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EFFECT OF PLASTIC DEFORMATION ON THE
STABILIZATION 0F MARTENSITE IN AN FE-NI ALLOY

presented by

Eileen Jeannette Bryk

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EFFECT OF PLASTIC DEFORMATION ON THE STABILIZATION
OF MARTENSITE IN AN FE-NI ALLOY

BY

Eileen Jeannette Bryk

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Metallurgy, Mechanics and Materials Science

1985

ABSTRACT

EFFECT OF PLASTIC DEFORMATION ON THE STABILIZATION
OF MARTENSITE IN AN FE-NI ALLOY

BY

Eileen Jeannette Bryk

A study of the effect of plastic deformation on the
thermal stabilization of the austenite-martensite
transformation has been conducted using an Fe-32.4Ni-.08C
alloy. Experiments were designed to study the influence of
1) Amount of plastic deformation prior to martensite
formation, ii) Strain rate used in deformation and iii)
Aging time. Other related studies included the effects of
austenitizing temperature and of prior deformation on Ms
temperature, and the effect of deformation on the
temperature at which 55% martensite was obtained.

The degree of stabilization was minimum at 0%
deformation, rose as deformation increased to 1%, and
remained relatively constant thereafter. With a fixed 2%
deformation, higher strain rates produced dramatically
lowered stabilization. Also, studies indicate that longer

aging time lowers the degree of stabilization.

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to Visiting
Professor S. C. Das Gupta and Professor K. Mukherjee for
their interest in this work and for the ideas and discussions
we shared. The many hours of assistance in experimental work
provided by Professor Das Gupta were especially appreciated.

Thanks are also due my family for their support.

ii

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . .

LiSt 0f Figures 0 I O O O O O O O O O O O O O O O O

I.

II.

III.

IV.

Introduction . . . . . . . . . . . . . . . .

Martensitic Transformations . . . . . . . . .

2.1.
2.2.

2.3.
2.4.
2.5.

Solid-Solid Phase Transformations . . .
General Characteristics of Martensitic
Transformations . . . . . . . . . . . .
Athermal and Isothermal Transformations
Autocatalysis and Partitioning Effects

Theories of Martensite Nucleation . . .

Prior work 0 O O O O O O O O O O O O O O O O

3.1.
3.2.
3.3.
3.4.
3.5.

Isothermal Martensite Formation . . . .
Burst Formation of Martensite . . . . .
Stabilization . . . . . . . . . . . . .
Theories on Stabilization . . . . . . .
Effect of Plastic Deformation on

Martensitic Transformation . . . . . .

Experimental Procedure . . . . . . . . . . .

4.1.
4.2.
4.3.

Material Composition and Preparation .
Austenitizing Conditions . . . . . . .

Method of Deformation . . . . . . . . .

iii

Page

vi

10
23
23
26
27
32

34
40
40
40
41

Page
4.4. Determination of Ms and the Extent of
Transformation . . . . . . . . . . . . . 41
4.5. Stabilization Treatment . . . . . . . . . 43
V. Results and Discussion . . . . . . . . . . . . 48
5.1. Effect of Austenitizing Temperature on Ms
Point . . . . . . . . . . . . . . . . . . 48
5.2. Effect of Prior Deformation on Ms Point . 50
‘5.3. Effect of Deformation on Tssg . . . . . . 52
5.4. Effect of Deformation on Stabilization . 55
5.5. Effect of Strain Rate on Stabilization . 56
5.6. Effect of Aging Time on Stabilization . . 58
VI. Conclusions . . . . . . . . . . . . . . . . . . 63
Appendix . . . . . . . . . . . . . . . . . . . . . . 64

List of References . . . . . . . . . . . . . . . . . 67

iv

LIST OF TABLES

Page
Table 1. Degree of stabilization as a function

of time of aging . . . . . . . . . . . . . 60

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

3.

4a.

4b.

4c.

4d.

6.
7.
8.

LIST OF FIGURES

Schematic diagram of martensitic plate

or embryo . . . . . . . .
Knapp and Dehlinger model
site embryo . .'. . . .
Experimental apparatus .
Time/temperature sequence
tion treatment . . . . .
Time/temperature sequence
tion treatment . . . . .
Time/temperature sequence
tion treatment . . . . .
Time/temperature sequence

tion treatment . . . . .

of the marten—

Effect of austenitizing temperature on

Ms point . . . . . . . .

Effect of prior deformation on Ms point

The effect of prior deformation on T55%

Electrical resistance vs.

plot 0 O O O O O O O O 0

temperature

Degree of stabilization as a function of

amount of prior deformation . . . . . .

vi

Page

11

18
42

44

45

46

47

49

51

53

54

57

Page
Figure 10. The degree of stabilization as a

function of strain rate . . . . . . . 59

vii

I. Introduction

The transient suppression of the austenite-to-
martensite transformation which results from aging treatment
is called thermal stabilization. Aging treatments, as well
as the cooling procedures, affect the course of the
transformation. The transformation from austenite to
martensite may be arrested and the partially transformed
specimen subjected to an aging treatment at or above the
arrest temperature. Upon subsequent cooling, the
transformation does not recommence, generally, at the arrest
temperature, but at a temperature below the arrest
temperature. Thermal stabilization is measured by the
temperature interval needed to cause the transformation to
recommence. Thus, the degree of stabilization, 6, is the
difference between the temperatures of arrest and of the
restarting of the transformation after a stabilization
anneal. It may be noted that the degree of stabilization is
sensitive to the time and temperature of the aging
treatment. Also, it is likely that at a given temperature
in the transformation range, a specimen which has undergone
stabilization will have a lower martensite content than one

which has been cooled without interruption.

2

As a result, stabilization (as well as cooling rate, in
some cases) may affect the proportion of retained austenite
in martensitic steels. It is desirable to avoid
stabilization, and thus retained austenite, in hardened
steels because the result may be a softer material subject
to structural and dimensional instabilities. This could
lead to mechanical failures and a reduction in safety. Thus
stabilization is an important phenomenon not only because of
its relationship to heat treatment but because of its

scientific implications in the martensite transformation.

II. Martensitic Transformations

2.1. Solid-Solid Phase Transformations

Solid state phase transformations may be separated into
two categories:

(1) Nucleation and growth type (diffusion controlled)

(2) Displacive or shear type (diffusionless)

In typical nucleation and growth transformations, the
new phase grows at the expense of the parent phase as the
interphase boundary slowly migrates. Neighboring atoms
interchange position as they cross the interphase boundary,
and they move independently at a rate which varies greatly
with temperature. Transformations of this type may occur
isothermally, and the amount of new phase increases with
time. The transformed region usually has a different
volume than that of the original phase.

Displacive on shear transformations are unique in that
the new phase forms from the parent phase by cooperative
movements of many atoms such that the region of
transformation undergoes a change in shape and thus a
transformation strain is produced. The change in shape
occurs through the additive effect of the translational
movements of the matrix. Homogeneous distortion, typical of

martensitic transformation, results due to the shifting of

4
the matrix to maintain as close a coherence as possible with
the growing plate. The cooperative movement of atoms occurs
with a velocity which approaches 1/3 the velocity of soundl.
Whereas nucleation and growth transformations may be
completed isothermally, the shear transformation usually
comes to a halt when the rate of heating or cooling is
brought to zero, the transformation resuming only when the
temperature change is recommenced. It may be noted that
some incidences of isothermal shear (martensitic)
transformations have been reported.

Surprisingly, martensitic transformations, which are of
the shear type, may propagate at temperatures near absolute
zero. Kulin and Cohen2 studied this phenomenon in iron-
nickel and iron-nickel-carbon alloys and noted that the
interface mobility did not seem hindered by the conspicuous
absence of thermal activation. Apparently, the interface
motion is coordinated, not an atom-by—atom process as in
nucleation and growth. In fact, each atom moves less than
one lattice spacing relative to its neighbors as a result of

transformation.

2.2. General Characteristics of Martensitic Transformations

(l) A plate-like or a needle-like morphology is typical of
martensites; the ratio of thickness to other dimensions
is small. The plates become thin toward the

extremities and so have a lenticular cross section.

