Journal of Minerals and Materials Characterization and Engineering, 2013, 1, 293-300
Published Online November 2013 (ht t p:/ / www.scirp.org/journal/jmmce )
http://dx.doi.org/10.4236/jmmce.2013.16044
Open Access JMMCE
Influence of Martensite Volume Fraction on Mechanical
Properties of High-Mn Steel
Rashid Khan, Tasneem Pervez, Sayyad Zahid Qamar
Mechanical & Industrial Engineering Department, College of Engineering Box 33, Sultan Qaboos University, Muscat, Oman
Email: rashidkhan.ned@gmail.com
Received September 10, 2013; revised October 12, 2013; accepted October 21, 2013
Copyright © 2013 Rashid Khan et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Elastic-plastic deformation behavior of austenitic, martensitic, and austenitic-martensitic high-Mn steels is investig ated
by using crystal plasticity theory. The development of expandable pipes made of two-phase steel for oil and gas well
applications is needed for improved and efficient recovery of hydrocarbons from difficult reservoirs. The current re-
search is aimed at improving the down-hole post-expansion material properties of expandable pipes. A mathematical
model is first developed based on finite-deformation crystal plasticity theory assuming that slip is the prime mode of
plastic deformation. The developed model is then numerically implemented by using the finite element software
ABAQUS, through a user defined subroutine. Finite element simulations are performed for austenitic, martensitic, and
austenitic-martensitic steels having different proportions of martensite in an austenite matrix. Three primary modes of
loading are considered: uniaxial tension, compression and simple shear. The variation in yield strength, hardening pat-
tern and dissipated energy is observed and analyzed.
Keywords: Dual-Phase Steel; Martensite; Mechanical Properties; Micromechanical Modeling; Finite Element Method
1. Introduction
As easily recoverable hydrocarbon resources are deplet-
ing, the oil and gas industry focuses more on exploring
oil and gas from ultra-deep, tight and pocketed reservoirs.
Not only are these recoveries difficult and costly, but they
require the development of new technolog ies and materi-
als. There is a significant d emand for n ew metallic materi-
als for well drilling and construction that can meet strin-
gent requirements regarding operation in sub-surface en-
vironment, elevated strength, immunity to sulphur crack-
ing, and light in weight. This poses new challenges for
materials scientist and engineers. This need is a driving
force for the development of modern innovative steel
grades for oil and gas well applications. One of the new
applications in well drilling and remediation is expand-
able pipes, requiring the development of modern steels.
These expandable pipes go through large diametral ex-
pansion at depths of several kilo meters in onshore or off-
shore wells. The resulting large permanent deformation
alters the post-expansion mechanical properties of the
pipes and can lead to premature failure during operations.
Since the first innovative development of expandable
pipe [1] to the recent developments [2], the challenge of
getting all desired material properties is still a distant
reality. Although a series of innovative steels (TRIP,
TWIP, Dual-phase, Martensitic steel, etc.) are being de-
vel o p ed in v ar io u s r esearch laboratories around the wor ld ,
the applications are focused more on transportation in-
dustry [3]. The idea revolves around complex thermo-
operations to obtain sophisticated microstructures with
combinations of different size grains, multiphase micro-
structure, and preferred crystallographic orientation. The
changes in grain morphology and the distribution of pha-
ses at the micro level during manufacturing lead to sig-
nificant improvement in material properties at macro
level. Dual-phase steel is categorized as advanced high-
strength steel (AHSS), and has two phases; either austen-
ite and ferrite, or austenite and martensite. Austenite is a
high temperature phase but in case of dual-phase steels,
metastable austenite exists at room temperature which
may transform into martensite upon application of ther-
mo-mechanical loads [4]. The existence of retained aus-
tenite plus the transformation mechanism may give rise
