The dissipation of energy during plastic deformation

THE

DISSIPATION

OF

ENERGY

DURING

PLASTIC

DEFORMATION*

J. F. NICHOLASt Various mechanisms are considered for the dissipation as heat of the mechanical energy of deformation in a cold-worked metal. Since dislocations seem to account for at most 10 per cent of this dissipation, the creation and annihilation of point defects is considered in some detail. It is concluded that the dissipation may be accounted for satisfactorily in this way.

LA DISSIPATION

DE L’ENERGIE

AU COURS

DE LA DEFORMATION

PLASTIQUE

Divers mecanismes sont envisages pour expliquer la dissipation sous forme calorifique de l’energie justifier mecanique de deformation dans un metal Bcroui. Comme les dislocations ne semblent qu’environ 10% de cette dissipation, la creation et la suppression de defauts ponctuels sont considerees en detail. L’auteur conclut que cette fapon de voir rend bien compte de la dissipation Btudiee.

DIE

ENERGIEDISSIPATION

WAHREND

PLASTISCHER

VERFORMUNG.

Fur die Uberfiihrung der mechanischen Verformungsenergie eines kaltverformten Metalls in W&me werden verschiedene Mechanismen betrachtet. Da Versetzungen fur hijchstens 10% dieser tfberfiihrung aufzukommen scheinen, wird die Erzeugung und Vernichtung van Punktfehlstellen ausfiihrlich betrachtet. Es folgt, dass die Dissipation auf diese Weise befriedigend erklart werden kann.

1. INTRODUCTION

Most of the energy expended is dissipated

Then, since clW = T ds where r is the stress and E the

in deforming

a metal

as heat and only a small proportion

stored in the metal.

Considerable

strain, we have

is

dE/de = r.

research has been

carried out on the manner in which this latter energy is stored(l) but no satisfactory picture has yet been given

for

the

mechanism

mechanical

energy

possibilities

are considered

that,

by

which

is converted

at high strains,

to

most heat.

in this paper.

the only

process

dissipating

sufficient

energy

is the

annihilation

of point

defects,

although

and annihilation

of dislocations

of

In order to treat dE/ds in terms of dislocations, we will consider a crystal consisting of a set of parallel

the

slip planes of equal area, deforming

Various

of

It appears capable

creation

of

of a crystal

the

PROBLEM

(1)

energy

when

unit

unit distance

(2)

7, AUGUST

of

of unit length of

(6)

density

of dislocations,

the distance L through which the dislocation moves in dissipating an energy equal to its energy of formation is given by L = NE,

Scientific and of Melbourne, 1959

E,

length as

where ,u is the shear modulus and K is a factor rather

* Received

September 23, 1958. t Division of Tribophysics, Commonwealth Industrial Research Organization, University Australia.

lines per

(5)

EF = Kpb2, less than 1 for a reasonable

dW = dE.

VOL.

of

moves through

Since the energy of formation dislocation line is given by

specific surface energy, and A is the surface area per unit volume. In general, at high strains,

so that we can write, as a good approximation,

these

(4)

dE/N dx = br.

ydA
If

Then, from (3) and (4), we can calculate

dissipation

dislocation

work, d W, done on ydA,

b.

move an average distance dx, the strain

where N is the length of these dislocation

where E is the energy dissipated as heat, S is the stored energy that can be liberated on annealing, y is the

METALLURGICA,

by the movement

vector

unit volume and certain geometrical factors have been

as

dW=dE+dX+

ACTA

Burgers

dE = Nb dx,

the creation

neglected. OF

of

and

may make a significant

We can express the incremental unit volume

dislocations

dislocations is(s)

contribution. 2. STATEMENT

(3)

dx/dE = K,ublr.

(7)

Some values of L are given in Table 1 where the stresses are typical of those operating in a 644

NICHOLAS:

THE DISSIPATION

OF ENERGY

heavily-worked specimen and X has been taken as one. These values are, in effect, upper estimates for L, since K will usually be less than one for the density of dislocations in such a specimen. It can be seen that each dislocation dissipates an energy equal to its energy of formation in about 1000 k or less. 3. CONSIDERATION

