Optimisation of Preform Temperature Distribution

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: [email protected]; [email protected], [email protected]

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.

1 INTRODUCTION Among the techniques devoted to the manufacture of PolyEthylene Terephtalate (PET) bottles, the twostage Stretch-Blow-Moulding (SBM) process is probably the most popular. This process involves the manufacture of structurally amorphous semiproducts, 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

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.

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).

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. Fig. 1. Wall thickness distribution of the bottle. The error bars show ± 1 standard deviation for a set of 3 trials.

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

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 costfunction. A simple way to proceed is to define the objective function F as the standard deviation of the computed thicknesses, as following: r F( x ) =

1 n ∑ th i − th n − 1 i =1

(

)

2

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.

(1)

r where x represents the set of optimization variables, n is the number of nodes along the bottle height, th i 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 realvalued 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

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.

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

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 nonuniform thickness distribution for the bottle, as illustrated by Fig. 5.

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

Fig. 5. Thickness distribution of the bottle before and after optimization.

1. 2. 3. 4. 5.

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

6. 7. 8.

G. Venkateswaran et al., Adv. Polym. Tech., 17, (1998). D. K. Lee, S. K. Soh, Polym. Eng. Sci., 36, 11 (1996). F. Thibault et al., Polym. Eng. Sci., 47, 3 (2007). M. Bordival et al., Proc. of the Int. Conf. ESAFORM 9, p. 511-514 (2006). M. Bordival et al., Proc. of the Int. Conf NUMIFORM 7, Porto, (2007) F. N. Fritsch and R. E. Carlson, SIAM J. Numerical Analysis, 17, (1980). J. C. Lagarias et al., SIAM J. Optimization, 9, 1 (1998). M. A. Luersen and R. Le Riche, Proc. of the Int. Conf. on Eng. Comp. Tech., p 165-166 (2002).

Measurement of Heat Transfer for Thermoforming Simulations H.L. Choo1, P.J. Martin1, E.M.A. Harkin-Jones1 1

School 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: [email protected] 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/m2-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

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.

Besides conduction, there are also convection and radiation heat transfer from the sheet to the surrounding. Temperature

TA TB

Interface Higher T

Distance Lower T

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): hc =

q [W/m2-K] T A − TB

(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.

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.

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]

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, q = −λ

dT dx

(2)

where dT/dx = temperature gradient in the specimens.

128 mm

38.1 mm 38.1 mm

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 ( (T A + TB ) 2 ) 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.

98 mm 100

Flux meter

90 80

3.2 Materials and methods

70 Temperature (°C)

Fig. 3. Apparatus for measurement of TCC

PVC

60 50 40

PTFE

30

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/mK) 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

20

Flux meter

10 0 0

20

40

60

80

100

120

140

160

Distance (mm)

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

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.

pressure on the TCC of polymer pairs will be investigated in the near future. ACKNOWLEDGEMENTS

100

Flux meter

90

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.

80

PP

Temperature (°C)

70 60 50 40

Hytac-B1X

30

REFERENCES

20 10

Flux meter

0 0

20

40

60

80

100

120

140

160

1.

Distance (mm)

Fig. 5. Temperature distribution along the apparatus for HytacB1X and PP at heater temperature of 90 °C

2.

40 35

3.

TCC (W/m 2-K)

30 25 20

4.

15 10 5

5.

0 40

45

50

55

60

65

70

75

80

Average Interface Temperature (°C)

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.

6.

7.

8.

9.

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

10.

11.

12.

13.

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 P. Collins, P. Martin, E. Harkin-Jones and D. Laroche, ‘Experimental investigation of slip in plug-assisted thermoforming’, In: ANTEC 2001 Conference Proceedings, (2001) D. Laroche, P. Collins and P. Martin, ‘Modelling of the effect of slip in plug-assisted thermoforming’, In: ANTEC 2001 Conference Proceedings, (2001) 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) 361369 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 M.G. Cooper, B.B. Mikic and M.M. Yovanovich, ‘Thermal contact conductance’, International Journal of Heat and Mass Transfer, 12, (1969) 279-300 B.B. Mikic, ‘Thermal contact conductance: theoretical considerations’, International Journal of Heat and Mass Transfer, 17, (1974) 205-214 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 C.J. Yu, J.E. Sunderland and C. Poli, ‘Thermal contact resistance in injection molding’, Polymer Engineering and Science, 30, (1990) 1599-1606 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 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 E.E. Marotta and L.S. Fletcher, ‘Thermal contact conductance of selected polymeric materials’, Journal of Thermophysics and Heat Transfer, 10, (1996) 334-342 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

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: [email protected]; [email protected]; [email protected] 2

Laboratoire d’Automatique, de Mécanique et d’Informatique industrielles et Humaines Le Mont Houy - 59313 Valenciennes Cedex 9 e-mail: [email protected]

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

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.

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 upsettingsliding. 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].

Sample

Contactor

Fig. 2. Scheme of principle of a HUST.

3 THERMAL INSTRUMENTATION Power supply

Frame

Hot upsettingsliding test

Handling arm

Hydraulic group

Induction furnace

Cross-piece control Jack

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.

Frame

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.

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. 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.

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.

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. 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.

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: ϕ g = ϕ e + ϕ c = 1,6.10 7 + 1,3.10 7 = 2,9.10 7 W / m²

β= −λ

∂Tc ∂x

+ β .ϕ g = 0

ϕc = 0,45 ϕg

Tc − Te ⇔ RTC = 6,7.10 −6 m².°C / W RTC

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]. Be β effusivity = = 0,49 Be + Bc where Bi = λi .ρ i .Cpi 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. ϕ g = β TQ .µ .σ N .V = 2,8.10 7 W / m² 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.

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.

2.

3.

4.

5.

6. 7.

8.

5 CONCLUSIONS 9.

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

10.

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. 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. T. H. C. Childs, K. Maekawa, T. Obikawa & Y. Yamane, Metal machining - Theory and applications, Elsevier Publishing (2000). T. Kato, H. Fujii, Energy partition in conventional surface surface grinding, Journal of Manufacturing Science and Engineering 121 (1999), 393-398. 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. M.C. Shaw, Metal cutting principles, Clarendon Press, Oxford (1984), 206-240. 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). 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). 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). 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 e-mail: [email protected] URL: www.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 2 EXPERIMENTAL PROCEDURE 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).

