Modelling the constitutive behaviour of aluminium alloy B206 in the as-cast and artificially aged states

Author’s Accepted Manuscript Modelling the constitutive behaviour of aluminum alloy B206 in the as-cast and artificially-aged states S.M. Mohseni, A.B. Phillion, D.M. Maijer

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S0921-5093(15)30455-X http://dx.doi.org/10.1016/j.msea.2015.09.118 MSA32844

To appear in: Materials Science & Engineering A Received date: 26 August 2015 Accepted date: 30 September 2015 Cite this article as: S.M. Mohseni, A.B. Phillion and D.M. Maijer, Modelling the constitutive behaviour of aluminum alloy B206 in the as-cast and artificially-aged s t a t e s , Materials Science & Engineering A, http://dx.doi.org/10.1016/j.msea.2015.09.118 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modelling the Constitutive Behaviour of Aluminum Alloy B206 in the As-cast and Artificially-aged States S.M. Mohsenia , A.B. Philliona,∗, D. M. Maijerb,∗ a

b

School of Engineering, The University of British Columbia, Kelowna, Canada Department of Materials Engineering, The University of British Columbia, Vancouver, Canada

Abstract The constitutive behaviour of the aluminum-copper casting alloy B206 has been investigated in both the as-cast and artificially aged states. For the as-cast material, a unified plastic flow stress model has been developed using experimental data acquired from compression tests performed using a Gleeble 3500 thermo-mechanical simulator. The unified model considers different constitutive models, each for a specific temperature interval, and the transition between these models at an intermediate temperature based on the material’s strain rate sensitivity. For the artificially aged material, a linear-fit yield strength evolution model as a function of the heat treatment parameters has been developed using experimental data from tensile tests and hardness measurements. The yield strength model takes advantage of an independently-developed microstructure model that specifically describes the precipitation kinetics of the material based on differential scanning calorimetry measurements. Evaluations of the developed models show good fits between the predicted strengths and data from experiments as well as the literature. Keywords: aluminium alloys, B206, constitutive modelling, precipitation kinetics modelling, Differential Scanning Calorimetry, Kissinger method 1. Introduction The aluminum alloy B206 is a recently-developed high-strength casting alloy that has strong potential for use in automotive, aerospace, and energy applications [1]. However, casting defects like hot tears and embrittling secondary phases currently limit this alloy’s application to components of simplified geometry [2]. Hot tears are prevalent in B206 due to its long freezing range, ≈ 100 ◦C, and thus increased usage of this alloy requires precise casting process design [3]. In addition to casting, the heat treatment and machining stages can also lead to variations in material properties that affect a component’s in-service performance. To take advantage of the inherent high strength of B206 while addressing this material’s propensity to form defects during processing, a through-process modelling (TPM) approach is being developed for this alloy at The University of British Columbia. TPM is a recently-developed materials engineering design approach to enhance component performance by simulating the evolution of microstructure, temperature, stresses and strains, and macro/micro scale defects at each stage of the manufacturing process [4].

∗

Corresponding author, [email protected], phone: 250-807-9403, fax: 250-807-9850

Preprint submitted to Elsevier

October 6, 2015

Process simulations developed for each stage of a TPM of a component require broad knowledge of the underlying material properties. For the case of B206, mechanical models are required to predict the alloy’s response to thermal and mechanical loading during processing over a large temperature range between room temperature and the semisolid state. As metallic alloys typically have low yield strains, in order to predict component shape and residual stress following casting, there is a need for constitutive models describing the in-elastic deformation behaviour in the as-cast state. In addition, thermally-activated evolution of the precipitates causes a significant transition in the yield strength of the aluminum-copper alloys [5] such as B206 during artificial ageing. Knowledge of the yield strength is critical for the success of a TPM, since this parameter is required in order to predict the in-service fatigue life of the fabricated component. Thus, a second constitutive model is required that expresses the yield strength of the material as a function of the heat treatment parameters (temperature and heating duration).

