Influence of predeformation and agEing of an Al–Zn–Mg alloy—II. Modeling of precipitation...

INFLUENCE OF PREDEFORMATION AND AGEING OF

AN Al±Zn±Mg ALLOYÐII. MODELING OF

PRECIPITATION KINETICS AND YIELD STRESS

A. DESCHAMPS1,2{ and Y. BRECHET1

1LTPCM/ENSEEG, Domaine Universitaire, BP 75, 38402 St Martin d'HeÁ res Cedex, France and2Pechiney±Centre de Recherches de Voreppe, France

(Received 20 February 1998; accepted 22 August 1998)

AbstractÐWe present a model for precipitation kinetics integrating nucleation growth and coarsening. Thismodel allows the description of competitive kinetics of heterogenous and homogeneous precipitation. Theprecipitation model is coupled to a model for structural hardening predicting the yield stress. This struc-tural hardening model includes the possibility of taking into account the distribution of precipitate size andstrength. The model is applied to experimental data on age hardening of Al±Zn±Mg alloys, and allows todescribe the e�ect of a prestraining on the subsequent aging behavior. # 1998 Acta Metallurgica Inc. Pub-lished by Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Modeling of precipitation kinetics and of mechan-

ical properties has always been an area of intense

activity in physical metallurgy.

Most of the modeling has been concerned with

simple and idealized situations, leading for instance

to the well-known LSW equation for coarsening

kinetics [1, 2]. In the case of precipitation hardening,

both precipitation shearing and by-passing are well

understood.

More complex situations have been described by

``integrated modeling'', i.e. modeling of a sequence

of situations which can individually be described by

simple equations but which require some simpli®ca-

tions and often numerical solving in order to be

treated continuously. Integrated models have

described isothermal precipitation kinetics [3, 4], or

non-isothermal precipitation in welding of Al alloys,

for instance [5, 6].

This type of approach is particularly useful in the

di�cult task of modeling the phenomena involved

in industrial processes of aluminium alloys, which

often show situations involving for instance both

phase transformations and plastic deformation, and

non-isothermal aging.

We have described experimentally such a situ-

ation in part I of this study. This experimental

study of precipitation kinetics and related hardening

investigated the response of an Al±Zn±Mg alloy to

a simpli®ed industrial thermomechanical treatment

involving deformation of the solid solution, fol-

lowed by isothermal aging with di�erent heating

conditions.

The aging of these alloys involves a sequence

consisting of metastable GP zones formed at room

temperature followed by metastable Z' precipitates,providing the maximum strength, which eventually

transform into stable Z precipitates. In order to

fully understand the e�ect of predeformation on the

aging sequence and the resulting mechanical proper-

ties, it is necessary to take into account several fea-

tures which di�er from the ideal case of isothermal

precipitation of a single equilibrium phase, such as:

Ðthe rate of formation of GP zones during natu-

ral aging, and its modi®cation induced by predefor-

mation;

Ðthe nucleation of Z' precipitates on these GP

zones, as controlled by the heating rate to the aging

temperature and the annihilation of vacancies;

Ðthe competition between precipitation on dislo-

cations and homogeneous precipitation;

Ðthe spatial heterogeneity and the variation in

precipitate size distribution which result from these

three phenomena.

The aim of the present paper is to describe these

situations in a simpli®ed manner in terms of pre-

cipitation kinetics (variation of the microstructure

parameters with aging time) and precipitation hard-

ening (in¯uence of these parameters on the yield

strength). A number of modeling tools are available

as a basis for this task.

Analytical models for homogeneous precipitation

kinetics are usually restricted to very limited situ-

ations such as the description of precipitate nuclea-

tion, growth or coarsening [3]. However, several

authors have successfully modeled precipitation in

an integrated manner [5±8]. Among these, the ana-

lytical model by Kampman and Wagner [7, 8],

Acta mater. Vol. 47, No. 1, pp. 293±305, 1999# 1998 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/99 $19.00+0.00PII: S1359-6454(98)00296-1

{To whom all correspondence should be addressed.

293

solved numerically, is a good basis for developing ahomogeneous precipitation model in our situation.

Comparatively, precipitation in the presence ofdislocations has received much less attention.Nucleation of precipitates on dislocations has been

treated in detail by LarcheÂ [9], and coarsening ofprecipitates linked by arrays of dislocations hasbeen modeled by a number of authors [10±12].

Growth of precipitates on dislocations has been sel-dom studied, apart from a study applied to Al±Zn±Mg alloys by Allen and Vander Sande [13]. On a

more phenomenological point of view, the in¯uenceof deformation on aging in Al±Zn-Mg±Cu has beenstudied by Poole and co-workers [14]. However,none of these studies consider explicitly the compe-

tition between homogeneous precipitation and pre-cipitation on dislocations, which has proven to bedetermining in our case.

Finally, extensive studies of precipitation harden-ing have led to a relatively good description of theshearing and by-passing modes [15±17]. The tran-

sition between shearing and bypassing has seldombeen investigated. Shercli� and Ashby [5, 6] haveproposed a simple way of obtaining a smooth tran-

sition between the two regimes. We propose here torelate this smooth transition to the size distributionof precipitates.

2. GENERAL OUTLINE OF THE MODEL

This model aims at describing quantitatively thevarious relevant parameters of the precipitationprocess (such as the mean precipitate radius or the

remaining solid solution) and the mechanical prop-erties (yield stress), in the various situationsexplored experimentally in paper I.The outline of the model is as follows: we ®rst

derive a reliable model for homogeneous precipi-tation, from the nucleation stage to the coarseningstage. Then we combine this model for homo-

geneous precipitation with a model for precipitationon dislocations. The two types of precipitation arecompeting in the material due to a solute ¯ux

towards dislocations. Finally, we derive a model forprecipitation hardening which considers the tran-sition between the precipitate shearing and by-pas-sing modes.