OF

POSSIBLE

MECHANISMS

Four possible mechanisms for dissipating the energy will be considered, viz, (1) the moving dislocations acquire a large kinetic energy that is released when the dislocation is stopped; (2) the dislocations dissipate energy continuously through thermoelastic damping, radiation damping, and scattering of sound waves; (3) large lengths of dislocation line are created and a~ihilated during the deformation; and (4) large numbers of point defects are created and annihilated during the deformation. The first two possibilities have been considered in detail by various authors and found to provide inadequate dissipation of energy unless the dislocations are moving at speeds comparable to that of sound.(“) The evidence available at present suggests that dislocations do not move at such high speeds, so that we will henceforth ignore possibilities (1) and (2). The third possibility implies that, in a heavily deformed metal, the average displacement of a dislocation between creation and annihilation, here called the slip distance, is of the order of L (~10~ 8). The best measurements of slip distances under these conditions are those of Seeger, Diehl, Mader and Rebstock.(Q These workers strained copper heavily (E = 0.6), polished it, strained it further (he .= 0.05), and then took electron micrographs. These showed main slip-lines, 24 x lo4 A in length, together with shorter cross-slip lines. Since the cross slip of a dislocation out of a pile-up is simply a continuation of its movement, the mean slip distance of a dislocation here appears to be of the order of a few microns, i.e. ~10L. Thus, this ~lecha~sm accounts for rather less than 10 per cent of the dissipated energy.

A1’ $

2.8 25 215

____-.____~~.~-2

2.8 x lo3 7.5 x 103 12 x 10% __

_ _--___

1

15 :: ______.

510 950 1000 ____-

DURING

PLASTIC

DEFORMATION

545

This last conclusion would be modified if individual slip-lines actually consisted of short lengths of line, each length arising from a different dislocation. In this case, the slip distance of an individual dislocation might be of t.he order of L. However, for slip lines to form in this way, the operating sources would have to be grouped together in neighbouring planes and we might then expect, if the surface is typical of the interior, to see slip lines at certain stages composed of short discrete lengths of line. This last effect has not been observed. The fourth possibility, creation and annihilation of point defects, has not been considered previously from the present point of view but is treated in some detail below. The results suggest that this mechanism could dissipate sufficient energy. 4. CREATION

AND ANNIHILATION OF POINT DEFECTS

The mechanism envisaged here is that the moving dislocations, on cutting through the Cottrell forest, become jogged and then create trails of point defects behind them during their further movement. Nearly all of these defects then disappear rapidly either by absorption at sinks or by recombination with other defects. Such recombination may take the form of an~h~ation or the creation of clusters or stacking faults. 4.1

Creation of defects

In the general case, the dislocation lines will be of complex shape and not amenable to exact treatment. However, two extreme cases, which are mathematically tractable, will be treated in detail here and a more realistic case then considered in the light of these results. In this section, consideration is restricted to copper since this is the only metal where reasonable estimates have been given for the energies of point defects. The oases to be considered are: (1) the dislocation lines remain straight and point defects are formed individually; (2) the moving dislocations loop around the ~tersecting dislocations and produce rows of defects when the loop reunites, after which the jogs glide conservatively out of the crystal; and (3) a more realistic case.

We suppose that, at a given stage, the jogs which are capable of producing defects are spaced an average distance li apart along the moving dislocations and that interstitials are being produced with twice the

ACTA

646

frequency

of vacancies

(cf.

METALLURGICA,

Cottrell(5)).

Then

the

VOL.

7,

1959

n, = 40. In this estimate, we have retained the value

energy necessary to form defects during a dislocation

of 4 eV for U,, since, although a row of vacancies will

movement

have less energy than the same number

dx is given by dE =

U,N ax/l, 6,

(8)

where U, = &lJ, + #Ui and U,, Ui are the energies of formation

of vacancies

In order to explain given by equation

taking

Ud/rb2.

(9)

U, = 4 eV, we find lj = 500 A.

speed v and that jogs when formed oscillate along the applied acquires

with a mean speed vj, in the absence of an

stress.

It is shown there that a dislocation

effectively

after moving

its equilibrium

forward

a distance

and that the equilibrium

number

of jogs

of less than 100 &

value of 1, is given by

lj = (vJvp)l’2, where p is the number

(10)

of dislocations

area of the glide plane and create, dislocations,

jogs

capable

On substituting

Neither

more realistic ca8e of the above

simple

that cut unit on the gliding

of producing

defects.

lj = 500 a into equation

(lo),

we

Vi/V = 3 x lo-lip.

