Journal of Materials Processing Technology 153–154 (2004) 122–127

Constitutive modelling of primary creep for ageforming an aluminium alloy

K.C. Ho∗, J. Lin, T.A. DeanMechanical and Manufacturing Engineering, School of Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Abstract

The age forming technique has been used in aerospace industry to manufacture panels components with improved mechanical propertiesand reduced fabrication cost. Age forming is based on the stress relaxation phenomenon due to creep, which occurs during the artificialageing of a metal. Different from other metal forming techniques, ‘ageing–creep’ deformation takes place well below yield point andthe amount of plastic deformation is directly related to ageing time and temperature. The aim of this paper is to model the ageing–creepbehaviour of an aluminium alloy in industrial age forming process. Ageing–creep tests are carried out at 150◦C for a solution treatedand quenched aluminium alloy. A new set of physically based, unified ageing–creep constitutive equation is developed based on the‘unified theories’ and ageing kinetics. The material constants within the equations are determined from experimental results and evo-lutionary algorithm (EA) based optimization package. Close agreement between the predicted and experimental creep curves has beenachieved.© 2004 Elsevier B.V. All rights reserved.

Keywords:Ageing; Constitutive equations; Creep; Stress relaxation

1. Introduction

In aerospace industry, there are increasing demands formanufacturing better performance aircraft panels includ-ing improved strength and toughness, lower weight andincreased resistance to fatigue and corrosion[1–8]. Ageforming has recently been selected as the favoured manu-facturing process as it can produce components with highstrength while maintaining good stress corrosion resistance.Examples of age forming application include the upperwing skins of Gulfstream GIV, B-1B long-range combataircraft and Airbus A330/340 manufactured by Textron aswell as the Hawk upper skin produced by British Aerospace[2,5].

Age forming is referred to a combined mechanical andheat treatment process[1–7], which takes place in an auto-clave. In age forming process, a solution treated, quenchedand cold stretched material sheet is heated to a temperaturesufficient for it to age, creep and allow stress relaxation tooccur. It is then loaded onto the tool by vacuum baggingtechnique. Once the sheet contacts the tool, it is held inplace by pressure for a controlled amount of time to allowstress relaxation to relieve the stresses produced by forming.

∗ Corresponding author.E-mail address:kch036@bham.ac.uk (K.C. Ho).

During this period, the constituents of the metal precipitate,altering the microstructure of material, e.g. increasing theyield strength of the material. When the vacuum force is re-leased, the material sheet springs back to a shape somewherebetween its original shape and the tool shape. Springbackoccurs due to the limited holding (ageing) period which isrequired to achieve proper mechanical properties is not suffi-cient to fully fix the shape of the component, i.e. not enoughplastic strain to cause permanent deformation. However,the final aged component has lower residual stresses[1–5]compared to structures formed by conventional formingprocess such as roll forming, brake forming, shot peeningor stretch forming. Therefore, the long-term performanceof the component has been improved due to its resistanceto both fatigue and stress corrosion cracking has beenenhanced.

Different from other metal forming techniques, whereelastic–plastic deformation of material is dominant, the‘ageing–creep’ deformation takes place at low stress leveland the amount of plastic deformation is directly related toageing time and temperature[4]. Furthermore, the mate-rial behaviour related to age forming is more complicatedthan the conventional creep/stress relaxation behaviour dueto the precipitation hardening, which takes place simulta-neously with forming process, enhances the material butdecreases the creep rates. For example, the yield strength

0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2004.04.304

K.C. Ho et al. / Journal of Materials Processing Technology 153–154 (2004) 122–127 123

Fig. 1. Comparison of creep curves for an as-quenched and a pre-agedAA7010 at 150◦C.

of the material increases up to 15–25% due to ageing heattreatment[5,7–9]. Referring to the two creep curves plottedin Fig. 1, the pre-aged material is stronger; exhibits littlecreep deformation and settles in secondary (steady) creepstage. Meanwhile, the as-quenched material, which is usedin industrial age forming, shows more creep deformationsand a great deal of hardening, i.e. more creep strains aredeveloped and primary creep period is longer. This kind of‘primary creep’ behaviour is important since it introducesmore of creep (plastic) deformation to retain the shape ofcomponent after forming.

The aim of this paper is to develop a new set of physicallybased, unified ageing–creep constitutive equations which aresuitable to model physical and mechanical behaviour of ma-terial in age forming, with following features:

• The constitutive equations enable both creep deformationand age-hardening behaviour including the effects of con-ventional work hardening and age hardening to be cap-tured.