In order to keep the model at a reasonable levelof simplicity and number of parameters, some sig-ni®cant simpli®cations need to be made. Among

them, two are particularly important:ÐWe consider that the alloy is quasi-binary,

which means that the precipitate composition is a

®xed combination of Mg and Zn and that one candescribe the di�usion of solute atoms by an equival-ent di�usion coe�cient for this combination of

species.ÐOnly one nature of precipitates is considered.

This means that we do not consider directly thesequence GP zones 4Z' 4 Z. However, the in¯u-

ence of the presence of GP zones during the nuclea-tion of Z' is taken into account through the value of

the activation energy for Z' nucleation. The trans-formation of Z' into Z is actually not taken intoaccount, since it has never been shown that it was

of any signi®cance.Our model is a di�erential model: at time t we

calculate the rate of variation of any parameter as a

function of the current value of all parameters. Thenumerical integration of these equations is then per-formed using a Runge±Kutta method.

3. MODEL FOR HOMOGENEOUS PRECIPITATIONKINETICS

The model for homogeneous isothermal precipi-

tation is partly based on the model by Langer andSchwartz [7], modi®ed by Kampmann andWagner [3, 8] (MLS model). In this model, the

authors treated simultaneously the three stages ofprecipitation (nucleation, growth and coarsening).However, considering simultaneously these threestages of precipitation led to some unsatisfactory

assumptions regarding the precipitates which radiusis lower than the critical radius for dissolution.Thus we consider two stages for precipitation, the

®rst involving simultaneously nucleation andgrowth, and the second involving simultaneouslygrowth and coarsening. A simple criterion ruling

the cross-over from one regime to another is pro-posed.

3.1. Thermodynamics of the model

As a ®rst step we need to de®ne the driving forcefor precipitation at any given time of the aging pro-cess. If we consider an ideal solution model, the ac-

tivity of the atomic species are given by theirconcentrations and when the precipitates consist ofpure solute (Cp=1) the driving force can be

written [18]:

Dg � ÿ kT

vatln

C

Ceq

!�1�

where vat is the atomic volume (considered as con-

stant for all species), Ceq the equilibrium solute con-centration of the matrix, and C is the current soluteconcentration of the matrix.

From this driving force we can derive a criticalradius R* for the precipitates at a given matrix con-centration C:

R� � ÿ 2gDg

which can be written

R� � R0

ln C=Ceq

ÿ � with R0 � 2g vatkT

: �2�

This critical radius is identi®ed both with the criti-cal nucleation radius, and with the radius above

DESCHAMPS and BRECHET: INFLUENCE OF PREDEFORMATION AND AGEINGÐII294

which, in the coarsening regime, precipitates growand below which they dissolve.

The corresponding activation energy for nuclea-tion writes:

DG� � 16

3p

g3

Dg2or DG� � DG0

ln2 C=Ceq

ÿ �with DG0 � 4

3pR2

0 g: �3�

Actually, this activation energy corresponds to purehomogeneous nucleation, which cannot occur at the

aging temperatures considered: usual values fornucleation in an Al±Zn±Mg alloy would give acti-vation energies of the order 6 eV at 1608C, which is

obviously much too high. In order to describe theheterogeneous nucleation of Z' on GP zones, wethus consider DG0 to be an adjustable parameter

characteristic of this nucleation (thus re¯ecting theGP zone structure at the time of Z' nucleation).

3.2. Nucleation and growth

In the ®rst stage of precipitation we consider sim-

ultaneously nucleation and growth. The variation ofthe precipitate density is given by the nucleationrate, and the evolution of the mean precipitate

radius is given by the combination of the growth ofexisting precipitates and the arrival of new precipi-tates at the nucleation radius R*.

ÐFor the nucleation rate we take a standardBecker±DoÈ ring law [19, 20]:

dN

dt

��nucleation

� N0 Zb� exp ÿDG�

kT

� �exp ÿ t

t

� ��4�

N is the precipitate density, N0 is the number ofatoms by unit volume (=1/vat), Z is Zeldovich's

factor (11/20),

b� � 4pR�2DC0

a4�5�

and

t � 1

2b�Z�6�

is an incubation time.ÐThe evolution of the precipitate radius is given

by:

dR

dt� D

R

Cÿ Ceq exp R0=R� �1ÿ Ceq exp R0=R� �

� 1

N

dN

dta

R0

ln C=Ceq

ÿ �ÿ R

!: �7�

The ®rst term corresponds to the growth of existingprecipitates [21, 22]. The second term corresponds

to the arrival of dN new nuclei in the existing popu-lation of precipitates: if dN precipitates nucleateduring dt at radius R*, the radius variation duringdt writes:

dR � dN

NR� ÿ Rÿ �

: �8�

The numerical factor a in equation (7) accounts forthe fact that nucleated precipitates can grow only iftheir radius is slightly larger than the nucleation

radius. The precise value of a is of no consequenceon the results of the model. We have takena= 1.05 [3] in the following.

