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Constitutive behaviour of aluminium alloy B206 : in the as-cast and artificially aged states Mohseni, Seyyed Mohammad 2015

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Constitutive Behaviour ofAluminium Alloy B206In The As-cast and Artificially Aged StatesbySeyyed Mohammad MohseniB.Sc., Sharif University of Technology, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)August 2015c© Seyyed Mohammad Mohseni, 2015AbstractThe constitutive behaviour of the aluminum foundry alloy B206 has been investigated inboth the as-cast and artificially aged states by combining compression, tension, hardnessand calorimetry testing. Aluminum alloy B206 is a recently-developed high-strengthfoundry alloy that has strong potential for use in automotive, aerospace and energyapplications. The results of the compression tests, performed on as-cast material andconducted on a Gleeble 3500 thermo-mechanical simulator over a broad range of temperature(50-530 ◦C) and strain rates (10−3-1 s−1), were used to develop an inclusive constitutiveplastic flow behaviour model. A new unified constitutive model was introduced thatcombines a Ludwik model for flow stress description at low temperatures (50-300 ◦C)with a Zener-Hollomon model at high temperatures (300-530 ◦C) while accounting forthe strain rate dependency of the transition between these two models. The results ofthe tensile tests, performed on the artificially aged material (ageing performed over atemperature range of 150-250 ◦C and a heating duration range of 1-24 h), were combinedwith the calorimetry experiments to develop a model that predicts yield strength. Tofit the experimental data into a linear-fit type model that states the yield strength asa linear summation of the effective parameters, temperature and time dependency wereintroduced through a microstructural variable, precipitation fraction. The precipitationkinetics of B206 were described by an Avrami model that utilized the experimental dataacquired from non-isothermal calorimetry followed by an analysis based on the Kissingermethod. The remaining material constants in the linear-fit model were found by fittingthe model against the tensile test results explaining the yield strength evolution up tothe peak-aged state. The model predictions were then qualitatively compared to theresults of the hardness measurements, restating the fast precipitation kinetics of B206 atiiAbstractthe temperature range used during artificial aging and predicted by the Avrami model.Together, the developed models of B206 in the as-cast and artificially aged states can beused as part of a Through-Process model to optimize the performance of castings madefrom this alloy.iiiTable of ContentsAbstract iiList of Symbols vAcknowledgements viiiChapter 1 Introduction 11.1 B206, An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Through Process Modelling (TPM) . . . . . . . . . . . . . . . . . . . . . 31.3 Constitutive Behaviour of B206 Alloy . . . . . . . . . . . . . . . . . . . . 4Chapter 2 Literature Review 62.1 Plastic Flow Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Ludwik Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Johnson-Cook (JC) Model . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Voce-type Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.4 Arrhenius-type Equation . . . . . . . . . . . . . . . . . . . . . . . 152.1.5 Advances in Constitutive Modelling . . . . . . . . . . . . . . . . . 182.2 Yield Strength Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Constitutive Model . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Precipitation Kinetics and Modelling . . . . . . . . . . . . . . . . 23Chapter 3 Scope and Objectives 28Chapter 4 Experimental Methodology 29iv4.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Compression Tests on As-Cast Samples . . . . . . . . . . . . . . . . . . . . 314.2.1 Plastic Deformation Determination . . . . . . . . . . . . . . . . . 334.3 Heat Treatment Experiments . . . . . . . . . . . . . . . . . . . . . . . . 344.3.1 Tensile Measurements . . . . . . . . . . . . . . . . . . . . . . . . 364.3.2 Thermal Analysis and Hardness Testing . . . . . . . . . . . . . . . 37Chapter 5 Results and Discussion 395.1 Constitutive Behaviour of As-cast B206 . . . . . . . . . . . . . . . . . . . 395.2 Constitutive Equation for As-cast B206 . . . . . . . . . . . . . . . . . . . 425.2.1 Ludwik Model Development . . . . . . . . . . . . . . . . . . . . . 445.2.2 Zener-Hollomon Model Development . . . . . . . . . . . . . . . . . 475.2.3 Unified Constitutive Model . . . . . . . . . . . . . . . . . . . . . . 495.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Constitutive Behaviour of the Artificially Aged B206 . . . . . . . . . . . 555.4 Constitutive Behaviour Modelling of the Artificiality Aged B206 . . . . . 595.4.1 Modelling of Precipitation Evolution . . . . . . . . . . . . . . . . 595.4.2 Yield Strength Modelling . . . . . . . . . . . . . . . . . . . . . . . 635.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Chapter 6 Conclusion 666.1 Constitutive Plastic Flow Behaviour Modelling . . . . . . . . . . . . . . . 666.2 Constitutive Yield strength Modelling . . . . . . . . . . . . . . . . . . . . 686.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69References 71vLIST OF TABLESList of TablesTable 1.1 Comparison between composition of different registered variants of206 casting alloys (Major and Sigworth 2006). . . . . . . . . . . . . . . . 2Table 4.1 Comparison between the registered composition for B206 and thefinal composition of the material cast at CMAT lab. . . . . . . . . . . . . 30Table 4.2 Eutectic percentage and grain size measurements for coupons takenat different locations in the hub ring. . . . . . . . . . . . . . . . . . . . . . 31Table 4.3 Results of yield point variation with strain rate. Yield point is foundto be a minor variant of the strain rate value. . . . . . . . . . . . . . . . . 37Table 5.1 Ludwik model coefficients for B206 . . . . . . . . . . . . . . . . . . 44Table 5.2 Accuracy of the developed Ludwik model at each temperature . . . 46Table 5.3 Comparison between the Zener-Hollomon parameters found for theB206 alloy and average values reported for age hardenable alloys (McQueenand Ryan 2002). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Table 5.4 Material constants in unified model for B206 . . . . . . . . . . . . 52Table 5.5 Experimentally determined variation in peak heat flow temperaturesas a function of heating rate. . . . . . . . . . . . . . . . . . . . . . . . . . 60Table 5.6 Heat treatment time to peak-aged state at different temperatures,based on tensile test results. . . . . . . . . . . . . . . . . . . . . . . . . . 60Table 5.7 Comparison between activation energy found for B206 (4.72wt%Cu)in this work and values reported in the literature for other aluminum-copperalloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Table 5.8 Linear-fit model parameters for B206. . . . . . . . . . . . . . . . . 64viList of FiguresFigure 2.1 Variation in Ludwik parameters with temperature for A356 (Royet al. 2012). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.2 Comparison between results from the Ludwik model (σLH) andthe Kocks-Mecking (σKM) at (a) high and (b) low temperature ranges forA356 (Roy et al. 2012). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.3 Stress-strain curves for AA6061-Al2O3 composite, obtained fromcompression tests (Davies et al. 1997). . . . . . . . . . . . . . . . . . . . . 17Figure 2.4 Results of the modified Zener-Hollomon model for 2124 (Lin et al.2010) at a) high strain rate, b) low strain rate and A356 (Haghdadi et al.2012) at c)high strain rate, d) low strain rate. . . . . . . . . . . . . . . . 18Figure 2.5 Variation in the experimental (σexpt) and predicted (σtot) hardnessof A356 as a function of aging time at 150 and 180 ◦C (Colley 2011). . . . 22Figure 2.6 Variation in the heat flow peaks with heating rate, from DSC datafor AA8090 (Starink and Gregson 1995). . . . . . . . . . . . . . . . . . . 26Figure 4.1 Demonstration hub component cast by CMAT. Red dots show therelative location of the coupons extracted for metallography. Samples wereextracted from 4cm beneath the top surface of the hub. . . . . . . . . . 30Figure 4.2 Typical microstructure of B206 with Cu-rich eutectic phase on grainboundaries and higher concenteration of coper around grain boundaries.Intermetalics were also mainly detected in areas close to the grain boundaries. 31Figure 4.3 Sketches of the (a) compression and (b) tensile testing samples. . 33viiLIST OF FIGURESFigure 4.4 Typical microstructure of B206 in (a) as-cast and (b) heat treatedstates. A uniform distribution of solute is obtained by heat treatment. . . 36Figure 5.1 Plastic flow stress variation with temperature at (a) 0.01 and (b)1 s−1 strain rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 5.2 Variation of the plastic flow stress with strain rate at differenttemperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 5.3 Variation of the strength factor (K parameter) with temperatureshowing higher strength within temperature range of 150 - 200 ◦C. . . . . 43Figure 5.4 Variation of the (a) K, m and (b) n coefficients in the Ludwik modelwith temperature. Highlighted coefficients were excluded in the polynomialfit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 5.5 Comparison between the experimental flow stress (full lines) andthe Ludwik model prediction (dashed lines) at 175 ◦C, ∆σ ≈ 20 MPa. . . 45Figure 5.6 Comparison between the experimental data (full lines) and theLudwik model prediction (dashed lines) at different strain rates and temperatures. 47Figure 5.7 Calculation of the (a) n1, (b) β, (c) nst and (d) Ezh parameters inZener-Hollomon equations based on the average slope of the plots at eachcurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 5.8 Correlation between the experimentally measured data and theZener-Hollomon model’s prediction over the entire range of strain, strainrate and temperature. and temperature. . . . . . . . . . . . . . . . . . . . 51Figure 5.9 Variation of the averaging coefficient with the  parameter, specifyingthe width of the transition range. . . . . . . . . . . . . . . . . . . . . . . 52Figure 5.10 Comparison between the experimental data (full lines) and theLudwik model prediction (dashed lines) at 300 ◦C. . . . . . . . . . . . . . 53Figure 5.11 Unified model prediction of the flow stress for as-cast B206 at strainrates of (a) 10−4 and (b) 10 s−1. Note: these strain rates were not used infitting the model coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . 54viiiLIST OF FIGURESFigure 5.12 Comparison between the experimental data (full lines) and theLudwik model prediction (dashed lines) at strain rate of 10 s−1. . . . . . 55Figure 5.13 Room-temperature tensile test results from samples heat treated atdifferent aging time and temperatures. . . . . . . . . . . . . . . . . . . . 56Figure 5.14 SEM imaging of a typical fracture surface after tensile testing ofheat treated B206. The observed granular surface is a sign of brittle fractureduring tensile testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 5.15 (a) Elongation, (b) UTS and (c) yield strength variation as a functionof ageing time for temperatures between 150 and 250 ◦C. The lines representtrend-lines while dots show the experiemntal data points. . . . . . . . . . 58Figure 5.16 Example isochronal heat flow curve taken from DSC experimentheated at 15 K min−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 5.17 Predicted (Avrami) precipitation fraction at different temperaturesfor B206. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 5.18 Vickers microhardness as a function of heat treatment temperatureand duration. The lines represent trend-lines. Note that the value at 24 hfor 175 ◦C heat treatment is an estimate based on the remaining data points. 64Figure 5.19 Comparison between the results of the Linear-fit model (full lines)and the experimental data at 150 and 175 ◦C(dot points). . . . . . . . . . 65ixList of SymbolsNote: Units are not defined for material coefficients and fitting parameters.Symbol Definition Unit Symbol Definition UnitAJC Material coef. in Johnson-Cook [–] C ′i=1,..,3 Material coef. in modified Johnson-Cook[–]As1,As2 Material coef. in Voce-Kocks [–] D Diameter [m]Ast Material coef. in Zener-Hollomon D0 Initial diameter [m]Appt Total area under the heat flow curve,up to the peak-aged point[J] E Elastic (Young) modulus [GPa]ai=0,...,4 Fitting coef. in Ludwik [–] EA Activation energy in precipitatecoarsening model[J]a Material coef. in precipitationstrength model[–] Eex Experimentally acquired data [MPa]BJC ,B′JC Material coef. in Johnson-Cook Est modelling/estimated data [MPa]Blsw Constant coef. in precipitatecoarsening model[–] Ezh Activation energy in Zenner-Hollomon [J]B′′ Fitting parameter in Zener-Hollomon Ej Activation energy in Avrami model [J]b Burgers vector [m] F Force [N]bj Material coef. in modified Ludwik [–] Fstr Obstacle strength [MPa]C Local solute concentration [molm3 ]FpeakFZHObstacle strength at peak-aged statePlastic stress-dependent function[–][–]C0 Nominal initial concentration [molm3 ]f Precipitates volume fraction [–]C¯t Solute concentration in time t [molm3 ]fpeak Volume fraction of the precipitates atpeak-age state[–]CJC Material coef. in Johnson-Cook fr Relative precipitate fraction [–]C ′′ Fitting parameter in Zener-Hollomon [–] g Reaction kinetic model(Sa´nchez-Jime´nez et al. 2008)[–]CagCssMaterial coef. in precipitation modelMaterial coef. in solid solution model[–][–]g0 Material coef. in Voce-Kocks [–]Ci=1,..,3 Material coef. in Johnson-Cook [–] K Strength index [MPa]xList of SymbolsKb Boltzman constant [ JK ]kj Avrami coef. [–] Tmelt Melting temperature [K]kjo Proportionality constant [–] T ∗ Homologous temperature [K]L Length [m] T˜ Normalized temperature [K]L0 Initial length [m] Ttrans Transition temperature [K]Lobs Average obstacles spacing [m] t Time variable [s]Llsw Particle growth rate in overaged state [m3s ] tpeak Time to peak-aged state [s]M Taylor factor [–] Z Zener-Hollomon parameter [–]m Strain rate sensitivity [–] α Averaging parameter in Unifiedconstitutive model[–]mag Exponent in extended precipitationstrength model[–] αfr Fraction of the migrated solute, fromthe solid solution to the precipitates[–]n Work hardening coef. [–] αst Material coef. in Zener-Hollomon [–]n1 Fitting parameter in Zener-Hollomon [–] β Transformed material fraction [–]nj Avrami exponent [–] βst Material coef. in Zener-Hollomon [–]nst Material coef. in Zener-Hollomon [–] γi=0,..,6 Material coef. in Voce-Kocks [–]P Material coef. in Cheng-Nematmodel[–]  Material coef. in Unified constitutivemodel[–]pi=1,..,3 Material coef. in modifiedJohnson-Cook[–] ε Plastic strain [–]Q Heat flow [J/s] εb Breaking strain [–]q Material coef. in Cheng-Nematmodel[–] εL0 Material coef. in modified model [–]R Universal gas constant [J/mol .K] εr Relaxation strain [–]r Particle size variable [m] εtrue True strain [–]ri=1,..,3 Material coef. in modifiedJohnson-Cookε˙ Plastic strain rate [s−1]rpeak Average obstacle radius at thepeak-aged state[m] ε˙0 Proportionality constant [s−1]s Material coef. in modified Ludwik [–] ε˙L0 Material coef. in modified Ludwik [s−1]T Absolute temperature [K] ε˙ref Reference strain rate [s−1]Tref Reference temperature [K] Θ0 Initial strain hardening rate [MPa]Tm Temperature at the maximumreaction