1990 shercliff (actamet) a process model for age hardening of

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Acta metall, mater. Vo l. 38, No. 10, pp. 1789-1802 , 1990 0956-7151/90 $3.00 + 0.00

Printed in Gre at Britain. All rights reserved Copyright © 1990 Pergamon Press plc

O V E R V I E W N O . 9 0

A PR O C E S S M O D E L F O R A G E H A R D E N I N G O F

A L U M I N I U M A L L O Y S - - I . T H E M O D E L

H . R . S H E R C L I F F a nd M . F . A S H B Y

Ca m b r id g e U n iv e r s it y En g in e e r in g D e p a r tm e n t , T ru m p in g to n S tr ee t, Ca m b r id g e CB2 IPZ , U .K .

Recewed 6February 1990)

Abst rac t - -Process m odel l ing techniques a re used to descr ibe the changes in y ield s t rength due to age

hardening o f hea t - t rea tab le a lum inium a lloys. A model for the i so thermal age ing curve i s deve loped. This

is demonst ra ted for a n um ber of a l loys an d the success of the approach is assessed. Ap pl ica t ions and a

new diagram, showing the var ia t ion of s t rength wi th tempera ture and t ime , a re descr ibed in an

a c c o m p a n y in g p a p e r.

R ~ s u m ~ ) n u t il ise les techniques de la mod61isa tion pour d~cri re les modif ica t ions de la l imi te 61astique

caus6es par le vieil l issement dans des all iages d 'alu m ini um sensibles au traitem ent thermique . On

d6veloppe un mod61e de la cou rbe d u vieil l issement isotherme. On d6m ontre la validit6 et le succ6s de ce

mod61e dan s le cas de plusieurs all iages. D ans l 'a rt icle suivant, on d~crit les applicatio ns du mod61e et un

nouveau d iagramme montrant la var ia t ion de la r6sis tance m6canique en fonc t ion de la tempera ture e t

du temps.

Zusammenfassuag--Die

durch Auslagerun gsh/ i r tung entstehenden ,~nd erungen in d er Fl ie l3fest igke it yon

wfirmebehandlungsffihigen Alu min ium legie rung en werde n mittels Verfahren der ProzeBmodellierung

beschrieb en. Es wird ein Mo dell ffir die isotherme Alter ung skurv e entwickelt. Dieses Mod ell wird an einer

Reihe yon Legie rungen veranschaul ich t ; de r W ert d iese r N/ iherung wird darge legt. An wen dung en und e in

neues Diagramm , welches d ie Ande rung en der Fest igke it mi t der Tem pera tur un d der Ze i t darste l len ,

werden in einer begleitenden Arbeit beschrieben.

1 . I N T R O D U C T I O N

T h e a g e - h a r d e n in g a l u m i n i u m a l lo y s , th e b a c k b o n e

o f t h e a i r c r a f t i n d u s t r y , h a v e b e e n t h e s u b j e c t o f

n u m e r o u s s c i en t if ic s t u d ie s . T h e t h e r m o d y n a m i c s a n d

k i n e t i c s o f s o l u t i o n h e a t t r e a t m e n t , p r e c i p i t a t i o n ,

c o a r s e n i n g a n d r e v e r s i o n a re e s t a b l is h e d [ I - 4 ] ; a n d

t h e i n t e r a c t i o n s b e t w e e n d i s l o c a t i o n s a n d t h e

s t r e n g t h - g i v i n g p r ec i p i ta t e s h a v e b e e n e l u c i d a t e d

[ 4 - 8 ] . Y e t t h e r e i s n o o v e r a l l process model f o r a g e

h a r d e n i n g , d e s p i t e t h e o b v i o u s a d v a n c e s t h a t i t c o u l d

b r i n g . B y t h i s w e m e a n : a m a t h e m a t i c a l r e l a t i o n

b e t w e e n t h e p r o c e s s v a r i a b l e s ( al l o y c o m p o s i t i o n , a n d

t h e h e a t t r e a t m e n t t e m p e r a t u r e a n d t i m e ) a n d t h e

a l l o y s tr e n g t h o r h a r d n e s s , b a s e d o n p h y s i c a l p r i n -

c i pl e s ( th e r m o d y n a m i c s , k i n e t ic t h e o r y , d i s l o c a t i o n

m e c h a n i c s a n d s o o n ) .

P r o c e ss m o d e l s , w i d e l y u s e d b y c h e m i c a l e n g i n e e rs ,

a r e s t i l l r a r e i n t h e p r o c e s s i n g o f s o l i d m a t e r i a l s .

T h e r e a r e s e v e r a l r e a s o n s f o r t h i s : r e a c t i o n s , p h a s e

t r a n s f o r m a t i o n s a n d d e f o r m a t i o n s t a k i n g p l a c e i n

t h e s o l id s t a t e a r e m o r e c o m p l e x t h a n t h o s e i n g as e s

o r l i q u i d s ; a n d s e n s o r s , e a s i l y i m p l a n t e d i n a g a s

s t r e a m t o m o n i t o r t h e p ro c e s s , c a n n o t r o u t i n e l y b e

s t u c k i n t o s o l id s u n d e r g o i n g s e v e re t h e r m a l a n d

m e c h a n i c a l t r e a t m e n t s . P r o g r e s s h a s t h e r e f o r e b e e n

s l o w , b u t s u c c e s s o f f e r s c o n s i d e r a b l e a t t r a c t i o n s :

a s w e l l a s t h e g a i n i n u n d e r s t a n d i n g ( t h e g o a l

h e r e ), a s u c ce s s f u l p r o c e s s m o d e l a l l o w s o p t i m i z a t i o n

o f b o t h p r o c e s s a n d p r o p e r t i es , b e t t e r q u a l i t y

c o n t r o l , f a s t e r p r o c e s s d e v e l o p m e n t , a n d t h e p o s s i -

b i l it y o f a u t o m a t e d , i n t e l l i g e n t c o n t r o l o f c o m -

p l e x p r o c e s s e s s u c h a s w e l d i n g a n d s u r f a c e h e a t

t r e a t m e n t .

T h i s p a p e r d e s c r i b e s a f ir s t a t t e m p t t o a s s e m b l e a

p r o c e s s m o d e l f o r t h e a g e i n g o f t h e s i m p l e r o f t h e

a g e - h a r d e n i n g a l u m i n i u m a l l o ys . T h e p r i n c i p l e s ar e

s t r a i g h t f o r w a r d , b u t th e i m p l e m e n t a t i o n o f t h e m o d e l

( i ts c a l i b r a t i o n t o a g i v e n a l l o y ) is n o t . W e h a v e

s t r i v e n t o k e e p t h e p r o c e d u r e a s t r a n s p a r e n t a s

p o s s ib l e . T h e m o d e l g i v es a g o o d d e s c r i p t i o n o f s o m e

2 0 0 0 a n d 6 0 0 0 s e r i e s a l l o y s , b u t i t i s n o t y e t c o m p r e -

h e n s i v e e n o u g h t o d e s c r i b e t he m o r e c o m p l i c a t e d

7 0 0 0 s e r i e s a l l o y s i n w h i c h t h e p r e c i p i t a t i o n s e q u e n c e

i n v o l v e s m a n y s t ag e s . O f m o r e i n t e r es t i s t h e o v e r a l l

a p p r o a c h , w h i c h h a s g e n e r a l i ty (i t c a n a d a p t e d t o a n y

a g e - h a r d e n i n g s y s t e m ) a n d w h i c h c a n b e e l a b o r a t e d

t o i n c l u d e n e w o r m o r e s o p h i s t i c at e d t r e a t m e n t s o f i t s

c o m p o n e n t s .

A p p l i c a t i o n s o f th e m o d e l a r e d e s c r i b e d i n a c o m -

p a n i o n p a p e r . S y m b o l s a n d u n i t s a r e d e fi n e d i n

A p p e n d i x 1 .

AM 3S t0--A 17 89



1790 SHERCLIFF and ASHBY: OVERVIEW NO. 90

2. AGE HARDENING

The microstructural changes which take place dur-

ing age hardening are well documented [1-3]. Coher-

ent GP zones nucleate and grow from the quenched

solid solution, depleting it; they then coarsen by

competitive growth at constant volume fraction. The

zones are the precursors of particles which evolve

through various intermediate stages, each governed

by its own metastable phase diagram, towards a final

equili brium precipitate. The degree of coherency falls

as the particles coarsen, until the widely-spaced,

incoherent particles of the overaged state are

reached.

This structural evo lution is associated with changes

in the yield strength of the alloy which rises to one or

more maxima, and then falls. Dilute alloys of 2000,

6000 and 7000 series aged at higher temperatures

show a single peak, while two-stage ageing is found

in more concentrated alloys aged at lower tempera-

tures. The way in which the structural evolution

changes the yield strength has been much studied [see,

for example, Ref 4--8]. It is generally thought that

single-peak ageing curves (to which we limit this

discussion) are dominated by a single precipitate

type. While the particles are small, coherent and

closely spaced, they are cut by moving dislocations;

because a dislocation has some flexibility, the numbe r

of particles it touches per un it length increases as the

particles grow and become stronger; this increase

more than offsets the increasing particle spacing, and

the strength rises (the Friedel effect [9]). As coars-

ening proceeds the particle strength and spacing

increase further, until the dislocati on flexibility allows

it to bulge between particles ( Orowan bowing [4])

and escape without cutting them; the stress required

to sustain this decreases as the particle spacing in-

creases, and the strength falls. Superimposed on the

precipitate hardening is a contribution from the

alloying elements in solution (which decays as the

precipitates form), and one from the intrinsic strength

of the matrix (plus any cold work applied to the

material after the solut ion treatment). Figure 1 shows

this scheme.

The aim in this paper is to assemble models for

structural evolution, and those for the dependence of

strength on structure, to give the framework of a

process model for the ageing of alumi nium (and

other) alloys. Most of the components of the model

exist in the experimental results, kinetic models and

dislocation-interactionstudies contained in the litera-

ture; we rely heavily on these in what follows. Oddly,

there does not seem to have been an attempt to draw

them together in the way we do here.

tit is not possible to list every contribution to the develop-

ment of the models on which we draw. Many are now

so widely accepted as to be textbook material; when so,

we cite recent reviews or books in which they can be

found.

-r-

(3:

LLJ

x

NET A G E tN G / / / ~ /

X / PRECIPITATIONTRENGTH

'~. DUE TO DISLOCATIONS:

'~ / ~ ' ( o } SHEARINGPARTICLES

// / - ( b } BYPASSING ARTICLES

/ ~',

C U R V E ~ " ~ \

/ / * ~ NET PRECIPITATION

~ / / / ~ / S T R E N G T H S O UD OLUTIGN

/ , / ~ " "-~. / '~ ~ STRENGTH

/ / / < , ~ / , N T R I N ~ C

/ 1 ~ \ . ~ - / STRENGTI

/ / -,- /

I

LOG (AGEING TIME)

Fig. 1. A schematic diagram of the relative contributions

to the full ageing curve of the intrinsic strength, solid

solution strength, and precipitation hardening due to shear-

able and non-shearable particles; note that the effective

mechanism of precipitation hardening is the one requiring

the least shear stress, with a smooth transition between the

two.

3 . E X P E R I M E N T A L W O R K

Ageing curves were generated for alloy 6082, a

member of the 6000 series; its composition is given

in Table 1. Specimens 9.5 x 10 x 17ram were cut

from the original plate. The specimens were

solution heat-treated at 570°C for 30 min, quenched

in cold water, and immediately aged in fluidized

beds and air furnaces (depending on ageing

time and temperature) at 40°C intervals from 140 to

460°C. Samples were stored in a freezer between

ageing and testing. Vickers hardness tests were

made with a 5 kg load, using several duplicate

specimens and averaging multiple indents to reduce

scatter.

