Pa.s VS 4.810 5 Pa.s) which showsthat the self heating has a important effect on the PETbehaviour. Therefore, a coupled themo-viscohyperelastiquesimulation of the biaxial elongation tests must be used to adjustthe parameters of the thermo viscohyperelastic model used torepresent the PET behaviour.

NUMERICAL IMPLEMENTATION OF THE MODEL

Considering the specific geometry of a PET perform, weimplement an axi-symmetric version of the visco-hyperelasticmodel coupled to temperature in order to simulate accuratelythe ISBM process. Figure 4 shows the boundary conditions andthe load applied to the circular sheet discretized by rectangularelements. The circular sheet is submitted to a radial velocity V on radius L and this case represents the equi-biaixal tensioncase: =rr . In order to compare the experimental results,the radius of the cylinder is 38 mm and the thickness is0.75 mm, which is representative of the PET specimen size(76mm76mm1.5mm) of the test.

The related elongation rate & can be given as thederivative of stretch ratio with respect to time t . So the strainrate is: &&= .

FIGURE 4. The cylinder under the radial velocity

The viscohyperelastic model is implemented in the Matlabenvironment using a finite element approach. An axi- symmetricformulation involving 4 fields (global velocity V , Lagrangemultiplier p associated with the global incompressibilitycondition, and multiplier q associated with the incompressibilityof the elastic part, the elastic left Cauchy Green tensor B) hasbeen performed using rectangular elements. Some standardmanipulations of equations 2, 4 and 6 lead to the weak form inthe following 4 equations:

( ) 0:

2::

**

**

=+

+

F

dS F V d q p I D

d D Dd BG D

d e

ne (13a)

0:*

= d D I p (13b)01det* = d Bq e (13c)

0::2

:2:

**

**

=+

+

d B BG

Bd D B B

d B Bd B B

eeeee

eeee

&

(13d)

where the superscript * denotes test quantities and F d denotesthe prescribed traction field over the boundary F where theloads are imposed.

This strongly nonlinear problem (finite elasticdisplacements, elastic left Cauchy Green tensor B, non constantshear modulus G and viscosity ), is solved using a classicalNewton-Raphson iterative procedure. A consistent linearizationis achieved with the Gteaux operator (see Appendix).

The domain is approximated by a set of 8-nodesisoparametric rectangular elements. In the case of the classicalincompressible problem (a mixed velocity and pressureformulation), the finite element methods are not stable, some ofthem showing spurious pressure oscillations if velocity andpressure spaces are not chosen carefully. To be stable, a mixedformulation must verify consistency. The well-known inf-supcondition or the LBB condition [5] guaranties the stability of a

finite element velocity - pressure calculation. For our nonclassical four-fields approach ( V , p , q and Be), we testeddifferent interpolations to choose the accurate solution. Wesolved our problem considering different shape functions andexamined the convergence in regrads of the element size. Theerror is calculated between numerical and analytical results ofstress in the isothermal equi-biaxial elongation case. We obtainthe best choice interpolation when velocity V and the elastic Be tensor are quadratic and for p and q linear.

The numerical simulation of the stretch-blow mouldingprocess with the proposed visco-hyperelastic model consists insolving the set of thermo-mechanical equations on section of

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a preform. The mechanical part is written in Eq. 13 and thethermal part is:

( ) ( )

* *

* *

0

:

0q

pC T Td k T Td

T D d h T T T dS

T T at t

+

=

= =

&

(14)

The conductivity term is introduced in this numericalsolution.

SIMULATION OF THE THERMO-MECHANICALBIAXIAL PROBLEM

As the self heating effect is not neglectible and produces animportant effect on the viscous part of the model, the firstestimation of the parameters obtained from our isothermalidentification must be adjusted. An optimisation procedure inthe Matlab Optimization Toolbox was chosen: the fminuncfunction was used for this unconstrained nonlinear optimization.This function chooses the large-scale algorithm which is asubspace trust region method and is based on the interior-reflective Newton method described in [8,9]. Each iterationinvolves the approximate solution of a large linear system usingthe preconditioned conjugate gradients method. Almost ahundred simulations must be achieved to obtain the adjustmentwanted. Finally, the characteristics of the PET for these visco-hyperelastic model expressions to represent conveniently thebiaxial experimental tension tests [4] are listed in the table 2.