rate[K] Θ1, Θ2 Material coef. in modifiedJohnson-Cook[–]xiList of Symbolsκ1, κ2 Material coef. in modifiedJohnson-Cook[–]Λ Material coef. in modified Johnson-Cook[–]λ1, λ2 Material coef. in modifiedJohnson-Cook model[–]µ Shear modulus [GPa]µ0 Absolute shear stress [GPa]ν Poisson’s ratio [–]νi=0,..,5 Material coef. in Cheng-Nematmodel[–]ξ Material coef. in modifiedJohnson-Cook model[–]σ Plastic stress [MPa]σtrue True stress [MPa]σyield Yield stress [MPa]σ0ss Strength at as-quenched state [MPa]σi Intrinsic strength [MPa]σb Breaking stress [MPa]σvs Voce saturation stress [MPa]σppt Precipitation strengthening [MPa]σss Solid solution strengthening [MPa]σry Reference yield stress [MPa]φ Heating rate [K/s]φm Heating rate at the maximumreaction rate[K/s]xiiAcknowledgementsI truly appreciate the help and support from all my colleagues who made the completionof this thesis possible and would like to express my sincere gratitude to them:My supervisor, Dr. Andre´ Phillion, who gave me the opportunity to pursue my Master’sdegree training and research in his group. I most appreciate his respect for students’ ideas,kind supports through nurturing them and wise directions all through the way.My co-advisor, Dr. Daan Maijer, who I enjoyed having weekly meetings with him andfavoured his constructive advices.Earnest colleagues Hamid Reza Zareie Rajani and Nima Haghdadi (PhD students atUBC-Okanagan) for their hours spent on experimental help and technical discussions aswell as all the entertaining moments we had together.Dr. Lukas Bichler and his research group who shared moments with our research groupduring weekly seminars, and provided me access to their research facilities such as hightemperature furnace and optical microscopy.Jannik Eikenaar for his help on improving the literature of this thesis.My caring beloved parents that in spite of living far from home for the past 6 yearsnever let me feel alone.Finally, I would like to acknowledge the Natural Sciences and Engineering ResearchCouncil of Canada for funding my research and studies at UBC-Okanagan.xiiiChapter 1IntroductionThrough Process Modelling (TPM), by studying and optimizing the entire fabricationprocess, is an advantageous tool for effectively predicting and enhancing the final perfor-mance of a component. This approach is being employed in a project at The Universityof British Columbia to improve the castability and usage of B206, a high-strength castingaluminum alloy that suffers from reduced casting yields because of its long freezing range(≈100 ◦C) and propensity to form defects. The demonstration component selected forthis project is a hub for a hydro-kinetic energy system. In the present chapter, a briefintroduction of the B206 alloy and TPM approach is provided, followed by a descriptionof the application of TPM for a turbine hub. Then, the essential constitutive materialproperty models for developing this TPM toolset for B206 are introduced. The developmentof these constitutive models is the main contribution of this thesis.1.1 B206, An OverviewThe industrial application of aluminum alloys in the manufacturing of high-quality castingshas been at the center of attention for the past few decades because of their lightweight and environmental advantages such as fuel consumption reduction. In additionto this, the possibility of achieving a wide range of strength and ductilities for variousapplications, through different heat treatment trials, has specifically drawn attentionto cast aluminum-copper alloys. In this family, the 206 alloys are well-known for theirconsiderable mechanical properties, and development of different variants has allowed fortheir application in the aerospace and automotive industries (Keshavaram et al. 2000).In A206, the composition of Fe and Si is controlled (Major and Sigworth 2006) as these11.1. B206, An Overviewelements weaken the alloy’s ductility by promoting the needle-like Al5FeSi (β) phaseformation in the matrix (Kamga et al. 2010). However, this variant still suffers fromdefects, most notably a high susceptibility to hot tearing. This issue has been alleviatedby further modifications in the composition (Sigworth and DeHart 2003), specifically alowering of the limit of Ti in the as-cast form. This allows for grain size refinement byincreasing the alloy’s response to Ti-B based grain refiners, which reduces the hot tearingsusceptibility. The registered composition of this new variant, B206, and prior 206 castingalloys are shown in Table 1.1 (Major and Sigworth 2006).Overall, B206 alloys have higher strength and better ductility in comparison to theconventional A356 and Al-12Si-Mg alloys (Major and Sigworth 2006). However, in spiteof all the improvements in the chemistry of the alloy and the resulting mechanicaladvantages, B206 still remains sensitive to casting defects (Sigworth and DeHart 2003)since these defects are inherited from the dendritic characteristic of the aluminum-coppermicrostructure (Backerud et al. 1990). However, it has been shown that casting defectsare dependent on different parameters such as grain size (D’Elia et al. 2012, Li et al. 2013),Fe/Si content (Kamguo Kamga et al. 2010), and cooling rate (Easton et al. 2006). Hence,controlling these parameters through an accurate process design seems to be the mosteffective approach in reducing the probability of casting defects formation. Furthermore,the importance of process design is critical when fabricating near net-shape castings sinceno further major processing occurs after casting, except a small amount of machining andheat treatment. Although B206 is mainly prone to casting defects, the possibility of otherissues arising during different stages of a component’s fabrication process also should bethoroughly inspected. All these show that numerous different interrelated design variablesshould be each carefully specified that require accurate design and production guidelines.21.2. Through Process Modelling (TPM)1.2 Through Process Modelling (TPM)The development of Through Process Modelling (TPM) in the past decade has provideda great tool for high precision design and production. The TPM approach is based onstudying the evolution of descriptive variables such as microstructure, generated/transferredheat, macro/micro scale defects, and stress/strain field during each stage of a manufacturingprocess. The underlying assumption is that descriptive variables are universal parameters,and their evolution through the whole fabrication process can be studied using multi-physicsmodels corresponding to each stage of the process such as solidification models for thecasting step. Using this precise knowledge as input, the macro-scale process models canthen be applied to predict and optimize the in-service performance of the material (Hirsch2006). This concept has been used by Li et al. (2007) to study and quantify the interactionsbetween processing and in-service loading, as the basis of the fatigue life prediction foran aluminum alloy A356 automotive wheel. Specifically, four main process models weredeveloped (casting, T6 heat treatment, machining and cyclic loading) and validated, andthen combined with empirical equations relating descriptive variables to porosity formationand SDAS (secondary dendrite arm spacing) in order to predict the fatigue life. A goodfit between the predictions and the experimental data was reported by the authors.A precise simulation of the material behaviour using each process model requiressubstantial knowledge about basic material properties. Depending on the process models,these properties can include material’s density, thermal expansion coefficient, elastic/plasticproperties, elongation, etc. Obtaining this knowledge, essential for developing an accurateTable 1.1: Comparison between composition of different registered variants of 206casting alloys (Major and Sigworth 2006).Alloy Si Fe Cu Mn Mg Ti Zn Others206 0.10 0.15 4.2-5.0 0.20-0.50 0.15-0.35 0.15-0.3 0.10 each totalA206 0.05 0.10 4.2-5.0 0.20-0.50 0.15-0.35 0.15-0.3 0.10 0.05 0.15B206 0.05 0.10 4.2-5.0 0.20-0.50 0.15-0.35 0.10 0.1031.3. Constitutive Behaviour of B206 AlloyTPM, may not be difficult in the case of materials with well-developed databases, butit demands complementary studies in the case of recently-developed materials such asB206. In these cases, the constitutive behaviour of the material needs to be studied andmodelled prior to the process modelling and TPM development.1.3 Constitutive Behaviour of B206 AlloyThe proposed TPM toolset for a hub component in a hydro-kinetic system using B206consists of three process models: casting, heat treatment and machining. The final in-servicemodel predicts the fatigue life of the hub component under the cyclic/static loads, consideringthe accurate information on design variables’ status that have arisen through processmodelling. In the light of the provided discussion about TPM, the formation of defectsis a critical aspect of a material’s microstructure that results from the evolution of theheat and stress/strain fields. Hot cracks occurring during casting (Pellini 1952), distortioncaused by heat treatment (Yu et al. 2012) and surface cracks forming during machiningprocess (Umbrello et al. 2010) are examples of such defects. Hence, while studying defectsformation a proper constitutive mechanical model is essential to predict the B206 alloy’sresponse to residual/thermal and mechanical loads during processing. In B206, and anymetallic system, the focus of constitutive modelling is on plastic deformation since theyield point is reached at very low strains. However, in the case of aluminum-copper alloys,thermally-activated evolution of the precipitates causes a significant transition in theyield strength (Byrne et al. 1961). The high temperatures during cooling after casting,the thermal processing during heat treatment and the heat generated during machiningcan all provide the required activation energy to modify the yield strength of the B206alloy. Hence, a constitutive model is required to express the yield strength of the materialas a function of the heat treatment parameters, i.e. temperature and heating duration.Although other material properties, such as elongation, may also vary as a result of the41.3. Constitutive Behaviour of B206 Alloyheat treatment, the knowledge of yield strength evolution is a critical element of a TPMtoolset in order to predict the fatigue life during in-service usage of the component.In order to develop accurate models, considerable experimentation is needed to providethe relevant data for fitting model adjustable parameters and to formulate the finalequations. Possible experiments include compression testing for acquiring the data in theas-cast state, and then heat treatment followed by tensile testing to acquire the data forpredicting yield strength. In the forthcoming chapter, the literature on various approachesfor modelling the constitutive behaviour of materials is reviewed.5Chapter 2Literature ReviewIn the first part of this literature review, the constitutive models for describing flow stressare presented, focusing on models that have been previously used for aluminum alloys.Examples of the use of these models in other material systems are also given in orderto introduce the broad application of constitutive models and to demonstrate the mainideas behind the specific modifications suggested to improve the model predictions fora specific material. It is worth noting that although each material contains a uniqueconstitutive behaviour, the general behavioural trends such as hardening, softening andstrain rate/temperature dependency, are the same for all materials.In the second part, constitutive modelling for yield strength evolution is reviewed.The constitutive models and methodologies presented in this section focus specifically onstudies of aluminum alloys.2.1 Plastic Flow ModellingThe flow behaviour of materials is a complex function of different parameters, includingintrinsic variables such as microstructural characteristics and state variables such astemperature or strain rate. A number of constitutive laws have been developed to integratesuch parameters, and their interrelations, into a mathematical formulation. However, itis an extremely complicated task to account for all the involved parameters in a singlemodel. Thus, different models exist based on specific assumptions and for certain conditions.Generally, based on their origin and parameters involved, constitutive plastic flow modelscan be placed in one of three categories (Lin and Chen 2011):62.1. Plastic Flow Modelling1. Phenomenological models. The flow behaviour predictions provided by these modelsis exclusively a function of state variables resulting from empirical observations thatare later embedded within mathematical functions. These models are appropriatefor use in the same range as the collected experimental data used to develop them.The low number of parameters makes them easy to calibrate and manipulate forsimulation purposes.2. Physically-based models. These models are built on theories that explain a material’sphysical properties, and the respective intrinsic variables of these theories. Contingenton the nature of the model, these theories can include thermodynamics, microstructuralevolution, dislocation density/kinetics and/or slip kinetics. Although such modelsare supported by established theories, simplifications and assumptions are oftenmade due to difficulties in measuring specific physical properties.3. Artificial neural network (ANN) models. The ANN method provides a great toolfor modelling highly non-linear systems. However, because it does not provideany physical insight or clear mathematical formulations, these models are highlydependent on the quality of the provided experimental data. This, together withcomplications in manipulating the resulted data, limits the applicability of ANNapproach. Hence, due to these difficulties ANN models will not be further discussed.It should be noted that the classifications above are based on the origins of the constitutivemodels. In one sense, all of the models can be considered phenomenological since they aredeveloped by fitting against the experimental data gained from mechanical measurements.Below, some of the common phenomenological and physically-based constitutive flowmodels are reviewed.2.1.1 Ludwik EquationThe Ludwik equation has been repeatedly used to model the constitutive behaviour ofaluminum alloys (Alankar and Wells 2010, Van Haaften et al. 2002). This phenomenological72.1. Plastic Flow Modellingexpression takes into account both the strain hardening and strain rate sensitivity effectsas follows:σ = Kεn(ε˙ε˙0)m (2.1)In the equation above, ε and ε˙ represent plastic strain and plastic strain rate, respectively.Furthermore,K, n and m are material parameters that are found by fitting experimentallymeasured data against the model so that a best fit is achieved. The empirical characteristicof the Ludwik model requires careful consideration of the range of the selected experiments.In other words, the model’s prediction would not be accurate if the modelling range atwhich the model is expected to predict the constitutive behaviour does not match to therange of the provided experimental data used to develop them (Chaudhary 2006). Thetemperature effect may be implicitly imposed through temperature-dependent materialcoefficients, i.e. K = K(T ), m = m(T ) and n = n(T ). Figure 2.1 shows the variation ofthe material constants