1 INTRODUCTION

Among the techniques devoted to the manufacture of

PolyEthylene Terephtalate (PET) bottles, the two-

stage Stretch-Blow-Moulding (SBM) process is

probably the most popular. This process involves the

manufacture of structurally amorphous semi-

products, called preforms, made by injection

moulding of PET resin. A reheating step is necessary

to heat the preforms to the appropriate temperature

distribution above the glass transition, which is

typically around 80°C for PET. This stage is

generally performed using infrared (IR) heaters,

taking advantage of the semitransparent behaviour

of PET submitted to IR radiation. In a second stage,

the preforms are stretched using a cylindrical rod,

and blown using two levels of air pressure. Then, the

bottles are cooled down by a mould whose

temperature is regulated using cooling channels.

The heating conditions, that control the preform

temperature distribution, strongly affect the blowing

kinematics (stretching and inflation), and

consequently the thickness distribution of the bottle.

Temperature also affects the orientation induced by

biaxial stretching, which in turn, affects mechanical,

optical and barrier properties of bottles [1].

Regarding to BM, Lee and Soh [2] presented a FE

optimization method to determine the optimal

thickness profile of a preform, given the required

wall thickness distribution for the blow-moulded

part. More recently, Thibault et al. [3] proposed an

automatic optimization of the preform geometry

(initial shape and thickness) and operating

conditions, using the nonlinear constrained

algorithm Sequential Quadratic Programming

(SQP). The robustness of the method was discussed

through a comparison with experiments performed

within industrial conditions. SQP was also used in

order to optimize heating system parameters [4]. The

ABSTRACT: This study presents an optimization strategy developed for the stretch-blow moulding process.

The method is based on a coupling between the Nelder-Mead optimization algorithm, and Finite Element

(FE) simulations of the forming process developed using ABAQUS®. FE simulations were validated using in

situ tests and measurements performed on 18.5g – 50cl PET bottles. To achieve that, the boundary conditions

were carefully measured for both the infrared heating and the blowing stages. The temperature distribution of

the perform was predicted using a 3D finite-volume software, and then applied as an initial condition into FE

simulations. Additionally, a thermodynamic model was used to predict the air pressure applied inside the

preform, taking into account the relationship between the internal air pressure and the enclosed volume of the

preform, i.e. the fluid-structure interaction. It was shown that the model adequately predicts both the blowing

kinematics and the thickness distributions of the bottle. In a second step, this model was combined to an

optimization loop to automatically compute the best perform temperature distribution, providing a uniform

thickness for the bottle. Only the last part will be fully detailed in this paper.

Key words: Stretch-blow moulding, PET bottles, heat transfer, finite element method, optimization.

Optimisation of Preform Temperature Distribution For the Stretch-Blow

Moulding of PET Bottles

M. Bordival, Y. Le Maoult, F.M. Schmidt

CROMeP, Ecole des Mines Albi, Campus Jarlard, 81000 Albi, France URL: www.enstimac.fr e-mail: lemaoult@enstimac.fr; schmidt@enstimac.fr, bordival@enstimac.fr

objective was to homogenize the temperature along

the preform length, by modifying the process

parameters related to the IR oven. It is interesting to

point out that authors questioned the relevance of the

objective chosen for the optimization.

In this work, we propose a numerical optimization

strategy for SBM. For that, we developed an

iterative procedure allowing to automatically

compute the best temperature distribution along the

preform length, providing a uniform thickness for

the bottle. We solve the optimization problem by

coupling FE simulations to the Nelder-Mead

optimization algorithm (nonlinear simplex). Results

were validated by careful in situ tests and

measurements performed on 18.5g – 50cl PET

bottles. To achieve that, special attention was given

to the measurement of boundary conditions required

for both the infrared heating stage, and the blowing

stage.

2 OPTIMIZATION OF PREFORM

TEMPERATURE

The performance of a bottle manufactured by SBM

is drastically affected by its thickness distribution. In

order to achieve bottles with appropriated thickness

distributions, it is more desirable to adjust the

process conditions, and to use the same design of

preform for making different shapes of bottles. This

approach aims to minimize the cost associated with

the design of a new perform (especially the

manufacture of a new injection mould). Determining

adequate operating conditions remains nevertheless

costly and time consuming. Different approaches are

possible, such as trial-and error methods, or design

of experiments. Both of them require a large number

of experiments (or simulations), especially when the

parameters are strongly interdependent. As a

consequence, they become inadequate and

impracticable for complex problems. In contrast, the

optimization algorithms make the optimization

process fully automatic, and from this point of view,

yield a significant assist in the development cycle.

In this section, we propose to couple an optimization

algorithm to FE simulations in order to optimize the

temperature distribution along the preform length.

The goal will be to provide an homogeneous

thickness for the bottle. Infrared heating and blow

moulding numerical models have been fully detailed

in a previous paper [5]. A typical blow moulding

simulation is presented in figure 1 where we

compare the thickness distribution computed, and

the thickness profile measured. Measurements were

averaged on a set of three trials. We observe a good

agreement along most part of the bottle (less than 15

% error on the mean thickness).

Fig. 1. Wall thickness distribution of the bottle. The error bars

show ± 1 standard deviation for a set of 3 trials.

2.1 Parameterisation and constraints

In order to describe the temperature distribution

along the preform length, we consider three

optimization variables. They correspond to three

temperatures located at different heights of the

preform, as illustrated by figure 2.

Fig. 2. Temperature distribution along the preform length -

Optimization variables.

The whole temperature distribution is then deduced

using the Piecewise Cubic Hermite Interpolating

Polynomial (PCHIP) method [6]. To provide an

accurate interpolation, an additional temperature is

added on the preform neck. This fourth temperature

is not optimized, but fixed to 80°C, which

corresponds approximately to the glass transition of

PET. Indeed, throughout the reheating stage, the

preform neck is generally protected from IR

radiation in order to prevent its temperature from

exceeding the PET glass transition. This approach

aims to prevent any deformation of the bottle neck

during the forming process. Finally, to simplify the

problem, the temperature is assumed to be uniform

through the preform thickness. The optimization

variables are constrained using lower and upper

bounds, corresponding respectively to the PET glass

transition temperature, and to the PET crystallization

temperature. These two physical limits have been

naturally chosen to prevent serious strengthening of

the structure from appearing, in which case, any

deformation would be proscribed during the forming

stage. Let us note that neither linear nor nonlinear

constraint is required.

2.2 Objective function

In this application, we attempt to provide a uniform

thickness for the bottle. This objective must be

mathematically formulated by an appropriate cost-

function. A simple way to proceed is to define the

objective function F as the standard deviation of the

computed thicknesses, as following:

( )∑=

−−

=n

1i

2

i thth1n

1)x(F

r (1)

where xr

represents the set of optimization variables,

n is the number of nodes along the bottle height,

ith is the thickness at the node “ i ”, and th is the

mean thickness. The nodal thicknesses are computed

using a Python script that we have developed into

ABAQUS® FEM software. Such a function is null

for a bottle with perfectly uniform thickness.

2.3 Choice of an Algorithm

The choice of the optimization algorithm is closely

related to the type of cost function. In our

application, we attempt to minimize a nonlinear real-

valued function, subject to bound constraints. In

addition, strong mechanical and geometric

nonlinearities could induce significant numerical

instabilities, making the objective-function noisy,

and therefore non-differentiable. As a consequence,

the gradient-based algorithms might not be adapted

to this type of problem. In contrast, the direct search

methods (which do not require the computation of

the cost-function gradient) remain particularly

adapted to the non-derivative optimization. Among

this family of methods, the Nelder-Mead simplex

algorithm is probably one of the most popular.

However, this local method provides relatively slow

convergence rates [7]. Nevertheless, when the

derivatives can not be explicitly written, this method

can save a significant amount of computation time

compared to gradient-based methods. Indeed, the

computation of the cost-function gradients can

become strongly time consuming when they are

approximated using the finite-difference method.

This is particularly true when the number of

optimization variable is large.

On the other hand, the Nelder-Mead simplex

algorithm is restricted to unconstrained problems. In

this work, we used the method proposed by Luersen

et al. [8] in order to add bound-constraints into the

Nelder-Mead simplex algorithm available in

Matlab® .

2.4 Results and discussion

All numerical results reported in the sequel were

obtained on a 2.8 GHz-512 Mo Pentium 4. Figure 3

displays the decrease of the objective-function value in terms of the number of optimization iterations.

Fig. 3. Objective-function value versus iterations.

We observe that the objective function is reduced by

60% of its initial value after the first iteration, and

by more than 80% at the end of the optimization

process. Consequently, the thickness distribution of

the formed bottle is 80% more uniformed after

optimization. The algorithm converges after 5

iterations, which involves only 10 objective-function

evaluations (that is to say, 10 FE simulations).

On average, one cost-function evaluation requires 26

min CPU. Thus, the total CPU time required for the

optimization is approximately 3h20min. Figure 4

illustrates the temperature distribution along the

preform length before and after optimization.

Fig. 4. Initial and optimized temperature distributions

along the preform length.

Initial conditions were chosen in order to apply a

uniform temperature (100°C) on the preform. Such

temperature distribution leads to a strongly non-

uniform thickness distribution for the bottle, as

illustrated by Fig. 5.

Fig. 5. Thickness distribution of the bottle before

and after optimization.

After optimization, there is a temperature gradient

along the perform length, which provides a more

uniform thickness and a full blowing of the bottle.

Fig. 4 also illustrates the optimal temperature

distribution determined by Logoplast Company

using an experimental trial-and-error method. This

result has been obtained using the same preform, but

with a different shape of mould. However, we can

notice that there is a good agreement in the trends

between the temperature profile experimentally

determined within industrial conditions, and the

temperature distribution computed using our

optimization method.

3 CONCLUSION

For SBM optimization, we have proposed a practical

methodology to numerically optimize the

temperature distribution of a PET preform, in order

to provide a uniform thickness for the bottle.

Encouraging preliminary results have shown the

viability of our approach. However, it would

probably be more desirable to directly optimize the

process parameters of the heating systems. But to do

so, both the infrared-heating simulation and the

blowing simulation would need to be included into

the optimization loop, resulting in further

complications essentially due to long computation

times.

Nevertheless, this approach would implicitly

account for the influence of the temperature

distribution through the preform thickness, which is

of prime interest.

ACKNOWLEDGEMENTS

This study was conducted within the frame of 6th EEC

framework. STREP project APT_pack; NMP – PRIORITY 3.

www.apt-pack.com. Special thanks to Logoplast Technology

for manufacturing the preforms, and QUB for their

collaboration.

REFERENCES

1. G. Venkateswaran et al., Adv. Polym. Tech., 17, (1998).

2. D. K. Lee, S. K. Soh, Polym. Eng. Sci., 36, 11 (1996).

3. F. Thibault et al., Polym. Eng. Sci., 47, 3 (2007).

4. M. Bordival et al., Proc. of the Int. Conf. ESAFORM 9,

p. 511-514 (2006).

5. M. Bordival et al., Proc. of the Int. Conf NUMIFORM 7,

Porto, (2007)

6. F. N. Fritsch and R. E. Carlson, SIAM J. Numerical

Analysis, 17, (1980).

7. J. C. Lagarias et al., SIAM J. Optimization, 9, 1 (1998).

8. M. A. Luersen and R. Le Riche, Proc. of the Int. Conf.

on Eng. Comp. Tech., p 165-166 (2002).

1 INTRODUCTION

Thermoforming is a polymer processing technique

in which an extruded sheet is heated to its softening

temperature and subsequently formed to the required

shape by mechanical stretching and applying a

pressure. Mechanical stretching is carried out using

a plug, which could be manufactured from materials

such as wood, polymer or metal. The plug material

is usually chosen based on its low thermal

conductivity and low friction. The most common

materials are polymer. Polymer composite known as

syntactic foam has also been used. It is a composite

of polymer with hollow spheres provided by hollow

particles such as glass.

The wall thickness distribution of formed products is

often used as a gauge to determine the quality of the

thermoformed products. A more uniform wall

thickness distribution represents higher quality

product. The influence of various factors such as

plug temperature, speed, geometry and materials,

and sheet temperature and materials on the final wall

thickness distribution of thermoformed products

have been studied by various researchers [1-5].

Many recognised that the heat transfer between plug

and sheet material plays an important role in

thermoforming.

Fig. 1. Schematic showing the plug and sheet contact during a

thermoforming process

Figure 1 shows a schematic of contact between plug

and sheet during a thermoforming process. Heat

flows from the higher temperature sheet to the plug

at a lower temperature. This heat flow is affected by

the TCC between the plug and the sheet material.

ABSTRACT: Thermoforming is a polymer processing technique in which an extruded sheet is heated to its softening temperature and then deformed through the application of mechanical stretching and/or pressure into a final shape. The stretching operation is often performed by the movement of a mechanical plug, which contacts some areas of the sheet. During contact it is known that conductive heat transfer between the plug and sheet materials is an important factor in determining the process output. Attempts are currently being made to build realistic simulations of the thermoforming process and it is therefore extremely important that these heat transfer effects are included. However, the measurement of thermal contact conductance (TCC) between polymer pairs is extremely difficult in practice and there are no published values in literature. In this study, an axial conductive heat flow test rig has been developed and used to measure the TCC between contacting polymer pairs. Preliminary results between PVC and PTFE have shown that the value of thermal conductance is very small compared to published values for polymer/metal interfaces. Tests have also been carried out on Hytac®-B1X and PP. The TCC values were found to lie between 28.2-34.4 W/m

2-K for

average interface temperature between 45-75 °C. There was a slight increase in TCC with increasing interface temperature up to a temperature of about 70 °C. However, more experiments will be required to ascertain this. Further tests are being carried out to measure the TCC between different polymer pairs, and to assess the effects of variables such as surface roughness and contact pressure. These results will then be used to develop realistic models for contact heat transfer in thermoforming simulations.

Key words: thermoforming, thermal contact conductance, polymer, heat transfer

Measurement of Heat Transfer for Thermoforming Simulations

H.L. Choo1, P.J. Martin

1, E.M.A. Harkin-Jones

1

1School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Ashby Building, Stranmillis Road, Belfast, BT9 5AH, United Kingdom. URL: www.qub.ac.uk e-mail: hchoo01@qub.ac.uk

Besides conduction, there are also convection and

radiation heat transfer from the sheet to the

surrounding.

Temperature

Distance Interface

Higher T Lower T

TA TB

Fig. 2. Temperature drop across the interface of two similar

materials due to TCC

Thermal contact conductance, hc, can be defined as

the ratio of heat flux density to the temperature drop

across the interface (refer to figure 2):

BA

cTT

qh

−= [W/m

2-K] (1)

where q = heat flux density and (TA – TB) =

temperature drop across the interface.

Presently, there is no literature on TCC between two

polymeric materials. Therefore, the aim of this

research is to measure this value at various interface

temperature, pressure, and surface condition

primarily for use in thermoforming simulations.

