Rheinisch-Westfälische Technische Hochschule Aachen
Institut für Eisenhüttenkunde
- Materials Science -
Study Integrated Thesis
Micromechanical modelling of ductile fracture of high strength multiphase steel XA980 based on real microstructures
M. Sc. Nithin Sharma
Matr.-Nr. 328059
Conducted in the department of Materials Characterization
From 1st September, 2014 to 15
th December, 2014
Supervisor: Univ. Prof. Dr.-Ing. W. Bleck
Dipl.-Ing. A. Rüskamp
Disclaimer i
I hereby assure that I have written the work in hand independently, that I have not made
use of other sources and means than listed and that I have indicated all citations.
Nithin Sharma
I hereby permit insight into my work by others than my examiner after handing it over.
Nithin Sharma
Acknowledgement ii
I would like to express my gratitude to Prof. Dr.-Ing. W. Bleck and group leader
Prof. Dr.-Ing. G. Heßling for giving me an opportunity to work in this project in IEHK,
RWTH Aachen.
I would take this opportunity to thank my supervisor Dipl.-Ing. A. Rueskamp for his
guidance and support throughout the study period. Without him this research wouldn’t
had been a success. His encouragement motivated me and made my work more
enjoyable.
I would also like to thank MSc. A. Serafeim, MSc. J. Lian and huge thanks to my dear
colleague Nachiket Deshmane for their valuable inputs to the research.
Abstract iii
Dual phase steels (DP) are among the most important advanced high strength steel
(AHSS) products recently developed for the automobile industry. A DP steel
microstructure has a soft ferrite phase with dispersed islands of a hard martensite phase
and hence has an excellent combination of high strength and formability. In the present
study, a commercially used XA980 steel for automotive applications was studied here.
This steel has an addition third phase, tempered martensite and its influence on the flow
behaviour of the steel is studied. Both experimental and numerical methods were
employed to investigate the mechanism of failure in XA 980 steel. Numerical simulation
were carried out using the SEM microstructures of fractured tensile specimen. A finite-
element micromechanical model was simulated by means of two dimensional
representative volume element (RVE) approach. Through numerical simulation new
understanding of the deformation localization was gained. Deformation localization,
which causes severely deformed regions in XA980, is most probably the main source of
rupture in the final stages of the failure. The SEM images after failure is the validation to
the results obtained from the simulation. The flow behaviour of single phases was
modelled using a Taylor-type dislocation based work-hardening approach. Flow
behaviour of XA980 was derived under uniaxial tensile loading and provided a good
approximation with the experimental results. In this study the formability of the XA980 is
studied, the RVE simulation is executed with different boundary conditions uniaxial,
biaxial and plain strain. From the equivalent plastic strain (PEEQ) contours, predictions
of damage initiation and failure under the above said boundary conditions are made.
Void nucleation in martensite grains and between martensite-tempered martensite
islands is the major damage mechanism in XA 980. Forming limit diagram (FLD) is
plotted. The simulation results is then compared with the experimental results.
List of Symbols iv
Symbol Definition Unit
σt True Stress MPa
σm True Stress of Martensite MPa
σF True Stress of Ferrite MPa
εt True Strain -
n Strain Hardening Component [Eq1] -
k Constant [Eq1, 5] -
Vm Martensite Volume Fraction %
εc True Uniform Strain for Composite -
εm True Uniform Strain for Martensite -
εF True Uniform Strain for Ferrite -
σGB Grain Size Strengthening MPa
σi Initial Yield Strength MPa
d Grain Diameter m
σ0 Peierl’s Stress MPa
Δσ Stress by Precipitation Hardening MPa
Cfss Carbon in Ferrite Solid Solution Wt %
Cmss Carbon in Martensite Solid Solution Wt %
α Constant [Eq 5] -
M, MT, M’ Taylor Factor -
μ Shear Modulus MPa
b Burgers Vector m
kr Recovery Rate m-1
L Dislocation Mean Free Path m
ΔV Volume Expansion %
n* Critical Number of Dislocations at the Phase Interface
[Eq 10]
-
λ Mean Spacing Between Slip Lines at Phase Boundary m
ρ Dislocation Density m-2
Index v
1. Introduction ........................................................................................................................................ 7
2. Literature Review .............................................................................................................................. 9
2.1. Dual Phase steel ....................................................................................................................... 9
2.1.1. Effect of Tempering on Martensite Phase..................................................................... 11
2.2 Fracture and Damage Mechanisms ........................................................................................ 12
2.2.1 Introduction to fracture mechanics ....................................................................................... 12
2.2.2 Cleavage Fracture .................................................................................................................. 13
2.2.3 Ductile Fracture ....................................................................................................................... 14
2.2.4 Void growth and coalescence ............................................................................................... 16
2.3 Damage Mechanisms of Dual Phase (DP) Steel .................................................................. 20
2.4 Forming Limit Diagram (FLD) ................................................................................................... 22
2.5 Micromechanical Modelling ................................................................................................. 24
2.5.1 Modelling Damage in Dual Phase steels ...................................................................... 24
2.6 Representative Volume Element (RVE) Approach ......................................................... 25
2.6.1 Boundary Conditions ........................................................................................................ 26
2.6.2 Modelling of Single Phase Flow Curve .......................................................................... 27
3. Methodology .................................................................................................................................... 29
4. Experimental .................................................................................................................................... 30
4.1. Material ...................................................................................................................................... 30
4.2. Heat Treatment ........................................................................................................................ 31
4.3. Microstructural Analysis ...................................................................................................... 31
4.4. Mechanical Testing ................................................................................................................ 33
5. Micromechanical Modeling Technique ..................................................................................... 35
5.1. RVE Generation ....................................................................................................................... 35
5.2. Flow curve of Single Phases ............................................................................................... 35
5.3 Modeling of Numerical Tensile Test ....................................................................................... 37
6. Results and Discussions .............................................................................................................. 40
6.1. Microstructural Analysis ...................................................................................................... 40
6.2. Material Characterization of XA980 ................................................................................... 40
6.2.1. Tensile Testing of XA980 ................................................................................................ 40
6.2.2. SEM Analysis of Fractured Zone in Tensile Specimen ............................................... 42
6.2.3. Nakajima Test Results for XA980 .................................................................................. 43
6.3. Flow Curves of Single Phases ............................................................................................ 44
Index vi
6.4. Numerical Tensile Test ......................................................................................................... 45
6.4.1. Flow Curve of Experimental and Numerical Tensile Test of XA980 steel ............... 45
6.4.2. Evaluation of Microstructural Development during Numerical Tensile, Biaxial and
Plain Strain Test ................................................................................................................................ 46
6.5. FLD Prediction ......................................................................................................................... 48
7. Conclusions ..................................................................................................................................... 49
8. References ....................................................................................................................................... 51
7
1. Introduction
Until now conventional steel was the main material in the automobiles. Due to an
increase in the demand of reducing the weight of the car and thereby reducing the cost
and energy. Leading to the use of new advanced materials like High strength steels
(HSS) and Ultra High Strength Steels (UHSS) .In recent times, Advanced High Strength
Steels (AHSS) have been utilized widely in industry due to their good mechanical
properties. This includes TRIP (Transformation Induced Plasticity), DP (Dual Phase), CP
(Complex Phase), TWIP (Twinning Induced Plasticity) and Martensite steels. Due to
their multiphase microstructure characteristics, with phase transformation and additional
strengthening by deformation mechanism. They possess a combination of high strength
and high ductility allowing for good formability resulting in wide application in automotive
industry.
