Latent hardening in iron single crystals

LATENT

HARDENING

IN

Y. NAKADAt

IRON

SINGLE

CRYSTALS*

and A. S. KEHt

The ls,tent hardening in iron single crystals was studied as a function of Burgers’ vector combination, amount of prestrain, deformation temperature, and strain rate. It was found that the latent hardening ratio varies between 1.2 and 1.4, relatively independent of the afore-mentioned variables. The temperature dependence of the flow stress is higher on the latent system than on the primary system down to 7YK; but the strain rate sensitivity is similar for both systems. It is concluded that at high temperatures (above 300”K), latent hardening in iron is due to either the elastic interaction between glide and forest dislocations or the lack of mobile dislocations in the latent system or both; while at low temperatures (below 300°K) latent hardening is due to the lack of mobile dislocations in the latent system. LE DURCISSEMENT

LATENT

DE MONO-CRlSTAUX

DE FER

Les auteurs ont etudie les durcissements latents des monoeristaux de fer, en fonction des combinaisons du vecteur de Burgers, des temperatures de deformations anterieures, du teux et de la vitesse de celles-ci. 11s ont trouve que le rapport du durcissement varie entre 1,2 et I,4 ce qui le rend relativement independant des variables mentionnees ci-dessus. L’influence de la temperature sur la tension de glissement est plus importante vis Q vis du systeme de glissement second&ire que vis B vis du systeme de glissement primaire; cette observation se verifie jusqu’b une temperature de 7’7°K. La sensibilite des deux systemes de glissement vis B vis de la vitesse de deformation est identique. Les auteurs concluent qu’aux temperatures &levees (au dessus de 300°K) le durcissement latent resulte soit des interactions Blastiques entre les dislocations mobiles dans le systeme de glissement latent soit, enfin, des deux raisons precitees. A basse temperature (au dessous de 300°K) le durcissement latent resulta du nombre trop restreint de dislocations mobiles. LATENTE

VERFESTIGUNG

IN EISEN-EINKRISTALLEN

Die latente Verfestigung in Eisen-Einkristallen wurde in AbhLngigkeit von der Kombination der Burgersvektoren, dem Betrag der Vorspannung, der Verformungstemperatur und der Verformungsgeschwindigkeit untersucht. Es stellte sich heraus, dass das latente Verfestigungsverhiiltnis zwischen 1,2 und 1,4 variierte und zwar relativ unabhhngig van den zuvor genannten Grossen. Die Temperaturabhkngigkeit der Flieljspennung ist bis hinunter zu 77°K im latenten System grosser als im primiiren System; der Einfluss der Verformungsgeschwindigkeit jedoch ist in beiden Systemen iihnlich. Hieraus wird geschlossen, dass bei hohen Temperaturen (tiber 300°K) die latente Verfestigung in Eisen entweder durch die elastische Wechselwirkung zwischen Versetzungen im Gleitsystem und Waldversetzungen, oder durch die geringe Anzahl beweglicher Versetzungen im latenten System oder durch beides verursacht wird; bei tiefen Temperaturen (unterhdb 300°K) dagegen tritt die latente Verfestigung infolge des Mangels an beweglichen Versetzungen im letenten System auf.

INTRODUCTION

of the latent

It is generally agreed that the work-hardening crystal is caused by dislocation various

work-hardening

interactions

are more

interactions.

theories

of various dislocation

principle,

be studied by activating

and then measuring second

slip system

system.

Ideally,

second

(latent)

interactions

both

the first (primary)

slips should

however,

to operate a

the previously

this condition

one always observes an appreciable

hardening

useful information In f.c.c. crystals, deformation Therefore,

structure,

be classified

In

is not realized, and amount of second-

ary dislocations in a so-called “single slip” deformation in which the major portion of the strain is caused by

between

into

of

reactions

during

slip systems.

can be grouped

Similarly, in crystals with

the dislocations

belong to one

In this case, the reactions two groups,

orthogonal

dislocations.(2)

generated

to one of twelve

all dislocation

and the

still yield

importance

interactions.

into five different classes.(l) the rock-salt

should

the relative

any dislocation

belongs

of six slip systems.

is produced.

behavior

about

the various dislocation

operated

be ideal single slips in

which only one kind of dislocation practice,

could, in

a single slip system

the stress necessary through

However,

disagree on which The relative im-

important.

portance

of a

namely

dislocations

may

the reaction

and nonorthogonal

On the other hand, because iron does

not slip on one particular slip plane, one cannot classify the dislocation systems.

interactions

according

to their slip

They can be classified, however,

to the Burgers vectors of the participating

according

dislocations.

the dislocations of the primary slip system. On the other hand, as long as one realizes this fact, a study

The dependence of the latent hardening on the Burgers

* Received September 23, 1965; revised November 8, 1965. t Edgar C. Bain Laboratory for Fundamental Research, United States Steel Corp., Research Center, Monroeville, Pennsylvania.

dence”.

ACTA METALLURGICA,

VOL. 14, AUGUST

1966

vector combination [ill]

as the primary

in Fig. 1. 961

will be called the “system

In this paper,

we shall arbitrarily

slip direction

depenchoose

in iron, as shown

ACTA

962

METALLURGICA,

VOL.

14,

mechanisms.

By studying

lat’ent system, thermally

1966

these dependences

it is possible

activated

of the

to infer what

type of

is controlling

the flow

mechanism

stress of the latent system. Because latent hardening is the result of interactions involving

secondary

kinds of secondary

dislocations, dislocations

mary slip may be important

the

amount

and

during pri-

produced

variables

of the latent

hardening. Although papers

there has been a considerable

published

ening,o-15)

on

the

subject

of

number of

latent

by Basinski and Jackson,03*14) Kocks(i*il) FIQ. 1. Specimen

and its stereographic

the definition

of the magnitude

hardening varies from one investigator

of latent

to another, we

have chosen the ratio of the flow stress of the latent system

after a given amount

of primary

system

at the end of the

prestrain as the magnitude

of latent hardening.

is referred to as the latent hardening In the following,

the symbols

This

ratio (L.H.R.).

