Aluminium alloys study

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A model for predicting fracture mode and toughnessin 7000 series aluminium alloys

D. Dumont a,b , A. Deschamps a, * , Y. Brechet a

a LTPCM ENSEEG, CNRS UMR 5614, Domaine Universitaire de Grenoble, INPG, BP 75, Saint Martin dH eres Cedex 38 402, Franceb Pechiney Centre de Recherches de Voreppe, 725 rue Aristide Berg es, BP 27, Voreppe Cedex 38 341, France

Received 19 December 2003; received in revised form 30 January 2004; accepted 31 January 2004Available online 5 March 2004

Abstract

A model is proposed, which predicts the toughness of 7000 series aluminium alloys in a variety of situations, including two alloycompositions, different quench rates from the solution treatment temperature and ageing states from underaged to overaged. Themodel is derived in three steps. The energy dissipated by transgranular fracture is rst calculated, using a simplied cohesive zoneapproach. The energy dissipated by intergranular fracture is then calculated using a critical strain criterion, and the total dissipatedenergy is then estimated using an averaging by the respective area fractions of the two modes, which are themselves dependent on therespective energies of the two main fracture mechanisms. The model input parameters are the materials mechanical properties suchas yield stress and strain-hardening rate, geometrical features and related properties of the precipitate-free zones, and area fractionof the grain boundaries covered with precipitates. The model predicts all the main features of the evolution of the toughness/yieldstrength compromise with changing quench rate or ageing treatment. It allows to predict the evolution of toughness when dramaticchanges in the occurrence of fracture modes are observed. Finally, using the model it is possible to predict the effect of changes inindividual parameters on the overall fracture behaviour.

2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Toughness; Modelling; Precipitation; Aluminium alloys; 7000 series

1. Introduction

Modelling the fracture toughness in industrial alu-minium alloys, in a broad range of situations (includingdifferent alloys, states of ageing and quench rates for in-stance) is a difficult task, owing to the complexity of themicrostructure and the fact that various fracture modesmay contribute simultaneously to the nal rupture.

In situations of practical relevance, the microstruc-ture contains a number of heterogeneities, which cancontribute both to the localisation of plastic ow, andto the initiation and propagation of failure. In order of decreasing size, these heterogeneities include interme-tallic particles, quench-induced intergranular and in-

tragranular precipitates, and precipitate-free zonessituated at the grain boundaries.

The heterogeneity of the microstructure, as well asthe plastic properties associated with the different zonesof the material, induces a complex competition betweendifferent failure modes, i.e. intermetallic decohesion,intergranular fracture and transgranular fracture.

A large number of analytical models exist in the lit-erature, which describe the fracture toughness of a ma-terial in the case where one fracture mode is dominant.These different approaches can be reviewed as follows:

The fracture controlled by void nucleation at sec-ond phase particles has been described by Rice andJohnson [1], Burghard [2], and later by Hahn and Ro-seneld [3]. These models have yielded the dependenceof fracture toughness on the volume fraction of secondphase particles

K IC / f 1=6v ; 1

* Corresponding author. Tel.: +33-4-76-82-66-07; fax: +33-4-76-82-66-44.

E-mail address: [email protected] (A. Deschamps).

1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.actamat.2004.01.044

Acta Materialia 52 (2004) 25292540

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which has been conrmed by several experimentalstudies (see, e.g. [3] or [4]). However, the proportionalityfactor predicted by these models did not correspond wellto the experimental facts. It implies a positive correlationbetween toughness and yield strength, opposite to whatis generally observed experimentally with 7xxx alloys.

An other approach, by Garett and Knott [5], renedby Chen and Knott [6] has aimed to describe the fracturetoughness in terms of a critical strain to failure at thecrack tip, associated with a critical opening displace-ment (COD) (related to the fracture of second phaseparticles). This analysis yielded, at constant secondphase particle distribution, to a dependence of thefracture toughness with the plastic properties of thematerial (yield stress r y, hardening exponent n), andsome characteristics of the intermetallic particles (dec-ohesion stress r c, spacing k):

K IC / n

ffiffiffiffiffiffiffiffiffiffiffiffir yr ck

p : 2

This proportionality was found to be consistent with anumber of experimental observations in different com-mercial alloys [7], but again the trend predicted for theinuence of the yield stress is opposite to experimentalndings.