2 LITERATURE REVIEW

There is no literature on TCC between polymeric

pairs to date. All of the studies on this subject were

carried out for metal/metal interfaces and

polymer/metal interfaces. Cooper et al. [6]

summarized and compared existing TCC models for

metallic pairs in a vacuum with some experimental

data. Profiles of mating surfaces and approximation

from deformation theory were used to determine the

parameters required for heat transfer. Mikic [7]

investigated the effect of mode of deformation on

TCC by deriving equations for 3 cases: pure plastic

deformation, pure elastic deformation and plastic

deformation of the asperities and elastic deformation

of the substrate. An excellent review on the

experimental technique to measure TCC was given

by Fletcher [8] along with some experimental work.

TCC in injection moulding between polymer and

mould cavity was first investigated by Yu et al. [9]

for cooling simulation. It was found that TCC is

dependent on the process conditions, material

properties, and part thickness and that TCC is

important in simulation of the cooling time. Bendada

et al. [10] also examined the TCC between polymer

and mould wall in injection moulding. An infrared

waveguide pyrometer and a two-thermocouple probe

were used to measure the surface temperature of the

polymer in the cavity and the heat flux across the

polymer-mould interface respectively. The same

approach was used by Bordival et al. [11] to

measure the TCC between polymer and mould wall

in stretch-blow moulding process.

Marotta and Fletcher [12] did a study on the TCC

between various polymeric materials and

aluminium. The authors examined the effect of

interface pressure on the TCC and found an increase

of TCC with increasing pressure. The authors also

conducted the tests at two temperatures, 20 and 40

°C, and found a general increase in TCC with

increasing temperature except for UHMW PE.

Experimental values were compared with 2 existing

models, Mikic and CMY models, and it was

concluded that the models did not correlate well.

Research on TCC between PS and mould steel was

carried out by Narh and Sridhar [13]. They

mentioned that the transition temperature, Tg, of

polymers is an important parameter in defining a

TCC model.

3 EXPERIMENTAL

3.1 Measurement technique

An axial heat flow heat transfer rig has been built to

measure the TCC between two polymeric materials.

It consists of two heat flux meters, a heater, a heat

sink, and plug and sheet materials to be tested

(figure 3). All specimens have diameter of 30 mm

and height of 38.1 mm. The heat source, heat sink,

and heat flux meters have diameter of 30 mm and

were made from Aluminium 2011 T3 with a thermal

conductivity, λ, of 151 W/m-K. Heat source was

provided by 3 cartridge heaters while cooling was

achieved using a copper coil wound around the

aluminium heat sink. Chilled water was circulated in

the copper coil. Temperature distribution in the

apparatus was measured by K-type thermocouples

placed at equal intervals along the length of the

apparatus. The whole rig was enclosed in a vacuum

chamber to minimise heat loss to improve the results

obtained. In addition, the specimens in the chamber

were insulated with glass wool and aluminium foil

(not shown) to minimise heat loss via convection

and radiation.

The temperature distribution in the flux meters and

test specimens was approximated with 1-D Fourier’s

equation,

dx

dTq λ−= (2)

where dT/dx = temperature gradient in the

specimens.

Fig. 3. Apparatus for measurement of TCC

3.2 Materials and methods

For preliminary testing, PTFE (plug material, λ =

0.25 W/m-K) and PVC (sheet material, λ = 0.15

W/m-K) was tested at a temperature of 90 °C. The

pressure at the interface was about 5 kPa, and is due

entirely to the weight of the sheet material specimen,

heat flux meter, and the heater. This test was carried

out to evaluate the workability of the rig and was not

carried out in the vacuum chamber in figure 3.

Another series of tests were carried out on Hytac®-

B1X (syntactic foam plug material, λ = 0.18 W/m-

K) and PP (sheet material, λ = 0.22 W/m-K) in the

vacuum chamber as shown in figure 3. Tests were

carried out at temperature between 80-180 °C. The

interface pressure was calculated to be around 1.5

MPa.

Linear least square fit was used to acquire the

temperature distribution in the specimens and heat

flux meters with the assumptions of 1-D heat flow.

TA and TB were obtained by extrapolating the

temperature distribution to the sheet-plug interface.

4 RESULTS AND DISCUSSION

4.1 Preliminary testing (PTFE and PVC)

Figure 4 shows the temperature distribution along

the length of the apparatus for PVC and PTFE. The

average PVC-PTFE interface temperature

( ( ) 2BA TT + ) was 42 °C and the TCC at the

interface was calculated to be 30 W/m2-K. There is

currently no literature to confirm the validity of this

value. However, comparison with the TCC value for

PVC-Aluminium interface (306.7-408.3 W/m2-K)

measured by Marotta and Fletcher [12] showed that

this value is 10 times lower. Nevertheless, this test

showed that the apparatus is capable of measuring

TCC for polymer pairs. The heat loss in the

apparatus was quite considerable in the preliminary

test. It was estimated that about half of the heat has

been lost before reaching the plug material.

Therefore, a vacuum chamber has been built to

reduce heat loss due to convection.

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100 120 140 160

Distance (mm)

Temperature (°C)

Fig. 4. Temperature distribution along the apparatus for PTFE

and PVC

4.2 Hytac®-B1X and PP

Figure 5 shows a typical temperature distribution

curve for Hytac-B1X and PP at heater temperature

of 90 °C. Figure 6 shows the graph of TCC at

average interface temperatures between 45-75 °C for

Hytac-B1X and PP. TCC values was found to lie

from 28.2-34.4 W/m2-K. There was a trend that the

TCC increased with increasing interface temperature

up to about 70°C, where the TCC dropped to about

28 W/m2-K. Similarly in this case, there was no

literature value to compare with. Marotta and

Fletcher [12] measured the TCC between PP and

Aluminium and found the value to lie between

276.7-324.0 W/m2-K. This value is again about 10

PVC

PTFE

Flux meter

Flux meter

128 mm

98 mm

38.1 mm

38.1 mm

times larger than the measured value here. Due to

the limited data points that are available, no models

have been fitted to the measured values.

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100 120 140 160

Distance (mm)

Temperature (°C)

Fig. 5. Temperature distribution along the apparatus for Hytac-

B1X and PP at heater temperature of 90 °C

0

5

10

15

20

25

30

35

40

40 45 50 55 60 65 70 75 80

Average Interface Temperature (°C)

TCC (W/m

2-K)

Fig. 6. TCC at various average interface temperature

Comparison between the TCC for PVC/PTFE and

PP/Hytac-B1X showed that the measured values did

not vary much between the two different material

pairs. This could be because the thermal

conductivities of these materials are quite similar.

However, more tests would be required to confirm

the behaviour of these materials.

5 CONCLUSIONS

Preliminary testing showed that the apparatus is

capable of measuring TCC between polymer pairs.

The vacuum chamber has successfully reduced heat

loss considerably in the apparatus. However, more

work has to be carried out to assess the accuracy and

repeatability of the results obtained. Measurements

thus far have shown that the TCC of PP/Hytac-B1X

pair lies between 28.2-34.4 W/m2-K.

Work is currently on the way to study the TCC

between different polymer pairs at thermoforming

temperature. Other factors such as surface

roughness, interface temperature, and interface

pressure on the TCC of polymer pairs will be

investigated in the near future.

ACKNOWLEDGEMENTS

This work is part of the PlugIn project funded by the 6th

framework programme of the European Union. The authors

would like to thank all the project partners for their help and

support.

REFERENCES

1. P. Collins, J.F. Lappin, E.M.A. Harkin-Jones and P.J.

Martin, ‘Effects of material properties and contact

conditions in modelling of plug assisted thermoforming’,

Plastics, Rubber and Composites, 29, (2000) 349-359

2. P. Collins, P. Martin, E. Harkin-Jones and D. Laroche,

‘Experimental investigation of slip in plug-assisted

thermoforming’, In: ANTEC 2001 Conference

Proceedings, (2001)

3. D. Laroche, P. Collins and P. Martin, ‘Modelling of the

effect of slip in plug-assisted thermoforming’, In:

ANTEC 2001 Conference Proceedings, (2001)

4. P. Collins, E.M.A. Harkin-Jones and P.J. Martin, ‘The

role of tool/sheet contact in plug-assisted

thermoforming’, Int. Polym. Process. 17, (2002) 361-

369

5. R. McCool, P.J. Martin and E. Harkin-Jones, ‘Process

modelling for control of product wall thickness in

thermoforming’, Plastics, Rubber and Composites, 35,

(2006) 340-347

6. M.G. Cooper, B.B. Mikic and M.M. Yovanovich,

‘Thermal contact conductance’, International Journal of

Heat and Mass Transfer, 12, (1969) 279-300

7. B.B. Mikic, ‘Thermal contact conductance: theoretical

considerations’, International Journal of Heat and Mass

Transfer, 17, (1974) 205-214

8. L.S. Fletcher, ‘Experimental techniques for thermal

contact resistance measurements’, In: Proceedings of the

3rd World Conference on Experimental Heat Transfer,

Fluid Mechanics and Thermodynamics, (1993) 195-206

9. C.J. Yu, J.E. Sunderland and C. Poli, ‘Thermal contact

resistance in injection molding’, Polymer Engineering and

Science, 30, (1990) 1599-1606

10. A. Bendada, A. Derdouri, M. Lamontagne and Y. Simard,

‘Analysis of thermal contact resistance between polymer

and mold in injection molding’, Appl. Therm. Eng. 24,

(2004) 2029-2040

11. M. Bordival, F.M. Schmidt, Y.L. Maoult and E. Coment,

‘Measurement of thermal contact resistance between the

mold and the polymer for the stretch-blow molding

process’, In: AIP Conference Proceedings, 1, (2007)

1245-1250

12. E.E. Marotta and L.S. Fletcher, ‘Thermal contact

conductance of selected polymeric materials’, Journal of

Thermophysics and Heat Transfer, 10, (1996) 334-342

13. K.A. Narh and L. Sridhar, ‘Measurement and modelling

of thermal contact resistance at a plastic metal interface’,

In: ANTEC 1997 Conference Proceedings, 2, (1997)

2273-2277

Flux meter

PP

Flux meter

Hytac-B1X

1 INTRODUCTION

At the tool-workpiece interface, on the high-speed machining, the mechanical energy is converted in heat into the shear zones. We are particularly interested by the secondary shear zone which is responsible of the raising of the tool temperature. This zones is located at the contact between the chip and the tool. The thermomecanical coupling is very strong here, since dissipations by friction are owing to the strain field at the interface, himself conditioned by the thermal level through the behaviour law of the material. Strains are not uniformly distributed along the chip - tool contact [1-3]. Hence at the tip tool, the normal strain attains a gigapascal and presents a null value at end contact, on the unsticking of the chip. This distribution involves that it exists a critical strain value from which the thermal contact becomes poor. In this zone, an imperfect contact have to be considered, what implicates the knowledge of the thermal contact resistance (regarded RTC) to describe the condition to the interior boundary. This condition is a no-homogeneous condition of third kind. Remind that this no-homogeneity appears as

the product of the generated heat flux (regarded ϕg) by the partition coefficient (regarded β) of the generated heat flux. To study those contact parameters, an original measurement principle is used. Some superficial parameters are estimated separately in each sub-domains (the workpiece and the tool) what are instrumented by thermocouples [4-6]. That prevents estimation difficulties induced by the strong correlation between β and ϕg in the interface equations. The estimations are done by means of inverse heat conduction technique and temperature records. Those estimated values allow to determine the thermal contact condition and to compare the results with the most useful hypothesis. The present study is compound by three sections. In the first section, the experimental device is presented; the employed machine reproduces the HSM conditions encounter at the tip tool. The machine, developed by the LAMIH of Valenciennes [7], does hot upsetting-sliding (HUS) tests which are friction tests. The second section is about the thermal instrumentation of each solids. The last section discuss of temperature experiments records and of the analysis of those results.

ABSTRACT: An hot upsetting-sliding device is used in order to reproduce thermal and mechanical conditions encounter in HSM at the tool tip. This device coupled with different estimation methods permit to determine the contact condition. This study presents an analyse of a friction test between a sample made in AISI 304L and a contactor made in AISI M2. The thermal instrumentation allow accurate estimations and the mechanical and thermal results are in good agreement. The common hypothesis employed in HSM models are validated, the thermal contact can be considered as perfect and the partition of the heat flux is in the thermal effusivity ratio of the contact materials.

Key words: Experimental, conduction, contact and friction

Estimation of thermal contact parameters at a worpiece-tool interface in a HSM process

E. Guillot1,*, B. Bourouga1, B. Garnier1, L. Dubar2

1 Laboratoire de Thermocinétique de Nantes, Ecole Polytechnique de l’université de Nantes Rue Christian Pauc - BP 50609 - 44306 Nantes Cedex 3 e-mail: emilien.guillot@univ-nantes.fr; brahim.bourouga@univ-nantes.fr; bertrand.garnier@univ-nantes.fr 2 Laboratoire d’Automatique, de Mécanique et d’Informatique industrielles et Humaines Le Mont Houy - 59313 Valenciennes Cedex 9 e-mail: laurent.dubar@univ-valenciennes.fr

2 EXPERIMENTAL PROTOCOL

The experimental device, used to characterise the contact condition between the chip and the tool at the tip of the latter, is a machine of hot upsetting-sliding. This put in touch the two pieces under high pressure and moves of one of them. The devices, presented in the figure 1, is compound of six elements which are: an oven to induction, a HUS machine, a contactor, a sample, a thermal recorder station and a mechanical recorder station The latter element is used by colleagues from LAMIH. The effort is coupled with the measure of the penetration and it permits to estimate parameters of the mechanical boundary condition between the contactor and the sample, this is-to says the strains and the friction coefficient at the interface [8].

Induction furnace Cross-piece

control

Frame Jack

Frame Power supply

Handling arm

Hot upsetting-sliding test

Hydraulic group

Fig. 1. Presentation of the HUTS device.

The tribometer is composed of a V support on which is put the hot sample heated in the induction furnace. This support is bounded to the hydraulic cross-street permitting so to displace the sample. The speed of the cross-street can attain a maximum speed of 0,4 m/s and it is imposed at 0,200 m/s in the experiments. The effort between the two solids is limited by the maximum value of stress measure sensor to 10 kN. The depth of penetration is about 0,1 mm and it is measured by means of a profilometer after each experiment. The contactor is fixed during the experiment. The schema of principle of the experiment of HUS is shown in the figure 2.

Sample

Contactor

Fig. 2. Scheme of principle of a HUST.

3 THERMAL INSTRUMENTATION

The shape of the contactor and the sample has been developed in order to reproduce conditions of a HSM contact and to permit mechanical and thermal studies during tests. So, the radius of the friction surface of the contactor has been obtained from 2D mechanical models in which the normal strain at the interface sample-contactor is similar to that of HSM proceeding. The width of the contactor and the sample is large enough to neglect, in their central plane, thermal losses with the environment. The contactor is instrumented by means of four varnish thermocouples of K-type of 80 µm diameter. They are planted in two groups of two at the beginning and the end of the friction area.