T and y denote resolved

shear stresses and strains, which are referred by the suffixes p and 1 to the primary and latent slip systems, respectively.

In this terminology

latent hardening

the magnitude

of

is expressed as:

(-‘-2 i If crystallographic

Priestner(15) on silicon-iron

is the only one on b.c.c.

metals. EXPERIMENTAL

The

raw

Ferrovac

material

E iron

Corporation.

PROCEDURE

used

was

supplied

The chemical

shown in Table

1.

analyses

The procedure

(wt.

%)

S ~ 0.005

Si 0.006

Ni ~ 0.005

Cr __ 0.001

v _ 0.004

W ~ K.D.

MO 0.001

Cu 0.002

Sn ~ 0.003

Pb 0.0003

Co 0.004

N __ 0.0003

0 _ 0.007

the single crystals was basically that of Stein and Low,(la) as described fully in a previous publication.(17) of the specimens slip system

the

of

planes

for

primary

and

latent

However, if this is not possible, then because magnitude

with various

of the frictional

possible

lation may not be strictly true.

slip planes, However,

had average

this re-

procedure

content

by choosing

decreased

can

the latent hardening useful in the

analysis of latent hardening is the difference between flow stress of the primary and latent systems at a given primary strain. This is expressed as:

specimen

the carbon

activated

of the flow

eliminated

complicate

room

treatment to 0.0016

temperature

the interpretation

behavior.

were carefully

of

The surfaces of each

polished,

specimen was used in three ways. to a predetermined

and then unloaded.

dependences

by an annealing

first mechanically,

then chemically in a 2 : 1 mixture of orthophosphoric acid (85 ‘A) and hydrogen peroxide (50 %). The large prestrained

%l)yB

of thermally

in.

This decarburizing

aging which would

flow stress are results

7

of 24 hr at 850°C in dry hydrogen.

(211) or (321)), this anisotropy

The strain-rate and temperature

1 x

in wet hydrogen

the slip planes to be near one of the closer packed

AT I--p = (71 -

x

for 120 hr at 760°C

stresses

planes (i.e. (llO), be minimized.

which is sometimes

of 0.05

The crystal was then decarburized

wt. o/0 and completely

quantity

so

case had a

The finished single

dimensions

followed

were controlled

in every

Schmid factor very close to 0.5.

Another

of this iron is

used for growing

P 0.002

crystal

associated

of

Metals

Mn __ 0.001

This relation is also true for iron crystals if we choose systems.

stock

C __ 0.004

The orientations

of the different

bar

the Vacuum

TABLE 1. Chemical enrtlyses of Ferrovac E iron (D79)

that the primary type

1 in.

by

identity between slip systems could

be assumed (as for f.c.c. crystals and ionic crystals),

same

in

Of all previous papers on latent hardening,

as far as the authors are aware, the work of Hu and

slip to the

flow stress of the primary

and Hu and

Priestner,05) they will not be discussed individually

projection.

this paper. Although

hard-

because these works are well summarized

In one case, it was

stress or strain value

Small specimens (1

x

0.1

x

0.04

in.) were then cut from the large crystal in different directions to study the latent hardening caused by

NAKADA

dislocations

LATENT

AND KEH:

The relation

of different Burgers’ vectors.

HARDENING

IN

in length of an inscribed gage mark, using the travelling

of the small specimens to the large one is schematically

stage

of a microscope.

shown in Fig. 1.

from

77°K

Point

A on the stereographic

pro-

jection shows the surface normal to the flat specimen.

to 473°K

temperature

Test temperatures were obtained

liquid baths.

Point P represents the primary tensile direction using [lli]

as the Burgers vector.

P direction,

After prestraining

small crystals

TABLE 2. Controlled

in the

with tensile directions

of

Temperature

Q, R, S, and T were cut out. The Burgers vectors for the slip systems [ll l] and [lil] small crystals

in these crystals respectively.

were [ill],

Also, at t’he same time,

parent crystal to determine

to the

the effects of unloading,

handling during spark-cutting,

unavoidable

room tem-

perature aging and the different specimen

size on the

yield

stress after reloading

work-hardening fully

behavior.

on a Servo-Met

spark intensity

#5.

and on the subsequent The cutting was done care-

spark-cutting After

gage section was produced

machine

spark-cutting,

using

a reduced

by chemical polishing.

In the other two cases, the large specimen into four specimens of intermediate

size (3.5 x 0.5 x

0.05 in.), and these were either prestrained

to a given

at the

during

strain rate (either 1.39 x 10-3/sec or 2.78 x 10-3/sec).

crystals and used in a

on the magnitude

this way four crystals obtained

of latent hardening.

of identical

work hardening tation degree. latent

from the orientation relationship.

Since the

of iron single crystals is quite orien-

dependent,07)

hardening

included in the calculation.

were

of the stress-strain it

hardening

ideally

deformation.

shear stresses on secondary

tensile

because

care was taken to minimize the

Because axis

Burgers’ vectors

deformation

and the latent

of this inability

for the primary

systems (and hence the orientation latent hardening

However,

this was not always accom-

resolved

tests.

is not eliminated)

to obtain and latent

dependence

of the

the interpretation

of the system dependence of the latent hardening is not clear-cut,. However, if one realizes this limitation one can still obtained meaningful information about the system dependence of the latent hardening. All specimens were strained on a table model Instron tensile machine 104/sec.

using a tensile strain rate of 2.78 x

Elongation

has reported that at room temperature

the

the (111) zone axis which bears the maximum resolved

Nevertheless,

same

Keho’)

to a certain

plished.

the

1. General observations

axis for the

triangle as the tensile

hardening

RESULTS

dependent

tation within the stereographic

during both the primary

EXPERIMENTAL

were

the tensile

the

have the same orien-

of the specimen geometry,

and the area

changes during the deformation

slip plane of iron single crystals is the one containing

that

tests should

axis for the primary

strain calculations,

latent

is likely

is also orientation Therefore,

All shear stress-shear

AT, were done by a computer,

In

could

dependence

including

and orientation

orientation

be studied without complications

changes in stress were always used in evalu-

of

and the effects of various parameters

The

was studied by cycling between

the base strain rate (2.78 x 10m4/sec) and a higher

then cut from the intermediate prestrain

The temperature

a test was no more than &l”C.

strain-rate sensitivity

Upward

and amount

placed tluctu-

ating AT.

of temperature

were measured

thermocouple

center of the gage section.