A second type of approach describes the toughnessas controlled by the heterogeneity of plastic ow insidethe material, corresponding to the intense shear bandsobserved notably in underaged states [8,9]. Such ap-proaches have a limited applicability, since the caseswhere the transgranular shear band fracture mode isdominant are quite rare for 7xxx alloys since they are

mainly used in T6 or T7x tempers. An important groupof models has concentrated on the fracture toughnesscontrolled by intergranular ductile fracture, which is aprominent fracture mode in high strength metallurgicalstates (e.g., T6-treated material). The rst model takinginto account the localisation of deformation in theprecipitate-free zone (PFZ) is due to Embury and Nes[10], and predicts a relationship between fracturetoughness and the area fraction of grain boundarycovered by precipitates f GB , through an expression forthe critical strain to failure in the PFZ:

ePFZ

ffiffiffiffiffiffiffiffiffiffiffiffi1= f GB

p 1

=2;

3

K IC E r R ePFZ 1=2: 4

This model has been successfully applied to a number of experimental situations, including the results of Unwinand Smith [11] and Vasud evan and Doherty [12].

Hornbogen and Gr

af [13] have adapted the rela-tionship developed by Hahn and Roseneld [3] for bulkplastic properties to the plastic properties of the PFZ.They developed an expression of fracture toughness as afunction of the strength, hardening exponent and widthof the PFZ, as well as the grain size D, valid for pureintegranular fracture, which yields a D 1=2 dependence,

coherent with the experimental results of numerousauthors [1416]. The Horbogen and Gr

af model, like theEmbury and Nes approach, relies on an expression of the critical strain to failure in the PFZ ePFZ . An alter-native expression for this parameter has been proposedby Kawabata and Izumi [17], which depends on several

parameters of the PFZ (precipitate size, density on thegrain boundaries, width of the PFZ).More recently, Li and Reynolds [18] compared these

approaches and found them very close in terms of theirability to describe experimental results.

In the case where several fracture modes are si-multaneously present (which is the general case in allpractically relevant situations), a number of authorshave proposed some mixture rules between the fracturetoughness predicted for each of the individual fracturemodes alone. Simple linear averaging has been proposedby Hornbogen and Gr

af [13], and more recently bySugamata et al. [19], however the linear averaging of fracture toughness lacks a clear physical meaning, asmentioned by Kamat and Hirth [20], since the quantitiesto be averaged have to be extensive ones, such as thedissipated energies in the various fracture processes, andthus the averaging should at least be performed on thesquare of the fracture toughness. Such quadratic lawshave been used recently by Deshpande et al. [21] andGokhale et al. [22], in order to predict the inuence of the fraction of recrystallisation fraction on the fracturetoughness.

Another strategy to approach the modelling of frac-ture toughness in ductile materials is to consider it as a

limiting case, for very high triaxiality, of a damage ac-cumulation process. In this approach, the description of the dissipated energy is obtained through the analysis of void nucleation and growth, preferentially at secondphase particles. This mechanistic approach is based onmaterial-related void nucleation laws [2325]. A numberof models are available for the growth of voids, eitherisolated in the material or interacting, the most popularapproach being the one by Gurson [26], later rened byTvergaard [27]. Finally, failure models based on thecoalescence of voids enable to predict the failure of thematerial (see, e.g. [28]). Although these approaches havenot reached the point where they can predict quantita-tively the fracture toughness as a function of micro-structural parameters and plastic properties, they havelead for instance to signicant advances on the under-standing of the competition between intergranular andtransgranular fracture depending on the microstructuralfeatures and on the state of triaxiality [29].