Thermocouples

Thermocouples

Fig. 3. Thermal instrumentation of the contactor.

Two K-type thermocouples of 100 µm diameter, sheathed with silk glass, are used in the sample. The thermocouples are located at 25 mm far from the leading edge. Hot junctions are welded without contributions in

the end of electroerosion drilled holes in the plane of symmetry of each solid. The distances between the interface and the hot junction are 0,9 mm and 1,9 mm.

4 RESULTS AND DISCUSS

The temperature recording of HUS test is shown in the figure 4. The two hotter records are provided by the sample thermocouples, the four another records correspond to the contactor thermocouples. The friction is more important at the beginning of the contact length then the two first contactor thermocouples rise a greater temperature than the two last thermocouples. The friction in a HUS test is intense because each thermocouple records increases of more than 100 °C in less of 0,5 s.

Fig. 4. Thermal records of HUS test.

The thermal superficial conditions (temperature and heat flux density) are estimated separately on either side of the contact interface by means of two different inverse conductive methods [7-8]. In the sample sub-domain, the estimates are presented on the figure 5 and figure 6.

Fig. 5. Heat flux estimation density on the sample side.

The heat flux density profile looks like a crenel, this maximum value is equal to 16 MW/m² and the crenel period is about 60 ms.

Fig. 6. Superficial temperature on the sample side.

The estimates of the superficial parameters in the contactor sub-domain are shown in the figure 7 and 8.

Fig. 7. Heat flux density estimation on the contactor side.

The estimates is noisier than on the sample side because the 2D inverse conduction technique is more complex. The heat flux density looks like steady from the time 0,17 s until 0,35 s at a value of 13 MW/m². The time need to stabilize the heat flux density is about 0,1 s that is matched at a length of 20 mm. So, the thermocouple located in the sample is in the thermal steady state area.

Fig. 8. Superficial temperature on the contactor side.

The maximum temperature on the contactor is really important and explain the wear phenomena observed. At this temperature, the oxidation and thermal diffusion wear are liable for a quick loss of matter and breaks. From the superficial parameters estimations, the

thermal contact condition is obtained. The generated heat flux, the partition coefficient and the thermal contact resistance have for value in this HUS test:

²/10.9,210.3,110.6,1 777 mWceg =+=+= ϕϕϕ

45,0==g

c

ϕϕβ

WCmRR

TT

x

TTC

TC

ecg

c /².10.7,6. 6

0

°=⇔−

=+∂

∂− −ϕβλ

The value of the thermal contact resistance is really small like it expects. The strain at the interface is huge and the flow strain is weak at high temperature, then the real contact surface is almost the geometric surface. The microconstrictions at the spot contacts are insignificant. The partition coefficient is compared with the partition ratio of heat flux between two solids suddenly put in touch without displacement. In other words, it is supposed that the heat flux generated by friction is divided, between the two solids, in the ratio of their thermal effusivities [4,10].

49,0=+

=ce

eeffusivity BB

Bβ

where iiii CpB ..ρλ=

This partition value is near of the experimental estimate. Therefore the displacement allows to use the effusivity ratio for the partition coefficient in the steady state friction area. The mechanical work due to friction is converted into heat in the ratio of Taylor-Quinney coefficient. The heat flux density generated at the interface can be calculated from mechanical estimates of the mechanical contact condition obtain by the LAMIH.

²/10.8,2... 7 mWVNTQg == σµβϕ

The coefficient of Taylor-Quinney (regarded βTQ) is supposed equal to 0,9 for alloy steel material, this value is not accurate but it permits to calculate a generated heat flux density with a good agreement with the experimental value.

5 CONCLUSIONS

Experiment of hot upsetting-sliding is realised between a sample made in AISI 304L and a contactor made in AISI M2. This test reproduces thermal and mechanical conditions obtained at the tip tool during a HSM proceeding. The contactor and the sample are instrumented by means of

thermocouples, the temperature records at some chosen locations and inverse conduction methods allow to estimate the thermal condition at the contact interface: the thermal contact resistance, the heat flux partition coefficient and the heat flux density generated at the interface. The estimates have check common hypothesis used in HSM. The thermal contact resistance can be neglect and the partition coefficient is near to the thermal effusivities ratio of the solids in contact. Moreover the experiment have a good agreement between the thermal and the mechanical analyses.

ACKNOWLEDGEMENTS

The authors would like to thanks the Fondation de France / CETIM for the financial support for this study.

REFERENCES

1. N. Zorev, Inter-relationship between shear processes occurring along tool face and shear plane in metal cutting, ASME International Research in Production Engineering (1963), 42-49.

2. L.C. Lee, X.D. Liu & K.Y. Lam, Determination of stress distribution on the tool rake face using a composite tool, International Journal of Tools & Manufacture 35 (1995), 373-382.

3. T. H. C. Childs, K. Maekawa, T. Obikawa & Y. Yamane, Metal machining - Theory and applications, Elsevier Publishing (2000).

4. T. Kato, H. Fujii, Energy partition in conventional surface surface grinding, Journal of Manufacturing Science and Engineering 121 (1999), 393-398.

5. W. Grzesik, P. Nieslony, A computational approach to evaluate temperature and heat partition in machining with multilayer coated tools, International Journal of Machine Tools & Manufacturing 43 (2003), 1311-1317.

6. M.C. Shaw, Metal cutting principles, Clarendon Press, Oxford (1984), 206-240.

7. E. Guillot, B. Bourouga, B. Garnier, and L. Dubar, Experimental Study Of Thermal Sliding Contact With Friction : Application To High Speed Machining Of Metallic Materials, Proc. ESAFORM congress (Zaragoza 2007).

8. E. Guillot, B. Bourouga, B. Garnier et J. Brocail, Measurement of the thermal contact parameters at a workpiece – tool interface in a HSM, ESAFORM congress (Lyon 2008).

9. J. Brocail, M. Watremez, L. Dubar & B. Bourouga, High Speed Machining: A New Approach To Friction Analysis At Tool-Chip Interface, ESAFORM congress (Zaragoza 2007).

10. P. Vernotte, Thermodynamique générale, Ministère de l’ air (1961).

Shrinkage kinetics and thermal behaviour of injection moulded polymers

C Nicolazo, P. Vachot, A. Sarda, R. Deterre

Operp - Nantes University - IUT Nantes BP 539 44475 Carquefou Cedex URL: www.univ-nantes.fr e-mail: remi.deterre@univ-nantes.fr

ABSTRACT: We measure the kinetics of shrinkage at the time of the cooling of injected polymers during a cycle of moulding. Dynamic measurement of the shrinkage of the castings after ejection is taken by the method of photomechanical in a thermally controlled environment representative of a workshop production one. The technique of photomechanical used allows measurement without contact of the shrinkage in the principal plan of the part. It is shown that the profile of temperature quickly becomes uniform in the thickness of the part after the phase of ejection and that the kinetics of shrinkage is controlled by the kinetics of cooling for the PS as for PP. One shows the influence of certain processing parameters on the evolution of the shrinkage, in particular, the dependence of the shrinkage according to the temperature of the mould and the packing pressure. Finally one notes the presence of an anisotropy of shrinkage in the case of PP.

Key words: injected polymers, cooling, kinetics of shrinkage, post-demoulding dimensional behaviour, photo mechanics

1 INTRODUCTION

Injection moulding process of polymers is very widespread and constitutes a stake of industrial importance. At the time of the process of moulding, polymer temperature and pressure evolutions influence the shrinkage and warpage of the castings [1]. The dimensional stability of the injection moulded parts strongly depends on the evolution of the process settings. We propose to measure the kinetics of shrinkage at the time of cooling of an amorphous polymer (PS) and a semi-crystalline polymer (PP) during a cycle of moulding and identify the influential parameters on its evolution (Table1). Table1. Polymers used for the mouldings

Polymer Producer Reference PS Total Petrochemicals Lacqrène 1541 PP Solvay HV 252

2 EXPERIMENTAL PROCEDURE

Dynamic measurement of the shrinkage of the castings after ejection is carried out by the method of photomechanical in a thermally controlled environment which is representative of a production workshop environment.

2.1 Device of measurement

The mould cavity used for our work is equipped with temperature and pressure sensors. A flying temperature sensor established in the depth of the part makes it possible to follow the change of the temperature of polymer from the phase of filling to the complete cooling of the part [2].

2.2 Technique of photomechanical

The technique of photomechanical used for the measurement of the shrinkage is based on the correlation between an image of reference and an

image of the deformed part. The treatment of the two images enables to detect the displacement of singular points (speckle) identified on the part [3]. The casting is inserted in an assembly (a) Fig. 1) which makes it possible to photograph it from the moment of ejection to complete cooling at the ambient temperature. The device permits the part (moulded part Fig. 1) to retract in its principal plan (x,y Fig. 1) by preserving a motionless reference position (point A Fig. 1) without undergoing friction thanks to the maintenance by rollers assembled on springs (roller Fig. 1) and guiding by slipping supports (support Fig. 1).

a) b)

Fig. 1. device of photomechanical measurement

The cooling conditions are controlled by the insertion of the device (a) Fig. 1) within a fluid vein equipped with a ventilator and a temperature regulation which generates a constant air flow at controlled temperature (b) Fig. 1) which can be connected at the ambient temperature of a workshop of production. The moulded polymers were compounded with a small quantity of powder of insoluble and infusible rubber (3% in mass) mixed with granulated of raw material before the operation of moulding. The presence of such an additive does not modify the process of moulding nor the polymer properties.

3 SHRINKAGE AND TEMPERATURE EVOLUTION

The polymers used are a PS (amorphous, transparent) and a PP (semi-crystalline, translucent). We traced on the same graph the change of the temperature and the dimensional evolution of the parts after ejection (Fig. 2).

Fig. 2. thermal and dimensional evolution of PS and PP

moulded parts

One notes that the shape of shrinkage evolution fits perfectly the temperature curve. Former work [4] showed that the profile of temperature in the thickness of the casting is flattened rather quickly after the phase of ejection : the signal of temperature of the flying probe within the casting joins rather quickly that of the optical pyrometer on the surface of the part (Fig. 3).

Fig. 3. change of the temperature on the surface and within the

moulded part

The measurement of shrinkage by photomechanical begins at the same time as the appearance of the signal from the optical pyrometer (a) Fig. 3). This measurement is mostly carried out on a part whose temperature is almost homogeneous and decreases gradually in the course of time. The total shrinkage of the part was measured using a micrometer, the measured value is in agreement with the values of the literature [1]. This value of shrinkage determines the asymptote position of the kinetics of shrinkage

measured by photomechanical (b) Fig. 3). It is noted that the shrinkage measured by photomechanical corresponds to half of the total shrinkage and that the extrapolation of the tendency curve of the shrinkage kinetics indicates the shrinkage starting location close to the moment where the pressure release in the mould cavity (c) Fig. 3).

4 INFLUENCE OF THE PROCESS SETTINGS ON THE SHRINKAGE

We varied the injection process parameters such as the temperature of the mould and the pressure of maintenance, according to the values indicated in Table2.

Table2. Process settings

Process settings Value Unit Injection temperature 240 °C Mould temperature 40 – 50 °C Packing pressure 6 - 18 - 30 – 42 MPa

4.1 Influence of the mould temperature

We carried out series of mouldings while varying the temperature of the mould of 40 °C with 50°C the other process parameters being fixed. The results of measurements are deferred on the Fig. 4.

Fig. 4. evolution of the shrinkage according to the mould

temperature

The difference of 10°C on the temperature of the mould involves an identical temperature difference on the average temperature of the part at the ejection moment. It follows a shift between the curves of shrinkage proportional to the initial variation in temperature. This confirms the strong dependence of the shrinkage with the temperature noted previously.

4.2 Influence packing pressure

We traced the kinetics of shrinkage for various values of the pressure of maintenance (Fig. 5).

Fig. 5. evolution of the shrinkage according to the packing

pressure

One observes the reduction in the shrinkage with the increase in the packing pressure. This result is in conformity with many former results [1]. However, the pressure of maintenance does not seem to modify the kinetics of the shrinkage.

5 ANISOTROPY OF PP SHRINKAGE

The gate of injected parts is located on the point A of the Fig. 6

Fig. 6. gate location of injected parts

During the cooling of polypropylene parts one notes often, the appearance of a difference in shrinkage between x direction and the y one (Fig. 7).

Fig. 7. anisotropy of the shrinkage of PP

This anisotropy was already mentioned by various authors [5]. We noted that this anisotropy varies according to process parameters (temperature of the mould and packing pressure).

6 CONCLUSIONS

We developed a device devoted to the shrinkage measurement of injection moulded parts. This measurement without contact uses the technique of photomechanical under controlled cooling conditions which are representative of industrial conditions. We took measurements on injection moulded parts of PS and PP equipped with temperature sensors in-situ. We noted that the

kinetics of shrinkage was dictated by the change of the temperature of the part. We showed the influence of the conditions of moulding on the kinetics of shrinkage. In the case of PP, we confirm the presence of a variable anisotropic shrinkage according to process parameters.

REFERENCES

1. H. G. Potsch ’Prozessimulation zur abschatzung von schwindung und verzug thermoplastischer spritzgussteile’, Doktor-Ingenieurs von der Fakultat fur Maschinenwesen der rheinish-westfalischen technischen hochschule, Aachen, 10 januar 1991.

2. Y. Farouq, C. Nicolazo, A. Sarda, R. Deterre, ‘Temperature measurements in the depth and at the surface of injected thermoplastic parts.’ Measurement 38 (2005) 1-14.

3. J. Réthoré, A. Gravouil, F. Morestin , A. Combescure, ‘Estimation of mixed-mode stress intensity factors using digital image correlation and an interaction integral’ International Journal of Fracture 132 (2005) 65–79.

4. Y. Farouq, C. Nicolazo, A. Sarda, R. Deterre ‘Analyse du comportement thermique et mécanique des pièces injectées lors de la phase de maintien et après éjection’ Congrès SFT 2006 Ile de Ré.

5. K.M.B. Jansen ‘Measurement and prediction of anisotropy in injection moulded PP products’, International Polymer Processing, 13 (1998) 309–317

.

1 INTRODUCTION

Our work is focussed on heat treatment process which is used to improve product mechanical properties (e.g. hardness, strength). These heat treatments may cause undesirable dimensional changes. Finishing operations are then needed; hence distortions control could help minimizing the rise in cost. Distortions prediction is however complex because depending on many parameters (thermo mechanical, metallurgical), which are, mostly, experimentally estimated. In this paper, we propose a distortions control strategy using metrological and simulation analyses. The first part reviews some literature methods to identify heat treatments distortions. Then, we introduce our method with data processing coming from measurements and simulations. The third part presents first results for an ASCOMETAL steel grade.