Smaller crystals

of the effect

Temperatures

about 0.5 in. from the surface of the specimen

strains at room temperature. study

baths

Liquid nitrogen Freon 12 and liquid nitrogen coil Methyl alcohol precooled with dry ice Methyl alcohol precooled with dry ice Air Boiling water Hot silicone oil Hot silicone oil

by means of a copper-constantan

strain at various temperatures or prestrained to various were

temperature Bath

atures and liquids used.

ation

was cut

varying

by controlled-

Table 2 lists the temper-

(“K)

77 148 201 248 298 373 423 473

[Iii],

(P in Fig. 1) were cut parallel

963

Fe

was measured from the increase

shear stress.

In most cases, this was found to be true down to 77’K. However, it must

in this investigation be remembered planes, (llO),

that at least one of the closer packed

(211) or (321), was always very near the

plane which bore the maximum Occasionally,

resolved shear stress.

the expected

system did not operate, or

it shared the deformation

with another unexpected In these cases

system having a smaller Schmid factor. one particular jection

secondary

of its Burgers’

dimension

slip system having the provector

of the cross section

This observation

parallel

to the short

was always

ations reported by Wu and Smoluchowski,(ls) et aLog) and Nakada et aZ.‘20) As pointed

favored.

agrees with several similar observ-

out by Kocks,o)

one cannot

Hikata force

a

particular secondary system to operate by orienting a specimen so that the system has the maximum resolved determine

shear stress.

Therefore,

which system actually

it is essential

to

operates in a latent

ACTA

964

METALLURGICA,

VOL.

14,

1966

Fe-42

0

a04

0.06

0.12 SHEAR

0.16

0.20

0.26

0.24

STRAIN

FIG. 4. Latent hardening behavior of Fe-42. FIG. 2. Inhomogeneous slip in latent system.

hardening

test.

This was accomplished

ously by two-surface In all except pected

trace analyses of the slip lines.

two crystals

slip systems

77°K showed

strained

operated.

about

The

unambigu-

at 77”K,

The two crystals

equal amounts

on the latent and primary

the exat

of slip activities

be exceeded

systems.

flow stress for the primary before yielding

larger than ml/m,, Basinski hardening

and

system may

on the latent system.

other words, the latent hardening

with an inhomogeneous

that

is always

slip pattern.

crystals, some indications typical

distinct

curve

of

three-stage

by Keh(l’) previously. treatment,

the

primary

hardening,

system

as reported

Because of the decarburization

there was no yield drop associated

virgin yielding

with

of the primary system, and no Liiders

strain was observed.

However,

the yielding behavior

latent

associated

Although

of inhomogeneous

this

strain (see Figs. 7 and 8). The effects of unloading, and room temperature experiments

the subsequent

Therefore,

always

cut

slip were

control

specimens

on the latent systems. Figure 2 shows a However, no particular correlation

was found between this type of slip and the relation-

parallel

smaller size

of the latent

relies on the assumption

do not affect hardening.

spark-cutting,

aging were carefully

the interpretation

in iron

example.

studied

hardening

that these factors

flow stress and work

several control specimens were to the parent

specimen.

were then processed

the same way as the latent hardening

The and tested in

specimens.

all cases, the flow stress and the work hardening of the control specimen were within *.!i’A

ship between the Burgers vectors of the primary and

800

I

600

5 z R B

400

% x 200

0

0

0.05

1 0.15

I

/

/

0.25 030 020 SHEAR STRAIN

0.35

1

0.40

I

a45

FIG. 3. Latent hardening behavior of Fe-48.

1

050

I 0.10 0

/ 0.20 0.10 SHEAR

0.30 0.20

In rate

of the final

the latent systems.

I 010

and

in some cases showed a distinct yield drop and Liiders

because

where m is the Schmid factor.

was not a general feature of latent hardening observed

In

ratio rI/~, may be

Jackson’13s14) reported

in Cu single crystals

stress-strain

of the latent system was very system dependent

It is interesting to note that due to latent hardening the previous

showed

0.40 0.30

STRAIN

FIG. 5. Latent hardening behavior of Fe-58.

(3.:i0

NrZKADA

I

I 0.10

0

SHEAR

FIG. 6. Latent

HARDENING

IN

I

0.40

0.50

STRAIN

hardening behavior

of Fe-66. I

flow

stress and work hardening

crystal.

0

rate of the parent

several virgin crystals of the small size. The resulting curves

were

almost

identical

to

the

stress-strain

curves of the larger crystals of the same

orientation.

Thus, any latent hardening

a genuine property due to variations

observed

of the latent system

is

and is not

in specimen preparation.

Figures 3-8 illustrate

SHEAR

Table

Burgers vector combinations

3 gives the various

used in Figs. 3-8 and in

Primary

Fe-42

11111

(Fig. 4)

Fe-47 Fe-48

[lli]

(Fig. 3)

[iii]

#2: #3: #4:

[ill] [lil] [ill]

#l: #2: ff3: $4:

11.

reproducibility

#g”; f:;;;

Fe-69 Fe-66

(Fig. 11) (Fig. 6)

riii]

R: [ill] L: [Iii] #2: [ill]

[Iii]

of data.

I

I

were

tested

up to four to

check

The reproducibility

the

was very

good, the scatter being no more than &5x. 2.1 Efect

L.H.R.

of Burgers vector combinutioru.

Figures

As shown in these figures, the

(latent hardening ratio) at a given stress level

of the primary dependent. because

systems seems to be not very system

Actually,

the

as it was pointed

orientation

dependence

dependence

out before,

of the

latent

may result from the can-

system

because the apparent system is consistently small, it is quite unlikely dependences

always

Thus, it is inferred here that the

dependence

crystals.(llJ4) havior

depend-

However,

cancel each other.

is

rather

small,

On the other hand,

(the shape of yield point)

as

in

f.c.c.

the yielding

is strongly

be-

system

dependent.

and [lli]

Curve No.