4. THE COMPONENTS OF THE PHYSICAL

MODEL

The components of the model--best called the

sub-models--are set down in this section. All are

taken from established sources, and simplified

as far as possible while maintaini ng adequate

precisiont.

Table 1. Nominal composition (wt %) of the

aluminiumalloy 6082

Element 6082

Si 0.7-1.3

Fe <0.5

Cu <0.1

Mn 0.4-1.0

Mg 0.6-1.2

Zn <0.2

Cr <0.25

Ti <0,1

Zr <0,05

AI balance



SH E R CL IFF and ASHBY: OVE RVIE W NO. 90 1791

T h e y i n c l u d e e x p r e s s i o n s f o r :

( a ) T h e g r o w t h i n v o l u m e f r a c t i o n o f p r e c i p i ta t e

a n d d e c r e a s e i n s o l u t e c o n c e n t r a t i o n w i t h t im e d u r i n g

t h e i n i t ia l s t a g e s o f p r e c i p i t a t i o n ;

( b ) T h e d e p e n d e n c e o f t h e e q u i l i b r i u m v o l u m e

f r a c t i o n o f p r e c i p i t a t e o n a g e i n g t e m p e r a t u r e ;

( c ) P r e c i p i t a t e c o a r s e n i n g b y c o m p e t i t i v e g r o w t h ;

( d ) T h e c o n t r i b u t i o n o f t h e s o l i d s o l u t i o n t o t h e

s t r e n g t h ;

( e) T h e c o n t r i b u t i o n o f s h e a r a b l e p r e c i p i t a t e s t o t h e

s t r e n g t h ; a n d

( f ) T h e c o n t r i b u t i o n o f n o n - s h e a r a b l e p r e c i p i t a t e s

t o t h e s t r e n g t h .

E q u a t i o n s d e s c r i b i n g t h e s e s u b - m o d e l s c o m e n e x t .

T h e y a r e c o m b i n e d t o d e s c r i b e t h e f u l l a g e i n g c u r v e

in Sec t ion 5 .

4. I . Prec ip i ta t ion f rom supersa tura ted so l id so lu t ion

T h e q u e n c h t o w h i c h a g e h a r d e n i n g a l l o y s a r e

s u b j e c t e d a f te r s o l u t i o n h e a t t r e a t m e n t p l u n g e s t h e m

f a r b e l o w t h e s o l v u s t e m p e r a t u r e . T h e s u p e r s a t u r a -

t i o n i s g r e a t , a n d n u c l e a t i o n o f t h e n o n - e q u i l i b r i u m

p h a s e w i t h t h e l o w e s t s u r f a ce e n e r g y - - a l m o s t a l w a y s

o n e w h i c h i s c o m p l e t e l y c o h e r e n t w i t h t h e m a t r i x - - i s

c o p i o u s a n d r a p i d . T h e n u c l e i g r o w b y d r a i n i n g

s o l u t e fr o m t h e s u r r o u n d i n g m a t r i x . D e t a i l s o f n u -

c l e u s lo c a t i o n , p r e c i p i t a t e s h a p e a n d s u r f a c e s t r u c tu r e

a f f e c t t h i s e a r l y s t a g e , b u t t h e y n e e d n o t c o n c e r n u s

b e c a u s e m o s t o f t h e a g e i n g c u r v e i s c o n t r o l l e d n o t b y

t h i s b u t b y c o a r s e n i n g k i n e t i c s [ 1 0 ] . T h e i n i t i a l p r e -

c i p i t a t i o n k i n e t i c s a r e a d e q u a t e l y d e a l t w i t h i n t h e

m a n n e r o f S h e w m o n [ 1 1 ]: t h e m e a n s o l u t e c o n c e n -

t r a t i o n i n t h e m a t r i x 6 d e c a y s e x p o n e n t i a l l y w i th t i m e ,

t , r a i s e d t o a p o w e r c l o s e t o u n i t y :

6 t ) = co +

( c: - c 0 ) e x p ( - I / T I ) ( 1)

w h e r e c i i s t h e i n i t i a l s o l u t e c o n c e n t r a t i o n , Co t h a t a t

e q u i l i b r i u m a t t h e a g e i n g t e m p e r a t u r e , a n d T z i s a

t e m p e r a t u r e - d e p e n d e n t t i m e c o n s t a n t . T h e v o l u m e

f r a ct i on , .f ~ o f p r e c i p i t a t e i s d i r e c tl y p r o p o r t i o n a l t o

s o l u t e l o s s c ~ - ? ( t ) , t e n d i n g t o a f i n a l e q u i l i b r i u m

va lue , f0 , wh en ? = Co; thus :

/ ( t ) c , - e ( t )

- - - - 1 - e x p ( - t / z l ) . (2 )

)Co c i - Co

4.2 . The dependence o f Jo on tem perature

T h e f i n al v o l u m e f r a c t i o n o f p r e c i p i t a t e , f 0 , d e -

p e n d s o n h o w f a r t h e a g e i n g t e m p e r a t u r e T li e s b e l o w

t h e ( m e t a s t a b l e ) s o l id s o l v u s t e m p e r a t u r e , T s . T h e

s o l v u s b o u n d a r y ( F i g . 2 ) i s d e s c r i b e d b y [ 1 2 ] :

c, = ,40 e x p ( - Q J R T s ) (3)

w h e r e A 0 i s a c o n s t a n t , a n d Q ~ i s t h e f r e e e n e r g y o f

s o l u t i o n o f t h e s o l u te . T h e e q u i l i b r i u m c o n c e n t r a t i o n

o f s o l u t e a t a t e m p e r a t u r e T < T ~ i s :

Co = A 0 e x p ( - Q ~ / R T ) . (4 )

D i v i d i n g t h i s e q u a t i o n b y t h e p r e v i o u s o n e g i v e s a n

T e . . . . . . . . . . . . . .

A S I N G L E I

P H A S E I

v : (3 1 0N S O L V US

bJ " BOU DARY

Q:

c:c: T s . . . . . . . . . . . . . . ~ d

LI.I

:~ %..

I L l f M E T A S T A B L E

~ -- T . . . . . ~ S O LV U S

/ BOUNDARY

I

0 i

0 CO C

t C O N C E N T R A T I O N ,

PURE A I . ( w t * / , )

Fig. 2 . A schematic view of the a luminium-rich end o f a

pseudo-bina ry phase diagram for a co mp onent which has

some solubility in aluminium and forms a pre cipitate in an

aluminium-rich matrix.

e q u a t i o n f o r Co i n t e r m s o f t h e s o l v u s t e m p e r a t u r e a n d

t h e h e a t o f s o l u t i o n :

1 )

Q , ( ( 5 )

o = c s exp - R \T

T h e e q u i l i b r i u m v o l u m e f r a c t i o n o f p r e c i p i t a t e a t

t e m p e r a t u r e T is g iv e n b y :

So ,6,

w h e r e f m ,x i s th e m a x i m u m p o s s i b l e v o l u m e f r a c t i o n

p r e c i p i t a t e d a t a b s o l u t e z e r o , w h e n t h e m a t r i x i s p u r e

a l u m i n i u m . S u b s t i t u t i n g f r o m e q u a t i o n ( 5 ) :

Q , 1 1

f 0 = f r , a x l l - e x p - - - ~ - ( ~ -- ~ ) ] . ( 7)

4.3. Part ic le coarsening

O v e r m o s t o f t h e a g e i n g c u r v e , p a r t ic l e s co a r s e n a t

a l m o s t c o n s t a n t v o l u m e f r a c t i o n [ 1 0] , l o s i n g c o -

h e r e n c y a n d c h a n g i n g s t r u c t u r e a s t h e y d o s o . W e

m a k e t h e s i m p l i f y in g a s s u m p t i o n t h a t f o r a s i n gl e -

p e a k a g e i n g c u r v e , t h e c o a r s e n i n g k i n e t i c s c a n b e

a p p r o x i m a t e d b y a s i n g l e k i n e t i c e q u a t i o n , a n d u s e

t h e c u b i c c o a r s e n i n g l a w , d e s c r i b e d [ 1 3 , 1 4 ] a n d r e -

c o n f i r m e d m a n y t i m e s i n th e l a s t 3 0 y r [ 1 5 - 1 7 ] , r e l a t-

i n g t h e m e a n p a r t i c l e r a d i u s r a t t i m e t t o t h a t , r 0 , a t

t ime t = 0 :

QA

r3 (t ) - r03 = ~-~ ex p - ~ (8)

w h e r e c~ i s a k i n e t i c f a c t o r w h i c h d e p e n d s o n t h e

m a t r i x c o m p o s i t i o n a n d Q A i s t h e a c t i v a t i o n e n e r g y

f o r v o l u m e d i f f u s i o n o f a t o m s b e t w e e n p a r t i c l e s .

4.4 . So l id so lu t ion s t rengthening

S o l u t e a t o m s o b s t r u c t d i s l o c a t i o n m o t i o n , c o n -

t r i b u t i n g t o t h e s t r e n g t h o f a n a l l o y . T h e gl ide

res i s tance

c a u s e d b y a s o l u te a t a m e a n c o n c e n t r a t i o n



1792 SH E R CL IFF and ASHBY: OVE RVIE W NO. 90

6 , e x p r e s s e d a s a c o n t r i b u t i o n t o t h e y i e l d s t r e n g t h ,

Aas s , i s g iven [8 , 9 , 18 , 19] to a n ad eq ua te ap pro x i -

m a t i o n b y :

Aas~ = c262/3 (9)

w h e r e c z i s a c o n s t a n t r e l a t e d t o t h e s i z e , m o d u l u s a n d

e l e c t r o n i c m i s m a t c h o f t h e s o l u t e i n c l u d i n g v a r i o u s

r e s o l u t i o n f a c t o r s , a n d 6 i s g i v e n b y e q u a t i o n s ( 1 ) a n d

(5).

4.5. Precipi tat ion shearing

W h e n p a r t i c le s a r e s m a l l t h e y a r e s h e a r e d b y

m o v i n g d i s l o c a t io n s . T h e i r c o n t r i b u t i o n t o t h e

s t r e n g t h o f t h e a l l o y i n v o l v e s a c o n v o l u t i o n o f t h e

r e s is t a n c e to s h e a r o f o n e p a r t i c le , th e i r p o p u l a t i o n ,

a n d t h e f l e x i b i li t y o f t h e d i s l o c a t i o n s w i t h w h i c h t h e y

i n t e r a c t, k n o w n a s t h e F r i e d e l e f f ec t [ 9 ] . M a n y

s t u d i e s h a v e b u i l t o n F r i e d e r s i d e a , a n d t e s t e d i t

e x p e r i m e n t a l l y [7 , 17 , 2 0 , 2 1 ]. F o r p r e s e n t p u r p o s e s , i t

i s a d e q u a t e t o a c c e p t t h a t t h e c o n t r i b u t i o n t o t h e

y i e l d s t r e n g th , A a A , o f a v o l u m e f r a c t i o n f o f s h e a r -

a b l e p a r t i c l e s o f r a d i u s r h a s t h e f o r m :

~

= C3 m n 1 O)

w h e r e c 3 , m a n d n c o n s t a n t s ; f o r m o s t d i s l o c a t i o n -

p a r t i c l e i n t e r a c t i o n s , b o t h m a n d n h a v e t h e v a l u e 0 .5

[ 20 , 2 1 ]. S e n s i t i v i t y a n a l y s i s o f t h e c o m p l e t e p r o c e s s

m o d e l i n S e c t i o n 5 s h o w s i t t o b e i n s e n s i t i v e t o t h e

v a l u e s o f m a n d n i n t h e v i c i n i t y o f 0 . 5, s o t h i s v a l u e

i s u s e d b e l o w , w i t h g a i n s i n s i m p l i c i t y .

