Relationship between extrinsic stacking faults and mechanical twinning in F.C.C. solid solutions with low stacking fault energy

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Relationship between extrinsic stacking faults and mechanical twinning in F.C.C. solid solutions with low

stacking fault energy

J.F.M. Vergnol, J.R. Grilhe

To cite this version:

J.F.M. Vergnol, J.R. Grilhe. Relationship between extrinsic stacking faults and mechanical twinning in F.C.C. solid solutions with low stacking fault energy. Journal de Physique, 1984, 45 (9), pp.1479-1490.

�10.1051/jphys:019840045090147900�. �jpa-00209887�

Relationship between extrinsic stacking ^faults and mechanical twinning

in F.C.C. solid solutions with low stacking ^{fault energy}

J. F. M. Vergnol and J. R. Grilhe

Laboratoire de Métallurgie Physique (*), 40, ^{avenue du Recteur} Pineau, ⁸⁶⁰²² Poitiers, ^France

(Reçu ^le 18 janvier ^1984, accepti ^{le 12 avril} 1984)

Résumé. 2014 Une analyse ^détaillée de la nucléation et de la croissance des macles, applicable ^aux matériaux C.F.C.

à basse énergie ^{de défaut} 03B3, aboutit à la conclusion suivante : la nucléation des macles n’est possible que si des conditions déterminées, relatives à l’orientation, ^{au sens} de la contrainte appliquée, ^{au taux de} déformation, ^sont

satisfaites. L’étude de ces conditions et de leur interdépendance ^montre que le maclage mécanique ^est gouverné

par la nucléation et le développement ^{de défauts} d’empilement ^{de nature} extrinsèque. ^{Le calcul de} l’énergie ^néces-

saire à la nucléation des macles aboutit à une relation quantitative ^{entre la taille} des macles et l’énergie ^{de défaut.}

L’analyse ^{est effectuée dans le cas où l’axe de traction} (ou ^de compression) ^est [100] ^ou [111]. ^Les hypothèses ^sont

en bon accord avec les résultats expérimentaux ^{relatifs à de} nombreuses solutions solides C.F.C.

Abstract. ²⁰¹⁴ A detailed analysis ^{of twin} nucleation and growth ^is performed ^{for the case of} face-centred cubic materials with low stacking fault energy 03B3. It is found that twin nucleation _may occur only ^{if certain conditions} concerning ^the crystallographic orientation, sign ^{of the} applied ^stress and deformation level, ^are satisfied. Each of these conditions is analysed ; ^their interdependence ^{leads to} the conclusion that the occurrence of twinning ^is

controlled by the nucleation and the growth of extrinsic stacking faults. This analysis concentrates on cases where the tensile (or compressive) axis is either [100] ^or [111]. ^The discussion of the _energy of twin nucleation leads to a quantitative relation between twin size and stacking fault energy 03B3. These assumptions ^{are in} good agreement with experimental ^{data on} many face-centred cubic solutions.

Classification

Physics ^Abstracts

62.20F

1. Introduction.

From the first theoretical

analysis

of twin nucleation in face-centred cubic materials

by

Cottrell and

Bilby [1],

mechanical

twinning

^was considered as

being improbable,

because of

geometrical

^{reasons deduced}

from the

crystallography

^{of this} structure. However, after the first identification of deformation twins in

single crystal

copper

(Blewitt,

Coltman and Redman

[2]),

many

experimental

observations have shown that mechanical

twinning

^is

quite possible

in face-centred cubic metals or

alloys,

^{if their}

stacking

fault energy y is low enough

(Venables, [3]).

But

twinning

mechanisms are not yet

completely

understood, ^and many

experimental

results differ from the theoretical

predictions

^of

Mahajan

and Williams

[4]. ^Various aspects ^{of twin} nucleation, the kinetics of twin

propagation,

^the critical shear stress for

twinning

and the relation between

slip

^and

twinning during

deformation are still under discussion.

The _purpose of the present ^{work is to}

provide

^a

(*) L.A. 131 du C.N.R.S.

detailed

analysis

of the conditions which allow the nucleation and the

growth

of mechanical twins,

without _any

hypothesis

^{about the different}

possible

mechanisms which _may

produce

the twin nucleus.

