materials and design - spiral: home · al-cu-mg [14], al-mg-si [15], al-zn-mg [16] and al-li [17]...

Materials and Design 183 (2019) 108121

JMADE-108121; No of Pages 15

Contents lists available at ScienceDirect

Materials and Design

j ourna l homepage: www.e lsev ie r .com/ locate /matdes

Experimental investigation and modelling of yield strength and workhardening behaviour of artificially aged Al-Cu-Li alloy

Yong Li, Zhusheng Shi ⁎, Jianguo LinDepartment of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK

H I G H L I G H T S G R A P H I C A L A B S T R A C T

• A constitutive model with microstruc-ture, strength, work hardening sub-models is developed for artificial ageingof AA2050.

• Themodel has for the first time success-fully predicted ageing behaviour cover-ing under-ageing to over-ageing.

• Yield strength and work hardening be-haviour from under-aged to over-agedconditions has been characterised andmodelled.

• Shearing-to-bypassing strengtheningtransition does not occur immediatelywhen reaching peak-ageing state.

⁎ Corresponding author.E-mail address: [email protected] (Z. Shi).

https://doi.org/10.1016/j.matdes.2019.1081210264-1275/© 2019 The Authors. Published by Elsevier Ltd

Please cite this article as: Y. Li, Z. Shi and J. Lartificia..., Materials and Design, https://doi.o

a b s t r a c t
a r t i c l e i n f o
Article history:Received 25 June 2019Received in revised form 27 July 2019Accepted 11 August 2019Available online 12 August 2019

The yield strength and work hardening properties of an Al-Cu-Li alloy AA2050 after artificial ageing havebeen experimentally investigated and modelled in this study. Uniaxial tensile stress-strain curves of thealloy artificially aged for up to 500 h have been acquired and evolutions of main precipitates during ageinghave been summarised to elucidate the underlying mechanisms of the observed mechanical properties,such as yield strength and work hardening behaviour. Work hardening analysis with Kocks-Meckingplots has been performed to analyse the shearing-to-bypassing transition progress of the aged alloy andit has been found that the transition does not occur at the peak-ageing state. A new mechanism-based uni-fied constitutive model, comprising three sub-models, has been developed to simultaneously predict theevolutions of microstructures, yield strength and work hardening properties of the artificially agedAA2050. It is the first unified model covering a wide range of artificial ageing conditions from under-ageing to over-ageing, providing an effective tool for performance prediction of the aged alloys for indus-trial applications. The model has the generic feature and could be applied to artificial ageing of other2xxx series aluminium alloys with dominant T1 precipitates.

© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Keywords:Artificial ageingConstitutive modellingAA2050Strengthening mechanismYield strengthWork hardening

. This is an open access article under

in, Experimental investigatiorg/10.1016/j.matdes.2019.10

Nomenclature

Vc,

the

n a812

ariables

CC BY lice

nd mod1

Unit

nse (h

ellin

Specification
c0, ca, cs wt% Solute concentration in the matrix, its initial value, equilibrium
values at ageing temperature and solution heat treatmenttemperature respectively

(continued on next page)

ttp://creativecommons.org/licenses/by/4.0/).

g of yield strength and work hardening behaviour of

http://crossmark.crossref.org/dialog/?doi=10.1016/j.matdes.2019.108121&domain=pdf

http://creativecommons.org/licenses/by/4.0/


https://doi.org/10.1016/j.matdes.2019.108121

mailto:[email protected]

Journal logo



http://www.sciencedirect.com/science/journal/

www.elsevier.com/locate/matdes


c,f,

faf

f

h

h

h

Nqrdrdrdrctεpθ,ρ

ρρ

ρ

ρρσ

σσσ

σ

σ

σ

2 Y. Li et al. / Materials and Design 183 (2019) 108121

Pa

c0, ca

lease cirtificia..

–

te this., Mate

Normalised c and corresponding values for c0 and ca
fn, fd – Total volume fraction and volume fraction components from T1
precipitate and dissolving clusters respectively
– Equilibrium volume fraction at ageing temperature
, f n , f d
– Normalised total volume fraction and components from T1precipitate and dissolving clusters respectively
0, f n0,

f d0

–
Initial values of f , f n and f d
, hc
nm Thickness of T1 precipitate and its critical value to be fullynon-shearable respectively
, h0
– Normalised thickness of T1 precipitate and its initial valuerespectively
tran ,

htran2

–
Normalised thickness of T1 precipitate when it starts to benon-shearable and becomes fully non-shearable respectively
–
Strength contribution exponent – Aspect ratio of precipitate
, rn
nm Radius of dissolving clusters and T1 precipitate respectively , rn – Normalised rd and rn
0, rn0
– Initial values of rd and rn
nm
Critical radius of T1 precipitate at peak-ageing state h Time
, εu
– Plastic strain and uniform elongation respectively θ0 MPa Work hardening rate and its initial value respectively , ρ0, ρs m−2 Dislocation density, its initial value in the as-received alloys and
the maximum value in the alloy during ageing respectively
i m−2 Dislocation density in the alloy after SHT and water quenching ss, ρgn m−2 Statistically stored dislocation density and geometrically
necessary dislocation density respectively
ssm,ρgnm
m−2
Maximum values of ρss and ρgn
, ρ0
– Normalised ρ and its initial value in the as-received alloy
ss , ρgn
– Normalised ρss and ρgn t MPa Combined contribution to yield strength from precipitates and
dislocations
d MPa Contribution from dislocations to work hardening f, σw MPa Flow stress and stress increase due to work hardening dis, σp,σss
MPa
Hardening contributions from dislocation, precipitate and solidsolution to yield strength respectively
p−d,σp−n

MPa
Contribution to precipitation hardening from dissolving clustersand T1 precipitates respectively
r
MPa A radius related factor in precipitation hardening equation ofσp−n
shear,σbypass

MPa
Contribution to precipitation hardening from shearableprecipitate and non-shearable precipitate respectively
y, σUTS
MPa Yield strength and ultimate tensile strength σ
1. Introduction

The recently developed third generation Aluminium-Lithium (Al-Li)alloys, also termed as Aluminium-Copper-Lithium (Al-Cu-Li) alloys, arelightweight materials and currently attracting strong interest in aero-space applications, as they overcome the limitations of low toughnessand high anisotropy in predecessor generations, while retaining thehigh modulus and high strength-to-weight properties [1,2]. Artificialageing is essential for the Al-Cu-Li alloys to achieve the high strength re-quirement for aerospace products. A complex precipitation sequencewith particular precipitates, such as T1 (Al2CuLi), has been reported dur-ing artificial ageing of these ternary system alloys [3,4], whichwill affectthe dislocation-precipitate interaction mechanism during plastic defor-mation and result in particular mechanical properties of the aged alloys[5,6]. Understanding and predicting the relationships between precipi-tation andmainmechanical properties during plastic deformation of ar-tificially aged Al-Cu-Li alloys are not only of scientific interest but also ofgreat practical importance to enhance their applications in the aero-space industry.

The precipitation behaviour of Al-Li alloys has been widely investi-gated. Different precipitates have been observed during ageing of Al-Lialloys, including GP zones, θ′ (Al2Cu), δ′ (Al3Li), S′ (Al2CuMg) and T1(Al2CuLi) [3,7]. The Li content has been reported to play a decisive rolein precipitation of Al-Li alloys [4]. With high Li contents (N2%) in the1st and 2nd generation Al-Li alloys, δ′withminor S′ plays the dominantrole in strengthening, while with low Li contents (b2%) in the 3rd

article as: Y. Li, Z. Shi and J. Lin, Experimental investigatiorials and Design, https://doi.org/10.1016/j.matdes.2019.10

generation Al-Li alloys investigated in this study, T1 is the dominantstrengthening precipitate, together with minor θ′ [1,4].

T1 precipitate provides a high strengthening effect and its strength-ening mechanism has been the subject of a number of studies. Previ-ously it was believed that T1 precipitate is a strong non-shearableparticle [8,9], while recent studies [10,11] have characterised it as ashearable precipitate that can be sheared only once at the same location.The thickening of T1 precipitates has been indicated as the main factorthat facilitates the strengthening mechanism transition from shearingto bypassing during plastic deformation [5,11,12]. Evolution of T1 pre-cipitate during artificial ageing of AA2050-T34 has been studied re-cently [13], whose thickness shows an increasing trend after 300 hageing at 155 °C, however, no mechanical property results have beenprovided. The effect of T1 precipitate on the work hardening behaviourof an Al-Cu-Li alloy during plastic deformation has been studied [5], andit was indicated that T1 precipitate can generate a higher strain harden-ing rate than other shearable precipitates, such as δ′, due to its single-pass shearing property.

Yield strength models based on detailed precipitation behaviourhave been developed for ageing of different aluminium alloys, such asAl-Cu-Mg [14], Al-Mg-Si [15], Al-Zn-Mg [16] and Al-Li [17] alloys.Most of these models utilised either shearing or Orowan bypassingmechanisms to characterise yield strength behaviour of alloys with ei-ther under-aged or over-aged conditions [18]. Shercliff and Ashby [19]have introduced a harmonic mean equation to approximately combineshearing and bypassing contributions to yield strength in the sameequation. For the Al-Cu-Li alloy investigated in this study, Li et al.[20,21] have proposed a yield strength model with simplified morphol-ogy of T1 precipitates and successfully predicted yield strength evolu-tion of AA2050-T34 alloy up to the peak-aged state. Dorin et al. [12]have proposed a yield strength model for AA2198 by considering inter-facial and stacking fault energy, however, overestimation of yieldstrength has been observed after the peak-ageing state. Hence, existingmodels are not sufficient to accurately capture the complicated precip-itation and yield strength evolutions of the aged Al-Cu-Li alloys rangingfrom under-ageing to over-ageing.

