A process model for on-line quenching of aluminium extrusions

A Process Model for On-Line Quenching of Aluminium Extrusions NIKLAS JARVSTRAT and STIG TJOTTA A complete process model for the cooling of aluminium extrusions is presented. It is capable of predicting both thermally induced distortions and possible strength reductions. The model consists of three parts: a thermal part, a metallurgical part, and a mechanical part. The thermal part includes heat-transfer and heat-conduction models and generates the temperature history needed as input to the other two parts. The metallurgical part consists of a kinematic model for the precipitation of nonhardening particles during cooling, and it predicts the resulting strength after subsequent aging. Finally, the mechanical part comprises the usual compatibility and consistency equations, as well as a unified material model that is very accurate both for rate-dependent material behavior at high temperatures and for the virtually rate-independent behavior at low temperatures. Water-cooling experiments have been performed, and finite element simulations were executed using the process model. The heat-transfer coefficient for water quenching is shown to be extremely sensitive to geometry and other cooling conditions. In addition, the cooling characteristics and the material model are factors of equal importance in the prediction of distortions. I. INTRODUCTION ALUMINUM alloys of the AIMgSi type obtain their good strength through a high density of fine Mg2Si precipitates. The fine precipitates are obtained by first dissolving large, nonhardening MgzSi particles at a high temperature, then allowing the fine precipitates to form through a separate aging operation at a lower temperature. During extrusion of these alloys, the hot-working temperature is normally high enough for dissolution of the Mg2Si particles. Thus, through an on-line cooling to room temperature followed by aging, the separate solution treatment becomes superfluous.Vl However, the applied cooling rate is restricted by two phenomena. On the one hand, too slow cooling leads to heterogeneous precipitation of large Mg2Si particles during cooling. This reduces the concentration o f Mg and Si in solid solution so that less of the fine precipitates can form during ageing. Significant reduction in properties can therefore be the outcome.tEl On the other hand, a too rapid cooling leads to large temperature gradients causing thermally induced deformations and high residual stresses. This makes it difficult to remain within geometrical tolerances. The optimum cooling conditions are obtained by cooling through the so-called quench window between the two limitations, as illustrated schematically in Figure I. Reduced properties are usually experienced when using air cooling, while distortions are mainly a problem with water quenching. The thermally induced distortions are strongly influenced by section shape. For instance, when the section has a difference in wall thickness, thin parts will cool faster than NIKLAS J,~RVSTRAT, formerly with the Research and Development center, Hydro Aluminium, is with the Military Engines Division, Volvo Aero Corporation, S-461 81 Trollhiittan, Sweden. STIG TJOTTA is with the Research and Development Centre, Hydro Aluminium, N-4265 Havik, Norway. Manuscript submitted September 26, 1994. METALLURGICAL AND MATERIALSTRANSACTIONS B thick parts, and a banana-shaped extrusion results. The same effect can also occur if the heat transfer is uneven around a shape. This is the case when the nozzles are asymmetrically positioned. The section shape can also affect the strength. For example, an internal ligament of an extrusion cools more slowly than external walls, which may lead to a reduced strength in the ligament. In addition, the sensitivity to precipitation during cooling is highly alloy dependent. Thus, the width of the quench window depends on factors such as alloy, size and shape of the extrusion, and nozzle position. For complex extrusion shapes, optimal quenching requirements and an appropriate quenching method are not easily established under production conditions. Mainly by using empirical knowledge, the operator must choose process parameters and quench medium. The most commonly used quench media are forced air, water spray, and water. Moreover, asymmetrical shapes often require asymmetric cooling that further complicates the empirical approach. The present article deals with a mathematical model that accounts for both of the limitations on the cooling rate. The model consists of three parts: a thermal part, a metallurgical part, and a mechanical part. In Section II, the mathematical models employed are presented; in Section Ill, an example of model application is