Starting

^from

experimental

^{data on}

single

crystal

copper

alloys

which have been

subject

^{to stress under}

widely varying

conditions

(Vergnol

^{and Villain,}

[5];

Vergnol [6])

^and

following

^{a scheme}

previously

sug-

gested (Fontaine [7]),

^we have divided the

twinning

mechanism into three stages :

-

development

^{of an initial} large

stacking

fault,

- nucleation of a microtwin

by superposition

^of

extended faults in the

neighbouring planes,

- extension of this nucleus

by propagation

^{of the}

twin

boundary through

the strained material.

We have

analysed

^{the nature} of the faults

(intrinsic

or

extrinsic)

^induced

by

deformation in relation to three variables :

-

crystallographic

direction of the

applied

stress,

- sign ^{of the stress}

(tension

^or

compression),

-

origin

^{of the} partial dislocations

bounding

^the

faults.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045090147900

Thus, ^{we have}

computed

^the energy

required

^to

nucleate a twin, which varies with the nature of the

stacking

faults. This

computation

indicates that the nucleation and then the

growth

of twins is favoured

or

impeded by

^{the nature} of the faults which are formed with

increasing

^strain.

Finally,

^{we have}

compared

the conclusions of this

analysis

^with

experimental

^data,

especially

^{our data}

for Cu-Al

alloys.

2. Formation and

growth

^{of an extended}

stacking

^faults

Any perfect

dislocation with Burgers ^{vector b} may be dissociated into two

partials,

^for

example

^two

Shockley

partials

b1

^and

b2.

^The

applied

^{stress a}

generally

exerts on the

partials

^{two different} forces which deter- mine the sense of

displacement

^{of each} dislocation and

consequently

^{the nature of the}

stacking

^{fault ribbon.}

Moreover, for low

enough

values of S.F.E. y, the width of this ribbon _may increase

enough

^{to reach}

values

giving

^an extended fault bounded

by indepen-

dent

partial

dislocations. The

density

^of

independent partial

dislocations and therefore the

density

^{of extend-}

ed faults increases with the strain. This facilitates different mechanisms of twin nucleation. We have examined the

possible configurations

for tensile and

compressive ^stresses

applied along 1 1 1 > and

¹⁰⁰ ),

which are two stable

crystallographic

directions.

2.1 APPLIED STRESS ALONG

A 111 >

DIRECTION. ^-

Referring

^{to the}

Thompson

tetrahedron of

figure

1 a, the

glide planes

^are

(a), (b)

^and

(c)

^for

perfect

^dislo-

cations with Burgers ^vectors AD, ^{BD and CD.}

2.1.1 Tensile

applied

^stress.

a) Dissociated dislocations

gliding freely

^{in their}

slip plane.

^{Let us} consider the dislocation loop ^BD

after dissociation :

the resultant forces on the

partials

per unit

length

^are

(Fig.

^{1 a, and}

1 b) :

Fig. ^{1. - Glide} planes ^and dissociated dislocations with a

tensile stress in the ( 111 ) direction, referring ^{to the} Thomp-

son tetrahedron.

We note that the two partials ^{aD and Ba}

glide

ⁱⁿ

the same direction. The tensile stress a makes the diameter of the dislocation loop ^BD increase with stress.

The

leading

partial ^is aD, ^because

faD

⁼

2fB.,

^and

behind is formed a ribbon of intrinsic fault, ^{but the}

stacking

fault is removed by ^the partial ^Ba.

b) ^Reaction

junction

^{in a}

glide

plane.

For example, ^{such a reaction} may ^occur between BD

(a)

^{and CD}

(a)

dislocation

loops

dissociated in the

glide plane (a) :

The two underlined

leading partials

^{are attractive}

because

they

^{have the same} Burgers ^{vector and} oppo- site

sign (Fig. 2b).