In conclusion, a large number of analytical expres-sions are available, which relate the fracture toughnessto microstructural parameters and mechanical proper-ties of the constituent material. A special interest hasbeen given to the microstructural heterogeneities such asPFZs at the boundaries. However one obvious missing

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element of the present state of the art is the descriptionof the fraction of the various fracture modes, fromwhich the appropriate averaging has to be performed.Moreover, very little work exists, which describe theevolution of fracture toughness of a single alloy withvariable heat treatments in a coherent framework en-

compassing situations ranging from dominant inter-granular fracture to dominant transgranular fracture.The aim of the present study is to develop a simple

analytical model for situations with competing fracturemodes, both ductile, either intergranular or transgran-ular. It has to be applicable to a wide range of situations,from very low to very high toughness values, corre-sponding to different proportions of the two fracturemodes. This model is based on existing equations for theindividual mechanisms, leading to specic expressionsfor the dissipation energies of the different failuremodes. This model will also propose a description of thecompetition between the transgranular and the inter-granular failure modes, and its relationship with therelative dissipation energies.

The results of the model will be compared to an ex-tensive experimental study, which details have beenpublished elsewhere [30,31]. This experimental studyprovides the measurement of tear resistance (using theKahn Tear Test) in a wide range of situations (two al-loys, different quench rates and ageing states). On thesame alloys, quantitative values for microstructural pa-rameters, plastic properties and fractions of failuremodes have been measured, which allow a direct com-parison with the present model.

In Section 2, we recall briey the main experimentalndings. In Section 3, we present the model and itscomparison with the experimental results.

2. Main experimental results

2.1. Alloys and heat treatments

Two alloys have been studied in terms of the rela-tionship between their microstructure, plastic propertiesand tear resistance: alloys AA7050 and AA7040 of re-spective compositions 6.33% Zn, 2.46% Mg, 2.2% Cu,0.11% Zr, 0.1% Fe, 0.08% Si and 6.51% Zn, 2.01% Mg,1.64% Cu, 0.11% Zr, 0.08% Fe, 0.05% Si (all in wt%).The latter has been specically developed for thick plateapplications, a slightly lower alloying content aiming atreducing the quench sensitivity and thus improving thetoughness/yield strength compromise. One aim of thepresent study is to determine which microstructuralparameters control this compromise, and what shouldbe the direction in which additional alloy developmentshould go in order to improve it further.

These alloys were subjected to different quench ratesafter solution treatment at 483 C, which initial rate

(from the solution treatment temperature down to200 C) ranged from 800 K s 1 (fast quench, F) to 7 K s 1

(slow quench, S), representative of the quench rateexperienced at mid-thickness of a 150 mm thick platequenched into cold water. Following this quench, thealloys have been subjected to a conventional ageing se-

quence, including natural ageing, and two step articialageing at 120 and 160 C.

2.2. Mechanical properties

The yield strength and strain-hardening characteris-tics were measured by conventional tensile tests; the tearresistance was measured on Kahn Tear Test samples[30], the relevant parameters being the unit initiationenergy, UIE (i.e. the energy dissipated before crackpropagation). Alloy AA7040 was tested in the TL di-rection, and alloy AA7050 in the LT direction, result-ing in an intrinsically higher toughness for the latter.Toughness tests (on CT specimens) on selected statesshowed a good linear relationship between the square of the critical intensity stress factor and the UIE [30]. Fig. 1

0

50

100

150

350 400 450 500 550

7040F

7040S

U I E ( N

. m m

- 1 )

U I E ( N

. m m

- 1 )

Yield strength (MPa)

350 400 450 500 550

Yield strength (MPa)

OA

PAOA

UA

UA

PA

(a)

0

50

100

150

2007050F

7050S

OA

PAOA

UA

UA

PA

(b)

Fig. 1. Plots of unit initiation energy (measured form Kahn Tear Tests)vs. yield strength after a fast quench (F) or a slow quench (S) for(a) AA7040 and (b) AA7050. UA stands for the underaged state, PAfor the peakaged state and OA for the overaged state.

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shows the main results concerning the alloy and quenchdependence on the yield strength/toughness compromiseduring ageing. The classical dependence of toughnessduring ageing is found: highest in underaged states,lowest in peakaged states and intermediate in the over-aged state. When the quench rate is lowered, a dramatic

decrease in fracture toughness is observed for all ageingstates, whereas the decrease in yield stress is almostnegligible. The dependence of toughness on the ageingstate is reduced as compared to the fast quench.