(2)

(3)

(4)

(5)

(6)

(7)

5
There exists a definite orientation relationship be-
tween the crystal lattices of the original and new
phases.
The growth rate of martensite plates is temperature—
independent and is of the order 104-105 cm/sec.
If other variables (mechanical and thermal history,
grain size) are fixed, the extent of transformation
will be dependent on temperature. Martensite begins
to form spontaneously on cooling at a temperature known
as Ms. Upon further cooling, the percentage of material
transformed increases until the transformation is
completed; this occurs at a temperature known as Mf.
In some cases, spontaneous transformation never reaches
completion.
Martensitic transformations are usually athermal
but may also be isothermal; both of these types of
transformation may occur in the same material. Whereas
the athermal mode exhibits a high nucleation rate with
no temperature dependence, the isothermal mode has a
rate of nucleation dependent on temperature.
Martensite plates have a regular internal structure;
often plates are internally twinned or contain
dislocations.
The transformation strain characteristic of martensitic
transformations reveals itself in a macrodeformation,
or relief, on the plane surface where the plate is

formed. The transformation strain consists of a shear

(8)

(9)

6

component lying parallel to the interface (or habit)

plane and a dilational component normal to it.
Martensitic transformations are reversible; the
original atomic ordering may be restored and destroyed
repeatedly. Consider a single crystal of the parent
phase subjected to cooling and transformed into several
crystals of martensite. When heated to some
temperature which is always above Ms (except for
thermoelastic martensites), the martensite will revert
to a single crystal with the same size, shape, and
orientation as the parent crystal. Upon subsequent
transformations, martensite plates that form during
cooling have the same size, shape, and location
possessed in earlier transformations. Reversibility
occurs in almost all martensitic transformations,
although there are exceptions. Where reversibility is
not seen, interfering secondary events are
responsible. An example is the iron-carbon alloys,
where the martensite phase is thermodynamically
unstable, decomposing into stable phases before the
reverse transformation begins.

Plastic deformation may have various profound effects
on martensitic transformations, for example, it may
induce martensite at a temperature too high for
spontaneous transformation (ie, above Ms). The highest
temperature at which martensite forms under stress

spontaneously is Md.

(10)

7

When stresses are applied within the transforma-
tion temperature range, the amount of transformation is
increased and the reaction may go to completion.
Different results are obtained when the plastic
deformation is administered significantly above Ms,
where the parent phase is stable. In such instances,
transformation at any temperature is reduced, and the
Ms temperature is depressed (or in rare instances,
raised).

In single crystal experiments, the martensitic
transformation may be hindered or aided by an
appropriately oriented applied stress.

Cooling regimes may be administered so as to have a
negative effect on the transformation. Consider a
specimen cooled to some temperature within the
transformation range and held for some period of time.
The transformation will not usually begin as cooling
recommences; the temperature must first be lowered by
some increment. This phenomenon is called
stabilization, and the extent to which it occurs in a
specimen tends to increase with the amount of time for
which the temperature is arrested. A general
concensus has not been reached as to whether stopping
the cooling schedule above the Ms temperature produces
stabilization. At all lower temperatures, the
percentage transformation tends to be lower than had
the specimen been cooled directly to the given tempera-

ture o

8
2.3. Athermal and Isothermal Transformations

As mentioned previously, martensitic transformations
are usually of an athermal-kinetic type of behavior. A
fraction of the total volume becomes martensite at a
temperature below Ms at a rate that is independent of
temperature. It has been observed that more transformation
occurs with decreased temperature and that its extent is a
function of temperature. Bunshah and Mehl3 have found that
martensite plates in an Fe—29.5Ni alloy propagate at about
105 cm/sec, and have a formation time of 0.05-0.5 usec.

They also found that the propagation velocities of athermal
and isothermal martensites were equal.

The isothermal component of martensite transformations,
on the other hand, is either not operative or is obscured by
the predominant athermal component and hence is not often
observed. Due to the fact that martensitic transformations
are strain sensitive and autocatalytic, athermal transforma—
tion may promote the nucleation of isothermal martensite
when cooling is stopped below Ms. The transformation
kinetics are completely different from that of the athermal
transformation; a normal C-curve behavior is seen on a TTT
diagram. This C-curve behavior would imply that the
isothermal transformation is thermally activated. Quite the
contrary: the plate formation time and linear growth
velocity are the same as those of athermal martensite.

Also, isothermal transformation increases the volume

fraction of martensite due to the formation of new plates,

9
not because of growth of existing plates. With these
observations, we can conclude that isothermal transformation
is dependent on the thermally activated triggering of
embryos. Growth does not require further thermal activation

once embryos are triggered.

2.4. Autocatalysis and Partitioning Effects

After a martensitic transformation begins, either
athermally or isothermally, its course may be affected by
autocatalysis. When this phenomenon occurs, the initial
rate of transformation is higher and preferred sites for
nucleation form in the parent phase due to the perturbation
around existing plates. A chain reaction may take place and
plates may form end to end.

Further complicating the reaction is the partitioning
of the parent phase, which contributes to the subsequent
decrease in transformation rate. Nucleation occurs in
progressively smaller volumes of austenite, therefore, the
volume fraction becoming martensite per nucleation event is
decreased and the size of newly formed fully grown plates
decreases. A formal theory of this process is given by
Fisher“.

The effects of autocatalysis and partitioning must be
considered in the transformation rate equation. A
semiempirical rate equation of the following form can be

used5.

df

3E -

where:

(Ni + pf - Nu) (l - f) vexp ('AWa/RT) mg (l - f)

10

1
“a (1)

f = fraction transformed

t = time

Ni = number of pre-existing nucleation sites

p = number of autocatalytic embryos per unit volume

Nu = number of martensitic plates per unit volume

v = lattice vibration frequency

AWa = activation energy of nucleation at
temperature T

m = thickness-to-diameter ratio of plates

q = average volume per grain of austenite

This equation can be numerically integrated using

metallographically determined values of q and m and unknown

quantities p and AWa which are obtained by curve fitting.

The equation predicts experimental results well when less

than 10% transformation has occurreds.

2.5.

Theories of Martensite Nucleation

As mentioned previously, the kinetics of martensitic

transformations are dependent mainly on nucleation, not

growth, since each plate grows rapidly to its final size.

We might begin our review of nucleation theories by

considering a simple model of a martensite p1ate5. The

plate is shown in Figure l and has an oblate spheroid

shape.

11

Volume 3 4/3 17’ch

d

Surface Area = 21(1'2

 

 

 

 

Figure 1. Schematic diagram of martensitic plate or embryo.

12
The plate radius is r and semi-thickness is c, where
r>>c. Its interfacial free energy per plate is:
VAg 9’” = 2nr2x (2)
Where V is the plate volume, AgsP+M is the surface energy
per unit volume, and l is the interfacial energy.
Linear elasticity theory allows us to compute the

strain energy for the plate:

VAgP+M = ‘2 anC (fig) (per plate) (3)
e 3 r
In this case, V = '% anC = plate volume, and (Ac/r) is the

strain energy per unit volume. The factor A is of the form:

A = («(2 - v)/8(I - v)} uroz + % ueoz

(3a)
or

A = u(r02 + 802) (3b)

where v = Poisson's ratio, u = shear modulus, and r0 and so
are shear and dilatational strains respectively that are
associated with shape deformation. The elastic constants
for the parent and martensitic phases are assumed equal.
Designating Ach’M as the chemical free energy per

unit volume of martensite, we have:

V AchTM ='% nrzc Ach+M (per plate) (4)

13
Note that below To, AchTM is negative. The total free

energy change due to the formation of a plate is:

AWP+M = - g anC AchTM + hr2 x = % anZA (5)

Classical nucleation theory allows us to determine the

free energy of nucleation W* corresponding to a critical

radius r* and critical semi-thickness C*. To do this we

 

 

set:
AWP+M _ o
r -
and (6)
MI
A _
c - 0
This yields:
c* = ZA/Ach+M (7a)
r* = 2Ac*/Ach*M (7b)
* _ 4 *2 *
V - -§wr C (7C)
w* = 32nA2 x3/3 (Agcp+M)4 (7d)
C*Z/r* = x/A (7e)

If the nucleation is completely random, each atom is a
potential nucleation site and the random nucleation rate N
is given by:

N = (No/vm) v exp (-W*/kT) (8)
where (No/Vh) = number of atoms per unit volume, v is the

lattice vibration frequency (1012/sec).

14

The nucleation kinetics predicted by equation (8)
should follow an isothermal C-curve behavior, with the
maximum nucleation rate being at the knee of the C-curve.

To summarize, in the classical model, a martensite
embryo becomes a critical size at some temperature and
becomes a nucleus. The classical model of homogeneous
nucleation may be used to compute V*, c*, r* and W* by using
experimental values of Ach*M and theoretical values of A
and 1.5

Various experimental results and theoretical
postulations supply evidence that the homogeneous classical
nucleation model does not apply to martensitic
transformations. In fact, the evidence suggests that the
reactions are heterogeneous.

Metallographic studies provide the strongest evidence
that martensite nucleates at preferred sites and not in a
random manner. Beta brass was selected for study7 on the
merit that its martensitic transformations are conveniently
reversed. It was found in repeated transformation cycles
that the position and the sequence in which individual
plates formed remained almost unchanged. It appears, from
this experiment, that the parent phase contained preferred
nucleation sites.