After their annihilation, ^{the intrinsic}

fault surface is delimited

by

^the

partial

dislocations Bat and Ca which are

repulsive,

^{but the}

applied

^stress

a exerts

opposite

^{forces on them.} Hence, ^{a reaction} is

possible,

either for a sufficient value of a, ^or with thermal activation

(we

^note that the energy

required

to

temporarily

^{form the}

perfect

dislocation

BC(a) by

recombination is the same as the energy for ^cross-

slip) :

where a is the lattice parameter. ^The

applied

^{stress a}

has no effect on the

perfect

dislocation

BC(a),

^{but it}

exerts

opposite

^{forces on the}

partials

^{Boc and Ca which}

bind an extrinsic fault Moreover, ^these

partials

^are

repulsive

^{and each of them can react} with the internal

partial

of the initial loop

(Fig.

2, stages ^{d and}

e)

^to

give

^{a new}

partial

^aD.

Fig. ^{2. - Reaction between two} dislocations initially ^dis-

sociated in a type (a) plane. ^An extrinsic fault takes place by dissociation of the junction dislocation BC.

Hence, ^{the final}

configuration resulting

^{from a}

junction

reaction in a

glide

plane ^{is a}

ring

of intrinsic faults

surrounding

^a large ^area of extrinsic

stacking

faults which widens with

increasing applied

^stress

(Fig.

2, stage f).

c) ^{Reaction on} the intersection of two

glide planes.

Two kinds of reaction _{may occur,}

resulting

^{in the}

nucleation of an extrinsic fault

The

first possibility,

^for example ^between type

(a)

and

(b) glide

planes, ^{leads to an extrinsic}

stacking

fault in a type

(a)

plane.

The

following

^reactions

(with

^{the two}

leading partial

dislocations

underlined)

^are consistent with

figure

^{3 :}

The

leading partial

dislocations aD and

DB

^are

attractive and their reaction

gives

^{rise to the stair-rod}

czp (Fig. 3b).

^This stair-rod, repulsive ^{for the}

partial

dislocation Ba but attractive for the

partial RC,

^reacts

with the latter and

gives

^the

partial

dislocation aC.

Hence, ^{we find}

again

^the

previous configuration

with two

repulsive Shockley

dislocations on both sides of an intrinsic fault

By

recombination,

they

may

reverse their mutual

positions (Fig.

3d) ^and

give

^rise

to an extrinsic fault which extends with

increasing a :

As

previously,

the last recombination stage needs ^an activation energy.

The second

possibility

results in the formation of a

Cottrell barrier between type

(a)

^and

(b) glide planes (Fig.

4, stage

a).

However, ^such barriers may be des-

troyed,

^either

by

the action of dislocations

piled-up

in

glide planes

behind the barrier, ^or

by

^{thermal acti-} vation, ^as in the reaction

(2) :

Figures

^{4b and 4c summarize the two}

possibilities

which result in the extension of two « half loops » ^of

extrinsic fault in the initial

glide planes.

^{In the first}

case, the Frank dislocation BB

lying along

^{the line CD}

of the Thompson tetrahedron, is sessile and

gives

rise,

by

dissociation, ^{to a first extrinsic} fault surface deli- mited

by

^the

Shockley partial

^Ba

(Fig. 4b),

^{then to a}

second one

by

^a similar dissociation of the Frank

partial

^aA.

The second

possibility

^{is the} recombination of the

perfect

dislocation BA, which,

by

dissociation, gives

the same configuration ^as

previously.

We note that the _energy needed for reaction

(2)

^is

a2/18

whereas it is

a2/12

for reaction

(1).

Thus, ^{it is} easier to

destroy

^{a Cottrell} barrier than to form a

perfect

dislocation from two

Shockley

dislocations,

Fig. ^{3. - Reaction on the} intersection of type (a) ^and type (b) slip planes ^{with a tensile stress in} the 111 > ^direction.

Fig. ^{4. - Formation of «} half-loops >> ^of stacking ^faults

from a Cottrell barrier, ^{with a tensile stress in the} ( 111 >

direction. After stage a, two possible ^different reactions lead to the same final configuration.

and such destructions ^{are more}

likely

^{to occur.}

Finally

it _appears that a tensile stress

applied

^{in the} ( 111 )

direction

gives

^{rise to} intrinsic faults in the initial stage of deformation. But as soon as these faults _grow, all the interactions between them nucleate extrinsic faults which extend

easily

^with

increasing

^{tensile stress.}

2.1.2 Compressive ^{stress. - When the}

sign

^{of the}

applied

^{stress is} reversed, the resultant forces ^on the

perfect

^and

partial

dislocations are also reversed This

means that the

partial

dislocations are

exchanged

ⁱⁿ

any dissociation, ^and

consequently

^the

stacking

^faults

formed are of

opposite

^{nature. From this evidence,}

it is easy ^to show that, ⁱⁿ every case, the

previous

conclusions for

elongation

tests must be reversed for the case of

compression

^tests.