to the phenomenon of transformation induced plasticity
(TRIP), which ultimately enhances the strength and for-
mability of an alloy [5]. It is envisaged that the down-
R. KHAN ET AL.
294
hole expansion of pipes in a well can be done in a way to
exploit the interaction between micro-phases to elevate
its properties after expansion [6]. The elevated properties
will enhance structural integrity of the well, collaps e and
burst strength of the pipe, as well as providing a safe-
guard against possible mechanical failures such as buck-
ling etc.
Currently, high-Mn steel is used to manufacture ex-
pandable pipes consisting mainly of two phases, austenite
and ferrite with some traces of martensite. Presence of a
reasonable amount of martensite phase will greatly en-
hance the mechanical properties of the pipe. In other
words, if during permanent deformation process of pipes,
the cubic austenite phase transforms to tetragonal mart-
ensite phase due to the spontaneou s shear deformation of
crystal lattice by mechanical stressing, then the resulting
property of the expanded pipe will be better than pre-
expanded pipe. Extensive research has been done on
martensitic transformation (MT) of steel, both theoretic-
cally and computationally, but no work has been done
with oil well applications in mind . Different types of MT
models are developed which span from single to poly-
crystalline, atomic to continuum, and microscopic to ma-
croscopic [7]. Whether these models are simple or com-
plex, one solution approach is based on the finite element
method (FEM). Crystal plasticity FEM is one such tech-
nique which is developed to define the complete trans-
formation and deformation characteristics of single or
poly-crystal to de form and/or transform fro m one type of
crystal lattice to other. In this work, crystal plasticity
theory based on finite element formulation is used to
predict the deformation behavior of dual-phase high-Mn
steel.
An elastic-plastic deformation behavior of high-Mn
steel is analyzed when subjected to external mechanical
load. Different proportions of martensite in this steel are
considered. The main objective is to estimate the defor-
mation pattern of high-Mn alloy steel having different
volume fractions of martensite phase in an austenite ma-
trix. Initially, a mathematical model based on the crystal
plasticity theory is presented in order to estimate the elas-
tic-plastic deformation. Then, crystal plasticity model
using the time integration procedure is implemented as a
user defined material subroutine in the co mmercial finite
element software ABAQUS [8]. Finally, finite element
simulations are performed to analyze the stress-strain
behavior of high-Mn steel having different percentage of
austenite and martensite.
2. Problem Formulation
The elastic-plastic deformation behavior of single crystal
is modeled using crystal plasticity theory, assuming that
slip constitutes the dominant part of the plastic deforma-
tion. The kinematics of a single crystal based on the finite
deformation theory can be expressed using multiplicative
decomposition of total deformation gradient [9] as given
by
,
ep
F
FF (1)
where
F
represents the total deformation gradient, while
e
F
and p
F
are the elastic and plastic deformation gra-
dients, respectively. The elastic deformation gradient can
be further decomposed into symmetric left stretch tensor
and orthogonal rotation tensor:
ee p
F
UR F
,
where is the left stretch tensor while is the or-
thogonal rotation tensor. The elastic deformation gradient
e
Ue
R
e
F
describes the deformation of a crystal due to elastic
stretch and rigid body rotation, while the plastic deforma-
tion gradient p
F
represents only the deformation due to
crystallographic slip, which results due to dislocation
movement on preferred crystallographic planes termed as
slip planes. Crystallographic slip deformation does not
alter the lattice structure and thus the elastic properties
remain unchanged during the course of deformation. The
plastic deformation gradient must satisfy plastic incom-
pressibility condition . Therefore the determi-
nant of total deformation gradient will be greater than zero,
. The rotation and plastic deformation gradient
tensors can be com bined into a plastic rigid r otation tensor
det 1
p
F
det
*
0F
F
as
*
,
eep
F
UF FRF