2 DISTORTIONS IDENTIFICATION METHOD

2.1 Existing methods

Within the framework of heat treatments, two main and complementary approaches are found in the literature.

2.1.a Deductive approach It consists in checking distortions predictive models via experimentations and simulations. A knowledge based system could be used [1]. Distortions are quantified on parts families for which are associated potential distortions (e.g. banana effect) and distortions generating factors such as part geometry, metallurgical properties, place in the furnace, cooling fluid characteristics. One can also find other foretelling models using these distortions factors but individually considered; as the part geometry [2] for which a correlation method was developed. However, finite element models (f.e.m.) are the most used because they help us to understand distortions origin through metallurgical and thermo mechanical phenomena analyses. Like all numerical models, the results accuracy greatly depends on the quality of the input data. In addition to simulation based approach, it is thus necessary to perform experiments to corroborate preliminary results. In that way, distortions simulation accuracy was improved on carburizing-quenching gear by experimentally determining materials thermal

ABSTRACT: Heat treatments could create local or global distortions on workpieces. Finishing operations, often costly, are then necessary to respect the required functional tolerances. In the long term, our objective is to optimize first, steel grade and heat treatment, then to adjust the numerical simulation models. In that way, the heat treatment distortions on C-ring test parts obtained for an ASCOMETAL steel grade, vertically gas quenched are qualified and quantified by a dimensional analysis. In this article, we focus on part measurement and data processing strategies. Then we present an approach to correlate the experimental results with simulations ones.

Key words: Distortions, Heat treatments, Metrology, Dimensional analysis, Numerical simulation

Dimensional control strategy and products distortions identification

C. Nicolas1, C. Baudouin1, S. Leleu2, M. Teodorescu3, R. Bigot1

1 Laboratoire de Génie Industriel et Production Mécanique (LGIPM – IFAB) Ecole Nationale Supérieure d’Arts et Métiers - C.E.R de Metz - 4, rue Augustin Fresnel - 57078 Metz Cedex3 http://www.metz.ensam.fr/ cyril.nicolas, cyrille.baudouin, regis.bigot@metz.ensam.fr; 2 Laboratoire de Métrologie et Mathématiques Appliquées (L2MA) Ecole Nationale Supérieure d’Arts et Métiers - C.E.R de Lille - 8, boulevard Louis XIV - 59046 Lille Cedex http://www.lille.ensam.fr/ stephane.leleu@lille.ensam.fr; 3ASCOMETAL CREAS - BP 70045 - 57301 - Hagondange Cedex m.teodorescu@ascometal.lucchini.com

properties [3].

2.1.b Inductive approach It consists in proposing models to identify distortions coming from experiment, mainly from metrology. So, we need to define the number and distribution of points so that significant surfaces for the optimization criterion would be measured with the smallest possible loss of information [4]. The number of points is subject to part size, tolerance specifications and measurement uncertainty. The theoretical smallest sampling size can be used for ideal geometric shape identification. But, if the form deviation is unknown (that is our case), we have to increase the sampling size so that estimated value of the measurement converges to the “true” value of form error [5]. As for points distribution, for not creating local weighting disturbing the optimization method, it is important to dispose them uniformly. Next, we need an optimization criterion to resolve this “data-overabundant” system. In our case of warped surfaces identification, a point-per-point comparison between measured and theoretical geometry is needed [6]. We chose the least squares criterion because it gives a very stable result and is less sensitive to asperities effects [7].

2.2 Developed method

2.2.a Introducing C-ring test part The C-ring type sample is currently used because its geometry allows us to amplify distortion which is suitable for the metrological analysis [8] [9]. We use a 100 mm long C-ring with a 16 mm wide opening. Outer and inner cylinders (respectively Ø70 and Ø45 mm) are 11 mm off-centered. These dimensions authorize thermocouples without disturbing thermal flow; they minimize side effects and allow obtaining required cooling velocity [10]. First, a metrological analysis occurs at three manufacturing states: after machining, after stress relieving and after high pressure gas quench. Then, we present a method to correlate C-ring measured mesh with f.e.m. one in order to compare simulate distortions field with measurement’s one.

2.2.b Experiment approach

Measurement strategy In order to characterise the distortions significant types, we define a fine mesh for the C-ring. We keep in mind that many measurement points may increase the appearance of abnormal points but we could detect them via a graphic interface (see section 3).

Points are evenly distributed on all surfaces (figure 1), but not on the putting plane (bottom one).

Upper plane

Outer cylinder

Inner cylinder

Bottom plane

Right line

Left line38 points

110 points

21 points

21 points

1722 points

1722 points

Upper plane

Outer cylinder

Inner cylinder

Bottom plane

Right line

Left line38 points

110 points

21 points

21 points

1722 points

1722 points

Fig. 1.Fine mesh for points probing

Data processing strategy Our mathematic approach is to optimize, for each point i, the measure error εi between theoretical point Ti and measured point Mi, orthogonally projected onto theoretical normal Ni (equation (1)).

( )( )

1

.:

j,i,framereferenceIn

1

=ΝΝΟΜ−ΟΤ=

=Ο

i

iiii

n

i

withε

ε

ε

ε

εM

Mrrr (1)

We also define the matrix Mph of optimization phenomena (equation (2)). Its scalars correspond to the phenomena elementary amplitude effect (ePhnj) on the theoretical points written in column. The description unit of phenomena would be close to their estimated value, so the millimetre.

yy

xx

mmyxyx

jj

jj

n

jph

nn

jj

..e

..e

1 with

eee

eee

eeePh

jy

jx

nn

jn

1n11

T

T

PhTyTx

PhTyTx

PhTyTx1

n

Ν=ΝΤ=

Ν=ΝΤ=

=Τ=Τ

ΟΤ

ΟΤ

ΟΤΤΤ

=Μ

KMKMMM

KMKMMM

K

K

(2)

To dissociate phenomena, those must be mathematically independent and linearly superimposable. In metrology, the size order of the defects is often small compared with nominal dimensions of the measured part and so, we will use the small displacements assumption. Finally, equation (3) gives the residual r to minimize using the least square criteria thanks to any solver.

)nPh ..., ,iPh (phenomena sdistortion and ntsdisplacemet significan of n) ..., c, b, (a, quantities elementary with

Ph....Ph.2Ph.1Ph. nnicbar −−−−−= ε (3)

Experimental device Our coordinate measuring machine (CMM) has an indication error for length measurement defined by E=(3.5+L/350)µm, L in mm.

C-ringExtension bar

Star stylus

TP20 probe

Positioning device

Fig. 2. Experimental control device

We use a dynamic probe (TP20) with a star stylus and an extension bar (figure 2). Combined with probe head rotations, we can access all surfaces. But, even with rotations, we do not obtain a uniform distribution of points for the inner cylinder. We thus checked the accuracy of our device compared with a single TP20 probe (without extension bar and star stylus) in measuring a 44.99mm standard ring (table1). To succeed, we optimize the 108 points of CMM with Tx-Ty and dilatation vectors. Table1. CMM measurements of a standard ring diameter Values in mm Single TP20 Our TP20 device Diameter 44.989 44.989 Range 0.006 0.005

2.2.c Processing of finite element simulation We use Forge 2005 software to make 3D-quench simulations for which the input data are partly experimentally determined (the phase transformation kinetics, the heat transfer coefficient) and partly from the literature (the mechanical data) [12].

Data processing strategy For a direct comparison between experimental and simulation results, we use the method previously presented. For instance, this method is applied for only the cylinders. By using “2D-cutting planes”, we export sections’ boundary geometry at the same heights as for CMM measured C-ring. However, we must keep in mind that CMM measurement points are obtained by probing the surface following the theoretical normals. But during simulation, nodes move on all directions. So, we have to find the intersection point S12 between theoretical normals T1 and finite element frontier (S1S2).

Fig. 3. Intersection point S12 between (S1S2) and D lines

Uncertainty due to the method The method generates a quantifiable error εs due to the linear approximation between two consecutive nodes of the finite element frontier (figure 3). To minimise this error, we look for the 2 simulation points (S1 and S2) whose angular coordinates are as close as possible to theoretical point (T1) ones. In figure 4, we show that higher is the sampling size; higher is the accuracy. This last also increase when the diameter size is smaller for the same sampling size.

Evolution of diameter identification errorEvolution of diameter identification error

0,000

0,010

0,020

0,030

0,040

82 91 121 181 362 723

Number of points from a cylinder section

0,000

0,010

0,020

0,030

0,040

82 91 121 181 362 723

Number of points from a cylinder section

Err

or in

mm

bet

wee

n th

eore

tical

and

id

entif

yed

diam

eter

s Outer section

Inner section

Case 1: Simulation nodes are theoretical Case 2: Simulation nodes are theoretical points of outer and inner diameters points of dilated diameters (0.5mm)

Fig. 4. Uncertainty due to data processing of simulation points

Initial geometry for quench simulation We must take into account C-ring dilation due to heating (at 930 °C). We consider for the studied steel grade a linear dilatation value of 22.3 µm/m/°C in austenitic phase. Then, we create the same mesh as for experimental analysis, consisting of 98 sections with 400 nodes for each cylinder’s perimeter. This fine mesh aims to minimize the uncertainty of our method.

3 FIRST RESULTS

A software based on the described method was developed. Thus, numerical and 3D-identification of distortions phenomena can be calculated. For a better visual understanding, a scale factor is applied on the errors onto the theoretical normals (figure 5). Theses measure errors include displacements phenomena, like translations, rotations and off-centring of the inner diameter (due to machining). By deleting them, we did what is often called best

fit, to better visualize distortion and to write their 3D-mathematical expression (figure 6).

Measured p art grade 1, at 20°C

Scale factor: 50 Theoretical points Quenched points

Simulated p art grade 1, at 20°C

Pincers’ opening

Fig. 5. Graphic comparison of best-fitted quenched geometries

All phenomena have physical origins, as the pincers’ opening which comes from a gradient of thermal stresses induced by the thickness and temperature gradients and the steel phase transformations.

Phenomena Mathematical expressions Vectors effects on errors onto theoretical normals

Ovalization (also dilation)

For each section :

2

2

jy

jx

e

..e

joval

jjjoval

y

xx

Ν=Ν=ΝΝ=

x

y z

Theoretical points Scale factor = 5

Pincers’ opening (or

closing)

For each section :

−=

−=

).;.(Atan22

cos1ej

xOTyOT jjj

jopen

α

α

x y

z α -α

Theoretical points Scale factor = 5

Barrel (or bobbin)

For each generatrix :

[ ]

∈×+−=−=

20,0,1.011e 2

j

kkzz

j

jbarrel x

z

Theoretical points Scale factor = 5

Banana

×Ν−=<×Ν=≥

jj

jj

ee 0; yIfee 0; yIf

2ci

2ci

barrelbanana

barrelbanana

y

y

z

x xcez xci

yce

yci

Theoretical points Scale factor = 5

Fig. 6. Linear and independent 3D-distortion phenomena

A comparison between experimental and simulation results (figure 7) show that distortions values of the outer cylinder are greater than inner ones (except for barrel effect). Pincers’ opening is the most important phenomenon in experiment just as barrel and banana effects in simulation.

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

Ele

men

tary

qua

ntiti

es in

mm

Inner cylinderOuter cylinder

Exp. Simu. Exp. Simu. Exp. Simu. Exp. Simu. Exp. Simu.

X-Ovalization

Y-Ovalization

Pincers' opening

Barrel Banana

Fig. 7. Results for 3D-distortions identification from experiment and simulation

The chosen vectors for phenomena explain the main distortions as the final residual is small (figure 8).

020406080

100

Initial Tx, Ty, Rx,Ry, Rz

+ x,y Off-centring

+Pincers'opening

+ Barrel + x,yovalization

+ Banana

Res

idua

l in

perc

ent

Measured part

Simulated part

Fig. 8. Residual evolution in function of cumulated phenomena

4 CONCLUSIONS

We developed a method for identifying distortions based on the optimisation of the measure errors by the mean of distortions significant phenomena. In this first approach, tendencies of these phenomena are quite the same in experiment and simulation but we have to refine input data of the models for a better quantitative evaluation. Further work will focus on many experiments and on simulations including mechanical experimental data.

REFERENCES

1. P. Lamesle, E. Vareilles and M. Aldanondo, Towards a KBS for a Qualitative Distortion Prediction for Heat Treatments, In: Proc. 1st Int. Conference on Distortion Engineering, Bremen (2005) 39-47.

2. C. Andersch, M. Ehlers, F. Hoffmann and H.-W. Zoch, ‘Systematic Analysis of the Relation between Part Geometry and Distortion due to Heat Treatment’, Mat.-wiss. u. Werkstofftech., 37, (2006) 23-28.

3. R. Mukai and D.-Y. Ju, ‘Simulation of Carburizing-Quenching of a Gear. Effect of Carbon Content on Residual Stresses and Distortion’, Journal de Physique IV, 120, (2004) 489-497.

4. A. Weckenmann and M. Knauer, The Influence of Measurement Strategy on the Uncertainty of CMM-Measurements, Annals of the CIRP, 47, (1998) 451-454.

5. G. Lee, J. Mou and Y. Shen, ‘Sampling Strategy Design for Dimensional Measurement of Geometric Features Using Coordinate Measuring Machine’, Int. Journal of Machine Tools and Manufacture, 37, (1997) 917-934.

6. C. Baudouin, R. Bigot, S. Leleu and P. Martin, ‘Gear Geometric Control Software: Approach by Entities’, Int. Journal of Advanced Manufacturing Technology, 2007.

7. H.T. Yau and C.H. Menq, ‘A Unified Least-Squares Approach to the Evaluation of Geometric Errors using Discrete Measurement Data’, Int. Journal of Machine Tools and Manufacture, 36, (1996) 1269-1290.

8. R.A. Hardin and C. Beckermann, Simulation of Heat Treatment Distortion, In: Proc. 59th Technical and Operating Conference, Chicago, (2005).

9. Z. li, B.L. Ferguson, X. Sun and P. Bauerle, ‘Experiment and Simulation of Heat Treatment Results of C-Ring Test Specimen’, In: Proc. 23rd ASM Heat Treating Society Conference, Pittsburgh, (2005) 245-252.

10. M. Teodorescu, J. Demurger and J. Wendenbaum, Comprehension of Cooling Distortion Mechanisms by the mean of F.E. Simulation, In: Proc. 15th IFHTFE, Vienne, (2006).

