Reply to comment on “the effect of carbon on the strain rate sensitivity of iron single crystals”

ACTA

150

METALLURGICA,

5. H. CONRAD and W. HAYES, Trans. Am. Sac. Metals 56, 249, 944 (1963). 6. H. CONRAD, N.P.L. Symp. Relation between Structure and Strength ia Metals and Alloys, p. 475. HMSO (1963). 7. H. CONRAD, J. Metals, N.Y. 16,582 (1964). 8. H. CONRAD and W. WIEDERSICH, Acta Met. 8, 128 (1960). 9. J. J. MICHALAK, Acta Met. 13, 213 (1965). 10. B. JAOUL and D. GONZALEZ, J. Mech. Phys. Solids 9, 16 (1961). Acta Met. 12, 547 (1964). 11. k. A&ENAULT, 13. 12. Z. S. BASINSKI and CHRISTIAN. Au&. J. Phws. u --I 299 (1960). Phys. 31, 362 (1960). 13. D. F. STEIN and J. R. Low, J. a&. 14. H. CONRAD, Phil. Mug. 5, 745 (1960). 15. D. F. STEEN. J. R. Low and A. U. SEYBOLT. Acta Met. 11. 1253 (1963): 16. D. S. TOMELIN rend D. F. STEIN, Trans. Am. Inst. Min. metall. Engrs 233, 2056 (1965). 17. J. T. MICHALAK, Iron and Its Dilute Solid Solutions, p. 319 (1963). 18. B. EDMONDSON, Proc. R. Sot. 264A, 176 (1961). A. SEEGER, Phil. Mug. 1, 651 (1956). 8:: J. DORN and J. RAJNAK, Trans. Am. Inst. Min. metall. Engrs 230, 1053 (1964). R. KOSSOWSKY and K. BROWN, Acta Met. 10,131 (1966). 38: J. MEAKIN and A. LAWLEY, Private communication. * Received

April

VOL.

15,

measurements

to support the Peierls-Nabarro

theory

of yielding. Also attributing

the effect to a change in the pre-

exponential term is not nearly as straight-forward as is implied by Conrad. The expression for the strain rate given by Conrad is: 9 = $ Abv* exp If the experiment

(1)

-

is done at two different

rates with a constant

structure,

should be [H(T*)].

A strain rate change experiment

in which the strain rate is changed quickly from one value to the other should satisfy this condition well.

Using the definition

and Haasen,@)

[H(T*) = Ho -

VT*], where V is the

volume and T* the effective stress, relation

(1) reduces

to the following

strain

rate

Conrad’s four

sensitivity objection

of the paper

is concerned under

be adequately

single thermally Conrad associated

with

discussion(l)

equations

at different

the

with the role of impurities

strain

and

of thermally

activated

yield

possible).

with stress is independent

interstitial

content

These results support overcoming stress as the rate controlling 300°K.”

Similar

and

of

distribution.

the Peierls-Nabarro

mechanism

statements

kT

(3)

(4)

confusion

The following is quoted from Conrad and Frederick :t6) “ . . . Also the variation of the activation energy volume

(H, -- VT~*)

-

(2)

(5)

and strain on a

(not the only thermally activatedmechanism

and activation

kT

Aln)i-&AT

on the basis of a

activated process. concern with

exp

1 1

P, - J’T~*) ~~

-

states

of these iron crystals

explained

model

p2/l,*A,b,v,*

conclusion

which

expresses

Peierls-Nabarro

f2=

of iron single crystals”*

that the strain rate sensitivity cannot

yl = pl/Zl*Alb,v,* exp

on “The effect of carbon on the

fairly

of H(T*) given by Mordike

strain rates :

Reply to comment

strain

the only parameter

to be affected on the right hand side of the equation

activation

15, 1966.

1967

have

made

in

if the pre-exponentials volume

is directly

do not change, related

measured

in a strain rate change

argument

that Conrad has presented

premise

that

the pre-exponential

in the following

to the AT

experiment. changes

discussion

The

is based on the

a high purity iron and the less pure irons. shown

in iron below

been

Therefore, the activation

between

It is easily

that this has no

bearing on the question. The strain-rate

change

measurement

is made

on

other publications.(7J3)

the same purity of iron crystal, the pure crystal was

In this discussion, Conrad states that if the PeierlsNabarro stress is rate controlling that it is obvious

the strain rate was changed.

that the measured activation

rate equations

volume will vary with a

change in impurities and structure. stated

that a constancy

In one paper it is

of activation

volume

with

strain and interstitial content supports overcoming of the Peierls-Nabarro stress as rate controlling and in a second paper a variable is taken

as support

activation

volume

of the Peierls-Nabarro

stress.