4 of Fig. 4 shows

an extremely

high

work hardening rate. double

I

and sometimes

system

that large system and orientation

ililj

[iii] [I ii]

I

of one

dependence

and [ill]

riiii

(Fig. 8) (Fig. 5)

9001

At least two

crystals

ences.

[ill! [ill]

Fe-52 Fe-55

STRAlhl

celling effect of large system and orientation

Qs: P111 Q4: [JlJl

[lli]

(Fig. 7)

Fig.

slight system

Latent

#I:

I

0.50

hardening is not eliminated in these tests, the apparent

TABLE 3. Burgers vector combinations Specimen

I

0.40

FIG. 8. Latent hardening behavior of Fe-52.

on latent hardening.

some of the results of latent

experiments.

0.50

3 and 4 show the effect of Burgers vector combinations

2. Latent hardening behavior hardening

0.20

0.10

The size effect was also tested by straining

stress-strain

965

Fe

I

I

0.30

I

0.20

LATENT

KEH:

AND

I

I

for

slip, even

single

hardening

slip.

This was caused by unexpected though the crystal was orient’ed

In

this

of the intended

case,

because

latent system

the

latent

was higher

than that of a neighboring latent system, it had to share the deformation with a neighboring system of lower latent hardening, even though the Schmid factor of the intended system was higher. 2.2 Resolved shear stress on the latent system. If latent hardening is caused by dislocation interactions between the primary system and the latent system, then the extent of dislocation motion and multipli-

loo%0

I 010

I 0.20

030 SHEAR

FIG. 7. Latent hardening

I OLO STRAIN

behavior

I 0.50

I 0.60

of Fe-47.

070

cation

on

deformation

the

latent

may

be

system an

during

important

the

primary

variable.

An

extreme example of this is found in ionic cryst’als, where

ACTA

966

the resolved

METALLURGICA,

VOL.

shear stress on the latent system is zero

during the primary deformation.(2)

14,

8001

1966

I

I

I

To study the effect 700

of this variable, a crystal was so oriented that the latent

c

system had no resolved shear stress during the primary slip. Also, the primary system had no resolved shear st#ress during the latent test. Fig. 5.

The magnitude

of latent hardening

smallest of all crystals tested. origin

coincide

on the

strain

was the

axis

without

with that for curve

Q.

Q is very

shifted

to

The shape of the

stress-strain

curve

stress-strain

curve Q,, of the latent system, and they

similar

to the virgin

and Jackson hardening

and Kocks’ll)

of the coplanar

system.

2 t

400

ti

300

.

Basinski

0

hardening

ratio is essentially

was tested

characteristic [lli]

behavior.

(101) during

hardening

Therefore,

is shown

such a

also showed

this

Fe-66 was chosen to slip on

primary

slip and on [ill]

during the latent hardening temperature

one.

to see if iron

test.

in Fig.

(101)

The result at room

6*.

Although

latent

was not zero, the value, 1.2, was one of the

lowest observed

in our experiments.

strain curve of the latent

Also, the stress-

system rather quickly

joined

the continuation

of the primary

curve.

The same experiment

re-

stress-strain

was done at 150°C.

The

result was very similar to that at room temperature, the L.H.R.

being 1.25 at 150°C.

2.4 Effect of strain.

Figures 7 and 8 illustrate

effect of strain on latent hardening. very

strong

Because to

function

7p increases

decrease

with

characteristics prestrain.

AT~_~ is not a

of the amount with strain,

increasing

of prestrain.

the L.H.R.

strain.

are also insensitive

The

tends yielding

to the amount

Figure 7 shows a good example

drop upon reloading

the

of

of a yield

in the latent system.

Figures 7

and 8 also include the virgin stress-strain

curves of

the latent system. 2.5 Summary of room temperature 9 shows the results from all crystals temperature,

plotted

results.

Figure

.

9,

:

~~‘1.1

~~+56.5

kg/cm2

CORRELATION

v

I

I

100

200

FACTOR = 0.977 I

I

500

600

I

300

400

~~(kg

line represents

the hypothetical

very small amounts

700

/cm*)

stress of latent system primary system.

FIG. 9. Flow

much different from all other systems, and the latent system

/

I :

showed that the latent

systems in f.c.c. crystals is

A

::

200-

can in fact be made to coincide by a vertical translation of Q0 by about 100 kg/cm2. 2.3 Latent hardening of a coplanar

I

-5co!/ :

prestrain.

has been

I.

!-. :: i i

“E

Q,, is the virgin stress-

strain curve of the latent system Its

. .:

The result is shown in

.L4 .. /

versus flow stress of

latent hardening

of prestrain,

extrapolated

initial flow stress of the iron crystals.

It is interesting

to note that in all cases TV is larger than -rP. The spread

of

points

the primary the

at

system

manifestation

of

latent hardening.

a

particular

stress

is not experimental slight

system

A least-squares

using 76 data points.

level

scatter

dependence

The line can be represented

by

an equation 71 = 1.1 7D + 56.5 kg/cm2. The correlation

factor for this equation was 0.977.

The L.H.R.

as a function

of the stress

primary system was calculated plotted value

on the

from the equation and

in Fig. 10. This ratio first rises sharply to a of

saturation

1.4

and

2.6 Temperature pendence

then

gradually

decreases

to

of

dependence.

latent

The temperature

hardening

was

studied

deusing

2.01 3 \

d

0

a

value of about 1.2.

The dotted

* It is important to realize that the experiments represented in Figs. 5 and 6 are not the same kind even though the tensile axesTobr the latent systems are in the same stereographic triangle. In Fig. 5, the slip plane for the latent system is not the same as that for the primary system because of the pencil glide in iron. On the other hand, in Fig. 6, the latent hardening specimen was oriented in such a way as to make the slip plane for the latent system the same as that for the primary system.

of

line was calculated

tested at room

system.

of but

as flow stress of latent systems

versus ilow stress of the primary

for

to the

100

200

FLOW STRESS

300 OF PRIMARY

400 SYSTEM

500

600

700

rP h/C&

FIG. 10. Latent hardening ratio versus flow stress of primary system.