4.6 . Prec ip i ta te bypass ing

T h e s p a c i n g o f p r e c i p i t a t e s in c r e a s e s a s t h e y g r o w .

T h e r e c o m e s a s p a c i n g a t w h i c h t h e s t r e s s r e q u i r e d t o

b e n d a d i s l o c a t i o n , s o t h a t i t b o w s b e t w e e n n e i g h -

b o u r i n g p a r t i c l e s i n i t s s l i p p l a n e [ 4, 7 ], b e c o m e s l e s s

t h a n t h a t r e q u i r e d t o s h e a r t h r o u g h t h e m . T h i s

b o w i n g s t r e s s i s :

c G b

Aa a = l (11)

w h e r e G i s t h e s h e a r m o d u l u s , a n d b t h e B u r g e r s

v e c t o r . T h e p a r t i c l e s p a c i n g , l , i n t h e s l i p p l a n e o f t h e

d i s loca t ion i s [17] :

r

1 = c - - ( 1 2)

f ~/2

S u b s t i t u t i n g t h i s i n t o e q u a t i o n ( 1 1 ) g i v e s :

f l/2

A t ra = c 4 - - ( 1 3 )

r

w h e r e c4 c o n t a i n s a l l t h e c o n s t a n t s i n e q u a t i o n s ( 1 l )

and (12) .

5 . T H E P R O C E S S M O D E L

W e n o w c o m b i n e t h e e q u a t i o n s o f t h e l a s t s e c t io n

t o g i v e a p r o c e s s m o d e l f o r a g e i ng . A s t h e e q u a t i o n s

a r e c o m b i n e d , t h e m o d e l g r o w s i n c o m p l e x i t y a n d a

d i ff i cu l ty m u s t b e f a c ed ; t h e s u b - m o d e l s a r e k n o w n t o

d e s c r i b e t h e i r p a r t o f t h e p r o c e s s p r o p e r l y , w i t h t h e

r i g h t d e p e n d e n c e o n m a c r o s c o p i c v a r i a b l e s : c o n c e n -

t r a t i o n , t e m p e r a t u r e , t i m e a n d s o f o r t h , b u t t h e y

c o n t a i n

m icroscopic

c o n s t a n t s : f r e q u e n c y f a c t o r s f o r

a t o m i c d i f f u s i o n , a c t i v a t i o n e n e r g i e s f o r c o a r s e n i n g ,

d i s l o c a t i o n - s o l u t e i n t e r a c t i o n c o n s t a n t s a n d s o o n .

T h e s e c a n n o t b e p r e d i c t e d f r o m f i rs t p r i n c i p l e s w i t h

t h e p r e c i s i o n w e n e e d . S o m e c o u l d b e m e a s u r e d b y

e l a b o r a t e e x p e r i m e n t a t i o n ( t ra n s m i s s i o n m i c r o s c o p y ,

m i c r o - a n a l y s i s , n e u t r o n d i f f r a ct i o n ) b u t t h a t i s n o t

t h e r o u t e t o t a k e i f t h e p r o c e s s m o d e l i s to b e

p r a c t i c al . T h e a l t e r n a t i v e is t o d e r i v e c o n s t a n t s f r o m

t h e a g e i n g c u r v e s t h e m s e l v e s , u s i n g a s u b s e t o f t h e

d a t a t o

cal ibra te the m ode l .

W e b e l i e v e t h a t t h i s i s a

n e c e s s a r y f e a t u r e o f a n y s u c c e s s f u l p r o c e s s m o d e l ,

a n d m u c h o f t h i s a n d t h e n e x t s e c ti o n a r e d e v o t e d t o

i t . A g e i n g c u r v e s f o r h a r d n e s s o r y i e l d s t r e s s m a y b e

c a l i b r a t e d e q u a l l y w e l l , a s s o m e o f t h e c o n s t a n t s

h a v e t h e d i m e n s i o n s o f y i e l d s t r e n g th , a n d m a y t a k e

v a l u e s i n a n y a p p r o p r i a t e u n i t s . I n w h a t f o l l o w s

t h e c o n s t a n t s

C t - C 4

r e l a t e d i r e c t l y t o t h e c o n s t a n t s

c ~- c4 o f t h e l a s t s e c ti o n , b u t t h e y n o w i n c l u d e v a r i o u s

n e w c o n s t a n t s . T h e i r a b s o l u t e v a l u e s a r e n o t i m -

p o r t a n t , a s t h e y w i l l b e c o m b i n e d t o f o r m o t h e r

c o n s t a n t s w h i c h w i l l b e e v a l u a t e d i n t h e c a l i b r a t i o n

p r o c e d u r e .

5 .1 . T h e p r o c e s s m o d e l f o r a f i x e d v o l u m e f r a c t i o n

C o n s i d e r f i r s t i s o t h e r m a l a g e i n g a t f i x e d ( e q u i -

l i b r i u m ) v o l u m e f r a c t i o n , )Co ( a c o n d i t i o n w h i c h i s

r e l a x e d l a t e r ) . T h e n t h e s h a p e o f t h e a g e i n g c u r v e i s

c o n t r o l l e d b y p a r t i c l e c o a r s e n i n g . I n s p e c t i o n o f

e q u a t i o n ( 8 ) s h o w s t h a t , a t a n y f i x e d t e m p e r a t u r e T ,

t h e e x t e n t o f a g e i n g i s g o v e r n e d b y t h e

t e m p e r a t u r e -

correc ted t im e , P ,

d e f i n e d b y :

t Q A

P = -~ ex p - R--T (14 )

P m e a s u r e s th e n u m b e r o f k in e t i c j u m p s t h a t h a v e

t a k e n p l a c e i n t i m e t ; i t is e s s e n t ia l l y t h e s a m e a s t h e

k i n e t i c s t r e n g t h d e f i n e d b y I o n

et al.

[22] . For

p r a c t i c a l a g e i n g t i m e s , r 3 ~ r03 i n e q u a t i o n ( 8 ) , w h i c h

b e c o m e s :

r ( t ) = C 1pt/3. (15)

T h e t h r e e b a s i c c o n t r i b u t i o n s t o y i e l d s t r e s s o r

h a r d n e s s d e r i v e f r o m t h e s o l i d s o l u t i o n , f r o m t h e

p r e c i p i t a t e s a n d f r o m t h e i n t r i n si c s tr e n g t h o f t h e

m a t r i x i t s e l f . T h e i n i t i a l p r e c i p i t a t i o n f r o m s o l i d

s o l u t i o n l e a v e s a c o n c e n t r a t i o n c o i n s o l u t i o n g i v e n b y

e q u a t i o n ( 5) . T h e s o l i d s o l u t i o n c o n t r i b u t i o n t o y i e ld

s t r e n g t h , A c q s , i s t h e n f r o m e q u a t i o n ( 9 ):

Aa, = C2c~/3.

(16)

T h e c o n t r i b u t i o n f r o m s h e a r a b l e p r e c i p i t a t e s i s g i v e n

b y e q u a t i o n ( 1 0 ) i n w h i c h w e s e t m = n = -~ .



SH ERC LIFF and ASHBY: OVERVIEW NO. 90 1793

Combin ing th i s wi th equa t ion (15) g ives :

A o A = C 3 f ~ / 2 e 1/6 (17)

w h e r e C 3 i n c lu d e s t h e c o n s t a n t s o f t h e s e t w o

e q u a t i o n s . T h e c o n t r i b u t i o n f r o m t h e b y p a s s i n g o f

non-shea rab l e p rec ip i t a t e s i s g iven by equa t ion (13) ;

combin ing th i s wi th equa t ion (15) g ives :

C f ~ / 2 ( 1 8 )

A O ' B - - 4 pl /3

Prec ip i t a t e shea r ing and bypass ing a re a l t e rna t ive

processes ; t he con t r i bu t ion , fo r a g iven va lues o f P ,

wi ll be c lose t o t he l esser o f t he two . W e inc lude t h i s

b y t a k i n g t h e i r h a r m o n i c m e a n , d e f i n i n g t h e n e t

c o n t r i b u t i o n o f p r e c i p it a t io n t o t h e s t r e n g th b y :

AO' p p t = + .

(19)

The peak in t he age ing curve l i e s ve ry c lose t o t he

p o i n t w h e r e t h e t w o c o n t r i b u t i o n s a r e e q u a l ,

AaA = A~zB. Def ine t he va lue o f P cor r e spo nd in g to

t h e p e a k a s P p . T h e n a t t h e p e a k :

C t ' 1 /2 1 ~1 6 - - C f ~1 2

3 J O a p - - 4 p i / 3

p

giv ing the re l a t i onsh ip :

C = C3 p~/2 .

(20)

N ote t ha t Po i s a c o n s t a n t , t h e s a m e ( in t h e f r a m e w o r k

of t h i s mo de l ) fo r a ll age ing curves ; i t s va lue (needed

la t e r ) i s found f rom equa t ion (14) by subs t i t u t i ng

v a l u es f o r t h e t i m e s t p c o r r e s p o n d i n g t o t h e p e a k s o f

t h e a g e i n g c u r v es a t s e v e ra l t e m p e r a t u r e s a n d t a k i n g

t h e m e a n . W e n o w i n t r o d u c e a p a r a m e t e r S 0 , t h e

p e a k p r e ci p i t a ti o n s tr e n g t h d e f in e d b y :

_ C3 ,e I/2e I /6

S 0 - - ~ - : 0 p . ( 2 1 )

I t is th e m a x i m u m c o n t r i b u t i o n o f p r e c ip i t a t io n t o t h e

t o t a l s t r e n g t h , a n d d e p e n d s o n t e m p e r a t u r e i n a w a y

d e s c r i b e d b e l o w . N o t e t h a t b e c a u s e P p i s c o n s t a n t ,

So oc f0 , a f ac t we m ake use o f i n t he ca l i b ra t i on

process . Su bs t i t u t i ng So in to equ a t ions (17) and (18) ,

us ing equa t ion (20) , g ives :

AO - A __--2 S o ( p * ) l / 6

2s0

A a B = ( p , ) l / 3 (22)

w h e r e P * =

P / P p ,

t h a t i s , n o r m a l i z e d t e m p e r a t u r e -

cor rec t ed t ime . Subs t i t u t i ng t hese i n to equa t ion (19)

gives:

2 S 0 ( P * ) 1 / 6

A°'pPt 1 + ( p , ) m (23)

A t t h e p e a k ( P * = 1 ) t h e p r e c ip i t a te s t r e n g t h i s S o .

5 .2 . T h e t i m e d e p e n d e n c e o f f a n d c a t c o n s t a n t

t e m p e r a t u r e

I t ha s been a ssumed thus fa r , fo r s impl i c i t y , t ha t

t h r o u g h o u t a g e i n g t h e v o l u m e f r a c t i o n o f p r e c ip i t a te

is c o n s t a n t a t f 0 , a n d t h e s o l i d s o l u t io n c o n c e n t r a t i o n

c o n s t a n t a t c 0 . M o d e l l i n g w o u l d i n d e e d b e s i m p l e i f

t h i s were so , bu t i t is no t ; b o th f and c evo lve wi th

t ime in t he ways desc r ibed by equa t ions (1 ) and (2 ) .