Fig1.1. Example of different steel types used in a car body 74% DP and 3% TRIP
[Image Source 1]
DP steels as the name says has two phases, ferrite and martensite. The soft ferrite
which has a BCC crystal structure provides the formability to the steel, whereas the fine
dispersed hard martensitic islands imparts the material with high strength. During the
heat treatment of this type of steel, a transformation of austenite to martensite occurs
accompanied along a shear mechanism and increase in volume of martensitic fraction.
This induces mobile dislocations at ferrite-martensite interfaces to compensate for the
volume change, also better known as Geometrically Necessary Dislocations (GNDs).
8
Fig 1.2. Schematic representation of the microstructure of a dual phase steel.
[Image Source 1]
In the current study DP steel (HCT980XA) consists of Ferrite, Martensite and an
additional third phase Tempered Martensite. Conventionally the debonding of the ferrite-
martensite phase boundaries and the martensite inner-cracking are considered as the
main damage mechanisms of the DP steel. But, he third phase drastically affects the
mechanical properties and is of particular interest to the study. The material properties
are characterized with tensile testing and Nakajima test.
Quantitative determination of the dominant damage mechanism that contributes to the
fracture is not yet clear. A reliable prediction of the damage behaviour of the DP steel is
necessary. To relate the microstructure and mechanical properties a physical
microstructure-based model is required. The microstructure-based employs
representative volume element (RVE) technique, so the individual mechanical properties
and distribution of different phases could be considered. The flow behaviour of single
phases was modelled using a Taylor-type dislocation based work-hardening approach.
In XA980 the presence of third phase tempered martensite significantly changes the
plastic behaviour of the steel. It has to be noted that, no research whatsoever has been
previously conducted to model the phase and hence it is a major challenge to predict its
flow behaviour. This study focuses on modelling the effect of tempered martensite on
DP steel. By comparing the experimental and numerical studies, the present work is
aimed to gain a better understanding of the damage mechanisms of XA980 steel and
give a better prediction of formability.
Hard Martensite
9
2. Literature Review
2.1. Dual Phase steel
Dual-phase (DP) steels represent the most important AHSS grade. DP steels contain
primarily martensite and ferrite, and multiple DP grades can be produced by controlling
the martensite volume fraction (MVF) [2]. As per Liedl [3] these materials show an
excellent combination of ductility and strength and due to their high work – hardening
rate during initial plastic deformation they gained considerable interest in the automotive
industry. The ferrite gets additional strength due to induced dislocations during cold
working or with GNDs generated at ferrite-martensite (FM) interface during austenite to
martensite transition. These areas of high dislocation densities are responsible for the
continuous yielding behavior and the high initial work hardening rate according to
Uthaisangsuk [4]. From Leslie [5] the strength of martensite shows a linear dependence
to its carbon content. It was investigated that an increase of carbon content in martensite
from 0.2 to 0.3 wt. % causes an increase of yield strength from 1000 to 1265 MPa.
Foresaid by Speich and Miller [6] the tensile strength and ductile properties of DP steels
are attributed to volume fraction and distribution of martensite and amount of carbon in
martensitic phase. During deformation mobile dislocations are formed at FM interface
and twinning is observed in martensite. Contributing to higher elongation and higher
yield stress [5]. DP steels display high ultimate tensile strength (800 – 1000 MPa) and
high ductility (30 – 40%). The strength of dual phase steels is a function of percentage of
martensite in the structure. Figure 2.1 [7], illustrates the elongation vs. strength curve
and relative strength of DP steels along with other categories.
10
Figure 2.1: Illustration of Dual phase steels with other categories [Image Source 7]
DP steels can be obtained by hot and cold rolling. In hot rolled DP steel the dual phase
structure is achieved by controlled cooling from austenising temperature, Figure 2.2 [8].
In case of cold rolled steel the specimen is heated to intercritical temperature between
A1 & A3 where austenite is partially formed. The austenite transforms to martensite after
quenching.
Figure 2.2: Production of dual phase steel by Hot Rolling and Cold Rolling [Image Source: 8]
11
Percentage of martensite in DP steel depends on its carbon content, annealing
temperature and hardenability of austenitic region. Higher martensitic fraction results in
higher Yield Strength (YS) and UTS values in microalloyed DP steel. Hardenability is
promoted by addition of alloying elements, and thus facilitating formation of martensite at
lower cooling rate during quenching. High ductility in ferrite can be obtained by removal
of fine carbides and low interstitial content.
From the understanding of the results by Sayed et.al [9] by tempering the DP steel up to
200°C, YS increases slightly. This increase is due to volume contraction of ferrite grains
accompanied by tempering and rearrangement of dislocations in ferrite. Strengthening is
further enhanced by pinning effect created by diffusing carbon atoms or formation of iron
carbides in ferrite. But at higher temperatures, a drop in YS and TS is observed. At
higher tempering temperatures martensite softens and losses it’s tetragonality along with
precipitation of є carbides. The matrix structure of martensite finally transforms to body
center cubic (BCC) and carbon concentration of tempered martensite approaches to that
of ferrite. Hence, the strength difference between ferrite and tempered martensite is
reduced.
2.1.1. Effect of Tempering on Martensite Phase
Aforesaid Sayed et.al [9] investigated the change in morphology of martensite with
increasing tempering temperature in dual phase steel (0.21% C, 1.18% Mn) with 31%
martensite. It was observed that microstructure of specimen, which was tempered at
200°C is slightly different from un-tempered sample. Prior to tempering pure martensitic
islands are observed in the ferrite matrix. At 400°C tempered martensite and ferrite is
observed. At 500°C the carbon diffuses to form facetted carbides and edges of
martensite islands become smooth. The martensite island is thus covered with white
carbide particles. At 600°C the martensite phase is completely transformed into
cementite and ferrite. They also stated that the change in morphology of martensite
during tempering process is a result of combined multiple processes namely:
Recovery of defect structure, precipitation of carbides and transformation of
retained austenite.
12
Figure 2.3: SEM micrograph of DP steel (a) As-quenched (intercritical temperature:
760˚C; holding time: 0.5 h, quenched in water) (b) Specimen tempered for 1 h at 200°C
(c) Specimen tempered for l h at 400 °C (d) Specimen tempered for 1 h at 500°C [Image
Source: 9].
2.2 Fracture and Damage Mechanisms
2.2.1 Introduction to fracture mechanics
When a material is subjected to external force, stresses build up within the material. If
the exerted external force is larger than the effective bonding strength within the
material, then it begins to fracture. A macroscopic separation of the material occurs in
the region where there is the lowest bonding strength [10].
The damage behaviors in the materials are divided into cleavage and shear fracture.