Cech and Turnbull8 studied the transformation kinetics
of an Fe-29.2 Ni alloy, and their results also shed some

light on this subject. Particles of the alloy, having

15
diameters of 37-100u, were austenitized and quenched to the
martensitic transformation temperature range. Temperatures
at which the martensitic transformation began varied widely
among the particles. Some of the particles underwent no
transformation even at the lowest temperatures attained.
The researchers concluded that structural singularities, or
heterogeneties, governed the nucleation in this alloy.
Evidently, some of the particles contained fewer effective
sites (heterogeneities) than others, and therefore required
further supercooling. It is hypothesized that the
probability that an effective heterogeneity exists increases
with particle size. Experiments by Huizing and Klosterman9
on single crystal Fe-Ni spheres have yielded similar
results.

In the classical models of nucleation, the Boltzmann
probability factor [exp(-AW*/kT)] appears in the nucleation
rate equations of the type of Eq. (8). This practice has
been criticized by Crussardlo. The Boltzmann factor arises
from the supposition that atoms constitute a system of
separate oscillators possessing a characteristic frequency.
Crussard disputes its use, contending that the statistical
reinforcements of elastic waves supplies the energy for the
thermally activated process being considered. Using quantum
theory, he proposed an alternative probability factor,
which predicts a finite rate of nucleation at 0°K and an
operative homogeneous mode. The ramifications are that

isothermal and athermal transformations may be explained in

16
this way and that heterogeneous nucleation need not be
considered. However, as mentioned previously, experimental
observations at very low temperatures suggest the
homogeneous nucleation aspect included in Crussard's
modification is deficient.

Numerous other theories of nucleation share Crussard's
perspective that the nonclassical View is a valid one. In
contrast with the classical nucleation model in which an
embryo reaches a critical size at a given temperature and
becomes a nucleus, Cohen11 has proposed a ”reaction~path”
model, which is non-classical. In this model, within the
volume where nucleation occurs, the lattice progresses
through a series of states which leads the parent phase to
become martensite. The primitive atom movements are thought
to take place in a synchronized manner, the lattice strain
generating a sequence of intermediate structures in any
given region before propagating out like a strain wave. The
series of states is visualized as a reaction path possessing
an energy barrier between the initial and final states.

From this perspective, activation of the embryos is due to
fluctuating atomic configurations, not embryo size. The
embryo associated with the reaction path model could be

a strain center composed of lattice imperfections such as
dislocation arrays. The strain center is assumed to be part
of the way along the ”reaction-path", and above Ms, it is a

region of high free energy. Hollomon and Turnbull12 have

17
added interfacial energy considerations to the reaction path
model.

Frank'sl3 model of the austenite-martensite interface
was used by Knapp and Dehlingerl“ in their formulation of
athermal martensite kinetics. They took into account the
free energy balance of the embryo. The martensite embryo is
viewed as a thin oblate spheroid with dislocation loops in
the interface; it is thought that the loops contain mainly
screw dislocations and that short edge components join the
positive and negative screws. The embryo model is shown in
Figure 2. Embryo growth in the [110]Y and [225]Y directions
is achieved by the dilation (or growth) of the dislocation
loops. Growth in the [554]Y directions entails the creation
of new loops. As the synergetic motion of the loops allows
growth of the martensite embryo, the dislocation interface
sweeps through the austenite.

The energy required to create and expand the
dislocation loops, the interfacial energy, and the strain
energy, is supplied by the chemical driving force AgcY*“’.
The embryo, as viewed by Knapp and Dehlinger, will be
triggered when the chemical driving force is greater than
the required interfacial and strain energies. This begins
to occur when the total free energy change per unit volume

becomes zero. By writing AWY*“' as'% AWY*°’ and maintain-

ing our established notation for surface and strain

energies, we write:

18

 

 

 

 

 

 

 

 

 

Figure 2. Knapp and Dehlinger model of the martensite
embryo.

19

’ I I

Y+a Y+a + Ag = 0 (9)

AWY+a = + Agee

-Agc

For spontaneous triggering of the embryo, then,

I

Y*0

y+a’ y+a’)
C

Ag = (Age + AgS (10)

The last two terms in eqn. (10) can be written as Ag non-
chem and equations (2) and (3) for an oblate spheroid may be

used to yield:

_ AA __
Ag non-chem - 2c + (ll)

where r, c, A, and A retain their previous meanings. For a
specified radius, Ag non-chem can be minimized relative to c
by setting 39 non-chem/ac = 0. The minimum value of

Ag non-chem is calculated to be

1/2

Ag non-chem = (ééé) = g1 (12)
minimum
when
(3131/2 (13)
2A

Equations (12) and (13) may be used to calculate
Ag non-chem as a function of embryo size. Equation (12)
shows that for a fixed value of c, Ag nonfchem decreases
minimum
with increasing radius, r. The implications are two fold:

(a) the non-chemical energy barrier decreases as the embryo

size increases, and (b) after the embryo starts to grow,

20
this barrier becomes smaller, enabling rapid interface
movement. Below Ms, AgCY*°’ > Ag “32;:232 for the larger
embryos, causing a net driving force which acts as a stress
on the dislocation interface. The interface is thus swept
outward, creating martensite from the embryo. With this
model, the approximate size of the largest embryo at Ms may
be found by using experimental and estimated values of the
various parameters. Raghavan and Cohen15 developed this
type of calculation further.

The model described above has been modified by
Kaufman16 so that an equivalent dislocation loop (on the
habit plane) encircling the oblate spheroid embryo is
considered. The interfacial energy is within the dislocae
tion loop. The loop's radius is r and its equivalent
Burger's vector, r, is given by cb/d; c is the semithick-
ness, b is the Burger's vector of the lattice, and d is the
distance between the loops in Knapp and Dehlinger's model
(Figure 2). The loop has a line tension of r = urz/Z; u is
the shear modulus. If a net shear stress T is imposed on
the loop, the accompanying increase in free energy (starting
from zero size) is

W1 = Zurf - ":2 CT (14a)
where

P = uC/Z (14b)

The shear stress T is equal to the chemical driving force
(-Agc) minus the strain energy per unit volume (Ac/r). We

then have

21
A
r=-AgC-;E=-Agc-g- (15)

where A is the surface energy. Noting from eqn.(7e) that
c2/r = A/A and substituting from eqns. (15) and (14b) into

eqn. (14a) yields:

1/2
1 = "WW3 1?. .1:
W p d [Ad 4» Age (A) + 1] (16)

 

maximizing W1 in terms of r, we have

rc = 35-;- (17)
A9
and
2 3
WCl = 9wA 2 (18)
2A9

In this case, rc is the critical radius of the loop at
which it may expand, thereby lowering its energy. This
growth model, proposed by Kaufman, resembles aspects of
mechanical twinning or kink band formation. Comparing

with eqn. (7b):

rc =3:- r* (19)

Because rc>r*, AW* has a negative value at re.

A proposal has been made by Machlin and Cohen17 that
embryos with radius r, where r*<r<rc (r* is the radius in
the previous model) are activated during isothermal
transformation. It may be noted that when r<r*, W* is of

such a large magnitude that thermal activation is impossible

22
and also that embryos with r>rc are already triggered by the
athermal process.

There are still other models of heterogeneous
martensite nucleation to be considered. Olson and Cohen18
proposed a model in which part of a wall of dislocations
sustained faulting on close-packed planes, yielding the
close-packed phase in the fcc to hcp transformation. In
order for the fcc to bee transformation to occur, a second
shearing is required. An activation barrier is not included
in their thermodynamic equations. However, it has been
proposed by Magee19 that a barrier exists due to the
thermally activated motion of the dislocations associated
with the internal slip.

It might be mentioned at this point that the free
energy change for nucleation of a martensite embryo within
an expanding dislocation loop was calculated by Easterling
and Tholenzo. In the case of internally twinned martensite,
there was no apparent nucleation barrier. They chose 20
ergs/cm2 for the interfacial energy of a martensite embryo;
this contrasts greatly with the Kaufman and Cohen16 estimate
of 200 ergs/cm2 which would be a prohibitively large
nucleation barrier in their model.

It has been suggested21 that heterogeneous martensite
nucleation may be initiated at dislocation pile-ups, and
also that an embryo at a pile-up location may experience the

classical free energy barrier to nucleationZI.

III. Prior Work

3.1. Isothermal Martensite Formation

Most martensitic transformations are classified as
athermal; they are of a temperature-dependent nature in that
the reactions proceed only when the temperature is changing.
Fletcher, Averbach, and Cohen“!23 in 1948-49 found a time-
dependent component to be present in some cases. Although up
to 5% isothermal martensite formation occurred in their work
on plain carbon and low alloy steels, the isothermal nature
was not recognized as such. It was mistakenly thought to be
a “tailing-off" effect that usually appears at the beginning
and final stages of athermal transformations.

Kurdjumov and MaximovaZ‘N25 treated isothermal
martensite as a separate phenomenon. They were able to
achieve 25% isothermal formation of martensite in an
Fe-0.6C-2Cu-6Mn alloy. They did this by rapidly cooling
their specimens to -180°C which completely suppressed the
transformation, then reheating and holding isothermally at
various temperatures between -80° and -160°C. The
transformation kinetics thus obtained had a C-curve behavior
on a TTT diagram. At one of the experimental temperatures,

the initial isothermal transformation rate was constant and

23

24
gradually decreased. There was a temperature at which the
greatest amount of isothermal martensite was formed.