with temperature, for the aluminum alloy A356 (Roy et al. 2012).Often, the Ludwik model adequately describes the plastic flow stress behaviour by definingsolely temperature dependent material coefficients, especially in the case of aluminumalloys. However, in a study by Cheng et al. (2008) on the mechanical behaviour of theAZ31 magnesium alloy sheets, the material constants were found to be functions of bothtemperature and strain rate:n = 0.031 + 0.013 log ε˙+ 97.1TK = −156.4 + 9.1 log ε˙+ 244980.4Tm = −145.263T + 0.377(2.2)Using the formulation in Equation 2.1, the plastic stress is equal to zero when ε = 0,i.e., at the beginning of the plastic deformation. This causes a discontinuity when applyingsuch models to an FEM analysis. To address this issue and to have non-zero yield stress atthis critical point, the Ludwik model has been modified as follows (Hannart et al. 1994):σ = K(ε+ εL0)n(ε˙+ ε˙L0)m (2.3)82.1. Plastic Flow ModellingFigure 2.1: Variation in Ludwik parameters with temperature for A356 (Roy et al.2012).In the equation above, ε˙L0 and εL0 are small values, in the range of 10−5, added fornumerical purposes to eliminate the discontinuity seen in Equation 2.1. It is worthmentioning that assigning a zero value to the strain hardening coefficient, n, yields theNorton law, whereas, eliminating the strain rate sensitivity, m = 0, gives the Hollomonlaw.While the Ludwik model is mostly used to describe the effects of strain and strain ratehardening mechanisms in materials, Zhang (2003) included a softening term to accountfor the recrystallization softening effects that occur during hot deformation of magnesiumalloy AZ31 at high strain and temperature :σ = Kεnε˙m exp(bjT + sε) (2.4)Here, s and bj are material coefficients. As the authors acknowledged, the added softeningterm has no physical meaning. Furthermore, while the added term improved the flow92.1. Plastic Flow Modellingstress prediction during the softening stage, it significantly decreased the accuracy of themodel in the initial hardening stage, especially in the low temperature range.By modelling the constitutive behaviour using the Ludwik model, Alankar and Wells(2010) evaluated this model’s application to the case of as-cast aluminum alloys AA3104,AA5182 and AA6111. The results in this study were compared to experimental datacollected form constant -temperature compression tests. The results showed that theLudwik equation reasonably replicated the experimental observations. Furthermore, toverify the developed model’s predictability in conditions close to DC casting, the authorscompared the results of a deformation simulation at continuous cooling conditions withthe corresponding experimental data. The deformation simulation used ABAQUS andapplied a Ludwik model as a part of the material property definition. The results of thesetrials indicated that while the Ludwik model offers a good fit for AA5182 and AA3104,a poor fit was achieved in the case of AA6111. The model’s weakness in the case ofthe AA6111 alloy was attributed to the complex chemistry in this alloy resulting in thecomplex precipitation reactions during cooling. Dynamic precipitation and recovery havebeen repeatedly studied in the literature in the case of the 6xxx series alloys (Kablimanand Sherstnev 2013, Roven et al. 2008). Clearly, the Ludwik model is not able to accountfor all of the physical aspects of hot deformation in metals.2.1.2 Johnson-Cook (JC) ModelJohnson and Cook (1983) proposed a constitutive model for metals under large strain, highstrain rate and high temperature conditions. This well-known phenomenological modelhas been developed to integrate the effects of strain hardening, strain rate hardeningand thermal softening in a single model. Generally, this model is based on the idea thatflow stress at elevated temperatures and strain rates can be determined by scaling theconstitutive behaviour at low temperature and low strain rates. This characteristic makesthe model favourable for engineering applications where it is challenging to accurately102.1. Plastic Flow Modellingcalibrate the model constants at very high strain rates (Liang and Khan 1999). In itsoriginal form, the JC model is presented as a product of three terms as:σ = (AJC +BJCεr2)(1 + CJC lnε˙ε˙0)(1− T ∗r1 ) (2.5)where,T ∗ =T − TrefTmelt − Tref(2.6)In these equation, T ∗, Tref and Tmelt are homologous, reference and melting temperatures,respectively; whileAJC ,BJC andCJC represent material coefficients and ε˙0 is a proportionalityconstant. The one major drawback of the original JC model is that it assumes that thermalsoftening, strain hardening and strain-rate hardening are independent phenomena, i.e.thermal and strain-rate histories are ignored (Lin and Chen 2011). In comparison, the K,n andm coefficients in the Ludwik model are temperature dependent. To address this issue,different modifications have been suggested. For example, Zhang et al. (2009) modified theoriginal JC model for IC10, a Ni3Al-based superalloy, to include a temperature-dependentstrain hardening coefficient as:σ = [AJC(1− T∗r1 ) +B′JC(T∗)εr2 ](1 + CJC lnε˙ε˙0) (2.7)Here, all the parameters are the same as the original equation form, except:B′JC(T∗) =σb(1− T ∗r3 )− σry(1− T ∗r1 )(εb(1 + p1T ∗ − p2T ∗p3 ))r2(2.8)In equations above, σry represents the reference yield stress, σb and εb stand for breakingstress and strain, and ri, pi, i = 1, .., 3 are fitting coefficients.Another difficulty regarding the JC model that limits its application, especially in thecase of aluminum alloys, is related to the strain rate sensitivity of the model. Due tothe simple strain rate hardening term in this model, most ductile materials harden morerapidly with strain rate than as described by the JC model. To address this issue in the112.1. Plastic Flow Modellingcase of the aluminum alloy 7075 and some other ductile materials, Rule and Jones (1998)enhanced the strain rate sensitivity by revising the original model as:σ =(AJC +BJCεr2)(1 + CJC lnε˙ε˙0+ C ′1( 1C ′2 lnε˙ε˙0−1C ′3))(1− T ∗r1)(2.9)where, C ′i=1,..,3 are material coefficients. The main intention of developing the revised JCmodel was to simulate the behaviour at strain rates higher than 103 s−1, specificallyimpact-type loadings. In the proposed formulation, the entire strain rate sensitivityenhancement term approaches zero for strain rates of less than unit magnitude. In otherwords, for strain rate values equal to unity or less, the revised JC model is identical tothe original model; hence, it does not improve the predictions in the case of low strainrates. As a more general remedy for the same issue, Zhang et al. (2015b) suggested fittingthe experimental data for 7075 to a modified JC model with a complex hardening term(CJC parametr in Equation 2.5) as a function of the strain rate. The results in their papershow that the modified model adequately describes the plastic behaviour of the materialin a wide strain rate range of 10−4 to 1200 s−1.Roy et al. (2012) used the Johnson-Cook model to characterize the constitutivebehaviour of the as-cast A356 aluminum alloy. The results in their work showed thatthe original JC model needs modifications to be able to adequately describe the hardeningbehaviour of the alloy at all strains. For this, they have substituted the expressions inthe first bracket of the Equation 2.5 with a hyperbolic Voce-type saturation stress model(discussed in following section) as:(AJC +BJCεr2)→((σvs)2 +((σry)2 − (σvs)2)exp(−Θ0εσvs))1/2(2.10)In this equation, σvs is the so-called Voce-type saturation stress and Θ0 is the initial strainhardening rate. To capture the softening effect more accurately, the authors suggested122.1. Plastic Flow Modellingexpanding the softening term in Equation. 2.5 as below:(1− T ∗r1)→ 1− κ1T∗λ1 + κ2T∗λ2 (2.11)In this equation, the material coefficients κ1 and λ1 apply to the low temperature whileκ2 and λ2 cover the elevated temperature range. Finally, they offered a temperaturedependent strain rate coefficient as:CJC = CJC(T∗) = θ1T∗Λ + θ2 (2.12)with material coefficients θ1, θ2 and Λ found by plotting CJC versus T ∗. However, after allmodifications, the JC model still did not effectively describe the experimental flow stress.According to the discussion above, different modifications are required while usingthe JC model for each specific material. Through these modifications, different materialconstants are added to the model, increasing the number of adjustable fitting parametersthat need to be calibrated versus related experimental data. Specific methodologies arerequired to acquire the model coefficients, as reviewed in (Gambirasio and Rizzi 2014).Furthermore, revisions of the JC model are based on individual case studies, and it hasbeen repeatedly shown that due to the lack of generality, they are not applicable to newstudies. For further information on this matter, the reader is referred to a review paperby Lin and Chen (2011). The JC model has been frequently employed in studies focusingon high strain rate applications (e.g. (Rule and Jones 1998)), however, case studies on itssuccessful use at low strain rates are rare.2.1.3 Voce-type ModelsAs the most common physically-based approach, Voce-type models have been developedto explain the constitutive behaviour based on the assumption that the work hardeningrate approaches zero at a specific stress, termed saturation stress. The basic formulation132.1. Plastic Flow Modellingfor these types of models was originally proposed by Voce (1948):σ = σvs +[(σry − σvs) exp(−εεr)](2.13)where, εr is the relaxation strain. This equation became the starting point for thedevelopment of different physically-based models, including Voce-Kocks (Kocks 1976)and Kocks-Mecking (Kocks and Mecking 2003) that are discussed below, respectively.Kocks (1976) considered the variation of the saturation and reference yield stresses inthe Voce equation as a function of temperature and strain rate:σvs = γ0(ε˙γ1)(KbT/As1), As1 = µb3γ2σry = γ3(ε˙γ4)(KbT/As2), As2 = µb3γ5(2.14)where, Kb stands for Boltzman constant and γi=0,1,.. are fitting coefficients. Then, therelaxation strain was defined as:εr(ε˙, T ) =[σvs(ε˙, T )− σry(ε˙, T )]θ0(2.15)Puchi et al. (1997) used this model, known as the Voce-Kocks (VK) model, to explainthe constitutive behaviour of commercially pure aluminum. The authors attributed thesignificant deviation in the modelling results to the wide range of the experimental data,provided in a temperature range of 293 - 673 K and at strain rates between approximately0.1-200 s−1.Kocks and Mecking (2003) found that the experimental data for many materials betterfits a phenomenological form of the saturation stress equation, i.e.:σvsµ=γ0µ0{1−( 1g0KbTµ3ln(γ6ε˙))}2(2.16)142.1. Plastic Flow Modelling(a) (b)Figure 2.2: Comparison between results from the Ludwik model (σLH) and theKocks-Mecking (σKM) at (a) high and (b) low temperature ranges forA356 (Roy et al. 2012).In equation above, µ is the shear modulus while µ0 represents its absolute value and g0is a material coefficient. The authors stated that the physical meaning of the model isrestricted to monotonic deformation at low temperatures, T < 0.6Tmelt for aluminum,where the dynamic recovery is not affected by recovery in conjunction with self-diffusionand dislocation climb. Roy et al. (2012) used this model to develop a constitutive law forthe A356 alloy. The reported results show that the model offers a better fit at elevatedtemperatures and high strain rates. Also, the fact that the model has few fitting parametershas been noted as a strength of the Kocks-Mecking model. Figure 2.2 shows a comparisonbetween the Ludwik-Hollomon (LH) model and the Kocks-Mecking (KM) model at lowand high temperature ranges. It can be seen that, in opposite to the good fit offered bythe LH model, the KM model underestimates the plastic flow at low temperatures whena low strain rate is applied. Also, both models overestimate the flow behaviour at hightemperature and high strain rate.Voce-type constitutive models are well-known for the few number of fitting parameters.This characteristic reduces the experimental effort to develop them. Generally, thesemodels are used for high strain rate applications. The recent improvements in this family ofmodels were also focused to enhance the predictions at very high strain rates. For example,152.1. Plastic Flow Modellingthe Lin-Voce model has been particularly developed for modelling metal’s behaviour duringforming that involves high strain rates in the range of 2.5-150 s−1 (Zhang et al. 2015a).Hence, for low strain rate studies such as in casting simulations, a proper prediction ofthe constitutive behaviour is not guaranteed by using Voce-type models.2.1.4 Arrhenius-type EquationZener and Hollomon (1944) introduced a physical model to define the temperature andstrain-rate dependency of the stress/strain state using only a single parameter. Byassuming that just one type of rate with a specific activation energy (Ezh) affects theisothermal behaviour, they proposed this parameter, so-called Zener-Hollomon parameter,as:Z = ε˙ exp(EzhRT)(2.17)where, R is the universal gas constant and Ezh is an activation energy. Following this work,Sellars and McTegart (1966) suggested the equation below to have a good correlationwith the hot deformation data:ε˙ = Ast(sinh(αstσ))nst exp(−EzhRT)(2.18)with Ast, αst and nst being material coefficients. Clearly, this equation accounts for astrain - independent behaviour that is seen in aluminum alloys under deformation at hightemperatures. Under these conditions, the material shows a creep-based or strain-rate-dependentmechanical response. A statement for calculating the plastic flow stress can be achievedby extending the sinh term in Equation 2.18 and writing the flow stress as a function ofthe Zener-Hollomon parameter:σ =1αstln{( ZAst)1/nst +[( ZAst)2/nst + 1]1/2}(2.19)162.1. Plastic Flow ModellingFigure 2.3: Stress-strain curves for AA6061-Al2O3 composite, obtained fromcompression tests (Davies et al. 1997).It should be noted that Equation 2.18 can be stated in a more descriptive manner:ε˙ = AstFZH(σ) exp(−EzhRT)(2.20)where,FZH(σ) =σn1 αstσ < 0.8exp(βstσ) αstσ > 1.2 , αst = βst/n1(sinh(αstσ))nst for all σ(2.21)Here, n1 and βst are material coefficients.In a study by Davies et al. (1997) on the constitutive behaviour of an AA6061-Al2O3composite, the strain-independent plastic behaviour of the material at temperatures higherthan 300 ◦C, shown in Figure 2.3, was properly simulated by fitting the experimental datato a Zener-Hollomon model. The Zener-Hollomon parameters found in this work arereported to match well with the values in the literature for similar composites such as6061-SiC.Although the Arrhenius-type equations have been developed for strain-independentbehaviour, Lin et al. (2008) studied the strain-dependency of the activation energy and172.1. Plastic Flow Modellingthe material constants in the Zener-Hollomon model. For this purpose, the parameterswere calculated at a limited number of fixed strains. Then, the strain dependency of theparameters was included by curve fitting to quintic functions. The results of this approachwhen applied to the 2124 (Lin et al. 2010) and A356 alloys (Haghdadi et al. 2012) areshown in Figure 2.4. In both cases, the constitutive plastic flow model fit well against theexperiential data. By using the same approach in another study on AA1070 (Ashtianiet al. 2012), the modified Zener-Hollomon model offered a precise estimate of the plasticflow behaviour of the material with a correlation coefficient of 99.2%. In spite of theprecise replication of the experimental data using this phenomenological approach, it isnot desirable for simulation and TPM development purposes. This is because, in mostcases, the behavioural trends under investigation can be properly described by usingsimpler models such as the Ludwik model.2.1.5 Advances in Constitutive ModellingA number of conventional constitutive flow stress models including their advantages anddisadvantages have been discussed in Section 2.1.1−2.1.4. Nevertheless, the applicationof these models is restricted to a monotonic behaviour path, e.g. continuous hardening/softening or complete strain independent behaviour. Hence, they fail to consider theprobable transition in the constitutive behaviour of the materials resulted from dynamicphenomena that occurs within the microstructure during the hot deformation. Thedynamic precipitation and its effect on the mechanical behaviour of materials has beenwidely investigated in the literature (Kabir et al. 2013, Kubin and Estrin 1991). In a studyon the rate-dependent deformation of austenitic iron at elevated temperatures (Anand1982), it has been shown that dynamic recrystallization in the material’s matrix leads todeviation in the developed constitutive model’s prediction.The dynamic effects are more significant in the case of materials with complex microstr-ucture, such as steels, than aluminum alloys. Hence, most of the literature on consideringthese effects in constitutive behaviour modelling are dedicated to this group of materials182.1. Plastic Flow Modelling(a) (b)(c) (d)Figure 2.4: Results of the modified Zener-Hollomon model for 2124 (Lin et al.2010) at a) high strain rate, b) low strain rate and A356 (Haghdadi et al.2012) at c)high strain rate, d) low strain rate.