C o n s i d e r f ir s t t h e e v o l u t i o n o f t h e v o l u m e f r a c t io n f

t ow ards i t s (me ta s t ab l e ) equ i l i b r ium va lue )c

[equa t ion (2 ) ] . The re i s ev idence [23] t ha t t he cub ic

c o a r s e n i n g l a w o f e q u a t i o n ( 3) h o l d s f r o m t h e v e r y

beg inn ing o f p rec ip i t a t i on , even whi l e the vo lum e

f rac t i on i s chang ing . Assuming th i s t o be so , i t i s

s i m p l y n e c e ss a r y t o r e p l ac e f 0 b y f ( t ) i n e q u a t i o n ( 21 ),

g iv ing a cor re spond ing va r i a t i on i n $2 :

S2 (t) = S02[1 - exp - ( t / z~)] . (24)

W hen t ~> f i , f ha s evo lved to f0 , an d S has reached

the pea k p rec ip i t a t e s t r eng th , S O Sim i l a rly t he ev o l -

u t i o n o f th e s o l u t e c o n c e n t r a t i o n i s i n c l u d e d b y

rep l ac ing Co in equa t ion (16) by J ( t ) , g iven by

equ a t ion (1 ) . Subs t i t u t i ng f rom equ a t ion (16), t he

e v o l u t i o n o f t h e c o r r e s p o n d i n g s o l id s o l u t i o n s t r e n g th

i s t hen desc r ibed by :

A as s( t ) = [A a ss03/2+ ~__rAa8~, - ~v̂ ~3/21sso

× e x p ( - - t / z t ) ] 2 /3

(25)

where t he subsc r ip t s , a s be fore , deno te i n i t i a l and

f inal va lues o f t he so l id so lu t i on con t r i bu t ion . Expe r -

i m e n t s s h o w t h a t t h e v o l u m e f r a c t i o n a n d s o l u t e

concen t ra t i on se t t l e t o t he i r equ i l i b r ium va lues a t a

t im e w h i c h is a c o n s t a n t f r a c t i o n o f t h e t i m e t o r e a c h

p e a k s t r e n g t h [ 2 4 ] . T h e t i m e c o n s t a n t z , o f e q u a t i o n

(2) mus t t he re fo re sca l e a s t p :

Zl = KI

tp

or us ing t he de f in i t i on o f Pp :

~ 1 = K I P p T e x p ( Q A / R T ) . (26)

T h e v a l u e s o f Q A , P p , S o , a n d K , a r e d e t e r m i n e d b y

the ca l i b ra t i on p rocedure desc r ibed in Sec t ion 6 .

5 .3 , T h e t e m p e r a t u r e d e p e n d e n c e o f f o , S o a n d A as s

T h e p e a k p r e c i p i t a te s t r e n g t h S o d e p e n d s o n t h e

a g e i n g t e m p e r a t u r e b e c a u s e t h e v o l u m e f r a c t i o n f 0

d o e s s o [ e q u a t i o n ( 7 ) ] . C o m b i n i n g t h is w i t h e q u a t i o n

(21) gives:

[ ) l

g(r)=(s0)Lx

l - e x p R \ T ~ ( 27 a)

( So ) . . . Q s a n d T s a r e f o u n d b y c a l i b ra t i o n . C o m b i n -

ing equa t ions (24) and (27a ) g ives :

x [ l - e x p - ( t / z l ) ]. (2 7b )



1794 SHE RCL IFF and ASHBY: OVERVIEW NO. 90

Simi l a r ly t he so l i d so lu t i on con t r i bu t ion t o t he

s t r e n g t h d e p e n d s o n t h e a g e i n g t e m p e r a t u r e b e c a u s e

i t de t e rmines t he a m ou nt o f so lu t e l e ft i n so lu t i on . In

the g ross ly ove raged cond i t i on , t he coa rse p rec ip i t a t e

c o n t r i b u t e s v i r t u a l l y n o t h i n g t o t h e s t r e n g t h , w h i c h

d e p e n d s o n l y o n t h e m a t r i x s o l u t e c o n c e n t r a t i o n .

Using equa t ions (5 ) and (16) , t he t empera tu re -va r i -

a t i o n o f t h e e q u il i b ri u m s o l id s o l u t i o n c o m p o n e n t o f

the yie ld st rength i s:

2 e . L _ l )

( A % 0 ) r = ( A % o ) r , e x p - - 3 R \ T ~ ( 28 )

where t he sub sc r ip t s T and Ts i nd i ca t e t he t empera -

tu re conce rne d . Th e ave ra ged s t r eng th , a0a i s t he sum

of t he so l i d so lu t i on s t r eng th an d the i n t r i ns i c

s t reng th o f pure a lu min ium (which we ca ll tr i) :

( O ' 0 a ) T ~ - O +

(Aa~0)r . (29)

At T = T~ the conce n t ra t i on o f t he ma t r ix equ a l s t he

a l l oy con cen t ra t i on , a nd the re i s no p rec ip i t a t e . In t he

a s - q u e n c h e d c o n d i t i o n t h e a l l o y is a u n i f o r m s o l u t i o n

o f t h e f u l l a l l o y c o n t e n t . H e n c e t h e a s - q u e n c h e d

s t reng th , aq , i s g iven by :

~q = a~ + (A a~ ) r . (30)

C o m b i n i n g t h e s e t h r e e e q u a t i o n s g i v e s t h e v a r i a t i o n

o f t h e o v e r a g e d s t r e n g t h w i t h t e m p e r a t u r e :

' )

2 Q s ( ,

O 0 a = O + ( O ' q - -

o'i)ex p

3 R \ T . (31)

U s i n g k n o w n v a l u e s o f

aq

a n d a ; t h e o v e r a g e d

s t reng th can be ca l cu l a t ed .

5 .4 . T h e s u m o f t h e c o n t r ib u t i o n s t o t h e a g e i n g c u r v e

The f i na l s t ep i n a ssembl ing the p rocess mode l i s

t h a t o f c o m b i n i n g t h e c o n t r i b u t i o n s t o t h e y i el d

s t reng th [17, 19]. At t he l evel o f appro x im a t ion a imed

a t he re , i t i s adequa t e t o i den t i fy t he y i e ld s t r eng th

w i t h t h e s u m :

a t ) = a i + A % + Atrpp,. (32)

T h e c o n t r i b u t i o n s t o t h e s t r e n g t h a r e s h o w n i n F i g. 1 .

B o t h ? a n d f v a r y w i t h t i m e , r e a c h i n g s t e a d y v a l u es

b e f o r e t h e p e a k o f t h e a g e i n g c u r v e ( s h o w n a s a s o l id

line) is reached.

6 . T H E C A L I B R A T I O N P R O C E D U R E

T h e e q u a t i o n s h a v e b e e n s t r u c t u r e d s o t h a t a l l th e

u n k n o w n c o n s t a n t s t h e y c o n t a i n c a n b e c a li b r a te d b y

u s i n g d a t a f r o m t h e a g e i n g c u r v e a lo n e , t h o u g h s o m e

i t e ra t i on may be necessa ry . Pub l i shed age ing curves

a r e a l w a y s f o r t h e t e m p e r a t u r e r a n g e b e l o w t h e

me ta s t ab l e so l i d so lvus , a s above i t t he re i s gene ra l l y

l i t t l e ha rden ing . Our a im, howeve r , i s t o desc r ibe

age ing a t a ll t empera tu re s up t o t he on se t o f me l t i ng ,

s o t h e p o s i t i o n o f t h e m e t a s t a b l e s o l v u s a n d t h e

b e h a v i o u r a b o v e m u s t a l s o b e d e t e r m i n e d . S o m e

a l loys con ta in more t han one d i s t i nc t p rec ip i t a t e ,

r e q u i r i n g a m o r e c o m p l e x c a l i b r a t i o n p r o c e d u r e . N o

a t t e m p t i s m a d e t o d o t h a t h e r e , t h o u g h c o m m e n t o n

t h e m e t h o d i s g i v e n i n A p p e n d i x 2 .

C e n t r a l t o t h e c a l i b r a t i o n p r o c e d u r e i s t h e u s e o f

t h e p r e c i p i t a t i o n c o n t r i b u t i o n t o t h e p e a k h a r d n e s s

(AO'ppt)p , and the a ssoc i a t ed t ime tp. Tw o imp or t a n t

t empera tu re -dependenc i e s i n t he mode l a re i nvo lved .

The f i r s t i s t ha t fo r p rec ip i t a t e coa rsen ing . I t was

s h o w n t h a t t h e p e a k t e m p e r a t u r e - c o r r e c t e d t i m e i s

i n d e p e n d e n t o f t e m p e r a t u r e [ e q u a t i o n (1 4 ) w i th

t = t p ] :

P p = t p / T ) e x p - Q A / R T ) = cons t an t . (33)

T h e n t h e a p p r o p r ia t e p l o t o f

l o g ~ t p / T )

v s

1 / T

gives

the ac t i va t i on ene rgy Q A. Th e con s t an cy o f Pp is a te s t

o f t he app l i cab i l i t y o f t he mod e l . I t i s eva lua t ed a t

e a c h t e m p e r a t u r e , a n d t h e m e a n v a l u e r e c o r d e d

(examples la ter) .

T h e s e c o n d i m p o r t a n t t e m p e r a t u r e - d e p e n d e n c e i s

t ha t o f the equ i l i b r ium vo lum e f rac t i on , f0 . I t i s

i nves t i ga t ed us ing measured va lues o f So s ince (a s

shown ea r l i e r ) S ~ o c f o , an d (Atrppt) = S 0. T his en-

ables u s to f ind va lues fo r (S0)m~x, Qs and Ts in

e q u a t i o n ( 27 a) . T h e p e a k p r e c i p i ta t i o n s t r e n g t h c o n -

t r i bu t ion , (Atrpp t)p, i s fou nd by su b t rac t i ng t he o th e r

con t r ib u t ion s ( so l i d so lu t i on A % and in t r i ns i c t r i)

f r o m t h e m e a s u r e d p e a k s t r e n g t h , s o t h e s e m u s t b e

f o u n d f i rs t. A t p e a k h a r d n e s s , t h e s u m o f t h e se o t h e r

con t r ib u t ion s i s t he ove rag ed s t r eng th , a s f ha s

r e a c h e d i t s e q u i l i b r i u m v a l u e b e f o r e t h e p e a k . T h e

o v e r a g e d s t r e n g t h d e p e n d s o n t h e v a l u e s o f Q s

a n d T s ( a s y e t u n k n o w n ) b u t w e m a y m a k e a f i rs t

e s t ima te a s fo l l ows: i t mus t l i e be tween the a s -

quenched s t r eng th aq (when a l l t he so lu t e i s i n

so lu t i on) , and the i n t ri ns i c s t r eng th a i (when n o so lu t e

i s le f t i n so lu t i on , t ha t i s, t he s t r eng th o f pure

a lumin ium ) . A f i rs t e s t ima te fo r t he o th e r con t r i -

b u t i o n s t o p e a k s t r e n g t h [ d e s c r i b e d b y e q u a t i o n

(29)] is:

( O ' 0 a ) e s t = ( / ~ O ' s s 0 + O ' i ) p • ( O ' q + o'i)/2.