• The contributions to the increase of the material’s yieldstrength that is associated with ageing are identified.

• The overall age-hardening behaviour can be characterisedby particle coarsening (growth of precipitates radius) atconstant volume fraction during isothermal ageing.

The physically based, unified ageing–creep constitutiveequations for aluminium alloy 7010 at 150◦C have beenfully determined from the creep experimental results.

Fig. 2. Schematic representations of material at different ageing stages: (I) supersaturated solid solution (SSSS); (II) nucleation of GP zones (coherent);(III) precipitate coarsening (semi-coherent) and (IV) overaged condition (incoherent).

2. Ageing–creep mechanism

2.1. Microstructural evolution of ageing mechanism

The ageing behaviour of 7000 aluminium alloys un-der isothermal heat treatment has been examined in detailby several investigators[8–11]. The generally acceptedsequence is

αsolid solution→ α + spherical GP zones→ α + η′ → α + η

whereα is the aluminium matrix, GP zones are Guinier–Preston zones,η′ is a transition phase andη is the equilibriumphase, MgZn2. Fig. 2shows the schematic representation ofthe microstructural evolution of an 7000 aluminium alloyundergoing ageing (precipitation) process.

Ageing (precipitation) mechanism basically is dividedinto several stages as mentioned above. Upon ageing, thequenching process retains the supersaturated solid solutionwithin the aluminum matrix (refer toFig. 2, stage I). At theearly stage of ageing, ordered and solute rich clusters, socalled GP zones form. GP zones are only one or two atomplanes in thickness and retain the similar crystal structure asthe Al matrix and are coherent. Because of this, their inter-facial energy is low, making their nucleation easy. GP zoneformation increases the hardness of the alloy; but becausethe GP zones are small and coherent, they can be cut bydislocations. During this stage, the number of precipitatesincreases drastically with time. The difference in atomicsize of zinc and aluminium strains the lattice. Hardeningis, therefore, due to the increased work required to movedislocations through the strained lattice and work requiredfor dislocations to pass through the GP zones. This morethan counteracts the decrease in solid solution strengtheningas the zinc concentration in the aluminium decreases. Theoverall strength and hardness of the material continues toincrease in the initial stages of coarsening as the particlescontinue to increase in size at constant volume fraction.

Eventually the GP zone itself is replaced by the morestableη′ phase, i.e. stage III inFig. 2. Their typical sizesare between 1 and 10 nm[10], and are semi-coherent withthe Al matrix. During this stage, the size of the precip-itates increases and the composition of the matrix phasesteadily decreases, approaching the equilibrium value. Asthe precipitates become larger, coherent particles will lose

124 K.C. Ho et al. / Journal of Materials Processing Technology 153–154 (2004) 122–127

their coherency and the dislocation–precipitate elastic inter-action will be diminished while the precipitate spacing be-come larger. During coarsening, the number of precipitatesdecreases and precipitates spacing increase further, wherethe equilibriumη phase forms as large incoherent particles(stage IV). These particles are large enough to be bypassedby dislocations by the Orowan bowing process without be-ing cut. At longer ageing times, the strength of the materialfalls. The alloy is now in the overaged condition.

2.2. Contributions to yield strength

Ageing mechanism improves the yield strength of the ma-terial. Fig. 3 shows a schematic diagram of relative contri-butions to the yield strength of an aluminium alloy 7010 andthe corresponding ageing stages. At time= 0, there has beeninsufficient time for precipitate to nucleate in the quenchedaluminium alloy. The initial yield stress would reflect con-tributions from the intrinsic stress and solute hardening. Theyield strength of material starts to increase as the precipi-tates start to nucleate and coarsen. As the coarsening processproceeds, the supersaturation in the matrix phase decreases.This decrease in the concentration of solute atoms in thematrix results in a decrease in the solute hardening. How-ever, the decrease in solute hardening is more than offsetby the increase owing to dispersion hardening. Therefore,the overall strength of the material continues to increasewith time as the precipitates continue to increase in size atconstant volume fraction. Eventually, there is no further de-crease in strength owing to a decrease in the concentrationof solute in the matrix as it has reached it equilibrium value.As the particle becomes larger, the coherent precipitates willlose their coherency and the dislocation–precipitate elasticinteraction will be diminished. In addition, the decreasingnumber of precipitates and the increase in mean inter centerspacing between precipitates cause a decrease in the parti-cle strengthening and an overall decrease in the strength ofthe material. Thus, the strength of the material will reach amaximum value and begin to decrease. As coarsening con-

Fig. 3. Contributions of solute hardening and dispersion hardening to yieldstrength of the aluminium alloy AA70110 as a function of ageing time.

tinues, the material’s strength will continue to decrease (seeFig. 3 for illustration).

