This work presents a neural networks approach for finding the effective activation energy and modeling the dissolution rate of hardening precipitates in aluminium alloys using inverse analysis. As way of illustration, a class of multilayer perceptron extended with independent parameters is applied for that purpose to aluminium alloys AA-7449-T79, AA-2198-T8 and AA-6005A-T6

Agelet de Saracibar, Carlos

Chiumenti, Michele

Ducoeur, B.

López, R.

Meester, B.

English

Scipedia

DOI 10.1007/s12289-008-0 -# Springer/ESAFORM 2008Modeling Precipitate Dissolution in Hardened Aluminium Alloysusing Neural NetworksR. Lopez1, B. Ducoeur2, M. Chiumenti1, B. de Meester2, C. Agelet de Saracibar11International Center for Numerical Methods in Engineering (CIMNE), Barcelona, SpainURL: www.cimne.upc.edu, E-mail: rlopez@cimne.upc.edu, chiument@cimne.upc.eud,agelet@cimne.upc.edu2Universite´ catholique de Louvain-la-Neuve, Louvain-la-Neuve, BelgiumURL: http://www.uclouvain.be, E-mail: benjamin.ducoeur@uclouvain.be,bruno.demeester@uclouvain.beABSTRACT: This work presents a neural networks approach for finding the effective activation energy andmodeling the dissolution rate of hardening precipitates in aluminium alloys using inverse analysis. As way ofillustration, a class of multilayer perceptron extended with independent parameters is applied for that purposeto aluminium alloys AA-7449-T79, AA-2198-T8 and AA-6005A-T6.KEYWORDS: Neural networks, Multilayer perceptron, Inverse analysis, Microstructural modeling1 INTRODUCTIONIn heat-treatable aluminium alloys, the dominant mi-crostructural effects are due to changes in the precip-itation state of the material, which provide the princi-pal strengthening mechanism. Modeling the dissolu-tion rate of hardening precipitates is therefore of greatimportance for predicting the hardness after reheat-ing of the base metal. In this regard, various semi-empirical approaches to model the dissolution rate ofhardening precipitates in aluminium alloys have beenproposed in the literature.The dissolution model of Myhr and Grong [3] [4]contains a single independent variable, the time, and asingle state variable, the volumetric fraction of hard-ening precipitates. This model provides a basis forobtaining quantitative information of the reaction ki-netics through simple hardness measurements, ratherthan through microscopic observations of the precipi-tates fraction.In this regard, Shercliff et al. [5] applied the disso-lution model of Myhr and Grong to several aluminiumalloys, and applied their results to friction stir weld-ing. They also showed that this does not apply tosofter tempers.Here a neural networks approach is proposed forthe identification of metallurgical properties for pre-cipitates dissolution in precipitation hardenable alu-minium alloys. The microstructural modeling of alu-minium alloys is formulated as a variational problemincluding independent parameters.The novel methodology is here applied to threedifferent aluminium alloys: AA-7449-T79, AA-2198-T8 and AA-6005A-T6. The experimental hard-ness data is taken through independent measurements,and the final software implementation is performedwithin the Flood library [2].2 DISSOLUTION MODELFirst, assuming that the effect of solid solution hard-ening is small compared to precipitate hardening at allstages of heat treatment, the following linear relation-ship between the Vickers hardness and the volumetricfraction of precipitates can be established [3] [4]ff0(HV ) =HV −min (HV )max (HV )−min (HV ) , (1)where f/f0 is the ratio of the current volume fractionof hardening precipitates to the initial volume frac-tion of hardening precipitates, HV is the actual hard-ness, max (HV ) is the initial hardness in the hardenedstate, and min (HV ) is the final hardness after com-plete dissolution of the precipitates.On the other hand, the model developed by Myrhand Grong [3] for cylindrical precipitates describes139 4Int J Mater Form (2008) Suppl 1:1291–1294the kinetic of transformation of hardening precipitatesin aluminium alloys at constant temperature by thefollowing expression,ff0(t) = 1−√tt∗(T ), (2)where t is the time and t∗(T ) is the time for com-plete dissolution at temperature T . The full dissolu-tion time t∗(T ) can be calculated as [3]t∗(T ) = tR exp[QR(1T− 1TR)], (3)where R is the universal