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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