3.3. Growth and coarsening

In late stages of precipitation, the nucleation ratein the desaturated solid solution progressivelydecreases, and the alloy experiences a combination

of growth and coarsening. When the mean radiusof precipitates is much larger than the critical radiusfor dissolution R*, one can consider that equations

describing pure growth are valid:

dR

dt

��growth

� D

R

Cÿ Ceq expÿR0=R

�1ÿ Ceq exp

ÿR0=R

�dN

dt

��growth

� 0:

8>>>><>>>>: �9�

When the mean radius and the critical radius areequal, the conditions for the standard LSW law are

ful®lled [1, 2]:

dR

dt

��coars

� 4

27

Ceq

1ÿ Ceq

R0D

R2

R � R� � R0

lnÿC=Ceq

�

8>>>><>>>>: : �10�

Di�erentiating the second equation, one can deducethe variation of the matrix solute concentrationwith time in pure coarsening:

dC

dt� ÿR0C

R2

dR

dt: �11�

Finally one can write the solute balance:

C 1ÿ 4

3pNR3

� �� C0 ÿ 4

3pNR3: �12�

The di�erentiation of this equation gives the rate ofvariation of the density of precipitates in pure coar-sening:

dN

dt

��coars

� 4

27

Ceq

1ÿ Ceq

R0D

R3

R0C

R�1ÿ C�3

4pR3ÿN

� �ÿ 3N

" #: �13�

In order to go continuously from the growth stage tothe coarsening stage, we de®ne a coarsening fraction:

dR

dt� ÿ1ÿ fcoars

� dRdt

��growth

� fcoarsdR

dt

��coars

dN

dt� fcoars

dN

dt

��coars

8>>>><>>>>: : �14�

DESCHAMPS and BRECHET: INFLUENCE OF PREDEFORMATION AND AGEINGÐII 295

We veri®ed that changing the fcoars function does notchange the ®nal results as long as fcoars=0 when

R>>R* and fcoars=1 when R= R*. We assume forfcoars the following expression which ful®ls theserequirements:

fcoars � 1ÿ erf 4R

R0ln C=Ceq

ÿ �ÿ 1

� � !: �15�

3.4. From (nucleation and growth) to (growth andcoarsening)

A progressive transition from the nucleation/growth regime to the growth/coarsening regime

takes place. A simple criterion for the transition iswhen the diminution of the precipitate density bycoarsening is larger than the increase of the precipi-

tates density by nucleation, i.e.

ÿdN

dt

��growth�coars

>dN

dt

��nucleation

: �16�

The application of the model will be shown in

Section 6.

4. PRECIPITATION ON DISLOCATIONS

4.1. Outline

From the experimental observations of paper I,

we can describe the main features of the hetero-geneous precipitation on dislocations and its inter-actions with the surrounding matrix:

(i) Dislocations are heterogeneous nucleation sites,and thus need a speci®c nucleation law.

(ii) Dislocations are fast di�usion paths, which

induces a faster coarsening of the precipitateslocated on them.

(iii) Dislocations attract solutes from the matrixbecause of their elastic interaction. Thus a

solute depleted zone around dislocationsdevelops, leading to a precipitate-free-zone.

Fig. 1. Evolution of the action radius of the dislocation between time t and time t + dt. The corre-sponding amount of solute dissolved during dt is shaded.

We will on the one hand consider a heterogeneousprecipitation model on the dislocations (including

nucleation, growth and coarsening), and on theother hand we will de®ne an ``in¯uence radius'' Ri

around the dislocations which is depleted of solutes

and grows with time.The driving force for the evolution of this in¯uence

radius is the concentration gradient between the

solute concentration in the bulk and the solute con-centration at the dislocation core (C(t)ÿCd(t), over adistance Ri(t)) [see Fig. 1]. The amount of solute

which has to di�use to the dislocation in order toincrease the in¯uence radius is calculated from thedi�erence between the mean concentration of thematrix C0 and the concentration Cd at the dislocation

core.Irrespective of any di�usion mechanism, the evol-

ution of the solute depleted zone width has to obey

a mass balance equation which can be written inthe cylinder of radius Ri(t + dt):

at time t :

�c � 4

3pR3

d �t�Nd�t� � pR2i �t�Cd�t�

� 2pRi�t�dRi C0 �17�where Rd is the mean radius of precipitates lying ondislocations and Nd is the density of precipitates on

dislocations (number of precipitates per unitlength). We write Ri(t+ dt)ÿRi(t) = dRi. Thesame solute balance can be made at time t+ dt:�

c � 4

3pR3

d �t� dt�Nd�t� dt�

� pR2i �t� dt�Cd�t� dt�: �18�

Equaling the equations (17) and (18) above yields:

R2i

dCd

dt� 4

3R3

d

dNd

dt� 4R2

dNddRd

dt

� 2RidRi

dtC0 ÿ Cd� �: �19�


4.2. Evolution of the in¯uence radius

For the sake of simplicity, we will consider that thein¯uence radius varies only when the matrix is in thegrowth + coarsening regime: when the matrix is in

the nucleation + growth regime, the solute atoms ex-perience a strong chemical force which prevents themfrom di�using massively to the dislocation.

