On calculations of the crss of alloys containing coherent precipitates

ON

CALCULATIONS

OF THE CRSS OF ALLOYS COHERENT PRECIPITATES* V.

CONTAINING

GEROLDT

Two recent estimates on the critical resolved shear stress of alloys containing coherent precipitates are critically compared. The differences between both estimates are discussed in detail. In both cases t)he interaction between dislocation and precipitates is assumed to occur via the elasticstrain field surrounding t,he coherent particles. Both estimates are performed in three steps: one calculates (i) the maximum repelling interaction force of the coherency strain field on the dislocation as a function of the position of (ii) The average distance I; between repelling strain fields along the dislocation line; and the slip plane; It is shown that the most. important difference between both (iii) there follows an averaging procedure. estimates is introduced by the averaging procedures. CALCCL

DE

LA

CISSIOX

CRITIQUE DES

RESOLUE

PRECIPITES

POUR

DES

SLLIAGES

CONTEXAST

COHERENTS

Deux &valuations r&entes de la cission critique r&solue pour des alliages contenant des pr&ipit& coh&ent,s sont soigneusement comparBes en vue d’une Etude critique. Les diffbrences entre les deux &valuations sont disc&es en d&ail. Dans les deux cas on suppose que l’interaction dislocations-p&&pit& se produit par l’interm&diaire du champ de deformation Qlastique entourent les particules cohhrentes. Les deux &ah&ions comprennent trois stades: on calcule (i) d’abord la force d’interaction repulsive maximum exe&e par le champ de coh&zion sur la dislocation, comme une fontion de la position (ii) ensuite la distance L moyenne entre les champs de contraintes de &p&ion 1e du plan de glissement; L’auteur montre que (iii) enfin suit une n&thode de calcul des moyennes. long de la ligne de dislocation; la diffhrence la plus importante entre les deux &valuations vient des m&hodcs de calcul des moyenn(~s. irBER

DIE

BERECHSUNG

DER MIT

KRITISCHEN

KOHBRENTER

SCHUBSPANNUNG

T’OS

LEGIERVSGES

AUSSCHEIDUNG

Es werden zwei Abschktzungen iiber die kritische Schubspannung 70 von Legierungen mit kohiirent,er Ausscheidung miteinander verglichen und ihre IJnterschiede eingehend diskutiert. In beiden Fiillen wird angenommen, dalj die Wechselwirkung zwischen Versetzung und Ausschaidung iiber das elastische Spannungsfeld erfolgt, das die kohiirenten Teilchen umgibt. Beide Abschhtzungen werden in 3 Schritten durchgefiihrt: Berechnung (i) der maximalen abstoI3enden Kraftwirkung K des koh&renten Spannungsfeldes auf die Versetzung als Funktion der Gleitebenenlage, (ii) dos mittleren Abstandes L zwischen abstoBenden Spannungsfeldern liings der Versetzungslinie und (iii) des Mittelwertes von T,,. Es wird

gezeigt, da13 die groDen Unterschiede Mit,telwertbildung hervorgerufen werden.

zwischen

1. INTRODUCTION

In a recent publication discusses stress

the increase

(CRSS)

two-phase

due

alloy.

to

of the critical

resolved

rewritten

in a slightly

the difference

and

between

st,rain fields

in a

strained coherent

different

way.

particle

G is

value

parameters

of

of the

R is the

and the matrix.

average radius of the spherical precipitated particles and f is their volume fraction. b is the length of the Burgers

vector.

formula

has

The line t,ension T of the original

been

taken

as G. b2/2. Equation

ACTS

dependence Authors

METALLITRGICA,

VOL.

16, JCNE

1963

between

both

formulae

is the

ArO

with the volume fraction. I used a straight line approximation in their

estimate. Contrary to this Gleiter claims this approximation to be invalid. He used an optimising method

for his estimate

II.