(11)

No reliable estimate of VJV is available

of case (l), a true value of vJv would probably of 10, rather than 40. unrealistic

in that

break-through assumes

However,

at

intersections,

indefinitely

continued

the

second motion

of a jog along the whole length of a dislocation. A more realistic case would be one where the defects are formed in rows (as in case (2)) but where the jog, after it has been formed, glides only a short distance and is then arrested. Such arrests will arise as soon as the direction direction

of the dislocation

of conservative

the dislocation

line deviates from the

motion of the jog, i.e. when

line no longer lies in the plane defined

by the Burgers vectors of the two dislocations which intersected to form the jog. It is clear that any given

but we would

it either glides out of the crystal or is annihilated a jog of opposite

type.

energy dissipation

in the form of equation

purposes that

7 = 20 kg/mm2 should

of

the

used above.

be noted

insufficient

vi/v < 3 x lo-lip,

Case (2)-Defects

discussion

we

for the value

This gives vJv = 6.

that,

energy

energy dissipation

present

p = 2 x loll cm-2 if

is dissipated

this theory

would

whereas, predict

if

more

than is found experimentally. produced in rows:

jog subsequently

glides out of crydal this

case,

each

time

a

moving

dislocation

intersects a stationary dislocation of the correct type, a row of defects is produced and this process is independent average

intersection,

of any other intersections. number

of

defects

the energy dissipated

Then, if n, is produced

Thence, by elimination (12), we find

between

(12) equations

(5) and

nB = rb/pU, . copper,

taking

per

is given by

dE = pN dx n,U, .

p = 2 x 1011 cm-2,

(13) we

produced

4.2

VJV > 3 x lo-lip,

find

interpret

ng. as the

by an “average”

Annihilation

by

We can again express the

of 40 for n, becomes reasonable.

the

will assume

For

instantaneous

while

conservative

For

the

both these cases are

the first assumes

now

In

exceed

6, while in case (2) n, is more likely to be of the order

certainly expect it to exceed one and this is consistent with equation (11) if p is of the order of 1011 cm-z.

then

quite

jog can then give rise to several rows of defects before

find

It

cases provides

enough energy dissipation since, under the assumptions

In the appendix, li is estimated on the assumption that the dislocation moves forward with a mean dislocation

row of interstitials. Case (3)---A

of the dissipation

(5), we need lj =

For copper,

and interstitials.

the whole

of isolated

the reverse may well be true for a short

vacancies,

total number jog.

(12) if we of

defects

On this basis, a value

of defects

The above discussion has been concerned

with the

creation of the defects but,, in order to dissipate energy as heat, these defects must subsequently

disappear or

aggregate

No detailed

theoretical

into clusters consideration

of low energy.

has been given to this part

of the process since there is such uncertainty the distribution

about

of the defects as formed.

On the experimental

side, however,

the available

evidence (see discussion by Clarebrough, Hargreaves, and Westt6)) show that, in copper, some point defects are present after deformation but

none

after

deformation

at sub-zero temperatures at room

temperature.

Furthermore, those that are present after the deformation at low temperatures disappear on annealing at, or below, room temperature. Thus, in deformation at room temperature any point defects formed should be sufficiently mobile to annihilate themselves and hence to dissipate their energy as heat during the formation.

NICHOLAS:

THE

DISSIPATION

OF

On the other hand, in nickel after deformation room temperature, The theory

some vacancies

given

here implies

ENERGY

at

are still present.

that these must

be

simply a residual concentration

(of the order of 1 per

cent)

formed

of

the

deformation.

total

number

during

why this concentration

is effectively

of

particular,

the

temperature

if the deformation

enough temperature, stored.

of

independent

of

deformation.

work on this line would be very

silicon provides

direct experimental

evidence

also points to a subsequent and hence a dissipation comparison

clustering

for the

His work

of the defects

of energy.

A quantitative

of his data with the present theory

be attempted

here

the materials

because

of

the

will

difference

of

the

dislocations

have shown that, within the

mechanisms

listed

in

Section

3,

alone cannot account for the dissipation of

during

However,

on

cold

work

theoretical

at

room

grounds,

temperature.

sufficient

point

defects should be created to account for the dissipation provided

they can subsequently

each other.