T h i s i s s u b t r a c t e d f r o m t h e p e a k s t r e n g t h a t e a c h

t empera tu re t o g ive f i r s t e s t ima te s fo r t he peak

precip i ta te c on tr ibu t ion , (Aappt)p(= $2o). T o ev aluate

(So) . . . Q~ an d T~ the f i rst s tep i s to plo t S 2 vs

t empe ra tu re . F igure 3 show s a schem a t i c o f t h i s p lo t ,

w i t h t y p i c a l d a t a p o i n t s . T h e e q u i l i b r iu m v o l u m e f r a c-

t i on t ends t o ze ro wi th i nc reas ing t empera tu re . The

i n t e r s e c t i o n o f t h e e x t r a p o l a t e d c u r v e w i t h t h e t e m -

pe ra tu re ax i s g ives an e s t ima te o f t he me ta s t ab l e

so lvus t empera tu re , Ts , fo r t he p rec ip i t a t e which i s

d o m i n a n t b e l o w t h a t t e m p e r a t u r e . D a t a p o i n t s c l o s e

t o t h e a p p a r e n t s o l v u s m u s t b e t r e a t e d w i t h c a u t i o n

a s t h e s e m a y b e d i s t o r t e d b y t h e p r e s e n c e o f a

d i f fe ren t , h ighe r t em pera tu re p rec ip i ta t e . F ig ure 3 has

t h e f o r m p r e d i c t e d b y e q u a t i o n ( 2 7a ). T h e v a l u e

o f S o t e n d s t o a c o n s t a n t ( S 0 ) ~ x a s t h e t e m p e r a t u r e

f all s, c o r r e s p o n d i n g t o f 0 a p p r o a c h i n g i t s m a x i m u m




I X E X P E R I M E N T A L

x POINTS

l ~ .D A T A P O I N T S E L O W

~ SECONDARYOLVUS

t THEORETCAL

S 2 ~ ./RELATIONSHIP

° ) m a ~ ~ I I ~ K . APPARENT

~ , P P A R E N T

I ~X~ IETASTABLE

S E C O N D A R Y - ~ S O L VU S

M E T A S T A B L r -

I

T E M P E R A T U R E .

S O L V U . l ~ ' / T s

T E M P E R A T U R E

F i g. 3 . T h e v a r i a t i o n o f S o ( w h i c h is p r o p o r t i o n a l t o t h e

e q u i l ib r i u m v o l u m e f r a c t io n ) w i t h t e m p e r a t ur e . T y p i c a l

e x p e r im e n t a l d a t a p o i n t s a r e s h o w n , s o m e o f w h i c h l ie a t

t e m p e r a t u r e s b e l o w a s e c o n d a r y s o l v u s . T h e s o l id l i n e i s t h e

t h e o r e t i c a l c u r v e u s e d t o m o d e l t h e r e l a t i o n s h i p .

p o s s i b l e v a lu e ,f m a x , a t a t e m p e r a t u r e o f a b s o l u t e z e r o .

I f t h e r e i s a s e c o n d m e t a s t a b l e s o l v u s a t a l o w e r

t e m p e r a t u r e , t h i s i s i n d i c a t e d b y a s h a r p s t e p i n t h e

d a t a t o h i g h e r v a l u e s o f S 2 b e l o w t h a t s o l v u s ( se e F i g .

3 ) . I n p r i n c i p l e , o n e c o u l d p e r f o r m a s e c o n d c a l i -

b r a t i o n f o r t h i s se c o n d m e t a s t a b l e p r e c i p i t a t e , b u t i n

p r a c t i c e t h e d a t a a r e s e l d o m s u f f i c i e n t .

T h e c o n s t a n t ( S0 )m ,x i s t h e l i m i t i n g v a l u e o f S o a t

l o w t e m p e r a t u r e s - - t h e p l a t e a u i n F i g . 3 p r o v i d e s a

f i r st e s t i m a t e . T h e e n e r g y o f s o l u t i o n Q ~ c a n t h e n b e

f o u n d f r o m t h e a p p r o p r i a t e A r r h e n i u s p l o t o f

e q u a t i o n ( 2 7 a ) o r b y i n s e r t i n g v a l u e s f o r S o , T , e t c .

a n d s o l v i n g f o r Q s . M o r e a c c u r a t e v a l u e s o f tr 0, m a y

n o w b e f o u n d , f o r e a c h t e m p e r a t u r e f r o m e q u a t i o n

( 3 1 ) u s i n g t h e f i r s t e s t i m a t e s f o r ( S o )

. . .

Q~, and T~

a n d t h e p r o c e s s i s r e p e a t e d u n t i l t h e f i t b e t w e e n

t h e o r y a n d d a t a i s a c c e p t a b l e .

A b o v e t h e m e t a s t a b l e s o l v u s , a d i f f e r e n t p r e c i p i t a t e

f o r m s , a n d a g e i n g p e a k s m a y n o t b e a v a i l a b l e t o

c a l i b r a t e t h e e q u a t i o n s . T o m o d e l t h i s r e g i m e , i t i s

a s s u m e d t h a t t h e a c t i v a t i o n e n e r g y , Q A , f o r a g e i n g i s

t h e s a m e a s t h a t b e l o w t h e s o l v u s , a n d t h a t t h e s a m e

e q u a t i o n s , w i t h a p p r o p r i a t e l y a d j u s t e d c o n s t a n t s , s t i ll

a p p l y . I f t h e p r e c i p i t a t e is t h e e q u i l i b r i u m o n e , t h e n

t h e t e m p e r a t u r e T e a t w h i c h i t d i s s o l v e s c o m p l e t e l y

c a n b e r e a d f r o m t h e e q u i l i b r i u m p h a s e d i a g r a m . I t

i s th e n n e c e s s a r y t o f i n d v a l u e s f o r ( S 0 ~) . . . Q e a n d

P ~ , w h i c h a r e t h e e q u i v a l e n t p a r a m e t e r s t o ( S o ) . . .

Q s a n d P p f o r t h e o t h e r r e g i m e . A n e s t i m a t e f o r Q ~ i s

o b t a i n e d a s f o l lo w s . A t t e m p e r a t u r e s a p p r o a c h i n g T ~,

t h e o v e r a g e d s t r e s s a 0 , i s r e a c h e d i n a p r a c t i c a l

e x p e r i m e n t a l t i m e s c a l e . A c a l i b r a t i o n v a l u e f o r a 0a a t

a p a r t i c u l a r t e m p e r a t u r e T m a y t h u s b e o b t a i n e d , a n d

e q u a t i o n ( 3 1 ) s o l v e d t o g i v e a n e s t i m a t e f o r Q e .

(S 0~ )m ,~ a n d P ~ a r e a d j u s t e d b y t r i a l - a n d - e r r o r . T h e

v a l u e s o f ( S 0) ma x a n d P p a r e u s e d a s i n i t i a l e s t i m a t e s

fo r (So¢)m, x an d P~ .

I t r e m a i n s t o d e t e r m i n e t h e t i m e c o n s t a n t r , . A s

e x p l a i n e d e a r l i e r , i t i s r e l a t e d t o t h e p e a k t e m p e r a -

t u r e - c o r r e c t e d t i m e , f o r b o t h p r e c i p i t a t e s , b y

e q u a t i o n ( 26 ). T h e c o n s t a n t K ~ i s a d j u s t e d b y t r i a l a n d

e r r o r t o g i v e t h e r e q u i r e d d e c a y ( a s i n F i g . 1 ) , g o i n g

t o i t s s t e a d y v a l u e a l i t t le b e f o r e t h e p e a k s t r e n g t h i s

r e a c h e d . T h e d e c a y o f t h e s o li d s o l u t io n c o m p o n e n t

o f t h e s t r e n g t h i s t h e n g i v e n b y e q u a t i o n ( 2 5 ) w i t h :

Ao'ss = O ' q - - O

a n d A a s s o

O ' o a - - O ' i .

T h e c a l i b r a t e d e q u a t i o n s a r e a s s e m b l e d t o c o n s t r u c t

t h e a g e i n g c u r v e i n t h e w a y d e s c r i b e d i n S e c t i o n 5 .

T h e a d d e d c o m p l e x i t y a s s o c i a t e d w i t h a g e i n g a b o v e

t h e m e t a s t a b l e s o l v u s , a n d w i t h t w o o r m o r e p r e -

c i p i t a t e t y p e s a r e d i s c u s s e d i n A p p e n d i x 2 , w h i c h

s u m m a r i z e s t h e c a l i b r a t i o n p r o c e d u r e .

7 . V A L I D A T I O N O F T H E M O D E L

T h e c a l i b r a t i o n p r o c e d u r e h a s b e e n a p p l i e d to a

n u m b e r o f a l u m i n i u m a l l o y s [ 25 ] . E x a m p l e s a r e g i v e n

h e r e , a n d t h e s u c c e s s o f t h e m e t h o d i s a s s e s s e d . A l l

r e q u i r e a v a l u e f o r t h e i n t r i n s i c s t r e n g t h ¢r~ o f a l u -

m i n i u m . T h e m e a s u r e d y i e l d s t r e n g t h o f p u r e a l u -

m i n i u m i s 5 3 M P a ; e x t r a p o l a t i o n o f t h e a s - q u e n c h e d

h a r d n e s s f o r b i n a r y A I - C u a l l o y s c o n t a i n i n g 2 , 3, 3 .5 ,

4 a n d 4 . 5 w t % c o p p e r [ 2 6] t o z e r o c o n c e n t r a t i o n g i v es

t h e s a m e r e s u l t [2 5 ]. W h e n c a l i b r a t i n g u s i n g h a r d n e s s

d a t a , t h e c o r r e s p o n d i n g v a l u e o f 1 5 V P N h a s b e e n

us ed .

7 . 1 . The pr oces s mode l app l i ed to a l loy 6061

T h e m o s t d e t a i l e d d a t a s e t a v a i l a b l e t o u s w a s t h a t

o f A n d e r s o n [ 2 4 ] f o r t h e a g e i n g o f a l l o y 60 6 1 a t 1 1

t e m p e r a t u r e s b e t w e e n 1 0 7 a n d 2 6 0 ° C . T h i s a l l o y h a s

t h e c o m p o s i t i o n 0 . 6 % S i , 1 % M g , 0 . 2 5 % C u , a n d

0 . 2 5 % C r . T h e c a l i b r a t i o n , f o l l o w i n g t h e n u m b e r e d

s t e p s o f A p p e n d i x 2 , i s a s f o l l o w s :

( i) T h e a s - q u e n c h e d y i e l d st r es s , % = 1 5 5 M P a ,

a n d t h e i n t r i n s i c s t r e n g t h a i = 5 3 M P a , a r e u s e d t o

g i v e a n i n i t i a l e s t i m a t e o f t h e o v e r a g e d s t r e s s :

( O ' 0 a ) e s t = ( O ' q - ] -

a i ) / 2 = 1 04 M P a .

( i i) P e a k d a t a ( t p, a p ) f r o m t h e a g e i n g c u r v e s a r e

l i s t ed in T ab le 2 (a ) .

( i i i ) F i g u r e 4 ( a ) s h o w s t h e p l o t o f

l o g e t p / T )

v s

I / T ,

f r o m w h i c h w e f i n d t h e a c t i v a t i o n e n e r g y

Q A =

1 50 k J / m o l .

( i v ) F r o m T a b l e 2 ( a ) t h e m e a n p e a k t e m p e r a -

t u r e - c o r r e c t e d t i m e , u s i n g t h i s v a l u e o f

Q A ,

i s

P 0 = 5 7 7 × 1 0 16 s/K .