Using 7000 aluminium alloys as model material, the cur-rent work concentrates on the early stage of the ageing mech-anism, i.e. before the strength of material drops.

3. Development of physically based unifiedageing–creep constitutive equations

3.1. Experimental program

Constant stress creep tests were carried out for aluminiumalloy 7010 at 150◦C. The total duration for each test is 24 hand the tests were repeated for selected stress levels, e.g.302.9, 325.2, 336.3 and 352.8 MPa. The creep specimenswere machined in rolling direction from aluminium alloy7010 plates, which were supplied by Alcoa Europe in W51condition (solution treated, quenched and stretched).

Creep tests were conducted using ESH creep machines atthe University of Birmingham. Each creep machine consistsof a closed furnace with three built-in thermocouples andwas fitted with Andrade–Chalmers profiled cams that cansustain a constant stress on uniformly deforming specimensup to a total strain of approximately 25%. Upon conductingtest, the specimen was fitted and aligned in the middle ofthe furnace and two additional thermocouples were wiredclose to the two end of specimen gauge length. The furnacewas heated until the temperature become steady at 150◦C.Then the extensometers were calibrated and the calculatedload was applied. The extension was measured using a RDPModular 600 system fitted with a number of dual transduceramplifier module 621 cards. Finally, the test results wereprocessed and used to evaluate the material constants in-volved in the proposed constitutive equations.

3.2. Unified ageing–creep constitutive equations

Due to the complications and limitations of classical plas-ticity theory, a difference approach called ‘unified theories’has been proposed and practiced[12–17]. In this approach,the classical separation of strain into time-independent plas-tic stain and a time-dependent creep strain is replaced bya total inelastic strain. All aspects of inelastic deformationsuch as plasticity, hardening, creep and recovery are treatedby a set of equations. These theories often make use of in-ternal state variables subjected to evolution rules and aresuitable for a broad range of applications.

Based on this ‘unified theories’ approach and ageing ki-netics, and assuming isothermal ageing and particle growthat constant volume fraction, a new set of physically based,unified ageing–creep constitutive equation is introduced asbelow.

εc = A sinh

{B(σ − σA)(1 − H)

(CSS

σSS

)n}(1)

K.C. Ho et al. / Journal of Materials Processing Technology 153–154 (2004) 122–127 125

H = h

σ0.1

(1 − H

H∗

)εc (2)

r = C0ε0.2c (Q − r)1/3 (3)

σA = CArm0 (4)

σSS = CSS(1 + r)−m1 (5)

σy = σSS+ σA (6)

whereA, B, h, H∗, n, Q, C0, CA, CSS, m0 andm1 are materialconstants.

Eq. (1)describes evolution of ageing–creep strain, whichinvolves several key features, i.e.:

(a) Creep rate is not only a function of stress,σ and dis-location hardening,H, but also the ageing precipitationhardening,σA and the solute hardening,σSS.

(b) The nucleation and growth of precipitates are related tothe creep deformation.

(c) CSS, σSS andn are responsible for softening of the ma-trix material due to particle coarsening.

The second equation characterises primary creep usingvariable H, which varies form 0 at the beginning of thecreep process toH∗ whereH∗ is the saturation value at theend of the primary period, and subsequently, maintains thisvalue for the remaining of the creep deformations. The stresssensitivity is reduced by applying lower order power uponthe stress term within the second equation.