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].

Table1. Polymers used for the mouldings Polymer PS PP

Producer Total Petrochemicals Solvay

Reference Lacqrène 1541 HV 252

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).

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

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).

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).

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

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 Injection temperature Mould temperature Packing pressure

Value 240 40 – 50 6 - 18 - 30 – 42

Unit °C °C MPa

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.

4.1 Influence of the mould temperature

5 ANISOTROPY OF PP SHRINKAGE

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.

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

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.

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).

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. 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).

2.

3.

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

4.

5.

.

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

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/ [email protected]; 3

ASCOMETAL CREAS - BP 70045 - 57301 - Hagondange Cedex [email protected]

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

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

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 110 points

Right line

Outer cylinder 1722 points

21 points

1722 points 21 points 38 points

Inner cylinder

Left line

Bottom plane

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)).  ε1    with : In reference r M frame ε = ε i ε i = ΟΤi − ΟΜ i . Ν i r r   Ο, i , j  M  Νi = 1 εn 

(

(

)

)

(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.

Μ ph

ΟΤ1 M = ΟΤ j M ΟΤn

Τx e Tx 1 M e Tx j M e Tx n

Τy e Ty 1 M e Ty j M e Ty n

K Ph n K e Ph n1 with Τx = Τy = 1mm K M e = Τx.Ν = Ν . x (2) j j K e Ph n j Tx j K M e Ty j = Τy.Ν j = Ν j . y K e Ph n n

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.

r = ε − a.Ph1 − b.Ph 2 − c.Ph i − ... − n.Ph n (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-ring Extension bar TP20 probe Star stylus

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.

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 error Evolution of diameter identification e Error in mm between theoretical and identifyed diameters

with elementary quantities (a, b, c, ..., n) of significant displacements and distortions phenomena ( Ph i , ..., Ph n )

0,040

0,040

0,030

0,030

0,020

0,020

0,010

0,010 0,000

0,000 82

91

121

181

362

723

Number of points from a cylinder section

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).

Outer section Inner section

82

91

121

181

362

723

Number of points from a cylinder 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 offcentring of the inner diameter (due to machining). By deleting them, we did what is often called best

Measured part grade 1, at 20°C

Residual in percent

fit, to better visualize distortion and to write their 3D-mathematical expression (figure 6). Simulated part grade 1, at 20°C

100 80 60 40 20 0

Measured part Simulated part

Initial Tx, Ty, Rx,+ x,y Off- +Pincers' + Barrel + x,y + Banana Ry, Rz centring opening ovalization

Fig. 8. Residual evolution in function of cumulated phenomena

4 CONCLUSIONS Pincers’ opening Quenched points

Theoretical points

Scale factor: 50

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. Vectors effects on errors onto theoretical normals

Phenomena

Mathematical expressions

Ovalization (also dilation)

For each section : eovalx j = Ν j .Ν j . x = Νx 2j eovaly = Νy 2j

y

For each section :

y

z x Theoretical points Scale factor = 5

j

αj  eopen j = 1 − cos  2 α j = − Atan2 (OT j . y; OT j . x )

Pincers’ opening (or closing)

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.

z x α -α

REFERENCES 1.

Theoretical points Scale factor = 5

z

For each generatrix : Barrel (or bobbin)

x

 e barrel = 1 − z j 2  z = −j 1 + 0 .1 × k , k ∈ [0 , 20 ]  j z

Banana

If y ci ≥ 0; ebanana j = Νy 2 × ebarrel j If y < 0; e 2 ci banana j = − Νy × e barrel j 

2. Theoretical points Scale factor = 5

yce x

yci z

xce xci

3.

Theoretical points Scale factor = 5

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

Elementary quantities in mm

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.

4.

5.

6.

0,6 0,5

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

7.

0,4 0,3 Inner cylinder Outer cylinder

0,2 0,1

8.

0,0 -0,1 -0,2

XOvalization

YOvalization

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).

9.

10.

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. 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. R. Mukai and D.-Y. Ju, ‘Simulation of CarburizingQuenching of a Gear. Effect of Carbon Content on Residual Stresses and Distortion’, Journal de Physique IV, 120, (2004) 489-497. A. Weckenmann and M. Knauer, The Influence of Measurement Strategy on the Uncertainty of CMMMeasurements, Annals of the CIRP, 47, (1998) 451-454. 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. C. Baudouin, R. Bigot, S. Leleu and P. Martin, ‘Gear Geometric Control Software: Approach by Entities’, Int. Journal of Advanced Manufacturing Technology, 2007. 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. R.A. Hardin and C. Beckermann, Simulation of Heat Treatment Distortion, In: Proc. 59th Technical and Operating Conference, Chicago, (2005). 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. 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 1

CROMeP - Ecole des Mines d’Albi, Campus Jarlard,8 1013 Albi, c´edex 9, France. URL: www.enstimac.fr/recherche/cromep e-mail: [email protected]; 2 Institut de Math´ematiques, Universit´e de Toulouse, UPS, 31062 Toulouse c´edex 9, France. URL: www.mip.ups-tlse.fr/ 3 ENSAM-Paris, 151 boulevard de l’hˆopital, 75013 Paris, France. URL: www.paris.ensam.fr/ ABSTRACT: Today, around 30% of manufactured plastic goods rely on injection moulding. The cooling time can represent more than 70% of the injection cycle. In this process, heat transfer during the cooling step has a great influence both on the quality of the final parts that are 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 and placement 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 practical methodology to optimize both the position and the shape of the cooling channels in injection moulding processes. We couple the direct computation with an optimization algorithm such as SQP (Sequential Quadratic Programming). First, we propose an implementation of the model reduction in the BEM solver. This technique permits to reduce considerably the computing time during the linear system resolution (unsteady case). Secondly, we couple it with an optimization algorithm to evaluate its potentiality. For example, we can minimize the maximal temperature on the cavity surface subject to a temperature uniformity constraint. Thirdly, we present encouraging computational results on plastic parts that show that our optimization methodology is viable. KEYWORDS: BEM, optimization, reduction model, injection moulding, SQP.