NAKADA

LATENT

KEH:

AND

HARDENING

IN

967

Fe

using a shear strain rate of 1.1 x lo-s/set,

and latent

hardening specimens of two different orientations

were

restrained at various strain rates (6.7 x 1O-5 N 2.7 x lOW/sec) to investigate latent hardening.

the strain rate dependence

Figure

13 shows the results.

yield stress increases with increasing when plotted

on a semilogarithmic

of The

strain rate, and scale the experi-

mental points for both the primary and the two latent orientations

can be represented

by two straight lines,

as shown by previous investigators.(sl)

The variation

of the latent hardening ratio as a function of the strain rate is not systematic and is rather small. 2.8 Strain rate sensitivity. The strain rate sensitivities of primary and latent systems were studied by I

I

0.20

0

I

I

0.40

cycling between a base rate and a higher rate. The strain rate ratio was usually 5 : 1 or 10 : 1. The cycling

I

/

0.60

0.80

was done immediately

SHEAR STRAIN

11. Effect of temperature

FIG.

hardening

on latent hardening.

was not destroyed crystals Fe-54, Fe-55, and Fe-69. were conducted.

Two kinds of tests

First, the parent crystals were pre-

strained at different temperatures, latent

hardening

specimens

prestrain temperatures. were prestrained

and then the small

were restrained

Secondly,

at the

the parent crystals

at room temperature,

and the latent

hardening specimens were tested at other temperatures. The latent

hardening

behavior

same for both kinds of tests. rized in Figs. stress-strain room

11 and 12.

the

latent

dependence 77°K. with

the

The results are summaFigure

11 shows typical

curves when a crystal is prestrained

temperature

and

lower temperatures. of

was essentially

then

restrained

at

at various

As one can see, the ilow stress

system

has

a

higher

At

system.

room

essentially systems

temperature, and

temperature

The latent hardening

as shown

ratio is essentially

in Fig.

12.

constant

at

all temperatures. 2.7 Strain rate dependence. to 16%

Crystal Fe-71 was pre-

shear strain at room temperature,

the

than

is constant

quantity

on the

in Fig. 14. A-r/Aln

9

is

and the latent At subzero strain.

with

A-r/A In 9 is slightly higher on the latent At lOO”C, on the primary system.

Ar/A In i, is about 3 times larger for the latent system than for the primary system initially. due to the initial nonhomogeneous latent

system.

However,

with

This might be

deformation

on the

increasing

strain,

AT/AIn i, of the latent system decreases gradually until it is the same as that of the primary system.

temperature

Thus, the flow stress difference, A-ra+ increases

deformation

plot is shown

the same for the primary

temperatures system

by excessive

A typical

DISCUSSION

than that on the primary system down to

decreasing

strained

latent

after yielding during the latent

test to insure that the effect of prestrain

1. General remarks In view of our experimental theory explain

of the latent hardening the following

hardening

results,

a successful

in iron crystals must

characteristics:

(1) The latent

ratio is always greater than one regardless

1000

10-5 100

200 TEMPERATURE,

FIG.

300 ‘K

12. Flow stress difference versus temperature.

10-4

‘:;;AIN TENSILE

400 FIG.

‘%t.-I RATE

13. Strain rate dependence

10-I 1

of flow stress.

1.0

ACTA

968

METALLURGICA,

VOL.

14,

1966

certain critical temperature

and which is proportional

to the square root of the concentration

Fe-55

impurities.

of interstitial

Of course, in both theories the long-range

stresses of dislocations

must be added to the impurity

effect. Thus, in both theories, the flow stress is the sum of (1) dislocation-impurity tion-dislocation distribution

interaction

interaction.

to be uniform

and (2) disloca-

Assuming

the impurity

in a deformed

crystal,

is difficult to see how the dislocation-impurity

it

inter-

action could be always larger for the latent system. The long-range internal stresses may come from two 0.05

0

0.15

0.10

0.25

0.20

E FIG. 14. Strain rate sensitivity versus strain.

of the Burgers vector combination,

deformation

perature,

of prestrain.

strain rate, and amount

The system dependence is surprisingly tation yield

of work

(3) The yielding point)

(2)

of the latent hardening ratio

small as compared

dependence

Keh.o’)

tem-

to the large orien-

hardening

behavior

reported

by

(the shape of the

of the latent

system

is highly

system

dependent.

(4) The latent

system

exhibits

a higher

temperature

dependence

primary system. exhibit

of flow stress than does the

(5) The latent and primary systems

similar strain rate sensitivity.

indicates the same activation

This in turn

volume for both systems

at a given stress level, as shown in Fig. 16.

It is convenient temperature

of temperature flow

namely

and the other

stress

increases

one at or above

below

rapidly

independent 300°K

with

where

decreasing

at or above 300°K.

the temperature

There are at least two proposed flow

stress

anisms

in this region.