Therefore,

one might

conclude that the differences of equation (1) and (2) are caused by these approximations. The purpose of tions result in a AT~ that differs only by a constant numerical

Institut

of

(2)

the present paper is to prove that both approxima-

(1)

*

Received October 25, 1967. t Max-Planck-Institut fiir Metallforschung, Metallkunde, Stuttgart, Germany.

Their result

AT,’ = 3G&3’2f1’2(R/b)1’2 The main difference

(1)

F is the positive

the lattice

untcrschiedliche

was

His result is

bhe shear modulus

durch

kornt2) (referred t,o as I in this paper).

shear

ATO” = 11 .SG,3’2fj’“( R/6)“2 when

Abschlltzungen

differs from a formula derived by Gerold and Haber-

Gleiter(l) (referred to as II) coherency

beiden

dependence

fiir

factor.

method of averaging. 823

The

obvious

on f, however:

differences

in the

have t,heir cause in the

L1C’l’X

824 2.

THE

GENERAL

THE

The

CONCEPT

of the

CRSS

interacting

dislocation

and

different If

the

syst’em:

the

at

force

K

the

strain

field.

used

particle

center

of

slip plane

a

as

with

gliding Here,

discussed

radius

Cartesian

of the

of three

between

are

spherical

the

R is

coordinate

dislocations

is taken

to be z = z,,. The general direction of the dislocation line is parallel to the y-axis. In the following discusIt’s is assumed. sion, an edge type dislocation position

at’ maximum

flmct’ion X,(y).

interaction

The force K(z,)

K(z,)

= b

is given

surrounding

the

by

the

is then defined as

-z TJ-UY), s -- m

where 7i is the internal strains

(3)

Y> %I dY

for

r2 < R2

xR% Ti(2,.Y, z) = A . 7

for

r2 > R2

distance,

where 4’ is the function

is not

curves X,(y,

He

equation

K”(z,/R)

(4)

In addition

(3).

average

centers

along

depends

on the radius

distance the

particles

is the calculation

L between

dislocation

repelling force This distance

line.

R and the volume

as well as on the

A+,)

fraction

bending

f

of the

= A . bR++,/R)

is still a function

value of A++,)

= K(z,)/bL

of z,,/R also plotted in Fig. 1.

the ratio 4”/+’ is shown in the

all three

The

authors

A,,.

I

used

a

simple

approximation

originally given by Fleischerc3) to calculate distance

L for single solute

In this, the distance

the average

aOoms in a dilute

alloy.

L depends on the bending

angle

28 of the dislocation.

For small values

of K where

0 Y K/2T P KlGb2

(5)

of the position

which is the CRSS

will review

of the slip

where

T is the line energy

is approximated

(8)

of the dislocation

which

by Gb2/2. The rc>sult is

The following

steps

as taken

by

I and II.

CALCULATION

OF SHEAR

The interacting For the estimate

dislocation y-axis. is

discussion,

(7

this angle is small

plane, zO. The third and final step takes an optimum

authors

values

results in

upper part’ of the diagram.

dislocation between the particles. From both steps a critical shear stress results

paragraphs

This estimate

3.2 Calculation of the average d&mace L step of the estimate

of the

which

“optimized”

to zOthe force K depends on the particle

radius R. The second

of the

r2 = x2 + y2 + 9.

and

line approxi-

calculated

z,,) which give rise to maximum

of the integral,

For further

position

in Fig. 1.

that this straight

valid.

with $I’ being a function A = ~.C:.E:

Z/R

of the normalized

(z,/R) . #II is plotted

Author II maintains

with

3.1

plane

I.

mation

acting on the dislocation:

,ri = 0

Slip

The dependence of the force functions 4’ and $I1 and their ratio #“/$I on the slip plane distance ZJR.

FIG.

coordinate

stress field due to coherency More precisely, particle.

it is bhc shear component

3.

19F8

of the maximum

t,he coherency

approximations

above. located

16.

OF

consists

The first is the calculation

repelling

J70L.