Therefore, it is expected that a certain amount of confusion will arise until a consistent position is arrived at by those who use activation volume

not interchanged

equations

with an impure

(2) and (3).

__=Yz

These equations

are combined

relation :

p$,*

Alblvl* exp [ -

Pzll,”

A&,%”

Examination

the strain

at the two strain rates are as given by

to give the following

Yl

crystal just before

Therefore,

1 1

(H, -- VT~*)

of the pre-exponential

kT

terms to find

LETTERS

which of them may be a function is held constant)

TO

THE

EDITOR

151

of stress (the purity

reveals:

If the slip system remains the same at the

b, = b,

different strain rates. The

lattice

coupled

frequency

moments

a weak function

and

therefore

the

would be expected

to be

of stress.

The density of dislocations at any instant of time should be a constant and since the strain rate is made quickly, not expect the number expected

one would

a change in density.

However

of mobile dislocations

may be

to increase with increasing stress.

Christian

has presented

evidence

that

t,he change in mobile density must be small. The

higher

stress

may

activate

some

400

shorter line segments. The area swept out per successful

fluctua-

Fm.

1

tion may be greater at a higher stress. Therefore,

it is possible

that small changes

in p, 1” and A resulting

occur

in a change

could in the

pre-exponential. Therefore,

the only possible

the difference

in activation

pure and impure

iron if both

measured

are controlled

in the pure

exponential

it

has

assumption strain

by the

paper under discussion was intended to deal with the

rate

of yielding

of

of stress (or at least

Nabarro

theory

that the pre-

is a function

do not suggest such a mechanism.

It is

may change (there

data

showing

in

the

previous

been

to be a

concerning

at only

that the pre-exponential

theory

and is also shown

reasonable

away

quite a different function of stress) in the impure material. Examination of the factors in the pregranted

Peierls-Nabarro

directed

material

stress, but it is not a function

the

by Christian

problem

same process, is to make the statement exponential

implied

for the

way to explain

volume

constant value of AT in a strain rate change experiment supports

change

discussion

experiments.

in a general

way

one interpretation

The

and not be

of the Peierls-

of yielding

and for this reason the

the variation

of AT with stress was

presented. It is agreed that more data are necessary vicinity

of room temperature

in the

for the iron containing

is no evidence one way or the other on the point) as a

40 ppm of carbon to determine the exact shape of the

function

yield stress vs. temperature

of purity,

but this has nothing

strain rate change experiment

to do with a

In Fig. 2 of the Comment

by Conrad, he purports

to show that the incremental

yield stress at absolute

zero above

that at room

for various

purity

plot

temperature

iron crystals.

of the same data in which

various irons into three distinct tical Stein

compositions and

accepted

Lowc4) is from

Conrad submitted

(The point

a paper but

1 shows

one separates

a

the

groups having iden-

and history.

for publication

is a constant

Figure

not

that

labeled

had

published

been before

his discussion.)

that these three curves extrapolate

that

Conrad

to the

has presented

that

are purported

favor a single thermally-activated mechanism of yielding. His first argument mechanism The

of yielding

parameters

activation

volume

and impurities.

that

in favor

to

Peierls-Nabarro

of a Peierls-Nabarro

has no basis in experiment. are

supposed

to

clearly are dependent

relate

to

upon strain

To now say that they only appear

to be, but really are not, is simply a way of avoiding experimental

This figure shows that there exists a very large difference in the yield stress at 20’K and it is highly improbable

curve at this point.

In addition, comments are in order on the arguments

on a given crystal.

contradiction.

The argument that observation lying along close packed directions supports

the Peierls-Nabarro

model

of dislocations in b.c.c. metals is very

super-

same value at absolute zero. The range of values given for the Tomalin and Steint2) iron is due to orientation

ficial.

effects and does not represent scatter in the data. While Conrad has not taken the position that a

which do not exhibit dislocations lying along close packed directions must support the position that the

First,

if one uses the argument

of the Peierls-Nabarro

in support

model, then those b.c.c. metals

ACTA

152

yielding is not controlled This is observed as other examined

pure iron

in the

directions

velocity

dislocation

between

Also, it is

of these

it appears that

in these materials

close packed

directions

it appears

that it is not difficult

motion is controlled

by other mechanisms.

be

Strong

of

disagreement

sented by Dorn

the

cited

by

in the

Peierls-Nabarro

theory.