In other experiments, they quenched a 1.6C steel in
alkaline iced water, forming approximately 20% martensite,
and then cooled the specimens to liquid nitrogen temperature.
Though only partial suppression of the transformation was
introduced, this technique ensured that the amount of
athermal martensite formed was fixed, and that upon reheating
the isothermal transformation could be studied.

Das Gupta and Lement26 encountered partial suppression
of the transformation in a Fe-lSCr-0.7C steel. Some athermal
martensite formed before the isothermal component was allowed
to form. They observed that the initial rate of isothermal
transformation increased with decreasing temperature, reached
a maximum at -110°C and then decreased below this tempera-
ture. Specimens cooled to liquid nitrogen temperatures so
that a constant initial amount of martensite was present were
up-quenched and held at various subzero temperatures.

Between about 3% and 8% isothermal martensite formed.

Machlin and Cohen27, using an Fe-29Ni alloy, found also
that the isothermal transformation followed partial athermal
transformation. The isothermal transformation rate was
determined to be a function of the amount of athermal
martensite present, the temperature and time of isothermal
holding, and the state of internal strain. The nucleation
of new plates, as opposed to the growth of existing ones, is

thought to be responsible for isothermal transformation.

25

Kulin and Speich28 found that isothermal martensitic
transformation occurs above the Ms temperature in an
Fe-l4Cr-9Ni alloy. A higher percentage of transformation.
occurred on holding isothermally at high temperatures than on
qUenching to low temperatures.

Cech and Holloman29 worked with an Fe—23Ni-3.7Mn alloy
and found that the rate of isothermal transformation
increased with decreasing temperature, reached a maximum at
-128°C, and decreased with further lowering of temperature.

It may be seen from the various studies mentioned that
isothermal martensite formation is not dependent on whether
athermal transformation has preceded it. At the same time,
if athermal transformation has not consumed too large a
portion of the parent phase, it may spur the nucleation of
isothermal martensite when cooling is stopped below Ms. This
is due to the strain sensitive and autocatalytic nature of
martensitic transformations. With such a relationship
between the two modes of martensitic transformation, it is
clear that isothermal transformations are best studied in the
absence of athermal martensite. As noted, however, sometimes
this condition is impossible to meet. At the same time, one
wishes to study isothermal kinetics without variable amounts
of athermal martensite obscuring them. Quenching below the
lowest isothermal level to be studied would yield a fixed
quantity of athermal martensite, solving the problem.
Temperatures could then be raised to study isothermal

kinetics.

26
The isothermal kinetics yield a C-curve behavior whether
there is no athermal martensite or a fixed amount is present.
Maximum transformation rates occur at some intermediate

temperature, typically between 100-150°K.

3.2. Burst Formation of Martensite

In some cases, the internal accomodation stresses from
the martensitic plates give an autocatalytic effect known as
"burst phenomenon". It manifests itself in a rapid, sudden
transformation of a large percentage of the specimen at or
below the Ms. Burst transformation may occur when
transformation resumes following an aging treatment and
thermal stabilization. It may also be seen in some cases
after long isothermal holding treatments, after sufficient
incubation period.

Upon metallographic examination, the martensitic plates
are seen to be appreciably wider than normal plates. They
form a zig-zag pattern in the specimen. Stresses produced by
one plate aid in the activation of another nucleus, and so
the burst transformation is like a chain reaction. The
implication is that the burst transformation is one in which
the stresses in the matrix are reduced. The plates formed in
a burst transformation to some extent comprise a self—
accomodating system of stress. They may be the reason
that the plates are wider than normal.

Working with Fe-Ni-C alloys, Entwisle and Feeney30
found that Mb, the temperature at which burst occurs, is

dependent upon the heat treatment given to the austenite. The

27
percentage transformation in a burst was seen to be a
function of austenite grain size and Mb.

There are several explanations offered for the burst
phenomenon. Machlin and Cohen31 feel that the plastic
deformation of the parent phase when martensite plates form
creates embryos, or that existing embryos become
supercritical at a given temperature.

Suzuki and Honma32 see burst transformation as analgous
to deformation twinning in that it requires the dislocations
encompassing the embryos to be able to multiply on favored
habit planes. This mechanism of multiplication is not likely
to happen on the{225}type of habit planes. In those cases,

the reaction proceeds by a mechanism similar to slip.

3.3. Stabilization

Harris and Cohen33 performed the first organized study
of stabilization using a 1.lC-l.5Cr steel. They found that
stabilization does not occur unless the temperature of
holding is below a certain temperature, we, below Ms. The
degree of stabilization, 6, increased linearly as the
temperature of holding was decreased.

Das Gupta and Lement3“ worked with an Fe-lSCr-0.7C steel
and found that stabilization against athermal and isothermal
transformation does not occur to a significant extent unless
some transformation has preceded. In fact, when the amount

of initial martensite exceeding what they referred to as the

28
"critical limit" was present before reaching the holding
temperature, stabilization is more pronounced. They found
that stabilization against isothermal transformation
increases with cycling temperature and time spent at the
cycling temperature. Also, they noted that for a fixed
amount of initial martensite, a higher intermediate cycling
temperature yielded a more permanent stabilization at the
subzero reaction temperature.

Morgan and K035 found that stabilization of austenite
occurs during continuous cooling and also with isothermal
holding both above and below Ms in steels with approximately
1%C and 0-5%Ni. The rate of stabilization was a function of
the austenite composition, and it increased with both
temperature and the presence of martensite. When
stabilization above Ms occurred, Ms was depressed and the
amount of austenite at a temperature near Ms was increased.

Edmondson36 studied the effect of aging time and
temperature on stabilization in an Fe-lONi-lC steel. The
extent of stabilization reached higher limiting values the
lower the aging temperature. At relatively higher
temperatures of aging, the value of a decreased on prolonged
aging after reaching a peak. The temperature dependence of e
is fairly pronounced in the Ni alloys.

WOodilla, Winchell, and Cohen37 studied stabilization
kinetics in an Fe-30.8Ni-.007C alloy. When they removed the

interstitial elements (C,N) by moist-hydrogen treatment,

29
stabilization would not occur. When .007C was present, the
activation energy for stabilization was commensurate to that
of the diffusion of C or N in ferrite (rather than in
austenite). They interpret this to mean that interstitial
diffusion governing the stabilization occurs within the
martensite embryos, not in the matrix of the parent phase.
According to them, the kinetics imply that the intersitials
diffuse from the embryo toward the surrounding matrix,
immobilizing the austenite/martensite interface and causing
stabilization.

In their work on the thermal stabilization of the
athermal martensite transformation in Fe-Ni—C alloys, Kinsman
and Shyne38 found that aging temperature had a distinct
effect on the character of stabilization; the results were
'markedly different for low vs. high aging temperatures.
Within each of the lower aging temperatures used, 9
increased, reached a maximum, then decreased with time.
Increasing the aging temperatures decreased the maximum
stabilization, emax, and the time of aging to reach emax.
Their results are similar to those of Odaka and 0kamoto39,
Glover“°, and Priestner and Glover“1, who also found that
after long aging treatment, beyond emax, 9 increases again.
In some cases the second increase of e is attributed to the
start of the bainite reaction. The transformation rate after
stabilization was consistently found to be higher than that
immediately preceding the transformation arrest and aging

treatment. When the percentage of martensite present before

30
aging was increased, the degree of stabilization was more
pronounced regardless of aging time and temperature.
Stabilization did not occur when the carbon content was less
than .002 wt%. Some experiments were done to test the
”reversibility” of the stabilization. In some control cases,
aging temperatures were held constant for the duration of
aging treatment. In other cases, the aging temperature was
changed during the treatment. For low temperature aging, the
extent of stabilization approached a value typical of the
last aging time and temperature used in the sequence, thus
the stabilization was termed ”reversible”.

When high aging temperatures were used, the results were
different. In partially martensitic specimens, the degree of
stabilization as a function of time was constant for a
sizeable interval, rising at longer aging times. The degree
of stabilization at any aging time tended to be higher when
the aging temperature was increased. Also, reversibility of
thermal stabilization was not observed, in contrast with the
reversibility of the low temperature results. In this case,
the value of 6 always apprOached the value associated with
the highest aging temperature in the sequence.

Glover"0 investigated the effect of aging time and
temperature of a partially martensitic 1.4%C steel on
subsequent transformations. Higher aging temperature yields
lower maximum values of 6. Regarding stabilization, Glover

compares the influence of nickel content to that of Mn“2 and

31
of Cr which was determined in a 1.5Cr-lC steel, and sees that
Mn and Cr greatly diminish the aging temperature dependence
of emax (the maximum stabilization which occurs). In these
cases, 9 increases with aging temperature and reaches a
limiting value which seems to be temperature-invariant.
Comparing the effects of Ni36 to those of Mn and Cr just
mentioned, he makes an interesting hypothesis. In light of
the fact that stabilization appears to involve carbon atoms,
the information suggests that its sensitivity due to aging
temperature is decreased when elements with an affinity for
carbon are present and increased when elements which do not
form stable carbides are present.