There are a number of possible interaction forcesbetween solutes and dislocations. We will considerhere only the elastic interaction. For an edge dislo-

cation, the solute ¯ux for a concentration di�erence(CÿCd) is given by [23]:

Jdisl � ÿ D

kTCÿ Cd� � @V

@R�20�

where V is the interaction potential between a

solute atom and the dislocation [24, 25]:

@V

@R� 2

p2mbDv

1

R2�21�

m is the matrix shear modulus and Dv is the relative

variation of volume associated with the soluteatom.Thus one can derive the solute ¯ux through a

cylinder of radius Ri and of unit length:

dn

dt

��Ri

� 2pRiD

kTCÿ Cd� � 2

p2mbDvR2

i

�22�

which needs to dissolve a quantity of solute:

C0 ÿ Cd� � dRi

dt2pRi: �23�

From equations (22) and (23) one can deduce the

evolution of the in¯uence radius:

dRi

dt� Cÿ Cd

C0 ÿ Cd

2

p2D

kT

mbDvR2

i

: �24�

4.3. Nucleation and growth on the dislocations

The nucleation of precipitates on dislocations canbe written as a standard heterogeneous nucleationlaw for sites with a linear density of 1/b. We will

consider that the nucleation radius of precipitateson dislocations is the same as in the bulk: growthand coarsening of precipitates on dislocations beingvery fast, the exact nucleation radius has little in¯u-

ence on the microstructure evolution during aging.Thus we can write:

dNd

dt� J0d

ln2 Cd=Ceq

ÿ � exp ÿ DG0d

ln2 Cd=Ceq

ÿ � !exp ÿ td

t

� ��25�

with

J0d � 4pZDC0 R20

a 4 b: �26�

The end of the nucleation stage can be estimated asfollowed:

ÐThe distance between two precipitates on thedislocation is d = 1/Nd. Thus the average time dt1to nucleate between two existing precipitates is:

dt1 � 1

ddNd

dt

� Nd

dNd

dt

: �27�

ÐOn the other hand, the time dt2 for a soluteatom to di�use on the dislocation to an existingprecipitate is approximately:��

Dd dt2p

� d

2�28�

Dd being the di�usion constant for the dislocationcore.

ÐNucleation on dislocation ends when both

quantities are equal, thus when:

N 3d �

1

4Dd

dNd

dt: �29�

Growth of precipitates on dislocations can be ap-proximated by a similar law as for homogenous

precipitation [equation (7)], representing the factthat precipitates on dislocations grow by pumpingsolute in the surroundings of the dislocation:

dRd

dt� D

Rd

Cd ÿ Ceq exp R0=Rd� �1ÿ Ceq exp R0=Rd� �

� 1

Nd

dNd

dta

R0

ln Cd=Ceq

ÿ �ÿ Rd

!: �30�

The evolution of the solute concentration at the dis-locations is then calculated by the mass balance de-rived in Section 4.1 [equation (19)]:

dCd

dt� 1

R2i

2RidRi

dtC0ÿCd� � ÿ 4

3R3

d

dNd

dtÿ 4R2

dNddRd

dt

� �: �31�

4.4. Growth and coarsening on dislocations

4.4.1. (a) Case of pure coarsening.Wewill ®rst con-sider the simple case where precipitates on dislo-cations are in the pure coarsening regime and thein¯uence radius of dislocations does not evolve (and

therefore there is no ¯ux of solute to the dislocation).In pure coarsening, the concentration Cd of

solute at the dislocations is in equilibrium with the

precipitate radius Rd. This can be written:

dCd

dt� ÿR0 Cd

R2d

dRd

dt�32�

combining equations (19) and (32), we can relatethe variation of radius and density of precipitates

on dislocations through the mass balance:

dNd

dt

��coal

� 3

4

R2i R0 Cd ÿ 4R4

d Nd

R5d

dRd

dt

��coal

: �33�


Finally, the evolution of the precipitate radius Rd isgiven by the superposition of a law for coarsening

through di�usion at the dislocation core [10] and alaw for classical volume di�usion coarsening:

dRd

dt

��coal

�g vat Ceq Dd R2core

25 kT

1

R4d

� 4Ceq R0 D

27 1ÿ Ceq

ÿ � 1

R2d

�34�

Rcore is the radius around the dislocation where theaccelerated di�usion coe�cient is valid. From theabove equation it appears that di�usion along dislo-

cations is most important when the precipitates aresmall, whereas when the precipitates are largevolume di�usion becomes dominant.

4.4.2. (b) Coarsening in the presence of solute dif-fusion towards dislocations. In a more realistic situ-ation the in¯uence radius of dislocations Ri variesduring the coarsening stage of precipitates on dislo-

cations, due to the solute ¯ux towards the dislo-cations. We will consider that this solute input tothe precipitates lying on dislocations superposes

with the isolated coarsening discussed above. Thenthe evolution of the density of precipitates on dislo-cations is still given by equation (33), and the evol-

ution of the precipitate radius is now obtainedthrough the mass balance [equation (19)] in whichnow dRi/dt$0:

dRd

dt�

2RidRi

dtC0 ÿ Cd� � ÿ 4

3R3

d

dNd

dt

4R2d Nd ÿ R2

a R0 Cd

R2d

: �35�

The applications of this model are shown inSection 6.

5. PRECIPITATION HARDENING MODELING

The aim of the model that we present here istwo-fold:

1. it aims at describing the complete agingsequence, and therefore notably at describing thetransition between shearing and by-passing ofprecipitates, which controls the peak strength of

the material. In order to complete this task, themodel will consider the size distribution ofhomogeneously distributed obstacles to dislo-

cations.2. it aims at describing microstructures which are

divided into several regions which show di�erent

properties (which is the case in the predeformedand aged material we studied), and thus todescribe the e�ect of the spatial distribution of

obstacles to dislocations.

5.1. Case of homogeneous precipitation

The case treated here concerns precipitates evenlydistributed in space, but presenting a size distri-bution. The precipitate size distribution is herede®ned by the function f(R) such as f(R).dR is the

probability that the size of a precipitate is betweenR and R+ dR.