(Cheng et al. 2001, Lindgren et al. 2008, Nemat-Nasser et al. 1999). However, a review onthe general advancements provides a scheme of possibilities for developing more effectivemodels in the case of aluminum alloys. Unified constitutive models have been introducedto describe dynamic phenomena and their effect on the plastic flow stress by combiningmechanical models with models for dynamic microstructure evolution. In an example inthe case of commercially pure titanium, Cheng and Nemat-Nasser (2000) developed aunified constitutive model to match the experimental observations showing anomalous192.2. Yield Strength Modellingtemperature dependency that has been attributed to the dynamic strain aging:σ = v0√CC0{1−[− KbTv1√C/C0(ln ε˙ε˙0)]1/q}1/p×{1 + v2[1− ( TTmelt )2]εv3}+ v5εv4(2.22)Here, C is the local solute concentration and C0 is its nominal initial value. Also, p, q andνi=0,1,.. are material coefficients.As the formulation above implies, the models that are introduced to consider complexmicrostructural attribute, typically include a large number of material coefficients thatmake the fitting and model development procedure difficult.2.2 Yield Strength ModellingIn this section, the methodology for modelling the evolution of yield strength due to heattreatment is discussed. The evolution of yield strength during industrial processing resultsfrom thermally-activated microstructure evolution. Thus, the kinetics of the microstructureevolution should be also independently investigated. In the case of the aluminium-copperalloys, precipitation is the most significant evolving microstructural factor that needs tobe studied (Sehitoglu et al. 2005). Also, since B206 is a dilute alloy with copper as themajor alloying element, it is expected that the precipitation of the Al2Cu is the mainmicrostructural variable of heat treatment process in this alloy.As shown in the literature (Gutie´rrez et al. 2014, Haghdadi et al. 2015, Kamga et al.2010), the transformation/formation of other phases in B206, specifically the complexintermetallics, occurs at temperatures above 500 ◦C. As this temperature is significantlyhigher than the temperature for artificially ageing of the alloy (100-250 ◦C), a microstructuremodel that only predicts precipitation of the Al2Cu phase is all that is needed for modellingyield strength evolution of B206. The validity of the whole microstructural modellingcan be then evaluated by comparing the results against the literature as precipitation202.2. Yield Strength Modellingin aluminium-copper systems is a well-studied subject. Moreover, the accuracy of thefinal yield strength model also can be interpreted as a sign of the legitimacy of themicrostructural model, as it is an important (or even only) variable in the mechanicalmodel. In the following sections, first, the mechanical model for yield strength evolution,and second the precipitation kinetics description are discussed.2.2.1 Constitutive ModelOne successful way to study different contributors to the total yield strength of metals(σyield) is to consider the summation of such parameters in a linear manner (Wang et al.2003):σyield = σppt + σss + σi (2.23)Below, each of the parameters in this formulation will be discussed.Intrinsic Strength (σi). Typically, intrinsic strength is considered as a constant, sinceits evolution due to the heat treatment is infinitesimal as compared to the contributionsof other parameters. The contribution due to the intermetallics, grain size effect and theeutectic phases may all be included in the intrinsic strength term (Colley 2011).Solid Solution Strengthening (σss). The relationship between the concentrationof the solute in the matrix and the solid solution strength, which is a major contributorto the yield strength, has been defined as (Esmaeili et al. 2003, Shercliff and Ashby 1990):σss = aC¯2/3t (2.24)By considering the fact that precipitation happens as a result of solute migration insidethe matrix, and also by assuming a binary system, the interrelation between the evolutionof the solid solution strengthening and the relative precipitation progression (fr) can be212.2. Yield Strength Modellingstated as: σss = Css(1 + αfrfr)2/3fr =ffpeak0 < αfr ≤ 1(2.25)In the equations above, αfr represents the fraction of the solute depleted from the matrixwhen the precipitation is fully developed and Css is a material coefficient. Also, thepeak-aged state, where precipitates volume fraction (f) is equal to fpeak, is defined as thepoint at which the precipitation strengthening is completed, and thereafter, by continuingthe heating, the material softens as a result of the overaging.Precipitation Strengthening (σppt). Based on theories related to interaction of thegliding dislocations with point obstacles (Ardell 1985), precipitation strengthening canbe stated as a function of obstacle, here precipitate particle, strength (Fstr) and spacing(Lobs): σppt = MFstrbLobsFstr = rrpeakFpeak(2.26)where, r is the average radius of precipitates; rpeak and Fpeak are precipitates average radiusand strength, respectively, at peak-aged state; M is the Taylor factor and b stands forthe Burgers vector. Assuming the strong obstacles theory (Murray 1985) to be dominant,the precipitates spacing can be stated as a function of their average diameter and volumefraction:Lobs =(2pif)1/2r (2.27)By substituting into Equation 2.26 and considering the definition of the relative precipitatefraction:σppt =MFpeakf1/2peakbrpeak(2pi)1/2× f 1/2r (2.28)Considering constant properties for the variables at peak-aged state in Equation 2.28and substituting the formulation for each parameter in Equation 2.23, the constitutive222.2. Yield Strength ModellingFigure 2.5: Variation in the experimental (σexpt) and predicted (σtot) hardness ofA356 as a function of aging time at 150 and 180 ◦C (Colley 2011).model for yield strength can be rewritten as:σyield = Cpptf12r + Css(1 + αfrfr)2/3 + σi (2.29)From the equations above, it is clear that the only microstructural parameter required inthe linear-fit model is the fraction of the evolved precipitates. Figure 2.5 shows the resultsof using the linear-fit approach in the case of A356, reported by Colley (2011). This modelis applicable only up to the peak-aged state, and additional considerations are requiredto predict the material’s strength in the overaged condition. For this purpose, Esmaeiliet al. (2003) developed a model assuming that the precipitate volume fraction remainsconstant (f = fpeak) during the overaging, since large particles grow at the expense ofsmall particles, and thus σppt is solely a function of the increasing particle size:σppt =MFpeakf1/2peakb(2pi)1/2 rmagpeak× rmag−1 = Cagrmag−1 (2.30)with mag and Cag being material coefficients. By assuming that at the overaged state, theyield strength reduction is solely a result of coarsening, a coarsening law can be developed232.2. Yield Strength Modellingto estimate the particle size (r) evolution at this stage. Based on the LSW theory (Lifshitzand Slyozov 1961, Wagner 1961) for coarsening, the particle size can be estimated as:r3 − r3peak = Llsw(t− tpeak)Llsw =BlswT exp(−EART )(2.31)In equation above, tpeak is the heating time to reach the peak-aged state,EA is an activationenergy, and Llsw and Blsw are material coefficients. To find these constants, Esmaeili (2002)suggested a back-calculation method using the data reported in the literature. In theabsence of a database, the constants can be found by fitting the model to the respectiveexperimental data.2.2.2 Precipitation Kinetics and ModellingAs discussed in the previous section, the modelling of the mechanical behaviour evolutiondue to the heat treatment is highly dependent on the microstructural development. Inthe case of dilute aluminum-copper alloys such as B206, the latter can be restricted to theevolution of the precipitation fraction. For this purpose, the Avrami model is commonlyused to explain isothermal transformation processes that involve nucleation and growthsuch as precipitation. One advantage of this method in comparison with more sophisticatednumerical models such as phase field (Hu 2004) and analytical models such as modifiedKampmann-Wagner model (Fazeli et al. 2008) is that it can be easily developed basedon straightforward experiments such as calorimetry. Furthermore, it has been repeatedlyshown in the literature that the Avrami model provides predictions that are reasonablyaccurate for mechanical behaviour modelling purposes (Esmaeili 2002, Shercliff and Ashby1990). Following this model, the fraction of the transformed material is given by:β = 1− exp(− (kjt)nj)(2.32)242.2. Yield Strength ModellingIn this equation, the Avrami coefficients, kj and nj, are found by fitting the model againstthe experimental data. Isothermal calorimetry and non-isothermal calorimetry are two ofthe frequently used methods for this purpose.Isothermal calorimetry is one of the common practises to measure the volumefraction of the precipitation (Esmaeili 2002). This method relies on the assumption thatthe precipitation will reach completion at the peak-aged condition, i.e. precipitates volumefraction (β) equals 1 at this point, and precipitate particles will solely grow in expenseof smaller particles, afterwards. It is worth mentioning that with this assumption, β isequivalent to fr in Equation 2.25. The volume fraction of the precipitation at the time tcan be found by dividing the total heat flow up to t by the total heat flow to reach thepeak-aged condition (Appt):β =t∫0dQdt dttpeak∫0dQdt dt=t∫0dQdt dtAppt(2.33)Here, dQdt is the rate of heat release derived from the heat flow curve. The time to reachthe peak-aged state, tpeak is calculated independently; tensile or hardness tests can beutilized for this purpose. After calculating the β values at each temperature, the Avramicoefficients can be simply found by drawing ln(ln( 11−β )) vs. ln(t) since:ln(ln(11− β)) = nj(ln(kj) + ln(t))(2.34)By determining kj and nj at various temperatures, temperature dependency of the kjconstant can be stated by an Arrhenius-type function:kj = kjo exp(−EjRT) (2.35)Here, kjo is a constant and Ej is an activation energy. It is worth mentioning thatthis activation energy, that is calculated using the slope of the ln(kj) vs. 1/T plot, is252.2. Yield Strength Modellinga combination of all nucleation and growth mechanisms, thus it really does not have aphysical meaning (Luo et al. 1993).Non-isothermal calorimetry with a constant heating rate, or so called isochronalcalorimetry, is another method for calculating the volume fraction of the transformedmaterial. One of the advantages of using non-isothermal calorimetry experiments forcalculating the Avrami coefficients is that, in contrast to the isothermal alternatives, theyare not time-consuming. Furthermore, the isochronal tests all start from room-temperature,followed by constant-rate heating up to a specific temperature. This equal condition for allthe measurements eliminates the error incurred by precipitation progression in the earlystages of heat-up, as compared to the isothermal tests where the samples are continuouslyheated in the chamber prior to stabilization at the testing temperature. The reaction ratefor a non-isothermal solid-state transformation can be represented as (Sa´nchez-Jime´nezet al. 2008):dβdt= kjo exp(−EjRT)g(β) (2.36)with g being a reaction kinetic model. By defining the heating rate as dTdt = φ and followingthe Kissinger method (Kissinger 1957), based on an assessment of the activation energyat the maximum reaction rate (d2β/dt2 = 0), Equation 2.36 can be rewritten as:EjφmRT 2m= −kjog′(β) exp(−EjRTm) (2.37)where, φm and Tm are the heating rate and temperature at which the maximum reactionrate occurs, respectively. By assuming a first order reaction, i.e. g′ = −1, Equation 2.37reduces to:ln(φmT 2m)= lnkjoREj−EjRTm(2.38)Here, the physical nature of the proportionality constant and activation energy are thesame as in the Avrami equation. A detailed description of this equivalence is providedby Mittemeijer (1992). By drawing a plot of the left hand side of the Equation 2.38 vs.1/T , the activation energy can be calculated as the slope of the plot, while the y-intercept262.2. Yield Strength ModellingFigure 2.6: Variation in the heat flow peaks with heating rate, from DSC data forAA8090 (Starink and Gregson 1995).gives the kjo parameter. To generate the required experimental data, differential scanningcalorimetry (DSC) can be used to obtain the maximum transformation rate, or peakpoints, for different heating rates. Figure 2.6 shows DSC curves from solution treatedaluminum alloy 8090 for various heating rates (Starink and Gregson 1995).It is not a straightforward task to calculate the Avrami exponent (nj) by directly usingthe non-isothermal measurement results, thus, additional consideration is required. Themethods reported in the literature for this purpose include hardness measurements (Esmaeiliet al. 2003), resistivity change measurements (Luo et al. 1993), XRD (Starink and Van Mourik1992) and direct TEM observations (Smith et al. 2000, Wang et al. 2003).It has been repeatedly reported that the results for the activation energy and theproportionality constant (kjo) in the Avrami model are identical using both the isothermaland isochronal calorimetry (e.g. Criado et al. 2005). Smith (1998) has shown this similarityby comparing the isothermal and isochronal calorimetry results for different transformationstages of the aluminum alloy 2124. However, an investigation by Starink and Van Mourik(1991) showed that in alloys with two major precipitating elements such as Al-1.3pctCu-19.1pctSi, the differential calorimetry peaks