Optimization of BEM-based Cooling Channels Injection Moulding Using Model Reduction

N. Pirc1, F. Schmidt1, M. Mongeau2, F. Bugarin1, F. Chinesta3

1CROMeP - Ecole des Mines d’Albi, Campus Jarlard,8 1013 Albi,cedex 9, France.URL: www.enstimac.fr/recherche/cromepe-mail: nicolas.pirc@enstimac.fr;

2Institut de Mathematiques, Universite de Toulouse, UPS,31062 Toulouse cedex 9, France.URL: www.mip.ups-tlse.fr/

3ENSAM-Paris, 151 boulevard de l’hopital, 75013 Paris, France.URL: www.paris.ensam.fr/

ABSTRACT: Today, around 30% of manufactured plastic goods rely on injection moulding. The coolingtime can represent more than 70% of the injection cycle. In this process, heat transfer during the cooling stephas a great influence both on the quality of the final parts thatare produced, and on the moulding cycle time.Models based on a full 3D finite element method renders unpractical the use of optimization of the design andplacement of the cooling channel in injection moulds. We have extended the use of boundary element method(BEM) to this process. We introduce in this paper a practicalmethodology to optimize both the position andthe shape of the cooling channels in injection moulding processes. We couple the direct computation with anoptimization algorithm such as SQP (Sequential Quadratic Programming). First, we propose an implementationof the model reduction in the BEM solver. This technique permits to reduce considerably the computing timeduring the linear system resolution (unsteady case). Secondly, we couple it with an optimization algorithm toevaluate its potentiality. For example, we can minimize themaximal temperature on the cavity surface subjectto a temperature uniformity constraint. Thirdly, we present encouraging computational results on plastic partsthat show that our optimization methodology is viable.

KEYWORDS: BEM, optimization, reduction model, injection moulding, SQP.

1 INTRODUCTION

Numerical simulations for designing injectionmoulds have become an important developement ininjection moulding processes. The location of thecooling channels is a major element in the design ofthe mould because the cooling time can represents upto 70 % of the injection cycle. We need efficient nu-merical simulations in order to optimize the processparameters, but models based on full 3D finite ele-ment method renders unpractical the use of optimiza-tion for this design and placement of cooling channelsin injection moulds.In this context, the Dual Reciprocity Method (DRM),introduced by Brebbia [1], is acknowledged to be oneof the most effective BEM techniques for transform-ing domain integrals into boundary integrals. More-

over, during the optimization, this method permits tocompute exact gradients, thereby avoiding the N di-rect computations per optimization iteration that areneeded by finite-difference gradient approximation(where N is the number of optimization variables).However, thermal models involved in the numericalmodelling in injection moulding processes a certainnumber of numerical difficulties such as size meshing,long simulations, or the necessity to define a homoge-nized thermal conductivity. In the first part of this pa-per, we present the use of boundary element method(BEM) and DRM applied to unsteady heat transferof injection moulds. The BEM software, developedat the CROMeP laboratory [2], was combined withan adaptive reduced modelling [3]. This procedurepermits to reduce considerably the computing timeduring the linear system resolution in unsteady prob-

1

lem. Then, we present a practical methodology to op-timize both the position and the shape of the coolingchannels in injection moulding processes. We cou-ple the direct computation with an optimization al-gorithm such as SQP (Sequential Quadratic Program-ming) [4].For the sake of simplicity, we will consider a potentialproblem defined in a 2D unbounded domain. The ca-pabilities of both the reduced order modeling and theboundary element method will be outlined.

2 BEM AND DRM APPLIED TO POISSONEQUATION

Using BEM, only the boundary of the domainhas to be meshed and internal points are explicitlyexcluded from the solution procedure. An interest-ing side effect is the considerable reduction in size ofthe linear system to be solved [5]. The transient heatconduction in a homogeneous isotropic bodyΩ is de-scribed by the diffusion equation [6], wherea is thematerial diffusion:

∀M ∈ Ω,−→∇2T (M, t) =

1

a

∂(M, t)

∂t(1)

We define the initial conditions and the boundary con-ditions as:

T (M, t) = T0 ∀M ∈ ΓP

φ(M, t) = λ.(T − TC) ∀M ∈ ΓC

T (M, t = 0) = T 0(M)(2)

WhereΓP is the boundary of the polymer andΓC the boundary of the channels. The temperature ofthe coolant isTC and the heat transfer coefficient,h,is related to the coolant flow rate (via Colburn cor-relation). Several strategies are possible to solve suchproblems using BEM. Matzig [7] propose to use spaceand time Green’s function. To express the domain in-tegral in terms of equivalent boundary integrals, weintroduce the DRM approximation [1]. The solutionis defined as a series of particular solutionsTk locatedin each boundary nodesNn, and each internal nodesNi. We obtain Eq (3), explain in detail by Mathey [2]

CiTi − a

∫

Γ

T.q∗dΓ −

∫

Γ

q.T ∗dΓ

=

Nn+Ni∑

k=1

βk

(CiTik +

∫

Γ

Tk.q∗dΓ −

∫

Γ

qk.T∗dΓ

)

(3)

Here,T andq denotes the temperature and theflux denotes, andCi is equal to 1 since the pointi isinside the domain and to 0.5 on its regular boundary.The following Green’s functionT ∗ andq∗ [1] denotesthe fundamental solution of this equation. The vectorβ is define such as:

β =1

aF−1T (4)

Matrix F consists of interpolation-function valuesf = 1 + r at each point.

3 REDUCED MODELING

Usual reduced models perform the simulation ofsome similar problem or the desired one in a shorttime interval. From these solutions, the Karhunen-Loeve decomposition [3] can be performed, allowingto extract the most relevant functions describing thesolution evolution.

3.1 The Karhunen-Loeve decomposition

We assume that the evolution of a certainfield T (x, t) is known. In practical applications,this field is defined at the spatial mesh nodesxi

(with i ∈ 1, · · · , N), and for some timetm =m.∆t with m ∈ 1, · · · , M. We introduce the no-tation Tm(xi) for defining the vector containing thenodal degrees of freedom (temperatures) at timetm.The main idea of the Karhunen-Loeve (KL) decom-position tell us how to obtain the most typical or char-acteristic structureφ(x) among theseTm(x) ∀ M .This is equivalent to obtaining a functionφ(x) maxi-mizingα defined as:

α =

∑P

p=1

[∑N

i=1φ(xi)T

p(xi)]2

∑N

i=1(φ(xi))

2(5)

This leads to:

P∑

p=1

[[N∑

i=1

φ(xi)TP (xi)

] [N∑

i=1

φ(xi)TP (xi)

]]

= α

N∑

i=1

φ(xi)φP (xi) ∀φ (6)

whereφ denotes the variation ofφ(x) which can berewritten under the form:

φTk.Φ = αφ

T.φ ∀φ

T⇒ k.φ = α.φ (7)

2

We define the matrix Q containing the discrete fieldhistory, and the vectorφ such that its i-component isφ(xi). This yields to the eigenvalue problemD =

Q.QT :

Q =

T 11 T 1

2 · · · T 1P

T 21 T 2

2 · · · T P1

......

. . ....

T 1N T 2

N · · · T PN

(8)

The functions defining the most characteristic struc-ture ofT P (x) are the eigenfunctionsφn(x) ≡ φn as-sociated with the largest eigenvalues.

3.2 A posteriori reduced model

We solve the eigenvalue problem defined by Eq(7) selecting the eigenfunctionsφn associated with theeigenvalues belonging to the interval defined by thelargest eigenvalue such asΦn’s sum is upper or equalto 99.9% ofΦN ’s sum. In practice,n is much lowerthanN . Let us now try to use these n eigenfunctionsφn for approximating the solution. LetB be the fol-lowing matrix:

B =

φ1(X1) φ2(X1) · · · φn(X1)φ1(X2) φ2(X2) · · · φn(X2)

......

. . ....

φ1(XN) φ2(XN) · · · φn(XN)

(9)

We express the linear system of equations resultingfrom the semi-implicit thermal-model discretizationas:

Tm+1 =

i=n∑

i=1

ζm+1

i φi = B.ζm+1 (10)

4 MOULD COOLING OPTIMIZATION

Each optimization iteration involves performinga BEM simulation and computing the objective andconstraint functions. The optimization method allowsupdating the cooling channel design parameters (sub-ject to the constraints) until a minimum of the costfunction is reached [5]. Figure 1 shows the couplingbetween the thermal solver and the optimization algo-rithm.

Figure 1: optimization procedure

The SQP method is designed for mono-objective optimization [4]. However, pratical opti-mization problems almost always involves at leasttwo objective functions. One way to proceed in such acontext is to consider as cost function a weighted sumof two objectives, but this method involves choosinga weighting parameter. We rather propose here usingone objective as optimization criterion, and the otheras a non-linear constraint. The first criterion involvesminiming the maximal temperature on the cavity sur-face. The second criterion aims at improving temper-ature uniformity. More precisely, we formulate ourproblem under the form:

minimizemaxi∈D

(Ti) (11)

subject to∑

i∈D

|Ti − Tmoy| ≤ σ (12)

where D is the set of discretization elements of theplastic part where the temperature Ti is measured,Tmoy is the average of the Ti’s, andσ is a user-definedtemperature uniformity tolerance, fixed here equal to4.

5 APPLICATION TO TWO-DIMENSIONALCASE

It is important to note that using reduced modelto optimize the cooling channel location is possiblesince the vectorφn(xN ) does not change when the

3

unsteady equation source term changes, i.e. even ifthe position of the cooling channels changed.Optimization variables are the coordinates (Xi, Yi) ofeach circle centeri, and the radius of them. This ge-ometry have 7 channels, thus we have 21 optimizationvariables in our problem. The coolant temperature isfixed asTC = 30C. Figure 2 displays the geometryused to validate our method. Dotted circles show theinitial configuration of the cooling channels, and boldcircles show the optimized position.

Figure 2: Channels configuration before and after optimization

Figure 3: Temperature before and after optimization

On average, one objective function evaluation re-quires 8 seconds of CPU time on a Macintosh 1.83GHz Intel Core 2 Duo, and 22 iterations and 534 eval-uations are necessary to reach convergence.

Table 1: CPU time comparisonMethod direct computation optimizationDRM 38.2 seconds 5.6 hoursreduction model 1 second 9.6 minutes

The reduced model permits to divide by more than40 the CPU of each direct computation, compared toDRM. We usedn = 17 in our simulations, whereasN = 288 (nodes number).

6 CONCLUSIONS

Our methodology uses BEM to solve the unsteadyheat transfer equation during the cooling step of theinjection moulding process. Simulation results areused in an optimization procedure to find the best ge-ometry and process parameters according to a givenobjective function. Reduce model technique involvesa Karhunen-Love decomposition leading to an opti-mal number of approximation functions, allowing toconsiderable CPU time savings (some times in the or-der of 40). Our preliminary test showed that our ap-proach is viable for optimizing the design of coolingchannels for injection moulding. Various objectivefunctions can be provided by the user (either directlyas a cost function or within constraints) . We presentlywork on more complex 3D moulds with more generalparameterizations of the cooling channels.

REFERENCES

[1] C. S. Chen, C. A. Brebbia, H. Power. Dual reciprocitymethod using compactly supported radial basis functionsCommunications in Numerical Methods in Engineering, ed-itorsJohn Wiley & Sons, Vol 15, 2 , pp. 137-150, 1999.

[2] E Mathey, L Penazzi, FM Schmidt, F Rond-Oustau. Auto-matic optimization of the cooling of injection mold basedon the boundary element method Materials Processing andDesign: Modeling, Simulation and Applications,Proc. NU-MIFORM”04, Vol 712, pp. 222-227, 2004.

[3] F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchchene,F. Boust. Alleviating mesh constraints: model reduction,parallel time integration and high resolution homogeniza-tion. Comput. Methods Appl. Mech. Engrg , editorsElse-vier, 2007.

[4] J. Nocedal and S. S. J. Wright Numerical optimization seriesin operation research, editorsspringer, 1999.

[5] N. Pirc, F. Schmidt, M. Mongeau, F. Bugarin,. BEM-basedcooling optimization for 3D injection molding. InternationalJournal of Mechanical Sciences ,Proc. ASMDO’07, Vol 48,4, pp. 430-439, 2006.

[6] S. Kenig, M.R. Kamal. Cooling molded parts, a rigorousanalysis. Soc. Plast. Eng. J., Vol 26, pp. 5057, 1970.

[7] B. A. Davis,P.J. Gramann, J.C. Matzig,T.A. Osswald. Thedual reciprocity method for heat transfer in polymer pro-cessing. Eng. anal. bound. elem., editor,Elsevier, vol. 13, 3,pp. 249-261, 1994.

4

Multiphysics welding simulation model

N. Poletz1, A. Francois1, K. Hillewaert1

1CENAERO - Bat. EOLE Rue des Freres Wright, 29 B-6041 Gosselies BelgiumURL: http://www.cenaero.be e-mail: nicolas.poletz@cenaero.be; arnaud.francois@cenaero.be;

koen.hillewaert@cenaero.be

ABSTRACT: The electron beam welding (EBW) process is extensively used for assembling titanium andother high strength components in the aircraft engine industry. For such applications, it is important to predictdistortions and residual stresses after the welding process. In welding simulation, identifying the main physicalphenomena is important to formulate reasonable hypothesis to capture the first order effect. A specific fluidflow model has been implemented in the in house CFD solver (ARGO). This model allows the simulation of themelt pool dynamics during welding by taking into account the influence of different convective terms. A singledomain approach with an enthalpy-porosity formulation has been used. The influence of each term on the finalmelt pool shape has been studied.

KEYWORDS: Welding, Fluid Flow, Marangoni Convection, Simulation

1 INTRODUCTION

Heat transfer during welding can strongly affect phasetransformations and thus the metallurgical structureand mechanical properties of the weld. In fusionwelding process, fluid flow in the melt pool is respon-sible for the melt pool shape and temperature distribu-tion in the workpiece. These factors have a close rela-tionship to the resulting material structure and prop-erties, such as microstructure, hardness and surfaceroughness. The melt flow is influenced by surface ten-sion gradients at the free surface (Marangoni effect)and thermal gradients in the melt pool (natural con-vection). Most work on heat transfer modelling andfluid flow during welding process has been devotedto the study of laser welding [3, 5, 6]. As beam pro-cesses deliver large amount of energy in a very smallregion of the workpiece, large temperature gradientsin melt pool are induced which give rise of importantconvective heat transport.

The different phenomena interfering in the meltpool have been identified and implemented in thein house CFD solver. The mathematical descriptionof the model and main assumptions are described.A comparative study of the influence of the differ-ent terms has been performed and the first resultsare presented. Several Phenomena have to be takeninto account to properly simulate fluid flow in meltpool. From energetic point of view, both conduc-

tive and convective heat transfer occur in melt pool.The temperature induced variation of the surface ten-sion on the melt pool produces thermocapillary ef-fect that combines with the buoyancy force and in-fluences the convection flow. These phenomena haverelevance in a wide range of applications with mov-ing front as crystal growth, solidification and weldingmaterial process. While in large systems buoyancyforces are the dominant driving mechanism, in smallscale systems surface tension forces at the liquid/airinterface play a significant role in determining the dy-namic of the flow. In addition to the convective terms,the heat loss due to radiation and convection has to beconsidered.The other fundamental phenomenon thattakes place during welding process is the Solid/Liquidphase change. Most alloys solidify with the formationof a two phase region known as mushy zone, whichis composed of solid dendrites and interdendritic liq-uid. A single-domain solidification model has beenused. This method overcome many of the limitationsof multidomain methods (e.g. of front-tracking meth-ods). This model consists of a single set of equationsfor momentum and energy which are applied in allregions (solid, mushy and liquid). It requires only asingle, fixed numerical grid and a single set of bound-ary conditions to compute the solution.