With a continuous increase in temperature from a low starting temperature, metals with
BCC or HCP crystal structure experience a change in their fracture behavior from
cleavage to ductile fracture. The reason for this difference in the fracture behaviors are
13
due to changes in the cleavage fracture stress and yield stress as a function of
temperature. Figure 2.4 [10], illustrates the effect of temperature on the fracture
mechanisms. As per W.Bleck [10], only ductile fracture can be in FCC metals.
Figure 2.4: Temperature dependency of fracture mechanisms [Image Source: 10]
2.2.2 Cleavage Fracture
When the largest normal stress σ1 subjected on a material, locally reaches the
macroscopic cleavage fracture stress σf*, cleavage fracture occurs. Cleavage fracture
can be usually observed in the BCC crystal structure material, when subjected to high
stress and low temperature conditions. The stages involved in this type of fracture are:
Formation of micro cracks around a grain,
Spreading of these micro cracks.
Figure 2.5 [10] depicts the cleavage fracture surface. Characteristic feature of this type
of fracture surface is metallic luster.
14
Figure 2.5: SEM investigation of cleavage fracture surface [Image Source: 10]
2.2.3 Ductile Fracture
According to T.L. Anderson [11], four common mechanisms in metals and metal alloys
may impose failure in a material. Three of them include cleavage, intergranular and
fatigue fracture. The fourth is the ductile fracture, which this thesis focuses on. Figure
2.6 [11] illustrates the general description of three of the mechanisms.
15
Figure 2.6: Three of the most common fracture mechanisms in metal alloys. Upper left:
ductile fracture, Upper right: cleavage, Lower: intergranular fracture. [Image Source: 11]
According to H.Lavengar [12] when a ductile material is loaded and strained towards the
ultimate capacity of the material, the strain hardening in the material evens out the
capacity lost due to reduction of cross section area. At one point the material will reach
its ultimate strength. Further straining will result in local instabilities in the material. After
the point of the ultimate strength, all the elongation of the material is localized in a small
region where the material is rapidly losing its load carrying capacity. This region is the
necking region, and after the creation of this region, all further deformation will be
localized inside it. Metals containing minimal amounts of impurities will have a more
sudden failure after necking, while materials containing larger amounts of impurities will
have a smoother behavior when approaching failure, but start to fail at lower values of
strain. Ductile fracture mechanisms can be summed up into three stages:
Formations of free surfaces around a particle inside the material, either by
interfacial decohesion or fracture in the particle itself.
Creation and growth of voids created around particles, due to plastic straining and
hydrostatic stress.
Coalescence of the growing voids that eventually lead to failure of the material.
16
Some materials have strong bonds between the particles in the material, and these
material’s properties regarding ductile fracture is controlled by the development of voids
around the particle. The strong bonds are usually caused by the material having
particles that are relatively uniform in size. When the voids first start to appear, the
stress inside the material is so great that the growth and coalescence of voids happens
quickly. This gives a material that is identified by a small amount of straining from the
point of ultimate strength and to fracture. Other materials where the bonding between
particles is weaker, usually because of larger differences in size between particles, the
voids easily develop around the large particle. The development of fracture is controlled
by the growing and coalescence of the voids around the larger particles that are evenly
spread, but have great distances between them. This gives a material with a softer
fracture behavior, but with large fracture straining, from ultimate strength to fracture as
per H.Lavengar [12].
Voids tend to appear around inclusions and so-called second-face particles that may be
in the material. The stress between the material matrix and the surface of the particle
increases until the interfacial surface starts to slip and a void is created between the
particle and the material matrix. It may also occur that the stress is so great that the
particle itself fractures, splitting into parts and creating voids when it is pulled apart by
the surrounding material.
2.2.4 Void growth and coalescence
After the creation of voids in a material, these voids start to grow and coalesce as the
load on the material increases. Before the voids are created, the stress in the material is
carried by the cross-section area of the specimen. As voids are created, the effective
cross-section that can carry the load is reduced, since only the material between the
voids is available to carry it. The plastic straining resulting from the increasing elongation
is concentrated in the walls between the voids. It is this effect that causes the necking
phenomenon of the material. Figure 2.7 [12] a schematic description of the void growth
and coalescence are shown.
17
Figure 2.7: Depicts the growth of voids is for the most part concentrated around the
larger particles in the material [Image source: 12].
18
Figure 2.7 depicts a state of high stress triaxiality caused by high values of hydrostatic
stress encourages the growth of voids around the larger particles, giving large voids that
are sparsely spread. A state of lower stress triaxiality will in addition give void growth
around the medium sized particles, giving a higher number of small voids evenly spread
in the material.
According to J.R.Lund [13] the growth of voids is governed by the increasing plastic
strain and the hydrostatic stress that acts on the material. In a specimen exposed to
straining, the volume at the center of the cross-section area carrying the load will
experience a higher amount of hydrostatic stress than the volume closer to the edge.
This higher hydrostatic stress results in a higher state of stress triaxiality in the center of
the cross-section area. This will increase the speed at which the voids around the larger
particles grow, and make the center of the loaded cross-section area fail before the
edges ([11] pg. 223). The shape of the center fracture caused by the growth of voids
around the larger particles is often circular shaped. (In a flat-bar specimen the circular
shape is compromised by the shape of the specimen’s cross-section area) Outside the
center fracture zone, the stress triaxiality has a lower value because the hydrostatic
stress is smaller. This has promoted the growth of smaller voids around the smaller
particles, resulting in a lower maximum amount of strain from ultimate strength to
fracture. When suddenly all of the loading is put onto this area because of the failure of
the center region, the void growth will happen quickly and failure will occur fast.
The fracture of the outer ring happens at an angle of about 45° on the direction of
loading, creating the characteristic cup and cone fracture surface, as illustrated in Figure
2.8 [11]. This, and the fact that the voids are so small that they are almost invisible,
even by microscope, makes it look like a shear fracture. The angle of 45° is created
because the maximum plastic strain occurs in this direction. The growth of voids along
these bands is enhanced and the coalescence of voids will happen at a faster rate along
this path, creating the angled fracture.
19
Figure 2.8: The figure shows how the cone and cup shape fracture developed in a
round-bar tensile specimen [Image Source: 11].
From Figure 2.8 it can be noticed that in the center of the cross-section area in the
necking area the high value of stress triaxiality causes voids to appear and grow around
the larger particles in the material. When they coalescence, the circular shaped center
fracture creates bands of high plastic strain at an angle 45° on the direction of tensile
loading. This encourages the growth of voids along these bands creating a fracture
surface at an angle of 45°
Figure 2.9: Microscopic images of the fracture zone of a ductile material [Image Source:
12].
20
In the figure 2.9 (a) is the rough surface of the center fracture can be seen surrounded
by the smooth surface of the angle fracture. In picture (b) depicts the close up on the
rough surface of the center fracture area
2.3 Damage Mechanisms of Dual Phase (DP) Steel
Aforesaid DP steels usually contain harder martensitic phases and softer ferritic phases,
the mechanical properties of these phases differ from each other. Many have
researched the damage mechanism of DP steels and many assumptions are proposed.