demonstrated; and in Section IV, some concluding remarks are given. II. COOLING MODEL The structure of the model is shown in Figure 2, with all possible couplings between the parts of the model indicated. The couplings included in the model are indicated by solid arrows, while neglected couplings are shown as dashed arrows. The coupling from the mechanical to the thermal part can be neglected, since during cooling, the thermally induced deformations are small, so the heat generation due to plastic dissipation is negligible. Also, the effect of latent heat due to possible precipitation is negligible due to the very small volume fraction of precipitates, VOLUME 27B, JUNE 1996---501 ,Temperature r cooling where p is the density, c the specific heat, k the isotropic heat conductivity parameter, and T the temperature. During normal extrusion conditions, the extrusion velocity is 0.1 to 1 m/s. For this velocity, it can be shown that if one follows a two-dimensional cross section of the extrusion through the cooling zone, the thermal gradient in the axial direction is negligible. Thus, axial heat conduction is also negligible, and the heat equation can be solved in two dimensions. The thermal boundary condition is given by -k Time Fig. l--Sketch of cooling speed limitations. I Thermal - Temperature l" . . . . . . . . .iMetallurgica - Strength ) Mechanical Stress 1 Strain Distortions - - - Fig. 2--Sketch of cooling model structure consisting of three parts. The solid arrows indicate couplings that are included, while dashed arrows indicate neglected couplings. being well under 1 pct. Furthermore, MgzSi precipitation does not cause any significant volume change.t3I It is therefore acceptable to neglect any stress due to this phase transformation, although some microscopic stresses obviously arise. The MgzSi precipitation does also influence the constitutive behavior and thermal properties of the alloy, but again, we choose to neglect this influence due to the very small fraction that actually precipitates during normal process conditions. Finally, we assume that the thermally induced stresses do not influence the kinematics of Mg2Si precipitation. With these assumptions, the only remaining couplings are those from the thermal part to the metallurgical part and to the mechanical part. The complete cooling model is solved numerically for an arbitrary extrusion shape by applying the ABAQUS* finite *ABAQUS is a trademark of HKS, Inc., Pawtucket, RI. element code. A. Thermal Model The thermal part of the model consists of the heat equation of the form pcT = k 502--VOLUME 27B, JUNE 1996 02 T Ox~ Oxi OT Oxi ni a (T - Tc) where nl is the surface normal vector, a the heat-transfer coefficient, and Tc the temperature of the cooling medium. It should be noted that the heat-transfer coefficient is not a constant property of the cooling medium but a very complex function of the conditions in the cooling medium, the cooling rate, the temperature, the geometry, and the surface condition (cf Reference 4). Air cooling is the least severe cooling method. It is often insufficient for obtaining optimum properties but only rarely leads to unacceptable distortions for usual extrusion shapes. When fans are used, the air velocities at different parts of the surface can be very different. This causes the surface heat flux to vary around the section, since the heat transfer is strongly dependent of the air velocity. However, intemal heat conduction is usually sufficient to ensure a reasonably homogeneous temperature, so that a homogeneous heat-transfer coefficient gives good simulation results. Cooling in water is one of the fastest cooling methods available. There are almost never any problems with reduced properties. However, the rapid cooling causes severe thermal gradients, and distortions are often unavoidable. The heat-transfer coefficient for water is strongly temperature dependent, the a vs T diagram shown as curve 2 of Figure 5 being typical. At high enough temperatures, a continuous steam film is formed instantly as the water evaporates at contact with the surface (the film boiling stage). The steam film insulates the surface from the liquid, and the heat transfer is dominated by radiation and gas convection. As the temperature decreases, however, the steam film breaks up, causing a dramatic increase in heat transfer (the nucleate boiling stage). Now, steam bubbles are formed from direct contact between water and the surface, and the evaporation energy is rapidly transferred from the surface by convection. Decreasing the temperature even further, below the saturation point, the dominating effect is free convection. The formation and breakup of the steam film are very important for the heat transfer but are particularly difficult