On the pile-up model for yielding

thermal

contribution

overcome

is the

I1 is a function

an apparent to either

temperature

the fact that

conducted

produced

alternative

stress

workers

dependence

in

upon

at 373°K.

up under

non-constant

stress

itself.

a barrier

due to the jth dislocation,

6, is the ext’ernal stress on the ith dislocation is the locks

maximum

value

the leading

of the

dislocation,

giveu bv the equilibrium

of

If oij is the

barrier

and ob

st,ress, which

the yield

condition

is

equations

are

Koppenaal(ll) of

t’he

former

determination

Although

of

an

this method is Other

are at’tractive.

and Conradds)

idea that H is temperature

favour

the

dependent and analyse their

This question

before a fully acceptable

or

against

finite strengt’h can be derived as follows:

of H may be due

favour

the results

accordingly.

685

EDITOR

stress and vice versa

the

such as Makino2)

results

stress

have point’ed out’ that,

of temperature

open to quest’ion,

the

test’s at low temperature

evidence

based

plastic

of whether

of temperature.

under a high effective

or t,o the effect

concerning

question

Conrad and Wiedersich(g)

effective

to

THE

stress on the ith dislocation

the main problem

mechanisms

enthalpy

has

is required

intersection.

In conclusion

both.

flow stress

TO

a longer range stress than in a dislocation-

dislocation flow

to

LETTERS

should be settIled

model can be formulated. B.

Department of Mining

where )L is bhe number assumed

in the pile-up

delta function.