Ahemed et al. [14] have identified three modes of void nucleation of DP steel, martensite
cracking, ferrite –martensite interface decohesion and ferrite- ferrite interface
decohesion. They observed that at low to intermediate martensite volume fraction (Vm),
the void formation was due to ferrite – martensite interface decohesion, while the other
two mechanisms are most probable to occur at higher Vm. M. Calcagnotto et al. [15]
analyzed the surfaces perpendicular to the fracture surface in order to illustrate the
preferred void nucleation sites. In the samples with coarse grains, the main fracture
mechanism is martensite cracking. While in the samples with ultra-fine grains, the voids
form primarily at ferrite-martensite interfaces and distribute more homogeneously.
Tamura et al. [16] presented pictures of the deformation fields in different DP steels.
They had reported that the degree of inhomogeneity of plastic deformation is extremely
influenced by the following factors: volume fraction of the martensite phase, the yield
stress ratio of the ferrite-martensite phase and the shape of the martensite phase. As
per Shen et al. [17], they had observed that, in general, the ferrite phase deformed
immediately and at a much higher rate than the delayed deformation of the martensite
phase. For DP steels with low martensite fraction, only the ferrite deforms and no
commendable strain occurs in the martensite particles; whereas for DP steels with high
martensite volume fraction, shearing of the ferrite-martensite interface occurs extending
the deformation into the martensite islands. According to Thomas et al. [18], they
considered that plastic deformation commences in the soft ferrite while the martensite is
still elastic, since the flow strength of ferrite is much lower than that of martensite. This
plastic deformation in the ferrite phase is constrained by the adjacent martensite, giving
rise to a build – up stress concentration in the ferrite. Thus the localized deformation and
21
the stress concentration in the ferrite lead to fracture of the ferrite matrix , which occurs
by cleavage or void nucleation and coalescence depending on the morphological
differences.
Experimentally its determined that the flow stress of HSLA and dual phase steels obey
the power law [19, 20, 21] given by:
𝝈𝒕 = 𝝐𝒕𝒏𝒌 (1)
σt is true stress, єt is true strain and k and n are constants.
Experimentally it is observed that stress component n is a function of Vm. Here n
decreases approximately linear with increasing percent martensite up to 50%
martensite. Davies further applied the composite theory [22] (change in uniform
elongation and tensile strength in composites of two ductile phases) to calculate change
in ductility with respect to the percent of the second phase in DP steels. The
assumptions of the theory are: 1) The tensile strength is a linear function of volume
fraction of second phase (mixture law) and 2) The uniform elongation of a composite is
less than indicated by law of mixtures. The relation between martensite fraction, Vm and
mechanical properties of two phases and composite is given by [22]:
𝑽𝒎 =𝟏
𝟏+𝜷𝝐𝒄−𝝐𝒎𝝐𝑭−𝝐𝒄
×𝝐𝒄𝝐𝒎−𝝐𝑭
(2)
Where, 𝜷 =𝝈𝒎
𝝈𝑭×
𝝐𝑭𝝐𝑭
𝝐𝒎𝝐𝒎
×𝒆𝝐𝒎
𝒆𝝐𝑭
σm and σF are the true tensile strengths of the martensite and ferrite respectively,
єc , єm ,єF are true uniform strains for the composite, martensite and ferrite respectively.
22
2.4 Forming Limit Diagram (FLD)
The forming limit diagram (FLD) provides a way to recognize and understand the plastic
instability of materials, in particular sheet metals.The maximum deformation degrees φ1
andφ2 for different samples under different loads are graphed against one another.
After connecting the each critical point, a forming limit curve (FLC) is plotted. The
resulting FLC represents the transition from safe material behavior to material failure for
a certain sheet metal at a given thickness. It is assumed that sheet failure by necking or
fracture is only determined by the plane stress state. Figure 2.10 [10] illustrates the
changes in a circular sheet element under different strain ratios and the resulting
degrees of deformation φ1 andφ2.
Figure 2.10 Common strain paths illustrated in FLD [Image Source: 10]
Figure 2.11 represents the typical V-Shape of FLC which is plotted from different
experimental data from a sheet metal [Image Source: 10].
23
Figure 2.11 V – shape of FLC [Image Source: 10]
As per W.Bleck [10], the level of the FLC is influenced primarily by the test conditions, as
in material properties, lubrication conditions, sheet metal thickness and diameter of grid
elements, which is illustrated in following Figure 2.12 [10]
Figure 2.12 Influences of experimental conditions on the level of FLC.[Image Source: 10]
Here, n-value comes from the Hollomon equation σ = Kφn, n exponent is a measure for
how far a material can be stretched without necking
24
2.5 Micromechanical Modelling
The flow behavior of DP steels – is predicted through Micromechanical Modelling using
numerical tensile test of a representative volume element (RVE). This method is
effective to analyze the stress and strain conditions prevalent in individual phases and
along the phase interface during deformation.
2.5.1 Modelling Damage in Dual Phase steels
Till date many researchers have worked in modelling of deformation behavior in dual
phase steels. As mentioned earlier in Section 2.2.3 ductile fracture is caused by plastic
instability of material. Plastic instability can be further classified into three categories:
(1) Instability triggered by initial geometrical imperfections: the classical Marciniak–
Kuczynski (M–K) model [23]
(2) Damage and void-growth: Predicted by cohesive zone model for ferrite-martensite
debonding and Gurson–Tvergaard–Needleman (GTN) model for ferrite degradation [24-
25]. For martensite cracking, three alternative approaches are available: Beremin local
criterion, Cohesive zone model and extended finite element method (XFEM).
(3) Material microstructure-level inhomogeneity: developed due to incompatible plasticity
between individual phases [26-27].
Particular to this thesis, Paul [26] approach has been adopted. As per this method a
micromechanical approach utilizing representative volume element (RVE) to predict the
flow behavior of DP steels based on localization of plastic strain in the matrix is follwed.
The approach applied by Paul did not employ any damage mechanism. The failure point
is determined by plastic strain localization approach. The approach signifies that plastic
instability in the material is a result of microstructure level inhomogeneity between
various constituent phases
25
2.6 Representative Volume Element (RVE) Approach
For a neat transition between the microscale and the effective material properties on the
macroscale an adequate definition of the RVE is necessary. Finite element (FE)
modelling is done on microstructural level using a real micrograph. Hence the light
optical or the high resolution SEM micrograph is first transformed to vectorial form. The
image is meshed forming grids termed as RVE. RVE defines for each phase separately
according to the microscopy of real microstructure; it is the statistical representation for
the entire material. RVE model of a material microstructure is used to calculate the
response of the corresponding macroscopic continuum behavior. RVE should have a
size large enough to represent enough heterogeneities and statistical
representativeness of all relevant microstructural aspects. Using RVEs of microstructure
is an important method for computational mechanics simulation of heterogeneous
materials such as DP steel. The reason for this being that the real material shows on
microscopic scale a complex heterogeneous behavior, in particular for multi phases with
differing strength. The stresses and strain show a distribution and partition on micro-
scale, which in turn affects the macro – behavior. There are methods to create 2D RVE.