We will make the approximation that thedislocation line has to pass through all theobstacles which are encountered in the slip plane in

order to cause macroscopic strain. In this case, thestress at which the particles are overcome can bewritten:

sp �M�F

bL� M

bL

�10

f �R�F�R�dR �36�

where F is the mean obstacle strength, F(R) is theobstacle strength of a precipitate of radius R, L isthe average particle spacing on the dislocation line,

and M is the Taylor factor.5.1.1. (a) Obstacle strength. The obstacle strength

depends on the mechanism of overcoming.

ÐIn the case of particle shearing, the obstaclestrength can have a number of di�erent dependen-cies on the particle radius, depending on the maininteraction mechanism between the shearable par-

ticle and the dislocation. In Al±Zn±Mg(±Cu) alloys,several mechanisms may account for the observedstrength, as for instance interface creation or coher-

ency, strains hardening [26]. Therefore we will notconsider in detail the mechanisms involved, buttake a more general formula as suggested by

Gerold [15]:

F�R� � kmbR: �37�where k is an adjustable parameter characteristic ofthe mechanism considered.

ÐIn the case of precipitate by-passing, the ob-

stacle strength is constant:

F�R� � 2G �38�where G is the line tension (=bmb2, where b is a

parameter close to 1/2).The critical radius for the shearing/by-passing

transition can be easily calculated as the radius forwhich the obstacle strengths are identical for the

two mechanisms:

Rc � 2bbk: �39�

5.1.2. (b) Particle spacing. In order to calculate

the particle spacing along the dislocation line it ispossible to consider several assumptions. One can®rst consider Friedel's statistics which applies best

for low obstacle strengths, and relies on theassumption of a steady-state for the number of pre-cipitates along the dislocation line in motion. Thenthe spacing is given by the calculation of the dislo-

cation curvature under the applied resolved shearstress t on the slip plane [27, 28]:

LF � 4p3fv

�R2Gbt

!1=3

�40�


where fv is the particle volume fraction and R is the

average particle radius.

One other statistic was given by Kocks on the

basis of computer simulations [29±31], for the case

of precipitate by-passing. In this case the particle

spacing along the dislocation line is larger by 15%

than the average precipitate spacing in the gliding

plane:

LK � 1:15

��2p3fv

s�R: �41�

5.1.3. (c) Calculation of the average obstacle

strength. The calculation of the integral in

equation (37) requires the de®nition of a particle

size distribution. In order to keep analytical sol-

utions, we will consider here that the particle distri-

bution obeys a gaussian law:

f �R� � K exp ÿ Rÿ �Rÿ �2

=D2h i

�42�

where R is the most probable radius and D the stan-

dard deviation of the distribution.

The condition�10 f �R� dR � 1 can be written

K � 2

D��pp 1

1� erf �R=Dÿ � : �43�

The corrective term arises from the absence of pre-

cipitates of negative radius. This will cause the true

average radius of the distribution to be slightly

di�erent from R, however in the following we will

consider that�10 f �R�R dR � �R.

The average particle strength can then be written:�10

f �R�F�R�dR ��Rc

0

km bR f �R�dR

��1Rc

km bRc f �R�dR: �44�

The two terms can be calculated in the following

way: �Rc

0

R f �R�dR � KD2

2

exp ÿ�R2

D2

!ÿ exp ÿ Rc ÿ �R

ÿ �2D2

0@ 1A24 35

� KD�R

��pp2

erf�R

D

!� erf

Rc ÿ �R

D

!24 35 �45�

�1Rc

f �R�dR � KD��pp2

1ÿ erfRc ÿ �R

D

!24 35: �46�

5.1.4. (d) Resulting contribution to the ¯ow stress.

Now we can calculate the contribution of precipi-

tates to the ¯ow stress, according to the two cases

stated above.

ÐFriedel statistics:

sFp �M��2Gp

b

��3fv2p

r1�R

�10

F�R� f �R�dR

264375

32

: �47�

ÐKocks statistics:

sKp �M

1:15b

��3fv2p

r1�R

�10

F�R� f �R�dR: �48�

In order to assess the validity of the above

equations, we can analyze them in the limiting cases

of pure shearing or pure by-passing modes.

ÐIf all precipitates are sheared, the two ex-

pressions yield:

sFp ��3

4pb

sk3=2Mm��

bp fvR� �1=2 and

sKp � kMm

��3

2p

rf 1=2v : �49�

As expected, the Friedel statistics shows the usual

dependency in fv and R, but not the Kocks stat-

istics.

ÐIf all precipitates are by-passed, the two ex-

pressions yield:

sFp ��6

p

rbMm b

f 1=2v

�R� 0:7M m b

f 1=2v

�R�50�

sKp ��6

p

rb

1:15Mm b

f 1=2v

�R� 0:6M m b

f 1=2v

�R: �51�

Both expressions give the same dependency in fvand R, and di�er only by the 15% length factor

introduced by Kocks. The Kocks expression gives

the usual calculation of the Orowan stress [15].

We have seen above that the Friedel statistics

give good results for the pure shearing mode, and

results very close from the Orowan stress for the

pure by-passing mode. Moreover, in such concen-

trated alloys, one should also take into account the

e�ect of ®nite obstacle size on the particle spacing.

A simple calculation shows that this e�ect can be

signi®cant in the late stages of overaging, leading to

a maximum decrease of 15±20% of the particle spa-

cing, and thus to a corresponding increase in the

contribution of precipitates to the ¯ow stress. Not

considering this correction actually compensates the

overestimation of the ¯ow stress in the by-passing

mode due to the use of the Friedel statistic.

Therefore we will consider this statistics throughout

the aging curve and will be able to describe the

mixed shearing + by-passing modes.