1

1.1 Constitutive equations

The differential equations governing the conservationof mass, momentum and energy are based on contin-uum formulation given by Chiang and Tsai [2]:

Continuity~5 ·

(ρ~V)

= 0 (1)

Momentum

~5·(ρ~V u

)= 5·

(µl

ρ

ρl

~5u

)−∂p

∂x−µl

K

ρ

ρl

(u−us) (2)

~5 ·(ρ~V v

)= ρg +5 ·

(µl

ρρl

~5v)− ∂p

∂y

−µl

Kρρl

(v − vs) + ρg [βT (T − T0)](3)

~5·(ρ~V w

)= 5·

(µl

ρ

ρl

~5w

)− ∂p

∂z− µl

K

ρ

ρl

(w−ws)

(4)Energy

~5·(ρ~V h) = ~5·(

k

cs

~5h

)+ ~5·

(k

cs

~5(hs − h)

)(5)

Where u, v, w are the velocities in the x, y, z direc-tions respectively. The subscript s and l refer to thesolid and liquid phases respectively; p is the pressure;µ is the viscosity; K is the permeability, which is ameasure of the ease with which fluid pass throughthe porous mushy zone; βT is the thermal volumetricexpansion coefficient; g is the gravitational accelera-tion; T is the temperature; the subscript 0 representsthe reference value for the natural convection in theBoussinesq approximation; h is the enthalpy; k is thethermal conductivity; c is the specific heat.

The third term on the right-hand side of Equations(2), (3) and (4) represents the drag force for the flowin the mushy zone. The last term on the right-handside of Equation (3) is the buoyancy force term whichis based on the Boussinesq approximation for natu-ral convection. The first two terms on the right-handside of Equation (5) represent the net Fourier diffu-sion flux. The last is the volumetric heat source use torepresent the energy flux from the beam. In Equations(1)-(5), the density, specific heat, thermal conductiv-ity, solid mass fraction, liquid are calculated from liq-uid and solid properties using a mixture law. Phasedproperties are assumed to be constant. However thephase enthalpies for the solid and the liquid can beexpressed as:

hs = csT, hl = clT + (cs − cl)Ts + Lm (6)

where Lm is the latent heat of fusion of the alloy.To model the fluid flow in the mushy zone, a per-

meability function is defined employing the Carman-Kozeny equation [1, 4]:

K =g3

l

cl(1− gl)2, cl =

180

d2(7)

where d is related to the dendrite dimension, which isassumed to be a constant and is on the order of 10−2

cm.The solid-liquid phase change is handled using the

continuum formulation. The last terms on the right-hand side in Equations (2)-(4) will dominate in solidphase since the liquid fraction gl tends towards 0;hence the velocity is forced to be equal to the solidvelocity. For the liquid region this term vanish be-cause gl = 1 and 1/K = 0. This term is only validin the mushy zone, where 0 < gl < 1. Therefore,the liquid region, mushy zone and solid region canbe handled by the same equations. During the fu-sion and solidification process, latent heat is absorbedor released in the mushy zone via the enthalpy for-mulation. Solidification shrinkage is handled by thedensity change between the liquid phase and the solidphase. This density difference induce fluid flow fromthe front part of the melt pool, where melting occurs,to the rear part of the melt pool where solidificationtakes place.

2 MODELLING CONDITIONS

In the following sections the assumptions are de-scribed as well as the different boundary conditions.

2.1 Assumptions

• The workpiece is initially at 293K. The heatsource is supposed to be fixed and the work-piece move in the positive z-direction with aconstant velocity equal to the process velocity.

• The surface of weld pool is flat.

• Thermophysical properties are supposed to beconstant in both liquid and solid phase.

• The density variation with temperature is takeninto account via the Boussinesq approximation.

• The flow is laminar and incompressible.

2

• The liquid volume fraction is assumed to fol-low a linear evolution versus temperature in themushy zone.

• The surface tension of the liquid phase is sup-posed to be linearly dependent with tempera-ture.

Figure 1: Description of the boundary conditions used in themodel.

2.2 Boundary conditions

The boundary conditions employed in this study, il-lustrated in Figure 1, are the following:

Top and bottom surfaces At the top free surface, sincetemperature distribution on the surface of the meltpool are always non uniform, surface tension gradi-ents will appear on the surface and affect the meltflow. The Marangoni shear stress at the free surfacein a direction tangential to the local free surface isgiven by :

µ∂~V · ∂~n

∂~n= − ∂γ

∂T

∂T

∂~n(8)

where s is a tangential vector, n a normal vector tothe local free surface and γ the surface tension of theliquid.

As we assume that the surface remain flat, the ve-locity component normal to the surface is set to 0. Incase of full penetration welding, the same momentumbalance is applied on both top and bottom surfaces.

3 RESULTS

imulation has been carried out on a TA6V workpiece10 mm wide, 30 mm long and 2.2 mm thick. A volu-metric cylindrical heat source with a gaussian power

distribution has been used to model the incoming en-ergy of the beam. The beam is considered to be fixat the position z = 2.75 mm, and the workpiece movewith a velocity of 30 mm.s-1. The preliminary com-putation carried out has permitted to define two differ-ent regions. First, we use a coarse grid in the regionwhere the metal is supposed to remain at solid state.Then, the mesh has been refined where complex fluidflow takes place.

First computation has been conducted with theporosity source term. Buoyancy force term , stand-ing for natural convection in liquid, has been had in asecond step.

Third step has been carried out taking into accountsurface tension effect with the Marangoni boundarycondition. Input energy from the beam on top sur-face of melt pool leads to large temperature gradients.The effect of surface tension variation induces largeflow, and temperature distribution near free surface isstrongly modified. Isotherm representation of temper-ature distribution in Figure 2 shows spreading of meltpool near the free surface on both top and bottom.Isotherm spacing at the vicinity of these surfaces islarger than in the centre of the melt pool because ofMarangoni effect. In this case the peak temperature islower since fluid flow redistributes a certain amountof beam energy.

Figure 2: Computed Temperature fiel in the weld pool.

The present fluid flow is mainly driven by theMarangoni shear stresses. The flow is directed radi-ally outwards from the hottest centre to the side of themelt pool as represented on . It can be seen that thehigh velocities occur in the vicinity of the top of themelt pool where temperature gradients are the mostimportant. The Marangoni term has a predominant

3

influence on the flow patterns. In case of full penetra-tion welding, Marangoni forces act on both top andbottom surface and leads to the formation of four vor-tices near these surfaces (c.f. Figure 3). In this flowregime, the convective heat transfer in the molten ma-terial plays a dominant role in the prediction of theweld pool shape. This phenomenon causes a spread-ing of the melt pool near the free surfaces and a nar-rowing in the bulk. Hence the melt pool cross sectionis strongly modified by introduction of surface tensioneffect.

Figure 3: Computed stream lines representation in melt poolwith both buoyancy and marangoni convection.

4 CONCLUSION

A single domain model using enthalpy porosity for-mulation has been implemented in the CENAERO inhouse CFD solver. This formulation allows the useof a single set of equations and boundary conditionsfor liquid, solid and mushy zone. The model takesinto account physical parameters change between liq-uid and solid phase, latent heat of fusion absorptionduring melting and release when solidification takesplace. Two different convective terms are considered.Natural convection is evaluated in the Boussinesq ap-proximation for incompressible flows. Surface ten-sion variation with the temperature on free surfacesof the melt pool gives rise to shear stress on these sur-faces (Marangoni effect).

The calculated weld shape is compared to a trans-verse macrosection of a electron beam welded joint inFigure 4. The simulated weld shape presents a spread-ing of the top and bottom part of the melt pool ofthe same order of magnitude than in the experimentalcase. The fluid model has permitted to have a betterprediction of the temperature distribution in the work-piece during beam welding, particularly for the fusionzone shape.

Figure 4: Comparison of transverse section of a electron beamwelded joint and computed melt pool shape.

The aim of this work is to used the predictedtemperature field as input for the in house finite ele-ment code Morfeo to predict distorsions and residualstresses after welding.

ACKNOWLEDGEMENT

The authors acknowledge the financial support fromVERDI (Virtual Engineering for Robust manufac-turing with Design Integration). VERDI is a re-search project within the European 6th FrameworkProgramme. http://www.verdi-fp6.org

REFERENCES

[1] P. C. Carman. Fluid flow through granular beds. ChemicalEngineering Research and Design, 15a:150–166, 1937.

[2] K. C. Chiang and H. L. Tsai. Interaction between shrinkage-induced fluid flow and natural convection during alloy solid-ification. International Journal of Heat and Mass Transfer,35(7):1771–1778, July 1992.

[3] L. Han and F.W. Liou. Numerical investigation of the influ-ence of laser beam mode on melt pool. Int. J. Heat MassTransf., 47(19-20):4385–4402, 2004.

[4] K. Kubo and R. D. Pehlke. Mathematical modeling ofporosity formation in solidification. Metall Trans B, 16B(2):359–366, 1985.

[5] J.F. Li, L. Li, and F.H. Stott. A three-dimensional numeri-cal model for a convection-diffusion phase change processduring laser melting of ceramic materials. Int. J. Heat MassTransf., 47(25):5523–5539, 2004.

[6] X.-H. Ye and X. Chen. Three-dimensional modelling of heattransfer and fluid flow in laser full-penetration welding. J.Phys. D: Appl. Phys, 35(10):1049–1056, 2002.

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1 INTRODUCTION Many articles ([1, 2], etc.) have shown POD (Proper Orthogonal Decomposition, also called Karhunen-Loeve expansion or Principal Component Analysis) method can help identify a shape. This method is used in this study to identify a deformation after heat treatment and to correlate it with a type of variation. In a previous article (see [3]), numerical simulations were performed to show that heat transfer coefficient, quenching temperature and carbon layer features have a major effect on deformation. This work focuses on the boundary conditions and their impact on distortions. First a set of nominal boundary conditions is defined to represent a nominal heat treatment process. Next, a POD of the FEM temporal solution is carried out to extract a POD basis of displacements. Taking into account all the displacements history, a more physical meaning of the eigenvectors is obtained (one or more vectors for dilatation for example). This basis is completed with other FEM computations for different boundary conditions. Directions are defined in the POD basis, each one linked to a variation of process parameter. Moving along a direction means that changes observed are

caused by the parameter variation. In parallel, other computations are carried out for different boundary conditions. The final results of these simulations are projected on the directions previously defined. Major component gives a paramount directions and so a variation of process parameter. To make measurements simple, tubular samples with different bores are used (Fig. 1). Important variations in the bore diameters are defined in order to cause large final distortions (see [4]).

Fig. 1: Sample (from [4]) The heat treatment process (heating and carburizing followed by quenching) is simulated in 2D using the software Sysweld. We focus in this work on variations of carbon layer concentration, on

ABSTRACT: It is well know that the heat treatment step of gearbox cogwheel induces distortions of the parts. In addition, some deviations of this deformation are often observed, due to unknown changes of process parameters. These deviations, and more precisely the detection of their origin, are the subject of this paper. We propose here a methodology based on the projection of measurements after heat treatment on a POD (Proper Orthogonal Decomposition) basis, extracted from FEM computations. This information about deviation origin can help correct incriminated process parameters.

Key words: Heat treatment, Gear, Distortion, Proper Orthogonal Decomposition basis, Model reduction

Detection of deviations origins in a heat treatment process using Proper Orthogonal Decomposition (POD) basis L. Vanoverberghe1,2, Lucia Garcia-Aranda1, David Ryckelynck3, Yvan Chastel2

1Renault – 67 rue des Bons Raisins, 92500 Rueil-Malmaison, France URL: www.renault.com e-mail: laurent.vanoverberghe@renault.fr;

lucia.garcia-aranda@renault.com 2Centre de Mise en Forme des Matériaux, CEMEF (ENSMP) – 1 rue Claude Daunesse, 06904 Sophia-Antipolis, France URL: http://www-cemef.cma.fr e-mail: yvan.chastel@ ensmp.fr 3Centre des Matériaux (ENSMP) - URL: www.mat.ensmp.fr e-mail: david.ryckelynck@mat.ensmp.fr

quenching temperature and heat transfer coefficient. Variations of these parameters are presented here and the other ones (CCT diagrams, mechanical model, etc.) are kept the same for all simulations.

2 THE DECOMPOSITION As the problem under investigation has a convenient number of degrees of freedom, a classic Snapshot POD [5] is used to find the POD vectors. All the displacements at each time step

tq are

placed in a same matrix Q. First, the covariance matrix is calculated:

QQC T= (1)

Eigenvalues and eigenvectors (matrix V) of this matrix are computed. Only eigenvectors with meaningful eigenvalues are taken into account. POD basis is given by:

VQ=Ψ (2) Each column of Ψ is a POD vectors. The coefficients Φ of these vectors are given by:

QTΨ=Φ (3) Initial displacements Q can be recovered with:

TQ ΦΛΨ= (4) where Λ is the identity matrix.

3 THE NOMINAL COMPUTATION A set of nominal values of the boundary conditions is defined. 3.1 Carbon layer depth The carburizing computation is performed with a carbon activity of 0.95. We obtain a maximum value of mass carbon concentration [C] of 0.87% at the sample corners. 3.2 Quenching temperature Nominal quenching temperature is 50°C. The quenching lasts 610 seconds. It is followed by an air-cooling to reach a temperature of 25°C in the part. 3.3 Heat transfer coefficient

Heat transfer coefficient h is considered as constant (1500 W/m²/K) on all surfaces.

4 THE PERTURBED COMPUTATIONS Simulations are performed for the set of parameters defined in Table 1. Table 1. Changes in boundary conditions to create the basis Parameter Values applied Carbon activity [-] 0.85, 0.9, 1, 1.05 Heat transfer coefficient [W/m²/K] 1350, 1425, 1525, 1550 Quenchant temperature [°C] 40, 45, 55, 60 Therefore the number of computations is the following: a nominal computation and twelve perturbed computations. Note that for heat transfer coefficient, only the heat transfer of the lower face (h*) is modified, as shown in Fig. 2. This would represent a modification of the quenchant media circulation for example.