RVE generation based on a real microstructure analyzed by light optical microscopy
(LOM). Thomser et.al [28] converted a light optical microscopy image of real
microstructure into 2D RVE by color difference between martensite and ferrite after
etching. RVE generation by electron back-scattered diffraction (EBSD) image. With
EBSD image, all grains and phases can be distinguished clearly. This helps in
description of phase distribution and phase fraction of martensite and ferrite in the 2D
RVE. Asgari et.al [29] used a meshing program OOF (Object Oriented Finite Element
analysis software [30]) to generate a 2D RVE from real high resolution micrographs. Sun
et.al [26] first processed the microstructure image in photo processing software to create
contrast, i.e martensite in white and ferrite in black. This image was subsequently
transformed from raster to vector form using ArcMap. The vectorized line image was
then imported to Gridgen, to generate a 2D mesh with triangular elements. Figure 2.13
[26] illustrates the method followed by Sun et.al [26]. Paul [27] used Hypermesh to
mesh the 2D RVE.
26
Figure 2.13: 2D RVE from a LOM image [Image Source: 26]
2.6.1 Boundary Conditions
The RVE is strained in FE based software Abaqus [31] under different loading
conditions. This requires the RVE to be constrained as per the loading condition which is
termed as Boundary Conditions. Two modes of boundary conditions are employed for
calculations: Homogeneous and periodic boundary conditions. Homogeneous boundary
condition simulates conditions close to tensile test. The left nodes (L) are restricted to
move in the direction of loading and right nodes (R) have same displacement in the
loading direction. Under tensile loading conditions displacement is applied at node point
3 or 2 or on complete right edge (R). The equations for Homogeneous boundary
conditions can be expressed as
𝑋𝑇 + 𝑋4
− 𝑋3 = 0
𝑋𝑅 + 𝑋2
− 𝑋3 = 0 (3)
In periodic boundary condition the RVE is spatially repeated to construct the whole
macroscopic specimen. Since RVE represents only a small part of the total tensile test
specimen, periodic boundary condition is also applied to perform numerical tensile test.
The equations for Periodic boundary conditions can be expressed as
𝑋𝑇 = 𝑋𝐵
+ 𝑋4 − 𝑋1
𝑋𝑅 = 𝑋𝐿
+ 𝑋2 − 𝑋1
𝑋3 = 𝑋2
+ 𝑋4 − 𝑋1
(4)
In equation 3 and 4: T, B, L and R, are notations for positive vector on top, bottom, left
and right boundaries of RVE respectively. And 1, 2, 3, and 4 are location of the position
vectors of the corner points as shown in Figure 2.14 [31].
27
Figure 2.14: Periodic boundary condition schematic diagram [Image Source: 31]
2.6.2 Modelling of Single Phase Flow Curve
Due to the difference in plastic properties and strain hardening behavior, the flow curve
is modelled for each individual phase. It depends on constituent/bulk properties, stress-
strain partitioning between two phases during deformation and chemical composition of
individual phase. During micromechanical modelling the flow behaviors of ferrite and
martensite are input as material properties.
Rodriguez Equation [33]:
The Rodriguez equation is used by Uthaisangsuk et.al [28], Ramazani et.al [32] and
Paul et.al [27] to predict flow curves of phases. The equation is based on Dislocation
Strengthening theory to model the flow curves of individual phases. This model is based
on the classic dislocation theory approach by Bergstroem [34] and Estrin and Mecking
[35], while the Peierl’s stress is based on the local chemical composition. The model
constants are derived from literature [28, 33]. The isotropic elasto-plastic hardening law
for single phases is given by
𝜎 = 𝜎0 + 𝛥𝜎 + 𝛼 ∗ 𝑀𝑇 ∗ µ ∗ √𝑏 ∗ √1−exp (−MTkrε)
kr∗ L (5)
Where σ is the flow stress at true strain ε. As displayed the equation comprises of three
terms. Frist term σ0 is the Peierls stress and is affected by elements in solutions.
𝜎0 = 77 + 750(𝑃) + 60(𝑆𝑖) + 80(𝐶𝑢) + 45(𝑁𝑖) + 60(𝐶𝑟) + 80(𝑀𝑛) + 11(𝑀𝑜) + 5000(𝑁𝑆𝑆) (6)
The second term Δσ is a measure of precipitation hardening or carbon in solid solution
[56].
Ferrite: ∆𝜎𝑓 = 5000 × (%𝐶𝑆𝑆𝑓) (7)
28
Martensite: ∆𝜎𝑚 = 3065 × (%𝐶𝑆𝑆𝑚) − 166 (8)
In equation 7 and 8, Cfss, Cm
ss denotes wt% of carbon in solid solution in ferrite and
martensite respectively. The third term [33-36] measures the dislocation strengthening
and work softening due to recovery. α is constant, MT is Taylor factor, μ is the shear
modulus, Burgers vector b, kr is the recovery rate: for ferrite kr= 10-5/dα, where dα is
ferritic grain size, L is dislocation mean free path, for ferrite Lf = dα. A dislocation mean
free path L is defined as the distance travelled by a dislocation segment of length l
before it is stored by interaction with the microstructure. L is proportional to the average
spacing of the homogeneously distributed dislocations, ρ-1/2 (ρ is dislocation density) or
can alternatively be derived by some microstructural parameter like the sub-grain size,
the interparticle spacing or the grain size.
Additionally to induce the residual stresses/back stresses by dislocation density along
ferrite-martensite interface, Paul [27] applied the volume expansion (ΔV %) of martensite
in the FE model
∆𝑉 = 4.64 − 0.53𝐶 (9)
Where C is carbon in wt%. Uthaisangsuk derived the long range back stress σs of
residual stresses by:
𝜎 =𝑀µbn∗
𝑑𝛼exp (
−λε
𝑏n∗) (1 − exp (−λε
𝑏n∗)) (10)
Where, n* is critical number of dislocations at the boundary and λ is mean spacing
between slip lines at phase boundary.
29
3. Methodology
The methodology of investigation is represented in the following Figure 3.1.
Figure 3.1 Methodology of Investigation
In the primary step includes the fundamental study of the chemical composition,
microstructure and mechanical properties of the XA980 steel. In the following step, RVE
model associated with the real microstructure is constructed, by a series of software
processing and followed by approximation of the numerical flow curve. Failure criteria
based on experimental results is assigned to the model and later subjected to boundary
conditions and is simulated. At the end the simulation results are extracted and is
compared with experimental observations, here the changes in the parameters like RVE
size, element size, mesh quality, flow curve is studied and analyzed.
30
4. Experimental
4.1. Material
HCT980XA dual phase steel was provided by Voestalpine Stahl GmbH and its
composition is illustrated in Table 4.1.
Table 4.1: Chemical Composition of investigated XA980 steel (in wt. %)
Material C Si Mn P S Cr Al Cu Nb B N
XA980 0.164 0.19 2.22 0.014 0.001 0.48 0.06 0.02 0.02 0.0002 0.0043
Davies and Magee [37], examined the effect of C content in DP steel, mainly the
mechanical properties at different carbon contents. The strength of the DP steel varies
linearly with the martensite amount, with no relation to the carbon content in martensite.