In the present model, contribution from ageing pre-cipitation to the increment of the yield strength,σy arecontributed by two sources,σA and σSS (also refer toFig. 3), which are described in terms of precipitate radius,r. The use of the precipitate radius simplifies the mod-elling of ageing mechanism as the precipitate evolves orgrows monotonically during isothermal ageing[18]. Theevolution of precipitates radius is described inEq. (3). Thestrengthening contribution from shearable precipitates canarise from a variety of the mechanisms such as chemicalhardening, coherency strain hardening, etc. However, theoverall strengthening contribution from various mecha-nisms is summarized inEq. (4), wherem0 is 0.5 andCAdescribes the interaction between dislocations and shearableprecipitates while assuming the coarsening is occurring atconstant volume fraction.Eq. (5) approximates the contri-bution from solid solution strengthening (solute hardening)where resistance is caused by solute atoms to obstructdislocation motion. Refer toEq. (5), CSS is a constant re-lated to the size, modulus and electronic mismatch of thesolute; m1 describes the depletion of solute into precipi-tate. As the concentration of the solute atom decreases, thesolid solution strengthening decreases, acting more or lesslike a ‘softening’ mechanism. By combiningEqs. (4) and(5), gives Eq. (6), the overall contributions to the incre-ment of yield strength for the material undergoing ageingprocess.

Table 1Values of the material constants for the constitutive equations at 150◦C

A (h−1) 3.97 × 10−7

B (MPa−1) 0.03n (–) 0.03h (MPa) 520H∗ (–) 0.2Q (nm) 120C0 (–) 1.26CA (–) 31.35CSS (–) 20m0 (–) 0.5m1 (–) 1.3

3.3. Determination of ageing–creep constitutive equations

The determination of the material constants within theageing–creep constitutive equations is carried out basedon the ageing–creep experimental data and the use of anoptimisation software package. The optimisation softwarepackage is developed using C++ based on evolutionaryalgorithm (EA) and the detailed descriptions are given byseveral researchers[15–17]. The unified ageing–creep con-stitutive equations are numerically integrated using fourthorder Runge–Kutta method and the time step for the inte-gration is automatically controlled. This ensures that thecreep curves can be accurately computed. Despite the equa-tions, the fitness function is computed based on the conceptdeveloped by Li et al.[16] and is implemented into the soft-ware package. The fitness function is referred to sum of thesquares of the shortest distances between the computed andexperimental data. The procedures to calculate the shortestdistance of an experimental data point to the correspondingnumerically integrated creep curve are briefly described asbelow:

(i) Identify two adjacent closest integrated data points onthe creep curve to the experimental data point;

(ii) Determine a straight line through the two integratedpoints;

(iii) Calculate the distance of the experimental data point tothe straight line.

By using above-mentioned procedures, the material con-stants used in the constitutive equations are determined andlisted inTable 1.

4. Modelling results

Having determined all material constants, the proposedset of constitutive equations is tested for its capability topredict the ageing–creep response for AA7010 at 150◦C fora number of different stress levels. The total ageing durationis 25 h.Fig. 4 shows the comparisons of the predicted andexperimental ageing–creep strains for four different stresslevels. It can be seen that very close agreement is achieved.Using Eq. (3), the evolution of the precipitate radius of the

126 K.C. Ho et al. / Journal of Materials Processing Technology 153–154 (2004) 122–127

Fig. 4. Comparison of predicted (solid line) and experimental (symbols)creep strain forσ = 302.9, 325.2, 336.3 and 352.8 MPa.

material is plotted inFig. 5 for stress level= 336.3 MPa.It can be seen that precipitate radius grows monotonicallyand reaches a value around 6 nm at the end of ageing,which is typical. For the same stress level, the overall re-sulting yield strength increment and its components areplotted in Fig. 6. From the figure, it can be observed thatσA increases monotonically with ageing time. Meanwhile,σSS decreases quickly at initial stage but slowly reachesits saturation point after 14 h. The net increment of theyield strength,σy is the sum ofσA and σSS. At the endof ageing–creep process,σy reaches a value of 79 MPa,which is approximately 20% of material original yieldstrength.

Fig. 5. Evolution of precipitate radius (forσ = 336.3 MPa).

Fig. 6. Contributions to the increment of yield strength (forσ= 336.3 MPa).

5. Conclusions

The set of physically based unified ageing–creep consti-tutive equations developed in the research enables age hard-ening and creep deformation during and after age formingto be predicted.

The equations describe hardening and softening effectsduring ageing–creep mechanism, which contribute to theyield strength of the material.

The determined constitutive equations will be used to pre-dict stress relaxation and springback in age forming.

Acknowledgements

The authors wish to thank Alcoa Europe (Birmingham)for supplying AA7010 in W51 condition and Peter Gloverfor his valuable discussion on AA7010’s properties. Theauthors would also like to acknowledge Simon Gray andMary Taylor in the Department of Metallurgy and MaterialScience, University of Birmingham, for their advice and helpin conducting ageing–creep experiments.

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