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Modelling the constitutive behaviour of aluminum alloy B206 in the as-cast and artificially aged states Mohseni, Seyyed Mohammad; Phillion, André; Maijer, Daan M. Jan 1, 2016

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Modelling the Constitutive Behaviour of Aluminum Alloy B206 in theAs-cast and Artificially-aged StatesS.M. Mohsenia, A.B. Philliona,∗, D. M. MaijerbaSchool of Engineering, The University of British Columbia, Kelowna, CanadabDepartment of Materials Engineering, The University of British Columbia, Vancouver, CanadaAbstractThe constitutive behaviour of the aluminum-copper casting alloy B206 has been investigatedin both the as-cast and artificially aged states. For the as-cast material, a unified plastic flowstress model has been developed using experimental data acquired from compression tests per-formed using a Gleeble 3500 thermo-mechanical simulator. The unified model considers differentconstitutive models, each for a specific temperature interval, and the transition between thesemodels at an intermediate temperature based on the material’s strain rate sensitivity. For theartificially aged material, a linear-fit yield strength evolution model as a function of the heattreatment parameters has been developed using experimental data from tensile tests and hard-ness measurements. The yield strength model takes advantage of an independently-developedmicrostructure model that specifically describes the precipitation kinetics of the material basedon differential scanning calorimetry measurements. Evaluations of the developed models showgood fits between the predicted strengths and data from experiments as well as the literature.Keywords: aluminium alloys, B206, constitutive modelling, precipitation kinetics modelling,Differential Scanning Calorimetry, Kissinger method1. IntroductionThe aluminum alloy B206 is a recently-developed high-strength casting alloy that has strongpotential for use in automotive, aerospace, and energy applications [1]. However, casting de-fects like hot tears and embrittling secondary phases currently limit this alloy’s application tocomponents of simplified geometry [2]. Hot tears are prevalent in B206 due to its long freezingrange, ≈ 100 ◦C, and thus increased usage of this alloy requires precise casting process design [3].In addition to casting, the heat treatment and machining stages can also lead to variations inmaterial properties that affect a component’s in-service performance. To take advantage of theinherent high strength of B206 while addressing this material’s propensity to form defects duringprocessing, a through-process modelling (TPM) approach is being developed for this alloy atThe University of British Columbia. TPM is a recently-developed materials engineering designapproach to enhance component performance by simulating the evolution of microstructure,temperature, stresses and strains, and macro/micro scale defects at each stage of the manufac-turing process [4].∗Corresponding author, andre.phillion@ubc.ca, phone: 250-807-9403, fax: 250-807-9850Preprint submitted to Elsevier September 30, 2015Process simulations developed for each stage of a TPM of a component require broad knowl-edge of the underlying material properties. For the case of B206, mechanical models are requiredto predict the alloy’s response to thermal and mechanical loading during processing over a largetemperature range between room temperature and the semisolid state. As metallic alloys typ-ically have low yield strains, in order to predict component shape and residual stress followingcasting, there is a need for constitutive models describing the in-elastic deformation behaviourin the as-cast state. In addition, thermally-activated evolution of the precipitates causes a sig-nificant transition in the yield strength of the aluminum-copper alloys [5] such as B206 duringartificial ageing. Knowledge of the yield strength is critical for the success of a TPM, since thisparameter is required in order to predict the in-service fatigue life of the fabricated component.Thus, a second constitutive model is required that expresses the yield strength of the materialas a function of the heat treatment parameters (temperature and heating duration).2. Modelling of Constitutive Behaviour in Aluminum Alloys2.1. Plastic Flow StressThe plastic deformation of aluminum alloys is dominated by strain hardening at low tem-peratures, and strain-rate dependent behaviour at elevated temperatures. A model often usedto described