(2)
Let 0
and
f
represent the reference (undeformed)
and current (deformed) configurations of a material point.
The decomposition of total deformation gradient can be
represented by two intermediate states between these two
end configurations, as shown in Figure 1. The two inter-
mediate configurations 1
and 2 represent the states
which the crystal experiences to attain the total deforma-
tion. The first intermediate state represents the con-
1
Figure 1. Kinematic decomposition of elastic-plastic defor-
mation of single crystal.
Open Access JMMCE
R. KHAN ET AL. 295
figuration with full plastic strain, which when subjected to
rigid rotation leads to the second intermediate state 2
.
The final deformed state of a material in current configu-
ration

f
is projected from second intermediate state
through stretch tensor .
2
During elastic-plastic deformation of a crystal in slip
mode, the major deformation mode is shear, resulting in
plastic flow. This rate of shear strain is termed as plastic
velocity gradient . The plastic deformation gradient
can be represented in term s of plastic velocity gradient as
e
U
p
L
,
p
pp
F
LF
(3)
In a crystal where slip activity happens on a number of
crystallographic planes (slip planes), the plastic velocity
gradient can be expressed as
 
1
11
,
SL SL
NN
ppp
LFFS mn
 





 (4)
where
represents a slip system , SL
is the total number of slip systems in a crystal,
1,, SL
N
N
is the
plastic shear strain rate on
-slip system, and S
is the
Schmid orientation tensor in first intermediate configura-
tion. The Schmid tensor is defined by the dyadic pro duct of
slip direction vector m
and area normal vector of slip
plane n
as Smn
. The Schmid tensor must be
updated in the second intermediate configuration to ac-
count for the lattice rotation effects on the deformation
behavior of the crystal. A forward mapping function is
used to estimate the Schmid tensor in the second interme-
diate configuration as

T,
ee
SRSR mn



(5)
where S
represents the Schmid tensor in second inter-
mediate configuration 2. The slip direction vector m
and area normal vector of slip plane n
can be estimated
as
,.
eeee e
m FmURm Rmn Rn

 

(6)
A list of slip directions and area normal vectors for face
centered cubic (FCC) and body centered cubic (BCC)
crystals can be found in [10]. The shear strain rate on
-sli p system in Equation (4) can be estimated using the
specific constitutive power law function [11], as given
by

1
0sin ,
m
s
 
 
 (7)
where 0
is the initial shear strain rate (constant for all
slip systems),
is the resolved shear stress (RSS),
s
is the slip resistance for
-slip system, and m is the rate
sensitivity parameter for the shear strain. The ratio of RSS
and slip resistance in the constitutive formulation of Equa-
tion (7) describes the slip activity, which is termed as slip
activity ratio, while the sensitivity parameter indicates the
response of a slip system subjected to the specific RSS
value. Two similar slip systems may not activate at the
same magnitude of RSS. The activity of slip system also
depends on the slip resistance, which is mainly governed
by the orientation of slip plane with respect to the loading
axis. Any slip system cannot be activated unless this ratio
becomes greater than unity.
The slip resistance
s
is responsible for the hardening
or softening of a slip system in a crystal at micro level and
affects material deformation at macro level. In addition,
the slip resistance of an individual slip system mainly
depends on the slip activity of other slip systems. For
example, the slip resistance of
-slip system
i
may increase or decrease if it interacts with another slip
system
,,1,, SL
kk iN
1, 1,i which is
inactive or more/less active at the same stress level. This
mechanism causes hardening or softening of
-slip sys-
tem. The interaction phenomenon becomes more complex
when
-slip system may interact with more than one
-slip systems. Furthermore, the variation in shearing
rate of
-slip systems may cause further complexity in
hardening mechanism, which may result in problems of
numerical solution convergence. These complexities can
be lessened by assuming self-hardening i.e. no interaction
among slip systems, and identical slip resistance for all
-slip systems. In this work, only identical slip resistance
is considered for all slip systems. The slip resistance for
-slip system can be expresse d as
1,
SL
N
sh

(8)
where h
is the strain hardening parameter of
-slip
system due to the interaction with
-slip system, and
is the shear strain rate of
-slip system. The strain hard-
ening parameter h
is estimated as ,hhq

where q
represents the hardening coefficient of
-
slip system. The value of represents self-
hardening 1.0q