processes to model, since both formation and breakup take some time (cf Reference 5). The rate of cooling is thus an important factor, and the "transition boiling stage," between film boiling and nucleate boiling, can cover most of the temperature interval experienced during a quench. Local effects such as surface roughness and wettability influence the stability of the film, and global effects such as geometry and temperature of nearby surfaces have an impact on the flow pattern of the liquid, thereby also influencing the film stability. The influence of geometry is especially proMETALLURGICAL AND MATERIALS TRANSACTIONS B nounced in the boiling regime (cf Reference 6). However, a quantitative description of these considerations is lacking, so we will use simple temperature-dependent heat-transfer coefficients. It will also be demonstrated in Section III that for some industrially important processes, the heat transfer is of lesser importance in determining the residual distortions. B. Metallurgical Model The possible strength reduction can be predicted by a socalled "quench factor analysis,"m using the temperature history of a material point, as predicted in the thermal part of the model, as input. The time at a constant temperature needed to reduce the strength by a certain amount is described by the time-temperature-property (TTP) diagram. The TTP curve can be described by the equationm t~(u'T)=ln(u)K~exP(RTKZK~('K3 --- T)~" ] exp ( R ~ ) --= In ( v ) f ( T ) where t~(u,T), the concentration time, is the time needed at constant temperature T to reduce the strength by a factor u. That is, the TTP curve for, say, 90 pct of maximum strength is obtained with v = 0.9. Here, K~ through K4 are materialdependent constants and R is the gas constant. The net strength reduction resulting from continuous cooling is found by adding the contributions at various temperatures. Hence, the resulting strength after a specific cooling history is found by (cf Reference 9) R,, = Rm . . . . (R,, . . . . mi,) e x p ( f df_~) R m where R m~"and RT,~ are the strengths obtained for, respectively, infinitely slow cooling and infinitely rapid cooling. The metallurgical model is implemented in ABAQUS as a user subroutine that is called during the thermal calculations. C. Mechanical Model but are confined to 3 degrees of freedom: extension/contraction and the two possible bending modes. This means that initially plane cross sections are assumed to remain plane. The thermal contraction is appropriately accounted for by allowing uniform contraction of the section; bending (caused, e.g., by uneven cooling) is correctly modeled, but the possibility of buckling or twisting instabilities is excluded. However, in most cases, these instabilities are not a major problem. The classical elastoplastic constitutive equations that rely on the accumulated plastic strain, fdep, to represent the thermomechanical history of the material raise some fundamental problems when both high-temperature and lowtemperature behaviors are of importance.U0,1, The reason for this is that dynamic recovery at high temperatures is not accounted for, causing an overestimation of the low-temperature strength. On the other hand, the alloys of interest are virtually rate independent below approximately 200 ~ making purely viscoplastic models unsuitable at lower temperatures. A unified material model suitable for the large temperature span in question has therefore been developed. The model is thoroughly described elsewhere,U q and in the present article, only a short outline is given. The plastic strain rate is described by C P = ~:o sinh ( ( ~ exp ( R = H R (T) ~p, where R and X,j are an isotropic and a kinematic hardening variable, respectively. Overbars are used to denote equivalent values of tensor entities, and HR and Hx are temperature-dependent hardening parameters. Furthermore, the temperature-dependent neutral yield limit, o0, and the ratesensitivity parameter, f, are o 0 (T) = min [o'a, o-, (T)] and The mechanical model is based on the equilibrium equation 0o'u _ 0 f(T) = Oxj where o-,j is the stress tensor. It is assumed that the strain rate, eu, is decomposed additively into an elastic part, an isotropic thermal part fl(T)Tru, and a viscoplastic part ~ . Applying Hooke's law to the elastic part then gives the following expression for the stress rate: O'i] = Lijk, (Ekt -- EPk, -- [3 (T)T ~k,) where Lou is the elasticity tensor, /3(T) the coefficient of thermal contraction, and 6e the second-order unit tensor. Because the axial thermal gradients are neglected, and there are no external load variations in the axial direction, the mechanical problem also is essentially two-dimensional. Thus, we solve the mechanical problem using the generalized plane strain