5.1.5. (e) Calculation of the yield stress. Whencalculating the yield stress, one needs ®nally to add

to the contribution of precipitation hardening thecontributions of dislocation hardening and of solutehardening. This will be done by a classical law of

mixtures [31] considering the various obstaclesstrengths:

stotal � s0 � ssol ��s2d � s2p

q�52�

s0 is the friction stress in aluminium (010 MPa).

The solid solution contribution to the yield stressis calculated as:

ssol � KC 2=3sol �53�

K= 840 MPa is a constant adjusted to the yieldstress of the material in solid solution.

The dislocation contribution is taken as:

sd �M a m b��rdp �54�

with a= 0.27, rd being the dislocation density.

5.2. Case of competitive precipitation: mixing law

When competitive precipitation occurs, a precipi-

tate-free-zone develops around dislocations. Thisresults in a ``natural composite'' with di�erentregions having di�erent yield strengths. Modeling

this situation is very complex, in particular due tothe topology of this composite: a connected path ofsoft regions imbedded in a hard matrix. We will

consider here a simple mixture law for the yieldstresses of the two media, averaged with their re-spective volume fractions as weighting factors. If wede®ne fi the volume fraction of the material con-

tained in the in¯uence radius around dislocations:

fi � 4pR2i rd: �55�

Then the average ¯ow stress can be calculated from¯ow stress contributions of the two media spv andspd by a simple rule of mixtures:

sp � 1ÿ fi� � spv � fi spd: �56�

6. APPLICATIONS OF THE MODEL

In this section we will show the application of this

model to the particular case of the aging of the Al±Zn±Mg alloy investigated experimentally in the com-panion paper [32]. We will ®rst consider the unde-

formed material, for which we have data on theevolution of precipitate radius and microhardnesswith time. In a second step we will show how we can

describe the aging response of the predeformed ma-terial as a function of the process parameters (amountof deformation and rate of heating).

6.1. General validation of the model

Let us take parameters which describe a typicalbehavior of our alloy (see Section 6.2). We have

represented in Fig. 2 the evolution of the various

microstructural parameters which are calculated bythe model, namely average precipitate radius and

particle density. From these two parameters one

can deduce other information, such as the criticaldissolution radius and the residual solute concen-

tration.

First, the average and critical radii are equal bothin the nucleation stage and in the coarsening stage.

It is only when signi®cant particle growth is presentthat the two radii are di�erent. Also, one can

hardly de®ne a pure growth stage: the material ex-

periences a constant particle density for only a veryshort time. This di�culty of de®ning a pure growth

stage in experimental data has been widely recog-nized in the literature [3].

Finally, the decrease in solute concentration is

relatively slow. Even when the material is well inthe coarsening stage, the residual concentration is

signi®cantly larger than the equilibrium concen-

tration. The validity of this solute evolution hasbeen checked for 10 h of aging at 1608C by tomo-

graphic atom probe measurements [33], which gaveCMg=0.9 at.% and CZn=0.54 at.%, thus

CMg + Zn=1.44 at.%, which is very close to the

value predicted by the model (Fig. 2).

Fig. 2. Evolution of microstructural parameters duringaging: (a) precipitate radius and critical radius for dissol-ution; (b) density of precipitates and matrix solute concen-

tration. Parameters: ®rst row of Table 1.


The second step is the validation of the mechan-ical model for the calculation of the yield stress.

Again, the value for the parameters of the micro-structure are adjusted on the Al±Zn±Mg alloy, sothat both the level of yield stress and the critical

radius are well described. Assuming that the sizedistribution width scaled as D = R/4, we adjustedthe parameters of the model to the yield stress of

the undeformed alloy. This yields:

k = 0.07 (where F(R) = kmbR)b = 0.43 (where G = bmb2)M = 2 (Taylor factor).

b is as expected very close to 1/2. The relativelylow value of M which is necessary to describe theexperimental data may seem surprising: self-consist-

ent modeling of these alloys, based on the measuredtexture, gives values closer to 3, close to the Taylorhypothesis. However, when considering the yieldstress (thus very low strains), the grains are not

fully constrained, thus deformations are not necess-arily compatible and a homogeneous stress hypoth-esis can be applied, leading to M12 [34]. A value

of 2 for the Taylor factor has already been con-sidered by Guyot and Cottignies [26] in modelingthe mechanical properties of Al±Zn±Mg±Cu alloys.

With these parameters, one can now calculate theevolution of the yield stress during the aging treat-ment, as a function of the size distribution of pre-cipitates (Fig. 3). It is often considered that this size

distribution scales with the average radius [3], andtherefore we assumed that the width D of the distri-bution is proportional to the average radius R.

When there is no size distribution (D = 0, the sizeof all precipitates is the average radius), the yieldstress experiences a singularity, corresponding to

the transition between shearing and by-passing.Thus, in order to describe the experimental data,which does not show such singularities, it is necess-

ary to take into account the width of the precipitatesize distribution. As expected, when the size distri-

bution widens the peak hardening decreases dra-matically, whereas the mechanical response in the

underaged and overaged states remains constant.This decrease in peak hardening with increasingwidth of the precipitate size distribution is a result

very similar to the calculated increase in mean ob-stacle spacing observed in peak aged alloys by someauthors [35, 36].

From the two above analyses, we can see thatthis model is well adapted to the description of theaging of Al±Zn±Mg alloys.

6.2. Description of the aging sequence of the unde-formed aging behavior

The main quantitative outcomes of the exper-iments described in part I for the undeformed ma-terial are the evolution of precipitate radius and

yield strength during aging. We will here evaluatethe in¯uence of the heating rate to the aging tem-perature on these two parameters.