The difference in strain distribution, where for high carbon content, strain is concentrated
in ferrite phase might be the cause. Si and Mn enhance the strength of the steel. As per
Rashid et al. [38], each percent of Si elevates the tensile strength by 150 MPa and 48
MPa, with each percent increase in Mn. Si profoundly improves the ductility, whereas
high content of Si affects the surface quality by formation of oxide layers, which is hardly
removed during hot rolling. S. Papaefthymiou [39] states, that Mn plays a significant role
in controlling phase transformation. It decreases austenite transformation temperature,
delays pearlite formation and increases ferrite hardenability. Davies [37] also studied the
effect of P and Al. P decreases the ductility faster than increasing the strength, while Al
rises the martensite start temperature. V enhances hardenability, Nb and Ti precipitate
as carbonitrides before and during annealing. Cr decreases the amount of dissolved C
content and Mo enhances both strength and ductility according to S. Papaefthymiou
[39]. Cu in particular is used in low concentration since it initiates cracks along the grain
boundaries by formation of low melting sulphides.
31
4.2. Heat Treatment
Steel slabs were initially hot rolled in a 7-stand four-high mill to a thickness of about 3.5
mm. After pickling in a hydrochloric descaling line the materials were cold rolled on a
five-stand four-high tandem mill to a thickness of around 1.45 mm. The hot-rolling and
cold-rolling parameters are listed in Table 4.2.
Table 4.2: Hot rolling and cold rolling parameters
FT
[°C]
CT
[°C]
Hot strip
thickness
[mm]
Cold rolling
reduction
[%]
Cold rolled
thickness
[mm]
910 630 3.6 60 1.45
The DP-grade was annealed (Tan= 800°C) near the A3-temperature (A3= 850°C and A1=
700°C) so that the microstructure was not fully austenitic. Rapid quenching started for
the DP-grade at a lower temperature of ~650°C so that amounts of ferrite can grow.
4.3. Microstructural Analysis
Conventional LOM was used for metallographic phase estimation and microstructure
formation. Figure 4.1 represents the LOM image of XA 980 after etched with Le-Pera
solution. Light blue in color in LOM is ferrite phase whereas the brown color suggests
Tempered Martensite. ImageJ photo processing software was used to measure the
phase content and from this 27% of the microstructure is ferrite. When viewed at higher
magnification, martensite islands are distinctively visible with two morphologies, virgin
and Tempered martensite respectively. Figure 4.2 illustrates the SEM image of XA 980.
Tempered martensite has carbide precipitates within the prior austenite grain boundary.
These precipitated carbides form parallel segments projecting inwards from the
austenite grain boundaries. To ascertain the phase fractions of the two different
martensite, the structure of tempered martensite was compared with the results from
Hernandey et.al [40]. It was approximated that the tempered martensite was at 46%,
whereas the martensite content was 27%.
32
Figure 4.1: LOM image of XA980 etched with Le-Pera.
Figure 4.2: SEM image of XA980.
33
4.4. Mechanical Testing
The tensile properties of XA980 are measured by employing tensile test. These tests
were executed on Universal Tensile Testing machine (Zwick – Z100), in IEHK
department at RWTH Aachen University. Figure 4.3 is a schematic figure which
illustrates the dimensions of tensile specimen. The specimen under study is studied at
three directions to the rolling direction: 0°, 45° and 90°. In each of this direction, three
numbers of specimens were tested. The maximum load capacity of the machine is 100
kN, and an optical measuring system (video extensometer) is employed. The tensile test
was carried out in accordance with DIN EN 10325 standard.
Figure 4.3: Tensile sample geometry, with geometrical data according to DIN EN 10002
(all dimensions in mm).
The forming limit curve was plotted with test results from Universal Sheet Testing
machine (Erichsen Model 142/40) with standard Nakajima test.
34
Figure 4.4: Flat sheet specimen geometry, with all geometrical data according to
ISO/DIS 12004-2 (all dimensions in mm).
Table 4.3 represents the specimen geometries (all dimensions are in mm). Three
samples per geometry were tested.
Table 4.3 Specimen geometries (all dimensions are in mm)
Base Width b 20 40 80 90 100 130 140 160 190
Head Width x 55 75 115 125 135 165 175 195 190
Width including Tolerance
59 79 119 129 139 169 179 199 190
35
5. Micromechanical Modeling Technique
To study the changes of individual phases during straining at micro level and the flow
behavior of multiphase steels, the microstructure under consideration is embodied in
Finite Element Analysis/ Mechanics (FEA/ FEM) framework of continuum mechanics
through Representative Volume Element (RVE). Aforesaid, the flow curves of individual
phases would be calculated and in turn the mechanical properties of heterogeneous
material will be predicted. With reference to this thesis, the modelling is based on three
phases, instead of conventional two phased Dual Phase steel. The presence of the third
phase tempered martensite influences the flow behavior of the steel and hence is a
major criteria in constructing RVE and modelling the individual phase flow curve. The
material plastic deformation would be investigated, by using 2D plane strain conditions
for the numerical tensile tests on RVE.
5.1. RVE Generation
The size and number of constituents in RVE along with its mesh size is analyzed and
same is implied here. RVE under study was selected with 20μm x 20μm with
representative 27% martensite and 46% tempered martensite based on required
constituent aforesaid in literature. The micrograph is obtained from SEM image of
XA980. Different phases were identified with different colors using image processing
software Photoshop. Image-J was used to measure phase fraction. To transform the
colored micrograph to a finite-element mesh OOF2 [30] was used. Quadratic elements
with parabolic form functions were applied. Based on the real microstructure of XA980,
the generated RVE is illustrated Figure 6.1.
5.2. Flow curve of Single Phases
In the current study Rodriguez approach was adopted, to study the flow curve. The
variables to this equation were obtained from Rodriguez [33] and Uthaisangsuk et.al [22]
and are given below in Table 5.1
36
Table 5.1: Variable values used in Rodriguez Equation 6
α Taylor
Factor
M
Shear
Modulus
µ (MPa)
Burgers
Vector
b (m)
Dislocation Mean Free Path
L (m)
Recovery Rate
kr
Ferrite Marten
site
Temp.
Mart.
Ferri
te
Martensite Temp.
Mart.
0.33 3 80,000 2.5E-10 3.0E-06
3.8E-08 9.0E-07 3.33 41 5
The values of L and Kr for tempered martensite (TM) is based on the fact that, during
tempering, martensite loses its tetragonality and tends to form BCC structure at high
tempering temperature. The approximated values of L and Kr will lie in the range
between martensite and ferrite. From the SEM images it is observed that TM contains
heavy carbide precipitates and disintegrated matrix. Hence it can be estimated that KrTM
will approach KrF. From [40] it is known that the bainitic laths are generally in the range
of 10-7m in width. The Kr value for bainite is determined as 10-5/ dγ where dγ is prior
austenitic grain size. Applying the same for TM it was deduced that KrTM is 2 μm where
prior austenite grain size was determined from SEM image as approximately 5 μm.
From the approximation made above and to achieve best fitting curve, the determined
values are KrTM = 5 and LTM = 9x10-7 m.
Peierls Stress (Equation 6) is derived based on chemical composition (Table 4.1) is
σ0 = 328.4 MPa.
The strengthening by precipitation hardening (Δσ) for individual phases was calculated
as given by Equations 7 and 8. The strengthening by tempered martensite ΔσTM is
calculated similar to that of martensite (Equation 8). The same are presented in Table
5.2. The carbon content was approximated based on previous literature sources and
mass balance calculations.
