The relation between the electrical resistivity and the yield strength of deformed copper

LETTERS

Crystal

Boundary

Deformed

Hardening

Metals

by Scratch Tuchschmido) stat,istical

has

analysis

measurements one obtain centration

pointed

out

of strain-hardening

that

only

by

microhardness aggregates may

regarding

STRETCHED

IRON

I

I

The

annealing

boun-

at

experiments resistivity

any con-

at the crystal

tests.

of publications.(l)

this increase of the resistivity

daries. He reviews the work of Boas and Hargreavesf2’ in this connection, but I should like to mention some earlier workf3) using scratch

of a number

four different effects.

many

information

EDITOR

reason for this interest is the possibility

Tests*

of a great

THE

subject

in

revealed

on strained crystalline

definite

TO

Fig.

different

as the sum of at least

temperatures. copper

We

20%

temperatures.

the resistivity

made

with a residual

of 6 x 10-l’ Rm in the annealed gives

main

One separates these effects by

on commercial

l(a)

stretching

The

of describing

at

state.

-195°C

after

and annealing for 30 min at different The dotted

curve was obtained

from

the results of Manintveld,c2) and the drawn curve was

data then

composed

after

our own

measurements.

Also

the

yield value (the stress at a plastic strain of 0.15%)

CRYSTALS

given (b). The resistivity

is

and the yield stress are measured on

the same wire in liquid nitrogen.

For every annealing

pulse of 30 minutes a new wire was used. From Fig. 1 it appears that in this case at 280°C the dislocations

disappear.

More information

is obtained

when

we plot

the

yield stress as a function of the resistivity for constant strain

and variable

annealing

temperature.

Fig.

2

shows the results for copper wires strained 5, 10, 15, 20, and 25%. About this figure we can make the following

remarks:

1. The horizontal part of the curves of Fig. 2 represents the three restoration steps (I, II, and III) FIG. 1.

obtained

on stretched

of Fig. 1. The resistivity

coarse-grained

stress in step IV when

iron have been

plotted in Fig. 1, and are independent suggesting that strain-hardening increases

evidence

yield stress and that in the first stage of mechanical

as boun-

recovery the decrease of the yield stress is proportional to the decrease

HUGH O’NEILL.

University College,

Fig. 3 shows these points.

1. TUCHSCHMID. Acta Met.. 3. No. 2, 215 (1955). 2. W. BOAS and M. E. HAR~RE~~Es, Prdc. R&.

Sm.,

A192,

stretching.

3. H. O’NEILL, Carnegie Schol. Mems. Iron and Steel Inst., 17, 109 (1928).

E and

the past five years the influence of metals

VOL.

4,

of cold-

1956

curves

geneity of the temperature

has been the

MARCH

The

like F-S-,4

give the same

temperature, inhomogeneous annealing of the wire might cause a larger decrease of the yield stress than of the resistivity. Experiments arranged in this direction showed that the influence of the inhomo-

The Relation between the Electrical Resistivity and the Yield Strength of Deformed Copper* on the resistivity

In this figure A-B-C-D-

relation during annealing. 3. As in the recrystallization region the yield stress and the resistivity vary much with the heating

July 29, 1955.

METALLURGICA,

point

E-F represents the yield stress at -195°C as a function of the resistivity of the dislocations during

89 (1948).

ACTA

The crossing

of the dislocations

the yield stress after 20% deformation. In this way five crossing points B, C, D, E, F are obtained. References

During

of step IV.

represents then the resistivity

Singleton Park, Swansea, Wales.

working

disappear.

2. We assume that step III does not influence the

daries are approached.

*Received

varies only with the yield the dislocations

211

was negligible.

ACTA

212

METALLURGICA,

4. Pry and Hennig’3) have shown that there exists a single-valued

relation between the extra resistivity

due to dislocations

and the yield stress.

result was obtained by Korevaar.c4)

The same

VOL.

and a part where the yield stress is

constant

attributed

\

-

1956

(S-A).

The

part

F-S

might

be

The recrystallization

to polygonization.

then takes place between S and A.

This dependence

J+

I:rl

(F-S)

strongly almost

4,

TEWPER*T”RE

,oc,

I

u

FIG. l(a). Remaining part of the extra-resistivity, after annealing 30 min, as a function of the annealing temperature. after the same heat(b). Yield strength at -195°C treatment.

clearly does not exist in the general obtain, e.g., at a dislocation any yield stress between

resistivity

case.

One can

For the sake of completeness

to

the

concentration

of

is

dislocations,

15 OF

20

25

DISLOCATIONS

FIG. 3. Yield strength at -195°C as a function of the resistivity of the dislocations (this figure is obtained from Fig. 2).

we note that we define

not influence the conclusions. 5. This result shows that, when the resistivity

10

5 RESISTIVITY

ti6”nm)

8 and 30 kg/mm2.

the resistivity of the dislocations in a way different from that of Pry and Hennig. This difference does

proportional

0 -

of 10 * lo-l1 Qm

This

view

was

supported

by

X-ray

diffraction.