this behaviour is the extended Ludwik model[6, 7]:σ = Kεnp (ε˙pε˙0)m (1)where σ, εp and ε˙p represent the flow stress, plastic strain and plastic strain rate, respectively,while K, n and m are temperature dependent material coefficients. ε˙0 is a constant with avalue of 1 s−1. Given its empirical nature, the flow stress predictions are accurate only in therange that matches the experimental data used to develop it. The extended Ludwik model canbe employed to explain the material’s plastic behaviour in the strain-rate dependent regime.However, fitting both strain hardening and strain-rate dependent behaviours to a single modelcauses an overall reduction in the accuracy of the model predictions over the whole temperaturerange of interest. This is mainly due to the averaging characteristic of the extended Ludwikmodel. In such cases, application of multiple models, each for a specific behaviour type, tendsto be advantageous.In the strain-rate dependent regime, the Zener-Holloman model is often used to describe theplastic flow stress:  σ =1αstln{(ZAst)1/nst+[(ZAst)2/nst+ 1]1/2},Z = ε˙p exp(EzhRT) (2)where Ezh, αst, Ast and nst are material coefficients. For both models, the coefficients can befound by fitting to data from mechanical deformation measurements over a range of tempera-tures using fitting/computational programs[8].As the formulations for the the extended Ludwik and Zener-Hollomon models show, theapplication of these models is restricted to a monotonic behaviour path, i.e., continuous hard-ening, or complete strain independent behaviour. Hence, they fail to consider the changes in2constitutive behaviour during testing resulting from microstructure evolution. Although in suchcases the use of multiple models in different ranges can be considered, the introduction of dis-continuities between regimes of behaviour often causes issues when using simulation tools suchas FEA. Furthermore, the material characteristics in the intermediate temperature range wherethe transition occurs would be improperly characterized. Although some of these issues have re-cently been overcome by [9], who used a temperature-dependent averaging constant to combinethe flow stress predictions between two different models selected for low and high temperatures,their approach failed to address any strain-rate dependency within the transition range. Atintermediate temperatures, metals tend to show strain hardening when high strain rates are ap-plied, but strain-rate dependency at low strain rates [10]. Thus, to have an accurate estimationof the plastic flow behaviour of B206 over a wide range of temperature and strain rates, aninclusive, continuous constitutive model must be developed that potentially consists of multiplesub-models while also accounting for any strain rate/temperature dependency within transitionranges.2.2. Yield Strength EvolutionThe evolution of yield strength during industrial processing results from thermally-activatedmicrostructure evolution. Typically, heat treatable aluminum alloys, like B206, are solutionizedand then artificially aged at elevated temperatures to optimize the yield strength while alsoensuring sufficient ductility for a given application. One successful approach for studying thedifferent contributors to the total yield strength of a precipitate-hardenable aluminum alloy isto consider the summation of such parameters in a linear manner [11]:σyield = σppt + σss + σi (3)where σyield represents the yield strength, σi represents the intrinsic strength including thecontributions due to intermetallics, grain size effects and the eutectic phases, and σppt and σssrepresent the contributions from precipitate and solid solution strengthening. Following theweak obstacles theory [12], σppt, and σss can be expressed as:{σppt = Cpptf1/2rσss = Css(1 + ηfr)2/3 (4)where Cppt and Css are material coefficients, fr represents the precipitate fraction in the matrix,and η represents the fraction of solute depleted from the matrix when precipitation has fullyoccurred. It should be mentioned that since Eq.(3) has been developed specifically for precip-itation hardening behaviour of the materials, it cannot account for the softening effects afterpeak-aged state.A close examination of Eq.