The strain-hardening behaviour of the two alloys isvery similar and has been shown to depend mostly onthe precipitate state inside the grain interiors (i.e. on thestate of ageing), whereas it is almost independent onthe quench rate. The strain-hardening rate is highest inthe underaged state, lowest in the peakaged state andslightly higher in the overaged state.

2.3. Microstructure

Both alloys contain intermetallic particles; howevertheir area fractions are quite different: 0.45% for alloy7040, and 0.69% for alloy 7050. The recrystallisedfraction is similar in the two alloys, about 20%; how-ever, due to the different rolling schedules, the grainaspect ratio is not identical in the two alloys.

After a fast quench, no intergranular precipitates canbe detected. During the ageing treatment,neprecipitatesdevelop in the matrix. In the later states of ageing, thegrain boundaries appear to be densely covered with pre-

cipitates coarser than the homogeneous precipitates; theiraverage size, measured on transmission electron micro-scope (TEM) images, is reported in Table 1. Fig. 2 showsthe microstructure at the grain boundaries in alloy 7040,after a fast quench and in an overageing heat treatment.

After a slow quench, an important fraction of coarse,

heterogeneous precipitation is present in the material.Grain boundary precipitates are present, which aremuch larger than the precipitates formed during ageingafter a fast quench. As a consequence, they are notobserved to evolve during ageing. Heterogeneous pre-cipitates are also present in the grain interiors, nucleatedon the Al 3Zr dispersoid particles (see Fig. 3). Their sizeis quite similar in the two alloys (see Table 2), whereastheir volume fraction is much larger in the 7050 alloy,owing to its higher quench sensitivity. A detailed anal-ysis of the microstructure as a function of the processroute can be found in [31].

2.4. Occurrence of fracture modes

The fracture modes observed on Kahn Tear Testsamples can be classied in four categories: fracture anddecohesion at second phase particles, transgranularductile fracture, transgranular shear fracture and inter-granular fracture (see Fig. 4).

The area fraction of the rst fracture mode has beendetermined by analysis of back-scattered electronimages of fracture surfaces in the SEM, using the highZ-contrast of Fe with respect to Al. The result of thesemeasurements is shown in Table 3. It is observed thatthe area fraction of second phase particles on the frac-ture surface is much larger than their area fraction onpolished surfaces (as expected), and that this parameteris totally independent on the state of ageing and thequench rate. This suggests that in the present situation,second phase particles act essentially as fracture initia-tion sites, and that their own fracture or decohesionoccurs independently to all other fracture events. In thisframework, their inuence can be separated from allother microstructural factors in terms of modelling.

Table 1Average largest dimension (in nm) of precipitates on grain boundaries,measured on TEM micrographs

Alloy Quench Underaged Overaged

7040 Fast 40Intermediate 50 50Slow 80 110

7050 Fast 40Intermediate 60 50Slow 70 80

SD of the measurements is of the order of 15 nm.

Fig. 2. Dark eld TEM micrograph of precipitates on a grain boundary in the AA7040 alloy after a fast quench, and an over ageing heat treatment.



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dissipation is separated in two contributions: one de-scribing the energy necessary to create two free surfaces,the other being related to the work in the plastic zoneahead of the crack tip.

The rst contribution, which we will call Ctrans , can becrudely estimated from the critical stress which inducesthe decohesion at second phase particles and their size.

Typical values of 600 MPa for the decohesion stress and10 l m for their size give a separation work:

Ctrans r decoh dcrit 6000 J m 2: 6In practice, this separation work will be estimated fromexperimental results in situations where only trans-granular fracture is observed (underaged states). The

Fig. 4. SEM micrographs of fracture surfaces showing typical occurrence of the four main fracture mechanisms: (a) fracture at intermetallic particles,(b) ductile transgranular fracture, (c) ductile shear transgranular fracture and (d) intergranular fracture.