assumption. With the generalized plane strain assumption, out-of-plane deformations are allowed METALLURGICAL R + o0(T)] 3 (o u'-Xu') ~c(-'-T) / 2 (ok, X~,) AND MATERIALS TRANSACTIONS B O VT<Tp } fmax ( T - Tp) k/T, Tp < T < I (<- r~) k f -~x (T,,,- T) Tc with o, (T) = (Qo/k) (1/T - 1~TIn) + o-,, The rate-sensitivity parameter, f, is zero below the temperature Tp, increases linearly to its maximum fmax at the temperature T,., then decreases to zero at the incipient melting temperature T,,. The term o a is the room-temperature yield stress and Om is the yield stress at incipient melting. It should be noted that the model reduces to the ordinary complementarity conditions of rate-independent plasticity: 9 ~5 = A (o-j - x ,"j )v o-,j- x,j___ R + o0 (T), A _ 0 VOLUME 2 7 B , J U N E 1996---503 9 Thermocouple .i 6mm I Flange, section A lOmm ~ jc. / ~,_ ~ ~ 3 m m / ":11/ /// 60cm , Temperature ~ C] 500 Coeff. 1 Coeff. 2 Point a Point b 400 300 ,! 6 cm / X 200 100 0 Z H R (T~P, Xij = H x (T~. when the strain rate sensitivity approaches zero. An experimental procedure for determining the material-dependent constants is described in Reference 11. The constitutive model is implemented into ABAQUS through a user subroutine. III. i . . . . 0.5 0 Fig. 3--Position of thermocouples during cooling experiment. The end ac enters the water first. = . . . . I . . . . ] I [seconds]] .5 Flange, section B Temperature ~'C] 5OO coo.. 31 E X P E R I M E N T S AND S I M U L A T I O N S As a validation of the model, water-cooling experiments were performed. In the experiments, two identical L-shaped sections in the alloy AA6082 were heated to approximately 540 ~ and held at that temperature for 5 minutes to ensure dissolution of the Mg2Si particles. Then, the sections were quenched by axial submersion into a water bath, at a velocity close to the extrusion velocity in order to reproduce the conditions during extrusion. The water was at room temperature and stirred by blowing compressed air into the bath about 1 m from the section. A simple section geometry was chosen to facilitate the finite element modeling (FEM). The section geometry, shown in Figure 3, was of a type with large variations in wall thickness, since this results in uneven cooling and thus often causes problems with distortions. The sections were instrumented with insulated, mantled thermocouples (type K) at the positions indicated in Figure 3. The geometry of the sections was measured with a Mitutoyo 192-653 measuring instrument on a measuring table (Mitutoyo, flatness 16/zm) before heating and after quenching. With water cooling, strength reduction is not considered a problem for this small section, so no strength measurements were performed. A. Temperature According to the finite element simulations, the temperature in the thin flange is reasonably homogeneous throughout the cooling. Thus, the readings from the thermocouples in the thin flange were used to calculate the heat-transfer coefficient, applying analytical results for an infinite plate. As shown in Figure 4, there was a considerable difference in the thermal history of the flange between the two sections 504---VOLUME 27B, JUNE 1996 ]00 ~ Ol "- ---~'~._~__ .... 0 t .... 0.5 ~ .... I 1 [seconds]1.5 Fig. 4--Measured and analytically calculated cooling curves in two identical sections. The time is zero for each point when the point enters the water. Point a is at the lower end of the section, entering the water before point b. The analytical curves were calculated assuming an infinite plate and using the heat-transfer coefficients in Fig. 5. The most likely reasons for the difference between the two sections are that the velocity with which the sections were lowered into the water was different and that the boiling of water at a hot surface is very sensitive to the conditions near the surface. One of the sections (referred to as section A) entered the water at an average velocity of 0.75 m/s, and the heat-transfer coefficient has the normal nucleate boiling peak shown in curves 1 and 2 in Figure 5. This is supported by the good fit of the calculated curves shown for section A. The average velocity 1 m/s of the other section (section B) apparently did not allow enough time for nucleate boiling to have an influence, and a constant heattransfer coefficient (curve 3 in Figure 5) gives the best fit for this section. Besides the time effect of the difference in velocity, a higher velocity would presumably also cause more turbulence, further delaying the nucleate boiling stage. Another possible reason for the differences is that the sections may have been submerged with a slight angle or at different positions relative to the movement of the stirred water. The experiments were simulated using the model deMETALLURGICAL AND MATERIALS TRANSACTIONS B [W/m2K)] Heat transfer coefficients Head, section A 20000 ,= 9 ... Coeff. 