1 INTRODUCTION

over, during the optimization, this method permits to compute exact gradients, thereby avoiding the N direct computations per optimization iteration that are needed by finite-difference gradient approximation (where N is the number of optimization variables). However, thermal models involved in the numerical modelling in injection moulding processes a certain number of numerical difficulties such as size meshing, long simulations, or the necessity to define a homogenized thermal conductivity. In the first part of this paper, we present the use of boundary element method (BEM) and DRM applied to unsteady heat transfer of injection moulds. The BEM software, developed at the CROMeP laboratory [2], was combined with an adaptive reduced modelling [3]. This procedure permits to reduce considerably the computing time during the linear system resolution in unsteady prob-

Numerical simulations for designing injection moulds have become an important developement in injection moulding processes. The location of the cooling channels is a major element in the design of the mould because the cooling time can represents up to 70 % of the injection cycle. We need efficient numerical simulations in order to optimize the process parameters, but models based on full 3D finite element method renders unpractical the use of optimization for this design and placement of cooling channels in injection moulds. In this context, the Dual Reciprocity Method (DRM), introduced by Brebbia [1], is acknowledged to be one of the most effective BEM techniques for transforming domain integrals into boundary integrals. More1

Here, T and q denotes the temperature and the flux denotes, and Ci is equal to 1 since the point i is inside the domain and to 0.5 on its regular boundary. The following Green’s function T ∗ and q ∗ [1] denotes the fundamental solution of this equation. The vector β is define such as:

lem. Then, we present a practical methodology to optimize both the position and the shape of the cooling channels in injection moulding processes. We couple the direct computation with an optimization algorithm such as SQP (Sequential Quadratic Programming) [4]. For the sake of simplicity, we will consider a potential problem defined in a 2D unbounded domain. The capabilities of both the reduced order modeling and the boundary element method will be outlined.

1 β = F −1 T˙ (4) a Matrix F consists of interpolation-function values f = 1 + r at each point.

2 BEM AND DRM APPLIED TO POISSON EQUATION

3 REDUCED MODELING

Using BEM, only the boundary of the domain has to be meshed and internal points are explicitly excluded from the solution procedure. An interesting side effect is the considerable reduction in size of the linear system to be solved [5]. The transient heat conduction in a homogeneous isotropic body Ω is described by the diffusion equation [6], where a is the material diffusion:

Usual reduced models perform the simulation of some similar problem or the desired one in a short time interval. From these solutions, the KarhunenLo`eve decomposition [3] can be performed, allowing to extract the most relevant functions describing the solution evolution.

− → 1 ∂(M, t) ∀M ∈ Ω, ∇ 2 T (M, t) = (1) a ∂t We define the initial conditions and the boundary conditions as:   T (M, t) = T0 ∀M ∈ ΓP φ(M, t) = λ.(T − TC ) ∀M ∈ ΓC (2)  T (M, t = 0) = T 0 (M)

We assume that the evolution of a certain field T (x, t) is known. In practical applications, this field is defined at the spatial mesh nodes xi (with i ∈ {1, · · · , N}), and for some time tm = m.∆t with m ∈ {1, · · · , M}. We introduce the notation T m (xi ) for defining the vector containing the nodal degrees of freedom (temperatures) at time tm . The main idea of the Karhunen-Lo`eve (KL) decomposition tell us how to obtain the most typical or characteristic structure φ(x) among these T m (x) ∀ M. This is equivalent to obtaining a function φ(x) maximizing α defined as: i2 PP hPN p i=1 φ(xi )T (xi ) p=1 α= (5) PN 2 i=1 (φ(xi ))

3.1 The Karhunen-Lo`eve decomposition

Where ΓP is the boundary of the polymer and ΓC the boundary of the channels. The temperature of the coolant is TC and the heat transfer coefficient, h, is related to the coolant flow rate (via Colburn correlation). Several strategies are possible to solve such problems using BEM. M¨atzig [7] propose to use space and time Green’s function. To express the domain integral in terms of equivalent boundary integrals, we introduce the DRM approximation [1]. The solution is defined as a series of particular solutions Tˆk located in each boundary nodes Nn , and each internal nodes Ni . We obtain Eq (3), explain in detail by Mathey [2] Z Z ∗ Ci Ti − a T.q dΓ − q.T ∗ dΓ Γ

=

NX n +Ni k=1



βk CiTˆik +

Γ

Z

Γ

Tˆk .q ∗ dΓ −

Z

Γ

This leads to: "" N #" N ## P X X X e i )T P (xi ) φ(x φ(xi )T P (xi ) p=1

i=1

i=1

=α

N X i=1

 ∗ qˆk .T dΓ

e i )φP (xi ) ∀φe (6) φ(x

where φe denotes the variation of φ(x) which can be rewritten under the form:

(3) 2

T T T φe k.Φ = αφe .φ ∀φe ⇒ k.φ = α.φ

(7)

We define the matrix Q containing the discrete field history, and the vector φ such that its i-component is φ(xi ). This yields to the eigenvalue problem D = Q.QT : 

  Q= 

T11 T12 .. . TN1

T21 · · · TP1 T22 · · · T1P .. . . . . .. . TN2 · · · TNP

    

(8)

The functions defining the most characteristic structure of T P (x) are the eigenfunctions φn (x) ≡ φn associated with the largest eigenvalues. 3.2 A posteriori reduced model

Figure 1: optimization procedure

We solve the eigenvalue problem defined by Eq (7) selecting the eigenfunctions φn associated with the eigenvalues belonging to the interval defined by the largest eigenvalue such as Φn ’s sum is upper or equal to 99.9% of ΦN ’s sum. In practice, n is much lower than N. Let us now try to use these n eigenfunctions φn for approximating the solution. Let B be the following matrix: 

  B= 

φ1 (X1 ) φ1 (X2 ) .. .