represent the simultaneous precipitationof two distinct phases. In such cases, the methodology for calculating the activationenergy using the isochronal calorimetry is not accurate. Another source of error forthe continuous heating method is where two calorimetrically opposed, endothermic and272.2. Yield Strength Modellingexothermic, processes occur simultaneously. Smith et al. (2000) have shown that in thealuminum alloy 339, initiation of the Si precipitation is concurrent with dissolution of40% of the S′ precipitates (Al2CuMg). Obviously, these are not a major concern in thecase of dilute alloys such as B206 with a single transformation phenomenon occurring intheir matrix at the specific range of temperature under investigation.28Chapter 3Scope and ObjectivesThe present study aims to investigate the constitutive behaviour of the B206 alloy anddevelop the relevant models in order to provide the essential description of the material’smechanical response as a function of the state and processing variables. These models area crucial part of developing a TPM toolset for a hub component in a hydro-kinetic energysystem, specifically where material’s constitutive mechanical properties are required topredict defects formation. Moreover, the constitutive mechanical behaviour modellingfor B206 has not been formerly investigated. Thus, the results of this study will alsocontribute to the community and future studies by enriching the knowledge of B206.Considering these and the literature reviewed in the previous chapter, the followingobjectives were identified to provide inclusive models for the constitutive mechanicalbehaviour of B206. Firstly, due to its importance in metallic systems and also theexperimental priority that utilizes the material for measurements in the as-cast state,development of a constitutive plastic flow stress model for B206 should be investigated.Two main features of this model are simplicity, to be convenient for FEM implementationtasks, and generalizability, to explain the behaviour in a wide range of experimentalconditions expected in the TPM toolset. These can be accomplished by combiningphenomenological constitutive modelling with mechanical compression testing. Secondly,a constitutive model should be established to simulate the yield strength evolutionas a function of the heat treatment. This can be accomplished by combining heattreatment experiments, non-isothermal calorimetry and tensile testing. The non-isothermalmeasurements are conducted in order to predict the precipitation evolution as requiredby the linear-fit mechanical model.29Chapter 4Experimental MethodologyCompression experiments on the as-cast samples and a combination of tensile, calorimetryand hardness tests on the heat treated samples were found as relevant methods forinvestigating the plastic flow stress behaviour and yield strength evolution of B206,respectively. In the following, details of the experiments conducted in this study andthe respective methodologies are discussed.4.1 MaterialsThe B206 material used in this work was provided in collaboration with the NaturalResources Canada - Materials Testing Laboratory (CMAT) in Hamilton, ON, with achemical composition shown in Table 4.1. To generate material with a relevant as-castmicrostructure, a demonstration component, shown in Figure 4.1, was produced andsamples were machined from sections of this casting. The demonstration casting is a hubdesigned for use in a hydro-kinetic energy system. Figure 4.2 shows a typical microstructureof as-cast B206 that includes copper-rich eutectic phases on the grain boundaries as wellas some dispersed intermetallics in the matrix. Also, a variation in composition of thecopper between the grain boundaries and grain cores was detected that in the micrographcan be seen as a transition from the dark region in the middle of the grains to the lighterregion close to grain boundaries. It is worth mentioning that the intermetallics were mostlydetected in the high concentration area close to grain boundaries, or light regions in themicrograph. The imaging and chemical analysis were performed using secondary electronmicroscopy (SEM) and X-ray spectroscopy.304.1. MaterialsFigure 4.1: Demonstration hub component cast by CMAT. Red dots show therelative location of the coupons extracted for metallography. Samples wereextracted from 4cm beneath the top surface of the hub.The microstructure consistency between samples is an important consideration forconstitutive property measurements. In the ring section of the hub, it is expected thatthe microstructure will not vary appreciably from one location to another due to castinggeometry. To check this assumption, the grain size and the area percentage of eutectic phasewere measured on six coupons taken from different locations in the ring, Figure 4.1. Aftermounting and polishing, the coupons were etched with Keller’s solution (1 % hydrofluoricacid, 1.5 % hydrochloric acid and 2.5 % nitric acid in distilled water) to reveal the grainboundaries and eutectic phases. The grain size and eutectic phase percentage were thenmeasured using a ZeissR©Axio optical microscope coupled with the OmnimetR©9.5 imageprocessing software. The line intercept method was used to measure the average grain sizeby superimposing four lines with random orientations and lengths on the micrographs.The results of the optical microstructure analysis are shown in Table 4.2. As can be seen,Table 4.1: Comparison between the registered composition for B206 and the finalcomposition of the material cast at CMAT lab.B206(Wt%) Si Fe Cu Mn Mg Ti Zn AlRegistered 0.05 0.10 4.2-5.0 0.20-0.50 0.15-0.35 0.10 0.10 Bal.CANMET 0.056 0.067 4.72 0.27 0.27 0.014 <0.0010 Bal.314.2. Compression Tests on As-Cast SamplesFigure 4.2: Typical microstructure of B206 with Cu-rich eutectic phase on grainboundaries and higher concenteration of coper around grain boundaries.Intermetalics were also mainly detected in areas close to the grain boundaries.grain size and eutectic fraction within the ring vary only minimally between differentcoupons.Table 4.2: Eutectic percentage and grain size measurements for coupons taken atdifferent locations in the hub ring.Sample # 1 2 3 4 5 6 AverageEutectic % 6.60 6.78 7.02 7.05 6.11 6.12 6.61±0.5Grain Size (µm) 110 114 124 76 142 151 120±454.2 Compression Tests on As-Cast SamplesUniaxial compression tests were performed to study the plastic flow behaviour of theB206 alloy in the as-cast state. These tests provided the required experimental datato develop a constitutive model. The experiments were performed on a GleebleR©3500thermo-mechanical simulator, located in the Advanced Materials and Process Engineering324.2. Compression Tests on As-Cast SamplesLaboratory (AMPEL) at UBC-Vancouver. Cylindrical samples with 10 mm diameter and15 mm length were extracted from the ring section of the hub casting (Figure 4.3). Themethodology used for the compression tests is outlined below:– A type-K thermocouple was attached to the sample to control and record changesin temperature during testing;– A layer of Nickel-based lubricant was applied to both ends of the sample. Then, agraphite foil was placed on the lubricant, to ensure a low-friction surface betweenthe sample and the anvils installed in the Gleeble system;– An extensometer was mounted on the center of the specimen to instantaneouslymeasure the diametral changes;– After aligning the sample in Gleeble system, it was pre-loaded, with a force of 0.5 kN,to hold it in the place;– The hydraulic actuator was retracted by 2 mm to allow for the thermal expansion;– Using Joule heating, the sample was heated to a target temperature at a rate of5 ◦C s−1, followed by a 60 s hold;– The sample was deformed up to a strain of 0.3, at a prescribed deformation rate;– During testing, data for load, length, diameter and temperature was measured forsubsequent analysis.Compression tests were performed at 14 temperatures between 50 and 530 ◦C, andat 4 strain rates, 10−3, 10−2, 0.1 and 1 s−1. Experiments were repeated for 9 differentcombinations of temperature and strain rate. Due to the variability of the mechanicalbehaviour in the intermediate temperature range, 200 to 350 ◦C, most of the repeats wereselected in this temperature range and for a variety of strain rates. The significance ofthe data in the intermediate temperature range will be discussed in Section 5.2.334.2. Compression Tests on As-Cast Samplesa(a) (b)Figure 4.3: Sketches of the (a) compression and (b) tensile testing samples.For most of the compression tests, a 100 kN load cell was used for measuring andrecording the loads during compression. However, for the highest temperatures, 500 and530 ◦C, a 5 kN load cell was used to enhance the accuracy of data acquisition.4.2.1 Plastic Deformation DeterminationThe diametral deformation data acquired during Gleeble tests was used to calculate thetrue strain as:εtrue = −2 lnDD0(4.1)With D representing the sample diameter, the initial diameter of the sample (D0) wasset to the diameter right before the start of the deformation. Thus, the initial diameterincludes the thermal expansion experienced by the sample due to heating to the testtemperature.Considering the recorded applied force (F ) data from the load cell, the true stressduring the compression tests was calculated as:σtrue =4FpiD2(4.2)Based on the true stress/strain data, the yield strength was then determined. The yieldpoint was selected based on the 0.2 % offset method. For this, a temperature-dependent344.3. Heat Treatment Experimentselastic modulus (E) was utilized (Frost and Ashby 1982):µ = 2.54× 104(1 + 300−T2Tmelt )E = 2µ(1 + ν)(4.3)Although Equation 4.3 was determined for pure aluminum, it has been shown previouslythat the elastic modulus of aluminum alloys is nearly insensitive to the solute content (Meyersand Chawla 2009). Note that for Equation 4.3, the melting temperature was set to 630 ◦C,matching the liquidus of B206 (Haghdadi et al. 2015) and the Poisson’s ratio (ν) was setto 0.33. Finally, the initiation of the plastic deformation region at each data point wasdetermined using the criteria explained above. Then, after removing the elastic strain,the plastic stress-plastic strain data was used for analysing the plastic flow variation withstate variables, temperature and strain rate, as well as for developing a constitutive model.4.3 Heat Treatment ExperimentsIn the second part of the present work, a constitutive model for yield strength of the B206alloy was investigated. Based on the literature reviewed in previous chapter, the yieldstrength of aluminum-copper alloys is a function of the heat treatment and subsequentprecipitate formation/growth. Thus, all the samples used for yield strength investigationswere initially heat treated that included a solution treatment and a quenching step followedby artificial aging. The solution treatment step eliminates the effect of any precedingnatural aging and dissolves all the second phase particles so that they can be re-precipitatedin a controlled manner. After quenching the samples, artificial aging is performed ata range of temperatures and heating times providing the samples for yield strengthanalyses. Finally, the yield strength evolution with heat treatment was evaluated byperforming tensile tests on heat treated samples. The experimental data provided bythe tensile testing was used to develop a constitutive model. To provide informationregarding the microstructural evolution with heat treatment, isochronal calorimetry tests354.3. Heat Treatment Experimentswere conducted using a DSC. This provides information on the precipitation kineticswhich was necessary to develop an Avrami model describing the microstructural evolution.In lieu of direct observation of the precipitates, the results of the developed Avrami modelwere qualitatively compared with hardness test results to verify the model predictions.The solution treatment and quench conditions selected for the alloy used in this studywas performed as follows: i)the furnace was preheated at 515 ◦C and held for 24 h; ii)samples were placed in the furnace at 515 ◦C for 2 h followed by an 8 h treatment at 525 ◦C(Chang et al. 2005). The temperature in the furnace was continuously recorded usingan OMEGA, OMB-DAQ-56 data acquisition system with a thermocouple placed near thesamples; iv) The process was followed by quenching in 50 ◦C water for 2 min; and v) Afterallowing the samples to cool down at room temperature for 4 min, they were relocatedand kept in a −80 ◦C freezer to prevent natural aging from occurring.For the artificial aging of the solution treated tensile samples, a low-temperaturemuffle furnace was preheated to the prescribed temperatures. Then samples were placedinside and held for the target timings, followed by quench in the 50 ◦C water. Twentycylindrical tensile samples, shown in Figure 4.3, with 19 mm gauge length and 4 mmgauge diameter (ASTM-B557 2010) were machined from the hub casting, solutionized,and then artificially aged at four different temperatures: 150, 175, 200 and 225 ◦C, for fivetimes: 1, 2, 5, 10 and 24 h. The prepared samples were kept in the −80 ◦C freezer beforeconducting the tensile measurements. It should be noted that prior to heat treatment andtensile testing, the samples were allowed to acclimatize at room temperature for 15 minafter being removed from the freezer. Figure 4.4 shows a comparison between the as-castmicrostructure and the microstructure of a sample heat treated for 2 h at 250 ◦C. It can beseen that although concentration of the eutectic phases on grain boundaries is relativelypreserved, heat treatment results in a uniform distribution of the solute within the grains.This can be interpreted from the variation of the color inside the as-cast sample’s grains,that is eliminated in the case of the heat treated sample.364.3. Heat Treatment Experiments(a) (b)Figure 4.4: Typical microstructure of B206 in (a) as-cast and (b) heat treatedstates. A uniform distribution of solute is obtained by heat treatment.4.3.1 Tensile MeasurementsTensile testing was employed to measure the room temperature yield strength, UTS andductility of the artificially aged samples. All tests were conducted using a 50 kN Instron5969 tensile tester located in the CRN lab at UBC-Okanagan. The deformation ratewas set to ≈0.02 mm s−1, which is equivalent to a strain rate of 0.01 s−1. Prior to theexperiments on the artificially aged samples, control tests were conducted on the solutiontreated samples at three different strain rates in order to verify the testing methodologyand to study the effect of the strain rate on the yield strength of the material at roomtemperature. The results, given in Table 4.3, show that the material’s yield propertieswere relatively insensitive to the strain rate at room temperature.During the tensile tests, the data from the load cell (F) and an extensometer (∆L)were recorded at a rate of 10Hz. The true stress and strain in the sample during each testwas then calculated. The true strain was calculated as:εtrue = ln∆LL0(4.4)374.3. Heat Treatment ExperimentsBy assuming constant volume, the instantaneous diameter can be calculated as D =D0√L0L . The true stress was then calculated by Equation 4.2. The yield point wasestimated by the 0.2 % offset method, assuming that the elastic modulus is given bythe linear part of the stress-strain curves.Table 4.3: Results of yield point variation with strain rate. Yield point is found tobe a minor variant of the strain