2. Modelling of Constitutive Behaviour in Aluminum Alloys 2.1. Plastic Flow Stress The plastic deformation of aluminum alloys is dominated by strain hardening at low temperatures, and strain-rate dependent behaviour at elevated temperatures. A model often used to described this behaviour is the extended Ludwik model[6, 7]: σ = Kεnp (

ε˙p m ) ε˙0

(1)

where σ, εp and ε˙p represent the flow stress, plastic strain and plastic strain rate, respectively, while K, n and m are temperature dependent material coefficients. ε˙0 is a constant with a value of 1 s−1 . Given its empirical nature, the flow stress predictions are accurate only in the range that matches the experimental data used to develop it. The extended Ludwik model can be employed to explain the material’s plastic behaviour in the strain-rate dependent regime. However, fitting both strain hardening and strain-rate dependent behaviours to a single model causes an overall reduction in the accuracy of the model predictions over the whole temperature range of interest. This is mainly due to the averaging characteristic of the extended Ludwik model. In such cases, application of multiple models, each for a specific behaviour type, tends to be advantageous. In the strain-rate dependent regime, the Zener-Holloman model is often used to describe the plastic flow stress: ⎧ 

1/2   Z 1/nst  Z 2/nst ⎪ 1 ⎨ σ= , ln Ast + Ast +1 αst (2) ⎪ ⎩ Z = ε˙p exp Ezh RT where Ezh , αst , Ast and nst are material coefficients. For both models, the coefficients can be found by fitting to data from mechanical deformation measurements over a range of temperatures using fitting/computational programs[8]. As the formulations for the the extended Ludwik and Zener-Hollomon models show, the application of these models is restricted to a monotonic behaviour path, i.e., continuous hardening, or complete strain independent behaviour. Hence, they fail to consider the changes in 2

constitutive behaviour during testing resulting from microstructure evolution. Although in such cases the use of multiple models in different ranges can be considered, the introduction of discontinuities between regimes of behaviour often causes issues when using simulation tools such as FEA. Furthermore, the material characteristics in the intermediate temperature range where the transition occurs would be improperly characterized. Although some of these issues have recently been overcome by [9], who used a temperature-dependent averaging constant to combine the flow stress predictions between two different models selected for low and high temperatures, their approach failed to address any strain-rate dependency within the transition range. At intermediate temperatures, metals tend to show strain hardening when high strain rates are applied, but strain-rate dependency at low strain rates [10]. Thus, to have an accurate estimation of the plastic flow behaviour of B206 over a wide range of temperature and strain rates, an inclusive, continuous constitutive model must be developed that potentially consists of multiple sub-models while also accounting for any strain rate/temperature dependency within transition ranges.

2.2. Yield Strength Evolution The evolution of yield strength during industrial processing results from thermally-activated microstructure evolution. Typically, heat treatable aluminum alloys, like B206, are solutionized and then artificially aged at elevated temperatures to optimize the yield strength while also ensuring sufficient ductility for a given application. One successful approach for studying the different contributors to the total yield strength of a precipitate-hardenable aluminum alloy is to consider the summation of such parameters in a linear manner [11]: σyield = σppt + σss + σi

(3)

where σyield represents the yield strength, σi represents the intrinsic strength including the contributions due to intermetallics, grain size effects and the eutectic phases, and σppt and σss represent the contributions from precipitate and solid solution strengthening. Following the weak obstacles theory [12], σppt , and σss can be expressed as: 

1/2

σppt = Cppt fr σss = Css (1 + ηfr )2/3

(4)

where Cppt and Css are material coefficients, fr represents the precipitate fraction in the matrix, and η represents the fraction of solute depleted from the matrix when precipitation has fully occurred. It should be mentioned that since Eq.(3) has been developed specifically for precipitation hardening behaviour of the materials, it cannot account for the softening effects after peak-aged state. A close examination of Eq.(3) shows that fr is the only microstructure variable within the model, and thus predicting σyield requires knowledge of the nucleation and growth of precipitates during artificial ageing. For this purpose, an Avrami model can be developed to describe the precipitation kinetics as a function of the temperature and heating duration:    β = 1 − exp − (kj t)nj (5) E kj = kjo exp(− RTj ) 3