Guimafges and Shyne“3, using an Fe-3lNi-.01C steel,
demonstrated that for a given temperature of arrest (prior to
aging treatment) the value of e is the same independent of
the extent of prior deformation. Deformation consisted of
up to 75% cold rolling.

Guimarges““ conducted another investigation on the same
alloy where some of the specimens were annealed and some had
25% deformation due to cold—rolling at room temperature. All
of the specimens were cooled to an arrest temperature of
-72°C, heated and aged, and subcooled to determine 6. The
result: the extent to which the annealed and the deformed
specimens underwent stabilization was equal, confirming the

results of Guimarges and Shyne“3.

32
Priestner and Glover"1 worked with an Fe-SNi-l.43C steel
whose Ms was near room temperature. Aging temperatures above
50°C were used and it was found that 6 decreased with time,

became negative, reached a minimum, and rose again.

3.4. Theories on Stabilization

There are numerous instances of stabilization reported
in the literature; all might not be attributed to the same
cause. Generally, theories of stabilization deal either with
change in the austenite phase or with the change in effect
that previously formed martensite has on the later transfor-
mation. It is widely thought that stabilization of austenite
results from stress relaxation of the parent phase in the
region of the martensite plates. The stress mentioned is
thought to promote nucleation in unstabilized austenite, and
its relaxation would have a stabilizing effect. Various
investigators propose different mechanisms by which this
relaxation takes place. The common element is that thermal
stabilization is recognized as a thermally activated process
controlled by interstitial diffusion (usually C or N)
kinetics. Models may be categorized in three general groups:
one37 attributes the effect to solute atoms pinning (locking)
the austenite/martensite interface, another"0 to the
diminished autocatalytic effect of existing plates, and the
last35 to strain aging that would strengthen the parent

matrix.

33

Das Gupta and Lement3'+ and Cohen, Machlin, and
Paranjpe“5 hypothesize that solute atoms diffuse to strain
centers in the parent phase. This lowers the driving force
for transformation in regions that might have otherwise been
preferred nucleation sites. Das Gupta and Lement3“ feel that
the driving force for C-diffusion from martensite to strain
embryos is given by the higher activity of carbon in
martensite as opposed to that in austenite. Carbon atoms
would be motivated to diffuse from the edges of martensite
plates to nearby strain embryos.

Another possibility"6 is that a condition analogous to
dislocation pinning may occur due to slight concentration
build-up at the embryo or plate interface, immobilizing it.

Hollomon, Jaffe, and Buffum“7 suggest that the coupling
between the austenite and martensite may be somewhat
diminished by time-dependent yielding on relaxation.

Morgan and K035 feel that stabilization may occur upon
strengthening of the austenite due to formation of Cottrell
atmospheres, which hinders displacements in martensitic
transformations. Crussard"8 feels that precipitate formation
may also have the same effect.

Kinsman and Shyne38 have developed a model which is
based on Knapp and Dehlinger's embryo model. They suggest
that stabilization results from the inability of the
semicoherent austenite/martensite nucleus interface to move
due to migration of intersitial solute atoms to the disloca-

tion array making up the interface. The kinetics of solute

34
segregation to the nucleus interface govern the time and
temperature dependence of stabilization. Thus, deriving the
time law for solute segregation enabled them to develop a
time law for stabilization.

Glover and Smith"2 feel that elastic stresses are
relaxed due to (carbon and nitrogen) interstitial diffusion
within the martensite, leading to the initial stages of
tempering and causing stabilization.

Maximova et al1+9 suggest that relaxation as well as the
crystal defects that it produces cause stabilization; the
defects interfere with plate growth.

KurdjumovSO suggests that a partially martensitic
specimen may experience readjustment of the stresses and
defects that would otherwise tend to enhance transformation.
This process yields a more stable structure which is thus

more difficult to transform.

3.5. Effect of Plastic Deformation on Martensitic

Transformation

Elastic stress fields and externally applied plastic
deformation influence martensitic transformations. It has
been observed that a specimen will transform spontaneously at
the Ms temperature under zero applied stress. When a stress
field is applied, however, it affects the shape deformation
of small regions of martensite, and alters the net driving
force for their expansion. Whether the effect is positive or
negative depends on both the nature of the stress field and

the plate orientation. The Ms temperature may thus be

35
increased or decreased if it is a function of this net
driving force. Frequently, the Ms temperature will be
raised because the driving force for some of the habit plane
variants will increase.

Sometimes, the Ms values may change by a much greater
extent when deformation occurs, i.e. stresses beyond the
elastic limit are applied. Transformation may occur above
Ms, the temperature at which it occurs rising as the applied
stress is increased. The highest temperature at which
transformation may occur in this manner is Md.

Stress applied to the parent phase above the Md
temperature may result in plastic deformation, and the matrix
usually becomes mechanically stabilized. Subsequent cooling
reveals a decreased Ms temperature in these cases.

It is possible that the defects introduced by plastic
deformation obstruct the growth of the martensite phase.
Sometimes the opposite occurs when small deformations are
administered; the Ms may be raised due to the formation of
internal stress concentrations promoting martensite
nucleation or nuclei growth.

Reed51 subjected austenitic polycrystalline
Fe-31Ni to plastic deformation at room temperature and found
on subsequent cooling that the Mb temperature was suppressed.
Bokros and Parker52 observed increased Mb temperatures in
plastically deformed Fe-3l.7Ni single crystals.

Brownrigg53 studied the relationship between plastic

deformation and isothermal martensite kinetics. Austenitic

36
specimens with small deformations (administered above Md)
exhibited slightly increased initial transformation rates,
which progressively decreased as the amount of prior
deformation increased. In all of the deformed specimens, at
all of the isothermal holding temperatures, further progress
of the transformation was retarded in comparison with
undeformed specimens. It is suggested that the enhanced
transformation rates are the result of a decreased activation
energy for nucleation. Higher amounts of deformation inhibit
(retard) the transformation; it is thought that the increased
number of dislocations interferes with the mechanism of
embryo propagation.

Many aspects of martensite kinetics are related to the
interface resistance to martensite plate growth, suggest
Magee and Paxtons“. The kinetics may also be related to the
ease of dislocation motion required in embryo formation in
heterogeneous nucleation. Whether nucleation or growth is
the controlling step, the researchers feel that the
transformation is profoundly affected by the resistance of
the parent phase to dislocation motion.

WOrking with Fe-Ni and Fe-Ni-C alloys, Patel and Cohen55
determined criterion that quantitatively predicted the effect
of applied stress on the Ms temperature. A mechanical work
term which is the product of the applied stress field and the
transformation strain is added to the chemical free energy
change which supplies the driving force for transformation.

Thus the change in temperature at which the critical

37
thermodynamic driving force is attained and the transforma-
tion intitated may be computed.

Using several alloy steels (including Airkool-S, with 5%
Cr), Breinan and Ansells6 found that when austenitic
specimens were deformed (above Md) to obtain greater flow
stresses, the Ms temperature was lower the higher the
austenite yield strength was. The decrease in Ms was linear
with increase in austenite flow stress. They suggested that
this was caused by a reduCtion in dislocation mobility. Such
a reduction might influence either the ease of nucleation or
the ability of martensite plates to propagate. Further
transformation was affected also, in the sense that the
percentage transformation at 25°C was lower, the higher the
austenite flow stress. When alloying increased the yield
strength, Ms decreased in a similar way. Similar work was
done by Ankara57, who subjected an Fe—30Ni alloy to
transformation cycling; this increased the austenite yield
strength. The result was a decrease in Ms with increased
yield strength. Ankara attributes this to a change in
dislocation density.

Plastic deformation also affects the morphology of
martensite subsequently formed. Bokros and Parker52 worked
with single crystals of Fe-37Ni, and found a large difference
in morphology between deformed and undeformed crystals. They
found that previously strained crystals exhibited increased
Ms temperatures and a decrease in the amount of

transformation occuring in a burst. This is also an apparent

38
reduction in the number of habit plane variants operative
during the burSt transformation. Because transformation
progresses along variants of favored groups, the morphology
is simplified. It was found that substantial regions of the
strained crystals underwent a burst transformation due to the
operation of four habit-plane variants whose poles group
about a common [110] direction. These favored groups are
those whose planes are almost perpendicular to the active
slip plane, but the group whose poles cluster about the
Burger's vector of the active slip system is not included.
Regions consisting of a single group were of larger size the
higher the amount of prior plastic strain.