In addition to yield strength, modelling of the strain hardening be-haviour during plastic deformation has also been conducted by manyinvestigators [14,22]. Themost common approach is to use internal var-iable models, based on the one-internal-variable model developed byKocks and Mecking [23,24]. Plasticity behaviour of alloys can be signifi-cantly affected by their ageing states and many studies have been fo-cused on this phenomenon [25–27]. Effects of microstructures, such assolutes [28] and precipitates [29], have been introduced as new internalvariables into strain hardening models for various aluminium alloys[30,31]. Most of the current models only considered either shearing orbypassing mechanisms during plastic deformation. Myhr et al. [32]have proposed a combined model considering both shearing andbypassing mechanisms for work hardening behaviour of Al-Mg-Si al-loys, which have different precipitation and shearing-to-bypassingprogress from those of Al-Cu-Li alloys [5]. To effectively facilitate the ap-plications of Al-Cu-Li alloys, it is of great importance to develop amodelto predict precipitate characteristics and basic plastic deformation prop-erties of the alloys from under-aged to over-aged conditions. However,such a model is still not available currently.

In this paper, the yield strength andwork hardening behaviour of anAl-Cu-Li alloy, AA2050-T34, after artificial ageing for large span of timeranging from under-ageing to over-ageing, has been revisited. Precipi-tate evolutions and work hardening rate analysis have been utilised toinvestigate the detailed relationships betweenmicrostructures andme-chanical properties of the alloy. Based on these, a set of mechanism-based constitutive equations has been proposed for the first time to in-corporate microstructural evolutions into yield strength and two-state-variable work hardeningmodels, with which, evolutions of microstruc-tures, yield strength andwork hardening behaviour of the Al-Cu-Li alloyfrom under-ageing to over-ageing have been successfully modelled.

n and modelling of yield strength and work hardening behaviour of8121


Table 1Main chemical composition ranges of AA2050 (wt%).

Al Cu Li Mg Ag Mn Zr

Balance 3.2–3.9 0.7–1.3 0.20–0.60 0.20–0.70 0.20–0.50 0.06–0.14

Table 2Summary of main precipitates during artificial ageing of AA2050-T34 at 155 °C,summarised from the results in [13,34–36].

Material state Time range Precipitates

As-received – Cu-rich clustersReversion 0–2 h Dissolving of clustersUnder-aged to peak-aged 2–18 h Nucleation and growth of T1 (minor θ′)Peak-aged to over-aged 18 h onwards Gradually coarsening of T1 (minor θ′)

3Y. Li et al. / Materials and Design 183 (2019) 108121

2. Experimental procedure

The material used in this study is a third generation Al-Cu-Li alloy,AA2050, whose main composition ranges are listed in Table 1. The as-received material was rolled plates of half inch thickness and hadbeen solution heat treated, water-quenched, pre-stretched and thennaturally aged for several months, leading to the T34 temper condition,as demonstrated in Fig. 1(a). The basic properties of the alloy have beenreported in detail in [2,33], which demonstrate a very good isotropic be-haviour inmechanical properties (b4% difference in yield strength alongthe rolling direction and along the 45 degree of the rolling direction).Hence, all the experiments in this study were performed along therolling direction as a representation.

Test specimens were machined from the centre of the as-receivedplate along the rolling direction. The detailed dimensions of the speci-mens for tensile tests are shown in Fig. 1(b). In addition, some cubicsamples with a dimension of 10 ∗ 10 ∗ 10 mm3 were also prepared forhardness tests. Artificial ageing tests were then performed for differentduration. During artificial ageing, specimens were placed in a furnace,heated to 155 °C and kept for different time (0, 2, 5, 12, 18, 24, 32, 70,150, 300 and 500 h). A thermocouple was attached on the specimensto record the temperature during the tests. After the required ageingtime, the specimens were taken out of the furnace and cooled to roomtemperature in atmosphere. As the cooling stage is much shorterthan the isothermal ageing period (the temperature decreases tobelow 80 °C within 10 min) and the temperature is continuously de-creasing, the effect of cooling phase on further evolution of precipitatesin the artificially aged alloy is not expected to be significant and hencewas not considered in this study. The heat treatment history is illus-trated in Fig. 1(a).

Fig. 1. (a)Heat treatment history of AA2050-T34 and subsequent artificial ageing tests and(b) dimensions of tensile test specimen (units: mm).

Please cite this article as: Y. Li, Z. Shi and J. Lin, Experimental investigatioartificia..., Materials and Design, https://doi.org/10.1016/j.matdes.2019.10

Room temperature tensile tests of the artificially aged specimenswere subsequently carried out in an Instron 5584 machine to obtainthe detailed yield strength and work hardening behaviour of the alloy.Strain data was obtained with an Instron 2630-107 extensometer at-tached to the gauge section of the specimens during tests. The strainrate used was 10−4 s−1. Tests for some selected ageing conditions (0,2, 18 and 150 h) have been repeated for three times, and the variationsof the yield strength results for the same aged condition were all within±5 MPa. In addition, Vickers hardness of the artificially aged cubicsamples was measured using a Zwick Roell hardness machine with aload of 1 kgf (HV1) and the reported value is an average of 10measurements.

3. Precipitate evolution during artificial ageing

The detailed precipitate evolutions of the naturally aged AA2050alloy during long-term artificial ageing at 155 °C have been investigatedpreviously [13,34–36], and themain sequence is summarised in Table 2.Cu-rich clusters are the main precipitates in the as-received T34 mate-rial, which will be dissolved during the first 2 h of artificial ageing.Meanwhile, nucleation and growth of the dominant T1 precipitatesoccur during artificial ageing until reaching the peak-ageing state at18 h [21]. Fig. 2(a) shows the transmission electron microscopy (TEM)image of the alloy at the peak-ageing state [35], and correspondinghigh resolution TEM (HR-TEM) image in Fig. 2(b) indicates a singlelayer structure of the T1 precipitate. After that, coarsening of T1 precip-itates occurs gradually with a very slow speed. T1 precipitate demon-strates a plate shape, which has been schematically illustrated in Fig. 2(c).

The detailed evolutions of T1 precipitate dimensions in AA2050-T34alloy, including average diameter and average thickness, during artifi-cial ageing at 155 °C from different studies are plotted in Fig. 3. The av-erage diameter values from different studies [13,34–36] are close toeach other and show a similar trend with ageing time (Fig. 3(a)), in-creasing with a high rate in the first 18 h and then remaining at a com-paratively stable level. The average thickness of T1 precipitates stays at astable level for a long time (single layer, b2 nm) during artificial ageingat 155 °C and starts to increase after about 300 h, as shown in Fig. 3(b).

4. Experimental results and discussion

4.1. Mechanical properties

Fig. 4 shows the hardness curve of AA2050-T34 after artificial ageingat 155 °C for different time. The results correspond well with the datafrom [36] for the same alloy aged at 155 °Cwithin 30h. The hardness ex-periences an initial decrease in the first 2 h due to the dissolution of Cu-rich clusters indicated in Table 2. A subsequently rapid increase of hard-ness is observed between 2 and 18 h, when nucleation and growth of T1precipitates occur. After that, the hardness keeps at a high level until300 h and a slight decrease of hardness is observed at 500 h.

Fig. 5 shows the true stress-strain curves of AA2050-T34 artificiallyaged at 155 °C. The yield strength of the material experiences thesame changing trend with the hardness data shown in Fig. 4. The as-received alloy shows a relatively high yield strength (272MPa)with ap-parent work hardening behaviour. Serrations occur at the late stage of



Fig. 2. (a) TEM image of AA2050-T34 after 18 h ageing at 155 °C, showing the T1 precipitates along111Al; (b)HR-STEM image showing the edge-on configuration of T1 precipitate [35]; and(c) schematic of a plate-shaped T1 precipitate.


the stress-strain curve, which is known as the Portevin-Le Chatelier ef-fect [37]. As the Portevin-Le Chatelier effect is generally caused by highlevel of solutes in alloys which restrain dislocation movement duringplastic deformation [37,38], the serrations indicate that a high level ofsolutes exists in the matrix of AA2050 alloy at T34 condition. The mostsignificant serrations occur in the specimen after 2 h artificial ageing, in-dicating the highest level of solutes in thematrix of the alloy after disso-lution of Cu-rich clusters. Serrations become less obvious after 5 h anddisappear after 8 h ageing because of the nucleation and growth of T1

Fig. 3. Evolution of (a) average diameter and (b) average thickness of T1 precipitates


precipitates. After peak-ageing, similar stress-strain curves are observedfor the alloy artificially aged from 18 to 300 h, while a decrease of yieldstrength is observed after 500 h in Fig. 5, which indicates a slight over-aged behaviour at this state.

Fig. 6(a) summarises the evolutions of yield strength (σy), ultimatetensile strength (UTS,σUTS) and uniformelongation (εu) from the tensiletest curves in Fig. 5. Yield strength and UTS share the same trend withthe hardness curve in Fig. 4. These strength values are related to themi-crostructural states in the alloy and can be explained according to the

in AA2050-T34 during artificial ageing at 155 °C. (Data comes from [13,35,36].)



Fig. 4. Hardness curve of AA2050-T34 after artificial ageing at 155 °C (square symbols –current study; diamond symbols – from [36]).

Fig. 6. (a) Evolutions of yield strength (σy), UTS (σUTS) and uniform elongation (εu) and(b) saturate work hardening stress (σUTS − σy) versus artificial ageing time for AA2050-T34 at 155 °C. The insert shows the relationship between εu and (σUTS − σy).


evolution of the dominant T1 precipitate shown in Fig. 3. As T1 precipi-tates grow only in diameter and the thickness remains constant beforepeak-ageing (18 h), it is reasonable to conclude that the increasing di-ameter of T1 precipitates is the main reason for the strengthening be-haviour of AA2050 from under-aged to peak-aged states. The highstrength remains stable after peak-ageing for a long period (from 18to 300 h), as the diameter of T1 increases only slightly and the thicknessremains thin and constant (b2 nm [13]), which can be sheared throughby dislocations. After 300 h of artificial ageing, the thickness of T1 pre-cipitates starts to increase, which leads to the start of the shearing tobypassing transition [11], and thus results in the over-aged behaviourafter 500 h ageing.