Consequently,

^a

compression

^{test in the} ( 111 )

direction

gives

^{rise to} extrinsic faults in the first stage of deformation, and reactions between these faults nucleate intrinsic faults which extend with

increasing applied

^stress.

2.2 APPLIED STRESS IN THE (

100 >

DIRECTION. ^-

As for the ( 111 ) direction whe have made ^a detailed

analysis

which is summarized here. The detailed reac-

tions and

figures

^are

given

ⁱⁿ

appendix

^1.

2 . 2 .1 Tensile stress.

1. Dissociated dislocations

gliding freely

^{in their}

planes :

^{the two}

Schockley partials

^{move in the same}

direction with a narrow ribbon of extrinsic fault between them.

2. Junction reaction in ^a

glide plane :

this results in the nucleation of an intrinsic fault which _grows with

increasing

^stress.

3. Reaction on the intersection of two

glide planes :

we have shown that if

only

^one dislocation is not

dissociated, ^{or has been} recombined, this results i4

the nucleation of ^an intrinsic fault

2.2.2

Compressive

^{stress. -For the} same reasons as

given previously,

^the above conclusions ^are reversed,

i.e.

compressive

^stress

develops

extrinsic faults.

2.3 CoNCLUSION. ^- The nature of the

stacking

^faults

induced in F.C.C.

crystals by

^an

increasing applied

stress on

( 111 )

^or

100 >

directions is as follows : 1. Stress in

the ( 111 )

direction :

- initial faults nucleated at the

beginning

^{of the} deformation, before any interaction between

gliding

dislocations, ^{are :}

+ intrinsic for ^a tensile test,

+ extrinsic for a compressive test,

these faults widen

slowly

^with

increasing

stress;

- in the next stages ^{of the} deformation, ^{the reac-}

tion either between

coplanar

dislocations, ^{or on the} intersection of

glide planes, gives

^{rise to new faults;}

for the most part, ^{the nature of these is :}

+ extrinsic for a tensile test,

+ intrinsic for ^a

compressive

test;

2. With ( 100 ) ^direction these conclusions are

reversed

3. Computation ^{of the} energy for twin nucleation The

following

computation

points

^out the influence of the nature of

stacking

^{faults on}

twinning

nucleation.

We consider an idealized twin with a

cylindrical shape.

^The planar ^{surfaces are coherent} boundaries,

for

example

^two

stacking

^{faults in}

plane (a),

^{that we}

name twin « faces ». The

cylindrical

surface, ^named

twin « front », ^is

composed

^of

piled-up

partials

(as

B0153, Ca ^or

Da) surrounding stacking

^{faults. The}

crystal

is

perfect

inside and outside this

cylinder, bounding

a twinned volume of matrix, and the energy stored is in consequence localized

only

^{on the surface of this}

cylinder.

3.1 STORED ENERGY ON THE TWIN FRONT. - We consider

stacking

faults of the ^same nature

(intrinsic

or

extrinsic)

^on

contiguous planes

^of type

(a)

^sur-

rounded

by partial

dislocations

(B0153,

^{Ca or}

Da) piled-

up in a

regular

array. Three

typical

^{cases are}

possible :

1. Intrinsic faults bounded

by

^{identical sets of three}

different

partial

dislocations.

This structure

(Fig. 5a)

^is

typical

ⁱⁿ

recrystallised

materials. The stored energy is low because each

triplet gives equivalence

^{with the}

perfect crystal :

2. Intrinsic faults bounded

by

^identical

partial

^dis-

locations.