Durlu and Christian58 observed that prior plastic strain
caused a decreased c/r (semithickness-to-radius) ratio for
plates formed in a temperature range below Mb in an
Fe-26.4Ni.-.24 alloy. Datta and Raghavan59 examined the
plate dimensions of martensite formed at Mb in prestrained
Fe-Ni alloys. They found that at any fixed transformation
temperature, the c/r ratio linearly decreases as the amount
of plastic deformation increases. Also, the variation of c/r
with plastic strain is more pronounced at higher
transformation temperatures.

Two types of martensite may form in austenitic Fe-Ni-C
alloys during plastic straining, according to Maxwell,
Goldberg, and Shyneso. It is thought that the two
martensites form by different mechanisms of transformation

because of differences in morphology, distribution, and

39
temperature dependence. One type is the same as that which
is formed spontaneously below Ms except that it is finer and
less regularly shaped; it is called stress-assisted
martensite. Strain-induced martensite formed on the {111}Y
slip bands of the deformed austenite, and appeared as sheaves
of parallel laths.

It has been illustrated that crystal defects and
imperfections have a profound effect on martensite
nucleation. Also, thermal stabilization is believed to occur
because of the rearrangement of crystal defects and the
pinning of martensite embryos. With this in mind, the
objective of this research is to examine thermal
stabilization in a appropriate steel, administering various

amounts of deformation.

IV. Experimental Procedure

4.1. Material Composition and Preparation

The composition of the alloy chosen for this study is
Fe-32.4%Ni-.08%C. Chemical analysis was done with
spectrographic equipment; the carbon percentage was
determined by wet chemistry.

The alloy was obtained in a cylindrical form, dimensions
being 200 mm in length and 7 mm in diameter. To insure
homogeneity, the material was first austenitized in an
evacuated quartz tube. Thin rectangular pieces were needed;
these were made by warm rolling in the austenitic condition
with intermediate anneals. A manual shear cutter was used to
obtain specimens of the following dimensions: 0.8 mm thick, 3

mm wide, and 45 mm long.

4.2. Austenitizing Conditions

After being sealed in evaculated quartz tubes, specimens
were austenitized and then cooled to liquid nitrogen
temperature; this was repeated twice with the intention of
having a uniform microstructure, grain size, etc., before
experimentation began. The austenitization was carried out
at 800°C for 1 hour; these conditions were selected upon
considerations of acceptable grain size and dissolution of
carbides. Afterwards, specimens were air-cooled to room

40

41
temperature inside the quartz tubes. In a series of
experiments, a single specimen was used repeatedly and
subjected to this treatment so that consistent results could
be obtained. Repeatedly austenitizing the specimen did not

have a significant effect on the Ms temperature.

4.3. Method of Deformation

Deformation in the uniaxial tensile mode was given by
mounting the austenitic specimens in a friction grip and
applying a load with an Instron Tensile machine. A strain
rate of 6.46 x 10"3 sec"1 was used, except for selected
experiments where the effect of strain rates between 3.23 x
10'3 and 64.6 x 10‘3 sec”1 were sought. The extent of
deformation ranged from 0.5% to 4%; in a few experiments, 8%
deformation was administered. It might be emphasized at this
juncture that room temperature, at which the deformation was

carried out, is above both Md and Ms.

4.4. Determination of Ms and the Extent of Transformation
The progress of the martensitic transformation was
studied by measuring the change in the specimen's electrical

resistance using a Kelvin double bridge. In order to
maintain an electrical resistance on the order of 10"2 to
10'3 ohms, the specimen size was manipulated to the
approximate dimensions listed in Section 4.1. Because the Ms
temperature of this alloy is below -65°C, a subzero
temperature atmosphere which allows control of the specimen

temperature is needed. For this reason, the specimen was

42

Ice Water
for thermocouple
reference junction

 

 

C]. P].

 

 

 

 

 

rlr

 

‘C2 P2

 

 

 

 

J. I

 

Kelvin Double Bridge ‘ Potentiometer

Thermal Flask —’H

Specimen holder ,)

i 3‘ Specimen
Thermocouple tip

 

 

 

 

 

Liquid nitrogen

 

 

 

 

 

 

Figure 3. Experimental Apparatus

43
lowered manually to different levels above liquid nitrOgen in
a tall Dewar flask, as shown in Figure 3 at a cooling rate of
about l.5°C per minute.

With this arrangement, temperatures in the range -25°C
to +195°C were easily obtained. The specimens were mounted
in a holder having knife-edged contact points for the current
and potential leads to the Kelvin double bridge. A
Leeds-Northrup potentiometer connected to a copper/constantin
thermocouple in contact with the specimen was used to monitor
temperature. Am empirical equation (see Appendix) was used
in conjunction with the temperature and resistance data to

calculate the extent of transformation.

4.5. Stabilization Treatment

After being austenitized and given fixed amounts of
deformation, specimens were transformed until specified
percentages of martensite were present. Then stabilization,
or aging treatment, was administered in a constant
temperature bath at 50°C for 1 hour and specimens were
remounted in the specimen holder. Subsequent cooling and
temperature and resistance measurements revealed the course
of the transformation as it recommenced. In selected
studies, the aging time was varied from 5 minutes to 2 hours.
Schematics of the time/temperature relationships in the

stabilization experiments are shown in Figures 4a-d.

44

h - heating

a - austenitizing
3' ac - aircooling
sc - subcooling

 

TEMPERATURE

 

 

TIME

Figure 4a. Time/temperature sequence for stabilization
treatment.

45

h - heating

a - austenitizing
3_ ac - aircooling
d - deformation
sc - subcooling

 

 

 

TE M PE RATURE

TIME

Figure 4b. Time/temperature sequence for stabilization
treatment.

46_

h - heating
a - austenitizing

 

 

 

 

ac - aircooling
3- sc - subcooling
T' s - stabilization
treatment
m h ac
a:
a ‘ s
<
m
E sc
5 SC
E-

 

TIME

Figure 4c. Time/temperature sequence for stabilization
treatment.

h - heating

 

 

 

 

a - austenitizing
ac - aircooling
3' d - deformation
sc - subcooling
s - stabilization
treatment
“’3 h
9‘ ac
g 3
£71
a. d
2 sc
ii] .
5.. SC

 

TIME

Figure 4d. Time/temperature sequence for stabilization
treatment.

V. Results and Discussion

5.1. Effect of Austenitizing Temperature on Ms Point

Specimens of the alloy Fe-32.4 Ni-.08C were sealed in
evacuated quartz tubes and austenitized for 1 hour at
temperatures between 700°C and 900°C. They were mounted in
the specimen holder in the undeformed state and subjected to
subcooling. The time/temperature sequence used was that
shown in Figure 4-a. Resistance and temperature measurements
were used to determine Ms. It was found that higher
austenitizing temperatures produced elevated Ms temperatures.
This trend is shown in Figure 5. Sastri and West61 and Maki
et al62 reported similar results. Others63 have found the Ms
increases as austenitizing temperature increases to a
limiting temperature, and then in some cases decreases.

It is likely that as increased numbers of crystal
defects are removed due to higher temperatures of treatment,
the energy needed for complementary shear during
transformation is reduced, thereby increasing the Ms. This
was the inference drawn by Ankara57 in his study of an
Fe-30Ni alloy. He found that the effect of austenitizing
temperatures on Ms was exaggerated by cooling immediately
after rapid heating, retaining the highest possible amount of

lattice defects.

48

49

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8 .v onsumsomfiofi wcwnfififimsa‘

nXHm Axum“ AXHN

 

d d d J I

l

A

M

L

.m musmwm
2..
2- m
L
m
E... .w
m.
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5..
m
1V
no. 0
Il\

 

50

The work of Entwisle and Feeney3o in high Ni steels
shows Mb rising with austenite grain size. One must not
infer, they say, that the larger austenite grain size raises
the Mb point; instead, that the increasing austenitizing
temperature causes growth of austenite grains and increased
Mb simultaneously.

A grain size relationship does seem to exist, however.
A martensite crystal will stop growing when it reaches the
grain boundary. Also, with smaller grain size, there is a
greater buildup of backstress when plates form, making
dislocation motion or shearing of the matrix difficult. In
light of these facts, it is thought that small grain size,
which will result from lower austenitizing temperatures,

stabilizes the austenite (decreasing the Ms temperature).

5.2. Effect of Prior Deformation on Ms Point

Austenitized and air-cooled samples (austenitic) were
deformed to various extents by tensile elongation. Amounts
of deformation ranged from 0.5% to 8%. The samples were
mounted in the specimen holder and subcooled, and Ms was
determined. It was found that Ms was not substantially
altered when 0% to 4% prior deformation existed; a depression
of Ms was noted when 8% deformation was given, however.
Average Ms values for differing amounts of deformation are
shown in Figure 6.

Deformation is expected to have an effect on Ms, and
these results may be explained by noting that opposing

effects are in action. Existing martensite embryos may be

51

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:ofimfisomom ammucooaom

 

 

 

 

 

m h o m w m N —.