Table 3Area fractions (in %) of intermetallic phases (Fe,Cu) on the fracture surfaces, as a function of quench rate and aging state for both alloys

Alloy Underaged Peakaged Overaged

Fast Slow Fast Slow Fast Slow

AA7040 3.7 3.4 4.1 3.7 3.9 3.9AA7050 2.2 2.2 2.2 2.3 2.1

These area fractions can be compared with the average area fraction measured on a polished surface, which is 0.45% for AA7040 and 0.69% forAA7050.

Table 4Area fractions (in %) of the three main fracture modes measured on the fracture surfaces (I, intergranular; S, transgranular shear and D, trans-granular ductile)

Alloy Underaged Peakaged Overaged

Fast Slow Fast Slow Fast Slow

40% I 10% I 50% IAA7040 35% S 40% S

65% D 20% D 90% D 100% D 50% D10% I 45% I 50% I 20% I 30% I

AA7050 15% S 5% S75% D 50% D 50%D 80%D 70% D



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value obtained will be checked for the correct order of magnitude.

Since this separation work depends directly on thevoid growth and coalescence between the second phaseparticles [34], it can be expected that its value is pro-portional to the spacing of these phases. Thus, for a

given size of the intermetallic particle, the dependencewith volume fraction is expected to be

Ctrans / f 1=3v : 7The second contribution to energy dissipation dependson the plastic zone ahead of the crack tip. In order todescribe this contribution, the plastic behaviour of thematerial will be described by a simple elasto-plasticpower-law:

r E e for r < gr y;

r y k e r E n

for r > gr y; 8the parameter g takes into account the stress state at thecrack tip, which itself depends on the sample geometry.The Kahn Tear Test samples thickness is 3 mm, thus wewill consider that plane stress is a good approximationto the stress state. In this case g 1:2 [32]. In the presentmodel, crack propagation occurs when the local stressequals the decohesion stress r decoh . The total energydissipation per unit area during fracture is then calcu-lated as follows:

E trans Ctrans for r < gr y;

Ctrans Up Rp for r > gr y; 9

where Rp is the size of the plastic zone and Up is the

plastic work per unit volume to failure, in the stress stateexperienced at the crack tip. In ductile fracture, only thesecond case is generally met. In plane stress conditions,the radius of the plastic zone has been calculated byIrvin [35]:

Rp E C trans

p r 2y: 10

Combining Eqs. (8)(10), one obtains the total dissi-pated energy in the transgranular mode:

E trans C trans 18>:

E p gr y

1 n=nr decoh

gr y1 k 2

435

1=n

n r decohgr y n 1 !9>=>;

: 11For the sake of simplicity, the decohesion stress r decohwill be taken as the fracture stress r fracture measured inuniaxial tensile tests. It can be observed that this ex-pression includes all the parameters characterising theplastic behaviour of the material: yield stress, workhardening rate, stress to fracture. Notably, the ratior fracture =r y plays a key role; this parameter, called not-

ched yield ratio, has been frequently shown to be rep-resentative of toughness [36,37].

3.2. Energy dissipation during intergranular fracture

In order to estimate the energy dissipated by pure

intergranular fracture, we will adapt the above-men-tioned modelling approaches, which take into accountthe characteristic dimensions of the relevant micro-structural parameters, to take also into account theenergy dissipation in the grain interiors. The simpliedgeometry describing the microstructure is shown inFig. 5. In the present approach, the grain boundaryenvironment is divided in two regions: the PFZ and thegrain interior, which have each their own plastic be-haviour. In order to obtain an analytical description, theplastic behaviour of both regions will be described by asimple linear hardening law:

r grain r yg hgeg; 12r GB r yb hb eb 13the respective sizes of the PFZ and the grain are noted d and D, and their ratio will be noted U d = D in thefollowing. The two regions are assumed to be loaded inseries, which means that r grain r GB at all times. Fol-lowing the approach of Embury and Nes [10], the crackis assumed to propagate in the grain boundary when acritical strain e is reached in the PFZ. To this criticalstrain, we can calculate the stress to fracture r R and thestrain to fracture in the grain interior eg:

r R r b hb e ; 14eg

r b r g hb e hg

: 15The macroscopic strain to fracture is nally calculatedby an average between the strains experienced by thePFZ and the grain interior (which is again coherent witha loading in series):

eR 1 U r b r g

hg hbhg

e Ue : 16This expression can be simplied in the usual case when

U 1 (the usual PFZ size is of the order of 2050 nm,whereas the grain size 1050 l m in diameter); integrat-ing the energy dissipated by plasticity until fractureyields:

W inter r 2R r

2g

2hg r b hbe

2 r 2g 2hg : 17This expression takes into account both the plastic be-haviour of the grain boundaries and the characteristicsof the grain interiors.

The value of the critical strain to fracture in the PFZwill be calculated from the Embury and Nes model [10],



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taking into account the area fraction of precipitates onthe grain boundaries:

e 12

1

ffiffiffiffiffiffiffi f GBp 1 : 18

Finally, the work to fracture of Eq. (17) needs to bemultiplied by the size of the plastic zone, given by Eq.(10), likewise the case of the transgranular fracture,leading to the total dissipated energy per unit area in theintergranular mode:

E inter E C inter2p r 20:2%hg r b hb

21

ffiffiffiffiffiffiffi f GBp 1

2

r 2g!:19

3.3. Description of the fracture toughness in mixed fracture modes

In a fracture mode where both intergranular and in-tragranular fracture coexist, the total dissipated energyis calculated using a rule of mixture between the energiesdissipated via each mode. For this rule of mixture to beapplied an essential ingredient is of course the respectivearea fractions of the two fracture modes. We have de-termined them experimentally in a variety of experi-mental situations, which will enable to test the modelpredictions. However, since no model exists, whichpredicts these fractions as a function of the materialsproperties, this inherently limits the applicability of themodel only to situations where the experimental deter-mination of the fracture modes (which is cumbersomeand somewhat subjective) can be performed. Therefore,we have sought a semi-empirical approach to relate thearea fractions of fracture modes to some other physicalparameters.

It can be expected that, given a constant grain mor-phology, the distribution of fracture between inter-granular and transgranular modes must be governed bythe respective energies dissipated by each of the twofracture modes. In extreme cases, when the dissipatedenergies are equal, there should be no reason for a crack

to propagate in a grain boundary, and if the intergran-ular fracture energy is zero, there should be no trans-granular fracture. It seems therefore reasonable toconsider that the fraction of each fracture mode iscontrolled by the ratio of the dissipated energies.

In the expressions calculated above for the two frac-ture modes, all parameters have been obtained fromtensile tests, except the two cohesion energies C trans andCinter . The values of these parameters for the differentquench and ageing states are listed in Table 5.

The transgranular cohesion energy has been rst es-timated for the alloy AA7040, fast quench, underaged,where fracture is completely transgranular:

C7040 FQtrans 2915 J m 2:The corresponding value for alloy AA7050 has been

obtained by considering that the main difference be-tween the two alloys is the difference in the area fractionof the fracture surface covered with intermetallic parti-cles (Table 3). Applying Eq. (1) then gives:

C7050 FQtrans C7040FQtrans f 7040inter f 7050inter

1=3

3500 J m 2:The cohesion energy of the grain boundaries is much

more difficult to estimate, since there is no experimentalsituation where fracture is purely intergranular. Thus,the value of Cinter has been chosen, such as to provideboth a reasonable description of the relationship be-tween E inter /E trans and the fraction of the intergranularfracture mode, and to provide a good description of thedata:

Cinter 500 Jm 2:This value, 1/6th of the transgranular cohesion energy, isof the right order of magnitude. Finally, the transgran-ular cohesion energies for the slow quenches have been

adjusted to get a good description of the global value of toughness. The values are:

C7040SQtrans 1670 Jm 2;C7050SQtrans 2000 Jm 2:In order to calculate the dissipated energy for the in-tergranular fracture mode, a nal parameter needed isthe area fraction of precipitates on grain boundaries.Following experimental observations, a constant valuefor both alloys and all ageing states for a given quenchrate has been considered: 20% for a fast quench and 40%for a slow quench.

grain

PFZ

D

Fig. 5. Geometrical conguration used for the intergranular fracturecalculation.