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

    

The SQP method is designed for monoobjective optimization [4]. However, pratical optimization problems almost always involves at least two objective functions. One way to proceed in such a context is to consider as cost function a weighted sum of two objectives, but this method involves choosing a weighting parameter. We rather propose here using one objective as optimization criterion, and the other as a non-linear constraint. The first criterion involves miniming the maximal temperature on the cavity surface. The second criterion aims at improving temperature uniformity. More precisely, we formulate our problem under the form:

(9)

We express the linear system of equations resulting from the semi-implicit thermal-model discretization as: T

m+1

=

i=n X

minimize max(Ti ) i∈D

subject to ζim+1 φi

= B.ζ

m+1

X

|Ti − Tmoy | ≤ σ

(11) (12)

i∈D

(10)

where D is the set of discretization elements of the plastic part where the temperature Ti is measured, Tmoy is the average of the Ti ’s, and σ is a user-defined temperature uniformity tolerance, fixed here equal to 4.

i=1

4 MOULD COOLING OPTIMIZATION Each optimization iteration involves performing a BEM simulation and computing the objective and constraint functions. The optimization method allows updating the cooling channel design parameters (subject to the constraints) until a minimum of the cost function is reached [5]. Figure 1 shows the coupling between the thermal solver and the optimization algorithm.

5 APPLICATION CASE

TO

TWO-DIMENSIONAL

It is important to note that using reduced model to optimize the cooling channel location is possible since the vector φn (xN ) does not change when the 3

unsteady equation source term changes, i.e. even if the position of the cooling channels changed. Optimization variables are the coordinates (Xi , Yi ) of each circle center i, and the radius of them. This geometry have 7 channels, thus we have 21 optimization variables in our problem. The coolant temperature is fixed as TC = 30◦C. Figure 2 displays the geometry used to validate our method. Dotted circles show the initial configuration of the cooling channels, and bold circles show the optimized position.

The reduced model permits to divide by more than 40 the CPU of each direct computation, compared to DRM. We used n = 17 in our simulations, whereas N = 288 (nodes number). 6 CONCLUSIONS Our methodology uses BEM to solve the unsteady heat transfer equation during the cooling step of the injection moulding process. Simulation results are used in an optimization procedure to find the best geometry and process parameters according to a given objective function. Reduce model technique involves a Karhunen-Love decomposition leading to an optimal number of approximation functions, allowing to considerable CPU time savings (some times in the order of 40). Our preliminary test showed that our approach is viable for optimizing the design of cooling channels for injection moulding. Various objective functions can be provided by the user (either directly as a cost function or within constraints) . We presently work on more complex 3D moulds with more general parameterizations of the cooling channels.

Figure 2: Channels configuration before and after optimization REFERENCES [1] C. S. Chen, C. A. Brebbia, H. Power. Dual reciprocity method using compactly supported radial basis functions Communications in Numerical Methods in Engineering, editors John Wiley & Sons, Vol 15, 2 , pp. 137-150, 1999. [2] E Mathey, L Penazzi, FM Schmidt, F Rond-Oustau. Automatic optimization of the cooling of injection mold based on the boundary element method Materials Processing and Design: Modeling, Simulation and Applications, Proc. NUMIFORM”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 homogenization. Comput. Methods Appl. Mech. Engrg , editors Elsevier, 2007. Figure 3: Temperature before and after optimization

[4] J. Nocedal and S. S. J. Wright Numerical optimization series in operation research, editors springer, 1999.

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

[5] N. Pirc, F. Schmidt, M. Mongeau, F. Bugarin,. BEM-based cooling optimization for 3D injection molding. International Journal of Mechanical Sciences , Proc. ASMDO’07, Vol 48, 4, pp. 430-439, 2006. [6] S. Kenig, M.R. Kamal. Cooling molded parts, a rigorous analysis. Soc. Plast. Eng. J., Vol 26, pp. 5057, 1970.

Table 1: CPU time comparison Method DRM reduction model

direct computation 38.2 seconds 1 second

optimization 5.6 hours 9.6 minutes

[7] B. A. Davis,P.J. Gramann, J.C. M¨atzig,T.A. Osswald. The dual reciprocity method for heat transfer in polymer processing. Eng. anal. bound. elem., editor, Elsevier, vol. 13, 3, pp. 249-261, 1994.

4

Multiphysics welding simulation model N. Poletz1 , A. Franc¸ois1 , K. Hillewaert1 1

CENAERO Bat. EOLE Rue des Fr`eres Wright, 29 B-6041 Gosselies Belgium URL: http://www.cenaero.be e-mail: [email protected]; [email protected]; [email protected]

ABSTRACT: The electron beam welding (EBW) process is extensively used for assembling titanium and other high strength components in the aircraft engine industry. For such applications, it is important to predict distortions and residual stresses after the welding process. In welding simulation, identifying the main physical phenomena is important to formulate reasonable hypothesis to capture the first order effect. A specific fluid flow model has been implemented in the in house CFD solver (ARGO). This model allows the simulation of the melt pool dynamics during welding by taking into account the influence of different convective terms. A single domain approach with an enthalpy-porosity formulation has been used. The influence of each term on the final melt pool shape has been studied. KEYWORDS: Welding, Fluid Flow, Marangoni Convection, Simulation

1

tive and convective heat transfer occur in melt pool. The temperature induced variation of the surface tension on the melt pool produces thermocapillary effect that combines with the buoyancy force and influences the convection flow. These phenomena have relevance in a wide range of applications with moving front as crystal growth, solidification and welding material process. While in large systems buoyancy forces are the dominant driving mechanism, in small scale systems surface tension forces at the liquid/air interface play a significant role in determining the dynamic of the flow. In addition to the convective terms, the heat loss due to radiation and convection has to be considered.The other fundamental phenomenon that takes place during welding process is the Solid/Liquid phase change. Most alloys solidify with the formation of a two phase region known as mushy zone, which is composed of solid dendrites and interdendritic liquid. A single-domain solidification model has been used. This method overcome many of the limitations of multidomain methods (e.g. of front-tracking methods). This model consists of a single set of equations for momentum and energy which are applied in all regions (solid, mushy and liquid). It requires only a single, fixed numerical grid and a single set of boundary conditions to compute the solution.