CALCULATIONS

calculation

steps.

METXLLURGICA,

THE

CRITICAL

K’(z,,

This quantity

depends

which is the effective

force K (zO) (I) assumed

line parallel

line approximation.) R) = A . bR - +‘(z,,/R)

Their

the

to the result (6)

on K(z,)

as well as on Az,,/R

range of influence of a particular

strain field perpendicular

of K the authors

line to be a straight

(Straight

RESOLVED

STRESS

to t,he glide plane.

Its value

will be discussed in the next section. The author based on calculations prefers a treatment Labusch,t4)

II of

which lead to similar results as calculations

* The equivalent equation of Fleischer‘s has an additional factor 1.23. For further discassion see equation (18).

bv Friedel.(@

and neglected

The result is R

(10)

L

(15).

If one uses equation

bet,ween the particles.

The range of influence of the

st’rain field perpendicular

to the glide plane is given

by the quant,ity h which corresponds

to AZ of equation

,4s it, will be shown in the discussion,

instead of equation bracket

which

to the second

Thus he came to equation

-

0.46(6fln)1!3]

(16)

(1). These equations differ by the

influences

the

result

considerably.

It reduces the value of AT* by 20-30% the dependence

(1).

(15) t,he final result is

AT” = 1.04A~,‘1[l

In this case the quantity RI.5 depends on AT, which is a measure of the curvature of the dislocation line

(9).

t$he t,erm corresponding

one in equation

and changes

on f.

equations

4. DISCUSSION

(9) and (10) are quite similar. As has been shown

3.3 The u.rerfqe over z.

From rcpations (6) and (9) the authors I got the result

the estimates authors

size and differ mainly AT(io) _=

This equation

(d!?‘)‘:“.($I)3i2(!y2

has to be averaged

in the following

which

was done

O.‘iR).

The

t,aken.

Then the range ALAR was increased until a

maximum

average

value

of

value (z,, N

[#13 * (d.~/R)]l/z

value for the average

was reached.

wa,s For

larger ranges AZ, the average decreases continuously, The maximum

average

was calculated to be 0.36. equation

(2).

independent

at AZ = 0.8R

occurred

and

With this value one obt*ains

The main point of thr volume

is that Az = 0.8R

is

fraction f of the precipi-

t#at,e. Author

II used a different estimate for AZ, i.e. the

quant,$v h in his notation.

He defines

(i)

shows

posi-

line are given in Fig. 2. The

the situation

for the slip plane

interaction

between

dislocation

and

The applied shear &ress pulls the dislocation

into the upward direction. the critical positions (ii) Attractive pulls

the

critical positions

Curves a’ and b’ then are

of a dislocation

medium (6’) interaction stress

of the slip

of critical

There are two possibilities :

Repulsive

part.icle.

for small values

for low (a’) and

forces.

interaction. dislocation

The applied shear downwards leading t,o

a” (for low interact,ion forces) or b”

(for medium forces). Curves I and II show the assumed critical positions used in the theories curves

are used forces.

by

for both

authors

I and II.

repulsive

and

These

attractive

The st,raight line approximation

(12)

which is half the mean spatial distance

between

the

Tt should be pointed out that, an addit,ional

dependence Inserting

part

interaction

h = (~/6~)1/3R

particles.

upper

z0 = 0.1 R.

by both

on t,he particle

Two examples

tions of the dislocation

way (see Fig. 1). A small range AZ/R

was selected where 4’ has its maximum

zO/B.

paragraphs

force K(zJ

show the same dependence

plane distance

(11)

in the preceding

of the interaction

on f is introduced

by t)his assumpt,ion,

equa’tions (7). (10) and (12) into equation (5)

t,he result, is zo= 0,89R

A--.