where Ej represents the activation energy for precipitation, R is the universal gas constant, and nj and kjo are the Avrami coefficients. Due to the difficulties in direct detection and measurement of the precipitates, isothermal calorimetry is typically used to determine Ej and kjo . However, a number of different studies [13, 14] have shown that an analysis based on non-isothermal calorimetry gives similar results while saving considerable experimentation time. The reaction rate for a non-isothermal transformation can be represented as [15]: dβ Ej = kjo exp(− )g(β) (6) dt RT = φ and following the where g is a reaction kinetics model. By defining the heating rate as dT dt Kissinger method [16], which is based on an assessment of the activation energy at the maximum reaction rate (d2 β/dt2 = 0), Equation 6 can be rewritten as: E j φm Ej  = −k g (β) exp(− ) jo RTm2 RTm

(7)

where φm and Tm are the heating rate and temperature at which the maximum reaction rate occurs, respectively. By assuming a first order reaction, i.e. g  = −1, Equation 7 reduces to:   φm kjo R Ej = ln − (8) ln 2 Tm Ej RTm Thus, Ej and kjo can be determined by performing a series of non-isothermal calorimetry experiments, determining the temperature at which the maximum reaction rate occurs, and then plotting the left-hand side of Equation 8 as a function of 1/Tm . Specifically, Ej is given by the slope of the plot and kjo is given by the intercept.

3. Materials and Experimental Methodology A B206 alloy, with composition Al-4.72Cu-0.27Mg-0.27Mn-0.067Fe-0.056Si-0.014Ti (wt.%), was used for the experiments. To generate material with relevant as-cast microstructure, a ring with dimensions 25 cm outer diameter, 15 cm inner diameter, and 14 cm height was cast in a sand mould at the Natural Resources Canada - Materials Testing Laboratory (CMAT) in Hamilton, ON. From this ring, samples were extracted for mechanical testing. The plastic flow stress behaviour of the material was investigated through compression testing conducted on a Gleeble 3500 thermo-mechanical simulator. The cylindrical compression samples were 10 mm in diameter and 15 mm in length. Compression experiments were performed at 14 temperatures between 50 and 530 ◦C, and at 4 strain rates, 10−3 , 10−2 , 0.1 and 1 s−1 . For verification purposes, tests were repeated for 9 different combinations of temperature and strain rate. To extract the plastic deformation of the material in each test, the yield point was selected using the 0.2% offset method. The temperature-dependent elastic modulus was calculated as [17]: μ = 2.54 × 104 (1 + 300−T ) 2Tmelt (9) E = 2μ(1 + ν) 4

where Poisson’s ratio, ν, and the melting temperature, Tmelt , were set to 0.33 and 630 ◦C. The yield strength variation with heat treatment of B206 was investigated through mechanical testing. Samples were initially solution treated at 515 ◦C for 2 h followed by an 8 h treatment at 525 ◦C [18], and then artificially aged at 4 distinct temperatures, 150, 175, 200 and 225 ◦C, for 5 time periods, 1, 2, 5, 10 and 24 h. Tensile tests were performed on cylindrical samples 4 mm in dia. and 19 mm in gauge length using a 50 kN Instron 5969 tensile tester. The yield stress for each heat treatment condition was estimated by the 0.2% offset method, assuming that the elastic modulus is given by the linear part of the true stress-strain curves. Hardness R  measurements were conducted on coupons 5×5×10 mm using a Wilson VH3100 Vickers microhardness tester. The coupons were heat treated with the tensile samples during the artificial ageing process to ensure identical heat treatment conditions. The VH1 data was extracted by averaging values from 20 readings on a 4×5 matrix, each with a holding time of 10 s. Following the non-isothermal calorimetry approach outlined above, differential scanning calorimetry (DSC) tests were performed using a NETZSC STA 449 F3 Jupiter thermal analyser. For this purpose, the DSC samples, prepared with an average weight of 25 mg, were solution treated using the same conditions as the tensile samples. The samples were then heated at one of 5 different heating rates: 2, 5, 10, 15 and 20 K min−1 up to 530 ◦C, in a nitrogen atmosphere while enclosed in an alumina (Al2 O3 ) crucible. The heat flow (W g−1 ) versus time (s) data acquired form the first heating cycle was corrected using a second heating cycle as the baseline.