37
Table 5.2: Phase fractions and carbon distribution
Phases Phase Fraction
(%)
Carbon Content
(wt. %)
Δσ
(MPa)
Ferrite 27 0.008 40.00
Martensite 27 0.260 635.90
Temp. Mart. 46 0.200 452.00
Back stresses on ferrite, developed during austenite to Martensite transformation with
volume expansion is adopted in this study and was formulated by Uthaisangsuk et.al
(Equation 10). The n and λ value are selected for best fitting curve. Value of n is 2.0x106
and λ is varied in the range 1.0x10-10 to 2.0x10-9. The stress values are added to ferrite
total stress. The resultant flow curves for individual phase obtained above are used as
input material property in Abaqus.
5.3 Modeling of Numerical Tensile Test
The commercial finite element code ABAQUS 6.13.1 [31] was used for the analyses. In
this study sheet specimens were tested. Since the sheet specimen is relatively thin as
compared to its in-plane dimensions and it is also subjected to in-plane loading during
uniaxial tensile tests, the sheet specimen is generally considered to be under plane
strain state. Therefore, two-dimensional plane strain CPS4 elements are adopted in this
study to simulate the in-plane failure modes of the XA980 sheet steel under tensile
loading. The mesh element type of RVE is assigned plane strain condition.
Both periodic and homogeneous boundary conditions were separately applied to RVE
and tested. Periodic boundary conditions (as defined in section 2.2.3) were applied to
RVE by an in-house FORTRAN based program 2d-Bounties. All the nodes of RVE
along the right edge have the same displacements in the x direction while they can
freely move in the y direction. Displacement along x direction was applied to bottom right
node. The bottom edge nodes were fixed in y direction and left edge nodes were fixed in
x direction. Alternatively, homogeneous boundary (HB) conditions were also applied. In
HB condition, displacement was measured on top right node of the RVE. The top edge
moment was constrained with respect to top right node. Similarly right edge moment
38
was constrained with respect to top right node. Following constraint equation was
manually entered to achieve HB condition and presented in Table 5.3.
Table 5.3: Constraint equation for Homogeneous Boundary condition, DOF: Degree of
Freedom
Coefficient Set Name DOF Coefficient Set Name DOF
1 Top Edge 2 1 Right Edge 1
-1 Top Right Node 2 -1 Top Right Node 1
The set “Top Edge” is a set of all elements on top edge of RVE (except the extreme right
edge where displacement is applied) and “Right Edge” is a set of all elements on right
edge except the node where displacement is applied. The equation takes the form “[1 x
Top Edge Nodes x 2] - [1 x Top Right Node x 2] = 0” and “[1 x Right Edge Nodes x 2] -
[1 x Top Right Node x 2] = 0” as referred in equation 3 (Section 2.6.1).
Figure 5.1: Boundary Conditions a) Homogeneous Boundary condition for displacement
at node 8 and b) Periodic Boundary condition for displacement at node 3.
No damage mechanism is induced and perfect bonding between the phases is
assumed. Hence no debonding at the grain boundary is modelled. The failure is
predicted as a result of incompatible plasticity between phases during straining. The
potential sites for strain localization can be viewed as the sites for void nucleation in the
conventional sense and, therefore the final shear failure is anticipated as a result of
growth and coalescence of these voids. The individual phase flow curves, detailed in
Section 5.2 are input in ABAQUS as material property. Additionally Poisson’s ratio 0.3
and Youngs Modulus 210 GPa was added for each individual phases as material
property.
39
The reaction force and displacement of right bottom node (for PB or on top right node for
HB) along x direction is obtained as the final output. Since the RVE is assumed to be the
representative in-plane microstructure for the XA980 steel under examination, the
macroscopic true stress (σt) is obtained by dividing the reaction force of the RVE in the x
direction with the initial area. Initial area is calculated as the product of initial thickness
and length of RVE for the plane stress model (Initial thickness of RVE 1, length of RVE
is 20µm). The true strain (εt) in the x direction is obtained by dividing the displacements
of the right bottom node with the initial length of this model (20 µm).
40
6. Results and Discussions
6.1. Microstructural Analysis
The micrograph of the XA 980 displays 27% ferrite, 27% virgin martensite and 46%
tempered martensite. ImageJ was used to measure the phase content. It is observed
that the carbides precipitate in tempered martensite within the prior austenite grain
boundary. It is determined that the grain diameter of ferrite is 3 μm and that of
martensite island is 0.8 to 5 μm respectively. Figure 6.1 represents the 2D RVE formed
from the actual SEM of XA980.
Figure 6.1: Image displays Generated RVE for XA980 with 27% ferrite.
6.2. Material Characterization of XA980
6.2.1. Tensile Testing of XA980
Figure 6.2 shows the experimental flow curves under tensile loading for XA980 steel.
Aforesaid, the mentioned samples were selected at 0˚, 45˚ and 90˚ to rolling direction. In
each direction three in numbers of samples were tested. The figure depicts the average
flow curve in each direction.
41
Figure 6.2: Experimental flow curve of tensile tested XA980 steel
The flow curve shows high tensile stress with reasonably short elongation as compared
to dual phase steels. One of the reasons attributed to this is due to high martensite
fraction present. A smooth yielding is observed which is characteristic of dual phase
steels. The elongation is observed highest along the rolling direction and a drop in
tensile stress is observed for specimen at 45˚ to rolling direction. Table 6.1 presents the
tensile test results.
Table 6.1: Tensile test results for XA980 (True stress strain curve)
Direction Rp02 [MPa] Rm [MPa] Ag [%] A [%]
0° 856.51 1193.92 5.74 9.32
45° 899.90 1130.51 4.31 8.82
90° 836.10 1188.47 5.45 8.32
0
200
400
600
800
1000
1200
1400
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Tru
e S
tress [
MP
a]
True Strain [-]
XA980_0deg
XA980_45deg
XA980_90deg
42
6.2.2. SEM Analysis of Fractured Zone in Tensile Specimen
For damage behavior the fractured tensile specimen (along rolling direction) was
analyzed under SEM. Void nucleation was observed at ferrite martensite interface and in
tempered martensite islands. Void initiation and crack within martensite was very less
visible. Voids are also observed at martensite and tempered martensite interface. It can
be inferred that void formation is mainly evident along the interfaces of martensite-
tempered martensite and ferrite-martensite. The main failure criterion is the ductile
damage.
Figure 6.3: Fracture tensile specimen along rolling direction is displayed with SEM
images a) at 8000X of the red square marked on specimen and b) 15000X of the red
square region marked in (a).
43
6.2.3. Nakajima Test Results for XA980
Figure 6.3 illustrates the averaged FLD of XA980 which is gained by conducting
Nakajima test with nine different geometries. Under biaxial condition the major strain
limit is high with approximately 2.7 strain. The FLC curve displays lower formability of
XA980 with regards to other dual phase steels.