The aI-cragc over z0 was done by integrating I A+

h AT

dz,

s0

fl4) Ic

The main part of t,his integral is ”

hlfl

+‘“(s)

ds = 0.97 -

Expected Curve a’: Curve b’: Curve au:

0.445R/h

s

for duthor

Criticalposltions of a dislocation line. line positions (shown only in the upper part). low repelling interaction. medium repelling interaction. low attracting interaction. Curve b”: medium attraction interaction. Approximated line positions: I: straight line approximation II: “optimized” line approximation. FIG. 2.

h/R > 1/(4/5)

II used the value of h defined in equation

(15) (12)

ACTA

X26

I b’

METALLURGICA,

VOL.

16,

1968

intermediate position of curves a’ and a” or The “optimized” approximation II 6”.

is an

and

drastically

favours

the

attractive

case

(curves

a”

and b”). There is a big difference between theoretical curves a” or b”. The calculated

curve II and the “real”

values of K, and K,, differ by 30%.

(See equations

(6), (7) and Fig. 1). The lower part of Fig. 2 shows the crit,ical positions plane

for the dislocation

z0 = 0.89R.

Here,

I and II are plotted.

only

line on the slip

the

values K, and KI1 differ by 3%. therefore, t,hat the big difference occurs

a,s a consequence

curve II.

of the extreme

Wiedersich(@

relating

the

curvature

internal

strain

field.

For

fits quite

approximation

the

equation

dislocation

values

With

to

the

its maximum

&R/b < 0.2

his

AZ = 0.8R

line

independent

I, equation

(6) which

is

values of &R/b,

increasing

however, he found smaller values for K compared with This deviation is contrary that approximation. to the deviation

of the estimate

seems to be more realistic. tion by Wiedersich, already

by author

According

the straight

line approximation

is an upper limit for the function

over, K depends more symmetric

K.

More-

also on the sign of zolR and is no as in approximations

I and II.

to be

0.3R < x0 < 1.1 R.

It is

from

of the volume fraction f of the precipitate. author

II

used

tional

to R and a function

of f.

results in a different dependence To explain this difference, are plotted

in Fig.

assumed constant. the averaging

3.

half

This assumption

of AT,,(~).

several functions

All

other

ATE

parameters

are

Both solid curves are calculated by

method used by authors I. They differ K(z,)

by the function

adopted,

AT,, = 1.567,‘).

equation

(7) for

equation

(1).

the

using either equation or equation

The dashed

lated by the averaging (10)

h being

a range

the average spatial particle distance which is propor-

curve,

AT,, = K/bL into equation

1

8

in %

This range was estimated

(6) (lower curve, AT, = AT,‘)

4.2 The particle distance L Tf one substitutes the final result is

value. reaching

In contrast,

II and

to the calcula-

fraction

FIG. 3. Xormalized CRSS AT~ as a function of the particle volume fraction f. Solid lines: averaging method I, linear and “optimized” approximations, equations (6) and (7). Dashed line: averaainn method II. equation (7). Dash-dot line: aveiagkg method 11,kquation (15).

of

numerically

well with the straight

of authors to R.

proportional

of

6

4

volume

case better.

calculated

force K via the differential

the interaction

K(z/R)

position

2

D

It is suggested, in the first case

The force K, fit the “real”

Very recently

curve

approximations

In this case, the calculated

method force

This equation,

(7) (upper

curve

is calcu-

of author

II, using

function, however.

as given

by

neglects

the

influence of the upper limits of the averaging integral, (17)

equation

(14).

The

dashed-dotted

line takes

account this influence given by equation which is quite similar to equation

(9) in the case of

Obviously,

the averaging

method

into

(15).

II always gives

there is an h = AZ. In spite of this similarity, important difference. In equation (10) ATE is the

lower values of ATE (dash-dot

line) as compared to the

averaging

solid line).

increase of the CRSS;

differences occur for low values of the volume fraction

R/L is also fixed. equations function

it is a fixed value;

In contrast,

therefore,

the quantity

K in

(9) and (17) is a function of zO, R/L is also a is therefore of zO. The above substitution

not quite correct.

main

differences

The largest

In that case, the ranges used for the averaging

procedures

show maximum

h, equation

differences.