4. Constitutive Plastic Flow Stress Model Development and Discussion Overall, 65 compression tests were conducted to investigate the plastic flow stress behaviour of the B206 alloy and to provide experimental data to develop the unified constitutive model. Based on the strain dependency of the results, the plastic flow behaviour was divided into 3 regions: low, high and intermediate temperatures. 4.1. Low Temperature (50 − 275 ◦C) At temperatures equal to, or below, 275 ◦C, B206 shows a significant amount of strain hardening together with low strain rate sensitivity, as can be seen in Figure 1 where flow stress data is plotted at each strain rate for a subset of four temperatures. To model this behaviour, the extended Ludwik model was utilized. The model coefficients at each temperature were determined using a least square error method. Then, the entity of the coefficients were fit to a polynomial function of temperature. The material coefficients found for the extended Ludwik model and all subsequent models in this paper are shown in Table 1. The model results are also given in Figure 1 for comparison purposes. As can be seen, the model properly replicated the plastic flow behaviour in the low temperature range. At each test temperature between (50 − 275 ◦C), the plastic flow stress exhibits complex strain-rate sensitivity for at least one tested strain rate. An example of this complexity can be seen in Figure 2, where the material was found to be softer for a strain rate of 10−2 as compared to 10−3 s−1 . The underlying microstructure cause of this behaviour has not been investigated in the present study, owing to the fact that the strain rate sensitivity is generally low at low 5

temperatures, so, it will not lead to a significant error within the final constitutive model. For example, at 175 ◦C, the maximum difference between the experimental data and predicted plastic stress at 0.01 strain rate is less than 20 MPa, which represents only about 7% of the experimental plastic stress value at the same point. The accuracy of the developed extended Ludwik model for B206 was verified by calculating the average absolute relative error (AARE) [19] at each temperature:  N  1   Mi − Pi  AARE(%) = × 100 N i=1  Mi 

(10)

where N represents the number of the acquired data at each data point, and M and P represent the experimentally measured and predicted flow stress values. Over the entire dataset at low temperatures, the average of the AARE was found equal to 5.38% with a standard deviation of 2.46%. Note that the calculation of the AARE considered all data points including those showing complex strain rate sensitivity. 4.2. High Temperature (350 − 530 ◦C) At temperatures equal to, or above, 350 ◦C, B206 shows significant strain-rate dependency and almost no strain dependency. The experimentally-measured flow stress curves along with the model predictions are shown in Figure 3. Figure 4 compares the yield strength extracted from the experimental data with modelling results. In this plot, each symbol represents a different temperature between (350 − 530 ◦C). As can be seen in both Figures 3 and 4, a good fit is achieved between the experimental data and the predictions given by the Zener-Hollomon model, especially at higher temperatures. To quantify the predictability of the developed model in this temperature regime, the Pearson Correlation Coefficient (PCC) [19] has been calculated and found to be 0.97. The PCC: i=1 ¯ )(Pi − P¯ ) (Mi − M PCC =  N (11) i=1 i=1 2 2 ¯ ¯ N (Mi − M ) N (Pi − P ) was used instead of the AARE in the high temperature regime since the strain-rate dependency at high temperatures is defined using only a single value for each flow stress. In addition to matching the experimental data, the Zener-Holloman coefficients for B206 (Table 1), fall within the range of values reported for other age hardenable aluminum alloys (e.g.[20]). Due to the physical basis of the Zener-Hollomon model, the material coefficients found for a specific aluminum alloy should be relatively close to other aluminum alloys within the same family. 4.3. Intermediate Temperature (275 − 350 ◦C) In the temperature range between 275 ◦C and 350 ◦C, a major transition in flow stress behaviour as a function of the strain and strain rate was observed. As can be seen in Figure 5, which shows the experimentally measured flow stress at 300 ◦C (full lines), the behaviour transitions from strain hardening at high strain rates to strain independence at lower strain rates. This behaviour cannot be described properly by the models developed for the high and low temperature ranges, since those models consider a single type of behaviour at a specific temperature. 6