The uniform elongation (εu) evolves oppositely to σy and σUTS. Thedifference between UTS and yield strength (σUTS− σy) is used to repre-sent the saturate work hardening stress of the alloy and its evolutionwith ageing time is shown in Fig. 6(b), which demonstrates the sametrend with εu and a very good linear relationship has been found be-tween them, as shown in the insert of Fig. 6(b). The solute contents inthe alloy can enhance the efficiency of dislocation storage during plasticdeformation, and thus, helping to increase the level of (σUTS − σy) andεu of the alloy [5]. As nucleation and growth of precipitates deplete thefree solutes, an opposite evolution trend for both (σUTS − σy) and εu isobserved to that of strength values, as shown in Fig. 6. In the over-aged states, geometric necessary dislocations are introduced due tothe existence of non-shearable precipitates [39],which could be the rea-son of the slight increase of saturate work hardening stress after 300 hartificial ageing shown in Fig. 6(b).

Fig. 5. True stress-strain curves of AA2050-T34 after artificial ageing at 155 °C for indicatedtime.


4.2. Work hardening behaviour

Work hardening analysis of artificially aged AA2050 is carried outwith the Kocks-Mecking plots [24], which demonstrates evolutions ofwork hardening rate (dσf/dε, where σf is the transient flow stress atstrain of ε) with work hardening stress (σf − σy) during tensile tests,as shown in Fig. 7(a). The initial work hardening rate (θ0) of eachcurve in Fig. 7(a) is defined according to the method proposed byCheng et al. [40], as demonstrated in Fig. 7(b), and corresponding re-sults are shown in Fig. 7(c). When only clusters or minor precipitatesexist in the early stage of ageing (within 5 h), similar work hardeningrate curves are observed with a high θ0 value, which can be explainedby the high level of free solutes in the alloy matrix [5]. θ0 drops withan increasing speed from 5 to 24 h artificial ageingwhen T1 precipitatessubstantially nucleate and grow. After peak-ageing, θ0 remains at a sta-ble level until 300 h ageing and the stable value of θ0 (between 1180 and1270 MPa from 24 to 300 h in Fig. 7(c)) is very close to that of purealuminium (about 1250 MPa) [41,42]. Similar work hardening behav-iour has been reported in Al-Cu-Li [5], Al-Zn-Mg [43] and Al-Mg-Si[44] alloys and has been explained as the occurrence of shearing of pre-cipitates during plastic deformation. Hence, these results indicate thatT1 precipitates in AA2050-T34 remain shearable duringplastic deforma-tion until 300 h artificial ageing at 155 °C, which corresponds well withthe microstructural results in Fig. 3(b) and tensile test results in Fig. 6(a). A slight increase of θ0 is observed in the specimen after 500 hartificial ageing and is attributed to the transition of shearing-to-bypassing strengthening mechanism in the alloy during plasticdeformation [43]. Hence, it can be concluded that the shearable/non-



Fig. 7. (a)Work hardening rate with thework hardening stress (σf− σy) (Kocks-Meckingplots) after artificial ageing of AA2050-T34 at 155 °C for indicated time, (b) definition ofinitial work hardening rate θ0 and (c) evolution of θ0 with yield strength.


shearable (shearing/bypassing) transition does not occur at the peak-ageing state (18 h) for AA2050 alloy.

5. Modelling

The relationships betweenmicrostructural variables andmechanicalproperties of AA2050-T34 during artificial ageing have beencharacterised in Sections 3 and 4, based on which, a set of unified con-stitutive equations is proposed in this section to model and predict theevolutions of microstructures, yield strength and work hardening ofthe alloy from under-aged to over-aged conditions.

The developed unified model comprises three sub-models:i)microstructure sub-model, which describes evolutions ofmainmicro-structural variables during artificial ageing of Al-Cu-Li alloys; ii) yieldstrength sub-model, which relates yield strength tomicrostructural var-iables during artificial ageing; and iii) work hardening sub-model,which, based onmicrostructural variables, predicts the work hardeningbehaviour of the artificially aged alloys.

5.1. Modelling of microstructures

According to Sections 3 and 4, the Cu-rich clusters, solid solutes,and new precipitates (mainly T1) generated during artificial ageingare the main microstructural constituents that affect the yieldstrength and work hardening behaviour of AA2050 during artificialageing. In addition, as T34 alloy has undergone pre-stretching, initialdislocations are present in the as-received material that can also af-fect the precipitation progress [45]. All these microstructures wereconsidered in the model.

5.1.1. DislocationsPrevious studies indicated that recovery of initial dislocations in the

as-received material occurs during artificial ageing [46] and the


evolution of corresponding dislocation density (ρ) can be modelled bythe rate form of a recovery model from [47,48]:

_ρ ¼ −Cpρm4 ð1Þ

where Cp andm4 arematerial constants characterising the recovery pro-cess. ρ is the normalised dislocation density defined as (ρ − ρi)/ρs,where ρi is the dislocation density in the alloy after SHT and waterquenching and ρs is the maximum dislocation density in the alloy dur-ing specific processes, such as plastic deformation and/or ageing. Theinitial dislocation density in the AA2050-T34 alloy (ρ0) used in thisstudy is much larger than ρi due to the pre-stretch (about 4%) thatwas performed after water quenching (ρ0 ≫ ρi). During artificial ageingprocess, the dislocation density experiences a decreasing trend due tothe recovery process [46] and the maximum dislocation density is atthe beginning of ageing (ρs= ρ0). Hence, the initial value of the normal-ised dislocation density of AA2050-T34 during the artificial ageing pro-cess investigated in this study is set as ρ0 ≈ 1.

5.1.2. Solute concentrationSolute concentration achieves its super saturated level after solu-

tion heat treatment (SHT) and water-quenching, and will decreasedue to precipitation until reaching an equilibrium level at the artifi-cial ageing temperature. In addition, dislocations in the alloy alsocontribute to the evolution of solute concentration (c) due to theirenhancing effect on diffusion and precipitation [34]. A rate evolutionequation for the normalised solute concentration (c) to consider botheffects has been developed in a previous study [21], which is derivedaccording to the classic equations for solute concentration from [19],in order to avoid possible numerical issues in solving or evaluatingcorresponding original equations. The rate evolution equation isused in this study, as:

_c ¼ −A1 c−cað Þ 1þ γ0ρm2

� � ð2Þ

where A1, γ0, m2 are constants, c is the normalised solute concentra-tion defined as c/cs and ca ¼ ca=cs is the corresponding normalisedvalue at the ageing temperature, in which cs and ca are respectivelythe equilibrium solute concentrations in the alloy at SHT tempera-ture and ageing temperature. In addition, some extra solutes becomefree due to the dissolution of Cu-rich clusters during artificial ageingof AA2050-T34 and an additional component characterising thiscompensation effect for solute concentration evolution is thenadded to Eq. (2), as:

_c ¼ −A1 c−cað Þ 1þ γ0ρm2

� �þ A2rd ð3Þ

where A2 is a constant and rd ¼ rd=rd0 is a normalised radius of Cu-rich clusters, where rd is the transient radius of clusters during artifi-cial ageing and rd 0

is the initial radius of clusters. The deduction ofthe compensation component of A2rd is introduced in Appendix 1.

5.1.3. PrecipitatesDuring artificial ageing of AA2050-T34, Cu-rich clusters dissolve in

the first couple of hours and new precipitates nucleate, grow and thencoarsen with increasing ageing time, leading to the particular strengthbehaviour shown in Fig. 6(a). The evolutions of both precipitates aremodelled in this section.

The total volume fraction of precipitates (f) in the alloy is directlyproportional to the solute concentration. The normalised total volume

fraction is defined as f ¼ f= f a, where fa is the equilibrium volume frac-

tion at the ageing temperature. f can be calculated from correspondingnormalised solute concentration (c), as [19]:

f ¼ ci−cci−ca

¼ 1−c1−ca

ð4Þ




where ci is the solute concentration at the initial state of the alloy and ciis approximately treated as cs here. The total volume fraction of precip-itates in AA2050-T34 during artificial ageing is comprised of volumefractions from Cu-rich clusters and new precipitates, as f = fd + fn.The normalised values of fd and fn are both defined with the same way

as the total volume fraction, as: f d ¼ f d= f a, f n ¼ f n= f a. Therefore:

f ¼ f d þ f n ð5Þ

Cu-rich clusters in the as-received AA2050-T34 material dissolve atthe early stage of artificial ageing, the evolution of the normalised radiusof these clusters during artificial ageing has beenmodelled based on thedissolution kinetics as [21,49]:

_rd ¼ −Cr1

rdð6Þ

where Cr1 is a constant. The cell assumption proposed by Reti and Flem-ings [50] can be used to relate the volume fraction to cluster radius forthe dissolving clusters during artificial ageing, as fd = (rd/rd0

)3 [30,51],based on which, the normalised volume fraction of the dissolving Cu-

rich clusters f d can be calculated as:

f d ¼ f df a

¼ f df d0

f d0f a

¼ f d0rdrd0

� �3

ð7Þ

where fd0is the initial volume fraction of the clusters and f d0 is the cor-

responding normalised value. At the initial state, f 0 ¼ f d0 in thematerial.

The newprecipitates include T1 and θ′, inwhich T1 to θ′ number ratiois around 25–30 [21], and thus T1 precipitates play the dominant role inprecipitation strengthening of AA2050 during artificial ageing. The evo-lution of T1 precipitate variables, including dimensions (radius rn andthickness h shown in Fig. 2(c)) and volume fraction fn, are modelled inthis study to represent the new precipitates.