INTRODUCTION

Heat transfer during welding can strongly affect phase transformations and thus the metallurgical structure and mechanical properties of the weld. In fusion welding process, fluid flow in the melt pool is responsible for the melt pool shape and temperature distribution in the workpiece. These factors have a close relationship to the resulting material structure and properties, such as microstructure, hardness and surface roughness. The melt flow is influenced by surface tension gradients at the free surface (Marangoni effect) and thermal gradients in the melt pool (natural convection). Most work on heat transfer modelling and fluid flow during welding process has been devoted to the study of laser welding [3, 5, 6]. As beam processes deliver large amount of energy in a very small region of the workpiece, large temperature gradients in melt pool are induced which give rise of important convective heat transport. The different phenomena interfering in the melt pool have been identified and implemented in the in house CFD solver. The mathematical description of the model and main assumptions are described. A comparative study of the influence of the different terms has been performed and the first results are presented. Several Phenomena have to be taken into account to properly simulate fluid flow in melt pool. From energetic point of view, both conduc1

1.1

where Lm is the latent heat of fusion of the alloy. To model the fluid flow in the mushy zone, a permeability function is defined employing the CarmanKozeny equation [1, 4]:

Constitutive equations

The differential equations governing the conservation of mass, momentum and energy are based on continuum formulation given by Chiang and Tsai [2]:

K=

Continuity 



~ · ρV~ = 0 5



!







~ ~ · ρV~ v = ρg + 5 · µl ρ 5v − 5 ρl µl ρ − K ρl (v − vs ) + ρg [βT (T − T 0)]

∂p ∂y

(3)

!

∂p µl ρ ~ · ρV~ w = 5 · µl ρ 5w ~ 5 − − (w − ws ) ρl ∂z K ρl (4) Energy 



!

!

k~ k~ ~ ~ ~ 5h + 5· 5(hs − h) 5·(ρ V~ h) = 5· cs cs

(5)

Where u, v, w are the velocities in the x, y, z directions respectively. The subscript s and l refer to the solid and liquid phases respectively; p is the pressure; µ is the viscosity; K is the permeability, which is a measure of the ease with which fluid pass through the porous mushy zone; βT is the thermal volumetric expansion coefficient; g is the gravitational acceleration; T is the temperature; the subscript 0 represents the reference value for the natural convection in the Boussinesq approximation; h is the enthalpy; k is the thermal 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 flow in the mushy zone. The last term on the right-hand side of Equation (3) is the buoyancy force term which is based on the Boussinesq approximation for natural convection. The first two terms on the right-hand side of Equation (5) represent the net Fourier diffusion flux. The last is the volumetric heat source use to represent the energy flux from the beam. In Equations (1)-(5), the density, specific heat, thermal conductivity, solid mass fraction, liquid are calculated from liquid and solid properties using a mixture law. Phase dproperties are assumed to be constant. However the phase enthalpies for the solid and the liquid can be expressed as: hs = cs T,

180 d2

(7)

where d is related to the dendrite dimension, which is assumed 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 righthand side in Equations (2)-(4) will dominate in solid phase since the liquid fraction gl tends towards 0; hence the velocity is forced to be equal to the solid velocity. For the liquid region this term vanish because gl = 1 and 1/K = 0. This term is only valid in the mushy zone, where 0 < gl < 1. Therefore, the liquid region, mushy zone and solid region can be handled by the same equations. During the fusion and solidification process, latent heat is absorbed or released in the mushy zone via the enthalpy formulation. Solidification shrinkage is handled by the density change between the liquid phase and the solid phase. This density difference induce fluid flow from the front part of the melt pool, where melting occurs, to the rear part of the melt pool where solidification takes place.

∂p µl ρ ~ ρV~ u = 5· µl ρ 5u ~ 5· − − (u−us ) (2) ρl ∂x K ρl 

cl =

(1)

Momentum 

gl3 , cl (1 − gl )2

2

MODELLING CONDITIONS

In the following sections the assumptions are described as well as the different boundary conditions. 2.1

Assumptions • The workpiece is initially at 293K. The heat source is supposed to be fixed and the workpiece move in the positive z-direction with a constant velocity equal to the process velocity. • The surface of weld pool is flat. • Thermophysical properties are supposed to be constant in both liquid and solid phase. • The density variation with temperature is taken into account via the Boussinesq approximation. • The flow is laminar and incompressible.

hl = cl T + (cs − cl )T s + Lm (6) 2

• The liquid volume fraction is assumed to follow a linear evolution versus temperature in the mushy zone.

distribution has been used to model the incoming energy of the beam. The beam is considered to be fix at the position z = 2.75 mm, and the workpiece move with a velocity of 30 mm.s-1. The preliminary computation carried out has permitted to define two different regions. First, we use a coarse grid in the region where the metal is supposed to remain at solid state. Then, the mesh has been refined where complex fluid flow takes place. First computation has been conducted with the porosity source term. Buoyancy force term , standing for natural convection in liquid, has been had in a second step. Third step has been carried out taking into account surface tension effect with the Marangoni boundary condition. Input energy from the beam on top surface of melt pool leads to large temperature gradients. The effect of surface tension variation induces large flow, and temperature distribution near free surface is strongly modified. Isotherm representation of temperature distribution in Figure 2 shows spreading of melt pool near the free surface on both top and bottom. Isotherm spacing at the vicinity of these surfaces is larger than in the centre of the melt pool because of Marangoni effect. In this case the peak temperature is lower since fluid flow redistributes a certain amount of beam energy.

• The surface tension of the liquid phase is supposed to be linearly dependent with temperature.

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

2.2

Boundary conditions

The boundary conditions employed in this study, illustrated in Figure 1, are the following: Top and bottom surfaces At the top free surface, since temperature distribution on the surface of the melt pool are always non uniform, surface tension gradients will appear on the surface and affect the melt flow. The Marangoni shear stress at the free surface in a direction tangential to the local free surface is given by : ∂γ ∂T ∂ V~ · ∂~n =− (8) µ ∂~n ∂T ∂~n where s is a tangential vector, n a normal vector to the local free surface and γ the surface tension of the liquid. As we assume that the surface remain flat, the velocity component normal to the surface is set to 0. In case of full penetration welding, the same momentum balance is applied on both top and bottom surfaces. 3

Figure 2: Computed Temperature fiel in the weld pool.

The present fluid flow is mainly driven by the Marangoni shear stresses. The flow is directed radially outwards from the hottest centre to the side of the melt pool as represented on . It can be seen that the high velocities occur in the vicinity of the top of the melt pool where temperature gradients are the most important. The Marangoni term has a predominant

RESULTS

imulation has been carried out on a TA6V workpiece 10 mm wide, 30 mm long and 2.2 mm thick. A volumetric cylindrical heat source with a gaussian power 3

influence on the flow patterns. In case of full penetration welding, Marangoni forces act on both top and bottom surface and leads to the formation of four vortices near these surfaces (c.f. Figure 3). In this flow regime, the convective heat transfer in the molten material plays a dominant role in the prediction of the weld pool shape. This phenomenon causes a spreading of the melt pool near the free surfaces and a narrowing in the bulk. Hence the melt pool cross section is strongly modified by introduction of surface tension effect.