This type of twin front

(Fig. 5b)

^is

promoted

^when

one

slip

system is activated

by

^the

applied

^{stress. The}

interactions between

piled-up

dislocations are at a

maximum, and the stored _energy is

proportional

^to b2.

3. Intrinsic faults bounded

by

^alternate

partial

^dis-

locations. This last structure is

possible

^{with two}

slip

systems

being

activated The

equivalent crystallo- graphic configuration

^is

given by

extrinsic faults

piled-up

^{at a distance of two}

planes,

^{since we have}

(Fig.

5c) :

We notice that

(Fig. 5d) :

The energy of the

dipole (b’, - b’)

^{can be}

^neglected

The

partial

dislocations

b3

^{= aD are}

equivalent

^to

imperfect

dislocations with

b" - 1 ^{b 3 Burgers}

^vector,

piled-up

^on

contiguous planes,

^{and the} stored energy is

proportional

^to

b2/4. Consequently,

^{the stored}

energy ^on the front of two identical twins is propor- tional to :

b2 with intrinsic faults

piled-up

^on

contiguous

planes,

Fig. ^{5. -} Structure of a twin front composed ^of piled-up

dislocations on type (a) planes. ^Three typical arrangements (a, b, c) ^of partial dislocations are drawn with two equivalent possibilities ^for (c) arrangement.

b2/4

with extrinsic faults

piled-up

^{on alternate} planes, ^{where b} =

b1

^{I =}

b2 = b3 .

Except ^{for the}

configuration (5a)

which is the most

interesting

but also the less

probable

^with

only

^one

or two

glide planes,

^{the last}

configuration

^{is the one}

which

requires

the least energy.

3.2 TOTAL ENERGY FOR TWIN NUCLEATION. ^- The final _energy for the nucleation of a twin

depends

^also

on the nature of the

piled-up stacking

faults. We shall

successively

examine the ^cases for the 111 > ^and 100 ) directions.

At first we determine the critical stress for a twin nucleus with extrinsic or intrinsic faults.

3 . 2 .1 Critical ^{stress on} the twin front. ^{- The critical}

stress (Ie for the

growth

^{of a} twin nucleus is reached when

equilibrium

^is achieved between the resolved force

f i on

^{the unit}

^length

^{of each} dislocation loop

constituting

the front of the nucleus, ^{and the back}

force

f2

resulting ^{from the} energy stored on the sur-

face of the nucleus

during

^its

growth.

^{This energy}

is

equivalent

^{to a} surface tension E and the back

lbrce on the unit surface of a

cylindrical

^{nucleus with}

radius r is :

In the

following

computation, ^we

only

consider the energy stored at the front of the twin nucleus

(for

^the

computation

with the energy of the total surface, ^see

§ 3.3.1).

From

previous

results of Kroner [8] and de Wit

[9],

we may write the energy of the twin front ^as follows :

where G and v are

respectively

^{the shear} modulus and the Poisson ratio, h the thickness of the twin, p the

density

^of dislocations

(with

Burgers ^vector

b)

^{on the}

front of the twin. If this front is constituted of n

piled-

up dislocations, ^{we have} p ⁼

n/h

and therefore :

The critical stress (J c is

proportional

^{to the back force}

f2

^{on the unit}

length

of dislocation at the front of the

twin, and this back force is

J2

⁼

F/n,

^{hence we} may write :

where A

depends

^on the radius r.

If we

neglect

the energy of the

dipoles (bi, - b’),

the

configuration given by

^{intrinsic faults}

piled-up

^on

contiguous planes

^is

equivalent

^{to that}

given by

extrinsic faults

piled-up

^{at a distance of two}

planes,

since each

configuration

^is

equivalent

^to dislocations with

Burgers

^vector

b/2

^on

contiguous planes (Fig. 5c).

Thus, the critical stress (J c does not

depend

^{on the}

nature of the

stacking

^faults

building

up the twin nucleus.

3.2.2

Applied

^{stress in the} 111 > direction. ^- The

following computation

may be

applied

^{both for}

the cases of a

compressive

^{or a tensile stress in the} ( 111 ) ^direction.

Let us consider a twin on n

contiguous planes.