“on

the other

data (the edge mobility

are

seriously

at

sections

‘Thermally

data

variance

of the single Peierls process

of this survey.

and

has been pre-

et aZ.u4) who state:

Conrad)

valleys

was also cited

with this point

hand, Stein and Low’s expectations

these

of edge dislocations

in support

the

from a single Peierls-Nabarro to overcome

to

along

Activated

with

described

Mechanisms’

predict

Therefore, locity

it is not

clear that

measurements

with any series mechanisms”.

the

support

the

dislocation

ve-

Peierls-Nabarro

parameters

argument

Conrad

support the Peierls-Nabarro volume

Peierls-Nabarro

presented

model is concerned

and energy.

model

has

to with

It is agreed that the

qualifies for consideration

of yielding.

but on the other

In conclusion,

it is agreed that the experiments

presented in the paper “The Effect of Carbon on the Strain Rate not

Sensitivity

completely

of Iron Single Crystals”

eliminate

Peierls-Nabarro

theory

the may

mechanism

of yielding

experiments

do demonstrate

results

component the

important

metals.

AT,

component

on

as supporting

Nabarro

mechanism

support

or the

the controlling

impurities)

evidence

of yielding

temperature

on

for this to be true, the activation of the nickel containing

carbon must be small;

have

impurity been

mechanism

since can be

on the basis of the pure metal behavior,

this small activation the

In order

volume for yielding

in this case the Peierls-Nabarro eliminated

the base pure

insensitive.

volume must be associated

strengthening.

made

Similar

by Hutchinson

Sb-Cu alloys and by Johnstonu7)

on LiF.

with

observations

and Bullen

on

Therefore,

it is clear that hardening by impurities in these systems results in a temperature

a

the

thermal

for

Peierls-Nabarro

mechanism

of

previously the

Peierls-

either are not valid mechanism

is not

of yielding in the tempera-

ture range studied. A test of a theory can be made by the predictions makes.

The

Peierls-Nabarro

theory

the effect of interstitials

not predict the strong orientation yield

stress

of

many

b.c.c.

does

on yielding,

which has resulted in a complete

single

not

it does

dependence

metal

of the crystals

breakdown

of a constant

of the

sensitive yield stress. A model

value of the

resolved shear stress at low temperature of impurity

content

independent

appear to be satisfied.

It could

be argued that the value at 0°K would be a constant, but it is clear that at a temperature

of 10°K or higher,

materials at reasonable temperatures.

nickel is relatively

as

What

constant

to pure nickel results in a yield stress which is very even though

the

is that the experimental

of

is of little

dependent,

do

that

of the yield stress, and independence

thermal

reported

be

in b.c.c.

constancy

(i.e.

possibility

this basis, but it is clear that impurity models must also qualify. Flinn 05) has shown that adding carbon temperature

clearly

However,

as well.

nor does the prediction

other

activation

the mechanism

mobility

critical resolved stresslaw at low temperatures,(2*4~1s-21)

mechanism. The

be

it might be interesting to examine the effect of impuri-

predict

dislocation

would

dependent

to more

ties not only on the pre-exponential,

as the temperature inconsistent

metals

temperature

need to be performed

understand

stresses that decrease

activated

a

It is most heartily agreed that dislocation

it

thermally

in b.c.c.

produce

experiments

energy for the higher constant

a trend that is wholly

to

hardening.

an apparent activation

of

1967

that interstitials

expected

In fact, Stein and Low’s data suggest decreases,

15,

of solution hardening presented by Fleischer(18) would

In addition,

lying

for the dislocation The mobility

screw

reveals that they are heavily

and disturbed therefore,

connecting

dislocations

VOL.

lie along close

the edge segments.

examination

valley;

range

is more likely due to the much higher

of the edge dislocation results in a rather straight

jogged

(as well

in both the thermal and athermal

regions. (12s13) Therefore,

temperature

force.

temperature

in Si-Fe that the dislocations

this alignment which

metals)

in the paper under discussion.

observed packed

by the Peierls-Nabarro

for relatively

b.c.c.

METALLURGICA,

the deviation

will be very consequence

strong,

to

the

and therefore

behavior

of

It is not surprising that the Peierls-Nabarro can be adjusted

to fit the experimental

expression is of the form Ae@‘r are

adjustable

parameters;

model

results.