1 ~ Temperature~ C] Coeff. 2 Coeff. 3 500 400 '~ ~- h .=_.,. clCoeff. oeff.PointpointCd2 . 300 Temperature ~C] 0 t I I 100 I 200 t I 300 t I 400 t I t 200 500 Fig. 5--Heat-transfer coefficients used for analytical and FEM calculations. Coeff. 1: Below 250 ~ bilinear. Above 250 ~ the empirical function ot = 1.4 ~/Ampmc,,, exp (0.32 ( T - T,=)/(To - T J ) + ~q~d + ~d,,~o.y2~ Coeff. 2: Empirical function featuring a distinct boiling peak. Coeff. 3: Constant heat-transfer coefficient in the entire boiling regime. 100 as] 0 .... 0.00 I .... 0.50 I .... 1.00 I .... 1.50 I ZOO Head, section B Temperature ~ C] i- 400 300 III1111111 ',', I ' ' Fig. 6--Mesh, consisting of 331 linear generalized plane strain elements used for all finite element simulations. 200 . 1130 0 METALLURGICAL AND MATERIALS TRANSACTIONS B ~ ................ .... 0.00 scribed in this article by employing the FEM mesh shown in Figure 6, using linear elements in both thermal and mechanical calculations. The heat-transfer coefficient was prescribed using the same temperature dependence for all surfaces. In the flange, where the heat-transfer coefficients were determined, the agreement with the experimental results was, of course, just as good with the numerical calculations as with the analytical. However, Figure 7 shows a larger difference between measured and numerically calculated values in the head of the section. The qualitative difference indicates that the temperature dependence of the heat transfer is different in the head than in the flange. The most probable reason is again the time factor of nucleate boiling. Because of the larger mass in the head, the surface cooling rate is lower. Thus, the time available for nucleate boiling to become established is longer. This is supported by the characteristic s-shape of some of the cooling curves in Figure 7 that is a manifestation of the nucleate boiling stage. Apparently, the nucleate boiling stage is delayed for up to 1 second by more or less unknown causes. Although it seems obvious that the heat-transfer coefficient does vary around the section, a quantitative description of this is lacking. Nevertheless, the presented model is a very powerful tool for studying the effects of different assumptions regarding the heat transfer. In this article, however, we have restricted ourselves to geometry-independent heat-transfer coefficients. -- Point c Point d I' 0,50 ''', .... 1.00 [sed6fids] 1 .... 1.50 1 2,00 Fig. 7--FEM calculated and measured cooling histories in the interior of the head of the section. B. Strength Reduction For the geometry studied here, water cooling is sufficient to avoid any significant strength reduction. For an application to another geometry, see Reference 13. C. Distortion To investigate the influence of the heat-transfer coefficient on distortions, we use two different thermal histories for the mechanical calculations. The thermal histories were calculated by using the heat-transfer curves 1 and 3 of Figure 5, respectively. The mechanical behavior was simulated by using both the unified material model and a classical plasticity model. The plasticity model was generated from the unified model, assuming a strain rate that was typical for the cooling experiment. Hence, the difference between the simulations with the two material models is, beside the strain rate effects, above all due to the use of state variables in the unified model. As can be seen in Figure 8, the section will first bend toward the thinner parts as thin parts are cooled more rapidly. When the temperature once again levels out, the section straightens out almost completely, leaving a small VOLUME 27B, JUNE 1996--505 Section A, x deflection x deflection [mm] [~m] 2 1.5 "- _.c,ol o.1 -6--8 Unified, coeff. 1 1 i Unified, coeff. 3 0.5 - - Plastic, coeff. 1 0 n -0.5 -- 20 40 60 axial position [cm] -10 -- Section A, y deflection [~] y deflection [mml 4 -- 2 -" Unified, coeff. 1 i -4-20'(:-01 ~ / [selc0onds] Unified, coeff. 3 -0_5 Plastic, coeff. 1 axial position [cm] Section B, x deflection [mm] Fig. 8 ~ a l c u l a t e d histories of bending (deflection of the center point), using the proposed unified model and using a conventional rateindependent plasticity model. The heat-transfer coefficients refer to Fig. 5. 