Yielding

is

to occur when t,he net’ stress on t,he leading

dislocation

exceeds the barrier stress ub. The external

stress crzincludes the applied stress and all other st,resses except

the mutual

in the pile-up. index

i for

summation

interactions

Equation

between

dislocations

(1) can be summed over the

1 ..< i :: n, a’nd since

~~~=

-oji,

the

gives nf7 = ob

ltUSSELL

and Metallurgical Engineering

of dislocations

and dij is the Kronecker

(2)

where

University of Qurensland Brisbanje, Australia References 1. T. J.

2. 3.

4. 5. 6. 7. 8.

9. 10. 11. 12. 13.

KOPPENAAL, Acta Met., this issue p. 681. B. RUSSELL, Acta Met. 13, 999 (1965). B. RUSSELL, Acta Met. 13, 11 (1965). VON MISES, 2. Anqew Math. Afech. 8, 161 (1928). R. E. SMALLMAN, K. H. WESTMACOTT and J. H. COLLEG, J. Inst. Metals 88, 127 (1959-60). J. SILCOX and P. B. HIRSH, A&. Phys. 1, 43 (1959). B. RUSSELL, Phil. Mag. 11, 139 (1965). J. FRIEDEL, Les Ilislocations. Gauthior-Villars, Paris

(1956). H. CONRAD and H. WJEDERSICH, Acta fifet. 8, 130 (1960). M. J. MAKIN, A.I.M.E. Conferenceon RadiationsEffects, Sept. 8-10th (1965), Asheville, N.C., in press. T. J. KOPPENAAL, Phil. Mag. 11, 1257 (1965). M. J. MAKIN, Phil. Mag. 10, 81 (1964). H. CONRAD, L. HAYS, G. SCHOEK and H. WIEDERSICH, Acta Xet. 9, 367 (1961).

and is the average

external

stress acting

on a dislo-

cat’ion in the pile-up. In general Z is difficult to evaluat’e when considering discrete positions be evaluated

of the dislocations.

by using a continuum

However,

in which the actual dislocation

distribution

by a continuous

distribution.

The dislocation

bution

f(x)

function

is defined

the number of dislocations The expression

ii can

approximation,‘l) is replaced

so that f(x)

in the interval

distridx gives

5 to x + dx.

for 5 is then given by

(3)

* Received Novrmher 22, 1965.

where L is the length

of the pile-up and o(x) is the

external

function

-or

f

stress) On the In dislocation assumed

that

pile-up

model

models

for

yielding*

for yielding

dislocations

pile

constant

stress against

a barrier. Non-constant stresses arising from sources such as inclusions or sub-boundaries, are neglected and the only external stress acting on the pile-up is the (constant) pile-up constant The

analysis

which

applied stress.

takes

into account

stresses will be considered condition

for yielding

g(x),

This

where

function

the integra,l inversion Their result for f (x) =

-

&

applied

stress

which accounts

(yield for the

stresses.

The distribution

f(x)

has the form a(x) =

ol/ is the

and g(x) is a function

non-constant

it is generally

up under

stress.

f(x)

procedure

can be obtained

by

of Chou and Louat.

is

(“-I;r--)“’

r

($Jl”

sx

(4)

at

A general the non-

when dislocations

where

D = ,ub/2mx, ,u is the

Burgers’

here. pile

(1 -

vector,

tc = 1 for

V) for edge dislocations,

shear

screw

modulus, dislocations

and v is Poisson’s

b

is

and ratio.

ACTA

6X6

On combining

equations

METALLURGICA,

(2), (3) and (4), the condition

VOL.

14,

evaluated

1966

by the method

of Pu’abarro and Cottrell.(5)

Their method gives

for yielding can be written as

nD@ sin 2np Oc,ll(z,y)

o(+(t)

Equation

for

pile-ups

under

any

kind

of

Once a(x) and ab have been determin.ed, the integrals in equation

stress.

solution of

(5) will give an explicit form

cash 2rq’ -

As an illustration

of the application

The sub-structure

linear array of edge dislocation misorientation

of this result a by an infinite

walls of height

2H,

The dislocations

will be assumed to pile up against

one of the walls on a slip plane perpendicular

to the

dislocation

to the

walls.

Yielding

case where the dislocations The

barrier

will correspond be

taken

stress required to push an edge dislocation wall of edge dislocations parallel slip planes. have a maximum

as the

through a

of the same sign and on

This stress has been estimated

to

cash 2rq and equation

(9) reduces to the sinusoidal

where B = SrD@qlcosh The external -ay

dislocations

The non-constant

(6)

stresses arise from the long-range

stresses of the walls. If these are neglected the external stress a(z) = -au into

equation

and is constant.

(5) and

combining

Substituting

this

the results

wit’h

equa,tion (6) gives

a(z)

into

equation

with equation ay2 -

(7)

where L is taken as d/2.

linear

array

approximately as follows:

between

must

be

two adjacent determined.

walls in This

has

For an edge dislocation

wall with the

origin taken at the center, the stress field is given byc4)

%all(z,y)

--

b

this form of the

results

(6) gives (11)

0.95 Ba, + 0.23 B2 = (2n-D2/b)(O/d)

where L is taken as d/2 and the results of Chou and parison of equations

the integrals.

of the non-constant

stresses

B will determine

modification.

A com-

(7) and (11) shows that inclusion modifies

the form

The magnitude

B is a decreasing

function

the significance of q for q > l/2,

of

of the of this

Choosing y # 0 will decrease B.

Also, and as

2 increases, i.e. closely spaced walls, B goes to zero. The

above

model

stresses will influence dependence. gives

illustrates

that

the form

of the yield

non-constant stress

Therefore any model where such stresses may

a general

not

be realistic.

solution

which

Equation

can be applied

pile-up models for yielding where non-constant The authors

(5) to

stresses

wish t’o express their appreciation

the many helpful discussions (P. C.)

acknowledges

National Aeronautics

the

(Y

+

by the under

of Technology. P. CHAUUHARI

x2+(y- 1 (8)

R. 0. SCATTERGOOD

Department of Metallurgy

H)

Ma,ssachusetts Institute of Technology

H)”