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1 é-
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52
stimulated by internal stress from plastic deformation or
from partial transformation, thus aiding transforma-
tion 5“,55. This plastic deformation, on the other hand,
strain-hardens the austenite matrix. Propagation of
martensite plates is difficult due to the obstacles formed by
dislocation accumulation. The Ms temperature is expected to
decrease when this occurs.

It is our belief that the opposing effects of embryo
stimulation and austenite strain hardening cancel each other
(negate each other) when small amounts (0 to 4%) of
deformation are administered, thus resulting in an Ms that is
neither raised nor lowered. Austenite strain hardening
effect predominated at higher deformation (8%), thereby

depressing Ms.

5.3. Effect of Deformation on T553

A series of experiments were done in which 55%
martensite was needed prior to stabilization treatment.
Austenitic specimens were given between 0% and 4% deformation
and subcooled. It was found that as the amount of
deformation increased, the temperature at which 55%
martensite had formed (denoted as T55%) also increased. The
relationship between deformation and average T55% is
illustrated in Figure 7. This data was compiled from
experiments whose time/temperature sequences followed those
of Figures 4a-d. A typical plot of resistance vs.

temperature is shown in Figure 8. This diagram shows that

53

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coflmfisomom mumpcoosom

m v. m N P

- 4 — q -

 

 

 

.5 musmwm

54

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.m ousmflm

 

333383.
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m2 . 9 .32
(1 a (J y
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ucoEumoup mcwmm ..
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55
martensite formation commences in this alloy with a large
burst, and manifests itself by a large decrease in electrical
resistance. Burst formation was accompanied by an audible
click and was of the order of 15% to 50% martensite. Amounts
of prior deformation did not have a marked effect on burst
size.

The formation of martensite may be assisted by plastic
deformation. Existing embryos may be stimulated by the
resulting internal or external stress5“r55. It is suggested
that the deformation administered did not have a significant
effect on Ms temperatures (except at 8% def.) because the
deformation did not alter the potency of the largest embryos
which would be the first to become martensite plates.
Possibly the smaller, less potent embryos were stimulated
such that they would transform at higher temperatures. The
statistical distribution of embryo size and potencies, then,
could have been condensed and the lower end of the potency
range shifted upwards with plastic deformation. This would
result in an increased amount of transformation (at
temperatures somewhat lower than Ms) resulting from increased
deformation. Thus, the temperature at which 55% martensite

is formed (T553) will increase with plastic deformation.

5.4. Effect of Deformation on Stabilization
Specimens were austenitized, deformed to various extents
at a rate of 0.02 cm sec“1 and subcooled. When prescribed

amounts of 45% or 55% martensite were obtained, specimens

56

were given stabilization (aging) treatment at 50°C for 1 hour
and again subcooled. The time/temperature sequences used are
shown in Figures 4c,d. The variation in the extent of
stabilization as a function of percentage prior deformation
is shown in Figure 9, where 55% martensite was present prior
to aging. Without deformation, the stabilization was
minimum; it increased with deformation until 1% deformation
was given, and above this it remained relatively constant.

Attempts to study the effect of varying amounts of
deformation and 45% martensite prior to aging on the
stabilization phenomenon were not successful. The
temperatures at which specimens achieved 45% martensite
(T45%) and the difference in temperature between TMs and T45%
varied widely. This presents inconsistencies in experimental
conditions.

It was found that as deformation was increased, 6
increased and then remained constant above 1% deformation.
It is suggested that higher deformation gives more
opportunity for C to diffuse, and therefore increased
stabilization may occur due to C destroying the embryo
potency. It is thought that the maximum effect of this

mechanism has occurred by 1% deformation.

5.5. Effect of Strain Rate on Stabilization

Austenitized and air-quenched specimens were deformed at
room temperature to 2% elongation with various strain rates.
The specimens were then subcooled until approximately 50%

martensite was formed whereupon they were removed, aged at

57

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com $8.539 owfico 0.8m

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q

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58
50°C for 1 hour, and cooled again. The effect of different
strain rates on stabilization has been presented in Figure
10. The time/temperature sequence followed for these
experiments is illustrated in Figure 4-d.

An increased degree of stabilization with decreased
strain rate was observed in the alloy Fe-32.4Ni-.08C. This
is in line with the proposed mechanism of C pinning the
martensite embryos. The potency of a martensite embryo is
destroyed when carbon diffuses to the site, and it is known
that C diffusion is a time and temperature dependent
function. With a slower rate of deformation, it will take a
longer time interval to reach a given amount of deformation,
allowing more time for the carbon to diffuse to the embryos
and destroy their potency. Destroyed potency would increase

the stabilization and the results show this.

5.6. Effect of Aging Time on Stabilization

Austenitized specimens were deformed to 2% elongation at
a strain rate of 6.46 x 10'3 sec'l, and subcooled until
approximately 50% martensite was present. Aging treatment at
a constant 50% was administered for various durations of
time. The treatment is shown in Figure 4-d. Data was
analyzed so that the difficulties presented by differing T50%
and TMs'T50% values would be eliminated. The data showed
that as the time of aging was increased, stabilization
decreased. While this is contrary to what is expected, the

trend was consistent. Results are reported in Table l.

59

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.oaanHm>m uoc ma moEflu ocwmm uconommwc Hmuo>om moansHocfi moon oanmumooom

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61

As mentioned, stabilization typically increases with
aging time. However, there are several cases in the
literature in which stabilization increases, reaches a
maximum, and then decreases as a function of aging time. If
this is the trend which the data (in Table 1) follows, then
it is possible that the shortest aging period used
represented the portion of the aging period vs. stabilization
curve in which stabilization was decreasing. Had it been
feasible to study the effect of shorter aging periods,
stabilization might have been found to first increase and
reach a maximum before decreasing.

Okamoto and Odaka57 studied the effect of aging time on
stabilization in an Fe-1.63Cr-l.06C steel. In specimens aged
at 100°C and at 200°C, stabilization increased, reached a
maximum, and then decreased. According to Nishiyama58,
sufficiently high aging temperatures allowed the interstitial
solute atoms to cluster, forming precipitates after their
segregation to nucleation sites. As a result of the
clustering, the interstitial content of the austenite phase
is lowered, consequently increasing Ms. According to
Nishiyama, this is why the degree of stabilization decreases
after reaching a maximum value. It might be noted that
Glover reported similar results for a 1.4%C steel.

The results of Priestner and Glover“1 are striking.
Studying an Fe-SNi-l.43C steel, they found that aging above
50°C caused stabilization (e) to increase with time, reach

a maximum, then decrease, even making a negative, before

62
increasing again. They attribute the negative 6 to softening
caused by overaging. A strain-aging model was used by these
authors to account for their overall results.

Kinsman and Shyne38 found that with relatively low aging
temperatures (0°C to 80°C), three Fe-Ni-C alloys also
exhibited a decreasing a after reaching a maximum. They
observed that the kinetics of thermal stabilization are
similar to those of precipitation hardening. Because in some
cases stabilization increases in a maximum before decreasing,
they feel that this behavior is analogous to overaging in
solid state precipitation. According to them, the fact that
stabilization occurs by the pinning of the martensite/
austenite interface means that the degree of stabilization
must be a function of the excess concentration of segregated
solute atoms at the interface. The time/temperature
dependence of interface carbon segregation was determined,
and it was found that the interface carbon concentration
increases to a maximum value and then decreases at longer
times, as does stabilization in many cases. This would seem
to support their contention.

It has not been possible, however, to conclusively
determine the cause of the increasing and then decreasing
stabilization in this research. As mentioned, data analysis
was quite difficult. Determining the cause of the observed
behavior would entail many further experiments and possibly

the use of more sophisticated equipment.

63

VI. Conclusions

Martensitic transformation begins in the Fe-32.4Ni-.08C
alloy with a burst producing between 15% and 50%
martensite.

As the austenitizing temperature is increased from 700°C
to 900°C, the Ms temperature increases.

Plastic deformation prior to transformation of austenite
has no effect on Ms until high amounts of deformation are
given.

In deformed specimens, after forming 55% martensite, the
degree of stabilization increases with deformation up to
1% deformation and then remains relatively constant at
higher deformations.

For specimens deformed 2%, transformed to 55% martensite
and aged, the slower the rate of strain, the higher the
degree of stabilization.

Specimens deformed 2% and transformed to 50% martensite
had a higher degree of stabilization if the aging period

was shorter.

APPENDIX

APPENDIX

Using resistance and temperature data, the amount of

martensite present at any point may be calculated. The

following procedure is used:

1)

An austenitic specimen is cooled from room tempera-
ture to liquid nitrogen temperature (-l95°C). Many
simultaneous readings of resistance and temperature are
taken. After maximum transformation has occurred
(at-195°C), the specimen is heated and the change of
resistance with respect to temperature is noted.