The

where both A and therefore

it

b.c.c.

there

B

is

a

very large range of results that could be satisfied by such an expression. For these many

reasons,

thermally-activated Nabarro

process

based

upon

a Peierls-

model of yielding is not an adequate

for yielding in b.c.c. metals. mechanism, support

it appears that a single

but

the

the position

theory

It may be an important

reasons

so far

advanced

to

that it is the only important

LETTERS

thermal

mechanism

many cases contrary

are simply

not adequate

to experimental

TO

and in

THE

153

EDITOR

vibrations

were assumed

results.

force f is proportional

D. F. STEIN General Electric Co.

known

References

* Received June 8. 1966.

that

expressed

of the Debye

of the role of atomic

Herein

as a basis for computing atoms

as a function

this equation,

is

well

dependent

energy of two r apart.

required

the atoms from the equilibrium and the constants

In

to separate

distance r0 to infinity,

m and ‘n (frequently

upon the particular

metal.

m and n, have been determined

integers) are The values of

for many metals by

Fiirthc2) and for the rare gases by Lennard-Jones.(3) M and C used equation

(4) to compute

the restoring

force inducing an atom to return to a position midway between

two

metallic

lattice

equation

(2). K

neighboring

atoms

in a linear

The expression for

small

in the crystalline

manner

expressed

for the restoring

displacements

by force

is given in This equation

(Isa) of the M and C paper.

in the

as a test of the validity

we furt,her test

the

Debye

Zegmn f.0” i

a simpli-

vibrations

the picture

to comput,e the Debye

known,

the potential

of their distance,

e, is the energy

g=-

of by

tempera-

ture Bo. As

of the

is repeated here

melting process of metals and used the computation

using the equations

body.

to reach an understanding

melting process, M and C used the Griineisen equation

and Chamberlain(l)

(hereafter referred to as M and C) presented

the picture.

by the equation

of pure metals*

In a recent paper, ~~~~chlan

temperatures

It is also force constant. Y of the oscillator is

frequency

In an attempt

constant

computations

the

where ml is the mass of the oscillating

equation

temperatures

(2)

where K is the restoring

1. D. F. STEW, Acta Met. 14, 99 (1966). 2. D. S. TOMALIN and D. F. STEIN, Trans. Am. Inst. Min. metall. Engrs 233, 2056 (1965). 3. D. F. STEIN, J. R. Low, JR. and A. V. SEYBOLT,Acta Met. 11, 1253 (1963). 4. D. F. STEIN and J. R. Low, JR., Acta Met., to be published. Proc. R. Sot. 2648, 176 (1961). 5. B. ED~~IOXDSON, 6. H. CONRAD and S. FREDRICK, Ada Met. 10,1013 (1962). 7. H. CONRAD and W. HAYES, Trans. Am. Sot. Met&. 58, 249, 844 (1963). H. CONRAD, J. Iron. Steel Inst. 1$8, 264 (1961). :: B. L. MORDIKE and P. HAWSER, Phil. Mug. 7, 459 (1962). 10. J. W. CHRISTAIN,Acta Met. 12, 1 (1964). 11. A. S. KEH and rS.WEISSMAN, Electron Microscopy and the Strength of Crystals, p. 231. Interscience, New York (1963). 12. J. R. Low, JR. tEndA. M. TURKALO, Acta Met. 10, 215 (1962). 13. A. M. TURKALO and J. R. Low, JR., G. E. Res. Lab. Report, 64-RL-3614M (1964). 14. J. E. DORN, J. MITCHELL and F. HAWSER, Exp. Mech. Nov. 1 (1965). in Solids, p. 17. 15. P. A. FLINN, Strengthening Mechaksms ASM, Metals Park, Ohio (1962). and F. P. BULLEN, Phil. Mug. 7, 1535 16. H. M. HUTCHISON (1962). 17. W. G. JOHNSTON, ,7. appl. P&s. 33, 2050 (1962). 18. R. L. FLEISGHER,J. appl. Phys. 33, 3504 (1962). 19. R. M. ROSE, D. P. FERRISand J. WOLFF, Trans. Am. Inst. Min. metall. Engrs 224, 981 (1962). 20. Y. NAKADA and A. KEH (New York Meeting of AIME, 1965). 21. D. F. STEIN A& Met. to be published.

of melting

to the displacement

Center

Schenectady, N. Y. 12301

fied picture

kind

forces such that the

f=-KX

Research and Development

Approximate

to be of the harmonic

where there are linear restoring

temperature

is

1 -

where f is the fractional

(m _t ?2,-

3)f

i

(5)

increase in r as the tempera-

ture is raised above zero. from the coefficient Substituting

K of equation

and substituting equation

Of course f can be obtained of expansion.

the resulting

(5) into equation

(3)

Y of equation (3) into

(1) while letting f equal zero gives,

defined by the equation

where h is Planck’s constant, k is the Boltzmann constant and v is the maximum frequency of the vibrating the

atoms about their equilibrium positions in In the Debye theory these crystal lattice.

e,

where, AH,,

=LI

AH,, .--

W

is the heat, of sublimation

at absolute