Let

Stroht2’) calculated slip systems locations

to see if they hardening

due to piled-up

dis-

in the case of a He concluded,

in a cross-slip system.

therefore, that the latent hardening systems due to the long-range dislocations

of latent

system and found it

less than one except

screw dislocation

are

behavior.

arrays of primary

to that of the primary

to be always

of secondary

slip

stresses of the primary

must be equal to or less than the workThis prediction

hardening of the primary slip system. is in direct

contradiction

observations,

which

to

indicate

all lat,ent hardening that

latent

hardening

Furthermore,

the

ratio of one is the coplanar system in f.c.c. crystals.(lm4) The long-range

stress theory

is 1.2-1.25,

us

which

tions may be important

Seeger’ss)

stress caused

by Snoek ordering

by a

disagrees

with

the

between the primary and forest disloca-

ture independent

in determining

flow stress.

flow stress according

the tempera-

The magnitude

of the

to this model isc26)

of inter7 = Gb l/p,

perature independent and proportional to the concentration of interstitial impurities. On the other hand, Fleischer(23) proposed that the flow stress is controlled by a short-range interaction of dislocations wit.h interstitial impurities which cause tetragonal distortion of t,he lattice. This theory predicts a flow above

a

(B = 2.5-10)

B

predicts a flow stress which is tem-

independent

also

Basinski(25) and Saada(26) have proposed that elastic

for the

and

a latent

Furthermore,

stress theory.

mechanisms

Schoeck

would predict

in b.c.c. iron the latent hardening ratio of a coplanar system

stitial carbon and nitrogen atoms around dislocations.

is temperature

latent

the ratio of hardening

interactions

frictional

which

the

region.

that the flow stress may be controlled

This mechanism

be considered

with

independent

proposed

stress

will now

compatible

long-range

2.1 Latent hardening

(1) the interactions

hardening ratio of + for that system.(s7)

temperature. first consider

by:

only latent system observed to have a latent hardening

300°K where the flow stress is essentially the

are caused

two or more primary dislocations, such as piled-up groups(24) and (2) the interactions between primary and forest dislocations.(25p26) These two mech-

effect) equal to or more than one.

to consider the latent hardening in regions,

They

between

ratio is always (except in the case of the Bauschinger

2. Latent hardening mechanisms two

sources.

where G is the shear modulus, b the Burgers vector and pr is the forest dislocation density. Kehd’) determined in iron

the ratios

of forest

dislocation

density

to

primary dislocation density at various points of the stress-strain curve by electron transmission microscopy and found them to vary from about 1:5 at low strains to 1:2 at high strains for bhe single slip

NAKADA

condition.

From

these

solely determined

KEH:

AND

data-if

LATENT

the tlow stress

by the forest interaction-the

is

latent

hardening ratios should be

where

22-14

pD is the dislocation

reduced

However,

because

the flow stress.

(2)

density

system and ps is the dislocation ary systems.

Fe

969

impurity content of the iron.

This does not necessarily is not contributing to

mean that the forest cutting

of the primary

density of all second-

these values

of the contribution

are somewhat of impurities

In this case, the L.H.R.

to

is

because

of the presence

coplanar tional

system

where 71i and 7gi are the impurity

contributions

the flow stresses of the latent and primary

and rlf and rDf are the forest dislocation

contributions primary

to

systems

to the flow stresses of the latent

systems,

respectively.

Taking

the

and

initial

yield stress as 7,; and the increase of flow stress due

However, hardening in the

of latent

(Fig. 6), we must consider an addi-

mechanism

of latent

hardening

which

coexist with the forest cutting mechanism.

may

Li(28) has

recently proposed that latent hardening may be caused by the lack of mobile dislocations

in the latent system

to maintain

the applied

strain rate.

the primary

dislocations

immobilize

of the latent (3)

respectively,

can still be

explained by the forest cutting mechanism.

-z= . .

TV

IN

the flow stress, and the latent hardening

-‘&

71 _

HARDENING

systems

In this model, the dislocations

by the process

of dislocation

tangling.

When a latent system does not have enough mobile dislocations to maintain the applied strain rate at a given stress level, the stress must be increased to increase

the dislocation

velocity.

As it will be

shown later, the increase of flow stress due to lack of mobile dislocations temperatures.

becomes more important

This mechanism

at lower

also predicts a rather

isotropic latent hardening because the tangling process

r.

to work hardening as -rPf, at low strains rDi w rZf and

would

at high strains

slight system dependence

distribution

rDi < -rDf. Because

of impurities

of the uniform

79i = -rli. From Keh’s data

on forest density, we have rlf = 2.27,, and 71f = 1.4~~~ at high strains.

at low strains

Therefore,

at low

strains,

involve

come from location

-rDf + 2.27,,

7P

+

7Pf

7Pf

(4)

and at high strains

the small differences

should

be strongly

interactions

systems. show

The

in the mobile

On the other hand, the subsequent location

3.2 = 1.6 2

dislocations.

of the latent hardening may dis-

density from one system to another.

behavior 72

all the secondary

of

yielding.

However,

apparent

system

on the dis-

between the primary and latent

This is consistent

a variety

work-hardening

dependent

rates

with our results, which of

work-hardening

after

it must be kept in mind that this

dependence

of the work-hardening

behavior may actually be the result of the orientation dependence These are in reasonable

agreement

with the experi-

mental

values given in Figs. 9 and 10.

would

predict

because

a rather

the elastic interaction

and forest

dislocations

particular

At room temperature,

Also,

suggest

hardening

the primary to the

of Burgers’ vectors. the strain rate sensitivity

of

volume is too

in terms of the forest cutting

the activation

with strain.

evidence

between

and the activation

small to be explained mechanism.

latent

is not very sensitive

combinations

iron is considerable,

change

isotropic

This model

Therefore,

volume

does not

these two pieces of

that the strain rate sensitivity

of

of the work-hardening

system dependence. more thoroughly capable

in the future.

of explaining

coplanar

rather than the

This point must be investigated the latent

This model

is also

hardening

of the

systems because dislocation

immobilize

the dislocations

of the coplanar

as well as those of other secondary Li’s

model

previously Results.

can explain mentioned

tests the previous

would systems

systems.

the interesting

in Section

We observed

tangling

Also,

observation

1 of Experimental

that during latent hardening

flow stress of the primary

system

may be exceeded before yielding occurs on the latent system.

Some dislocations

are generated in the latent

system in the pre-yield region of the latent hardening

iron at room temperature is not caused by forest cutting. In this case, the strain rate sensitivity and

test because of the higher Schmid factor on the latent

the temperature

system.

controlled

dependence

by the interstitial

of flow stress may impurities

be

in solid solu-

tion as proposed by Fleischer.(23) The activation volume is of the right order of magnitude for the

The dislocations

or multiply However, dislocations

cannot move long distances

sufficiently to cause macroscopic they can immobilize

the previously

yielding. mobile

of the primary system, thus preventing the

ACTA

970

METALLURGICA,

yielding of the primary system when the previous flow stress is reached. macroscopic

At a somewhat

higher flow stress

yielding takes place on the latent system.