A normalised precipitate radius (rn) is defined as rn ¼ rn=rc, where rnand rc are respectively the radius of the new precipitates and the criticalradius at the peak-ageing state. rn then evolves from 0 to 1 from under-ageing to peak-ageing, andwill coarsen to reach a saturate level eventu-ally after peak-ageing. A rate controlling equation of rn has been pro-posed to model this evolution progress during artificial ageing, fromunder-ageing to over-ageing, as [21]:

_rn ¼ Cr Q−rnð Þm1 ð8Þ

whereQ represents the saturate value ofrnduring artificial ageing and Crandm1 are constants. In addition, dislocations in the as-received mate-rial also affect the evolution of precipitates, especially for T1 precipitateswhich require high energy sites for nucleation and growth [3,4]. This ef-fect has been characterised by introducing a dislocation density control-ling part into the rate equation for rn, as [52]:

_rn ¼ Cr Q−rnð Þm1 1þ γ0ρm2

� �: ð9Þ

The thickness of newprecipitate (h) is also represented by a normal-

ised value (h) in this study, ash ¼ h=hc, where hc is the critical thickness.For Al-Cu-Li alloy AA2050, thickening of T1 precipitate occurs afterpeak-ageing, and before which, T1 precipitate remains as a single layer

structurewith a constant thickness ( _h ¼ 0). Thickening of T1 precipitateenables the transition of strengthening mechanism from shearing tobypassing, as discussed earlier. hc then represents the critical thickness

of T1 precipitatewhen it becomes fully non-shearable andh≥ 1 indicatesnon-shearable precipitates. Thickening of precipitates during ageing hasbeen modelled based on the diffusion mechanism in some previousstudies [53,54] and the corresponding equation is used to predict the


thickening of T1 precipitates after peak-ageing states (rn≥1) in thisstudy:.

_h ¼ 0 rnb1ð Þ_h ¼ 1

3βDh

¼ Ch

hrn ≥1ð Þ

8<: ð10Þ

where β = Ω/A is a dimensionless growth parameter, in which A is aconstant and Ω is the supersaturation of solid solutes in the materialand it can be regarded as a constant for a particularmaterial at a specificageing temperature.D is the diffusion coefficient of solute. Ch is then de-fined as a constant to include all these parameters.

As f and f d have been respectively calculated by Eqs. (4) and (7), the

normalised volume fraction of new precipitates f n can be obtained ac-cording to Eq. (5).

5.2. Modelling of yield strength

It is generally accepted that during artificial ageing of aluminium al-loys, the yield strength of the material is contributed by dislocation(σdis), precipitation (σp) and solid solution (σss) hardening [19], whichare all taken into account in the strength model detailed in this section.

5.2.1. Dislocation hardeningDislocation hardening has been well modelled according to the dis-

location density [23,55], as:

σdis ¼ A3ρn ð11Þ

where A3 is a constant and n is a coefficient generally treated as 0.5[23,55].

5.2.2. Precipitation hardeningShercliff and Ashby [19] have proposed a set of equations to model

the strengthening effects from shearable and non-shearable precipi-tates according to their changing radius and volume fraction, with aconstant aspect ratio assumption during ageing. For AA2050 investi-gated in this study, thickness of newprecipitates (T1) evolves differentlyfrom radius during long-term artificial ageing. The aspect ratio of radiusto thickness (q) is then no longer a constant during ageing and thusneeds be considered in the strengthening model. Zhu and Starke Jr.[56] have proposed a strengtheningmodel for plate shaped precipitatesby computer simulations of dislocation movements through linear ob-stacles, in which the strengthening effect from both volume fraction(f) and aspect ratio (q) has been obtained as a combined form of qαfβ

(α and β are constants). Zhang et al. [57] have simplified the form as(qf)α in a later study and demonstrated its effectiveness in strength pre-diction for aluminium alloys containing plate shaped precipitates.Hence, this relationship between the precipitation hardening and vol-ume fraction and aspect ratio of precipitates has been adopted to updatethe original strengthening model from [19] to include the thickness ef-fect of T1 precipitates in this study. By introducing this aspect ratio, thestrengthening equations for shearable (σshear) and non-shearable(σbypass) precipitates from [19] can be modified as:

σ shear ¼ c1 qf n� �ma

rnan

σbypass ¼ c2 qf n� �mb

=rnbnð12Þ

where c1, c2,ma,mb, na and nb are constants. The aspect ratio (q) of newprecipitates is defined as:

q ¼ rnh¼ rc

hc

rnh

ð13Þ




where rc and hc are respectively critical radius and critical thickness of T1precipitate and both are constants for AA2050 alloy investigated in thisstudy. Inserting q into Eq. (12), the precipitation strength equations be-come:

σ shear ¼ c01fma

n

hma

rn0a

n

� �

σbypass ¼ c02fmb

n

hmb

1

rn0b

n

! ð14Þ

where c1′ and c2′ are constants representing all the constants (c1, c2 andrc/hc) in Eqs. (12) and (13). Following [19], the harmonic mean ofshearable and non-shearable precipitate strength components is usedhere to approximately combine both shearing and bypassing strength-ening mechanisms in the same equation. With this method, strengthcomponents in Eq. (14) can be integrated as a simple formulation forhardening from new precipitates (σp−n):

σp−n ¼ σ shearσbypass

σ shear þ σbypass¼ σ r

fm5

n

hm6

ð15Þ

where m5 and m6 are constants. σr represents the strengthening effectfrom precipitate radius, which has been discussed previously and is de-scribed by the following equation [21,58]:

_σ r ¼ Ca_rm7

n 1−rm8n

� � ð16Þ

where Ca, m7 and m8 are constants.In the current naturally aged alloy, Cu-rich clusters are present and

will be dissolved during artificial ageing. The clusters in the as-received material are small and shearable during plastic deformation,hence the strengthening equation for shearing in Eq. (12) can be di-rectly used. As mentioned earlier, the volume fraction is satisfied witha cell assumption according to Eq. (7), the strength fromdissolving clus-ters (σp−d) can be simplified to:

σp−d ¼ Ca1rm9d ð17Þ

where Ca1 and m9 are constants.The total strength contribution from precipitation (σp), including

both new precipitates and dissolving clusters, is modelled according tothe classical law of mixtures [59]:

σp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2

p−d þ σ2p−n

q: ð18Þ

5.2.3. Solid solution hardeningThe solid solution strength is directly determined by the solute con-

centration and can be modelled as [19]:

σ ss ¼ CSScm10 ð19Þ

where CSS andm10 are constants, the latter is generally taken as 2/3 ac-cording to [19].

5.2.4. Overall yield strengthThe overall yield strength (σy) is composed of the strength compo-

nents mentioned above and can be modelled as [40]:

σy ¼ σ ss þ σ t ¼ σ ss þ σNdis þ σN

p

� �1N ð20Þ

where σt represents the combined strengthening effects from disloca-tions and precipitates. N is a parameter characterising interactions be-tween dislocation hardening and precipitation hardening, which


generally varies from 1 to 2. For weak obstacles, such as clusters andsomeweak precipitates in under-aged conditions, N=1. For strong ob-stacles, such as non-shearable precipitates in over-aged conditions, N isset as 2 [60,61]. N evolves from 1 to 2 to represent the transition prog-ress of precipitates from shearable to non-shearable and will bediscussed in more detail in Section 6.1.

5.3. Modelling of work hardening behaviour

Work hardening behaviour of aluminium alloys during plastic defor-mation is generally attributed to dislocation hardening [23]. Consider-ing the dislocation-precipitate interactions in the material, the totaldislocation contributed to work hardening can be divided into twoparts: the statistically stored dislocation and the geometrically neces-sary dislocation [39]. Statistically stored dislocations mainly comefrom interactions between dislocations and weak obstacles (shearableprecipitates) and alloying elements in solid solutions, while geometri-cally necessary dislocations are generally from the storage of disloca-tions around strong obstacles (non-shearable precipitates) [32]. Thecontribution from these dislocations to work hardening (σd) can bemodelled by two independent internal variables - statistically storeddislocation density (ρss) and geometrically necessary dislocation den-sity (ρgn) [23,39]:

σd ¼ αMGbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρss þ ρgn

p ð21Þ

where α is a constant,M is the Taylor factor, G is the shear modulus andb is the Burgers vector.

5.3.1. Statistically stored dislocationIt has been shown in Fig. 7 that similar work hardening rates have

been observed in AA2050 material with high free solutes or withweak obstacles (clusters and shearable precipitates before peak-ageing). Hence, it is reasonable to use the same equation to charac-terise statistically stored dislocation evolutions with either alloyingelements or clusters/shearable precipitates in AA2050 during plasticdeformation. Kocks [23] proposed an equation to model the evolu-tion of ρss by considering both statistic storage and dynamic recoveryeffects, as:

_ρss ¼ K1ρ1=2ss −K2ρss

� �_εp ð22Þ

where K1 is a constant determined by the alloy composition and K2 isa constant dependant on the solute concentration in the alloy [23].The actual statistically stored dislocation density in the alloy duringplastic deformation is very hard to obtain, but theoretically, statisti-cally stored dislocation density would accumulate and eventuallyreach a saturate level during plastic deformation in tensiletests [43]. A normalised value of statistically stored dislocation den-sity is used to represent ρss in this study, which is defined as ρss ¼ ρss

=ρssm, where ρssm represents the maximum value of the statisticallystored dislocation density in the saturate level during plastic defor-mation. ρss then will evolve from 0 at the initial state to the saturatevalue of 1 during plastic deformation. When ρss tends to 1, the statis-tically stored dislocation density reaches its saturate level and its

evolution rate _ρss becomes 0. Therefore, Eq. (22) can be transformedto another simple form while maintaining its original physical phe-nomenon (the mathematical transformation process has been intro-duced in [21,62]), as:

_ρss ¼ k1 1−ρssð Þ _εp: ð23Þ

ρss then increases continuously until reaching 1 during plastic defor-mation. k1 is used to represent the effect from constants K1 and K2 in



Table 3A summary of the unified constitutive model developed in this study.