(3) shows that fr is the only microstructure variable within themodel, and thus predicting σyield requires knowledge of the nucleation and growth of precipitatesduring artificial ageing. For this purpose, an Avrami model can be developed to describe theprecipitation kinetics as a function of the temperature and heating duration:{β = 1− exp (− (kjt)nj)kj = kjo exp(− EjRT )(5)3where Ej represents the activation energy for precipitation, R is the universal gas constant, andnj and kjo are the Avrami coefficients.Due to the difficulties in direct detection and measurement of the precipitates, isothermalcalorimetry is typically used to determine Ej and kjo. However, a number of different studies [13,14] have shown that an analysis based on non-isothermal calorimetry gives similar results whilesaving considerable experimentation time. The reaction rate for a non-isothermal transformationcan be represented as [15]:dβdt= kjo exp(− EjRT)g(β) (6)where g is a reaction kinetics model. By defining the heating rate as dTdt= φ and following theKissinger method [16], which is based on an assessment of the activation energy at the maximumreaction rate (d2β/dt2 = 0), Equation 6 can be rewritten as:EjφmRT 2m= −kjog′(β) exp(− EjRTm) (7)where φm and Tm are the heating rate and temperature at which the maximum reaction rateoccurs, respectively. By assuming a first order reaction, i.e. g′ = −1, Equation 7 reduces to:ln(φmT 2m)= lnkjoREj− EjRTm(8)Thus, Ej and kjo can be determined by performing a series of non-isothermal calorimetryexperiments, determining the temperature at which the maximum reaction rate occurs, and thenplotting the left-hand side of Equation 8 as a function of 1/Tm. Specifically, Ej is given by theslope of the plot and kjo is given by the intercept.3. Materials and Experimental MethodologyA B206 alloy, with composition Al-4.72Cu-0.27Mg-0.27Mn-0.067Fe-0.056Si-0.014Ti (wt.%),was used for the experiments. To generate material with relevant as-cast microstructure, a ringwith dimensions 25 cm outer diameter, 15 cm inner diameter, and 14 cm height was cast in a sandmould at the Natural Resources Canada - Materials Testing Laboratory (CMAT) in Hamilton,ON. From this ring, samples were extracted for mechanical testing.The plastic flow stress behaviour of the material was investigated through compression testingconducted on a Gleeble 3500 thermo-mechanical simulator. The cylindrical compression sampleswere 10 mm in diameter and 15 mm in length. Compression experiments were performed at 14temperatures between 50 and 530 ◦C, and at 4 strain rates, 10−3, 10−2, 0.1 and 1 s−1. Forverification purposes, tests were repeated for 9 different combinations of temperature and strainrate. To extract the plastic deformation of the material in each test, the yield point was selectedusing the 0.2% offset method. The temperature-dependent elastic modulus was calculated as[17]:µ = 2.54× 104(1 + 300−T2Tmelt)E = 2µ(1 + ν)(9)4where Poisson’s ratio, ν, and the melting temperature, Tmelt, were set to 0.33 and 630◦C.The yield strength variation with heat treatment of B206 was investigated through mechani-cal testing. Samples were initially solution treated at 515 ◦C for 2 h followed by an 8 h treatmentat 525 ◦C [18], and then artificially aged at 4 distinct temperatures, 150, 175, 200 and 225 ◦C,for 5 time periods, 1, 2, 5, 10 and 24 h. Tensile tests were performed on cylindrical samples4 mm in dia. and 19 mm in gauge length using a 50 kN Instron 5969 tensile tester. The yieldstress for each heat treatment condition was estimated by the 0.2% offset method, assumingthat the elastic modulus is given by the linear part of the true stress-strain curves. Hardnessmeasurements were conducted on coupons 5×5×10 mm using a Wilson R© VH3100 Vickers mi-crohardness tester. The coupons were heat treated with the tensile samples during the artificialageing process to ensure identical heat treatment conditions. The VH1 data was extracted byaveraging values from 20 readings on a 4×5 matrix, each with a holding time of 10 s.Following the non-isothermal calorimetry approach outlined above, differential scanningcalorimetry (DSC) tests were performed using a NETZSC STA 449 F3 Jupiter thermal analyser.For this purpose, the DSC samples, prepared with an average weight of 25 mg, were solutiontreated using the same conditions as the tensile samples. The samples were then heated at oneof 5 different heating rates: 2, 5, 10, 15 and 20 K min−1 up to 530 ◦C, in a nitrogen atmospherewhile enclosed in an alumina (Al2O3) crucible. The heat flow (W g−1) versus time (s) dataacquired form the first heating cycle was corrected using a second heating cycle as the baseline.4. Constitutive Plastic Flow Stress Model Development and DiscussionOverall, 65 compression tests were conducted to investigate the plastic flow stress behaviourof the B206 alloy