Fig. 2 : Heat transfer coefficient used for perturbed computations

5 THE POD BASIS CONSTRUCTION The goal is then to create a basis and a direction related to a change of boundary conditions. Emphasis is placed on comparisons with results of the nominal set. So it is relevant to create two POD bases: a “recurrent” one, which contains all common eigenvectors, and another “specific” one, with eigenvectors which are dedicated to the variations. The nominal and perturbed computations are used to create the POD basis. Matrix Q contains all displacements at each time step and for each case:

=

43421

KL

43421

K

nSimulation

BCt

BCt

Simulation

BCt

BCt

n

f

n

ifiqqqqQ

1

11 (5)

Simulation 1 is the nominal case. After performing the decomposition described in section 2, we obtain a matrix Ψ with all eigenvectors

and a matrix Φ with components of these vectors. Φ has the same structure as Q. A coefficient γjBCk representing the “intensity” of an eigenvector j for a simulation k is defined:

( )∑ Φ=i

BCji

BCj

kk2γ (6)

An eigenvector j is considered as specific if the following condition is fulfilled:

kk BCjk

BCjk γαγ maxmin < (7)

where α is a parameter to be determined.

6 DEFINITION OF THE DIRECTIONS Once the specific POD basis Ψs has been determined, the directions correlated to a change of boundary conditions have to be defined. Matrix Q is again projected but this time on the specific basis, to provide specific coefficients Φs, which fit best the results Q. For each perturbed simulation, difference between its projection and the nominal projection is used to create the direction in which the simulation moves in the specific basis. The projection variation is:

k

BCs

BCsBC

s BC

k

k

∆Φ−Φ

=∆Φ1

(8)

where ∆BCk is a weight attributed to the boundary conditions change. The components of the direction related to the change in boundary conditions ∆BCk are computed by:

( )∑ ∆Φ=i

BCjis

BCj

kk2δ (9)

The set of all these directions form the matrix δ. The matrix δT.δ shows the coupling between each direction. A high coupling means that directions are nearly similar and can not provide a unique information on a variation.

7 IDENTIFICATION OF CHANGE IN BOUNDARY CONDITIONS

Directions of some changes of boundary conditions have been defined. If some new perturbed results are projected on these directions, the major components will give the probable cause of the variation. The goal is to project the final result of the computation

or of an experimental measurement. Let us investigate in detail the projection of the final state. The variation of this projection is computed as in equation 8 but only with the final displacement Φtf (a column vector):

1BCttt fff Φ−Φ=∆Φ (10)

This variation is projected on the directions contained in the matrix δ:

ftTp ∆Φ= δ (11)

Each column of p gives the components of the variation on the directions previously computed. A direction will be considered as paramount if the absolute value of its component is strictly superior to 90% of the absolute value of the maximal component. This paramount direction provides information about the change of boundary conditions.

8 APPLICATION OF THE METHOD One creates a basis able to recognize three types of variations: carbon layer concentration, quenching temperature and heat transfer coefficient. Of course, the parameters α (see equation 7) needs to be tuned to define the separation between recurrent and specific basis. One computes the dimension of the specific basis and maximum coupling between directions as a function of α (Fig. 3).

Fig. 3: Evolution of the specific basis dimension and of the maximum coupling of directions with alpha

The best results are obtained when coupling is

minimum but in association with an adequate number of specific POD vectors. In our case, the best detection of the variation occurs with a value for parameter α of 0.17. Fig. 4 shows examples of POD vectors which were determined. Some of them can obviously be linked to physical phenomena (for example, the first one is a dilatation mode) but the major part represents an arbitrary deformations.

Fig. 4: Examples of normalized POD vectors (colours are

function of the displacement norm). In a first step, the final displacements of the 12 perturbed computations are projected on the three directions. In this case, one could expect a good detection since the same simulations results were used to create the basis. A really promising result is obtained: 10 variations are identified, one is undetermined and only one is false. In a second step, 6 other simulations were carried out, with the variations defined in Table 2. Table 2. Changes in boundary conditions to detect Parameter Values applied Carbon activity [-] 0.8, 1.1 Heat transfer coefficient [W/m²/K] 1175, 1725 Quenchant temperature [°C] 35, 65 The projection of the final displacements on the basis leads to successful identifications of 5 variations, one being undetermined. This result is excellent since this final displacement is nearly similar for all the computations (see Fig. 5). Furthermore, only displacements on the external surface are used. If one restricts to points of external diameter, the results are nearly as good (4 correct identifications and 2 errors).

a b c Fig. 5: Final displacements of three simulations (a: carbon

activity of 0.8; b: heat transfer coefficient of 1725 W/m²/K on lower surface ; c: quenching temperature is 65°C). The scale is

the same for the three figures.

9 CONCLUSIONS AND PERSPECTIVES This method seems to be promising since a source of variations can be detected with the mere projection of a final displacement on a numerical POD basis. Once the major deviations are identified and correlated with simulations, the creation of the basis is quite simple. An important point here is that the correlation can only be qualitative and not quantitative. In future works, experimental measurements will be projected on the POD basis, for the same sample. Finally, applications to tooth gear shape will be considered.

REFERENCES 1. Grigoriev, A. Ya. and Chizhik, S. A. and Myshkin, N.

K., Texture classification of engineering surfaces with nanoscale roughness, Int. J. Mach. Tools Manufact., Vol. 38 (1998) 719-724.

2. Pinowski B., Principal component analysis of speech spectrogram images, Pattren recognition, Vol. 30 (1997) 777-787.

3. Vanoverberghe, L. and Garcia-Aranda, L. and Ryckelynck, D. and Chastel Y., Anticipation of gears distortions during heat treatment, 10th International Conference on Material Forming (Esaform), Zaragoza (2007).

4. Lasserre, R. and Henault, E., Une méthodologie d’étude des deformations lors du traitement thermique : l’éprouvette de deformation, In: Proc. Journées franco-allemandes ATTT-AWT, Belfort (1997) 26-30.

5. Sirovich, L., Turbulence and the dynamics of coherent structures. Part 2: Symmetries and transformations. Part 3: Dynamics and scaling. Quartely of Applied Mechanics 45 (1987) 561-590.

Determination of the thermophysical properties of a CuCr1Zr alloy fromliquid state down to room temperature

J. Wisniewski1,3, J.-M. Drezet2, D. Ayrault3 and B. Cauwe4

1LG2M, Universite de Bretagne-Sud - 56321 Lorient, FranceURL: www.univ-ubs.fr ; e-mail: jean.wisniewski@gmail.com2LSMX, Ecole Polytechnique Federale de Lausanne - Lausanne, SwitzerlandURL: lsmx.epfl.ch ; e-mail: jean-marie.drezet@epfl.ch3DEN/DM2S/SEMT/LTA, CEA Saclay - 91191 Gif-sur-Yvette, FranceURL: www.cea.fr ; e-mail: danielle.ayrault@cea.fr4Le Bronze Industriel - 3 av. du general Leclerc, 51600 Suippes, FranceURL: www.lebronzeindustriel.com ; e-mail: bcauwe@lebronzeindustriel.com

ABSTRACT: Laboratory tests and inverse methods are used forthe determination of the thermophysical prop-erties of a CuCrZr alloy. The solidification path (temperature versus solid fraction curve) is determined usingthe Single Pan Thermal Analysis (SPTA) technique developedat LSMX. The temperature dependent thermalconductivity is identified by inverse analysis using temperature measurements in one dimensional solidifiedcasting. The thermophysical properties will be used as input data in numerical models of the laboratory testaiming at evaluating the hot cracking sensitivity of copperbased alloy in electron beam welding for the Inter-national Thermonuclear Experimental Reactor (ITER) project.

KEYWORDS: CuCrZr alloy, thermophysical properties, Single Pan Thermal Analysis SPTA, solidificationpath, thermal conductivity, one dimensional solidified ingot, inverse analysis.

NOMENCLATURE

Tm Melting temperature of pure coppermi Liquidus slope of alloying element iki Partition coefficient of alloying element ic0i Nominal composition of alloying element ifs Mass solid fraction

1 INTRODUCTION

The precipitation hardened CuCrZr alloy has been se-lected as a heat sink of the first wall components forthe future thermonuclear fusion reactor ITER [1] ow-ing to its good mechanical and thermal properties.The feedback from its use in Tore Supra [2] showedthat this alloy is very sensitive to hot tearing (solidi-fication cracking) during electron beam welding. Inorder to characterize the hot tearing susceptibility ofthe alloy and thus define acceptance tests of varioussupplies, a laboratory test, inspired by the work car-ried out at the Joining and Welding Research Insti-tute (JWRI) [3] is used. An electron beam weld seam

is performed on a thin rectangular plate instrumentedwith thermocouples. The welding parameters (speedand heat input) are fixed. As width of the plate de-creases, a crack appears. The test consists in deter-mining this specific width and is analysed with thehelp of numerical modelling, hot tearing criteria, thelocal Rappaz-Drezet-Gremaud (RDG) approach [4]and a thermomechanical criteria [5], will be evalu-ated. To carry out the finite element numerical anal-ysis, it´s necessary to know not only the mechanicalbehaviour from liquid state down to room tempera-ture but also the thermophysical properties of the al-loy. The missing physical properties are determinedby associating laboratory tests and numerical analy-sis.

2 EXPERIMENTAL METHODS

The Single Pan Thermal Analysis (SPTA) is used todetermine the solidification path of the CuCr1Zr alloy(EN 12163 CW106C [6]) alloy (Tab. 1).

1

Table 1: Chemical composition of CuCr1Zr alloy.Compo. (wt%) Cu Cr Zr Fe Si Othermin bal. 0.5 0.03 - - -max bal. 1.2 0.3 0.08 0.1 0.2

Contrary to Differential Thermal Analysis (DTA),SPTA permits the analysis of a huge volume of metal(cm3 order), thus reducing the effect of nucleationundercooling. Details of this method are availableelsewhere [7]. The experiments are conducted witha cylindrical sample (diameter 13.8 mm, height 15mm) using a high purity gas to minimise oxidation.The samples are subjected to heating and coolingcycles as follows: Room temperature→ heating to1200C (variable heating rate)→ isothermal holdingat 1200C for 3h→ cooling (5K/min) to room tem-perature.

For the determination of the thermal conductiv-ity, cylindrical sample of CuCr1Zr (diameter 40 mm,height 70 mm) is solidified under one dimensionalheat flow condition. Fives thermocouples are placedat various distances from the water cooled copperchill (4 mm, 19 mm, 33 mm, 48 mm, and 62 mm).Fig 1 presents the mould and the empty crucible to-gether with the five ceramic tubes in which the ther-mocouples are inserted. The measured temperaturehistories are then used in an inverse method to iden-tify the thermal conductivity of the alloy at selectedtemperatures.

Figure 1: Experimental set up for the 1D casting.

3 RESULTS AND DISCUSSION

3.1 Solidification path

Fig 2 shows the solid fraction versus temperature forthe CuCrZr alloy obtained by SPTA (cooling rate

5K/min). The liquidus temperature isTL = 1080C.The slope change of the curve at 1075C correspondsto the formation of an eutectic phase according to theCuCr binary phase diagram [8].

Figure 2: Solidification path of a CuCr1Zr alloy in equilibriumand non equilibrium solidification conditions.

Assuming that the system contains three major el-ements: chromium, zirconium and phosphorus (deox-idizer), the lever rule is considered [9, 10]:

T = Tm +3∑

i=1

mic0i

1 − (1 − ki)fs

(1)

This model calculates the solid fraction versus tem-perature for an equilibrium solidification assuming in-finite diffusion in solid and liquid. The coefficientsmi andki are unknown. The binary phase diagram ofCuCr alloy gives a rough estimation ofmCr andkCr

parameters:mCr = −3.5C/%wt andkCr = 0.1. Weconsider that the experimental cooling rate (5K/min)is near the cooling rate of equilibrium solidification.Iterative least squares fit is used in order to identifythe remaining unknown parameters. The method isbased upon a minimization of the error between thecalculatedfs(T ) curves obtained with the lever rule(1) and the measured curve. The lever rule doesn’ttake into account the formation of a new phase (i.e.change of slope at 1075C) so the minimization is ledbetween 1080C and 1075C. The obtained values ofthe coefficients after the minimization are:mZr =−5.65C/%wt, mP = −5.11C/%wt, kZr = 0.1,kP = 0.1.

In order to obtain the solidification path inquenched conditions, typical conditions encounteredduring welding, the estimated parameters are used in

2

the Scheil-Gulliver model [9, 10]. This model as-sumes that there is no solute diffusion in the solid (i.e.the cooling rate is infinite):

T = Tm +3∑

i=1

mic0i(1 − fs)ki−1 (2)

Experimental data, lever rule and Scheil-Gullivermodel results appear in fig 2.

The problem with the Scheil-Gulliver model isthat the lower limit of the solidification interval is notdefined. To solve this problem, the fraction of sec-ondary phase is estimated using transverse section ofelectron beam welding plate. Fig 3 presents scanningelectron micrography of a transverse section. Threedistinct regions are observed: the base metal (BM),the heat affected zone (HAZ), the melted zone (MZ).Detail of the melted zone is presented in fig 4.

Figure 3: SEM examination of a transverse section of a plateafter welding (secondary electrons).

Figure 4: SEM detail within the melted zone.

We mainly observe primary A and secondary Bphases and porosity. To determine the volume frac-tion of each phases, micrographs of the alloy areanalysed using the analySISTM image software. Thearea fraction is equal to the volume fraction of eachphase. Three images were used to calculate the frac-tions. The image analysis yields a mean value of 5%of secondary phase. Considering the Scheil-Gullivermodel, this percentage allow us to fix the non equilib-rium solidus temperature at 1048C (fig 2).

Therefore, the solidification interval is 21C inequilibrium solidification and 32C in non equilib-rium solidification.

3.2 Thermal conductivity

In order to apply the inverse method described byRappaz et al [11], the specific heat of CuCrZr pre-sented in fig. 5 is used. For temperatures below900C, the data come from [12]. For temperaturesgreater than 900C, a linear extrapolation is done. Thelatent heat of CuCrZr is assumed to be equal to thatof pure copper [13]:

LCu = 204kJ/kg (3)

The result of the inverse calculation is shown in fig. 5.The thermal conductivity of pure copper taken from[13] is also given for comparison.

Figure 5: Thermal conductivity and specific heat of a CuCr1Zralloy and pure copper.

At low temperatures, the thermal conductivity ofCuCrZr is two times smaller than the thermal conduc-tivity of pure copper. Indeed, alloying elements de-crease the thermal conductivity. In our case, the phe-nomenon is even more pronounced because Cr and Zr

3

remain in supersaturated solid solution during the fastcooling experienced in the 1D casting. The thermalconductivity of CuCrZr determined in the liquid stateis huge owing to the high convection experienced bythe liquid metal right after filling the mould (fig 1).

4 CONCLUSIONS

The solidification path and the thermal conductivityare determined for a CuCr1Zr alloy by associatinglaboratory tests and numerical analysis:

• the solid fraction versus temperature curve isobtained using the single pan thermal analy-sis (SPTA) technique developed at LSMX. Thisyields a better description of the solidificationpath of the alloy not only in equilibrium condi-tions but also in non-equilibrium conditions astypicaly encountered in electron beam welding.