TABLE 2. THE CHARACTERISTICS OF THEPET

G0 8 MPaG( e) 0.001 9.91a 2( )v f & m 0.20 8.4 MPa.sK 3.2h0 -0.21

N 0.42

( )vh vlim_ref 1.83

C1 1.880(T) C2 25.81oC

B1 0.07

( v, v & , T)

vlim(T) B2 111.88oC

With the identified parameters, we can obtain the stress-strain curves from the thermo-visco-hyperelastic model. Thestresses and the temperature are slightly heterogeneous in thespecimen. For example, under a 8 s -1 strain rate, when thenominal strain rate reaches 1.8, the final radial stress field ( rr)

and the temperature field are shown on figure 4. One can seethat stress varies from 14.2 MPa to 14.6 MPa (about 2.8%) andthat temperature varies from 96 oC to 99 oC. The conductivity

term was took into account in the numerical simulation. There isabout 3 oCs difference in the temperature field.

(a)

(b)

FIGURE 4. (a) The rr stress field; (b) the temperature field at final stage

Figure 5 shows the comparison between the stresses

obtained from this thermo-mechanical simulation and from theexperimental data.We listed the absolute errors between the thermo-

mechanical simulation and the experimental data in the table 3.The mean difference does not exceed 10% for each deformationrate.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

20

Nominal Strain

T r u e

S t r e s s

( M

P a

)

Model__1/sExperimental__1/sModel__2/sExperimental__2/sModel__4/sExperimental__4/sModel__8/sExperimental__8/s

Model__16/sExperimental__16/s

FIGURE 5. The experimental data (dots) and the thermo-

mechanical results (lines)

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TABLE 3. ERRORS BETWEEN THEEXPERIMENTAL AND THE RESULTS OF THE

MODEL

Strain Rate (/s) Absolute Error (%)1 5.522 7.894 5.838 5.28

16 5.75

Evolutions of the temperature and the viscosity are shown infigure 6. We focus on the temperature in the midline of thespecimen thickness.

We can notice that the increasing evolution of thetemperature versus strain is nearly linear. Furthermore, the selfheating of the specimen for the strain rates from 1s -1 to 16s -1 aresimilar, but for the higher strain rate 32s -1, the temperatureincreases more. It confirms that at very high deformation rate,the adiabatic heating due to the viscous dissipation, generates asignificant temperature rises. The slope of the curve for thestrain rate 32 s -1 is more important than the one for the otherstrains rates, therefore the strain hardening effect is not soconsiderable at 32 s -1. We can see in the figure 6b, the viscosityincreases when the temperature rises.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.890

91

92

93

94

95

96

97

Nominal Strain

T e m p e r a

t u r e

( o C )

strain rate 1/sstrain rate 2/sstrain rate 4/sstrain rate 8/sstrain rate 16/s

strain rate 32/s

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-4

-3

-2

-1

0

1

2

3

4

Nominal Strain

l o g

( V i s c o s

i t y

)

strain rate 1/sstrain rate 2/sstrain rate 4/sstrain rate 8/sstrain rate 16/s

strain rate 32/s

(b)FIGURE 6. (a) The evolution of temperature under

different strain rates; (b) The variation of viscosity withtemperature under different strain rates.

If we examine the high strain rate case, for example, at 32s -1 for 90 oC, the coupled themo-visco-hyperelastique model

simulation gives the true stress rr evolution plotted on figure 7which have only 7.83% errors with the experimental data. Onecan see that the stress evolution measured during the experimentsaturates and seems to decrease when approaching the 1.8elongation. This may be explained by the significantly highertemperature level coming from the self heating of the specimenfor this test. Our model does not predict the decreasing shapebut gives a pretty good estimation on the evolution with nearlyno strain hardening effect for this case.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

14

16

18

20

Nominal Strain

T r u e

S t r e s s

( M P a

)

Model__32/sExperimental__32/s

FIGURE 7. The experimental data (the points) and thethermo-mechanical results at 32s -1 for 90 oC

CONCLUSIONS

Experiments have been conducted to characterise thethermal properties of the PET in the range of the ISBMtemperature. Thermals imaging have been used to determine thesurface temperature distribution of the PET sheets which areheated by infrared lamps. The Monte Carlo method is used toprovide the parameters best fit from the temperature evolution.

The coupled thermo-visco-hyperelastic model proposed has

been used to manage a finite element simulation of the equibiaxial elongation test. The weak form of the model has beenimplemented in Matlab. It shows that the self heating effect hasan important effect on the viscous part of the model and that theidentification of the parameters has to be adjusted by non-isothermal simulations.