^If

it is

composed

of ni intrinsic faults bounded

by,

^for example, ^{the same}

partials

aD, ^we _{have ni} ^{= n and} the formation of this twin under the

applied

^{stress a}

induces a variation of

length hi

^{of the} crystal, ^where

(Fig.

1) :

The expended energy is :

If the same twin is

composed of ne

extrinsic faults bounded

by

^{the same}

partials (Ba

^or

Ca)

^{we have}

now ne ⁼

n/2,

^{and with}

these ne

^{faults, the}

changes

in length

he

^{and energy}

We

^{are :}

But a’ D ⁼ 2 6 a’, ^and

consequently :

Thus,

twinning

^{on n} planes ^{induces a} variation of the

length

which is four times smaller for extrinsic faults

as

compared

with intrinsic faults.

Alternately,

^{a defi-}

nite change ⁱⁿ

length

is induced

by

^a twin four times

as thick if

piled-up

^{faults are} extrinsic. We note that the

expended

energy is

nearly

^{the same for both cases,}

because the

energetic density

^{on the} twin front is four times as small with extrinsic faults.

As a result, ^{the nature of} the constitutive faults, ⁱⁿ

twins induced

by

^an

applied

^{stress on the} ( 111 >

direction, ^{cannot be}

predicted

from the consideration of

expended

energy of twin

growth.

However, ^{we note} that the twin nuclei with extrinsic faults will reach their critical size (for ^{a critical stress}

(J c)

before the other nuclei, ^{if we assume that this}

critical size is not

dependent

^{on the nature of the}

faults. Since further growth of the « extrinsic .nuclei » may ^occur for stresses lower than _a,

(Friedel [10]),

it can be

predicted

^that

only

^twins

composed

^of

extrinsic

faults

^will appear

during plastic

strain in the

I I I > ^direction.

3.2.3

Applied

^{stress in the} 100 > direction. ^- A similar

analysis (see appendix 2)

^{for two} twins of the

same thickness,

composed

either of ni intrinsic faults,

or

of ne

⁼

n; j2

^extrinsic faults,

gives

^the

following

relation between the

change

ⁱⁿ

length hi

^or

h.,

^{and the}

energy

W

^or

W e :

In consequence, the ^same energy ^W allows

twinning

on n

planes,

^either with ni ^{= n} intrinsic faults or

ne ⁼

n/2

^{extrinsic faults, with the same}

change

ⁱⁿ

length

^h

(for

^both tensile and

compressive stresses).

But the

energetic density

^{on the} twin front is four times

as low with extrinsic faults.

Consequently,

^for

plastic

strain in

the 100 >

direction, ^the

expended

energy promotes

twinning

^{with extrinsic faults. For a different}

reason this conclusion is the same as that with the stress in

the 111 >

^direction.

3.2.4 Conclusion concerning ^the occurrence

of ’ twin-

ning. ^{- The}

energetic

^{cost for the formation of a twin}

nucleus was determined

by

^the computation ^{of two}

typical

parameters :

- the

energetic density

^{on the twin} front,

- the thickness of the twin, ^{connected to the shear}

of the

crystal.

It appears that for the four cases considered

(tension

and

compression

ⁱⁿ the 111 ) ^and ( 100 )

directions)

twins are

preferentially

formed with extrinsic faults :

- in the 111 ) ^direction because, ^{for the same}

expended

energy, the critical size of the nucleus is more

rapidly

attained,

- in the 100 ) ^{direction because, for the same}

twin size the energetic ^{cost is lower.}

The greatest

probability

^for

twinning

^occurrence

corresponds

with the strain conditions which induces extrinsic faults in the

crystal.

^{Table I summarizes}

these conclusions.

3.3 MINIMIZATION OF THE ENERGY OF A TWIN NUCLEUS.

- Up ^{to this}

point,

^{we have}

compared

twins which

only

^differ

by

^{their thickness and the nature of their}

constitutive

stacking

faults, ^{and we took}

only

^into

account twin front energy. We ^now compute ^{the total} energy of ^a twin nucleus

composed

with extrinsic

faults,

according

^to the above conclusions.

The minimization of this term will

give

^the

optimum

distribution of the total energy between the twin front and the faces for each stage ^{of the}

growth.