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It is rst interesting to focus on the respective inu-ence of Ctrans and f GB . In fact, these are the two pa-rameters which are expected to change most when thequench rate is changed: a slow quench favours inter-granular precipitation, thus increasing f GB , and trans-granular coarse precipitation on the dispersoids, whichis expected to decrease Ctrans . According to the modelpredictions, these two parameters modify the overalltoughness/yield strength relationship in a very differentway: changing f GB results in an overall decrease of thetoughness level, whatever the ageing state, whereaschanging Ctrans results both in an overall decrease of toughness but also in a reduced inuence of ageing ontoughness. Actually this parameter had to be changedsignicantly from the fast to the slow quench in order todescribe correctly the experimental data.

The second point which could be surprising at rst isthe effect of Cinter . Changing this parameter not onlychanges the value of toughness in the states wherefracture is mainly intergranular, but also, and in thelargest extent, in the states where intergranular fracture

is initially very limited (like in the underaged state).This illustrates the complexity of the effect of micro-structure: changing the value of Cinter not only changes

0

10

20

30

40

350 400 450 500 550

350 400 450 500 550

F - experimentS - experimentF - modelS - model

G c ( 1 0 3 J . m

- 2 )

G c ( 1 0

3 J

. m - 2 )

Yield stress (MPa)

Yield stress (MPa)

(a)

AA 7040

Fast quench

Slow quench

0

10

20

30

40

50

607050 F7050 S7050 F7050 S

(b)

AA 7050

Fast quench

Slow quench

Fig. 7. Comparison between the experimental data (dashed lines) andthe model calculations (solid lines) in the G c vs. yield stress diagram,for both quenches and all ageing states. (a) AA7040; (b) AA7050.

0

50

100

150

200

0.25

1.251

0.750.5

2

4

10 Influence of trans

0

10

20

30

40

50

60

0.25

1.2510.75

0.5

2410

Influence of inter

0

10

20

30

40

50

60

0.6

1.21

0.8

1.41.6

Influence of f GB

G c ( 1 0

3 J

. m - 2 )

G c ( 1 0

3 J

. m - 2 )

G c ( 1 0

3 J

. m - 2 )

Yield stress (MPa)

Yield stress (MPa)

Yield stress (MPa)

(a)

(b)

(c)350 400 450 500 550

350 400 450 500 550

350 400 450 500 550

Fig. 8. Parametric evaluation of the inuence of various parameters onthe G c vs. yield stress diagrams. The reference curve is for alloyAA7050, after a fast quench. The parameter values are given norma-lised to this reference state (which always has the value 1). (a) Inuenceof the transgranular separation work C trans ; (b) inuence of the inter-granular separation work Cinter ; (c) inuence of the area fraction of thegrain boundaries covered with precipitates f GB .



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the energy dissipated by this fracture mode, but also theproportion of intergranular fracture, amplifying itsinuence.

4. Conclusions

A model has been proposed, which describes theenergy dissipated during ductile failure, using threeingredients:

(i) A model for transgranular failure, based on thesimplied cohesive zone approach [32], which in-cludes all the main materials parameters (intrinsiccohesive energy, yield and failure strength, workhardening exponent).

(ii) A model for intergranular failure, based on a clas-sical failure criterion [10], which includes the prin-cipal geometrical features and materials propertiesof the PFZ, as well as the plastic properties of thegrain interior.

(iii) A simple phenomenological model for estimatingthe proportion of the two failure modes as a func-tion of their respective dissipation energies, and alaw of mixture to estimate the materials dissipationenergy.

This model has been applied to an extensive data set,including two different aluminium alloys, quenchedfrom the solution treatment with two different quenchrates, and after various ageing treatments. This set of process parameters resulted in a large variety in micro-

structures, failure modes and consequently yieldstrength and toughness. A key point in describing thedata with the model has been the quantitative determi-nation of the respective proportions of failure modes asa function of process parameters.