Microstructure sub-model Yield strength sub-model Work hardening sub-model

_rn ¼ CrðQ−rnÞm1 ð1þ γ0ρm2 Þ

_h ¼ Ch

h_rd ¼ −

Cr1

rd_c ¼ −A1ðc−caÞð1þ γ0ρ

m2 Þ þ A2rd f ¼1−c1−ca

f d ¼ f d0 ðrd=rd0 Þ3

_ρ ¼ −Cpρm4

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

σdis ¼ A3ρn

σp−n ¼ σ r fm5

n

hm6

_σ r ¼ Ca_rm7

n ð1−rm8n Þ

σp−d ¼ Ca1rm9d

σ ss ¼ CSScm10

σp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2

p−d þ σ2p−n

q

σ t ¼ ðσNdis þ σN

p Þ1N

σy ¼ σ ss þ σ t

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

_ρss ¼ k10 � cn1 � ð1−ρssÞ _εp

ρgn ¼ k2ðh−htranÞ

rn½1− expð−εpÞ�

σd ¼ Cd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρss þ ρgn

qσ f ¼ σ ss þ ðσN

d þ σNt Þ

1N

σw ¼ σ f−σy

8>>>>>>>>>>><>>>>>>>>>>>:


Eq. (22), which has been reported to be depended on the solute concen-tration in the alloy [23,32] and is updated as:

k1 ¼ k10 � cn1 ð24Þ

where k10 and n1 are constants.

5.3.2. Geometrically necessary dislocationAshby [39] proposed an equation to model the evolution of geomet-

rically necessary dislocation around strong obstacles in thematerial. Forplate-shaped obstacles with a large constant aspect ratio, the equationis:

ρgn ¼ 4bγl

ð25Þ

where l is the length of precipitate and γ is the simple shear strain. Dur-ing the plastic deformation of real engineering materials, recovery ofdislocations also occurs to prevent further accumulation of geometri-cally necessary dislocations when shear strain is high enough [32,63].Hence, ρgn cannot increase infinitely with increasing strain as describedby Eq. (25) and will reach a saturate level when the strain is highenough. Considering the most commonly used uniaxial tensile strain(εp) and replacing the length of precipitate with radius, geometricallynecessary dislocation in Eq. (25) can be modified to the followingform to suit engineering materials:

ρgn ¼ 2brn

K 1− exp −εp� � ð26Þ

where K is a constant representing the maximum contribution fromstrain to geometrically necessary dislocation. During over-ageing ofAA2050 alloy investigated in this study, the thickness of T1 precipitatesgrows, while the radius remains at a comparatively stable level. There-fore the aspect ratio becomes smaller with increasing ageing time and

the effect from T1 precipitate thickness (h) on the evolution of geomet-rically necessary dislocation needs be considered. In addition, previousstudies [11,64] have reported that thickness of plate-shaped precipi-tates can control whether or not the precipitate would support Orowanloops for geometrically necessary dislocation storage. Hence, a thicknessfactor is added in Eq. (27) to include the T1 precipitate thickness effecton the evolution of ρgn for the material investigated in this study, as:

ρgn ¼ 0 h≤htran� �

ρgn ¼ k2rn

h−htran� �

1− exp −εp� �

hNhtran� �

8><>: ð27Þ

where ρgn ¼ ρgn=ρgnm is the normalised geometrically necessary dislo-cation density, in which ρgnm is the maximum value of ρgn in the alloy


during plastic deformation and can be treated as a constant for a specificalloy. k2 is a constant representing 2 K/b in Eq. (26). ðh−htranÞ is thethickness factor, in which htran represents the normalised thickness ofT1 precipitate when the transition from shearable (weak) to non-

shearable (strong) starts. When h≤htran, T1 precipitates are shearableand no storage of geometrically necessary dislocations exist around

the precipitates (ρgn ¼ 0). When hNhtran, T1 precipitates become non-shearable, and the thickness of precipitates contributes to the storage

of geometrically necessary dislocations by a factor of ðh−htranÞ inEq. (27).

As a result, the work hardening strength contribution from disloca-tions can be updated according to the normalised internal variables ofρss and ρgn, as:

σd ¼ αMGbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρssmρss þ ρgnmρgn

q¼ Cd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρss þ ρgn

qð28Þ

It is assumed that the maximum value of statistically stored disloca-tion density in the saturate level approximately equals the maximumvalue of geometrically necessary dislocation density in this study, i.e.ρssm= ρgnm, as it is plausible that themaximum level of dislocation stor-age depends mainly on the alloy compositions and temperature, andless on the type or configuration of dislocations [65]. Hence, Cd is a con-stant representing αMGb

ffiffiffiffiffiffiffiffiffiρssm

p.

The flow stress (σf) during plastic deformation in tensile tests can beobtained according to the same superposition law used in Eq. (20), as:

σ f ¼ σ ss þ σNd þ σN

t

� �1N: ð29Þ

Hence, the actual work hardening stress component (σw), which isdirectly shown in the tensile curves in Fig. 5 is:

σw ¼ σ f−σy: ð30Þ

5.4. Summary of the unified constitutive model

The comprehensive unified constitutive model for microstructures,yield strength and work hardening behaviour proposed above issummarised in Table 3 below.



Table 4Initial and equilibrium values of variables in the microstructure and yield strength sub-models for AA2050-T34.

rn0 (−) h0 (−) rd0 (−) c0 (−) ρ0 (−) f 0 (−) σdis0 (MPa) σp−n 0(MPa) σp−d0

(MPa) σss0 (MPa) ca (−)

0 0.25 1 0.978 1 0.032 100 0 52 120 0.316


6. Modelling results and discussion

6.1. Determination of materials constants

The unified constitutive model proposed in Section 5 is based onbasic physical equations related to aluminium alloys. The microstruc-tural variables were all modelled in a normalised way in this study, soas to avoid the significant difficulty in obtaining experimental data ofsome microstructures in the alloys, such as dislocations. Some materialconstants in the model keep their physical meanings in the originalequations, and can be physically determined directly. Meanwhile, assome equations in the model have been transformed by mathematictechniques to avoid numerical difficulties in solving or evaluating corre-sponding original equations, some related material constants may nothave strong physical meanings and need to be calibrated according tocorresponding experimental data or normalised data from basic physi-cal theories. The detailed determination and calibration processes ofall the material constants in the model for AA2050 are introducedbelow. The calibration process was completed by a numerical fittingmethod with the non-linear least square criterion [66,67].

6.1.1. Microstructure sub-modelThe normalised radius of new precipitates rn shows only minor in-

creases after peak-ageing in Fig. 3(a), hence, its maximum value Q inEq. (9) was determined as 1, the value at the peak-ageing state, for sim-plification in this study. Constants Cr and m1 in Eq. (9) were then cali-brated according to the normalised experimental data from Fig. 3(a).

The normalised thickness of T1 precipitates h remains stable before

peak-ageing (when rnb1), as shown in Fig. 3(b), during which h was

set as the initial value h0 which will be characterised in detail in the

next section. After the peak-ageing state (rn≥1), h evolves accordingto Eq. (10) and Ch was determined by the normalised experimentaldata in Fig. 3(b).

For the normalised radius of dissolving Cu-rich clusters rd, its initialvalue (rd0 ) was 1. Although no experimental data is obtained in thisstudy, its normalised value decreases monotonously from 1 at 0 h to0 at around 2 h according to the physical understanding obtained inSection 4, based on which, Cr1 in Eq. (6) was calibrated. It is well-known that the normalised dislocation density ρ decreases monoto-nously to a stable level during ageing, and this physical-based trendwas applied to the calibration of Cp and m4 in Eq. (1).

The initial value of the normalised solute concentration (c0) has beenobtained before, as 0.978 [21]. cwill experience a slight increase due tothe dissolution of Cu clusters and reach its maximum value (near butb1) at around 2 h. After that, it decreases continuously until reachingthe equilibrium state (ca) during artificial ageing. Constants in Eq. (3)

Table 5Summary of material constants used in the microstructure sub-model for artificial ageing of AA

Parameter Value Methods Pa

Q (−) 1 Determined by data in Fig. 3(a) A1

A2

Cr (h−1) 0.238 Calibrated Eq. (9) to data in Fig. 3(a) Cpm1 (−) 1.05Ch (−) 8E-5 Calibrated Eq. (10) to data in Fig. 3(b) mγ0 (−) 0.08 From [21] Crm2 (−) 1.28


were calibrated by these normalised data. The normalised volume frac-

tion f was then obtained according to Eq. (4).The initial and equilibrium values of these microstructural variables

are listed in Table 4. Table 5 gives a summary of the material constantsand their determination or calibration methods in the microstructuresub-model.

6.1.2. Yield strength sub-modelThe initial values of strength components, including dislocation

hardening (σdis), precipitation hardening (σp−n and σp−d) and solid so-lution hardening (σss), of AA2050-T34 have been determined from ex-perimental data in [21] and the results were directly used here, aslisted in Table 4. A3 in Eq. (11), Ca1 in Eq. (17) and CSS in Eq. (19) werethen directly calculated according to the initial values of correspondingstrength components and relevant microstructural variables listed inTable 4. In addition, n in Eq. (11) was 0.5, m9 in Eq. (17) was 2 andm10 in Eq. (19) was 0.67 for aluminium alloys from [19].