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

The aim of this work is to used the predicted temperature field as input for the in house finite element code Morfeo to predict distorsions and residual stresses after welding.

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

ACKNOWLEDGEMENT

4

The authors acknowledge the financial support from VERDI (Virtual Engineering for Robust manufacturing with Design Integration). VERDI is a research project within the European 6th Framework Programme. http://www.verdi-fp6.org

CONCLUSION

A single domain model using enthalpy porosity formulation has been implemented in the CENAERO in house CFD solver. This formulation allows the use of a single set of equations and boundary conditions for liquid, solid and mushy zone. The model takes into account physical parameters change between liquid and solid phase, latent heat of fusion absorption during melting and release when solidification takes place. Two different convective terms are considered. Natural convection is evaluated in the Boussinesq approximation for incompressible flows. Surface tension variation with the temperature on free surfaces of the melt pool gives rise to shear stress on these surfaces (Marangoni effect). The calculated weld shape is compared to a transverse macrosection of a electron beam welded joint in Figure 4. The simulated weld shape presents a spreading of the top and bottom part of the melt pool of the same order of magnitude than in the experimental case. The fluid model has permitted to have a better prediction of the temperature distribution in the workpiece during beam welding, particularly for the fusion zone shape.

REFERENCES [1] P. C. Carman. Fluid flow through granular beds. Chemical Engineering Research and Design, 15a:150–166, 1937. [2] K. C. Chiang and H. L. Tsai. Interaction between shrinkageinduced fluid flow and natural convection during alloy solidification. International Journal of Heat and Mass Transfer, 35(7):1771–1778, July 1992. [3] L. Han and F.W. Liou. Numerical investigation of the influence of laser beam mode on melt pool. Int. J. Heat Mass Transf., 47(19-20):4385–4402, 2004. [4] K. Kubo and R. D. Pehlke. Mathematical modeling of porosity formation in solidification. Metall Trans B, 16 B(2):359–366, 1985. [5] J.F. Li, L. Li, and F.H. Stott. A three-dimensional numerical model for a convection-diffusion phase change process during laser melting of ceramic materials. Int. J. Heat Mass Transf., 47(25):5523–5539, 2004. [6] X.-H. Ye and X. Chen. Three-dimensional modelling of heat transfer and fluid flow in laser full-penetration welding. J. Phys. D: Appl. Phys, 35(10):1049–1056, 2002.

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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 Renault – 67 rue des Bons Raisins, 92500 Rueil-Malmaison, France e-mail: [email protected]; URL: www.renault.com [email protected]

1

Centre de Mise en Forme des Matériaux, CEMEF (ENSMP) – 1 rue Claude Daunesse, 06904 SophiaAntipolis, France URL: http://www-cemef.cma.fr e-mail: yvan.chastel@ ensmp.fr 2

Centre des Matériaux (ENSMP) URL: www.mat.ensmp.fr e-mail: [email protected] 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. 3

Key words: Heat treatment, Gear, Distortion, Proper Orthogonal Decomposition basis, Model reduction caused by the parameter variation. 1 INTRODUCTION Many articles ([1, 2], etc.) have shown POD (Proper Orthogonal Decomposition, also called KarhunenLoeve 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

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

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.

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

2 THE DECOMPOSITION

Simulations are performed for the set of parameters defined in Table 1.

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 q t are

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

(1)

T

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

(2)

Ψ = QV

4 THE PERTURBED COMPUTATIONS

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.

Each column of Ψ is a POD vectors. The coefficients Φ of these vectors are given by: Φ=Ψ Q T

Initial displacements Q can be recovered with: Q = Ψ ΛΦ

T

(3) (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

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:

   BC1 BC1 BC n BC n  Q = q t K q t L q t K q t  i f f 4243 1i 4243  1Simulation  1 Simulation n  

(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 )

k γ BC = j

BC k 2 ji

i

(6)

An eigenvector j is considered as specific if the following condition is fulfilled: k k min k γ BC < α max k γ BC j j

(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:

∆Φ s

BC k

=

Φs

BC k

− Φs

BC1

∆BC k

(8)

where ∆BCk is a weight attributed to the boundary conditions change.

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):

∆Φ t f = Φ t f − Φ t f

BC1

(10)

This variation is projected on the directions contained in the matrix δ: p = δ ∆Φ t f T

(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).

The components of the direction related to the change in boundary conditions ∆BCk are computed by:

δ jBCk =

∑ (∆Φ ) i

BC k 2 s ji

(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

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.

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

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).

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.

2. 3.

4.

5.

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. Pinowski B., Principal component analysis of speech spectrogram images, Pattren recognition, Vol. 30 (1997) 777-787. 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). Lasserre, R. and Henault, E., Une méthodologie d’étude des deformations lors du traitement thermique : l’éprouvette de deformation, In: Proc. Journées francoallemandes ATTT-AWT, Belfort (1997) 26-30. 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 from liquid state down to room temperature J. Wisniewski1,3, J.-M. Drezet2 , D. Ayrault3 and B. Cauwe4 1

LG2M, Universit´e de Bretagne-Sud - 56321 Lorient, France URL: www.univ-ubs.fr ; e-mail: [email protected] 2 LSMX, Ecole Polytechnique F´ed´erale de Lausanne - Lausanne, Switzerland URL: lsmx.epfl.ch ; e-mail: [email protected] 3 DEN/DM2S/SEMT/LTA, CEA Saclay - 91191 Gif-sur-Yvette, France URL: www.cea.fr ; e-mail: [email protected] 4 Le Bronze Industriel - 3 av. du g´en´eral Leclerc, 51600 Suippes, France URL: www.lebronzeindustriel.com ; e-mail: [email protected] ABSTRACT: Laboratory tests and inverse methods are used for the determination of the thermophysical properties of a CuCrZr alloy. The solidification path (temperature versus solid fraction curve) is determined using the Single Pan Thermal Analysis (SPTA) technique developed at LSMX. The temperature dependent thermal conductivity is identified by inverse analysis using temperature measurements in one dimensional solidified casting. The thermophysical properties will be used as input data in numerical models of the laboratory test aiming at evaluating the hot cracking sensitivity of copper based alloy in electron beam welding for the International Thermonuclear Experimental Reactor (ITER) project. KEYWORDS: CuCrZr alloy, thermophysical properties, Single Pan Thermal Analysis SPTA, solidification path, thermal conductivity, one dimensional solidified ingot, inverse analysis.