Equations (6) through (10) describe this model Several investigators(29-sQ have shown quantitatively. that dislocation expressed as effective

a

velocity

in various

power function

crystals

1966

1.0 0.6 0.4

can be The

stress ra

internal stress ri. r* = -ra -

Then the velocity-stress

14,

of effective stress.

stress r* is defined as the applied

minus the long-range

VOL.

7i

0.06F

Iti

I

I

/

:

I

(6)

relationship may be written(3s)

as 9 =

bpc=

bp

m

c

(7)

(>70

where 9 is the shear strain rate, b the Burgers vector, p the

density

necessary

of mobile

dislocation,

to move a dislocation

at a given temperature characteristic perature;

r.

the stress

on tem-

this equation should hold true for both the

latent and primary systems.

Solving the equation

Tp

( kg /cm2)

FIG. 15. Ratio of mobile dislocation density versus flow stress of primary system.

metals.

This strong temperature

* = roJj$pp)l’m

?J

(8)

likely explanation necessary

activated

the Peierls force.(44-46) deformation

(!y*

p = pbsv* exp -

(9)

= Y exp since the strain rates are the same for the latent and The constant

direct etch-pit

process

The

may

be

written as

?$$LjpyL

a

has been

to date is the thermal activation

to overcome

thermally

and the latent hardening ratio is given by

either by

dependence

ascribed to several effects.(23*3s-44) However, the most

71* = roz(PJW1’”

primary systems.

600

400

for

rp* and rl*,

r

1 200

0

and m is an empirical constant

of the material and dependent

~

0.001

with a unit velocity

“m” can be obtained

technique

or by strain-rate

cycling experiments. (36,37) In the present investigation “m” was determined by strain-rate cycling between 2.78 x 10-4/sec and 2.78 x 10-3/sec and found to be about 8 at room temperature.

Assuming

the internal

stress to be the same on all systems we have

(AG(r*)/kT)

(11)

(AG(T*)/~T)

where f is the shear strain rate, p the mobile dislocation density, the number

6 the Burgers vector,

occur per unit length of dislocation out per successful vibration

of

s the product

of places where thermal

a

fluctuation,

dislocation

in Gibbs free energy. on a dislocation,

activation

of can

and the area swept

Y* the frequency

of

segment, AG(T*) the change

T* is the effective stress acting

defined

as 7A

-

TV,

the difference

between the applied shear stress and the long-range internal stress. The activation energy AH and activThe ratio (p,/p,)

was calculated

for the latent hard-

ation volume v = - (dG/dr*) the following equations :(a?)

can be obtained

from

ening ratio shown in Fig. 10. This is plotted in Fig. 15. This figure shows that the mobile density of the latent system must be about &N i&j of the primary mobile density at room temperature, if Li’s model is applicable. 2.2 Latent hardening between 77°K and 300°K. As the deformation temperature is lowered from room temperature, iron shows the pronounced temperature dependence

of flow stress characteristic

of all b.c.c.

~=-(g)~=kT(f$)~

(13)

Figure 16 and Table 4 show the results of calculations

NAKADA

I

I

AND

KEH:

LATENT

IN

Fe

971

systems, then any difference in the temperature dependenee of the flow stress of two systems must originate in the temperature dependence of the preexponential term Y = pbsv* in equation (11). In principle, the value of v = p&v* can be obtained from the following equation :(4O)

/aLATENT

600 500 ;

HARDENING

400 300 200

--

;

t-m

aIn p

100

0

500

1000

1500 2ooa 2

3

bml - Kg/cm2 Fro. 16. Activation volume versus stress.

for the activation volume and energy. These results are in good agreement with those of other investigators.(45,U) Figure 16 is particularly significant because it is seen that the activation volumes of both primary and latent systems have the same stress dependence. This implies that the rag-controlling mechanism is the same for both systems. The values of activation volume vary from v = 717 b3 at 320 kg/cm2 to v = 17 b3 at 2500 kg/cm2. The extremely small value of the activation volume at high stresses, together with the constancy of the strain-rate sensitivity with strain (see Fig. 14), leads us to believe that the Peierls mechanism is the rate controlling process at low temperatures for both the primary and latent systems.* One of the important experimental results is the stronger temperature dependence of the flow stress of the latent system as compared to that of the primary system in the t,emperature range of 300”K-‘77°K. This phenomenon is clearly shown in Fig. 12. If the activation energy IY(r*) is the same one for both * For the d&died discussionofthe rats controlling processes in bee deformation. see for example Ref. 42-48.

Table 4 shows the vaIues of In (r/j) calculated from this equation. Table 4 also shows that above 77°K Y is higher in the latent system. If we assume SV* to be the same for both the primary and the latent systems, this would imply that the mobile dislocation density is higher in the latent system. This is contrary to the hypothesis previously introduced that the latent hardening is caused by the lack of mobile dislocations. However, it is quite possible that sv* may not be the same for both the primary and latent systems, and hence the mobile dislocation density may not be determined in this manner. Because the cutting of forest dislocations is a thermally activated process, this process may result in the higher temperature dependence of the flow stress of the latent system. However, previous results in f.c.c. crystals(25) show that the increase in flow stress at 4°K from this process can be at most about 50% of the athermal flow stress. On t,he other hand, the extra temperature dependence of the flow stress of the latent system in iron at 77°K is about 2 to 3 times the athermal flow stress. Therefore, it is unlikely that the stronger temperature dependence is due to forest cutting. Returning to the empirical power relationship 9 K (T/To)“, the ratio of mobile densities in the primary and latent systems can be calculated as shown in equation (10) if we know m values at different temperatures. Table 5 lists nz values at different temperatures obtained by strain rate cycling. These

4. Activation energy H end pm-exponential factor 1)

TABLE

Primary system 21;:

0.182 23.2

WK) 7 - T373”K(kg/Cm2) H (ev) in(W)

77 2890 0.194 29.2

148

201 755 0.485 27.9 Latent system

q

148 1730 0.416 32.7

201 1020 0.542 31.0

248 1

298 80 0.63 24.5

373 0 1.23 87.6

248 450 0.76 35.4

298 70 1.05 41.7

373 0 0.53 47.5

ACTR

972

/

-8 50

100

200

250

x)0

Ratio of mobile dislocation density versus temperature.