Precipitate radii for short aging times weredirectly calculated from X-ray Small AngleScattering experiments, and for longer aging timeswere evaluated from TEM micrographs. Yield

strength was approximated as 3 times the Vickersmicrohardness. The validity of this relation waschecked by selected tensile tests in the underaged,

peak-aged and overaged conditions.Let us ®rst consider the simplest case of the fast

heating rate to the aging temperature. In this situ-

ation a large quantity of GP zones dissolve duringthe heating and one can consider that the precipi-tation of Z' precipitates is the dominant mechanism

during isothermal aging. A good description of theevolution of precipitate radius and yield stress canbe achieved with the set of parameters shown inTable 1 (Fig. 4). The precipitate radius is well

described throughout the aging process, and theyield strength is well described for aging times closeto the practically important peak hardening.

Although the model gives an accurate descriptionof the precipitate radius and volume fraction, itunderestimates the yield stress at short and long

aging times. The discrepancy observed in the earlystages of aging can be explained by the presence ofundissolved GP zones in the microstructure whichparticipate to the strength. The discrepancy in the

late overaging stage is more di�cult to understand.It may be due to an incorrect description of the linetension or of the spacing of precipitates along the

dislocation line.With this set of parameters, we can now consider

the case of slow heating rate (308C/h) to the aging

temperature. In order to describe the microstruc-tural evolution in this case, only one parameter canbe changed: the activation energy for precipitate

nucleation DG0, which is characteristic of thenucleation of Z' on existing GP zones. We can seein Fig. 5 that we can again describe the evolutionof precipitate radius by changing DG0 from 1.75 to

Fig. 3. In¯uence of the precipitate size distribution widthD on the evolution of yield stress during aging with the

same parameters as in Fig. 2.


1.56 eV, which corresponds to an easier nucleationof Z' on GP zones in the case of slow heating, as

observed experimentally.

However, in this case the precipitation hardeningin the underaged condition is drastically underesti-

mated. This is due again to the fact that the mech-

anical model does not take into account the GPzones surviving dissolution in the early stages of

aging. Thus, a model for describing the transitionbetween GP zones and Z' precipitates would need

an adjusted volume fraction of precipitates in the

early stages.

6.3. Description of the aging after deformation

The model for competitive precipitation in the

presence of predeformation needs to describe two

main features which come from the experimental

observations of paper I [32]:

ÐThe in¯uence of predeformation on peak hard-

ening depends dramatically on the rate of heating

to the aging temperature.

ÐThe apparent overaging kinetics is accelerated

by the presence of dislocations.

6.3.1. Slow heating rate. In the case of a slow

heating rate, experiments give us only a semi-quan-

titative description of the evolution of the micro-

structure, i.e. of the evolution of the precipitate

radius on dislocations, of the in¯uence radius of

dislocations or on the density of precipitates on dis-

locations. However, this semi-quantitative descrip-

tion is su�cient to obtain approximate values for

the parameters of the model for precipitation on

Table 1. Parameters of the model for the aging at 1608C of theternary alloy with the two di�erent heating rates investigated in

part I

Parameters D(m2 sÿ1) g(J mÿ2) DG0(eV) k b

Fast heating rate 5�10ÿ20 0.300 1.75 0.07 0.43Slow heating rate 5�10ÿ20 0.300 1.56 0.07 0.43

Fig. 4. Best ®t of the model to the experimental data onthe Al±Zn±Mg alloy [32] for aging at 1608C with a fastheating rate: (a) comparison of the predicted precipitateradius with Small Angle Scattering and TEM data; (b)comparison of the microhardness data with the predictedyield stress. The parameters of the mode are listed in

Table 1.

Fig. 5. Best ®t of the model to the experimental data onthe Al±Zn±Mg alloy [32] for aging at 1608C with a slowheating rate (308C/h): (a) comparison of the predicted pre-cipitate radius with Small Angle Scattering and TEMdata; (b) comparison of the microhardness data with thepredicted yield stress. The parameters of the model are

listed in Table 1.


dislocations, and thus to check for the physicalmeaning of their order of magnitude. The list of

parameters giving a good description of the micro-structure evolution is listed in Table 2. Some com-

ments can be made on these parameters:ÐThe low value of the incubation time for pre-

cipitation on dislocations and of the activation

energy for nucleation on dislocations simply meanthat we consider that nucleation on dislocations is

almost instantaneous as compared to nucleation inthe bulk and that it stops rapidly due to the di�u-

sion mechanism stated in Section 4.3.

ÐThe values of the di�usion coe�cient on dislo-cations and of the in¯uence radius of this di�usion

coe�cient were taken in order to obtain a reason-

able evolution of the precipitate radius on dislo-

cations. These values are fully in the possible range.

ÐThe initial value for the in¯uence radius Ri was

considered rather arbitrarily, but its value does not

in¯uence the ®nal result.

ÐThe volume di�erence between the solute

atoms and the aluminium atoms is very high: for

Mg, for instance, it could not be expected to be

more than 11% [37]. However, this value was

necessary to describe the experimental results. This

abnormal value means that the volume e�ect is not

the only interaction which exists between the solute

atoms and the dislocations, and that we encompass

all the mechanisms in our simple description.

Figure 6(a) shows the evolution of the di�erent

radiuses involved in the microstructure: it sows par-

allely the evolution of the homogeneous precipitate

radius, of the radius of precipitates on dislocations

and of the in¯uence radius around the dislocation.