NOMENCLATURE Tm mi ki c0i fs

is performed on a thin rectangular plate instrumented with thermocouples. The welding parameters (speed and heat input) are fixed. As width of the plate decreases, a crack appears. The test consists in determining this specific width and is analysed with the help of numerical modelling, hot tearing criteria, the local Rappaz-Drezet-Gremaud (RDG) approach [4] and a thermomechanical criteria [5], will be evaluated. To carry out the finite element numerical analysis, it´s necessary to know not only the mechanical behaviour from liquid state down to room temperature but also the thermophysical properties of the alloy. The missing physical properties are determined by associating laboratory tests and numerical analysis.

Melting temperature of pure copper Liquidus slope of alloying element i Partition coefficient of alloying element i Nominal composition of alloying element i Mass solid fraction

1 INTRODUCTION The precipitation hardened CuCrZr alloy has been selected as a heat sink of the first wall components for the future thermonuclear fusion reactor ITER [1] owing to its good mechanical and thermal properties. The feedback from its use in Tore Supra [2] showed that this alloy is very sensitive to hot tearing (solidification cracking) during electron beam welding. In order to characterize the hot tearing susceptibility of the alloy and thus define acceptance tests of various supplies, a laboratory test, inspired by the work carried out at the Joining and Welding Research Institute (JWRI) [3] is used. An electron beam weld seam

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

5K/min). The liquidus temperature is TL = 1080◦C. The slope change of the curve at 1075◦ C corresponds to the formation of an eutectic phase according to the CuCr binary phase diagram [8].

Table 1: Chemical composition of CuCr1Zr alloy. Compo. (wt%) min max

Cu bal. bal.

Cr 0.5 1.2

Zr 0.03 0.3

Fe 0.08

Si 0.1

Other 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 nucleation undercooling. Details of this method are available elsewhere [7]. The experiments are conducted with a cylindrical sample (diameter 13.8 mm, height 15 mm) using a high purity gas to minimise oxidation. The samples are subjected to heating and cooling cycles as follows: Room temperature → heating to 1200◦C (variable heating rate) → isothermal holding at 1200◦ C for 3h → cooling (5K/min) to room temperature. For the determination of the thermal conductivity, cylindrical sample of CuCr1Zr (diameter 40 mm, height 70 mm) is solidified under one dimensional heat flow condition. Fives thermocouples are placed at various distances from the water cooled copper chill (4 mm, 19 mm, 33 mm, 48 mm, and 62 mm). Fig 1 presents the mould and the empty crucible together with the five ceramic tubes in which the thermocouples are inserted. The measured temperature histories are then used in an inverse method to identify the thermal conductivity of the alloy at selected temperatures.

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

Assuming that the system contains three major elements: chromium, zirconium and phosphorus (deoxidizer), the lever rule is considered [9, 10]: T = Tm +

3 X

mi c0i i=1 1 − (1 − ki )fs

(1)

This model calculates the solid fraction versus temperature for an equilibrium solidification assuming infinite diffusion in solid and liquid. The coefficients mi and ki are unknown. The binary phase diagram of CuCr alloy gives a rough estimation of mCr and kCr parameters: mCr = −3.5◦ C/%wt and kCr = 0.1. We consider that the experimental cooling rate (5K/min) is near the cooling rate of equilibrium solidification. Iterative least squares fit is used in order to identify the remaining unknown parameters. The method is based upon a minimization of the error between the calculated fs (T ) curves obtained with the lever rule (1) and the measured curve. The lever rule doesn’t take into account the formation of a new phase (i.e. change of slope at 1075◦ C) so the minimization is led between 1080◦ C and 1075◦ C. The obtained values of the coefficients after the minimization are: mZr = −5.65◦ C/%wt, mP = −5.11◦ C/%wt, kZr = 0.1, kP = 0.1. In order to obtain the solidification path in quenched conditions, typical conditions encountered during welding, the estimated parameters are used in

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 for the CuCrZr alloy obtained by SPTA (cooling rate 2

We mainly observe primary A and secondary B phases and porosity. To determine the volume fraction of each phases, micrographs of the alloy are analysed using the analySIST M image software. The area fraction is equal to the volume fraction of each phase. Three images were used to calculate the fractions. The image analysis yields a mean value of 5% of secondary phase. Considering the Scheil-Gulliver model, this percentage allow us to fix the non equilibrium solidus temperature at 1048◦ C (fig 2). Therefore, the solidification interval is 21◦ C in equilibrium solidification and 32◦ C in non equilibrium solidification.

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

3 X

mi c0i (1 − fs )ki −1

(2)

i=1

Experimental data, lever rule and Scheil-Gulliver model results appear in fig 2. The problem with the Scheil-Gulliver model is that the lower limit of the solidification interval is not defined. To solve this problem, the fraction of secondary phase is estimated using transverse section of electron beam welding plate. Fig 3 presents scanning electron micrography of a transverse section. Three distinct 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.

3.2 Thermal conductivity In order to apply the inverse method described by Rappaz et al [11], the specific heat of CuCrZr presented in fig. 5 is used. For temperatures below 900◦ C, the data come from [12]. For temperatures greater than 900◦ C, a linear extrapolation is done. The latent heat of CuCrZr is assumed to be equal to that of 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 3: SEM examination of a transverse section of a plate after welding (secondary electrons).

Figure 5: Thermal conductivity and specific heat of a CuCr1Zr alloy and pure copper.

At low temperatures, the thermal conductivity of CuCrZr is two times smaller than the thermal conductivity of pure copper. Indeed, alloying elements decrease the thermal conductivity. In our case, the phenomenon is even more pronounced because Cr and Zr

Figure 4: SEM detail within the melted zone.