and to provide experimental data to develop the unified constitutive model.Based on the strain dependency of the results, the plastic flow behaviour was divided into 3regions: low, high and intermediate temperatures.4.1. Low Temperature (50− 275 ◦C)At temperatures equal to, or below, 275 ◦C, B206 shows a significant amount of strain hard-ening together with low strain rate sensitivity, as can be seen in Figure 1 where flow stress datais plotted at each strain rate for a subset of four temperatures. To model this behaviour, the ex-tended Ludwik model was utilized. The model coefficients at each temperature were determinedusing a least square error method. Then, the entity of the coefficients were fit to a polynomialfunction of temperature. The material coefficients found for the extended Ludwik model andall subsequent models in this paper are shown in Table 1. The model results are also given inFigure 1 for comparison purposes. As can be seen, the model properly replicated the plasticflow behaviour in the low temperature range.At each test temperature between (50 − 275 ◦C), the plastic flow stress exhibits complexstrain-rate sensitivity for at least one tested strain rate. An example of this complexity can beseen in Figure 2, where the material was found to be softer for a strain rate of 10−2 as comparedto 10−3 s−1. The underlying microstructure cause of this behaviour has not been investigatedin the present study, owing to the fact that the strain rate sensitivity is generally low at low5temperatures, so, it will not lead to a significant error within the final constitutive model. Forexample, at 175 ◦C, the maximum difference between the experimental data and predicted plasticstress at 0.01 strain rate is less than 20 MPa, which represents only about 7% of the experimentalplastic stress value at the same point.The accuracy of the developed extended Ludwik model for B206 was verified by calculatingthe average absolute relative error (AARE) [19] at each temperature:AARE(%) =1NN∑i=1∣∣∣∣Mi − PiMi∣∣∣∣× 100 (10)where N represents the number of the acquired data at each data point, and M and P representthe experimentally measured and predicted flow stress values. Over the entire dataset at lowtemperatures, the average of the AARE was found equal to 5.38% with a standard deviationof 2.46%. Note that the calculation of the AARE considered all data points including thoseshowing complex strain rate sensitivity.4.2. High Temperature (350− 530 ◦C)At temperatures equal to, or above, 350 ◦C, B206 shows significant strain-rate dependencyand almost no strain dependency. The experimentally-measured flow stress curves along withthe model predictions are shown in Figure 3. Figure 4 compares the yield strength extractedfrom the experimental data with modelling results. In this plot, each symbol represents adifferent temperature between (350 − 530 ◦C). As can be seen in both Figures 3 and 4, a goodfit is achieved between the experimental data and the predictions given by the Zener-Hollomonmodel, especially at higher temperatures. To quantify the predictability of the developed modelin this temperature regime, the Pearson Correlation Coefficient (PCC) [19] has been calculatedand found to be 0.97. The PCC:PCC =∑i=1N (Mi − M¯)(Pi − P¯ )√∑i=1N (Mi − M¯)2∑i=1N (Pi − P¯ )2(11)was used instead of the AARE in the high temperature regime since the strain-rate dependencyat high temperatures is defined using only a single value for each flow stress. In additionto matching the experimental data, the Zener-Holloman coefficients for B206 (Table 1), fallwithin the range of values reported for other age hardenable aluminum alloys (e.g.[20]). Dueto the physical basis of the Zener-Hollomon model, the material coefficients found for a specificaluminum alloy should be relatively close to other aluminum alloys within the same family.4.3. Intermediate Temperature (275− 350 ◦C)In the temperature range between 275 ◦C and 350 ◦C, a major transition in flow stress be-haviour as a function of the strain and strain rate was observed. As can be seen in Figure 5,which shows the experimentally measured flow stress at 300 ◦C (full lines), the behaviour tran-sitions from strain hardening at high strain rates to strain independence at lower strain rates.This behaviour cannot be described properly by the models developed for the high and low tem-perature ranges, since those models consider a single type of behaviour at a specific temperature.6To address this issue and also to provide a unified model for the whole temperature range