( v ) F i r s t e s t i m a t e s o f S o = ( A % p ,) p a t e a c h t e m p e r a -

t u r e a r e g i v e n i n T a b l e 2 ( a ) .

( v i ) T h e f i r s t p l o t o f S o v s t e m p e r a t u r e , s h o w n i n

F i g . 4 ( b ) , g i v e s a n e s t i m a t e o f t h e m e t a s t a b l e s o l v u s

t e m p e r a t u r e T s = 3 0 0 ° C .

( v i i) T h e e s t i m a t e d v a l u e o f (S o ) . . . f r o m F i g . 4 ( b ) ,

i s 1 80 M P a ; s o l v i n g e q u a t i o n ( 2 7 a ) f o r Q s , u s in g t h e

v a l u e s o f S o a n d t h e e s t i m a t e s f o r T s a n d ( S 0 ) . . . g i v e s

a n a v e r a g e e s t i m a t e d Q s = 2 8 k J / m o l .

( v ii i) A f t e r a n u m b e r o f i t e r a t i o n s t h e b e s t f i t o n t h e

S O - T

p l o t , s h o w n i n F i g . 4 ( c ) , i s a c h i e v e d w i t h t h e

fo l low ing va lues : T s = 270°C , (S0)max = 202 M Pa , a nd

Q s = 3 0 k J / m o l .

( ix ) T h e r e q u i r e d d e c a y o f t h e s o l i d s o l u t io n c o m -

p o n e n t i s a c h i e v e d u s i n g t h e v a l u e K ~ = 0 . 5 .



1796

SH E R C L IFF a n d A SH B Y : O V E R V IE W N O . 90

T a b l e 2 . T h e d a t a a n d e v a l u a t e d p a r a m e t e r s i n t h e c a l i b r a t i o n o f t h e a g e i n g c u r v e s f o r

a l loy 6061

( a ) D a t a f o r p e a k t i m e a n d y i e l d s t r e ss , a n d e v a l u a t e d p a r a m e t e r s , a t v a r i o u s t e m p e r a t u r e s

Tem pera tu re Tim e to pea k Pea k y ie ld s t ress 1016 pp (AO-ppt) (m So )

T ( 'C) t p ( s ) a p ( M Pa ) ( s / K ) ( M P a )

260 135 211 .I 5.05 107.1

232 577 245.6 3.48 141.6

218 2491 247.7 5.59 143.7

204 5557 258.1 4.36 154.1

191 20,700 263.6 5.79 159.6

171 142,000 266.6 7.20 162.6

163 360,000 266.3 8.82 162.3

149 780,000 271.2 5.00 167.2

135 4,330,000 266.3 6.63 162.3

( b ) A c c e p t e d v a l u e s f o r t h e c a l i b r a t i o n p a r a m e t e r s

I n t r i n s i c y i e l d s t r e s s a i 5 3 M P a

A s- q u e n c h e d y i e l d s t r e s s a q 1 55 M P a

A c t i v a t i o n e n e r g y f o r a g e i n g Q A 1 45 k J / m o l

M e t a s t a b l e so l v u s t e m p e r a t u r e 7 , 2 7 0 ° C

S o l v u s b o u n d a r y e n t h a l p y Q , 3 0 k J / m o l

Streng th par am ete r (So)m~~ 2 0 0 M P a

Pe a k t e m p e r a t u r e - c o r r e c t e d t i m e Pp 2 . 5 x 10-n5 s /K

Co n s t a n t r e l a t i n g z I t o t p K I 0 . 5

T h e c a l i b r a t i o n c o n s t a n t s w e r e fi n e - tu n e d b y c o m -

p a r i s o n w i t h t h e a g e i n g c u r v e s t h e m s e l v e s - - t h e f in a l

va lues fo r t h i s a l l oy a re l i s t ed i n Tab le 2 (b ) . F igure s

5(a ,b ) show the t heore t i ca l cu rves (con t inuous l i ne s)

a n d t h e d a t a ( s y m b o l s ) f o r a l l o y 6 0 6 1 - - t w o f i g u r e s

a r e p l o t t e d f o r c l a r i t y d u e t o t h e d e n s i t y o f t h e

d a t a p o i n t s . T h e f it a t t h e p e a k s o f t h e c u r v e s is g o o d .

T h e d i s p a r i t y b e t w e e n t h e o r y a n d e x p e r i m e n t a w a y

10

8

~

O

. - i

2

0

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1 8

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LLOY6 0 6 1 [

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x / /x

/

/

/

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o

o 3

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5 ___

100 200 300 400

A G E I N G T E M P E R A T U R E = C )

c )

5

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ff l 2

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0

0

~ k 6061

J / t o o (

Fig . 4 . Cal ibrat ion d iagrams for a l l o y 6061: (a ) l o g (base e ) o f t ime t o reach the peak over t emperature,

t p / T vs reciprocal o f temperature, I / T to give a gradient Q ^ / R ; (b) the f irst p lo t o f S02 vs temperature

using an est imate o f the ov eraged stress , a0. , to give est imates of the solv us temperature, T, , and the

strength parameter, (S0)m~x: (c) the f inal p lo t o f S02 vs temperature to determ ine accep ted v alues fo r T~,

(So) ,,~x and Q, .

i s =270*C

100 200 300 400

A G E I N G T E M P E R A T UR E ° C )




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oo j

sou c. o

U 3

1 0 0

h i

5 O I

Y I E L D S T R E S S , f f ~

' • " I N

R I N S I C

Y I E L D S T R E S S , (:r ~

1 0 1 02 1

A G E I N G

i0 @

T I M E s )

ALLOY

6 0 6 1

ANDERSON11959}

(D 218 C

• 191 *C

O 163ac

1 3 5 " C

1 0 7 " C

1¢ 107 1#

Fig. 5 . Data from A nderson [24] for a l loy 6061 com pared with the model for 10 ageing temperatures

(spread over two figures), using the calibration values listed in Table 2(b).

f r o m t h e p e a k s f o l l o w s t w o g e n e r a l t r e n d s : ( a ) t h e

p r e d i c t e d r i s e t o t h e p e a k i s g e n t l e r t h a n i s o b s e r v e d

a t h i g h t e m p e r a t u r e s ; ( b ) a t a l l t e m p e r a t u r e s t h e

h a r d n e s s f a ll s o f f b e y o n d t h e p e a k m o r e s l o w l y t h a n

p r e d i c t e d , t h e m e a s u r e d v a l u e s b e i n g u p t o 2 0 %

g r e a t e r .

7 2 The process model applied to alloy 6082

O u r o w n d a t a f o r t h e a g e in g o f 6 08 2 ( t he U K

n e a r - e q u i v a l e n t o f 6 0 6 1 ) a r e s h o w n i n F i g . 6, w i t h t h e

r e s u l t s o f t h e m o d e l ( c o n t i n u o u s l i n e s ) , t h i s t i m e

i n c l u d i n g t h e r e g im e a b o v e t h e m e t a s t a b l e s o l v u s f o r

t h e l o w - t e m p e r a t u r e p r e c i p i t a t e ( 2 8 2° C ) . T h e a l l o y

s h o w e d d i s t i n c t p e a k s w h e n a g e d a t 2 6 0 ° C a n d

b e l o w ; a b o v e 3 00 ° C o n l y t h e o v e r a g e i n g t a i l s a r e

a v a i l a b l e .

D e t a i l s o f t h e c a l i b r a t i o n a r e g i v e n e l s e w h e r e [ 2 5 ] ;

f i n a l v a l u e s f o r t h e p a r a m e t e r s a r e l i s t e d i n T a b l e 3 .

T h e s t e p s i n t h e h i g h - t e m p e r a t u r e p a r t o f th e c a l i b -

r a t i o n ( a b o v e t h e m e t a s t a b l e s o l v u s ) a r e a s f o l l o w s :

( i ) T h e a c t i v a t i o n e n e r g y f o r p r e c i p i t a t e g r o w t h ,

QA, a n d t h e d e c a y c o n s t a n t K j t a k e t h e s a m e v a l u e s

a s i n t h e l o w t e m p e r a t u r e r e g i m e : Q A = 1 30 k J / m o l ,

K t ----0.5.

( ii ) T h e s o l v u s t e m p e r a t u r e f o r t h e h i g h - t e m p e r a -

t u r e p r e c i p i t a t e , T e , i s r e a d f r o m t h e p s e u d o - b i n a r y

A I - M g 2 S i p h a s e d i a g r a m g i v e n i n R e f. [ 2] :

Te = 565°C .

( ii i) T h e o v e r a g e d h a r d n e s s a t a h i g h t e m p e r a t u r e

g i v e s a n e s t i m a t e f o r t h e h e a t o f s o l u t i o n Q , . A t

4 6 0 ° C , t h e o v e r a g e d h a r d n e s s w a s 3 8 V P N . T h e

e q u i l i b r i u m s o l i d s o l u t i o n p a r t i s t h u s t r 0 ~ - tr~ = 2 3

V P N . T h e a s - q u e n c h e d s o l id s o l u t i o n p a r t i s

t r q - t r ~ = 3 4 V P N . S u b s t i t u t i n g in t o e q u a t i o n ( 3 1)

w i t h t h e s o l v u s t e m p e r a t u r e r e p l a c e d b y T , , g i v e s

Q , --- 2 9 k J / m o l ( o f t h e s a m e o r d e r a s Q s )-

( iv ) In i t i a l e s t ima te s fo r (S0e )m, , and Ppe a re the

v a l u e s o f ( S0 )m ,, a n d P p . T r i a l - a n d - e r r o r a d j u s t m e n t

o f Q e , (S 0~ )m ,x a n d P ~ b y p l o t t i n g t h e t h e o r e t i c a l

a g e i n g c u r v e s w i t h t h e h i g h t e m p e r a t u r e d a t a



179 8 S H E R C L I F F a n d A S H B Y : O V E R V I E W N O . 9 0

Fig . 6 .

L L O Y

6 0 8 2

1 5 o o t . 6o = c o 2 6 o * c

X /.20'C 4D220 C

3 8 0 "C + 1 8 0 " C

O 3 / . 0 ' C A 1 / .0 " C ~ ^ * ~ 1 0 ~ 1 /.0 0C

Z 3 0 0 0 C z = : u

u

q u

u

0 3 1 0 0 - . 31.0% , 2 60 " C - - ~ l - l X J [ ~ , ~ / " , . + . . . / ~ = =

03 38,0 c/3 0 *c 1 . Y \

/ -,

_

~ A S - Q U E N C H E D ~ ' ~ " ~ ~ 3 - ~

H A R D N E S S .

crg

k l N T R I N S I C

H A R D N E S S . c q

0 i i i i I I

1 0 1 0 2 1 0a 1 0 4 1 0s 106 107

A G E I N G T I M E ( S )

D a t a f r o m t h e p r e s e n t s t u d y f o r a l l o y 60 8 2 c o m p a r e d w i t h t h e m o d e l f o r n i n e a g e i n g t e m p e r a t u r e s

u s in g t h e c a l i b r a t i o n v a lu e s l i s t e d i n T a b l e 3 .