gas constant, tR is the ref-erence time for complete dissolution at the referencetemperature TR selected for calibration, and Q is theeffective activation energy for precipitates dissolu-tion.It is easy to see that taking log (1− f/f0) as afunction of log (t/t∗) in (2), the resulting plot resultsin a straight line of slope 0.5.However, Myrh and Grong [3] and Shercliff etal. [5] experimentally observed that this theoreticalmodel overestimates the the dissolution rate at thelater stages. They proprosed to modify the model asfollowslog (1− ff0) = g(log (tt∗)), (4)where g is a function whose form is given by a “look-up table”. Here the early stages of dissolution followa straight line of gradient 0.5, but this slope steadilydecreases at the later stages.The formulation of Myrh and Grong and Shercliffet al. presents some drawbacks. First, the modelingprocess occurs in two steps, (i) estimation of the ef-fective activation energy and (ii) generation of the dis-solution model. This might result in poor accuracy.Second, a “look-up table” is an unsuitable result here,and an explicit expression for the dissolution model isa desirable aim. Third, the logarithmic representationin the y-axis produces spurious effects. Indeed, sep-arates the data at the lower stages of dissolution andjoins them at the later stages.The representation of the dissolution model pro-posed here is of the form1− ff0= g(log (tt∗)), (5)where the function g is to be found from a suitablefunction space.The dissolution modeling process consists in esti-mating an activation energy Q which provides mini-mum dispersion for the experimental data, while find-ing a function g(·) which provides minimum error.Mathematically, this can be formulated as a varia-tional problem including independent parameters.3 EXPERIMENTAL DATAExperimental tests are performed to get the isother-mal time evolution of Vickers hardness at differenttemperatures and for various aluminium alloys. Threematerials are used here for the isothermal heat treat-ments, AA-7449-T79, AA-2198-T8 and AA-6005A-T6.Figures 1, 2 and 3 depict the results of these Vick-ers hardness tests for the three aluminium alloys con-sidered. Note that, in all figures the Vickers hardnessdecreases with the time. This softening results alsofrom precipitate coarsening. However, the underly-ing mechanism is also controlled by precipitate dis-solution. Hence, this simple model is satisfactory forboth.100 101 102 103 104 10550100150200250AA−7449−T79time [s]Vickers hardness200ºC250ºC300ºC350ºC400ºC450ºC500ºCFigure 1: Vickers hardness test for AA-7449-T79.1292100 101 102 103 104 10560708090100110120130140AA−2198−T8time [s]Vickers hardness200ºC250ºC300ºC350ºC400ºC450ºC500ºCFigure 2: Vickers hardness test for AA-2198-T8.100 101 102 103 104 1052030405060708090100AA−6005A−T6time [s]Vickers hardness200ºC250ºC300ºC350ºC400ºCFigure 3: Vickers hardness test for AA-6005A-T6.Table 1 shows the parameters which are needed totransform from Vickers hardness to volumetric frac-tion of hardening precipitates in (1). Here m and Mare the minimum and maximum Vickers hardness; tRand TR are the reference time in seconds and the ref-erence temperature in Celsius.m M tR TRAA-7449-T79 66.6 207.5 10000 300AA-2198-T8 65 138 10000 350AA-6005A-T6 29.9 95.8 1000 350Table 1: Parameters for aluminium alloys AA-7449-T79, AA-2198-T8 and AA-6005A-T6.4 NUMERICAL RESULTSA variational formulation for neural networks is inthis section applied to find the optimal dissolutionmodel and the effective activation energy of alu-minium alloys AA-7449-T79, AA-2198-T8 and AA-6005A-T6.The first step is to choose a function space to rep-resent the dissolution model (1 − f/f0)(log (t/t∗))[2]. Here a multilayer perceptron with a sigmoid hid-den layer and a linear output layer is used. This neuralnetwork is very useful in inverse problems, since it isa class of universal approximator [1].Here the multilayer perceptron must have one in-put, log (t/t∗), and one output neuron, 1 − f/f0. Itis found that the optimal network architecture for thethree case studied considered here is that with 1 hid-den neuron. On the other hand, information about theeffective activation energy is required. Thus, an inde-pendent parameter Q must be associated to the mul-tilayer perceptron. This neural network spans a fam-ily V of functions (1 − f/f0)(log (t/t∗;α,Q), whichare parameterized by the biases and synaptic weightsvector α and the effective activation energy Q and hasdimension 4 + 1 = 5.The second step is to select an appropriate objec-tive functional, in order to formulate the variationalproblem [2]. The simple mean squared error betweenthe dissolution model and the volumetric fraction datais here used. The mathematical expression of this ob-jective functional isF [y(x;α,Q)] =1SS∑s=1(y(x;α,Q)(s) − y(s))2, (6)where S is the number of samples available, andy(x;α,Q)(s) is calculated by means of (1).Note that the pairs of values (x, y) in (6) are notfixed, since they depend on the particular value usedfor the effective activation energy Q. Moreover, theoptimal value for this independent parameter will pro-vide the minimum dispersion of the data.The variational problem can then be formulated soas to find a function y(x;α∗, Q∗) and an independentparameter Q∗ for which the objective functional in (6)takes on a minimum value.The third step is to choose a suitable training al-gorithm to solve the reduced function optimizationproblem [2]. To check for presence of local minima aquasi-Newton method with BFGS train direction andBrent optimal train rate methods [2] has been usedfrom different initial guesses. The results providedare in all cases the same, and no local minima seemto appear for these particular case studies.1293Equations (7),(8) and (9) are an explicit expres-sion for the dissolution model of AA-7449-T79, AA-2198-T8 and AA-6005A-T6, respectively. Here x de-notes log t/t∗ and y denotes 1− f/f0.y∗(x;α∗, 140kJ/mol) = 0.5[−0.075 + 0.888× tanh { 1.429 + 3.671((x + 6)/6− 1)}+ 1], (7)y∗(x;α∗, 159kJ/mol) = 0.5[ 0.015 + 0.870× tanh { 1.037 + 3.535((x + 6)/6− 1)}+ 1], (8)y∗(x;α∗, 176kJ/mol) = 0.5[ 0.044− 1.015× tanh {−1.298− 2.835((x + 6)/6− 1)}+ 1]. (9)Figures 4, 5 and 6 show the results provided by theneural networks for aluminium alloys AA-7449-T79,AA-2198-T8 and AA-6005A-T6, respectively. Com-paring the shape of these curves to those reported byMyhr and Grong [3] and Shercliff et al. [5], thereare similarities and differences. For medium and highstages of dissolution they are very similar. For earlystages the neural network fits the experimental data,while those curves reported in the literature do not.−6 −4 −2 0 2 4 600.20.40.60.81AA−7449−T79log(t/t*)1−f/f0200ºC250ºC300ºC350ºC400ºC450ºC500ºCModelFigure 4: Dissolution model for AA-7449-T79, withQ∗ = 140kJ/mol.−6 −4 −2 0 2 4 600.20.40.60.81AA−2198−T8log(t/t*)1−f/f0200ºC250ºC300ºC350ºC400ºC450ºC500ºCModelFigure 5: Dissolution model for AA-2198-T8, withQ∗ = 159kJ/mol.−6 −4 −2 0 2 4 600.20.40.60.81AA−6005A−T6log(t/t*)1−f/f0200ºC250ºC300ºC350ºC400ºCModelFigure 6: Dissolution model for AA-6005A-T6, withQ∗ = 176kJ/mol.5 CONCLUSIONSNeural networks were successfully used to find theeffective activation energy and model the dissolutionrate of hardening precipitates for different aluminiumalloys.Future work includes applying of these results toprocess simulation in the metallurgical industry, suchas friction stir welding.ACKNOWLEDGEMENTThis work has been supported under the 6th Framework Pro-gramme of the European Comission through the project “De-tailed Multi-Physics Modelling of Friction Stir Welding (DEEP-WELD)”, Ref. AST4-CT-2005-516134.REFERENCES[1] K. Hornik, M. Stinchcombe, and H. White. Multilayer feed-forward networks are universal approximators. Neural Net-works, 2(5):359–366, 1989.[2] R. Lopez. Flood: An open source neural networks c++ li-brary. www.cimne.com/flood, 2005.[3] O.R. Myrh and Ø. Grong. Process modelling applied to6082-t6 aluminium weldments - i. reaction kinetics. ActaMetallurgica et Materialia, 39(11):2693–2702, 1991.[4] O.R. Myrh and Ø. Grong. Process modelling applied to6082-t6 aluminium weldments - ii. applications of model.Acta Metallurgica et Materialia, 39(11):2703–2708, 1991.[5] H.R. Shercliff, M.J. Russel, A. Taylor, and T.L. Dickerson.Microstructural modelling in friction stir welding of 2000series aluminium alloys. Mecanique & Industries, 6:25–35,2005.1294

Modeling Precipitate Dissolution in Hardened Aluminium Alloys using Neural Networks

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Modeling Precipitate Dissolution in Hardened Aluminium Alloys using Neural Networks

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