foruse in an FEA model, a unified constitutive model is introduced. In this model the transitionfrom a strain-hardening behaviour explained by the extended Ludwik model, σL, to a strainindependent but strain-rate dependent behaviour explained by the Zener-Hollomon model, σZH,is accounted for through a simple averaging coefficient, α, along with a normalized temperature,T˜ :σ = ασL + (1− α)σZH{α = 12(1 + T˜)T˜ = T−TtransTmelt(12)where Ttrans is the temperature at which the material behaviour starts to change, the so-calledtransition temperature, Tmelt is the melting temperature set to 630◦C for B206, and  is a fit-ting constant representing the width of the transition region (i.e. the temperature region where0 < α(T) < 1). The effect of  is given in Figure 6, where a lower value for a given Ttrans leadsto a sharp transition and a higher value gives a wider transition.As discussed previously, the constitutive behaviour in the transition, and in fact the transitiontemperature itself is critically dependent on the strain rate. For this purpose, inspired byconstitutive behaviour modelling of polymeric materials [21], a strain rate dependency has beenincorporated into Ttrans as:Ttrans(ε˙) = ξ logε˙ε˙ref+ Tref (13)where ξ, ε˙ref and Tref are fitting constants.The fitting constants in the unified model were found for as-cast B206 by fitting the modelagainst the experimental compression test data at 300 ◦C and are shown in Table 1. By compar-ing the model predictions at 300 ◦C (dashed lines in Figure 5) to the experimental data, it can beseen that the unified model is able to replicate both significant strain hardening at high strainrates and the strain rate dependent behaviour at low strain rates. Note that for the minimumpossible strain rate (ε˙ = 0), Ttrans = 150◦C. However, for most of the parameters used in themodel, Ttrans ≈ 300 ◦C causing the averaging factor to approach unity at ≈275 ◦C and zero at≈350 ◦C. This ensures a smooth transition to the extended Ludwik and Zener-Hollomon modelsat these extents.5. Yield Strength Evolution Model Development and Discussion5.1. Experimental MeasurementsTensile test and micro hardness measurements on B206 samples artificially aged at differ-ent temperatures and heating durations are shown in Figure 7. The results indicate that yieldstrength and hardness increase with increasing artificial aging time at 150 and 175 ◦C, but de-crease with increasing time at 200 and 250 ◦C. These results are consistent with the resultsobtained from Rockwell tests in an independent study on B206 [22]. The apparent decrease instrength at the high temperatures occurs because the minimum ageing time was 1 h which islikely longer than the time to peak strength; shorter times would be needed to determine thepeak in the curve. Since both a continuous increase and decrease in material strength can be7observed in the experimental data (Figure 7), it can be assumed that the maximum precipitationrate occurs within this temperature range.5.2. Modelling of Precipitate EvolutionAs discussed in Section 2.2, knowledge of precipitation kinetics is essential for to calculatethe yield strength variation in B206, assuming that precipitation of the Al2Cu phases is themajor strengthening agent. To develop the required Avrami model, non-isothermal calorimetrytests using DSC were conducted at different heating rates. One example result from the DSCtests is shown in Figure 8 for a heating rate of 15 K min−1. As expected from a dilute alloy, theonly peak in the heat evolution was detected at a temperature of approx. 250 ◦C. Furthermore,precipitation initiates after ≈5 min of heating, when the temperature was ≈75 ◦C, however, themaximum reaction rate was reached at ≈250 ◦C.Based on the variation in the peak temperature with heating rate, shown in Table 2, Ejand kjo in Eq.(5) were determined and reported in Table 1. As shown in Table 3, a strongagreement is found between the activation energy of the Al2Cu phase calculated in this workand the values reported in the literature in other aluminum-copper alloys. This result confirmsthat precipitation of the Al2Cu particles is the dominant evolving microstructural feature duringartificial aging of the B206 alloy. The Avrami exponent, nj, was then calculated assuming thatprecipitation terminates at the peak aged state, i.e., β = 1. 