Table 3. The accepted values for the adjustable param eters in the calibration of the ageing curves

for alloy 6082

Intrinsic hardness

As-quenched hardness

Activation energy for ageing

Transition temperature

Metastable solvus temperature

Phase boundary solvus temperature

Solvus boundary enthalpy

Phase boundary enthalpy

Strength parameter (below Tc)

Strength parameter (above To)

Peak temperature-corrected time (below T,)

Peak temperature-corrected time (above T¢)

Constant relating ~ to tp

a i 15 VP N

% 49 VPN

QA 130 kJ/mo l

T, 269 C

Ts 282°C

T 565°C

Q~ 30 kJ/tool

Q~ 30 kJ/mo l

(So)m~~ 94 VPN

(S0¢)m=~ 64 V P N

Pp 5.5 x 10 14s/K

Pr~ l a x 10 13s/K

K~ 0.5

y i e l d s t h e f i n al a c c e p t e d v a l u e s : Q e = 3 0 k J / m o l ,

( S0 e) m ax = 6 4 V P N a n d P p e = 1 . 4 x 1 0 - 13 s /K .

( v) T h e t r a n s i t i o n t e m p e r a t u r e b e t w e e n t h e t w o

r e g i m e s i s e s t a b l i s h e d a s i n A p p e n d i x 2 , l o c a t i n g

t h e p o i n t a t w h i c h t h e p r e d i c t e d p e a k h a r d n e s s

i s t h e s a m e i n b o t h r e g i m e s - - t h e v a l u e o f T c i s

2 6 9 ° C .

F o r t h is a ll o y , th e a g r e e m e n t b e t w e e n t h e o r y

a n d e x p e r i m e n t ( F i g . 6) is g o o d t h r o u g h o u t . T h e

m o d e l f it s t h e e x p e r i m e n t a l d a t a t o w i t h i n 1 0 % ,

g e n e r a ll y s h o w i n g t h e s a m e t r e n d s a s w e r e f o u n d f o r

a l l o y 6 0 6 1 .

7 .3 . The process mode l app l i ed t o a lumin ium copper

al loys

H a r d y [ 26 ] g i v e s a g e i n g c u r v e s f o r f iv e b i n a r y

A I - C u a l l o y s c o n t a i n i n g 2 , 3, 3 .5 , 4 a n d 4 . 5 % c o p p e r .

F i n a l v a l u e s f o r t h e c a l i b r a t e d p a r a m e t e r s f o r a ll f iv e

a l l o y s a r e l i s t e d i n T a b l e 4 ( d e t a i l s i n R e f . [ 2 5 ]) . T h e

r e s ul t s a r e s h o w n i n F i g s 7 ( a - e) . T h e a g r e e m e n t

b e t w e e n t h e m o d e l a n d t h e d a t a i s g o o d f o r a ll fi v e

a l l o y s, w i t h a s i m i l a r d i s p a r i t y b e f o r e t h e p e a k a s w i t h

a l lo y 60 61 : th e c u r v e s s h o w a m o r e m a r k e d i n c u -

b a t i o n p e r i o d a t c o n s t a n t h a r d n e s s f o ll o w e d b y a

s t e e p e r r i se t o t h e p e a k t h a n i s p r e d i c t e d .

Table 4. The accepted values for the adjustable param eters f or the five AI42u alloys

Alloy copper conten t (wt % ) 2.0 3.0 3.5 4.0 4.5

Intrinsic hardn ess, a i (VPN ) 15 15 15 15 15

As-quenched hardness, aq (VPN) 39.5 50.5 56 63 69

Activation energy for ageing,

QA (kJ/ mo l) 128 128 128 128 128

Peak temperature-corrected time,

1013 x Pp (s/ K) 28.5 14.5 10.0 6.99 4.04

Metastable solvus temperature,

T~ CC) 235 245 250 255 235

Solvus boundary enthalpy,

Qs (kJ/m ol) 25 25 25 25 25

Strength param eter, (S0)m~x VPN ) 46 57 64 70 89

Con stant relating z I to t 0, K 0.5 0.5 0.5 0.5 0.5



S H E R C L I F F a n d A S H B Y : O V E R V I E W N O . 9 0 1 79 9

7 4 A t t e m p t s t o a p p l y t h e p r o c e s s m o d e l t o 7 0 00 se r i es

a l loys

W e a t t e m p t e d t o f i t t h e m o d e l t o a n u m b e r o f 70 0 0

s e r i e s a l l o y s ( d e t a i l s i n R e f . [ 2 5 ]) . T h e d a t a p a s s e d t h e

f ir s t t e s t o f v a l i d it y : t h a t t h e p e a k t e m p e r a t u r e -

c o r r e c t e d t i m e , P p , w a s c o n s t a n t ( a l l o w i n g a n a c -

t i v a t i o n e n e rg y , Q A , t o b e f o u n d ) . H o w e v e r t h e

m e t a s t a b l e p r e c i p i t a t e f o r m s a t s u c h l o w t e m p e r a -

t u r e s , w h e r e t h e a g e i n g i s s l o w , t h a t t h e s c o p e o f t h e

d a t a w a s n o t s u f f i c i e n t t o a l l o w a f u l l c a l i b r a t i o n .

F u r t h e r w o r k i s n e c e s s a r y h e r e .

( a }

15o

v 1 0 0

u')

¢/

w

z

D

-i- 5o

0

10

A[-2 , .oA , L o Y T ]

- "~A-'E6Y"Tgs~-qT ' -

+ 2 2 0 " C O 1 6 5 °C

• 1 9 0 * C 1 3 0 ' C

lg0*C 165=C 130 C

2 ~ o c / / I I

AS-QUENCHED

/ I _ - ~ ~ ~ ~ X ~ I

ARDNESS, o ~ ~ l~e-- . - O , L -

+ x ~ O 9 # q P - ~ p ~ x x ""

_ _ ~ N T R I N S J C

/

H A R D N E S S , o i

102 103 10¢ 105 106 107 108


Q .

v

c,o

co

iJJ

z

(:3

r r

r

1 5o ( b )

A [ - 3 % C u A L L O Y

- ~ y ' ~

+ 2 2 0 * C O 1 6 5 "C

• 190 *C

100 190"C 165=C

220 C

l 1

] o o

+ + l c O + U w u ~

, ~ I N T R I N S I C

HARDNESS,

0 I I I I I i

1 0 1 0 2 1 0 3 1 0 ~ 1 0 s 1 0 6 1 0 ? 1 0 e


S O l

is0 c)

z

Q .

v 100

<I: 50

" r

I

o

lO

A t - 3 . 5 % C u A L LO Y

- - H - ~ ' ~ " ( 1 9 5 1 ) I

+ 2 2 0 * C O 1 6 S * C I

1 9 0 * C

I

1 9 0 * C 1 6 5 " C

220'C

I

l

A S -Q U E N C H E D ~ ' - ~

~ , o ' O -

. . / INTRINSIC

HARDNESS. (~

i i I i I I

1 0 2 1 0 3 1 0¢ 1 0 5 1 0 1 0


+@

Fig . 7 . (a -c ) Caption overleaf



1800 SHERCLIFF and ASHBY: OVERVIEW NO. 90

Z

0 _

1 0 0

U 3

tO

U J

Z

C3 I

n -

50

"1-

15o d)

,

I -A -4

I ,

CuALLOY

. ..RDY.951

1¢2~o*c 19o*c

t + 2 2 0 ' C

0

10

r

220 C

/

2~.o*c ~ ~.....- =-~ t..=

l g o * c

~ A S - Q U E N C H E D

HAR DNE SS, o - o.

j . ~ I N T R I N S I C

H A R D N E S S , ~ t

102 103 1@ 10s 106 107 10


e )

i s° I

AI-4.S%CuALLOY

H A R D Y ( 1 9 5 1 )

+ 2 2 00C 19 0*C

Z

Z , ~ + +

~ 50 . AS-QUENCHED

r HARDNESS, (Tq

/ ~ I N T R I N S I C

" v H A R D N E S S ,

010 0 ' ' ' ' '

2 10a 10~ 105 106 107

A G E I N G TI M E ( s )

Fig. 7(d,e)

108

Fig. 7. (a) Data from Hardy [26] compared with the model for the binary AI-Cu alloys at selected

temperatures, using the calibration values listed in Table 4; this plot is for 2% Cu alloy; (b) data for 3% Cu

alloy compared with the model, as in (a); (c) data for 3.5% Cu alloy compared with the model, as in (a);

(d) data for 4% Cu alloy compared with the model, as in (a); (e) data for 4.5% Cu alloy compared with

the model, as in (a).

8, CONCLUSIONS

A process model has been developed to describe the

ageing of age-ha rdening alloys. It is based on simple,

established principles of phase equilibria, precipitate

coarsening, and dislocation-precipitate interactions.

These components are assembled into an overall

process model which is structured such that the

numerous microscopic parameters in the com ponent

equations can be combined and calibrated to the

ageing curves themselves. The model, though obvi-

ously simplistic in ma ny regards, describes the ageing

curves of binary alumini um alloys, and alloys of the

6000 series with fair success; with the more complex

7000 series alloys there remain unresolved problems.

Such a process model, once developed and vali-

dated, has numerous applications: the prediction of

the effect of heat -trea tmen t cycles an d of the st rength

of welds; process control in heat treatment and the

optimization of process cycles; and the presentation

of data in new ways and its compact storage in data

bases. Some of these are illustrated in the companion

paper.

The main conclusion then is this: that the approach

explored here has sufficient promise to be worth

pursuing further. Considerable work is needed but

the benefits are large, and we have shown that some

degree of success is assured.

Ackn o wled g emen t s T h e

financial support of the U.K.

Science and Engineering Research Council is gratefully

acknowledged. We also wish to thank Dr J. C. Ion for

helpful discussions, and Mr B. Butler for assistance with the

experimental work.

R E F E R E N C E S

1. D. A. Porter and K. E. Easterling,

Phase Transform

ations in Metals and Alloys.

Van Nostrand-Reinhold,

U.K. (1981).

2. I. J. Polmear,

Light Alloys.

Edward Arnold, London

(1981).



S H E R C L I F F a n d A S H B Y : O V E R V I E W N O . 9 0

1801

3. J . W. C hr i s t i a n ,

The Theory o f Transformations in

Metals and Alloys. P e r g a m o n P r e s s , O x f o r d ( 1 9 6 5 ) .

4 . A . K e l l y a n d R . B . N i c h o l s o n , Precipitation Hardening,

Progress in Material Science,

V o l . I 0 . P e r g a m o n P r e s s ,

London (1963) .

5 . A. H. Geis le r , in

Phase Transformations in Solids

(ed i ted

b y R . S m o l u c h o w s k i a n d W . A . W e y l ) . W i l e y , N e w

York (1951) .

6 . J . B . Newkirk , in

Precipitation from Solid Solution,

Am. Soc . , Meta l s Park , Ohio (1959) .

7 . L . M . B r o w n a n d R . K . H a m , i n

Strengthening Methods

in Crystals

( e d i t e d b y A . K e l l y a n d R . B . N i c h o l s o n ) ,

Chap . 2 . E lsev ie r . Amsterdam (1971) .

8 . F . R . N . N a b a r r o , Theory of Crystal Dislocations.

Oxford Univ . Press (1967) .

9. J. Fried el ,

Dislocations,

P e r g a m o n P r e s s , L o n d o n

( 1 9 6 4 )

1 0. P. H a a s e n , V . G e r o l d , R . W a g n e r a n d M . F . A s h b y ,

Proc. 2nd Acta -Scr ipta Int. Conf. , Decomposition o f

Alloys. the Earl) Stages. P e r g a m o n P r e s s , O x f o r d

( 1 9 8 3 )

11 . P . G. Shewmon,

Diffusion in Solids,

C h a p . 1 . M c G r a w -

Hi l l , New York (1963) .

12. R. A. Swalin,

Thermodynamics of Solids .

W i l e y , N e w

York 11962) .

13 . I . M. Li f sh i tz and V. V. S lyozov , J. Phys. Chem. Solids

19, 35 (1961).