The main conclusions of the model application on themicrostructural features controlling ductile failure in thepresent situation are:

The evolution of energy dissipation during ageing iscontrolled by a complex set of parameters, including theintrinsic plasticity of the grain interiors and the com-petition between deformation in grain interiors and thePFZ. This evolution is greatly reduced after a slowquench, largely due to a decrease of the intrinsic cohe-sive energy of the grain interiors.

The effect of quench rate on the energy dissipationis dramatic. The application of the model shows thatthis reduction is due in approximately equal proportionsto an increase in the area fraction of grain boundaryprecipitates and a decrease in the intrinsic cohesiveenergy of the grain interiors due to heterogeneousprecipitation.

In the framework of the present experimental situ-ation, a relationship between the materials averageproperties and microstructure ( E inter /E trans ) and the area

fraction of intergranular fracture ( f inter ) has been char-acterised. This relationship can be used to predict theeffect of individual process parameters, not only on en-ergy dissipation during fracture, but also on the occur-rence of the different fracture modes.

A parametric analysis of the behaviour predicted

by the model can provide a guideline for further alloydevelopment.One limitation of the present state of modelling is

that the above mentioned relationship E inter /E trans vs.( f inter ) is expected to depend signicantly on a number of microstructural features such as the grain morphologyand the proportion of recrystallised grains. In order tofurther improve the prediction of fracture toughness, anecessary step is therefore to model in a predictive waythe occurrence of failure modes in ductile materials as afunction of the relevant microstructural parameters.This is by no means an easy task, but some step forwardhas already been obtained through micromechanicalmodelling [29].

References

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Eng Fract Mech 1993;46(2):209.[8] Jata KV, Starke EA. Metall Trans 1986;17A:1011.[9] Roven HJ. Scripta Metall Mater 1992;26:1383.

[10] Embury JD, Nes EZ. Metallkunde 1974;65:45.[11] Unwin P, Smith G. J Inst Met 1969;97:299.[12] Vasud evan AK, Doherty RD. Acta Metall 1987;35:1193.[13] Hornbogen E, Gr

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Metall Mater 1991;39(5):807.[17] Kawabata T, Izumi O. Acta Metall 1976;24:817.[18] Li BQ, Reynolds AP. J Mater Sci 1998;33(24):5849.[19] Sugamata M, Blankenship CP, Starke EA. Mater Sci Eng A

1993;A163(1):1.[20] Kamat SV, Hirth JP. Acta Mater 1996;44(3):1047.[21] Deshpande NU, Gokhale AM, Denzer DK, Liu J. Metall Mater

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Trans A 1998;29A:1203.[23] Goods SH, Brown LM. Acta Metall 1979;27:1.[24] Argon AS, Im J, Safoglu R. Metall Trans A 1975;6A:825.[25] Needleman A. J Appl Mech 1987;54:525.[26] Gurson ALJ. Eng Mater Tech 1977;99:2.[27] Tvergaard V. Int J Mech Sci 2000;42:381.[28] Thomason PF. Ductile fracture of metals. Oxford: Pergamon

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[30] Dumont D, Deschamps A, Brechet Y. Mater Sci Eng A2003;356:326.

[31] Dumont D, Deschamps A, Brechet Y. Mater Sci Tech [in press].[32] Zehnder AT, Hui CY. Scripta Mater 2000;42(10):1001.[33] Tvergaard V, Hutchinson JW. J Mech Phys Solids 1992;40:1377.[34] Wei Y, Hutchinson JW. Int J Fract 1999;95:1.

[35] Irwin GR. In: Proceedings of the 7th Sagamore OrdonanceMaterial Research Conference, Syracuse, NY, vol. 4; 1961. p. 63.

[36] Kaufman JG, Knoll AF. Mater Res Stud 1964;4:151.[37] Senz RR, Spuhler EH. Met Prog 1975;107:64.[38] S. Ma

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itrejean, Ph.D. Thesis, Institut National Polytechnique deGrenoble, France, 2000.


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