1 This was done by first identifyingthe four peak-aged points in the tensile test results, highlighted in Figure 7a, then calculatingthe nj value for each (since it was the only remaining variable in Equation 6), then taking theaverage value of the four results.The Avrami exponent is a characteristic of the reaction being described by the Avrami equa-tion. Theory suggest that it equals unity for a reaction with diffusion-based growth of the nucleiin a 2D shape within a saturated matrix [23]. Given that solution treated B206 meets the super1This means that precipitate nucleation does not occur beyond the peak-aged state; only previously formedparticles grow at the expense of smaller particles.Table 1: Values of fitted coefficients in different models.Equa-tionModel Coefficients(1) Extended LudwikK = 4.5e-7T 4-8e-4T 3+0.53T 2-154.5T+17312.67n = -3.51e-8T 3+4.54e-5T 2-1.96e-2T -2.97m = 4.17e-11T 4-5.71e-8T 3+2.82e-5T 2-5.84e-3T+0.42(2) Zener-HollomonEzh = 274.94 kJ, lnAst = 35.72, αst = 0.03 MPa,nst = 4.01(12,13) Unified Model  = 8.7e-3, Tref = 623.5 K, ε˙ref = 1 s−1, ξ = 25 K(5) Avrami Equation Ej = 92.52 kJ, kjo = 1.03e7, nj = 1.1(3,4) Linear-fit Cppt = 60 MPa, Css = 135 MPa, σi = 13 MPa8saturated condition, the value found in this work, nj=1.1, agrees well with the theoretical valuefor the precipitation of the Al2Cu phase. It is worth noting that precipitation is a diffusion-basedphenomenon that in aluminum copper alloys leads to 2D plate-like precipitates [24].The evolution in precipitate fraction predicted by the Eq. (5) is shown in Figure 9a for thefour heat treatment temperatures between 150-250 ◦C. As can be seen, there is slow precipitateevolution at 150 ◦C, leading to a continuous increase in material hardness, and fast precipitateevolution at 250 ◦C, where the over-aged state is reached during the initial heating. Completeprecipitation is predicted to occur in ≈30 h at 150 ◦C but requires only ≈10 min at 250 ◦C5.3. Yield Strength ModellingThe linear-fit model, Equation 3, was used to describe the yield strength evolution of B206as a function of heat treatment. As discussed in Section 2.2, this model is only valid up to thepeak-aged point. This limits the applicability of the experimental results, to be used for modeldevelopment, to 150 and 175 ◦C, since at higher temperatures the over-aged state is reachedin the initial stages of the aging treatment. By assuming that all the solute atoms separatingfrom the solid solution join the precipitate structure, η in Equation 4 equals 1.0. Other mate-rial coefficients in the linear fit model were determined by fitting the experimental data againstthe model. A comparison between the predicted yield strength and the experimental data, inFigure 9b, shows that the model is able to replicate the yield strength evolution of B206 fairlywell. Over this dataset, the AARE was found to have a value of 6.1%, indicating a good matchbetween experimental data and yield strength model predictions.Table 2: Experimentally determined variation in peak heat flow point as a function of heating rate.φm (C/min) 2 5 10 15 20Tm (C) 220.9 234.5 250.7 264.3 269.4Table 3: Comparison between activation energy found for B206 (4.72wt%Cu) in this work and values reportedin the literature for other aluminum-copper alloys.Alloy Ej (kJ mol−1) ReferenceB206 92.52 Present studyAl-1.7% Cu 73.33 - 115.8 [25]Al-2.4% Cu 119.51 [26]Al-3.7% Cu 67.43 - 76.67 [27]Al-4.5% Cu 98.58 [28]6. SummaryThe constitutive behaviour of B206 has been studied in the as-cast and artificial aged states.1. A unified plastic flow stress model was developed from experimental data acquired fromcompression tests conducted on the as-cast material. The developed model predicts the9plastic flow stress via an extended Ludwik model at low temperatures and a Zener-Hollomon model at high temperatures. The intermediate transition regime is modelledthrough the use of an averaging coefficient that takes into account both the temperatureand the strain rate sensitivity of the material. This unified model is able to predict theflow stress of B206 with an accuracy of 94% over a temperature range of 50-530 ◦C and astrain rate range of 10−3-1 s−1.2. A yield strength evolution model for artificial ageing was developed from experimentaldata (tensile and hardness tests, and iso-calorimetry experiments). Although the yieldstrength model was constructed with only a limited set of experimental data points, anaccuracy of 93% was achieved, resulting from the precise description of the precipitationkinetics given by the developed Avrami equation, as it is the main variable in a linear-fitmodel.10References[1] G. K. Sigworth, J. F. 