14 . Z . Wagner ,

Z. Electrochem.

65, 581 ( in Germ an ) (1961) .

1 5 . G . W . G r e e n w o o d , i n

The Mechanism o f Phase Trans-

formations in Crystalline Solids .

I n s t i t u t e o f M e t a l s ,

London (1969) .

1 6. P . W . V o o r h e e s a n d M . E . G l i c k s m a n , Acta metall. 32,

2001, 2013 (1984).

17 . J . W. Mar t in ,

Microm echanisms in Particle-hardened

Alloys.

Cambr idge Sol id S ta te Sc ience Ser ies .

C a m b r i d g e U n i v . P r e s s ( 1 9 8 0 ) .

18 . R . Labusch ,

Physica status solidi

41, 659 (1970).

1 9 . U . F . K o c k s , A . S . A r g o n a n d M . F . A s h b y , Thermo-

dynamics and Kinetics o f Slip, Progress in Materials

Sciencr,

V o l . 1 9 . P e r g a m o n P r e s s , N e w Y o r k

(1975).

20 . E . A. S ta rke J r ,

Mater. Sci. Engng

29, 99 (1977).

21 . T . H. Sanders J r ,

Proc. Is t Int. Conf. Alu mi niu m -

Lithium Alloys, S t o n e M o u n t a i n , G a ( M a y 1 9 8 0 ) .

22 . J . C . Ion , K. E . Eas te r l ing an d M . F . Ashby , A c ta

metall.

32, 11, 1949 (1984).

2 3 . H . W e n d t , Z . L i u a n d P . H a a s e n ,

Proc . 2nd Ac ta-

Scripta Int. Conf. , Decomposition of Alloys: the Early

Stages.

P e r g a m o n P r e s s , O x f o r d ( 1 9 8 3 ) .

2 4 . W . A . A n d e r s o n , i n Precipitation from Solid Solution.

A m . S o c . M e t a l s , M e t a l s P a r k , O h i o ( 1 9 5 9 ) .

2 5 . H . R . S h e r c l i ff a n d M . F . A s h b y , C a m b r i d g e U n i v e r s i t y

E n g i n e e r i n g D e p a r t m e n t , T e c h n i c a l R e p o r t C U E D / C -

M a t . / T R I 5 6 ( 1 9 8 9 ) .

26 . H. K. Hardy ,

J . lns t . Meta ls

79, 321 (1951).

A P P E N D I X 1

N o m e n c l a t u r e

A0 = S o l u t e c o n c e n t r a t i o n c o n s t a n t ( w t % )

C ~ . . . C 4 = M a t e r i a l c o n s t a n t s , w h i c h a r e n o t e v a l u -

a t e d ( v a r i o u s u n i t s )

G = S h e a r m o d u l u s o f t h e a l l o y ( G P a )

K~ = C a l i b r a t i o n c o n s t a n t , c o n t r o l li n g t i m e

c o n s t a n t z j

P = T e m p e r a t u r e - c o r r e c t e d t i m e ( s / K )

P p , Pp ~ = P e a k t e m p e r a t u r e - c o r r e c t e d t i m e ( l o w a n d

h i g h t e m p e r a t u r e ) ( s / K )

P * = D i m e n s i o n l e s s t e m p e r a t u r e - c o r r e c t e d t i m e

Q A = A c t i v a t i o n e n e r g y f o r a g e i n g ( J / m o l )

Q ~ , Q , = F r e e e n e r g y o f s o l u t i o n ( h i g h a n d l o w

t e m p e r a t u r e s o l v i ) ( J / m o l )

R ~

s 0 , s 0 , =

T =

T 0 =

To Ts=

b =

C s

6 =

C I . . . C 4 =

f =

f0 =

I =

m , n =

U n i v e r s a l g a s c o n s t a n t ( 8 . 3 1 4 J / m o l K )

P r e c i p i t a t i o n s t r e n g t h p a r a m e t e r ( l o w a n d

h i g h t e m p e r a t u r e ) ( V P N o r M P a )

T e m p e r a t u r e ( ° C o r K )

T r a n s i t i o n t e m p e r a t u r e ( °C o r K )

S o l v u s t e m p e r a t u r e ( h i g h a n d l o w t e m -

p e r a t u r e ( ° C o r K )

B u r g e r s v e c t o r o f a d i s l o c a t i o n ( n m )

C o n c e n t r a t i o n o f t h e a l l o y ( w t % )

M e a n s o l u t e c o n c e n t r a t i o n i n t h e m a t r i x

( w t % )

M a t e r i a l c o n s t a n t s , w h i c h a r e n o t e v a l u -

a t e d ( v a r i o u s u n i t s )

V o l u m e f r a c t i o n o f p r e c i p i t a t e

F i n a l e q u i l i b r i u m v o l u m e f r a c t i o n o f p r e -

c ip i ta te

M e a n p a r ti c le s p a c i n g ( # m )

E x p o n e n t s o n f a n d r i n p a rt i c l e s h e a r in g

e q u a t i o n

r = M e a n p a r t i c l e r a d i u s ( p m )

t = Time (s)

t p = T i m e t o r e a c h a g e i n g c u r v e p e a k ( s )

t;r = In t r in s ic y ie ld s t reng th (VP N or M Pa )

a0 ~ = O v e r a g e d y ie ld s t r e n g t h ( V P N o r M P a )

trp = P e a k y i e ld s t r e n g t h ( V P N o r M P a )

t r q = A s - q u e n c h e d y i e l d s t r e n g t h ( V P N o r

M P a )

z , = E x p o n e n t i a l t i m e c o n s t a n t ( s )

A Gpp = N e t p r e c i p i t a t i o n s t r e n g t h i n c r e m e n t

( V P N o r M P a )

Atrss = Sol id so lu t ion in crem ent o f y ie ld s t rength

( V P N o r M P a )

AirA = P r e c i p i t a t i o n s t r e n g t h i n c r e m e n t ( p a r t i c l e

s h e a r i n g ) ( V P N o r M P a )

AirB = P r e c i p i t a t i o n s t r e n g t h i n c r e m e n t ( p a r t i c l e

b y p a s s i n g ) ( V P N o r M P a )

A P P E N D I X 2

S u m m a r y o f t h e C a l ib r a ti o n P r o c e du r e

T h e s t e p s i n t h e c a l i b r a t i o n p r o c e d u r e a r e s u m m a r i z e d

b e l o w f o r c l a r i ty . F i r s t , f o r t h e r e g i m e f o r w h i c h p e a k s a r e

ava i lab le :

( i) C h o o s e v a l u e s f o r t h e a s - q u e n c h e d s t r e n g t h , a q , a n d

the in t r ins ic s t reng th , t ry ; ca lcu la te the f i r s t es t im ate of a0a ,

m i d w a y b e t w e e n t h e t w o .

( i i ) E x a m i n e t h e a g e i n g c u r v e s , a n d r e a d t h e t i m e t o

r e a c h t h e p e a k , t p , a n d t h e p e a k s t r e n g t h , a p , f o r a s m a n y

t e m p e r a t u r e s a s p o s s i b l e .

( i i i ) Plot

log~(tp/T)

v s

1 / T

a n d m e a s u r e t h e g r a d i e n t ,

Q A / R ,

to g ive the ac t iva t ion energy QA.

( i v ) C a l c u l a t e P p , t h e a v e r a g e o f t h e v a l u e s o f t h e p e a k

t e m p e r a t u r e - c o r r e c t e d t i m e ( r e j e c t in g r o g u e v a l u e s ).

( v ) F o r e a c h t e m p e r a t u r e , e v a l u a t e t h e p e a k p r e c i p i-

t a t i o n h a r d e n i n g i n c r e m e n t , ( A tr pp ) p , a n d h e n c e e x p e r i m e n -

ta l va lues for So .

( v i) P l o t S g v s t e m p e r a t u r e ( a s i n F i g . 3 ), a n d e s t i m a t e

Ts.

( v ii ) E s t i m a t e ( S O ), ~x f r o m t h e l o w t e m p e r a t u r e p l a t e a u ,

a n d s o l v e f o r Q s u s i n g a l l t h e d a t a .

( v ii i ) F o r e a c h t e m p e r a t u r e , c a l c u l a t e tr0 ~ m o r e a c c u r a t e l y

us ing the fu l l equa t ion , then (At rppt ) (=S02) ; ad jus t Ts ,

( so)max and Q~ and rep lo t So vs T . Rep ea t un t i l the f i t

b e t w e e n t h e o r y a n d d a t a i s s a t i s f a c t o r y .

( i x) A d j u s t t h e c o n s t a n t K ~ s u c h t h a t t h e s o l id s o l u t i o n

p a r t d e c a y s w i t h a n a p p r o p r i a t e t i m e c o n s t a n t z I ( s c al e d b y

P p a n d T ) .

F o r i n c o m p l e t e c u r v e s w i t h o u t p e a k s a b o v e t h e s o l v u s ,

t h e f o l l o w i n g a d d i t i o n a l s t e p s a r e m a d e :

( i) U s e t h e s a m e v a l u e s f o r Q A a n d K I a s f o r t h e o t h e r

reg ime.



1 80 2 S H E R C L I F F a n d A S H B Y : O V E R V I E W N O . 9 0

u' )

u ' )

u. I

z

r,t"

<

" r

,,,,.

<

w

o. .

L O W T

R E G I M E

T R A N S I T I O N

~ TEMPERATURE,Tc

. . . . . . . ~ t R GM

i - Z

S - Q U E N C H E D

H A R D N E S S ~ /

T s T e

T E M P E R A T U R E - . - - -

Fig . A 1 . Schem atic d iagram of the va r ia t ion in theore t ica l

peak hardness vs tempera ture pred ic ted by the low and

high tempera ture ca l ibra t ions of the model . The por t ion

of each curve used ( shown so l id) is tha t which g ives the

higher va lue , with the t r ans i t ion tempera ture , To , be ing

the poin t where the two ca l ibra t ions pred ic t the same

value .

( ii ) F ind the so lvus tem pera ture To f rom the phase

diagram.

( ii i) Es t imate Qe f rom a h igh tem pera ture m easuremen t

of overaged s t rength .

( iv) Use (S0)m~ and Pp as initial estimates for (S~)max

and P ~ respectively, th en adjust these param eters an d Q~ by

tr ia l -and-er ror to f i t the ava i lab le par ts of the age ing curves.

(v) Select a transition temperature between the regimes.

A cr i te r ion for th is t r ans i t ion is the tempera ture where the

two reg imes predic t the same peak s t rength , as shown

schematica l ly in Fig . A1 ( the so l id par ts o f the curves show

the operative regime). The transition temperature, To, lies

s l igh t ly be low Ts . At the t r ans i t ion tempera ture the two

models pred ic t the same peak s t rength , bu t there is a

d iscont inui ty in s t r ength away f rom the peak due to the

dif fe ren t ca l ibra t ion va lues ; i t is apparent in the iso-y ie ld

diagram presented in the accom panying paper . In

rea l i ty the s t rength wil l vary smooth ly , p robably by mixed

prec ip i ta t ion .

The ca l ibra ted va lues may require fur ther small changes

when the theore t ica l curves a re compared with the ex-

perim ental da ta. A sensitivity analysis [25] helps in this

f ine- tuning process by ident i fy ing the sense and magnitude

o f th e c h a n g e b r o u g h t a b o u t b y s ma l l a d jus tme n t s o f e ac h

cons tan t in turn .

1990 shercliff (actamet) a process model for age hardening of

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