In the overall yield strength equation (Eq. (20)), a variable N is de-fined to characterise different strength contributions from weak andstrong obstacles in the material. As the transition of shearable to non-shearable properties of T1 precipitate is determined by its thickness

[5], the evolution of N can be modelled as a function of h, as [60]:

N ¼ 1:5þ 0:5 tanhh−hf

A4

!ð31Þ

where hf ¼ ðhtran2 þ htranÞ=2, in which htran and htran2 represent respec-tively the normalised thicknesses of precipitates when itstarts deviating from shearable and when it becomes fully non-shearable. The evolution of N value with thickness is illustrated in

Fig. 8. When h≤htran , N = 1, representing fully shearable precipitates;

when h≥htran2 , N = 2, indicating fully non-shearable precipitates;

when h evolves from htran to htran2, N increases from 1 to 2, which repre-sents the shearable-to-non-shearable transition progress of precipi-tates. It has been concluded in Section 4 that the transition fromshearing to bypassing starts between 300 and 500 h of artificial ageingat 155 °C for AA2050-T34 and it also has been shown in Fig. 3(b) thatthe thickness of T1 precipitate starts to grow at the same time range.

Hence, htran is approximately set as the initial thickness value of T1 pre-

cipitate (single layer) in AA2050, which equals to h0. Deschamps et al.[5] have reported that when T1 precipitate grows to a four-layer struc-ture, it becomes non-shearable during plastic deformation. Hence, thecritical thickness value of hc with fully non-shearable property, wasapproximately set as four times of the initial single-layer precipitate

(h0). As a result, htran ¼ h0 ¼ h0=hc ¼ 0:25, htran2 ¼ hc ¼ hc=hc ¼ 1 and

2050-T34.

rameter Value Methods

(h−1) 0.05 Calibrated Eq. (3) to theoretical normalised data(h−1) 0.045(h−1) 0.1 Calibrated Eq. (1) to theoretical normalised data

4 (−) 6.51 (h−1) 0.185 Calibrated Eq. (6) to theoretical normalised data



Fig. 8. Schematic showing evolutions of N value with the normalised thickness (h) of T1precipitate.

Fig. 9. Evolutions of normalised microstructural variables during ageing at 155 °C for

AA2050-T34: (a) radius (rn) and thickness (h) of new precipitates and radius (rd) ofdissolving clusters, and (b) solute concentration ( c ), dislocation (ρ ) and volume

fractions ( f , f n and f d). (Symbols represent normalised experimental data from Fig. 3and lines are modelling results.)


hf = 0.625 for AA2050 in this study. The value of A4 in Eq. (31) deter-mined by [60] is used in this study.

Finally, the material constants in Eq. (15) were calibrated by the ex-perimental data of yield strength evolution in Fig. 6(a). Table 6 is a sum-mary of the material constants and corresponding determination orcalibration methods in the yield strength sub-model.

6.1.3. Work hardening sub-model

htran in Eq. (27) for the geometrically necessary dislocation has beendetermined in Section 6.1.2. Eqs. (23) and (28) together predict thework hardening behaviour of the alloy before peak-ageing, and hence,relatedmaterial constants k10, n1 and Cdwere calibrated by correspond-ing experimental data of alloys before 24 h ageing in Fig. 5. To predictthe work hardening behaviour of the alloy after peak-ageing, Eq. (27)needs to be included, and related material constant k2 was then cali-brated by the experimental data of the alloy after 24 h ageing in Fig. 5.Three sets of experimental data from AA2050-T34 artificially-aged for0, 18 and 500 hwere selected for the calibration process and relatedmi-crostructural variables in these equations were obtained according tothe microstructure sub-model determined earlier. The other sets of ex-perimental data at different ageing timeswere used for the validation ofthe determined material constants. Table 7 summarises the materialconstants and their determination or calibration methods in the workhardening sub-model.

6.2. Results and discussion

6.2.1. MicrostructuresFig. 9 shows modelling results of evolutions of microstructural vari-

ables during artificial ageing of AA2050-T34 at 155 °C with calibrated

Table 6Summary of material constants used in the yield strength sub-model for artificial ageing of AA

Parameter Value Methods

A3 (MPa) 100 Calculated by σdis0

n (−) 0.50 From [60]m5 (−) 0.30 Calibrated Eq. (15) to data in Fig. 6(a)m6 (−) 0.05Ca (MPa) 36.3 Calibrated Eq. (16) to data in Fig. 6(a)m7 (−) 0.06m8 (−) 9.50

Table 7Summary of material constants used in the work hardening sub-model for artificial ageing of A


k10 (−) 1.80 Calibrated by data in Fig. 5 before 24 h ageingn1 (−) 1.57Cd (MPa) 425


constants in Table 5. The experimental data in Fig. 3 is also normalisedand plotted in Fig. 9 for comparison. The modelling results of normal-ised radius of new precipitate rn correspond well with all experimentaldata from previous studies [13,34–36], which increases continuouslywith ageing time and reaches the peak-ageing state (rn = 1) at around18 h. As rn was set to reach its maximum value at the peak-ageing statein the current model for simplification as stated in Section 6.1.1, a smalldiscrepancy betweenmodelling and experimental results is observed in

2050-T34.


Ca1 (MPa) 52 Calculated by σp−d0

m9 (−) 2 From [19]CSS (MPa) 120 Calculated by σss0

m10 (−) 0.67 From [19]A4 (−) 0.22 From [60]

hf (−) 0.625 Determined by data in Fig. 3(a)

A2050-T34.


k2 (−) 1.80 Calibrated by data in Fig. 5 after 24 h ageing

htran (−) 0.25 Determined by data in Fig. 3(a)




Fig. 9(a) after long term artificial ageing (N300 h). The modelling andexperimental results of normalised thickness also agree well with

each other. h remains as a single-layer structure (h = 0.25) for a longageing time and apparent thickening behaviour occurs after 300 h ofageing, where shearing-to-bypassing transition starts, as indicatedfrom experimental results in Section 4. The modelling results showthat the normalised radius of dissolving clusters rd decreases continu-ously and the clusters disappear between 2 and 3 h of ageing, which isconsistent with the microstructural observations summarised inSection 3.

The additional solutes in the matrix of the alloy released from dis-solving clusters at the first 2 h of ageing has been well modelled, asshown in Fig. 9(b). After that, solutes are significantly reduced due tothe fast nucleation and growth of newprecipitates, leading to the signif-icant decrease of normalised solute concentration c towards its equilib-

riumvalue. The normalised total volume fraction (f) shows the opposite

trend to c. The normalised volume fraction of clusters ( f d) shows thesame trend with rd, while the normalised volume fraction of new pre-

cipitates (f n) increases continuously due to the nucleation and growthof T1 precipitates during artificial ageing. For the normalised dislocationdensity ρ, a continuous recovery is predicted during artificial ageing ofthe alloy, as shown in Fig. 9(b).

6.2.2. Yield strengthFig. 10 shows the modelling results of strength components and

overall yield strength of AA2050-T34 during artificial ageing at 155 °Cwith the calibratedmaterial constants in Table 6. Strength contributionsfrom dislocations (σdis), solutes (σss) and dissolving clusters (σp−d)show similar evolving trends to the corresponding normalised micro-structural variables in Fig. 9 respectively. The strength contributionfrom new precipitates σp−n (T1 for AA2050-T34) demonstrates an ap-parent increasing trend after about 2 h of ageing, when fast nucleationand growth of T1 precipitates start. After the peak-ageing state (18 h),σp−n still shows someminor increases,which is attributed to the still in-creasing volume fraction predicted in Fig. 9(b). σp−n starts to decreaseslightly at the later stage of artificial ageing as thickening of precipitateoccurs. The yield strength data obtained from modelling agrees excel-lently with experiments. An initial decrease with subsequent rapid in-crease of yield strength σy is well predicted. A plateau with an almostconstant yield strength is successfully modelled between 18 and 300 hof ageing. After that, a decreasing trend of yield strength is predicted,which is determined by the combined effect from decreasing σp−n andthe transition of N value from 1 to 2 in the model, indicating the transi-tion of strengthening mechanism from shearing to bypassing. Hence,the current model can be used to predict evolutions of both

Fig. 10. Evolutions of strength components and yield strength during artificial ageing at155 °C for AA2050-T34 from experiments (symbols) and modelling results (lines).


microstructural variables and yield strength during artificial ageing ofAA2050-T34 from under-ageing to over-ageing.

6.2.3. Work hardeningBased on the microstructural variables obtained from the micro-

structure sub-model, the work hardening curves of AA2050-T34with different ageing time during plastic deformation (εp) were pre-dicted with the calibrated constants in Table 7. The results fromAA2050-T34 with some selected ageing times are plotted in Fig. 11.Apart from the data used for calibration (0 h in Fig. 11(a), 18 h inFig. 11(d) and 500 h in Fig. 11(h)), the predicted work hardeningcurves at other ageing times all demonstrate a very good agreementwith corresponding experiments, as shown in Fig. 11. These resultsindicate that the proposed model is capable to predict work harden-ing behaviour of AA2050-T34 after artificial ageing with differenttime, from short-term under-ageing to long-term over-ageingconditions.

6.2.4. Prediction of UTS and uniform elongationThe Considere's necking condition [42] is generally applied to obtain

the UTS (σUTS) and uniform elongation (εu) data through a true stress-strain curve, as:

dσ f

dεpjnecking

¼ σ f ð32Þ

As Eq. (32) is fulfilled at the necking point of a true stress-straincurve, the stress and strain values at this point are respectively σUTS

and εu for thematerial. By analysing the true stress-strain curves pre-dicted by the proposed model with Eq. (32), σUTS and εu values forAA2050-T34 artificially aged at 155 °C for different time were ob-tained and the results are plotted and compared with experimentalresults in Fig. 12. A good agreement has been achieved for both setsof results, which shows the effectiveness of the proposed model inthe prediction of work hardening behaviour of AA2050-T34 after ar-tificial ageing.