, while for hardening due to slip sys-
tems interaction
h, . The single slip
hardening param eter 1q

can be estimated as
01
b
hh ss


, (9)
where 0 is the initial value of hardening parameter, h
s
is the slip resistance of
-slip system,
s
is the satura-
tion value of slip resistance, and is the hardening sen-
sitivity parameter which depends on slip resistance. The
hardening parameters 0,
b
h
s
and b are assumed to be
constant for all slip systems. Considering the reference
configuration, the Green finite strain tensor E
can be
calculated from the elastic deformation gra dient as

T
*1.
2
ee
EFFI

(10)
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296
The elastic deformation gradient can be calculated using
Equation (1)

1
ep
FFF
(11)
Once Green strain tensor is calculated, the second Piola-
Kirchhoff stress T can be obtained using crystal plas-
ticity constitutive equation as given below
*
:,TE
*
(12)
where is the elasticity tensor. The effective elasticity
tensor, for current two-phase high-Mn steel is deter-
mined using the elasticity tensors of austenite
e
a
and
martensite based on the rule of mixtures, the most
common hom ogenization technique used in com putational
mechanics:
m
m

1,
emam


 



(13)
where m
represents the volume fraction of martensite.
The resolved shear stress on
-slip system of Equation (7)
can be estimated using Schm id law
*,
e
F
TS
(14)
where is obtained using Equation (12) and other two
parameters are already known from kinematic and hard-
ening descriptions. Finally, the true stress T (Ca uchy stress)
can be calculated as
T



T
*
det .
eee
TF FTF (15)
It is interesting to observe the changes in amount of
energy dissipated with increasing volume fraction of mar-
tensite during the elastic-plastic deformation of crystal.
The extended form of this dissipated energy is given in
detail in [12] and is summarized as
 


1
SL
N
aaa
Dw
 
 
 
 
(16)
where
is the thermal equivalent part of resolved shear
stress
, a is the parameter which describes the strain
(and depends on dislocation density), a
w
is the modulus
of rigidity, a
is the lattice defect energy parameter, and
is a parameter which depends on slip resistance of
-slip system. The three terms within the square bracket
of Equation (20) represent the contribution of mechanical,
thermal and lattice defect energies, respectively, to the
energy dissipated during the deformation.
The crystal plasticity model developed abov e is numeri-
cally solved using finite element method. The algorithm
followed is given in Table 1 and is implemented in
ABAQUS using a user define d subro utine.
3. Numerical Modeling
Finite element simulations are done for polycrystalline
austenitic, martensitic and two-phase high-Mn steel used
for pipes in well applications. The two-phases considered
are face centered cubic (FCC) austenite and body centered
cubic (BCC) martensite. Both crystals are modeled as a
single cubic finite element using 8-noded 3-D brick ele-
ment with reduced integration (C3D8R), using one point
integration rule. Each finite element represents 500 grains
with random texture. In all cases, three primary deforma-
tion modes are considered i.e. uniaxial tension and com-
pression, and simple shear. Figure 2 shows geometry of
one element with pr escribed loading and bou ndary condi-
tions for tension, compression and shear. The element is a
cube with each side of 1 mm . For tension and compression
modes, a displacement of ±0.25 mm is applied on the face
having normal in the 3
e
axis direction, and for simple
shear displacement is applied in the 1 direction on the
same surface (Figure 2). Planar symmetric boundary con-
dition is applied on three faces while two remaining sur-
faces are traction free. The material models for austenitic
and martensitic steels include material and hardening pa-
rameters. The material parameters are defined through
their respective elasticity tensors, and . The elas-
ticity tensor elements for austenitic and martensitic steels
are used in the current work as reported in [
e
a
m
c
e
m
13]: =
286.6, = 166.4, 44 = 145. 0 and 11 = 372.4, 12 =
345.0, 44 = 201.9 (in GPa). Here, superscripts a and m
represent austenite and martensite, respectively. The ma-
terial parameters for dual-phase steels are represented by
the effective elasticity tensor, as given in Equation
(13). The hardening parameters for austenitic and marten-
sitic steels are extracted from the calibration of experi-
mental results reported in [
11
a
cm
c
12
a
cm
c
a
c
14], and given in Table 2.
4. Results and Discussion
Uniaxial tension, compression, and simple shear simula-
tions have been performed in order to estimate the defor-
mation behavior of steel with varying volume fraction of
austenite and martensite phases. The simulations have
been done for dual-phase steel having 5, 10, 15, 20, 25,
and 50 percent of martensite. In uniaxial tension and
compression, Cauchy stress component 33
T is plotted
against logarithmic strain 33
L
as other stress compo-
nents are negligible. The results are shown in Figures
Figure 2. FE model geometry and boundary conditions: (a)
Tension/compression, (b) Simple shear.
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297
Table 1. Numerical integration algorithm for crystal plasticity model.
1) START: Given parameters n
F
, 1n
F
, p
n
F
, n
S
, ,
0
h0
, m,
s
, b
2) Initial trial
Elastic deformation gradient