According

^{to the}

hypothesis

^{of Sumino} [11], ^{we consi-}

der that this minimization determines the

dynamic equilibrium

of the twins

during

deformation at cons- tant strain rate.

3.3.1 Qualitative

features of

^{the model. - Let us}

consider a circular

loop (radius r 1)

^of

partial

^dislo-

cation

bounding

^{an extrinsic fault}

(area Sl, S.F.E.y).

The dislocation

loop

energy is a function

F(rl).

^The

total energy

Ei

^{of the set «} dislocation loop ⁺

stacking

fault » _may be written as :

Table I. ^-

Effect of

^{a tensile or} compressive ^{stress o}

applied

^{in the} 100 > or 111 > direction, ^{on the nature}

of

^nucleated

faults,

^and

resulting

twinning occurrence.

The variation of

sample

ⁱⁿ

length,

^induced

by

^the

growth

^{of this}

loop

^{is the same as that induced}

by n

identical loops,

coaxially piled-up, bounding

^{a stack-}

ing

^{faults, with}

Sn

⁼

S1/n and r.

⁼

r,l.,,In-.

^{The total}

energy

En

of this twin nucleus,

composed

^{of these n} faults, ^{is the} energy of the two faces

(with

^{a surface}

density y/2)

^and of the front :

With a determined value rj, i.e. for ^a definite strain of the

crystal,

the first and the second terms are res-

pectively decreasing

^and

increasing

^{functions of n.}

Then En

^{takes a minimum value}

EN

^{for a definite value}

N of n, ^as shown in

figure

^6.

3.3.2 The

physical

meaning ^and computation

of

^N.

Influence of

the S.F.E. _y. ^- From a mechanistic

point

of view, ^the

growth

^{of one extrinsic}

stacking

^fault

S,

is

exactly equivalent

^{to the}

growth

^{of n} identical extrin- sic fault loops, ^{such as}

Si

⁼

nS,,,

^{for all}

possible

^values

of n. But the

energetic

^cost varies with n and the most favourable mechanism is realized when n ⁼ N. Then,

starting

from its critical size, determined

by

^{N, nucleus}

twin

growth

^takes

place by

simultaneous extension of the N loops, ^{each of} them in its own

glide

plane.

The thickness of such a nucleus is 2

N d111,

^where

d 111

1 is the distance between the

{ 111 } planes

^{in the}

F.C.C. structure. Then,

assuming

^{that :}

we condense n

partial

dislocations b into one dislo- cation

(nb),

^and write, with G shear modulus :

With

aEn/an

^{= 0, we find :}

Fig. ^{6. -} Energy En ^{of a} twin nucleus vs. the number ⁿ of piled-up stacking ^faults.

A more accurate

computation (see appendix 3)

^based

on

previous

^results

(de

^Wit

[9])

^{leads to a relation}

which is

nearly

^{the same as that obtained with the}

above

approximation :

Figure

7 shows the

corresponding

^{curves. The most} important ^feature of this result is that the critical value N increases with y/G, i.e. with the S.F.E. _y

(generally

the modulus G does not vary very much in

comparison

^with

y).

Consequently

^we expect ^a

significant

^{variation of}

the thickness of twins when the S.F.E. _y varies

signi- ficantly.

3. 3. 3 Conclusions concerning ^the energetic ^cost

of

^the

twin nucleation. ^- These conclusions _may be summa-

rized in two

points :

a) whatever the conditions of nucleation

(sense

^and

direction of the

applied

^stress

a),

^the

energetic density

of the twin front is a minimum when the

piled-up

dislocations

constituting

this front delimit extrinsic

faults;

b)

^a definite shear _may be related to a

single

twin,

formed

by

^{a number n} of extrinsic faults which _may vary. The total _energy of this twin is lowest when n is

equal

^{to a} definite value N which

depends

^{on the}

material through ^the

S.F.E. y,

the modulus G, ^the Burgers ^vector b. This value N increases with the S.F.E. _y. In the next

paragraph,

^we compare these theoretical

predictions

^{with different}

experimental

results.

Fig. ^{7. -} Computation of the energy En ^{of a twin vs.}

number n of extrinsic stacking ^{faults and} stacking ^fault

energy y.