The axi-symmetric version of the visco-hyperelastic modelcoupled to temperature is implemented and may be used inorder to simulate the ISBM process.

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ACKNOWLEDGEMENT

This work could not possible without lamps IR given bySidel Company and without injection of the PET sheets madeby the LIM at Arts et Mtiers ParisTech. Special thanks to G.Menary of QUB for his study on the experiments.

REFERENCES

[1] L. Figiel, C.P. Buckley. On the modeling of highly elasticflows of amorphous thermoplastic. International Journal ofNon-linear Mechanics, 44, 389-395, (2009).[2] Luo Y.M., Chevalier L., Monteiro E., Identification of aVisco-Elastic Model for PET Near Tg Based on Uni andBiaxial Results, The 14th International ESAFORM Conferenceon Material Forming, Queens University , Belfast, Irlande duNord, April 27-29, 2011.

[3] B. Cosson, L. Chevalier, J. Yvonnet. Optimization of thethickness of PET bottles during stretch blow molding by using amesh-free (numerical) method. International PolymerProcessing, Volume 24, Issue 3, Pages 223-233, (2009).[4] G.H. Menary, C.W. Tan, E.M.A. Harkin-Jones, C.G.Armstrong, P.J. MartinBiaxial Deformation of PET atConditions Applicable to the Stretch Blow Molding Process.Polymer Engineering & Science, in press.[5] Modest , M ., Radiative Heat Transfer, Mc Graw-Hill,. NewYork, 1993.[6] M. Bordival, Modlisation et optimisation numrique deltape de chauffage infrarouge pour la fabrication de bouteillesen PET par injection-soufflage, Thse de doctorat, 2009.(in

French)[7] S. Monteix, F. Schmidt, Y. L. Maoult, R.B. Yedder, R.W.Diraddo, D. Laroche. Experimental study and numericalsimulation of preform or sheet exposed to infrared radiativeheating. J. Materials Processing Technology 119(1-3):90-97,(2001).[8] F. Brezzi, M. Fortin, Mixed and hybrid Finite ElementMethods, Springer :New York, 1991.[9] Coleman, T.F. and Y. Li, "On the Convergence of ReflectiveNewton Methods for Large-Scale Nonlinear MinimizationSubject to Bounds," Mathematical Programming , Vol. 67,Number 2, pp. 189-224, 1994.

APPENDIXThe four-fields approach is adopted for the numericalimplementation. The weak form is written in the equation 13.since its a strongly non linear problem, an iterative Newton-Raphson scheme is used. A consistent linearization is achievedwith the help on the Gteaux operator:

{ }

{ }

{ }

{ }

{ } +

=

=

=

=

d G D B D

d BG D R D

pd I D R D

Pd I D R D

d D D R D

nonlin Be

enonlin B

p

P

nV

e

:

:

:

:

:2

*

*1

*1

*1

*1

(1a)

{ }

{ }{ }{ }=

==

=

0

0

0

:

2

2

2

*2

R D

R D

R D

d D I P R D

e B

p

P

V

(1b)

{ }{ }{ }

{ } ( ) =

===

d B I B p R D

R D

R D

R D

ee B

p

P

V

e:1det

0

0

0

*3

3

3

3

(1c)

{ }

{ }{ }

{ }

+

++

=

==

=

d G

D B B B

d B BG Bd B BG B

d B D Bd B Bd B B R D

R D

R D

d D B Bd B B R D

nonlin

nonlin Beee

eenonlin

nonlineeenonlin

nonline

eeeeee B

p

P

eeeeV

:

::

:2:2:

0

0

:2:2

*

**

***4

4

4

**4

&

(1d)Once the linearization has been done, we can implement it in

the finite element code. In this work, we use a self made finiteelement code developed in the Matlab environment. The systemto solve takes the following matrix form:

{ }[ ] { }[ ] { } { }{ }[ ] [ ] [ ] [ ]{ }[ ] [ ] [ ] [ ]{ }[ ] [ ] [ ] { }[ ]

[ ][ ][ ][ ]

[ ][ ][ ][ ]

=

4

3

2

1

44

3

2

1111

00

000

000

R

R

R

R

B

p

P

V

R D R D

R D

R D

R D R D R D R D

e BV

V

V

B pPV

e

e

(2)

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