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Maier, Precipitate effects on the mechanical behavior ofaluminum copper alloys: Part I. Experiments, Metallurgical and Materials Transactions A36 (3) (2005) 749–761.[25] M. Starink, P. Van Mourik, Cooling and heating rate dependence of precipitation in anAl-Cu alloy, Materials Science and Engineering: A 156 (2) (1992) 183–194.[26] M. Fatmi, B. Ghebouli, M. Ghebouli, T. Chihi, M. A. Hafiz, The kinetics of precipitationin Al-2.4 wt% Cu alloy by Kissinger, Ozawa, Bosswel and Matusita methods, Physica B:Condensed Matter 406 (11) (2011) 2277–2280.[27] M. Fatmi, B. Ghebouli, M. A. Ghebouli, T. Chihi, E.-H. Ouakdi, Z. A. Heiba, Study ofprecipitation kinetics in Al-3.7 wt% Cu alloy during Non-Isothermal and Isothermal Ageing,Chinese Journal of Physics 51 (5) (2013) 1019–1032.[28] A. Hayoune, D. Hamana, Structural evolution during non-isothermal ageing of a diluteAl–Cu alloy by dilatometric analysis, Journal of Alloys and Compounds 474 (1) (2009)118–123.12List of Figures1 Comparison between the compression test experimental data and the extendedLudwik model’s prediction in the temperature range between (50 − 275 ◦C) andfor different strain rates. The experimental flow stress and model predictions aregiven by solid and dashed lines, respectively. . . . . . . . . . . . . . . . . . . . . 142 Experimentally-measured flow stress curves at 175 ◦C, showing complex strain-rate behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Comparison between the compression test experimental data and the Zener-Hollomon model’s prediction in the temperature range between (350 − 530 ◦C)and for different strain rates. The experimental flow stress and model predictionsare given by solid and dashed lines, respectively. . . . . . . . . . . . . . . . . . . 164 Correlation between the experimental data and model predictions for plastic stressvalues at high temperatures, for entire strain rate range. . . . . . . . . . . . . . 175 Comparison between the measured plastic flow stresses and the values predictedby the unified constitutive model at 300 ◦C. . . . . . . . . . . . . . . . . . . . . . 186 Variation in the averaging coefficient with the choice of  parameter, specifyingthe width of the transition range. . . . . . . . . . . . . . . . . . . . . . . . . . . 197 (a) Yield strength and (b) hardness variation as a function of ageing time andtemperature. The highlighted points show the maximum yield strength reachedat each temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Example non-isothermal heat flow curve, at 15 K min−1, taken from the DSCexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Results of the (a) Avrami model showing precipitation evolution kinetics and (b)Linear-fit model showing yield strength evolution. The experimental data is givenby dots for comparison purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . 2213(a) 1 s−1 (b) 0.1 s−1(c) 0.01 s−1 (d) 0.001 s−1Figure 1: Comparison between the compression test experimental data and the extended Ludwik model’sprediction in the temperature range between (50− 275 ◦C) and for different strain rates. The experimental flowstress and model predictions are given by solid and dashed lines, respectively.14Figure 2: Experimentally-measured flow stress curves at 175 ◦C, showing complex strain-rate behaviour.15(a) 1 s−1 (b) 0.1 s−1(c) 0.01 s−1 (d) 0.001 s−1Figure 3: Comparison between the compression test experimental data and the Zener-Hollomon model’sprediction in the temperature range between (350− 530 ◦C) and for different strain rates. The experimentalflow stress and model predictions are given by solid and dashed lines, respectively.16Figure 4: Correlation between the experimental data and model predictions for plastic stress values at hightemperatures, for entire strain rate range.17Figure 5: Comparison between the measured plastic flow stresses and the values predicted by the unifiedconstitutive model at 300 ◦C.18Figure 6: Variation in the averaging coefficient with the choice of  parameter, specifying the width of thetransition range.19(a) (b)Figure 7: (a) Yield strength and (b) hardness variation as a function of ageing time and temperature. Thehighlighted points show the maximum yield strength reached at each temperature.20Figure 8: Example non-isothermal heat flow curve, at 15 K min−1, taken from the DSC experiments.21(a) (b)Figure 9: Results of the (a) Avrami model showing precipitation evolution kinetics and (b) Linear-fit modelshowing yield strength evolution. The experimental data is given by dots for comparison purposes.22

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