rate value.Displacement rate (mm min−1) 0.13 1 5 12.6Yield Stress (MPa) 170.6 166.3 162.6 164.5Yield Strain 0.011 0.010 0.010 0.014The heat treatment and tensile tests were repeated for all data points in order to verifythe initial experiments. Results closely replicated the first set of tensile results.4.3.2 Thermal Analysis and Hardness TestingNon-isothermal calorimetry was used to study the kinetics of precipitation and to providethe experimental data needed to develop an Avrami model of the precipitation process.Differential scanning calorimetry (DSC) tests were conducted using a NETZSCHR©STA449 F3 Jupiter thermal analyser. The DSC samples were prepared with an average weightof 25 mg by using a diamond saw. After solution treatment, samples were enclosed in analumina (Al2O3) crucible and placed on a P−type sample carrier. A nitrogen atmospherewas used during the tests. The samples were heated at five different heating rates: 2, 5, 10,15 and 20 K min−1 up to 530 ◦C to provide the data necessary for analysis using Kissinger’smethod. The heat flow (W/g) versus time (s) data acquired form the first heating cyclewas corrected using the second heating cycle as the baseline. This baseline is proportionalto the specific heat of the sample under investigation and highlights the precipitationprogression, specifically at its maximum rate, occurring during the first cycle.Hardness measurements were conducted on coupons with approximate size of 5×5×10 mm using a WilsonR©VH3100 Vickers microhardness tester. The coupons were coupledwith the tensile samples during the heat treatment process, i.e., they experienced the384.3. Heat Treatment Experimentssame heat treatment conditions. The VH1 data, Vickers hardness with a loading equal to1 kg, were extracted by averaging values from 20 readings on a 4×5 matrix, each with aholding time of 10 s.39Chapter 5Results and DiscussionIn this chapter, the experimentally-measured as-cast flow stress behaviour of the B206alloy as a function of state variables, constitutive modelling of this behaviour and thecorresponding results will be discussed. Then, the results obtained by studying the yieldstrength evolution of the alloy as a function of heat treatment, and corresponding yieldstrength and microstructural models will be outlined.5.1 Constitutive Behaviour of As-cast B206As discussed in Section 4.2, 65 compression tests were conducted on as-cast B206 over awide range of temperatures and strain rates. Figure 5.1 shows the stress-strain data for eachof the four strain rates on different plots for a subset of six temperatures. Significant strainhardening occurs at low temperatures which gradually changes to strain-independent, butstrain rate dependent behaviour at higher temperatures. In general, lower temperaturesand higher strain rates resulted in higher flow stresses for a given strain. However, ananomaly was observed in the results at 150 ◦C. At this temperature, the flow stress for allstrain rates were lower in comparison to the results at 175 ◦C. An example of this behaviourcan be seen in Figure 5.1a. Because of the material’s behaviour at this temperature, theconstitutive models discussed in Section 2.1 are all expected to yield an overestimatedprediction of the plastic flow stress. This will be discussed in following sections.An understanding of the strain-rate dependency as a function of temperature isessential for constitutive modelling. To this end, the strain rate dependency of B206at 6 different temperatures is shown in Figure 5.2. Based on these results, the variationbetween the maximum and minimum flow stress (Figure 5.2a (I,II), Figure 5.2f (III,IV)),405.1. Constitutive Behaviour of As-cast B206(a) Strain rate 0.001 (b) Strain rate 0.01(c) Strain rate 0.1 (d) Strain rate 1Figure 5.1: Plastic flow stress variation with temperature at (a) 0.01 and (b) 1 s−1strain rates.at a fixed temperature and strain, increased from 11.5% of the minimum stress value at100 ◦C to 140% at 450 ◦C. This confirms that B206 shows minimal strain rate dependencyat low temperatures and high strain rate dependency at high temperatures, which isconsistent with the typical response of aluminum alloys.Although the general trend observed in Figure 5.2 is that the flow stress decreases withdecreasing strain rate, a few data points in the low temperature range (50 - 275 ◦C) showthe opposite trend. In these cases, the average flow stress increased with decreasing strainrate, Figure 5.2a, b, c. To verify these observations, a set of repeat tests were performedat strain rate/temperature combinations of 0.1/200, 0.001/200 and 0.1/225. The repeattests reproduced the initial results. An example of these results is shown in Figure 5.2c.415.1. Constitutive Behaviour of As-cast B206(a) 100 ◦C (b) 200 ◦C(c) 225 ◦C (d) 300 ◦C(e) 350 ◦C (f) 450 ◦CFigure 5.2: Variation of the plastic flow stress with strain rate at differenttemperatures.425.2. Constitutive Equation for As-cast B206The underlying microstructural cause of this behaviour has not been investigated in thepresent study, owing to the fact that the strain rate sensitivity is generally low at lowtemperatures, so, it will not lead to a significant error within the developed constitutivemodel. It is worth mentioning that the complex strain rate sensitivity of materials duringhot deformation has been attributed in other researches (Kubin and Estrin 1991) tothe concurrent dynamic phenomena that may occur in the matrix. However, in the caseof B206, presence of such phenomena has not yet been studied. At high temperatures(350-530 ◦C), where plastic flow behaviour of B206 is specifically strain rate dependent,no complex or abnormal stain rate dependency was observed.Another characteristic response detected in the compression results was the majortransition in the flow stress behaviour at intermediate temperatures between 275 - 350 ◦C.As the results in this temperature range show, by increasing the temperature, the strainhardening behaviour transitions to a strain-independent / strain-rate dependent behaviour.The complex strain rate dependency of the material response observed at 300 ◦C, Figure 5.2eand verified by repeated tests may be attributed to the fact that this temperature is inthe middle of the transition region between strain hardening and strain rate dependentbehaviour.5.2 Constitutive Equation for As-cast B206As discussed in Chapter 2, the Ludwik model can be used to predict the constitutive flowbehaviour of aluminum alloys, including the characteristic response seen in the B206 alloy,i.e. the transition from a high strain hardening at low temperatures to a strain-independentbehaviour at high temperatures. Thus, as a first step toward developing an inclusiveconstitutive flow stress equation, the possibility of applying this model to the whole rangeof the collected compression data was evaluated by calculating the Ludwik parametersat each testing temperature. The Ludwik coefficients (K, n, m) were determined usingthe least square error method available in the Solver utility within MicrosoftR©Excel.435.2. Constitutive Equation for As-cast B206Figure 5.3: Variation of the strength factor (K parameter) with temperatureshowing higher strength within temperature range of 150 - 200 ◦C.The temperature-specific results obtained for the K parameter are shown in Figure 5.3(red circles). The trendline based on a second order function of the K parameters ateach temperature, after excluding data for 150, 175, 200 and 225 ◦C, is also shown. TheK parameter is a measure of the material’s intrinsic strength. Bearing this in mind andconsidering the results of the variation seen in Figure 5.3, the B206 material shows a muchhigher strength than the average trend at temperatures around 200 ◦C. This variationin the strength factor seems to be phenomenologically related, perhaps to the abnormalstrain rate sensitivity also seen at these temperatures.The examination of the strength factor is a simple demonstrator of the fact that auniversal fit to a single Ludwik model could not be found even using high order functions,unless with significant compromises at different data points. Accordingly, it was concludedthat constitutive modelling at low and high temperatures should be performed separately.This further enabled the accurate modelling of the behaviour at high temperatures thatare in the center of focus in the present study, with regards to the application of the finalmodel in casting simulations within the TPM.The Ludwik model was selected to describe the constitutive flow stress behaviourat low temperatures (50 - 300 ◦C), because it was able to appropriately simulate thestrain hardening behaviour of B206 within this temperature range. For the results at high445.2. Constitutive Equation for As-cast B206temperatures the Zener-Hollomon model was selected to describe the strain independentresponse of the material.In the following, the material coefficients determined for the Ludwik model at lowtemperature and for the Zener-Hollomon model at high temperature are discussed, alongwith challenges encountered by using different models for specific experimental ranges.5.2.1 Ludwik Model DevelopmentThe Ludwik model was selected to model the flow stress at low temperatures. First, theLudwik parameters at each temperature were determined, then the parameter data waswas fit to a polynomial function of temperature. The resulting polynomial coefficients areshown in Table 5.1.In fitting the coefficients to these polynomials, the coefficients found for theK parameterat 150 ◦C and m parameter at 200 ◦C were excluded (highlighted points in Figure 5.4). Inthe former case, as discussed in Section 5.1, the experimental results were generally foundto be weaker than the material strength expected from the results at other temperatures.Hence, the corresponding strength factor was excluded in order to preserve the accuracyof the model at other temperatures. In the latter case, a jump was detected in the strainrate sensitivity of B206 at 200 ◦C. This behaviour was expected based on the complexstrain rate sensitivity of the alloy at low temperatures. However, considering the resultsin Figure 5.4, the strain rate sensitivity at low temperatures is generally low and the mvalue is not a significant contributor to the flow stress predictions within this temperatureTable 5.1: Ludwik model coefficients for B206(f = a4x4 + a3x3 + a2x2 + a1x+ a0)a4 a3 a2 a1 a0K 4.47E-07 -8.01E-04 0.53 -154.53 17312.67n 0.00 -3.51E-08 4.54E-05 -1.96E-02 2.97m 4.17E-11 -5.71E-08 2.82E-05 -5.84E-03 0.42455.2. Constitutive Equation for As-cast B206(a) (b)Figure 5.4: Variation of the (a) K, m and (b) n coefficients in the Ludwik modelwith temperature. Highlighted coefficients were excluded in the polynomialfit.Figure 5.5: Comparison between the experimental flow stress (full lines) and theLudwik model prediction (dashed lines) at 175 ◦C, ∆σ ≈ 20 MPa.range. This can be further explained by a comparison between the experimental data andthe developed model’s prediction, for example at 175 ◦C (Figure 5.5). At this temperature,opposite to the general behavioural trend, the material was found to be softer at the 0.01strain rate than 0.001. However, the maximum difference between the experimental dataand the predicted flow stress at a strain rate of 0.01 is less than 20 MPa, which representsonly about 7% of the experimental flow stress value.465.2. Constitutive Equation for As-cast B206Figure 5.6 shows a comparison between the experimentally determined flow stressand the predictions obtained from the developed Ludwik model at each strain rate andfor a subset of 5 temperatures. As can be seen, the Ludwik model was able to properlypredict the flow stress-strain behaviour of the B206 alloy in the low temperature region.The resulting effect of the previously discussed behaviours at 150 and 200 ◦C on theconstitutive flow stress model’s predictions can be seen in Figure 5.6b and c, respectively.The developed model over-predicts the flow stress at 150 ◦C for all strain rates. However,in the case of 200 ◦C, although the strain rate sensitivity of the material was excluded forfitting purposes, the Ludwik model properly simulates the experimental data at all strainrates, except at 0.01 s−1.For a quantitative evaluation on the accuracy of the developed model, the averageabsolute relative error (AARE) was calculated using the expression below:AARE(%) =1NN∑i=1∣∣∣∣Expi − EstiExpi∣∣∣∣× 100 (5.1)In this equation, N states the number of the acquired data at each data point, and Expiand Esti represent the experimental and modelling/estimated data, respectively. Thecalculated AARE at each data point, shown in Table 5.2, was less than 9% with anaverage of 5.38% over the entire data set, with the exception of the data at 150 ◦C, whichhad an error of 22.7%. This verifies the accuracy of the developed Ludwik model. Notethat the data for the compression tests at 150 ◦C has been removed because both the highvalue of the AARE percentage and the overestimation of the strength factor show that asystematic error occurred during testing at this temperature.Table 5.2: Accuracy of the developed Ludwik model at each temperatureTemperature(◦C) 50 100 150 175 200 225 250 275AARE% 5.96 4.01 22.70 3.70 8.05 4.21 5.73 5.94475.2. Constitutive Equation for As-cast B206(a) 1 s−1 (b) 0.1 s−1(c) 0.01 s−1 (d) 0.001 s−1Figure 5.6: Comparison between the experimental data (full lines) and the Ludwikmodel prediction (dashed lines) at different strain rates and temperatures.5.2.2 Zener-Hollomon Model DevelopmentTo describe the flow stress at temperatures higher than 300 ◦C, a Zener-Hollomon modelwas developed. The methodology used to find the material constants in this model starts bysubstituting the FZH(σ) term in Equation 2.20 with the descriptive terms in Equation 2.21:ε˙ = B′′σn1ε˙ = C′′ exp(βstσ)(5.2)485.2. Constitutive Equation for As-cast B206where, B′′ and C ′′ are fitting coefficients. By taking the logarithm of both sides of theabove equations:ln(σ) = 1n1ln(ε˙)− 1n1ln(B′′)σ = 1βstln(ε˙)− 1βstln(C′′)(5.3)The values of n1 and βst can be found as the mean slope of the ln(σ) vs. ln(ε˙) and σvs. ln(ε˙) plots, respectively. Then, the αst parameter can be found as αst = βst/n1. Byrewriting Equation 2.20 for all strain ranges:ε˙ = Ast(sinh(αstσ))nst exp(EzhRT)(5.4)The logarithm of the equation above gives:ln(sinh(αstσ))=ln(ε˙)nst+EzhnstRT−ln(Ast)nst(5.5)Based on the equation above, the average slope values of the ln(sinh(αstσ))vs. ln(ε˙) atdifferent temperatures and ln(sinh(αstσ))vs. 1/T at different strain rates yield nst andEzh parameters, respectively. Finally, the Ast value can be determined from the y-interceptof the aforementioned plots. Figure 5.7 shows the procedure used for evaluating theZener-Hollomon parameters for B206. As the results in this figure show, the data pointsfor 350 ◦C and 530 ◦C temperatures at the strain rate of 1 s−1 are missing. At 530 ◦C,the deformation force exceeded the prescribed limit for the 5 kN load cell during testing,making the corresponding data unreliable. At 350 ◦C, the material was still showing ahigh amount of hardening at 1 s−1 strain rate, which is incompatible with the strainindependent assumption of the Zener-Hollomon model.The Zener-Hollomon coefficients found in the case of B206 fit the range of valuesreported for age hardenable aluminum alloys (McQueen and Ryan 2002), shown inTable 5.3. To evaluate the predictability of the developed model, the Pearson CorrelationCoefficient (PCC) was