The stress at any point between

support

Contract NsG-496 with the Center for Space Research,

W2

(Y -

One of the authors

financial

and Space Administration

Massachuset’ts Institute +

of

w&h Drs. A. S. Argon,

M. B. Bever and K. C. Russell.

(Y + H) [ x2

combining

LouaU2) are used to evaluate

the

a sinusoidal form, which can be shown

DOx

Substituting

(5) and

are present,.

If the stresses of the walls are to be included, the

as a(x) =

The plus sign appears because the

in the wall are assumed to have the same

are neglected

au2 = (2rD2/b)(O/d)

stress at any point

2rq.

stress can now be written

+ B sin 2np.

parameter a,, = rD0/2b

(19)

accell(s,O) = B sin 2rp

the yield strength dependence.

value of(s)

H)/d;

cos 23rp G

form

can break through the wall.

stress will therefore

1 (9)

then cash 2rrq -

sign as those in the pile-up.

angle 0, and period d.

cos 2rrp

where p = x/d, q = (y + H)/d, and q’ = (y -

hardening will be considered.

will be approximated

cos 2rrp

if y = 0 and q > I/2.

for the yield stress. model for sub-boundary

4 [ cash 2rq -

(5)

(5) is the desired result and gives the yield

condition

6

4’

dx dt == ab

~ t-x

-

two adjacent

Cambridge, Massachussetts

walls,

which constitute a cell, is obtained by summing the contributions of all the walls. This infinite sum can be

References 1. G. LEIBFHIED, 2. T. T. CHOTJ

Z. and

Physik S.

LOVAT,

130, 214 (1951). .I. Appl.

Phys.

33, 3312 (1962).

LETTERS

TO

THE

687

EDITOR

3. J. C. M. LI, Electron Microscopy and Strengthof Materials, Interscience edited bv G. Thomas and J. Washburn. Publisheis, John Wiley, New York (1963). 4. F. R. N. NABARRO, Adw. Phys. 1, 269 (1952). 5. A. H. COTTRELL, Dislocations and Plastic 870~ in Crystals. Clarendon Press, Oxford (1953). * Received 1965.

September

24,

1965;

revised

Sovember

26,

Coefficient of linear expansion in different solid phases of cuprous selenide* It is well known that copper and silver chalcogenides have polymorfic perature

transformations

range between

appear

in the nonstoichiometric

temperature

of the transition

unit cells is considerably tion.

occurring in the tem-

100°C and 200°C.

As they

form Ai_,B”‘, between

dependent

on the composi-

show that after the tetragonal-cubic

sition the phase transformation temperature. dependent

continues

There is a temperature

arrangement

extends

In our previous behaviour

tran-

to the high

range where the

of atoms and lattice constant are strongly

on the temperature.

this interval

which

cuprous

For Cu,,,,Se

between

103°C and 200”C.(1.2)

work(3) we described

of the thermoelectric

occurs

samples

an anomalous

power

of

temperature

range.

In the present note dilatometric

examinations

of the

samples with the same composition The

thermal

measured.

are described.

of linear

expansion

The sample was a polycrystalline

prepared

by direct synthesis

and melted performed Weiss”

coefficient

with the Differential

in the temperature

was

cylinder

of Cu and Se at 400°C

in the proper form.

Measurements Dilatometer

were

“Bauart

interval from 20°C up to

400°C. The change of the sample length vs. temperat’ure is shown graphically

(Fig. 1).

In the low temperature

modification

the coefficient

of linear expansion

0°C

and

to

110°C

equal

measured

from high to low temperatures,

of the curve in the transition without

well defined

behaviour

connected

14 . 1O-6 deg-l.

At

to the nature

Such

expansion

of phase

is

trans-

format’ion in cuprous selenide. The authors are obliged to the staff of the physical department

of High Technical

School, Zagreb, for the

use of the dilatometer. Institute “Ruder

Boik&”

Z.

OGGRELECt

B.

C~ELUSTKA~

zag?&

Yugoslavia References 1. P. RAHLFS, %. Phys. Chem. B31, 157 (1936). 2. W. BORCHERT, Z. K&t. 106, 5 (1945).

3. Z. OGORELEC, and B. ?ELVSTKA, .J. Phys.

27,

Chrm.

Solids

217 (1966).

* Received November 9, 1965. Permanent addresses: j- Institute of Physics, Faculty of Science. 3 Institute of Physics, Faculty of Medicine, Zagrch, Yugoslavia.

University

of

from high

where a-Cu,,,,Se

exists the coefficient is

also constant

from 230°C to 400°C and has the value

22 . 1O-6 degpl.

Precipitation

where pure low tem-

of vanadium

carbide*

1. Observation of the stacking fault type precipitate Precipitation

interval

at 160°C.

of linear

(p-C$,+.Se)

is constant

temperatures

In the temperature

sharp change

of the coefficient

evidently

the shape

region was similar, but

Cu,,,,Se

in the mentioned

IN ‘C

FIN. 1

two types of

X-ray examinations of nonstoichiometric

selenides

TEMPERATURE

the

of vanadium

by Irani and Weiner(l)

carbide has been shown

to occur in association

with

perature phase transforms to the pure high temperature

stacking faults in a way analogous to that previously

one,

changes

observed

The value of it starts to decrease from

stacking

the

coefficient

irregularly.

of

linear

expansion

110°C to 130°C. At 130°C the coefficient changes in sign and at 160°C it changes sharply again into the positive

value,

interval

from 230°C to 400°C.

which

is a little higher than in the

efficient at temperatures deg-l .

If

the

change

The value of the co-

near to 160°C is -22 of

the

sample

length

. 1OW was

in

niobium-containing

steels.(2-4)

fault fringes were not observed,

their steel contained

The

but since

a very high supersaturation

of

vanadium (5 wt. ‘A), growth of the carbide particles was no doubt so rapid that the faults were quickly destroyed.

A steel containing

wt. o/o C has been examined precipitation

of

coarse

1.5 wt. % V and 0.13

at C.E.R.L.

M,,C,

occurs

In this steel at the grain