The model proposed in this study is for AA2050 with T34 initialtemper but has the generic feature for application to other tempersof the alloy with the change of initial conditions to be suitable to dif-ferent states. Details on the method of determining the initial valuescan be referred to a previous study [20]. Furthermore, the modelcomprehensively reveals and quantifies the evolution of main geo-metric parameters (thickness and radius) and volume fraction of T1precipitates during artificial ageing, and their particular effects onthe evolution of other microstructural features (dislocations and sol-utes). Themodel utilises a normalised concept for themicrostructuresub-model, which not only keeps the physical meaning of the model,but also largely reduces the requirement of time-consuming micro-structural observations for extended application of the model toother similar aluminium alloys. The yield strength and work harden-ing behaviour of the alloys after artificial ageing have also beencharacterised based on fundamental ageing models. Hence, themodel developed in this study has the generic feature and the poten-tial to be applied to artificial ageing of other 2xxx series aluminiumalloys, in which plate-shaped T1 precipitates play the dominantrole. In addition, it should be noted that the current model only con-siders artificial ageing under isothermal conditions, and may applyto other isothermal ageing temperatures with recalibration of mate-rials constants. However, the effect of changing temperatures on ar-tificial ageing behaviour is not included in the model, which needsfurther studies.

7. Conclusions

The yield strength and work hardening behaviour of AA2050-T34after artificial ageing at 155 °C have been investigated in this study.



Fig. 11. Comparison of true stress - strain curves in the plastic region of AA2050-T34 after artificial ageing at 155 °C for (a) 0, (b) 2, (c) 8, (d) 18, (e) 32, (f) 150, (g) 300 and (h) 500 h fromexperiments (solid lines) and modelling results (dashed lines).


The detailed relationships between microstructures and mechanicalproperties of the artificially-aged alloy, from under-ageing to over-ageing, have been analysed, based on which a unified constitutivemodel has been proposed and validated to simultaneously predict mi-crostructures, yield strength and work hardening behaviour of thealloy. The following conclusions can be drawn:

1) The shearing-to-bypassing transition does not occur immediatelywhen reaching the peak-ageing state (18 h) for AA2050 alloy andthe high yield strength stays up to 300 h ageing. Over-ageing ofAA2050 with a slight drop of yield strength starts between 300


and 500 h of artificial ageing, as thickening of T1 precipitates occursand the shearing-to-bypassing transition starts.

2) A unified constitutive model has been established based on funda-mental ageing and work hardening equations and physically-basedassumptions, comprising three sub-models for microstructure,yield strength and work hardening. The model successfully predictsthe shearing-to-bypassing transition of AA2050 during artificial age-ing by considering the thickening of T1 precipitates and the changingof dislocation-precipitate interactions.

3) The model developed in this study has successfully predicted evolu-tions of main microstructural variables (precipitate, solute



Fig. 12. Comparison of evolutions of UTS (σUTS) and uniform elongation (εu) of AA2050-T34 after artificial ageing at 155 °C from experiments (symbols) and modelling results(lines).


concentration and dislocation), yield strength and work hardeningproperties (including UTS and uniform elongation) of AA2050 afterartificial ageing for a wide range of time, from under-ageing toover-ageing, providing an efficient tool to characterise themainme-chanical properties of the aged alloys for industrial applications.

CRediT authorship contribution statement

Yong Li: Investigation, Writing - original draft, Visualisation.Zhusheng Shi: Conceptualisation, Project administration, Writing- review & editing. Jianguo Lin: Conceptualisation, Funding acquisition,Writing - review & editing.

Acknowledgments

The authors are grateful to ESI Group (France) for the financial sup-port and Embraer (Brazil) for the provision of the test material.

Declaration of competing interest

None.

Appendix 1

Similar with the normalised volume fraction equation for precipi-tates in Eq. (4), the volume fraction of clusters can also be representedby corresponding normalised solute concentration cd, as:

f d ¼ K1−K2cd ðA1Þ

where K1 and K2 are constants for a specific ageing temperature. Com-bining Eq. (A1) with Eq. (7), the following equation can be obtained:

K1−K2cd ¼ rdrd0

� �3

ðA2Þ

The time derivative of cd is then:

_cd ¼ −3K2

1

rd0ð Þ3rdð Þ2 _rd ðA3Þ

Replacing _rd in Eq. (A3) by Eq. (6), the compensation component ofsolute concentration evolution fromdissolving clusters can be obtained:

_cd ¼ 3K2

Cr1

rd0ð Þ3rd ¼ A2rd ðA4Þ

where A2 represents all the other constants in Eq. (A4).


References

[1] E.A. Starke, Chapter 1 - historical development and present status of aluminum–lithium alloys, in: N. Eswara Prasad, A.A. Gokhale, R.J.H. Wanhill (Eds.),Aluminum-Lithium Alloys, Butterworth-Heinemann, Boston 2014, pp. 3–26.

[2] P. Lequeu, K. Smith, A. Daniélou, Aluminum-copper-lithium alloy 2050 developedfor medium to thick plate, J. Mater. Eng. Perform. 19 (6) (2010) 841–847.

[3] K. Kumar, S. Brown, J.R. Pickens, Microstructural evolution during aging of anAlCuLiAgMgZr alloy, Acta Mater. 44 (5) (1996) 1899–1915.

[4] B. Decreus, A. Deschamps, F. De Geuser, P. Donnadieu, C. Sigli, M. Weyland, The in-fluence of Cu/Li ratio on precipitation in Al–Cu–Li–x alloys, Acta Mater. 61 (6)(2013) 2207–2218.

[5] A. Deschamps, B. Decreus, F. De Geuser, T. Dorin, M. Weyland, The influence of pre-cipitation on plastic deformation of Al–Cu–Li alloys, Acta Mater. 61 (11) (2013)4010–4021.

[6] E. Balducci, L. Ceschini, S. Messieri, S. Wenner, R. Holmestad, Thermal stability of thelightweight 2099 Al-Cu-Li alloy: tensile tests and microstructural investigationsafter overaging, Mater. Des. 119 (2017) 54–64.

[7] H. Sidhar, R.S. Mishra, Aging kinetics of friction stir welded Al-Cu-Li-Mg-Ag and Al-Cu-Li-Mg alloys, Mater. Des. 110 (2016) 60–71.

[8] C. Blankenship Jr., E. Hornbogen, E. Starke Jr., Predicting slip behavior in alloyscontaining shearable and strong particles, Mater. Sci. Eng. A 169 (1–2) (1993)33–41.

[9] C. Gao, Y. Ma, L.-z. Tang, P. Wang, X. Zhang, Microstructural evolution and mechan-ical behavior of friction spot welded 2198-T8 Al-Li alloy during aging treatment,Mater. Des. 115 (2017) 224–230.

[10] A.A. Csontos, E.A. Starke, The effect of inhomogeneous plastic deformation on theductility and fracture behavior of age hardenable aluminum alloys, Int. J. Plast. 21(6) (2005) 1097–1118.

[11] T. Dorin, F. De Geuser, W. Lefebvre, C. Sigli, A. Deschamps, Strengthening mecha-nisms of T1 precipitates and their influence on the plasticity of an Al–Cu–Li alloy,Mater. Sci. Eng. A 605 (2014) 119–126.

[12] T. Dorin, A. Deschamps, F. De Geuser, C. Sigli, Quantification and modelling of themicrostructure/strength relationship by tailoring the morphological parameters ofthe T1 phase in an Al–Cu–Li alloy, Acta Mater. 75 (2014) 134–146.

[13] Y. Yan, L. Peguet, O. Gharbi, A. Deschamps, C. Hutchinson, S. Kairy, N. Birbilis, On thecorrosion, electrochemistry and microstructure of Al-Cu-Li alloy AA2050 as a func-tion of ageing, Materialia 1 (2018) 25–36.

[14] Y. Lin, Y.-Q. Jiang, G. Liu, S. Qin, Modeling the two-stage creep-aging behaviors of anAl-Cu-Mg alloy, Mater. Res. Express 5 (9) (2018), 096514.

[15] S. Nandy, K.K. Ray, D. Das, Process model to predict yield strength of AA6063 alloy,Mater. Sci. Eng. A 644 (2015) 413–424.

[16] U. Curle, L. Cornish, G. Govender, Predicting yield strengths of Al-Zn-Mg-Cu-(Zr) al-uminium alloys based on alloy composition or hardness, Mater. Des. 99 (2016)211–218.

[17] N. Nayan, S. Mahesh, M. Prasad, S. Murthy, I. Samajdar, A phenomenological harden-ing model for an aluminium-lithium alloy, Int. J. Plast. 118 (2019) 215–232.

[18] H. Fan, A.H. Ngan, K. Gan, J.A. El-Awady, Origin of double-peak precipitation harden-ing in metallic alloys, Int. J. Plast. 111 (2018) 152–167.

[19] H. Shercliff, M. Ashby, A process model for age hardening of aluminium alloys—I.The model, Acta Metall. Mater. 38 (10) (1990) 1789–1802.

[20] Y. Li, Z. Shi, J. Lin, Y.-L. Yang, Q. Rong, Extended application of a unified creep-ageingconstitutive model to multistep heat treatment of aluminium alloys, Mater. Des. 122(2017) 422–432.

[21] Y. Li, Z. Shi, J. Lin, Y.-L. Yang, Q. Rong, B.-M. Huang, T.-F. Chung, C.-S. Tsao, J.-R. Yang,D.S. Balint, A unified constitutive model for asymmetric tension and compressioncreep-ageing behaviour of naturally aged Al-Cu-Li alloy, Int. J. Plast. 89 (2017)130–149.

[22] J. Balík, P. Dobroň, F. Chmelík, R. Kužel, D. Drozdenko, J. Bohlen, D. Letzig, P. Lukáč,Modeling of the work hardening in magnesium alloy sheets, Int. J. Plast. 76 (2016)166–185.