Figure 6.3: Nakajima test result for XA980 steel
0
0.05
0.1
0.15
0.2
0.25
0.3
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
Majo
r S
train
ϕ1
Minor Strain ϕ2
XA_Avg
44
6.3. Flow Curves of Single Phases
Figure 6.4: Single phase flow curves
According to the dislocation density model, discussed in section 5.2 the flow curves of
single phase (ferrite, martensite and tempered martensite) for XA980 are calculated and
plotted. The martensite shows higher stress values than soft matrix. After 0.02 true
strain, no work hardening can be seen in martensite flow curve. The work hardening rate
is high in martensite in initial deformation period, on the other hand gradual increase in
work hardening is observed in ferrite matrix. The tempered martensite flow curve shows
similar yielding to ferrite with gradual increase in strain hardening. The work hardening
rate in tempered martensite is high to about 0.2% and further gradually saturates at
about 0.6%. As phase fraction increases the carbon content in phase decreases as per
mass balance calculations. This in turn affects the stress level of individual phases as
depicted in the flow curves. Similar flow curve results were observed by Uthaisangsuk
et.al [41].
0
500
1000
1500
2000
2500
0.00 0.10 0.20 0.30 0.40 0.50
Tru
e S
tress [
MP
a]
True Strain [-]
Martensite_27%
Temp.Martensite_46%
Ferrite_27%
45
6.4. Numerical Tensile Test
As discussed in Section 5, numerical tensile test were carried out on the generated 2D
RVEs of DP steel. The evolution of stress and strain in the RVEs can be obtained from
the numerical tensile tests. The figure illustrated below compare the simulated flow
behavior to the experimental.
6.4.1. Flow Curve of Experimental and Numerical Tensile Test of XA980 steel
Figure 6.5: Stress vs Strain curve for experimentally and numerically tensile tested
XA980 sample
The comparison between experimental tensile tests on XA980 steel and predicted true
stress – strain curves is displayed in Figure 6.5. The above figure depicts the three flow
curves obtained from the three parallel uniaxial tensile tests in the rolling direction. The
simulated numerical test curve is equated based on dislocation density theory, afore
discussed in section 2.6. Both the periodic and homogeneous boundary condition is
considered to simulate flow curve. The simulated curve with homogeneous boundary
condition fit well with the initial experimental strain hardening region. A slight diversion is
0
200
400
600
800
1000
1200
1400
0 0.02 0.04 0.06 0.08 0.1
Tru
e S
tress [
MP
a]
True Strain [-]
Experimental_1
Experimental_2
Experimental_3
Simulated_Homogeneous Boundary Condition
Simulated_Periodic Boundary Condition
46
observed in the tensile strength. The elastic region of the tensile test is not predicted by
the curves. The Rodriguez approach is suitable to predict the strain hardening behavior
and the plastic region flow characteristics. It can be observed that there is a deviation in
2D simulation which may be caused by planar state of strain adopted in calculations.
However it is evident from the figure that onset of yielding is predicted adequately by
taking into account of residual stresses developed at FM interface during phase
transformation after cooling of DP steel.
6.4.2. Evaluation of Microstructural Development during Numerical Tensile,
Biaxial and Plain Strain Test
Failure generally is initiated at strain localization places. This is caused by the formation
of void and subsequent growth and coalition (or de-cohesion between ferrite –
martensite) at higher strained regions. The local strain increases drastically in the strain
localization zone. Therefore, the probable region for failure initiation can be considered
as strain localization zone. This is in accordance with Paul [27] and Ramazani [32, 42].
The localized high strain spots represent void initiation in accordance with Paul [27].
Void nucleation is observed between martensite grains. The failure initiation is observed
between martensite – tempered martensite grains and within tempered martensite
grains. Further straining in RVE causes the strain bands (or the crack) to propagate
along ferrite martensite interface. From the figures, it can be observed that ferrite has
elevated strains and formed large amount of shear bands under the plane strain
condition. The direction of these bands of localized plastic deformation is, on average
45° to the tensile direction. The localized plastic strain in ferrite might be
overestimated/predicted because of the plain-strain condition, whereas martensite
undergoes finite plastic strain. Stresses around ~2000MPa are noticed in martensite
grains.
47
(a)
(b)
(c)
48
Figure 6.6: Von Misses Stress distribution (left hand side images) and equivalent strain
distribution (right hand side images) at onset of mesh distortion a) with uniaxial test, b)
with biaxial test, c) with plain strain test
6.5. FLD Prediction
Figure 6.7 Comparison of experimentally measured FLC and the predicted results of
RVE under three different loading conditions
Comparing with the experimentally measured FLC, all the three results are relatively
lower. This difference may come from the influence of sample size; the experiment is in
millimeter scale while the simulation is in micrometer scale. Secondly lack of input
parameters (eg. carbon content in each phase, which is been approximated with the
help of literature study). Thirdly the homogenous boundary condition with plain strain
condition works well for the uniaxial test condition, but the same condition doesn’t
produce good results for biaxial and plain strain condition.
0
0.05
0.1
0.15
0.2
0.25
0.3
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
Flow Limiting Curve
Simulated FLC
Experimental FLCMajo
r S
train
Minor Strain
49
7. Conclusions
Ductile damage is displayed by traditional dual-phase steels. But, the strain localization
theory approach to deduce failure is applicable in ductile damage only. In the present
study, the steel XA980 exhibits ductile failure by void formation and growth as detained
in section 6.2.2. Thus, the plastic localization theory can be utilized in this study to
predict the ductile fracture of XA980. This type of approach primarily focuses on the
plastic strain resulted from the incompatible plastic deformation accumulated in the
phases. This results in very high strain concentration locally and finally triggers failure.
These localized strain sites can be cited as sited sites for void nucleation. Thus the
microstructure inhomogeneity forms the source of microscopic imperfection and failure is
predicted as an outcome of plastic instability between phases.
1. The numerical tensile test on 2D RVE of DP steel is well correlated with the
experimental results. The experimental test exerts a three dimensional strain and
hence the deviation from the simulated curves. But the numerical biaxial and plain
strain conditions on 2D RVE of DP steel, particularly doesn’t correlate well with
experimental results as the reason is briefed in section 6.4.3. Further studies can
be done, especially on these two numerical tests, to better the results. It is
suggested from the experience, that the factors like (Young’s Modulus, Type of
Mesh, and DoF in Constraint Equations) might help in obtaining better results.
2. Both boundary conditions; periodic and homogeneous boundary condition
provide satisfactory results. The periodic boundary condition however provides a
slight gradual and higher yielding.
3. Microstructural evaluation during deformation displays that ferrite has elevated
microscopic strain/stresses and develops shear bands at an angle of 450 to
tensile axis. High stress is observed in martensite islands.
4. The localized strained area in RVE during deformation is predicted as void
nucleation site. The results obtained are in accordance with Paul [27]. The void
nucleating sites are observed between martensite - tempered martensite grains
and within tempered martensite grains. The results are in correspondence to
Speich and Miller [6].
50
5. The actual SEM microstructure of XA980 in fractured region displays void to be
nucleated between martensite grains and within tempered martensite grains,
Figure 6.3 and Figure 6.6a (corresponding to RVE formed from actual SEM
XA980 microstructure) display similar results. Thus the adopted stain localization
approach in this study provides good prediction of failure initiation as compared to
experimental results.
51
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