Temperature (“K) m

77 47

148 14

201 9

300 8

values are in good agreement with those of Michalakc3’) and Keh et al.(m) Calculated values of pp/pE are shown in Fig. 17. Thus, we see that the ratio of mobile dislocation densities must change from about lo2 : 1 to X06:I from 3~‘Kto77’K if the lowtem~erature latent hardening is caused by the lack of mobile d~sloeations in the latent system. This hypothesis receives some qualitative support from the observation that for the latent system the tendency to show a yield drop increases as the deformation temperature is decreased, as shown in Fig. 11. However, because the tensile axes for the primary and latent systems were not identical with respect to the unit triangle, another possible explanation for the stronger temperature dependence of the Aow stress of the latent system is the orientation dependence of the temperature dependence of flow stress in iron. However, at this time there is not enough experimental data to support or reject this possibility. SUMMARY

AND

14,

1966

ACKNOWLEDGMENTS

*K

TABLE 5. “m” values at different temperatures

-~ -_._-.-

VOL.

Latent hardening at high temperatures (above 300”K), where the flow stress is very weakly dependent on temperature, may be due to either the elastic interaction between glide and forest dislocations or to the lack of mobile ~slocations in the latent system or both. Latent hardening at low temperatures (below 300°K) is most likely due to the lack of mobile dislocations in the latent system. Thus the stronger temperature dependence of the flow stress of the latent system as compared to that of the primary system must be due to the more rapid decrease of mobile dislocations with decreasing temperature in the latent system. However, the possibility that the orientation dependence of the temperature dependence may be causing the stronger temperature dependence must be left open.

/

150 TEM~RATURE,

Frc. 17.

METALLURGICA,

CONCLUSION

The latent hardening ratio in iron single crystals is always greater than one, regardless of the Burgers vector combination, deformation temperature, strain rate and amount of prestrain. The system dependence of the latent hardening ratio is small. The shape of the yield point of the latent system may be system dependent. The latent system exhibits a higher tem~)erature dependence of the flow stress than the primary system down to 77’K. The latent and primary systems exhibit similar st,rain rate sensitivities.

The authors wish to thank: Dr. Z. S. Basinski of National Research Council of Canada, J. C. M. Li and J. T. Michalak of this Laboratory for stimulating discussions; and T. Churay and R. Sober for their competent experimental assistance. REFERENCES 1. U. F. KOCKS, Trans. Met. Sot. AIME 230, 1160 (1964). 2. T. H. ALDEN, Trans. Met. Sot. AIME 230, 649 (1964). 3. E. W. EDWARDS,J. WASHBURNand E. R. PARKER,Trans. Met. Sot. AIM_?3 197. 1525 (1953). 4. E. H. EDWARDS-&~ J. WA&&RN, Tram. Net. SOC. AIME ZOO, 1239 (1954). 5. H. W. PAXTON and A. H. COTTREL~,Acta Het. 2,3 (1954). 6. H. REBSTOCK,~.~~e~~l~.~,~O6(1957~ 7.1%'. L. PEIILLIPS, JR. and W. D. ROBERTSOX,Truns. Met. See. AIM.73 212,406 (1958). 8. D. B. HOLT, Acta Met. 7. 446 (1959). 9. J. J. HAULER, Trans. ket. Sec. AiME 221, 305 (1961). 10. T. H. Atn~:rv, Acta Met. 11, 1103 (1963). 11. U. F. KOCKS, Acta Met. (to be published). 12. 0. KRISEMEEJT, G. F. DE VRIESand F. BELL, P?qs. Status Solidi 6. 73 (1964). 13. Z. S. E&IN&I and P. J. JACKSON,Appl. Phys. Lett. 6, 148 (1965). 14. Z. S. BASINSKI and P. J. JACKSON, Phys. Status Solidi 9, 805 (1965): ibid 10, 45 (1965). 15. H. Hu and R. PRIESTNER,J. Iron Steel Imt. (to be published). 16. D. F. &ETN end J. R. Low, JR., Trans. Het. Sm. RIME 221, 744 (1961). 17. A. S. KEN, P&Z. Maa. 12. 9 119651. 18. T. Wu end R. S~~O~&HC&SI& Ph&s. Rev. 78, 468 (1950). 19. A. HIKATA, B. CHICX, C. ELBAUAZand R. TEUELL, AppZ. Whys. Lett. 2, 5 (1963). 20. Y. NAKADA, U. F. KOCKSand B. CHALIMERS, Trans. Met. Sot. AIME 230, 1273 (1964). 21, W. C. LESLIE, J. T. MICHALAKand F. W. AUL, iron and Its Dilute Solid Solutions, p. 119. Interscience, New York I1 QR.71. \_““._,. 22. G. SCHOECKand A. SEEOER,Acta Met. 7,469 (1959). 23. R. L. FLEISCHER,Acta Met. 10, 835 (1962). 24. A. SEEQER, S. MADER and H. KRONX~~LLER, Electron. Microscopy and Strength of Crystals, p. 665. Interscience, New York (1963). 25. Z. S. XASINSKX, F’hiL Mag. 4, 393 (1959). 26. G. SAADA, _f%ctrolzMicroscopy and Strength of Cq#&, p. 651. Interscience, New York (1963). 27. A. P*‘.STROH, Proc. P&s. See. Bfj& 2 (1953). 28. J. C. M. Lr, J. Metals 16, 762 (1964). 29. W. G. JOHNSTON and J. J. GILMAN,J. Appl. Phys. SO, 129 ,lr&Rfl\ ,‘“““,.

NAKADA

AND

KEH:

LATENT

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HARDENING

IN

Fe

973

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