• the temperature dependent thermal conductiv-ity is deduced from inverse modeling usingtemperature measurements in one dimensionalsolidified ingot. It appears that the conductiv-ity is about two times lower than that of purecopper.

ACKNOWLEDGEMENT

The authors express their gratitude to J.-D. Wagniere and F.

Kohler who carried out the experiments at the Ecole Polytech-

nique Federale de Lausanne and to F. Castilan for the met-

allographical inspections at the Laboratoire des Technologies

d’Assemblage (LTA-CEA Saclay).

References

[1] U. Luconi, M. Di Marco, A. Federici, M. Grattarola, G.Gualco, J.M. Larrea, M. Merola, C.Ozzano, G. Pasquale.

Development of plasma facing components for the dome-liner component of the ITER divertor.Fusion Engineeringand Design, Vol. 75-79, 2005, pp. 271-276.

[2] M. Lipa, A. Durocher, R. Tivey, Th. Huber, B. Schedler,J. Weigert. The use of copper alloy CuCrZr as a structuralmaterial for actively cooled plasma facing and in vesselcomponents.Fusion Engineering and Design, Vol. 75-79,2005, pp. 469-473.

[3] M. Schibahara, H. Serizawa, H. Murakawa. Finite elementmethod for hot cracking analysis using temperature depen-dent interface element.Mathematical Modelling of WeldPhenomena, Vol. 5, 2001, pp. 253-267.

[4] M. Rappaz, J.-M. Drezet and M. Gremaud. A new hot tear-ing criterion.Metallurgical and materials transaction A,Vol. 30A, February 1999, pp. 449-445

[5] N. Kerrouault. Fissuration a chaud en soudage d´un acierinoxydable austenitique.These CEA-R-5953, mars 2001.

[6] EN 12163:1998 - number CW106C

[7] F. Kohler, T. Campanella, S. Nakanishi, M. Rappaz. Appli-cation of Single Pan Thermal Analysis to Cu-Sn peritecticalloys.To be published in Acta Materialia, 2008.

[8] T.B. Massalski, J.L. Murray, L.H. Bennet, H. Baker. Bi-nary Alloy Phase Diagrams Volume 1.American Societyfor Metals1986.

[9] W. Kurtz, D.J. Fisher. Fundamentals of Solidification,fourth revised edition.Trans Tech Publications, 2005.

[10] J.-M. Drezet. Direct Chill and Electromagnetic Casting ofAluminium Alloys: Thermomechanical Effects and Solid-ification Aspects.Thesis1996.

[11] M. Rappaz, J.-L. Desbiolles, J.-M. Drezet, Ch.-A.Gandinn, A. Jacot and Ph. Thevoz. Application of InverseMethods to the Estimation of Boundary Conditions andProperties.Modelling of Casting, Welding and AdvancedSolidification Processes VII10-15 sept. 1995.

[12] J. Wisniewski, E. Gautier and P. Archambault.Determination des proprietes thermophysiques et desevolutions microstructurales au cours du chauffage rapided’un alliage CuCrZr,Rapport de DEA, Ecole doctoraleEMMA, 2005.

[13] Metals Handbook. Properties and Selection: NonferrousAlloys and Special Purpose Materials,ASM InternationalHandbook Commities.

4

1 INTRODUCTION

The growing interest of aircraft industry in reducing

the weight of aerospace structure has led the

introduction of laser-beam welding into the

fabrication of aerospace structure with stiffeners,

instead of riveted joints. This development has two-

fold advantages. First, the considerable amount of

material added up in the form of rivets is no more

required; second, the welding process is extremely

fast and hence leads to high production rates. Yet,

the non-uniform distribution of residual stresses and

the distortions induced due to the local solid-liquid

transformations remain undesirable. It is, therefore,

believed that the information regarding the

distribution of these residual stresses and distortions

may assist in exercising better control over

unwanted aspects of the process.

In recent years, various researchers have

successfully used numerical simulation methods to

predict these residual stresses and distortions. C.

Darcourt et al. [1] and E. Josserand et al. [2] have

attempted to predict residual stresses and out-of-

plane displacements for the aeronautic aluminium

alloy while working on thin sheets. Comparison

between experimental and simulated results for

distortions is developed by Tsirkas et al. [3] who

used the commercial software SYSWELD for

simulation. Various heat source models, ranging

from Gaussian cone-shaped source to Goldak’s

double ellipsoidal [4] with volumetric distribution,

exist in literature that are primarily meant to apply

the heat flux in the finite element model as

accurately as possible. Ferro et al. [5] used one such

model of conical shape with an upper and lower

sphere to include the effect of ‘keyhole’ during

electron-beam welding process. Moreover, some

authors [6,7] have also studied the mathematical

modelling and simulation of keyhole formation.

2 EXPERIMENTAL WORK

To study the thermo-mechanical response of the

material 6056T4 a simple experiment was performed

in which a fusion pass was created in the middle of a

thin test plate of dimension 300 mm x 200 mm and

thickness 2.5 mm with industrially used boundary

conditions. These boundary conditions are complex

ABSTRACT: The gradual introduction of laser-beam welding in manufacturing aerospace structures has offered new challenges in terms of acquiring control over distortions and residual stresses. The aim of this work is to study the thermo-mechanical response of thin sheets made of an aluminium alloy 6056T4, which is used for fabrication of fuselage panels, to the laser-beam welding under the complex industrial boundary and loading conditions. A single pass fusion welding with laser-beam was performed on several test plates. Temperature histories were recorded using thermocouples. Weld bead geometry was examined by macrography while displacement fields were observed through 3D image correlation technique. An uncoupled thermo-mechanical analysis is then performed using Abaqus 6.6-1, and simulation results are compared with experimental results. Good accordance is found between the simulated and experimental results.

Key words: Laser-beam welding, thermo-mechanical analysis, distortions, residual stresses

Thermo-mechanical Analysis of Laser Beam Welding of Thin Plate with

Complex Boundary Conditions

M. Zain-Ul-Abdein1, D. Nélias

1, J.F. Jullien

1, D. Deloison

2

1LaMCoS, INSA-Lyon, CNRS UMR5259, F69621, France

URL: www.insa-lyon.fr e-mail: Muhammad.Zain-Ul-Abdein@insa-lyon.fr; Daniel.Nelias@insa-lyon.fr; Jean-Francois.Jullien@insa-lyon.fr 2EADS Innovation Works, 12 rue Pasteur, BP 76, 92152 Suresnes Cedex, France

URL: www.eads.net e-mail: dominique.deloison@eads.net

Aluminium Table

q conv+rad q conv+rad q conv+rad

q th cond + q forced conv

in a way that the test plate is held in position with

the help of air suction force applied through an

aluminium table on the bottom of the test plate. It is

assumed that forced convection is present at the

bottom surface of the test plate due to air suction.

Additionally, as the test plate comes in contact with

aluminium support, some heat loss takes place as a

result of conductance between the test plate and the

support. Geometry of the test plate with

thermocouple positions as TC1, TC2, TC3, TC4 and

TC5 on the upper surface and the experimental setup

of the aluminium support and the test plate with

installed thermocouples, LVDT sensors and speckle

pattern are shown in figures 1.a. and 1.b.

respectively. Moreover, the schematic sketch of

thermal boundary conditions is also shown in figure

1.c.

Fig. 1.a. Geometry Fig. 1.b. Experimental Setup

Fig. 1.c. Thermal boundary conditions

Temperature histories were recorded during welding

by thermocouples, while in-plane and out-of-plane

displacements were recorded using 3D image

correlation technique.

3 NUMERICAL SIMULATION

3.1 Finite Element Mesh

As the welding was performed in the middle of the

test plate, the selection of a symmetric model for

half of the test plate is a wise approach to

considerably reduce the degrees of freedom and

hence the computation time. The finite element (FE)

mesh consists of 8-nodes linear brick elements and

some 6-node linear prism elements totalling over

58,000 nodes and 50,000 elements. The mesh size

increases progressively across the test plate from

very fine in the fusion zone to very coarse at the far

end. The dimensions of the smallest element along

with the mesh of the symmetric model for test plate

and support are shown in figure 2. The FE code

Abaqus 6.6-1 is used to perform the simulation.

Fig. 2. Finite Element Mesh

3.2 Heat Source Model

The heat source model used to apply the thermal

load consists of a conical part with Gaussian

distribution and an upper hollow sphere with linear

distribution of volumetric heat flux. The schematic

sketch of the heat source model is shown in figure 3.

Fig. 3. Heat Source Model

Equation (1) presents the mathematical model of the

above heat source which is programmed in

FORTRAN as DFLUX subroutine.

(1)

TC2

TC1 TC3

TC5

TC4

30

0 m

m

200 mm

LVDT SENSORS

THERMOCOUPLES

ALUMINIUM SUPPORT

TEST PLATE WITH

SPECKLE PATTERN

0.31

0.32

0.5

Plate

Support

Axis of

symmetry

zsu

zeu

res

ris

ze

zi

zel

Sphere

Cone

Z

X re

ri

Qs Qc

s3

ises

2

c

2

2

iie

2

eie

3v

drr4

f1P3

r

r3

rrrrzz

1

e1

fP9Q

⋅−

−+

−⋅

++−⋅

−=

−

)(

).(

exp))(()(

.

πη

πη

))()(()( 4

abs0

4

abs0convradconv TTTTTThq −−−+−=+ σε)( 0convforcedconvforced TThq −=

)( TThq scondthcondth −=

Here, Qv is the total volumetric heat flux in W/m3, P

is the laser beam power in Watts, η is the efficiency

of the process, f is the fraction of heat flux attributed

to conical section, rc is the flux distribution

parameter for the cone as a function of z and ds is the

flux distribution parameter for the hollow sphere

such that its value is 1 at ris and 0 at res. The

remaining parameters are shown in figure 3. An

efficiency (η) of 37% is used for the thermal

analysis with the power (P) of 2300 W. The

remaining parameters are adjusted to obtain the

required weld pool geometry.

3.3 Thermal Analysis

An uncoupled thermo-mechanical simulation is

performed, where the thermal analysis is first carried

out to calculate the temperature fields with the

boundary conditions (BC) as shown in figure 1.c.

The thermal BC are detailed below.

(2)

where T, T0, Tabs and Ts are the temperature of the

test plate, ambient temperature, absolute temperature

and temperature of the support respectively. The

values used for the heat transfer coefficients and

radiation constants are as follows:

• Convective heat transfer coefficient of air, h conv =

15 W/°C.m2

• Emissivity of speckle pattern, ε = 0.71

• Emissivity of aluminium, ε = 0.08

• Stefan-Boltzmann constant, σ = 5.68 x 10-8

J/K4

.m2

.s

• Convective heat transfer coefficient for air

suction, hforced conv = 200 W/°C.m2

• Thermal conductance, hth cond = 50 W/°C.m2 at

0 bar, 84 W/°C.m2 at 1 bar

DC3D8 and DC3D6 type elements with linear

interpolation between the nodes are used for thermal

simulation. The comparison of experimental and

simulated fusion zone is shown in figure 4.

Fig. 4. Experimental vs simulated weld pool geometry

The temperature histories recorded at thermocouple

positions TC1, TC2 and TC3 in figure 1.a. are

compared with simulated results in figure 5.

Time-temperature Curves - Exp vs Sim

0

20

40

60

80

100

120

140

0 1 2 3 4 5

Time (s)

Tem

perature (°C

)

TC1-EXP

TC2-EXP

TC3-EXP

TC1-SIM

TC2-SIM

TC2-SIM

Fig. 5. Experimental vs simulated time-temperature curves

3.4 Mechanical Analysis

Mechanical analysis is performed using

temperatures calculated in thermal analysis as

predefined fields. C3D8R and C3D6 type elements

with linear interpolation between the nodes are used

for mechanical simulation. The material is assumed

to follow an elasto-viscoplastic law with isotropic

hardening. A friction coefficient of 0.57 is used at

the contact surfaces of test plate and aluminium

support. A suction pressure of 1 bar was applied on

the bottom surface of test plate through the support.

Taking into account the possible leakage present at

the fine rubber joint between the test plate and

support, it is assumed that 80% of the actual

pressure was present between the test plate and the

support. Figures 6 and 7 present the comparison

between maximum and minimum out-of-plane and

in-plane displacements respectively measured

experimentally by 3D image correlation technique

and calculated numerically.

Out-of-plane Displacement across the weld joint - Exp vs Sim

0

0.2

0.4

0.6

0.8

1

1.2

0 25 50 75 100 125

Distance across weld joint (mm)

Vertical Displacement (m

m)

MAX-EXP

MIN-EXP

MAX-SIM

MIN-SIM

Fig. 6. Experimental vs simulated out-of-plane displacements

1.47

1.12

TC1

TC2

TC3

MAX MIN

In-plane Displacement across the weld joint - Exp vs Sim

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

-125 -100 -75 -50 -25 0 25 50 75 100 125

Distance across weld joint (mm)

Displacement across weld joint (m

m)

EXP

SIM

Fig. 7. Experimental vs simulated in-plane displacements

Results of only the symmetric part of the test plate

are shown in figure 6, while that of both sides of

weld joint are shown in figure 7.

Having obtained the good accordance between

experimental and simulated temperature and

displacement results, residual stresses can now be

predicted. Figure 8 shows the magnitude of

predicted residual stresses present in the upper

surface of the test plate. These stresses are presented

for the symmetric part of the test plate only.

Residual Stresses across the weld joint

-100

-50

0

50

100

150

200

250

0 2 4 6 8 10

Distance across weld joint (mm)

Stress (M

Pa)

Sigma_xx

Sigma_yy

Sigma_zz

Fig. 8. Predicted Residual Stresses

It is found that the longitudinal residual stresses (σxx,

stresses in the direction of welding) have the

maximum magnitude and are largely tensile in

nature, while the transverse stresses (σyy, stresses

across the weld joint) are mainly compressive in the

fusion zone and becomes tensile in the heat affected

zone (HAZ). The residual stresses in the thickness

direction, σzz, are negligible. The non-zero

magnitude of these stresses is because of the

interpolation of the values from integration points to

nodes. The test plate regions away from these non-

zero residual stress areas may be regarded as un-

affected base metal.

4 CONCLUSIONS

Based on the results of thermo-mechanical analysis,

following conclusions can be made.

1. Good accordance is found between experimental

and simulation results for temperature histories

and fusion zone geometry.

2. Good agreement is found for out-of-plane and

in-plane displacements between experimentally

measured and numerically calculated results.

3. Residual stress field is predicted and it is found

that the longitudinal stresses, σxx, are as high as

the yield strength of the material and will,

therefore, have the strongest affect upon the

failure of material.

4. As linear interpolation is used between the nodes

for mechanical simulation, improvement in

results may be expected with quadratic

interpolation.

ACKNOWLEDGEMENTS

The author would like to acknowledge the financial support

provided by EADS, AREVA-NP, EDF-SEPTEN, ESI Group

and Rhône-Alpes Région through the research program

INZAT4.

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