To address this issue and also to provide a unified model for the whole temperature range for use in an FEA model, a unified constitutive model is introduced. In this model the transition from a strain-hardening behaviour explained by the extended Ludwik model, σL , to a strain independent but strain-rate dependent behaviour explained by the Zener-Hollomon model, σZH , is accounted for through a simple averaging coefficient, α, along with a normalized temperature, T: σ  = ασL + (1 −α)σZH α = 12 (1 + T ) (12) trans T = T −T Tmelt where Ttrans is the temperature at which the material behaviour starts to change, the so-called transition temperature, Tmelt is the melting temperature set to 630 ◦C for B206, and is a fitting constant representing the width of the transition region (i.e. the temperature region where 0 < α(T ) < 1). The effect of is given in Figure 6, where a lower value for a given Ttrans leads to a sharp transition and a higher value gives a wider transition. As discussed previously, the constitutive behaviour in the transition, and in fact the transition temperature itself is critically dependent on the strain rate. For this purpose, inspired by constitutive behaviour modelling of polymeric materials [21], a strain rate dependency has been incorporated into Ttrans as: ε˙ ˙ = ξ log + Tref (13) Ttrans (ε) ε˙ref where ξ, ε˙ref and Tref are fitting constants. The fitting constants in the unified model were found for as-cast B206 by fitting the model against the experimental compression test data at 300 ◦C and are shown in Table 1. By comparing the model predictions at 300 ◦C (dashed lines in Figure 5) to the experimental data, it can be seen that the unified model is able to replicate both significant strain hardening at high strain rates and the strain rate dependent behaviour at low strain rates. Note that for the minimum possible strain rate (ε˙ = 0), Ttrans = 150 ◦C. However, for most of the parameters used in the model, Ttrans ≈ 300 ◦C causing the averaging factor to approach unity at ≈275 ◦C and zero at ≈350 ◦C. This ensures a smooth transition to the extended Ludwik and Zener-Hollomon models at these extents.

5. Yield Strength Evolution Model Development and Discussion 5.1. Experimental Measurements Tensile test and micro hardness measurements on B206 samples artificially aged at different temperatures and heating durations are shown in Figure 7. The results indicate that yield strength and hardness increase with increasing artificial aging time at 150 and 175 ◦C, but decrease with increasing time at 200 and 250 ◦C. These results are consistent with the results obtained from Rockwell tests in an independent study on B206 [22]. The apparent decrease in strength at the high temperatures occurs because the minimum ageing time was 1 h which is likely longer than the time to peak strength; shorter times would be needed to determine the peak in the curve. Since both a continuous increase and decrease in material strength can be 7

observed in the experimental data (Figure 7), it can be assumed that the maximum precipitation rate occurs within this temperature range.

5.2. Modelling of Precipitate Evolution As discussed in Section 2.2, knowledge of precipitation kinetics is essential for to calculate the yield strength variation in B206, assuming that precipitation of the Al2 Cu phases is the major strengthening agent. To develop the required Avrami model, non-isothermal calorimetry tests using DSC were conducted at different heating rates. One example result from the DSC tests is shown in Figure 8 for a heating rate of 15 K min−1 . As expected from a dilute alloy, the only peak in the heat evolution was detected at a temperature of approx. 250 ◦C. Furthermore, precipitation initiates after ≈ 5 min of heating, when the temperature was ≈ 75 ◦C, however, the maximum reaction rate was reached at ≈ 250 ◦C. Based on the variation in the peak temperature with heating rate, shown in Table 2, Ej and kjo in Eq.(5) were determined and reported in Table 1. As shown in Table 3, a strong agreement is found between the activation energy of the Al2 Cu phase calculated in this work and the values reported in the literature in other aluminum-copper alloys. This result confirms that precipitation of the Al2 Cu particles is the dominant evolving microstructural feature during artificial aging of the B206 alloy. The Avrami exponent, nj , was then calculated assuming that precipitation terminates at the peak aged state, i.e., β = 1. 1 This was done by first identifying the four peak-aged points in the tensile test results, highlighted in Figure 7a, then calculating the nj value for each (since it was the only remaining variable in Equation 6), then taking the average value of the four results. The Avrami exponent is a characteristic of the reaction being described by the Avrami equation. Theory suggest that it equals unity for a reaction with diffusion-based growth of the nuclei in a 2D shape within a saturated matrix [23]. Given that solution treated B206 meets the super 1