1
11
ep
nnn
FFF

Green strain tensor

T
*
111
1
2
ee
nnn
EFF

I
2nd Piola-Kirchhoff (P K ) stress tensor **
11
:
nn
TE
Slip resistance n
s
Resolved shear stress

T*
1111
ee
nnnn
FFTS
n

3) Iterative scheme to compute *
1
n
T
, 1n
s
3.1) Newton Raphson method to solve new estimate of *
1n
T
Compute shear strain rate

1
101 1
sin
m
nnnn
s
 



nn
Compute plastic deformation gradient 11
1
SL
N
pp
nn
F
IS


F

Compute elastic deformation gradient

1
111
ep
nnn
FFF

Compute Green strain tensor

T
*
111
1
2
ee
nnn
EFF

I
Compute 2nd PK stress tensor **
11
:
nn
TE
Check convergence: if *
12Tol
n
T
, GOTO step 3.2 ELSE 3.1
3.2) Newton Raphson method to solve new estimate of 1n
s
Compute slip resistance

10
11
SL b
N
nnn n
ssqh ss
  
1

 

Check convergence: if 12Tol
n
s
, GOTO step 4 ELSE 3.1
4) Update rotation tensor

11
1
exp skew
SL
N
ee
nnnn
R
tS


 
R
1
5) Update Schm id tensor

T
11 11
ee
nnnnnn
SRSR mn
 
 