The quantity

(la), then is large resulting in low values

A question arises about the validity averaging between

the

estimates,

equations (1) and (a), occur by the averaging procedures. An important r61e is played by the distance over which a particle influences a dislocation on a given slip plane. This sphere of influence is determined differently

I (upper

of the average force K acting on the dislocation

4.3 The average over z The

f.

method

by authors I and II.

a range AZ where the interaction

Authors

I assumed

force K(z,/R)

has

procedures.

Let us assume a hypothetical

interaction between a dislocation which is restricted to the particle equal

and a particle volume. In this

the range AZ or h is restricted

case, undoubtedly, values

line.

of the different

to or lower

than

to

R, i.e. the particle

radius. Here, one never would average over regions where no influence occurs. Therefore, h will be independent

of

f,

thus

leading

to

an

averaging

GEROLD:

procedure

HARDENING

similar to method I.

decrease

the

CRSS.

COHERENT

As

has

this should

been

proved,

Al-Zn

averaging method II leads ho lower values of AT, t,han method

1. From this, one can conclude

I gives the better estimate.

If ,rnl and precipitate

WITH

R=LOA /

(2) therefore,

on f. EXPERIMENTS

are tile zinc

mz

20

that method

Equation

should givr the right, dependence 5. COMPARISON

XL’i

PRECII’I’YATES

If we now assume an

intera~t,io~l reaching to farther distances, not

BP

con~ent~rations in the

and in Dhe ma.trix. respectively,

the volume

fraction f is dc+ined by

where

md is t,he zinc

According

concentration

of

to equat#ion (I), one finds (AT&~/~ -_.f -

whereas, according

-f-

equations

ment,,

recent

various

A-Zn

m2

(19)

mA -

alloys

CRSS 70 as a function

mz

(20)

of

are taken.

Haberkorn”)

on

He measured

t,he

of the particle radius R and of

t’hc alloy concentration ‘nz_4. To correct for the friction force in the depleted matrix 0.2 kg/mm2 were subtracted’s)

from

dist#inct particle t)o equations validity

(19)

T”. and

of oquat,ions

corresponding

AT,

values

R were

radius

(20) (I)

to equation

and

in Fig. (2).

belonging

plotted

to

4 testing Only

a

a,ccording the

the curve

20 is a straight, line.

It

9

7 at. %

FIG. 4. Relationsbi~)between ~~~and~lllo~ic~fl(!erltration, ns.4,according to equat,ions (19) and (20) and to rmm~wemmts.‘TJ

This

(19) and (20) with the experi-

measurements

3 5 zinc concentration,

int’ersects the concentration

to equat’ion (2) one expects

(Ar,$ To compare

mA -

1

t,he alloy.

corresponds

well

axis at. pn,e= 1 .T at. % Zn. t’o

analyzing small angle X-ray

m2 =

1.75

found

by

sca.ttering.‘g)

REFERENCES 1. H. GLEITER, %. anger. Phys. 23, No. 2, 108 (1967). 2. V. GERALD and H. HABERKORN, Php. Status soli& 18, 675 (1966). 3. R. L. E'LEISCHER, Th.e Strengthening of Hrfals, editrd by D. PECKNER. Reinhold (1964). 4. R. LABUSCH, 2. Phya. 167,452 (1962). 5. J. FRIEDEL, Les Dislocatdans. Gauthier-Viliars (1956). 6. H. ~~~EDERSICH, lnt. Conf. strength Hat& Alto,y, Tokyo, 1967. 7. H. HABERKORN, Phys. Status Solidi 15,I53 (1966). 8. H. HABERKORN and I". GEROLD, Phys. Stutus Solidi 15, 167 119661. 9. 1’. &ROLL), Snzall A@e S-ray Scatter&g, Proceedings of the Conference held at. Syracuse 1965: Gordon & Breach, p. 277, 1967.