calculated from the results of the Zener-Hollomon model. Thisstatistical parameter demonstrates the linear relationship between the experimental data495.2. Constitutive Equation for As-cast B206and the calculated values using the developed model. Figure 5.8 gives a visual demonstrationof this linear relation, showing that a good agreement has been obtained between theexperimental and predicted plastic stress values. PCC can be calculated as:PCC =∑i=1N (Expi − Exp)(Esti − Est)√∑i=1N (Expi − Exp)2∑i=1N (Esti − Est)2(5.6)with the overline stating the average of each respective parameter in whole data range. ThePCC and AARE (Equation 5.1) for the developed Zener-Hollomon model’s prediction ofthe plastic flow behaviour of B206, were found to be 0.97 and 9.51%, respectively. The lowvalues of the error indicators demonstrates that the developed model is able to accuratelydescribe the flow stress behaviour of the B206 alloy at high temperatures.Table 5.3: Comparison between the Zener-Hollomon parameters found for theB206 alloy and average values reported for age hardenable alloys (McQueenand Ryan 2002).Ezh(Kj) lnAst αst (MPa−1) nstB206 274.94 35.72 0.030 4.01(McQueen and Ryan 2002) 124 - 312 22.02 - 91.82 0.030 - 0.052 1.45 - Unified Constitutive ModelAlthough using multiple constitutive models gives an accurate prediction of the plasticflow stress over a wide range of temperatures, the discontinuity at the transition betweenthe constitutive models encumbers their application in the TPM toolset being developedor, in general, in any FEM analysis. Moreover, since the constitutive models in the presentstudy were specifically developed for low and high temperatures, the transition of thematerial’s behaviour from one model to another, in the intermediate regime, must beaccounted for. To address these issues, a unified constitutive model is introduced. In thismodel a simple averaging coefficient (α) accounts for the transition from a hardeningbased behaviour, explained by the Ludwik model (σL), to a strain independent behaviour505.2. Constitutive Equation for As-cast B206(a) (b)(c) (d)Figure 5.7: Calculation of the (a) n1, (b) β, (c) nst and (d) Ezh parameters inZener-Hollomon equations based on the average slope of the plots at eachcurve.based on the Zener-Hollomon model (σZH):σ = α(T)σL + (1− α(T))σZH (5.7)Here, 0 ≤ α(T) ≤ 1 and is defined as:α(T) =12(1 + T˜) (5.8)515.2. Constitutive Equation for As-cast B206Figure 5.8: Correlation between the experimentally measured data and theZener-Hollomon model’s prediction over the entire range of strain, strainrate and temperature. and temperature.In this equation,  represents the width of the transition range (i.e. where 0 < α(T) < 1),shown in Figure 5.9. T˜ , the normalized temperature factor, is defined as:T˜ =T − TtransTmelt(5.9)where Ttrans is the temperature at which the materials behaviour starts to change or theso called transition temperature and Tmelt is the melting temperature, set to 630 ◦C. Thepresent unified approach accounts for the temperature dependency of the change in theconstitutive behaviour of the B206 alloy. However, as discussed in Section 5.1, in thetransition temperature range, the constitutive behaviour is critically dependent on thestrain rate. This shows the necessity of considering the strain rate dependency of theaveraging coefficient, i.e., α = α(T, ε˙). For this purpose, inspired by constitutive behaviourmodelling of polymeric materials (Dupaix and Boyce 2007), a strain rate dependency hasbeen incorporated in the reference temperature as:Ttrans(ε˙) = ξ logε˙ε˙ref+ Tref (5.10)where ξ is a fitting constant and ε˙ref is a reference strain rate.525.2. Constitutive Equation for As-cast B206Figure 5.9: Variation of the averaging coefficient with the  parameter, specifyingthe width of the transition range.The material constants in the unified model were found for as-cast B206 by fittingthe model against the experimental compression test data at 300 ◦C and are shown inTable 5.4. The flow stress prediction using the unified constitutive model at 300 ◦C,shown in Figure 5.10, highlights the ability of the model to properly simulate the plasticdeformation at both high strain rate, incorporating significant strain hardening, and at lowstrain rate, where the material response is strain rate dependent. By fitting the materialconstants in the unified model, the averaging coefficient (α) approaches unity at 275 ◦Cand zero at 350 ◦C. Hence, using the unified model, the predictions of the Ludwik modelat low temperature and Zener-Hollomon model at high temperature remain unchanged.Nevertheless, these two models are combined into a single inclusive unified model. Inthe case of as-cast B206, the transition was found to occur around a single temperature(300 ◦C). However, as it is shown in Figure 5.9, the width of the transition region can becan be controlled by variation of the  parameter in the unified constitutive model.Table 5.4: Material constants in unified model for B206 Tref ε˙ref ξ0.0087 623.5 K 1 s−1 25 K535.2. Constitutive Equation for As-cast B206Figure 5.10: Comparison between the experimental data (full lines) and theLudwik model prediction (dashed lines) at 300 ◦C.5.2.4 DiscussionIn this section, the flow stress behaviour of the as-cast B206 obtained via compressionmeasurements will be compared with the constitutive model results and discussed. Twoseparate constitutive models were developed for low and high temperature ranges foran accurate prediction of the flow stress evolution in the material. These models werecombined into a single unified model that is not only convenient for implementationin simulation tools or FEM analyses, but also provides a precise description of thematerials behaviour at intermediate temperatures. It is worth noting that the Ludwikand Zener-Hollomon models were selected as they accurately represent the behaviour ofas-cast B206 under the limiting conditions; otherwise the unified constitutive model isa general approach that can be employed for describing the transition between differentbehavioural characteristics and their respective constitutive models.The extensibility of the developed model should also be evaluated, to ensure that themodel predictions are realistic outside of the range of strain rates assessed experimentally,in case future FEA simulations encounter such conditions. For this purpose, the modelpredictions at strain rates of both 10 and 10−4 s−1 were calculated. The predictions areshown in Figure 5.11. First, for the low strain rate, Figure 5.11a, the unified model gives a545.2. Constitutive Equation for As-cast B206(a) 10−4 s−1 (b) 10 s−1Figure 5.11: Unified model prediction of the flow stress for as-cast B206 at strainrates of (a) 10−4 and (b) 10 s−1. Note: these strain rates were not used infitting the model coefficients.reasonable description of the flow stress behaviour. This results from the dominance of theZener-Hollomon model at this strain rate. In this case, the transition temperature (Ttrans)has a value of 250 ◦C. The minimum transition temperature, at ε˙ = 0 s−1, is 150 ◦C. Themodel predictions from the high strain rate case, Figure 5.11b, give reasonable results atsome temperatures, but also show what appears to be erroneous behaviour. The erroneouspredictions occur in the intermediate temperature range, 250-300 ◦C where flow stressinversions with respect to temperature are taking place. This behaviour is related to thedominance of the Ludwik model at the high strain rates. Due to the phenomenologicalcharacteristic of the Ludwik model, the stress description at the temperature/strain ratesabove the experimental range are not reliable.Recently, high strain rate Gleeble compression tests (10 s−1) were performed on as-castB206 material, at the NRCan-CMAT laboratory, for another project. A comparisonbetween the model predictions and the experimental data is shown in Figure 5.12. Inspite of the inversion occurring around 300 ◦C, the developed constitutive model doesa generally good job in predicting the flow stress at a high strain rate. The agreementis better at high temperatures, where the prediction is based on the Zener-Holloman555.3. Constitutive Behaviour of the Artificially Aged B206equations, and worse at low temperatures, where the prediction is based on the Ludwikequation.Based on the above analysis, it can be concluded that the developed model is applicableto strain rates lower than those used for data fitting, i.e. strain rates equal to or lowerthan 1 s−1. Within the developed TPM toolset, this is favourable, since high strains, of1 s−1 or more, are unlikely to be seen during B206 casting processes. While the model isnot applicable at high strain rates, even then, the erroneous curve inversions only occurredat strains above ≈0.05. This magnitude of plastic strain is relatively large and would notbe expected to occur during casting.Figure 5.12: Comparison between the experimental data (full lines) and theLudwik model prediction (dashed lines) at strain rate of 10 s−1.5.3 Constitutive Behaviour of the Artificially AgedB206The age hardening behaviour of the B206 alloy at temperatures between 150-250 ◦C foraging timespans of 1-24 h was studied using room-temperature tensile measurements.Figure 5.13 shows the true stress-true strain data acquired from these experiments. Allfracture surfaces following tensile testing showed evidence of brittle and intergranularfracture without necking, as shown in Figure 5.14, indicating the material’s weakness at565.3. Constitutive Behaviour of the Artificially Aged B206(a) 150 ◦C (b) 175 ◦C(c) 200 ◦C (d) 250 ◦CFigure 5.13: Room-temperature tensile test results from samples heat treated atdifferent aging time and temperatures.grain boundaries. This weakness can be attributed to the eutectic phases presence ongrain boundaries, as previously discussed in Section 4.3. The elongation, ultimate tensilestrength (UTS) and yield strength properties of aged B206 were determined from the truestress-true strain curves. As shown in Figure 5.15a, the tensile elongation decreases asa function of both temperature and aging time. However, this decrease is sharper in thecase of lower temperatures, reaching a nearly invariant state at 250 ◦C. The UTS is alsoa decreasing function of temperature and time. Its drop with increasing aging durationremains sharp even at the initial stages of heat treatment at 250 ◦C. Both the presense of575.3. Constitutive Behaviour of the Artificially Aged B206Figure 5.14: SEM imaging of a typical fracture surface after tensile testing of heattreated B206. The observed granular surface is a sign of brittle fractureduring tensile testing.the intergranular fracture and the decrease in UTS and elongation can be related to theevolution of the precipitates, leading to weaker and more susceptible grain boundaries.Figure 5.15c shows the evolution of yield strength in aged B206 as a function ofheat treatment. The results indicate that lower material strength is achieved at agingtemperatures of 150 and 175 ◦C; while at these temperatures, the material’s yield strengthincreases with increasing aging time. In contrast, the strength at 200 and 250 ◦C wasfound to be a decreasing function of aging time, implying that the peak strength wasreached at the initial stages of aging. These observations indicate that the temperaturerange selected in this thesis, to study the effects of aging on yield strength, adequatelycovers the range over which precipitation is most active.585.3. Constitutive Behaviour of the Artificially Aged B206(a) (b)(c)Figure 5.15: (a) Elongation, (b) UTS and (c) yield strength variation as afunction of ageing time for temperatures between 150 and 250 ◦C. The linesrepresent trend-lines while dots show the experiemntal data points.595.4. Constitutive Behaviour Modelling of the Artificiality Aged B2065.4 Constitutive Behaviour Modelling of theArtificiality Aged B206Based on the discussion in the last section, the constitutive behaviour of B206 is highlysensitive to the aging conditions. Furthermore, as discussed in Section 1.3, the knowledgeof the material’s yield strength evolution as a function of heat treatment is an importantpart of the material description for the TMP toolset being developed. Hence, a constitutivemodel is required that expresses the yield strength of the material as a function of theheat treatment parameters, temperature and heating duration.As it has been discussed in Section 2.2, Al2Cu phase precipitation is the most significantevolving microstructural factor for B206 resulting the yield strength variation. Thus, amicrostructural model was developed to characterize the precipitation kinetics for theconstitutive mechanical model. For this purpose, an Avrami model was developed usinga series of experiments conducted to determine the material coefficients. Then, followingthe technique described in Section 2.2, a linear-fit model was developed to describe theyield strength evolution as a function of the input from the Avrami model. Developmentof these models and their results are respectively discussed in following sections.5.4.1 Modelling of Precipitation EvolutionThe microstructural evolution of the solution treated B206 was studied using non-isothermalcalorimetry. For this purpose, solution treated DSC samples were heated up to 530 ◦C atheating rates of 2, 5, 10, 15 and 20 K min−1. The baseline for analysing the DSC resultswas provided by re-running the experiments on heated samples. As expected for a dilutealloy, the only peak in the DSC results was detected around 250 ◦C (the highlighted area inFigure 5.16). A peak at this temperature range is related to the maximum activation rateof the Al2Cu phase precipitation in an aluminum-copper system (Fatmi et al. 2011). Thevariation of the peak temperature with the heating rate (φ) is shown in Table 5.5. Followingthe methodology discussed in Section 2.2.2, by calculating the slope and intercept of the605.4. Constitutive Behaviour Modelling of the Artificiality Aged B206ln(φmT 2m)vs. 1Tm plot, the activation energy and kjo parameter were found to be equal to92.52 kJ and 1.03× 107 s−1, respectively.Figure 5.16: Example isochronal heat flow curve taken from DSC experimentheated at 15 K min−1.Table 5.5: Experimentally determined variation in peak heat flow temperatures asa function of heating rate.