1 [] 0.5 residual deflection. The residual deflection may be in the same direction as the original bending or may be an overcompensation, as seen in Figure 9. The distortion at the final, homogeneous temperature is exclusively due to plastic deformations incurred during the cooling. Thus, the final direction of bending is determined by whether plastic deformations are more dominant during bending or during straightening. Because the total plastic deformations (-0.5 pet) are much larger than the ones contributing to the bending (-0.05 pet), the final shape is very sensitive to differences in the boundary conditions. For example, the predicted bending in the x direction is in opposite directions when different heat-transfer coefficients are used. The simulations with the conventional plasticity material model and the unified material model show considerable differences in the maximum deflection, although similar values for the bending are reached at room temperature. Figure 9 reveals that none of the simulations gave a prediction in complete quantitative agreement with experiments. It is clear, though, that the effect of differences in the heat-transfer coefficient is large enough to explain the discrepancy. Note that in all simulations, the heat-transfer coefficient was assumed to be described by the same function of temperature at all points of the surface. Better results would probably be obtained if the correct geometrical dependence was known or could be reasonably estimated. The slight buckling of section B, evidenced in Figure 9 by the 506--VOLUME 27B, JUNE 1996 0 -0.5 I - 0I- "~ ~ 60 axial position [cm] Section B, y deflection [~] 0.5 _2 . . . . . . . . . -0_.5 axial position [cm] Fig. 9--Calculated and measured deflection after quenching of the L section. Experimental values are denoted by markers. Thick lines are calculated using the unified material model, and thin lines are calculated using rate-independent conventional plasticity. For section A, heat-transfer coefficient 1 was used in the simulations, while coefficient 3 was employed for section B. (The heat-transfer coefficients are defined in Fig. 5.) x deflection of the flange, is considered to have a minor effect. METALLURGICAL AND MATERIALS TRANSACTIONS B [mml physical background gives more confidence in the unified model (cf Reference 11). 5 x deft., plastic 4 ..... 3 2 y deft., plastic x deft., unified 1 ---- 0 0 20 .. y deft., unified 40 [cm] 60 Fig. 10--Predicted deflection when the section was restrained against bending during cooling. -IOOMPz ]io IOOMPa (a) ~ i -IOOMPa ~~ 0 "(b) Fig. l l---Contours of the axial stress at room temperature but before removing the puller: (a) unifiedmodel and (b) plastic model. Figure 9 also demonstrates that the calculations are just as sensitive to the mechanical material model as to the heattransfer coefficient. It is not surprising that conventional plasticity gives as good an experimental agreement as the more detailed unified model, since the strain rate was known approximately from the simulations with the unified model, and yield strengths for this rate were used. However, the good agreement of the plastic model may only be a coincidence. Figure 8 shows that the initial bending in the y direction is much smaller than in the simulation with the unified model although the residual bending eventually reaches comparable values. This could well indicate that two different errors are made with the plastic model, which cancel in this particular case. For example, the neglected rate effects at high temperatures could be compensated by the overestimation of the low-temperature yield stress that is caused by high-temperature straining. Consequently, even though the comparison with experiment gives more or less similar results for the conventional model and the unified model proposed in this article, its METALLURGICALANDMATERIALSTRANSACTIONSB D. Distortion, with Puller Both the experiments and the simulations in Section III were performed with free and unloaded ends of the section. However, in the real process, a so-called puller loads the section in the axial direction during extrusion, preventing any bending during cooling. This will dramatically change the character of the distortion after unloading the puller force, as two degrees of freedom will be locked, making the problem strain governed instead of stress governed. The effect of the puller is simulated by applying a force of 1000 N in the axial direction and constraining bending of the generalized plane strain model during cooling. After cooling to room temperature, the force and the constraint are removed in a separate step. Because the material is not rate sensitive at room temperature, the length of this step is without significance. The predicted magnitude of the distortion is much larger with puller than without. This may seem surprising at a first glance, as the puller is supposed to keep the section straight. Doing so during the cooling, however, constrains the bending peak at 0.1 seconds, shown in Figure 8. Thus, the thermal strain is accommodated by an opposing elastoplastic strain. When the puller is released, the