With this information, the following parameters are
calculated:

Average resistance drop per 1°C in austenite = RY

Average resistance drop per 1°C in martensite = Ra

Resistance drop per 1% martensite formation = RM

Using x-ray diffraction, the maximum amount of
martensite was found to be 80%. Thus, RM could be
calculated in the following way:

RMS - R195 - Ra(MS+l95)
RM:

 

80
In this case, 3M3 and R195 are the resistances at Ms
and at -l95°C, respectively. To eliminate the size

effect between specimens, all of the above parameters

64

2)

65

were divided by RMs and the ratios Ry/RMS, Ra/RMS and

RM/RMs were calculated. These ratios were found to be

relatively constant from specimen to specimen despite

size differences.

Having obtained the necessary parameters, the

calculations performed in stabilization experiments are:

i)

ii)

iii)

iv)

An austenitic specimen is cooled to the Ms
temperature, and the average value of Ry is
calculated.

The resistance measured immediately prior to
transformation is denoted as RB and the corresponding
temperature, TB- The first data measured after
transformation commences consists of a resistance RAl
and temperature TAl- Resistance at Ms (RMs) may be
calculated: RMs = RB - Ry(TB-Ms).

The percentage of martensite at TAl is then

determined:

1 - RAl/RMS - RY/RMS(MS-TA1)
% martensite : ‘ = X1

RM/RMs

 

The amount of transformation at a lower temperature
(TAZ) may be determined using this formula: %

martensite at TAZ = X2 =

- - £1 x -
xl+RA1/RMs RAZ/RMS {Ry/Rus‘1oo’+Ra/Rus‘1'I60’}[TAz TAl]

 

RM/RMs

66
v) In this manner, the amount of martensite present with

successive temperature decreases may be calculated.

LIST OF REFERENCES

5.
6.

10.
11.

12.

13.
14.
15.

16.

LIST OF REFERENCES

G. V. Kurdjumov and O. P. Maximova, Doklady Akad. Nauk
Azerb. SSR, 66 (1948) 211.

S. A. Kulin and M. Cohen, Trans. AIME, 188 (1950) 1139.

R. F. Bunshah and R. F. Mehl, Trans. AIME, 197 (1953)
1251.

J. C. Fisher, Acta Met., 1 (1953a) 32; Trans. AIME 197
(1953b) 918.

M. Cohen, Trans. AIME, 212 (1958) 171. 212 (1958) 171.

L. Kaufman and M. Cohen, Progress in Metal Physics, 7
(1958) 207.

A. B. Greninger and V. C. Moordavian, Trans. AIME, 128
(1938) 337.

R. E. Cech and D. Turnbull, Trans. AIME, 206 (1956)
124; J. Metals, N.Y. 8, No. 2 (1956) 124.

R. Huizing and J. A. Klosterman, Acta Met., 14 (1966)
1693.

C. Crussard, Physica, 's Grav., 15 (1949) 184.

M. Cohen, E. S. Machlin, and V. G. Paranjpe,
Thermodynamics in Physical Metallurgy, ASM, (1949) 242.

J. H. Holloman and D. Turnbull, Progress in Metal
Physics, Pergamon Press, London, 4 (1953) 333.

F. C. Frank, Acta Met., 1 (1952) 15.
H. Knapp and U. Dehlinger, Acta Met., 4 (1956) 289.

Raghavan and M. Cohen, Acta Met., 20 (1972) 333; 20
(1972) 779.

L. Kaufman and M. Cohen, Progress in Metal Physics, 7
(1958) 165.

67

17.

18.

19.
20.

21.
22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.
34.

35.

68

E. S. Machlin and M. Cohen, Trans. AIME, 194 (1952)
489.

C. B. Olson and M. Cohen, Met. Trans., 7A, (1976) 1897;
(1976) 1905, (1976) 1915.

C. L. Magee, Phase Transformations, ASM, (1970) 115.

K. E. Easterling and A. R. Tholen, Acta Met., 24 (1976)
333.

M. Suezawa and H. E. Cook, Acta Met., 28 (1980) 423.

B. L. Averbach, M. Cohen, and S. G. Fletcher, Trans.
ASM, 40 (1948) 728.

B. L. Averbach and M. Cohen, Trans. ASM, 41 (1949)
1024.

G. V. Kurdjumov and O. P. Maximova, Doklady Akad. Nauk
Azerb. SSR, 61 (1948) 83. '

G. V. Kurdjumov and O. P. Maximova, Doklady Akad, Nauk
Azerb. SSR, 73 (1950) 95.

S. C. Das Gupta and B. S. Lement, Trans. AIME, 191
(1951) 727.

E. S. Machlin and M. Cohen, Trans. AIME, 184 (1952)
489.

S. A. Kulin and G. R. Speich, Trans. AIME, 194 (1952)
258.

R. E. Cech and J. H. Hollomon, Trans. AIME, 197 (1953)
685.

A. R. Entwisle and J. A. Feeney, Inst. of Metals,
Special Report No. 33, (1969) 156.

E. S. Machlin and M. Cohen, Trans. AIME, 191 (1951)
746.

H. Suzuki and T. Honma, J. Japan Inst. Met., 21 (1957)
263.

W. J. Harris and M. Cohen, Trans. AIME, 180 (1949) 447.

S. C. Das Gupta and B. S. Lement, Trans. AIME, 197
(1953) 530.

E. R. Morgan and T. Ko, Acta Met., 1 (1953) 36.

36.

37.

38.

39.

40.

41.

42.

43.

44.
45.

46.
47.

48.

49.

50.
51.
52.

53.

54.

69
B. Edmondson, Acta Met., 5 (1957) 208.

J. WOodilla, P. G. Winchell, and M. Cohen, Trans. AIME,
215 (1959) 849.

K. R. Kinsman and J. C. Shyne, Acta Met., 14 (1966)
1063.

R. Odaka and M. Okamoto, Nippon Kink. Gakk., 16 (1952)
81.

S. G. Glover, J. Iron Steel Inst., 200 (1962) 102.

R. Priestner and S. G. Glover, Physical Properties of
Martensite and Bainite, 181 Special Report 93, p33,
London (1965).

S. G. Glover and T. B. Smith, ”The Mechanism of Phase
Transformations,” Institute of Metals Monograph and
Report Series, No. 18, (1955) 265.

J. R. C. GuimarSes and J. C. Shyne, Met. Trans., 2
(1971) 2064.

J. R. C. GuimarSes, Scripta Metall., 9 (1975) 647.

M. Cohen, E. Machlin, and V. G. Paranjpe, Thermodynamics
in Physical Metallurgy, ASM, (1949) 242.

L. C. Chang, J. Appl. Phys., 23 (1952) 727.

J. H. Hollomon, L. D. Jaffe, and D. C. Buffum, J. Appl.
Phys., 18 (1947) 780.

C. Crussard, "The Mechanism of Phase Transformations in
Metals,” Institute of Metals Monograph and Report Series
No. 18 (1955) Discussion.

0. P. Maximova et. a1., Doklady Akad. Nauk Azerb. SSR,
134 (1960) 871.

G. V. Kurdjumov, J. Met., 11 (1959) 449.
R. P. Reed, Acta Met., 15 (1967) 1287.

J. C. Bokros and E. R. Parker, Acta Met., 11 (1963)
1291.

A. Brownrigg, Ph. D. Thesis, University of Sheffield,
Sheffield, England (1956).

C. L. Magee and H. W. Paxton, Thesis, Carnegie Institute
of Technology, Pittsburgh, PA (1966).

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.
65.
66.

67.

68.

70
J. R. Patel and M. Cohen, Acta Met., 1 (1953) 531.

E. M. Brienan and G. S. Ansell, Met. Trans., 1 (1970)
1513. ‘

A. Ankara, J. Iron Steel Inst., 208 (1970) 819.

T. N. Durlu and J. W. Christian, Proc. International
Conference on Martensitic Transformations, Cambridge,
MA, (1979) 65.

R. Datta and V. Raghavan, Scripta Metall., (1983) 615.

P. C. Maxwell, A. Goldberg, and J. C. Shyne, Met.
Trans., 5 (1974) 1305.

A. S. Sastri and D. R. F. West, J. Iron Steel Inst., 203
(1965) 138.

T. Maki, S. Shimooka, M. Umemoto, and I. Tamura, Met.
Trans., 2 (1971) 2944, Trans. J.I.M., 13 (1972) 400.

T. Araki and K. Shibata, Iron and Steel Inst. Japan,
Spring Meeting, (1972) 153.

G. V. Kurdjumov, J. Iron Steel Inst., 28 (1960).
M. Tokizane, Scripta Metall., 10 (1976) 459.

R. Pradhan and G. S. Ansell, Met. Trans. A., 9 (1978)
793.

M. Okamoto and R. Odaka, J. Jpn. Inst. Met., 16 (1952)
81.

Nishiyama, Martensitic Transformations, Academic Press,
New York, (1978) 321.

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