3

remain in supersaturated solid solution during the fast cooling experienced in the 1D casting. The thermal conductivity of CuCrZr determined in the liquid state is huge owing to the high convection experienced by the liquid metal right after filling the mould (fig 1).

Development of plasma facing components for the domeliner component of the ITER divertor. Fusion Engineering and 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 structural material for actively cooled plasma facing and in vessel components. Fusion Engineering and Design, Vol. 75-79, 2005, pp. 469-473.

4 CONCLUSIONS

[3] M. Schibahara, H. Serizawa, H. Murakawa. Finite element method for hot cracking analysis using temperature dependent interface element. Mathematical Modelling of Weld Phenomena, Vol. 5, 2001, pp. 253-267.

The solidification path and the thermal conductivity are determined for a CuCr1Zr alloy by associating laboratory tests and numerical analysis:

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

• the solid fraction versus temperature curve is obtained using the single pan thermal analysis (SPTA) technique developed at LSMX. This yields a better description of the solidification path of the alloy not only in equilibrium conditions but also in non-equilibrium conditions as typicaly encountered in electron beam welding.

[5] N. Kerrouault. Fissuration a` chaud en soudage d´un acier inoxydable aust´enitique. Th`ese CEA-R-5953, mars 2001. [6] EN 12163:1998 - number CW106C [7] F. Kohler, T. Campanella, S. Nakanishi, M. Rappaz. Application of Single Pan Thermal Analysis to Cu-Sn peritectic alloys. To be published in Acta Materialia, 2008.

• the temperature dependent thermal conductivity is deduced from inverse modeling using temperature measurements in one dimensional solidified ingot. It appears that the conductivity is about two times lower than that of pure copper.

[8] T.B. Massalski, J.L. Murray, L.H. Bennet, H. Baker. Binary Alloy Phase Diagrams Volume 1. American Society for Metals 1986. [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 of Aluminium Alloys: Thermomechanical Effects and Solidification Aspects. Thesis 1996.

ACKNOWLEDGEMENT

[11] M. Rappaz, J.-L. Desbiolles, J.-M. Drezet, Ch.-A. Gandinn, A. Jacot and Ph. Th´evoz. Application of Inverse Methods to the Estimation of Boundary Conditions and Properties. Modelling of Casting, Welding and Advanced Solidification Processes VII 10-15 sept. 1995.

The authors express their gratitude to J.-D. Wagni`ere and F. Kohler who carried out the experiments at the Ecole Polytechnique F´ed´erale de Lausanne and to F. Castilan for the metallographical inspections at the Laboratoire des Technologies d’Assemblage (LTA-CEA Saclay).

[12] J. Wisniewski, E. Gautier and P. Archambault. D´etermination des propri´et´es thermophysiques et des e´ volutions microstructurales au cours du chauffage rapide d’un alliage CuCrZr, Rapport de DEA, Ecole doctorale EMMA, 2005.

References

[13] Metals Handbook. Properties and Selection: Nonferrous Alloys and Special Purpose Materials, ASM International Handbook Commities.

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

4

Thermo-mechanical Analysis of Laser Beam Welding of Thin Plate with Complex Boundary Conditions M. Zain-Ul-Abdein1, D. Nélias1, J.F. Jullien1, D. Deloison2 1

LaMCoS, INSA-Lyon, CNRS UMR5259, F69621, France e-mail: [email protected]; URL: www.insa-lyon.fr [email protected]; [email protected]

2

EADS Innovation Works, 12 rue Pasteur, BP 76, 92152 Suresnes Cedex, France e-mail: [email protected] URL: www.eads.net

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

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 twofold 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-ofplane 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

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.

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. 0.5 0.31 0.32 Plate Support

THERMOCOUPLES

TC5

Axis of symmetry

TEST PLATE WITH SPECKLE PATTERN

300 mm

TC4

Fig. 2. Finite Element Mesh

3.2 Heat Source Model

TC2 TC3 TC1

200 mm

LVDT SENSORS ALUMINIUM SUPPORT

Fig. 1.a. Geometry

Fig. 1.b. Experimental Setup

q conv+rad

q conv+rad

q conv+rad

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.

Aluminium Table

Qs Q

q th cond + q forced conv

Z

res ris

c

Sphere

Fig. 1.c. Thermal boundary conditions

re

Temperature histories were recorded during welding by thermocouples, while in-plane and out-of-plane displacements were recorded using 3D image correlation technique.

Cone

X

zsu zeu

ze zi

ri

zel

Fig. 3. Heat Source Model

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

Equation (1) presents the mathematical model of the above heat source which is programmed in FORTRAN as DFLUX subroutine.

Qv =

 3r 2 9ηP. f 1  − 2 ⋅ ⋅ exp π (1 − e −3 ) ( z e − z i )(re2 + re ri + ri 2 )  rc

+

3ηP.(1 − f ) ⋅ ds 4π (res − ris ) 3

  

(1)

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 140

TC1

120 Temperature (°C)

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.

100

TC1-EXP TC2-EXP TC3-EXP TC1-SIM TC2-SIM TC2-SIM

TC2

80 60

TC3

40 20

3.3 Thermal Analysis

0 0

q conv + rad = hconv (T − T0 ) + σε ((T − Tabs ) − (T0 − Tabs ) ) 4

q forced conv = h forced conv (T − T0 )

4

(2)

qth cond = hth cond (Ts − T ) 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. 1.47 1.12

2

3

4

5

Time (s)

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 1.2 Vertical Displacement (mm)

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.

1

MAX

1

MIN

0.8

MAX-EXP MIN-EXP MAX-SIM MIN-SIM

0.6 0.4 0.2 0 0

25

50

75

100

125

Distance across weld joint (mm)

Fig. 4. Experimental vs simulated weld pool geometry

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

4 CONCLUSIONS

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

Displacement across weld joint (mm)

0.03

Z

Ref. line X

0.02

Y

0.01 0 -125

-100

-75

-50

-25

0

25

50

75

100

125

EXP SIM

-0.01 -0.02 -0.03 Distance across weld joint (mm)

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

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.

250

Z

Ref. line

200

REFERENCES

X Stress (MPa)

150

1.

Y Sigma_xx Sigma_yy Sigma_zz

100 50 0 0

2

4

6

8

10

2.

-50 -100 Distance across weld joint (mm)

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 nonzero residual stress areas may be regarded as unaffected base metal.

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