It appeared that the texture, which remains unchanged d uring deformation and the first stage of mechanical recovery

(F-S), changes markedly

between S and A.

FIG. 2. Yield strength at -195°C as a function of the extra resistivity during annealing.

FIG. 4. The total increase of the resistivity (P), the resistivity of the dislocations (Q), and their ratio (R) as a function of the strain.

the yield strength depends markedly on the position of the dislocations. ‘It is a good example of the

7. Fig. 4 (obtained from Fig. 3) gives the total increase of the resistivity P and the resistivity due to

difficulties of the theory of work-hardening. 6. The curves of Fig. 3 (e.g., F-S-A) can be divided in a part where the yield st.rength varies

dislocations Q as a function of the strain. Curve R gives the ratio of the effects. This ratio is almost constant (between 0.4 and 0.5).

LETTERS 8. The

resistivity

of

the

accord with the calculations

dislocations

is not

TO THE in

EDITOR

This reduces to

of Hunter and Nabarro.t5) Qmax;&,

To illustrate this, we take for example the dislocation resistivity

at a strain of 25%.

and Nabarro,t5)

According

this resistivity

we find a stored energy of 2.3 cal/cm3. of the work done in stretching.

by

where

8 = Young’s

randomly

as Lagerberg

This is about

voor Fundamenteel

b=

wetenschappelijk

Onderzoek”

x

(Z.W.O.).

A220, 542(1953). 6. L. M. CLAREBROUGHet al., September

Internal

Proc. Roy. Sm.,

20, 1955.

Friction

in a-Iron

Interstitial

Solutes*

Lagerberg the

due to

0.866

n/s so

& Joseffsono)

anelasticity

of

an

obtained aggregate

oriented crystals and of a wire with a (110)

they obtained approximate solutions only. Methods for obtaining exact solutions are given below. Where of Lagerberg

and

Joseffson

is used. (a) Wire with a randoom orient&ion-A similar expression was used by Rawlings and Tambini,c2) by

changing

viz., x2 + y2 = 8, .z2= 1 -

Qmax;,& wheref

to

polar

n/2 =

-0.866

=

I2

where

e(l -

tan w = x

-1.74

(1 + 22) dx m s o x4 + 2.40-f

-C

=

B = D =

t = 1.417

(2.7:-t s) + I,=

-1.74

toss v

a = S,, &,I

-

:&I}

sin2 q~)}

1)

1/2t.

s = 0.655,

and

1/28(1/t -

B -

x2 + sx + t

19= cos2 y, and y/x = tan y.

0 (1 -

b = 2{(fJ,, -

dW

A =

=: -1.74 =

dW

w s 0 0.4975 + 0.201 COG w + 0.3015 CO84

I,=

co-ordinates,

=

(0,~)

1

1

of

Being unable to integrate their expressions,

notation

-

ex-

randomly

is solved

from equation

dW

1.608F(w)

Then,

for

and

(1)

we have

0.866 1.866 + ---1.608F(w) -

(110) Qmax&2a, = __

A215,507(1952).

pressions

the

we obtain

(b) Wire with a (110) texture-starting

Substitute,

possible,

lo-l2 cm2/dyne

(18) of Lagerberg and Joseffson,

References 1. T. BROOM, Adv. Phys., 9, 26 (1954). 2. J. A. MANINTVELD, Thesis, Delft, 1954. 3. R. H. PRY end R. W. HENNIG, Acta Met., 2, 318 (1954). 4. B. M. KOREVAAR, private communication. 5. S. C. HUNTER end F. R. N. NABARRO, Proc. Roy. Sm.,

texture.

we calculate

The second part of the integral is

Holland.

Recently

of

1.216 x lo-l2 cm2/dyne

Laboratoriwm voor Technische Physica,

*Received

an aggregate

Qmax&&, = A x 0.647 x 1012

C. W. BERGHOUT. De&

for

Using the same constants

Taking 3 = 2.06 x lOi dyne/cm2,

OnderzoekderMaterie”

(F.O.M.), and was also made possible by financial support from the “Nederlandse Organisatie voor zuiver

and Joseffson, a = 0.757

It is about ten

This work is part of the research programme of the research group “Metals F.O.M.-T.N.O.” of the

modulus

oriented crystals.

times larger than the measured values.c6)

“Stichting

=

to Hunter

must be caused

1.2 x 1012dislocations cm-2. When the low value of 2 eV per atomic plane is the energy of a dislocation, 20%

213

: [In z+SX--f] ?1.74 x2 - sx + t 0

f$