Figure 6(b) shows the corresponding evolution of

the in¯uence volume as a function of dislocation

Table 2. Parameters used for the description of the aging in thepresence of dislocations

td DG0 d Dd Rcore Ri Dv

10 1 eV 10ÿ16 m2 sÿ1 10 AÊ (13b) 50 AÊ 30%

Fig. 6. Evolution during aging of the parameters of the deformed microstructure. (a) Homogeneousprecipitate radius, dislocation precipitate radius and in¯uence radius of dislocations; (b) volume fraction

in¯uenced by the dislocations for two di�erent dislocation densities. Parameters listed in Table 2.


density. Dislocation densities of 1013 and 1014 mÿ2

correspond to approx. 1 and 10% predeformation.

For the dislocation density of 1014 mÿ2 we can

observe that the material is totally free of homo-

geneous precipitates after approx. 1000 h of aging.

This corresponds well to the experimental obser-vation which showed that most homogeneous pre-

cipitates have disappeared after 700 h at 1608C.Now that the microstructural evolution in the

presence of dislocations is described in a semi-quan-

titative manner, we can investigate its e�ect on the

mechanical properties. We are concerned here with

the slow heating rate, where TEM observations

have shown that predeformation did not induce a

signi®cant change in the overall size distribution of

precipitates. Therefore we do not consider in this

case a change in precipitate size distribution in our

mechanical model.

In Fig. 7(a) we can see the in¯uence of the two

levels of predeformation on the aging curve. A ®rst

outcome of the model is that the yield stress in the

underaged condition is increased by predeforma-

tion. This comes from dislocation hardening andcorresponds to the experimental observations.

The in¯uence of predeformation on peak harden-ing appears to be negligible. This results mainlyfrom the mixing law that we chose for adding the

precipitates and dislocations contributions to the¯ow stress, and corresponds well to the experimen-tal observations for this slow heating rate. This

result means that in the peak aged condition, thefact that larger precipitates lie on dislocations doesnot in¯uence the achievable peak strength because

the volume fraction concerned is too small. Ofcourse, this result is only valid for relatively smalldislocation densities, and in the case of large pre-strains the predictions could be di�erent.

The in¯uence of predeformation on overagingappears to be a faster decrease of the yield stress.Moreover, this acceleration of overaging is faster as

the amount of predeformation is increased. Both ofthese features correspond to the experimental obser-vations presented in paper I.

6.3.2. Fast heating rate. In the case of the fastheating rate, it is not possible to describe the largedecrease of peak strength observed in the exper-

iments with a reasonable set of parameters fordescribing the microstructural evolution.However, the experimental observations can be

satisfactorily described by widening the precipitate

size distribution, while keeping the same microstruc-tural parameters as for the slow heating rate. Thiscorresponds well to the experimental observation

that in the case of rapid heating one can observe alarge precipitate size gradient from the homo-geneous precipitates far from dislocations to the

precipitates lying on dislocations. We can see inFig. 7(b) the e�ect of predeformation on the agingcurve when it induces a widening of the precipitatesize distribution from D = R/4 to D = 3R/4. The

e�ect on the peak strength is now very large, asobserved experimentally, and the e�ect of predefor-mation on overaging is again an acceleration, simi-

larly to the case of slow heating and toexperimental observations.

7. SUMMARY AND CONCLUSIONS

In this paper we have developed a simple model,

with a limited number of adjustable parameterswhich all have a transparent physical meaning. Thismodel allows to take into account simultaneously

nucleation, growth and coarsening of precipitates,and, in a simpli®ed manner, the precipitation reac-tion on dislocations competing with homogeneous

precipitation. This microstructural model is coupledwith a structural hardening model which allows toconsider the distribution in strength of the obstacles

to dislocation movement.This model of microstructure evolution has been

shown to describe accurately the behavior of thealloy when no drastic change of the phase precipi-

Fig. 7. E�ect of predeformation on the aging curve: (a)when no widening of the size distribution is induced bythe presence of dislocations; (b) when widening of the sizedistribution is induced by the presence of dislocations is

considered.


tating occurs during the aging sequence. In thissituation, it allows to model the e�ect of prestrain-

ing through its in¯uence on the size distribution ofprecipitates. When two phases are coexisting, suchas GP zones and the Z' phase, the model is, in the

current state, unable to predict the evolution of theyield stress. This indicates a natural development ofthe present approach. In the same way as we

described the competition between homogeneousand heterogeneous precipitation, a model for thecompetition between two phases should be devel-

oped. Another development of this model whichcould be considered is the precipitation kinetics in aconcentration gradient in the dendrite interior ofcast alloys.

More generally, the present approach re¯ects theneed for ``integrated modeling''. The elementarymechanisms governing many microstructural evol-

utions as well as structural hardening are now wellunderstood. The next step in this branch of physicalmetallurgy is to use this knowledge to investigate

the e�ect of non-isothermal heat treatments, ofcompeting precipitation reactions or of interactingphenomena (such as recrystallization and precipi-

tation). These situations are extremely frequent inindustrial situations, and the modeling of thesee�ects cannot be as precise as the modeling of theelementary mechanisms. It has to rely on the same

sound scienti®c grounds, which provide the formal-isms for elementary laws. Within this framework, ithas to process in a pragmatic manner, with just a

few adjustable parameters, and the couplingbetween quantitative experimental investigationsand a simple modeling approach is crucial. The pre-

sent paper gives an example of such a study for agehardenable aluminium alloys, but the method canbe applied to a wide variety of materials and pro-blems.

AcknowledgementsÐThe authors wish to thank Pechiney-Centre de Recherches de Voreppe for ®nancial support.We wish also to thank Dr F. Montheillet and Professor J.Driver for valuable discussions.

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