6) Update Cauchy stress tensor
 

T
*
11 111
det
ee
nn nnn
TF FTF
 
e
7) END: Output parameters *
111
,,
nnn
STT

Table 2. Hardening parameters for austenite and martensite.
Phase m

1
0s
0MPah
0MPas

MPas b
Austenite 0.02 0.001 280 165 340 2.5
Martensite 0.01 0.001 300 235 350 1.8
3(a) and (b). In both loading conditions, the two main
observations are that 1) stress-strain curves are higher,
and 2) hardening pattern (shape of the stress-strain curve)
varies with increasing volume fraction of m artensite in the
austenite matrix. Higher stresses for higher martensite con-
tent are obviously as expected. As for hardening behavior,
R. KHAN ET AL.
298
Figure 3. Stress strain behavior: (a) Tension; (b) Compression; (c) Shear; (d) Yield strength variation.
the hardening pattern (shape of the curve) of 5%-marten-
site dual-phase alloy is almost the same as that of austen-
ite; for higher martensite fraction (10% or more) the
curves are more like pure martensite. One very interesting
observation is that all dual-phase curves coincide at a
strain value of about 0.15.
Figure 3(c) shows the variation of von Mises stress with
logarithmic strain component 12
L
for austenitic, marten-
sitic and dual-phase steels in simple shear loading condi-
tion. At low magnitudes of strain, stress levels of steels
having 5% - 20% of martensite are somewhat higher than
pure austenite; but above the strain value of 0.2 these
dual-phase steels show almost the same stress magnitude
as that of austenitic steel. As observed for tension-com-
pression, dual-phase alloys with 25% or more martensite
show hardening behavior that is closer to martensite.
The variation in yield strength under all three loading
conditions (tension, compression, shear) is almost linear;
Figure 3(d). Yield strength values are almost identical
under tension and compression at martensite fraction of
up to 20%. For higher amounts of martensite, there is a
difference of about 20 MPa in yield strengths under ten-
sion and compression. An increase in yield strength of an
alloy with increasing martensite content (harder phase) is
as expected. However, the difference in yield strength
behavior under tension and compression for higher mart-
ensite content may be due to various reasons: 1) increase
in heterogeneity due to random distribution of harder
phase, 2) mismatch in the crystal structure from FCC to
BCC, which results in changing the slip deformation me-
chanism, 3) dislocation pile up at the grain and austenite-
martensite boundaries, which may delay the onset of
yielding, etc.
The magnitude of dissipated energy for austenitic,
martensitic and dual-phase alloys for the three deform ation
modes is illustrated in Figure 4. As expected, martensitic
steel (hardest and strongest phase) needs the highest
amount of energy req uired to pro duce the same ma gnitude
of permanent strain. The other steel alloys with different
volume fraction of martensite show intermediate values of
dissipated energy. It is interesting to note that under ten-
sion, Figure 4(a), the dissipated energy of dual-phase
alloys reduces after reaching equivalent strain of 0.2, ex-
cept for 5% martensite. This trend is not observed in the
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R. KHAN ET AL. 299
Figure 4. Dissipated energy variation of dual-phase steels: (a) Tension; (b) Compression; (c) Shear.
case of compression or shear. This behavior could be due
to the interaction among slip systems (latent hardening),
which is responsible for hardening and softening of a me-
tallic material during the course of deformation. In com-
pression and shear, there is a high possibility for slip sys-
tems to interact more than in tension because of the mate-
rial flow behavior. This produces more hardening effects
as a result of dislocation interlocking and dislocation pile-
up at the grain boundary or within the grain. The energy
required to deform the material permanently therefore
becomes higher as the deformation progresses under com-
pression and shear. It can also be seen (Figure 4) that
under tensile loading there is a significant difference in
dissipated energies of austenitic, martensitic and rest of the
dual-phase steels. This trend is less prominent in com-
pression and shear. These observations show that the ad-
dition of harder martensite phase in an austenite matrix
may give significant variation in material deformation
behavior subjected to diffe rent loading co ndi tions.
All of the above results can be crucial in the applications
of dual-phase alloy steels where large permanent defor-
mation is required. This investigation becomes even more
important when the material needs to be deformed using
less energy. Prime examples are the use of expandable
tubulars in the oil and gas industry, extrusion of seamless
steel pipes, rolling of aluminum thin sheets, etc.
5. Conclusion
A mathematical model has been developed for the elastic-
plastic deformation of austenitic, martensitic, and austen-
itic-martensitic steels, and a numerical investigation has
been carried out for their behavior u nder tension, compres-
sion and shear. The percentage of ma rtensite in dual-phase
alloy is varied from 5% to 50%. Under tension and com-
pression, 5% martensite steel behaves almost like austenite,
while two-phase steels having 10% or more martensite
exhibit hardening behavior (shape of stress-strain curve)
that is closer to martensitic steel. Loaded under shear,
stress levels of steels having 5% - 20% of martensite are
somewhat higher than pure austenite; but above the strain
value of 0.2, these two-phase steels show almost the same
stress magnitude as that of austenitic steel. Variation of
yield strength against martensite content is almost linear
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R. KHAN ET AL.
300
under all loading conditions. Being the hardest and strong-
est, martensitic steel needs the highest amount of energy
required to produce a given amount of permanent strain,
while two-phase alloys require intermediate values of
dissipated energy. Under tension, less dissipated energy is
required after reaching a strain value of around 0.2 - 0.3.
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