φm (◦C min−1) 2 5 10 15 20Tm (◦C) 220.9 234.5 250.7 264.3 269.4Table 5.6: Heat treatment time to peak-aged state at different temperatures, basedon tensile test results.Temperature (◦C) 150 175 200 250Time (h) 24 10 2 1615.4. Constitutive Behaviour Modelling of the Artificiality Aged B206To calculate the Avrami exponent (nj), it was assumed that precipitation ends whenthe peak aged state, i.e. β = 1, is reached. This assumes that there will be no nucleationof new particles and any particle growth that occurs will be at the expense of smallerparticles. Based on this assumption, the experimental peak-aged results from the tensilemeasurements, refer to Table 5.6, were used to find the nj parameter in the Avramiequation. Using this method, the Avrami exponent was found equal to 1.1. The valueof this exponent is generally ≈1 for precipitation processes in aluminum alloys (Starink2004).Table 5.7 shows a comparison between the activation energy found for the microstructureevolution in this work and in other studies of aluminum copper alloys. As can be seen, thevalue found for B206 is in agreement with the values found in the literature for Al2Cu phaseprecipitation in aluminum-copper alloys. This confirms the hypothesis that precipitationof the Al2Cu phase is the dominant evolving microstructural feature occurring duringartificial aging of the B206 alloy.The final results of the developed precipitation evolution model are shown in Figure 5.17.Due to the fast transformation rate at 250 ◦C, precipitation is almost terminated after10 min. However, for the samples at 150 ◦C, precipitation is predicted to take more than30 h of heat treatment to complete.Microhardness experiments were employed to qualitatively evaluate the developedmicrostructure model. The variation of the microhardness in B206 as a function of heatTable 5.7: Comparison between activation energy found for B206 (4.72wt%Cu) inthis work and values reported in the literature for other aluminum-copperalloys.Alloy Ej (kJ mol−1) ReferenceB206 92.52 Present studyAl-1.7% Cu 73.33 - 115.8 (Starink and Van Mourik 1992)Al-2.4% Cu 119.51 (Fatmi et al. 2011)Al-3.7% Cu 67.43 - 76.67 (Fatmi et al. 2013)Al-4.5% Cu 98.58 (Hayoune and Hamana 2009)625.4. Constitutive Behaviour Modelling of the Artificiality Aged B206treatment temperature and duration is shown in Figure 5.18. For the samples at 150 ◦C,hardness was found to increase continuously as a function of ageing time, while the oppositetrend was found at 250 ◦C, where a continuous decrease in hardness with ageing time wasobserved. At 175 ◦C, the hardness increased quickly within the first 1 h of heating, thenremained constant with further heating. It should be noted that the result for the 24 hheating duration at this temperature is estimated from the hardness variation in last 3heating durations (2, 5 and 10 h), due to experimental error that occurred at this condition.The same trend was detected for 200 ◦C where the hardness remained constant after aslight increase following 1 h of heating. The hardness measurements in the present studyreplicated the results obtained from Rockwell tests performed in another independentstudy on B206 alloy by Jean et al. (2009); i.e. continuous hardness increase at 150 ◦C,continuous decrease at 250 ◦C and constant hardness results in the temperature range of175-200 ◦C typically exhibiting a slight increase during the initial stages of heating.The hardness results are consistent with the developed microstructure model sincethey highlight the fast precipitation kinetics at temperatures higher than 200 ◦C wheresamples reach an overaged state (softening) during the initial stages of heat treatment.Also, the continuous increase in hardness at 150 ◦C demonstrates ongoing but slowerprecipitation kinetics that was expected based on the results from the precipitate fractionmodelling. Furthermore, the hardness results for the two intermediate temperatures arehigher than those observed for both the low and high temperatures. This also shows thefast precipitation kinetics at high temperatures where the sample hardness started todrop before 1 h heating, which was the minimum heating duration investigated in thepresent study. However, the constant hardness at the intermediate temperatures doesnot match the expected behaviour based on the Avrami model. This contrasts with thetensile experimental results, which showed a peak yield strength at intermediate heatingtime. As acknowledged by Jean et al. (2009), although hardness experiments are usefulindicators of the artificial aging process, they are not suitable for quantitatively studyingmechanical properties of B206.635.4. Constitutive Behaviour Modelling of the Artificiality Aged B206The results of the Avrami model indicate that the microstructural evolution in B206primarily occurs within the temperature range of 150-250 ◦C and that the material’smechanical properties will not vary significantly by heat treatment at temperatureslower or higher than this range. This is because at temperatures lower than 150 ◦C, theprecipitation kinetics are too slow to allow for significant age hardening. In contrast, whenthe temperature is higher than 250 ◦C, the material’s mechanical properties are dictatedsolely by the coarsening of precipitates. This can be further explained by the hardnessresults at these two temperatures that are evidently lower than the average hardness atother two intermediate temperatures, i.e. 175 and 200 ◦C.Figure 5.17: Predicted (Avrami) precipitation fraction at different temperaturesfor B206.5.4.2 Yield Strength ModellingA linear fit model has been developed to simulate the yield strength evolution of B206as a function of the heat treatment. According to Section 2.2, this model, without anyextension, is only able to explain the yield strength evolution up to the peak-aged state,i.e., it cannot account for softening which occurs in the overaged stage. This limits theapplicability in the present work to less than 175 ◦C, since for higher temperatures, samplesreach the peak-aged state during the initial stages of heat treatment (less than 2 h). It645.4. Constitutive Behaviour Modelling of the Artificiality Aged B206Figure 5.18: Vickers microhardness as a function of heat treatment temperatureand duration. The lines represent trend-lines. Note that the value at 24 hfor 175 ◦C heat treatment is an estimate based on the remaining datapoints.was assumed that all the solute atoms not in solid solution contribute to the precipitatestructure, i.e., αfr in Equation 2.25 equals 1.0. By fitting the experimental data againstthe linear fit model, the model coefficients were found as shown in Table 5.8. Figure 5.19shows the results of this model at different temperatures. It can be seen that in spite ofthe limited set of experimental data points utilized to find the coefficients in the linear-fitmodel, it accurately estimates the yield strength evolution obtained from the tensile tests.Table 5.8: Linear-fit model parameters for B206.Cppt σi Css60 MPa 13 MPa 135 MPa5.4.3 SummaryThe constitutive behaviour of artificially aged B206 has been experimentally investigated.The knowledge gained was then used to develop a model for yield strength evolutionas a function of the aging temperature and time. This model takes into account thedevelopment of precipitate fraction with time and temperature through an Avrami model.655.4. Constitutive Behaviour Modelling of the Artificiality Aged B206Figure 5.19: Comparison between the results of the Linear-fit model (full lines)and the experimental data at 150 and 175 ◦C(dot points).The developed Avrami model is expected to give reasonable predictions over a wide rangeof temperature and time. This is for two reasons. First, the heat treatment experimentswere selected and conducted in the temperature range at which the precipitation processis most active. Second, the exponential description provided by the Avrami model hasa physical basis that is expected to be applicable over a broad range of heat treatmentconditions. The microstructural model developed in the present study can be individuallyused in future studies because it was developed independently of the mechanical models.The resulting yield strength model developed in the present study is only applicableto the age hardening stage and is not able to account for the softening that occurs inthe overaged state. Hence, while using this model in a TPM toolset, another model willbe required to estimate material softening that may occur during ageing at locationsreaching an overaged state. These locations can be identified using the microstructuralmodel, showing the completion of the Al2Cu precipitation.66Chapter 6ConclusionThe objective of the present thesis was to study the constitutive behaviour of a recently-developed aluminum-copper alloy, B206, and develop a constitutive law to simulate itsmechanical behaviour. This constitutive law is to be used as part of a TPM toolsetfor fabricating a hub component for hydro-kinetic energy productions. The behaviouraldescription is expected to consider a wide range of material states in order to predictthe response to residual, thermal, or external stresses. As B206 reaches the yield pointat low strains, the simulation of the plastic flow characteristics of this alloy gives anextensive overview of the material’s response to applied stresses as a function of statevariables: strain, strain rate, and temperature. Furthermore, as shown from the tensileexperiments on artificially aged samples, the yield strength of B206 is highly sensitive toageing parameters. This highlights the importance of being able to predict the evolutionof yield strength as a function of heat treatment, especially for the in-service usage of thematerial. These considerations directed the present thesis towards two main scopes:First, to develop a constitutive model that predicts the plastic flow behaviour ofthe B206 alloy in the as-cast state.Second, to develop a strengthening model that predicts the yield strength of theB206 alloy as a function of heat treatment.6.1 Constitutive Plastic Flow Behaviour ModellingThe plastic flow behaviour of the B206 alloy was investigated through a series of compressiontests conducted using a Gleeble 3500 thermomechanical simulator located at UBC-Vancouver.676.1. Constitutive Plastic Flow Behaviour ModellingThese tests were performed over a wide range of temperatures, 50-530 ◦C, and strain rates,10−3 to 1 s−1. As expected from the general behavioural trend of aluminum alloys, B206showed a high rate of strain hardening at low temperatures followed by a strain ratedependent behaviour at high temperatures. In addition, some deviations from the generaltemperature and/or strain rate sensitivity as well as material hardening were observed atintermediate temperature range. Although the general transition from strain hardeningto strain rate dependent behaviour is expected to be adequately described by a Ludwikmodel, the aforementioned deviations encumbered the material coefficients fitting. Hence,it was concluded that multiple constitutive models are required for flow stress descriptionat different ranges of temperature. For this purpose, a Ludwik model was developed tosimulate the plastic flow stress at low temperatures (50-275 ◦C) and a Zener-Hollomonmodel was developed to describe the behaviour at the high temperature range (350-530 ◦C).Then, a unified model was introduced to combine the developed models into a singleformulation:σ = α(T)σL + (1− α(T))σZHα(T) = 12(1 + T˜)T˜ = T−TtransTmelt , Ttrans(ε˙) = ξ logε˙ε˙ref+ TrefσL = Kεn( ε˙ε˙0 )mσZH =1αstln{(ZAst)1/nst +[(ZAst)2/nst + 1]1/2}(6.1)Based on the developed constitutive model and a qualitative/quantitative analysis of themodel’s predictions, the following conclusions can be made:1. At low temperatures, B206 shows a complex strain rate sensitivity at some data pointsthat results in abnormal softening and/or hardening.2. The unified model represents a segmental approach to approximate the plastic flowbehaviour of the material with an accuracy of higher than 90%. In addition, as all theconstitutive models are combined into a single model, the unified model is a convenientapproach for introducing material properties in simulation tools or FEM analyses.686.2. Constitutive Yield strength Modelling3. Examination of the final model at strain rates higher, 10 s−1, and lower, 10−4 s−1that the measured data shows that while the model gives reasonable results at lowstrain rates, some erroneous predictions occur at high strain rates. This is becausethe plastic behaviour at low strain rates is mainly expressed by the physically-basedZener-Hollomon model, while the behaviour at high strain rates is dominated by thephenomenological nature of the Ludwik model.6.2 Constitutive Yield strength ModellingThe variation of the mechanical properties of B206 as a function of age hardeningparameters were studied using a combination of heat treatment trials, tensile tests,hardness measurements and differential calorimetry. The heat treated samples wereprepared by artificial aging at a temperature range of 150-250 ◦C and heating durationof 1-24 h, after being solution treated. Then, tensile experiments were conducted on agedsamples. Results showed that mechanical properties of B206 are highly sensitive to thermalprocessing. The material’s elongation and UTS were found to be a decreasing function ofboth temperature and heat treatment duration. However, in the case of the yield strength,the continuous age hardening at lower temperatures was followed by the continuoussoftening at high temperature, a signature of the overaged state. A linear-fit model was usedto model the yield strength evolution as a function of the thermal processing parameters:σyield = Cpptf12r + σ0ss(1 + αfr)2/3 + σi (6.2)A microstructural model based on the Avrami model was developed to estimate theprecipitation progression. The material coefficients in the Avrami equation were found byfitting the equation to the experimental data obtained from the non-isothermal calorimetryand following the Kissinger method. Furthermore, hardness measurements were conductedfor a qualitative comparison against the results of the developed Avrami model. Based on696.3. Future Workthe developed constitutive model and a qualitative/quantitative analysis of the model’spredictions, the following conclusions can be made:1. The precipitation activation energy found for the B206 matched closely to the valuesreported in the literature for other dilute aluminum-copper alloys, verifying thatprecipitation of the Al2Cu phase is the dominant microstructural variable duringthe artificial aging of B206. In addition, its fast kinetics in the temperature range of150-250 ◦C was predicted by Avrami model and supported by hardness results.2. The results provided by non-isothermal calorimetry (DSC experiments), can be usedto accurately investigate the kinetics of isothermal processes such as precipitation.3. The overaged state in B206 is reached after less than 30 min of heat treatment at250 ◦C and more than 30 h at 150 ◦C, demonstrating this temperature range as animportant factor to be considered in designing heat treatment trials for B206.4. The linear-fit model can accurately describe the yield strength evolution of the B206alloy in the age hardening stage. However, the softening effects within the overagedstate cannot be explained by this model.6.3 Future WorkBased on our research presented in this thesis, the following ideas are recommended tobe pursuit for future works:− The experimental data for the flow plastic behaviour of B206 at strain rates lower than10−3 s−1 is lacking. This data can be used to verify the low-strain-rate predictions ofthe unified model. For a TPM toolset of cast alloys, low-strain-rate accuracy is of highimportance.− Microstructural analysis trials using TEM or other tools could be employed for a directverification of the precipitation kinetic modelling / Avrami model of B206.706.3. Future Work− The linear-fit model should be extended in order to include the softening effects withinthe overaged state. The respective methodology for this purpose has been brieflydiscussed in the Literature Review chapter of the present manuscript.− The material elongation and UTS are both a decreasing variable of the thermalprocessing. Hence, developing simple models to describe these reductions as a functionof thermal processing temperature and duration is potentially useful in developmentof the TPM toolset for B206. Moreover, the plastic flow behaviour of the materialwas explored at the as-cast state. 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