This means that precipitate nucleation does not occur beyond the peak-aged state; only previously formed particles grow at the expense of smaller particles. Table 1: Values of fitted coefficients in different models.

Equation

Model

Coefficients

(1)

Extended Ludwik

K = 4.5e-7T 4 -8e-4T 3 +0.53T 2 -154.5T +17312.67 n = -3.51e-8T 3 +4.54e-5T 2 -1.96e-2T -2.97 m = 4.17e-11T 4 -5.71e-8T 3 +2.82e-5T 2 -5.84e-3T +0.42

(2)

Zener-Hollomon

(12,13) (5) (3,4)

Unified Model Avrami Equation Linear-fit

Ezh = 274.94 kJ, ln Ast = 35.72, αst = 0.03 MPa, nst = 4.01 = 8.7e-3, Tref = 623.5 K, ε˙ref = 1 s−1 , ξ = 25 K Ej = 92.52 kJ, kjo = 1.03e7, nj = 1.1 Cppt = 60 MPa, Css = 135 MPa, σi = 13 MPa

8

saturated condition, the value found in this work, nj =1.1, agrees well with the theoretical value for the precipitation of the Al2 Cu phase. It is worth noting that precipitation is a diffusion-based phenomenon that in aluminum copper alloys leads to 2D plate-like precipitates [24]. The evolution in precipitate fraction predicted by the Eq. (5) is shown in Figure 9a for the four heat treatment temperatures between 150-250 ◦C. As can be seen, there is slow precipitate evolution at 150 ◦C, leading to a continuous increase in material hardness, and fast precipitate evolution at 250 ◦C, where the over-aged state is reached during the initial heating. Complete precipitation is predicted to occur in ≈ 30 h at 150 ◦C but requires only ≈ 10 min at 250 ◦C 5.3. Yield Strength Modelling The linear-fit model, Equation 3, was used to describe the yield strength evolution of B206 as a function of heat treatment. As discussed in Section 2.2, this model is only valid up to the peak-aged point. This limits the applicability of the experimental results, to be used for model development, to 150 and 175 ◦C, since at higher temperatures the over-aged state is reached in the initial stages of the aging treatment. By assuming that all the solute atoms separating from the solid solution join the precipitate structure, η in Equation 4 equals 1.0. Other material coefficients in the linear fit model were determined by fitting the experimental data against the model. A comparison between the predicted yield strength and the experimental data, in Figure 9b, shows that the model is able to replicate the yield strength evolution of B206 fairly well. Over this dataset, the AARE was found to have a value of 6.1%, indicating a good match between experimental data and yield strength model predictions. Table 2: Experimentally determined variation in peak heat flow point as a function of heating rate.

φm (C/min) Tm (C)

2 220.9

5 234.5

10 250.7

15 264.3

20 269.4

Table 3: Comparison between activation energy found for B206 (4.72wt%Cu) in this work and values reported in the literature for other aluminum-copper alloys.