elastic part of the strain is relieved by the deflection shown in Figure 10. With the present geometry and the constraints caused by the puller force, it turns out that the problem is virtually insensitive to the heat-transfer coefficient. The difference in residual deflection caused by different heat-transfer coefficients is not even visible in Figure 10. Any of the coefficients in Figure 5 is large enough to make almost the entire section flow plastically. The only effect of different cooling rates is in the size of the very small elastic transition zone between tensile and compressive longitudinal yield. Although the elastic transition zone is much smaller at elevated temperatures due to the lower yield stress, large areas of homogeneous stress are still evident after cooling to room temperature, as shown in Figure 11. The relation between tensile and compressive yield is determined by axial equilibrium, so the location of the elastic transition zone is determined almost exclusively by the geometry and is thus not sensitive to the cooling rate. However, the distortions are presumably more sensitive to spatial differences in the heat transfer than to differences in magnitude. On the other hand, as a consequence of an incorrect assumption of conventional plasticity, there is a significant difference between the two material models, According to that assumption, all plastic strains contribute equally to the hardness, regardless of the temperature of the deformation. Now, because there are differences in the plastic flow at high temperature, the conventional plastic model will predict a distribution in the room-temperature yield stress. Since most of the material is yielding, this is reflected by the stress contour plot and can be seen in Figure 11 as more distributed stress gradients when the plastic model is used. Also, the increased elastic region before reverse yielding occurs allows a larger elastic bending when the puller is removed. With the unified model, the predicted stresses are lower, and consequently, the residual bending is smaller. This effect is especially pronounced for the y deflection, VOLUME27B,JUNE 1996--507 because during cooling, an equilibrium is established between high compressive stresses in the thin flange, high tension in the center of the head, and moderate compression at the top of the head. This equilibrium is redistributed when the puller is released, and the resulting distortion is sensitive to the current yield strength of the material. IV. CONCLUDING REMARKS A complete process model for the cooling of aluminum extrusions is presented, consisting of three parts: thermal, metallurgical, and mechanical. The model is capable of predicting strength reduction caused by insufficient cooling and the distortions occurring when the temperature gradients are too large. The two most important parameters in the simulation of a cooling process are as follows. presence of a puller makes the simulation much less sensitive to the heat-transfer coefficient, while the mechanical material model used is still of great importance. ACKNOWLEDGMENT The work was partially supported by the Nordic Industrial Fund. REFERENCES 1. T. Sheppard: Mater. ScL Technol., 1988, vol. 4, pp. 635-43. 2. O. Lohne and A.L. Dons: Scand. J. Metall., 1983, vol. 12, pp. 34-36. 3. Aluminium: Properties and Physical Metallurgy, J.E. Hatch, ed., ASM, Metals Park, OH, 1984. 4. D. Menzler and B. Repgen: Aluminium, 1994, vol. 70, pp. 360-64. 5. D.C. Groeneveld: Post Dryout Heat Transfer, Hewitt et al., eds., CRC Press, 1992, oh. 5. 6. R.A. Wallis: Heat Treat., 1989, Dec., pp. 26-31. 7. J.W. Cahn: Acta Metall., 1956, vol. 4, pp. 572-75. 8. J.T. Staley: Mater. Sci. Technol., 1987, vol. 3, pp. 923-35. 9. D.H. Bratland, 13. Grong, H.R. Shercliff, S. Tjotta, and O.R. Myhr: 1. The choice of mechanical material model: the proposed unified model is recommended because of its physical background. 2. The heat-transfer coefficient: more experimental work is needed to investigate the influence of geometry and cooling rate. Proc. 4th Int. Conf. on Aluminium Alloys--Their Physical and Mechanical Properties, Atlanta, GA, Sept. 1994, pp. 418-25. 10. E. Holm and A. Mo: ,/. Therm. Stresses, 1991, vol. 14, pp. 571-87. 11. N. J~irvstr~t and S. Tj~tta: ABAQUS Users" Conf. Proc., Newport, RI, It is demonstrated that using a puller to keep the section straight during cooling may dramatically increase the residual bending after the puller force is released. Further, the HKS, Inc., Pawtucket, RI, 1994, June 1994, pp. 307-16. 12. M. Bamberger and B. Prinz: Mater. Sci. TechnoL, 1986, vol. 2, pp. 410-15. 13. N. Jfirvstr~t and S. Tj~tta: J. Eng. Mater. Sci., 1995, in press. 508--VOLUME 27B, JUNE 1996 METALLURGICAL AND MATERIALSTRANSACTIONS B