[23] U. Kocks, Laws for work-hardening and low-temperature creep, J. Eng. Mater.Technol. 98 (1) (1976) 76–85.

[24] U. Kocks, H.Mecking, Physics and phenomenology of strain hardening: the FCC case,Prog. Mater. Sci. 48 (3) (2003) 171–273.

[25] J. Gracio, F. Barlat, E. Rauch, P. Jones, V. Neto, A. Lopes, Artificial aging and sheardeformation behaviour of 6022 aluminium alloy, Int. J. Plast. 20 (3) (2004)427–445.

[26] A. Kula, X. Jia, R. Mishra, M. Niewczas, Flow stress and work hardening of Mg-Y al-loys, Int. J. Plast. 92 (2017) 96–121.

[27] M. Khadyko, O. Myhr, O. Hopperstad, Work hardening and plastic anisotropy of nat-urally and artificially aged aluminium alloy AA6063, Mech. Mater. 136 (2019),103069.

[28] A. Deschamps, Y. Brechet, C. Necker, S. Saimoto, J. Embury, Study of large strain de-formation of dilute solid solutions of Al-Cu using channel-die compression, Mater.Sci. Eng. A 207 (2) (1996) 143–152.

[29] K. Kaneko, T. Oyamada, A viscoplastic constitutive model with effect of aging, Int. J.Plast. 16 (3–4) (2000) 337–357.

[30] N.Y. Zolotorevsky, A. Solonin, A.Y. Churyumov, V. Zolotorevsky, Study of work hard-ening of quenched and naturally aged Al–Mg and Al–Cu alloys, Mater. Sci. Eng. A502 (1–2) (2009) 111–117.

[31] M. Jobba, R. Mishra, M. Niewczas, Flow stress and work-hardening behaviour of Al–Mg binary alloys, Int. J. Plast. 65 (2015) 43–60.

[32] O.R. Myhr, Ø. Grong, K.O. Pedersen, A combined precipitation, yield strength, andwork hardening model for Al-Mg-Si alloys, Metall. Mater. Trans. A 41 (9) (2010)2276–2289.


http://refhub.elsevier.com/S0264-1275(19)30559-3/rf0005






















































































[33] R.A. Hafley, M.S. Domack, S.J. Hales, R.N. Shenoy, Evaluation of Aluminum Alloy2050-T84 Microstructure and Mechanical Properties at Ambient and CryogenicTemperatures, NASA/TM-2011-217163 2011.

[34] Y. Li, Z. Shi, J. Lin, Y.-L. Yang, B.-M. Huang, T.-F. Chung, J.-R. Yang, Experimental inves-tigation of tension and compression creep-ageing behaviour of AA2050 with differ-ent initial tempers, Mater. Sci. Eng. A 657 (2016) 299–308.

[35] T.-F. Chung, Y.-L. Yang, C.-N. Hsiao, W.-C. Li, B.-M. Huang, C.-S. Tsao, Z. Shi, J. Lin, P.E.Fischione, T. Ohmura, Morphological evolution of GP zones and nanometer-sizedprecipitates in the AA2050 aluminium alloy, Int. J. Light. Mater. Manuf. 1 (3)(2018) 142–156.

[36] V. Proton, J. Alexis, E. Andrieu, J. Delfosse, A. Deschamps, F. De Geuser, M.-C. Lafont,C. Blanc, The influence of artificial ageing on the corrosion behaviour of a 2050aluminium–copper–lithium alloy, Corros. Sci. 80 (2014) 494–502.

[37] P. McCormick, The Portevin-Le Chatelier effect in an Al-Mg-Si alloy, Acta Metall. 19(5) (1971) 463–471.

[38] H. Ovri, E.T. Lilleodden, Temperature dependence of plastic instability in Al alloys: ananoindentation study, Mater. Des. 125 (2017) 69–75.

[39] M. Ashby, The deformation of plastically non-homogeneous materials, Philos. Mag.21 (170) (1970) 399–424.

[40] L. Cheng, W. Poole, J. Embury, D. Lloyd, The influence of precipitation on the work-hardening behavior of the aluminum alloys AA6111 and AA7030, Metall. Mater.Trans. A 34 (11) (2003) 2473–2481.

[41] E. Nes, Modelling of work hardening and stress saturation in FCC metals, Prog.Mater. Sci. 41 (3) (1997) 129–193.

[42] A. Simar, Y. Bréchet, B. De Meester, A. Denquin, T. Pardoen, Sequential modeling oflocal precipitation, strength and strain hardening in friction stir welds of an alumi-num alloy 6005A-T6, Acta Mater. 55 (18) (2007) 6133–6143.

[43] G. Fribourg, Y. Bréchet, A. Deschamps, A. Simar, Microstructure-based modelling ofisotropic and kinematic strain hardening in a precipitation-hardened aluminiumalloy, Acta Mater. 59 (9) (2011) 3621–3635.

[44] W. Poole, X. Wang, D. Lloyd, J. Embury, The shearable–non-shearable transition inAl–Mg–Si–Cu precipitation hardening alloys: implications on the distribution ofslip, work hardening and fracture, Philos. Mag. 85 (26–27) (2005) 3113–3135.

[45] W. Cassada, G. Shiflet, E. Starke, The effect of plastic deformation on Al 2 CuLi (T1) precipitation, Metall. Trans. A. 22 (2) (1991) 299–306.

[46] Z. Ma, L. Zhan, C. Liu, L. Xu, Y. Xu, P. Ma, J. Li, Stress-level-dependency and bimodalprecipitation behaviors during creep ageing of Al-Cu alloy: experiments and model-ing, Int. J. Plast. 110 (2018) 183–201.

[47] J. Lin, Y. Liu, A set of unified constitutive equations for modelling microstructureevolution in hot deformation, J. Mater. Process. Technol. 143 (2003) 281–285.

[48] E. Nes, Recovery revisited, Acta Metall. Mater. 43 (6) (1995) 2189–2207.[49] M. Whelan, On the kinetics of precipitate dissolution, Metal Sci. J. 3 (1) (1969)

95–97.[50] A. Reti, M. Flemings, Solution kinetics of two wrought aluminum alloys, Metall.

Trans. 3 (7) (1972) 1869–1875.


[51] M. Starink, S.Wang, Amodel for the yield strength of overaged Al–Zn–Mg–Cu alloys,Acta Mater. 51 (17) (2003) 5131–5150.

[52] L. Zhan, J. Lin, T.A. Dean, M. Huang, Experimental studies and constitutive modellingof the hardening of aluminium alloy 7055 under creep age forming conditions, Int. J.Mech. Sci. 53 (8) (2011) 595–605.

[53] R. Doherty, K. Rajab, Kinetics of growth and coarsening of faceted hexagonal precip-itates in an fcc matrix—II. Analysis, Acta Metall. 37 (10) (1989) 2723–2731.

[54] M. Ferrante, R. Doherty, Influence of interfacial properties on the kinetics of precip-itation and precipitate coarsening in aluminium-silver alloys, Acta Metall. 27 (10)(1979) 1603–1614.

[55] Y. Estrin, Dislocation theory based constitutive modelling: foundations and applica-tions, J. Mater. Process. Technol. 80 (1998) 33–39.

[56] A. Zhu, E. Starke Jr., Strengthening effect of unshearable particles of finite size: acomputer experimental study, Acta Mater. 47 (11) (1999) 3263–3269.

[57] J. Zhang, Y. Deng, X. Zhang, Constitutive modeling for creep age forming of heat-treatable strengthening aluminum alloys containing plate or rod shaped precipi-tates, Mater. Sci. Eng. A 563 (2013) 8–15.

[58] J.-H. Zheng, J. Lin, J. Lee, R. Pan, C. Li, C.M. Davies, A novel constitutive model formulti-step stress relaxation ageing of a pre-strained 7xxx series alloy, Int. J. Plast.106 (2018) 31–47.

[59] U. Kocks, A. Argon, M. Ashby, Models for macroscopic slip, Prog. Mater. Sci. 19(1975) 171–229.

[60] F. Fazeli, W. Poole, C. Sinclair, Modeling the effect of Al3Sc precipitates on the yieldstress and work hardening of an Al–Mg–Sc alloy, Acta Mater. 56 (9) (2008)1909–1918.

[61] A. Foreman, M. Makin, Dislocation movement through random arrays of obstacles,Philos. Mag. 14 (131) (1966) 911–924.

[62] J. Lin, Selection of material models for predicting necking in superplastic forming,Int. J. Plast. 19 (4) (2003) 469–481.

[63] H. Proudhon, W.J. Poole, X. Wang, Y. Brechet, The role of internal stresses on theplastic deformation of the Al–Mg–Si–Cu alloy AA6111, Philos. Mag. 88 (5) (2008)621–640.

[64] J. Costa Teixeira, L. Bourgeois, C. Sinclair, C. Hutchinson, The effect of shear-resistant,plate-shaped precipitates on the work hardening of Al alloys: towards a predictionof the strength–elongation correlation, Acta Mater. 57 (20) (2009) 6075–6089.

[65] Q. Zhao, B. Holmedal, Modelling work hardening of aluminium alloys containingdispersoids, Philos. Mag. 93 (23) (2013) 3142–3153.

[66] B. Li, J. Lin, X. Yao, A novel evolutionary algorithm for determining unified creepdamage constitutive equations, Int. J. Mech. Sci. 44 (5) (2002) 987–1002.

[67] J. Lin, J. Yang, GA-based multiple objective optimisation for determining viscoplasticconstitutive equations for superplastic alloys, Int. J. Plast. 15 (11) (1999)1181–1196.


























































































materials and design - spiral: home · al-cu-mg [14], al-mg-si [15], al-zn-mg [16] and al-li [17]...

Documents