Experimental

cluster or annihilate

evidence

suggests that, in

copper at least, the defects are sufficiently do so. The calculations the dissociation

could be extended

of dislocations

line;

certainty

however,

and the glide of jogs at

in view

about dislocation

mobile to

by considering

an oblique angle to the average direction cation

of

the

of the dislopresent

un-

speeds, such refinements

do not seem worthwhile.

dn. ->

at

=

p

lj

for

we

and for his encouragement

can

2-

write

down

of extra jogs formed

ClX

(A.l)

incipient

k,nj2,

(A.1)

of annihilation

movement

of an existing type.

jog

by

a

The third term gives of opposite

along the dislocation.

pairs

Losses due to

of the jogs to the ends of the dislocation Strictly,

equation

by two equations,

vacancy-producing interstitial-producing purpose,

while the

to this to allow for the

the loss of jogs due to annihilation by movement

The first

gives the number

at new intersections,

jog of opposite

of

We have

klnjp - at

second term is the correction possibility

a

l/ii = the number

nj =

where t is the time and k,, k, are constants.

(A.l)

should

be

one for the number

of

jogs and one for the number

of

jogs.

However,

such a refinement

for the present

is unnecessary

and we

assume that half the jogs are of each type. Estimation

of k,

We consider the equilibrium value of nj in the case of i&=0.

Thenequation(A.l)gives

k,=(l/n,),,=(Zj)eq.

However, the equilibrium value of li should be equal, in this case, to the interaction cross-section for jogs, i.e. the length over which an existing jog will interact with a new one as soon as it is formed.

A reasonable

estimate for k, is therefore 3b. Estimation We

of k,

consider

unit length

Then the number

like to thank Dr. W. Boas for suggesting

the problem

throughout.

Thanks are also due to Drs. L. M. Clarebrough, M. E. Hargreaves, and A. K. Head for much helpful discussion

estimate

of dislocation

as being

divided by the jogs into nj intervals of random length.

ACKNOWLEDGMENTS I would

to

equation

jogs per unit length of dislocation.

replaced

5. CONCLUSIONS

energy

order

are neglected.

used.

The above arguments

547

OF Ii

term on the right of equation

creation of point defects as envisaged above.

limits

In

differential

is carried out at a low

It may be noted here that recent work by Dash(‘) on

between

APPENDIX-ESTIMATION

In

interesting.

not

DEFORMATION

the

almost all the energy should be

Experimental

PLASTIC

5. A. H. COTTREU, Dislocations and Mechanical Properties of Crystals, Conference at Lake Placid, p. 509. John Wiley, New York (1957). 6. L. M. CLAREBROUGH,M. E. HAR~REAVES and G. W. WEST, Proc. Roy. Sot. A232, 252 (1955). 7. W. C. DASH, J. Appl. Phys. 29, 705 (1958).

It does not help to solve the problem of

the amount of deformation.(s) It suggests, however, that the amount of energy stored should be a sensitive function

DURING

and criticism. REFERENCES

1. A. L. TITCHENER and M. B. BEVER, Progr. Met. Phys. 7, in press. 2. A. H. COTTREU, Dislocations and Plastic Flow in @yetale, p. 17. Clarendon Press, Oxford (1953). 3. A. H. COTTRELL,Dislocaticns and Plastic Flow in Crystals, p. 66. Clarendon Press, Oxford (1953). 4. A. SEEGER, J. DIEHL, S. MADER and H. REBSTOCK, Phil. Mag. 2, 323 (1957).

of intervals

of length

b is given

approximately by nj{l - exp (-nib)> znj2b, if njb < 1. However, if each jog is jumping v times per second the length of such an interval should change by b, 2v times per second and one half of these changes should reduce the length to zero, i.e. the number of intervals going to zero each second is vbni2. Since an interval

going to zero causes a loss of either two jogs

(if they are of opposite of the same type), cause is vbnj2.

the mean dislocation.

type)

or of none (if they are

the rate of loss of jogs from

Thus, k,=

this

vb = vj, say, where vj is

speed of movement

of a jog

along

the

ACTA

548

Equation

(A.1)

can

dxldt = v = constant

now

be

METALLURGICA,

solved

if we

put

reasonable

assumptions

that

p < 1012 cm-2, the solution becomes nj N (v~p/v~)l/~tanh [(v,~/v)~‘~z].

vj>

v and

7,

1959

Therefore,

and take the initial condition

asni=Oatx=O. On the

VOL.

(n&J=

l/(&?q=

(VP/ViP2.

Further,

nj reaches 0.9(nj)eq in a distance of about 1.5 (VUIPVJl12. Thus, even if the dislocation has no jogs initially, it has virtually its equilibrium number after

a distance of the order of 1OW cm.