Alloy B206 Al-1.7% Al-2.4% Al-3.7% Al-4.5%

Cu Cu Cu Cu

Ej (kJ mol−1 ) 92.52 73.33 - 115.8 119.51 67.43 - 76.67 98.58

Reference Present study [25] [26] [27] [28]

6. Summary The constitutive behaviour of B206 has been studied in the as-cast and artificial aged states. 1. A unified plastic flow stress model was developed from experimental data acquired from compression tests conducted on the as-cast material. The developed model predicts the 9

plastic flow stress via an extended Ludwik model at low temperatures and a ZenerHollomon model at high temperatures. The intermediate transition regime is modelled through the use of an averaging coefficient that takes into account both the temperature and the strain rate sensitivity of the material. This unified model is able to predict the flow stress of B206 with an accuracy of 94% over a temperature range of 50-530 ◦C and a strain rate range of 10−3 -1 s−1 . 2. A yield strength evolution model for artificial ageing was developed from experimental data (tensile and hardness tests, and iso-calorimetry experiments). Although the yield strength model was constructed with only a limited set of experimental data points, an accuracy of 93% was achieved, resulting from the precise description of the precipitation kinetics given by the developed Avrami equation, as it is the main variable in a linear-fit model.

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List of Figures 1

2 3

4 5 6 7

8 9

Comparison between the compression test experimental data and the extended Ludwik model’s prediction in the temperature range between (50 − 275 ◦C) and for different strain rates. The experimental flow stress and model predictions are given by solid and dashed lines, respectively. . . . . . . . . . . . . . . . . . . . . Experimentally-measured flow stress curves at 175 ◦C, showing complex strainrate behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison between the compression test experimental data and the ZenerHollomon model’s prediction in the temperature range between (350 − 530 ◦C) and for different strain rates. The experimental flow stress and model predictions are given by solid and dashed lines, respectively. . . . . . . . . . . . . . . . . . . Correlation between the experimental data and model predictions for plastic stress values at high temperatures, for entire strain rate range. . . . . . . . . . . . . . Comparison between the measured plastic flow stresses and the values predicted by the unified constitutive model at 300 ◦C. . . . . . . . . . . . . . . . . . . . . . Variation in the averaging coefficient with the choice of parameter, specifying the width of the transition range. . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Yield strength and (b) hardness variation as a function of ageing time and temperature. The highlighted points show the maximum yield strength reached at each temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example non-isothermal heat flow curve, at 15 K min−1 , taken from the DSC experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results of the (a) Avrami model showing precipitation evolution kinetics and (b) Linear-fit model showing yield strength evolution. The experimental data is given by dots for comparison purposes. . . . . . . . . . . . . . . . . . . . . . . . . . .

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(a) 1 s−1

(b) 0.1 s−1

(c) 0.01 s−1

(d) 0.001 s−1

Figure 1: Comparison between the compression test experimental data and the extended Ludwik model’s prediction in the temperature range between (50 − 275 ◦C) and for different strain rates. The experimental flow stress and model predictions are given by solid and dashed lines, respectively.

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Figure 2: Experimentally-measured flow stress curves at 175 ◦C, showing complex strain-rate behaviour.

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(a) 1 s−1

(b) 0.1 s−1

(c) 0.01 s−1

(d) 0.001 s−1

Figure 3: Comparison between the compression test experimental data and the Zener-Hollomon model’s prediction in the temperature range between (350 − 530 ◦C) and for different strain rates. The experimental flow stress and model predictions are given by solid and dashed lines, respectively.

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Figure 4: Correlation between the experimental data and model predictions for plastic stress values at high temperatures, for entire strain rate range.

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Figure 5: Comparison between the measured plastic flow stresses and the values predicted by the unified constitutive model at 300 ◦C.

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Figure 6: Variation in the averaging coefficient with the choice of  parameter, specifying the width of the transition range.

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(a)

(b)

Figure 7: (a) Yield strength and (b) hardness variation as a function of ageing time and temperature. The highlighted points show the maximum yield strength reached at each temperature.

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Figure 8: Example non-isothermal heat flow curve, at 15 K min−1 , taken from the DSC experiments.

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(a)

(b)

Figure 9: Results of the (a) Avrami model showing precipitation evolution kinetics and (b) Linear-fit model showing yield strength evolution. The experimental data is given by dots for comparison purposes.

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