1. Rheinisch-Westfälische Technische Hochschule Aachen
Institut für Eisenhüttenkunde
- Werkstofftechnik/Metallurgie -
Masterarbeit
des
cand.ing. Nithin Sharma
Matr.-Nr. 328059
Thema: Mikromechanische Modellierung des Deformations-
und Beschädigungsverhalten von DualPhase Stahl
mit Hilfe der Kristallplastizität-FEM Methode
Topic: Micromechanical modelling of the deformation and
damage behavior of dual phase steels by crystal plas-
ticity finite element method
Durchgeführt in der Abteilung Werkstoffmechanik
vom 15.07.2015 bis 26.02.2016
Betreuer: Univ. Prof. Dr.-Ing. W. Bleck
Dr.-Ing. J.Lian

2. Disclaimer
Hiermit versichere ich, dass ich die vorliegende Arbeit selbständig verfasst habe und
keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe, sowie Zita-
te kenntlich gemacht habe.
cand.-ing. Nithin Sharma
Hiermit erlaube ich, dass meine Arbeit nach der Abgabe durch weitere Personen als
meine Prüfer eingesehen werden darf.
cand.-ing. Nithin Sharma

3. i
Acknowledgements
I would like to express my gratitude to Prof. Dr.-Ing. W. Bleck for giving me an oppor-
tunity to work on this project in ZMB, RWTH Aachen.
I would take this opportunity to thank my supervisor Dr.-Ing. J. Lian for his guidance
and support throughout the study period. Without him, this research wouldn’t had
been success. His encouragement motivated me and made my work more enjoyable.
Personally, I would like to thank my parents. They have been the most important fac-
tor to support me through these three years of studying in abroad. I would also love
to thank my friends here in Aachen, for all the motivation and the good times we
spent together.

4. ii
Abstract
Dual phase steels (DP) are among the most important advanced high strength steel
(AHSS) products recently developed for the automobile industry. A DP steel micro-
structure has a soft ferrite phase with dispersed islands of a hard martensite phase
and hence has an excellent combination of high strength and formability. The aim of
this thesis was to material model on nanoindentation tests and fit the plasticity pa-
rameters of ferrite and martensite phase based on nanoindentation tests. With the
application of CPFEM for ferrite and J2 plasticity for martensite, nanoindentation
simulations were performed. In the present study, DP600 steel for automotive appli-
cations was used. Further, RVE simulations on artificial microstructure models were
investigated. This artificial microstructure model was constructed from RSA and Vo-
ronoi algorithm. In the present thesis, nanoindentation test on single grain was per-
formed for ferrite. From this test, the load displacement curves were recorded to
study the deformation mechanisms and the material strength. Nanoindentation simu-
lations were also performed in order to calibrate the parameters. By comparison of
the load displacement curves from nanoindentation tests and the corresponding
CPFEM simulations, the material parameters for single ferritic crystals were deter-
mined. Similar procedures were followed to determine the material parameters for
martensite phase, by the comparison of load displacement curves from nanoindenta-
tion tests and the corresponding J2 Plasticity simulations. A representative volume
element model with the crystallographic orientation as stated previously was utilized
to study the plasticity and damage behavior of the selected steel.

5. iii
Contents
Acknowledgements………………………………………………………….….……………i
Abstract……………….…………………………………………………………………...….ii
Contents………………………………………………………………………………………iii
1 Introduction.......................................................................................................... 1
2 Theoretical background....................................................................................... 4
2.1 Definition of DP steels ................................................................................... 4
2.2 Damage mechanisms of DP steel ................................................................. 6
2.3 RVE Approach............................................................................................... 8
2.3.1 Mechanical properties of single phase ...................................................... 9
2.3.1.1 Ferrite Phase.................................................................................... 9
2.3.1.2 Martensite Phase ............................................................................. 9
2.3.2 Homogenization and Boundary Conditions ............................................. 10
2.3.1.3 Homogeneous boundary condition (HBC’s) ................................... 10
2.3.1.4 Periodic boundary conditions (PBC’s) ............................................ 11
2.3.1.5 Homogenization strategy................................................................ 11
2.4 Crystal Orientation and Texture................................................................... 11
2.4.1 Rotation matrix, g .................................................................................... 12
2.4.2 Miller indices............................................................................................ 13
2.4.3 Euler angles ............................................................................................ 13
2.4.4 Texture .................................................................................................... 14
2.5 Deformation of single crystal ....................................................................... 15
2.5.1 Slip systems in FCC and BCC crystals.................................................... 15
2.5.2 Schmid’s law ........................................................................................... 16
2.5.3 Strain hardening effect ............................................................................ 17
2.5.4 Influence of strain rate on strain hardening ............................................. 18
2.6 Crystal plasticity finite element method ....................................................... 19
2.6.1 Kinematics............................................................................................... 20
2.6.2 Constitutive models ................................................................................. 23
2.6.3 Numerical model...................................................................................... 25
2.7 Nanoindentation test.................................................................................... 27
2.7.1 Nanoindentation test definition ................................................................ 27
2.7.2 Strain rate definition in nanoindentation tests.......................................... 27
2.7.3 Load displacement curves and pile up .................................................... 30

6. iii
3 Methodology.......................................................................................................33
4 Material...............................................................................................................34
4.1 Chemical composition ................................................................................. 34
4.2 Microstructure.............................................................................................. 34
4.3 Tensile properties........................................................................................ 35
5 Experimental and numerical investigation ..........................................................38
5.1 Grain size characterization by EBSD........................................................... 38
5.2 RVE model construction.............................................................................. 42
5.3 Nanoindentation simulations ....................................................................... 45
5.4 Parametric study of crystal plasticity parameters ........................................ 46
5.5 Parametric study of Swift law parameters ................................................... 48
6 Results and discussion.......................................................................................50
6.1 Results of CP parameters on nanoindentation ............................................ 50
6.1.1 Effect of strain rate sensitivity of slip, 𝑚................................................... 50
6.1.2 Effect of shear rate sensitivity of slip, 𝛾̇0 ................................................. 54
6.1.3 Effect of resolved shear stress on slip system, 𝜏0.................................... 57
6.1.4 Effect of slip hardening parameter,𝜏 𝑐
𝑠
...................................................... 60
6.1.5 Effect of slip hardening parameter, ℎ0 ................................................... 63
6.1.6 Effect of slip hardening parameter, 𝑎....................................................... 66
6.1.7 Effect of reference set of parameters for CPFE simulations.................... 69
6.1.8 Summary of the effect of CPFE parameters............................................ 71
6.2 Results of Swift law parameters on nanoindentation................................... 73
6.2.1 Effect of Swift law parameter, 𝑘............................................................... 73
6.2.2 Effect of Swift law parameter,𝜀0............................................................... 74
6.2.3 Effect of Swift law parameter, 𝑛............................................................... 75
6.2.4 Effect of reference set of parameters for J2-plasticity simulations........... 76
6.2.5 Summary of the effect of Swift law parameters ....................................... 77
6.3 Calibration result.......................................................................................... 78
6.4 RVE simulations .......................................................................................... 80
7 Conclusions........................................................................................................82
8 References.........................................................................................................84

7. 1
1 Introduction
Until now conventional steel has been the main material in the automobiles. Due to
increase in demand of reducing the weight of the automobiles, lead 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 transfor-
mation induced plasticity (TRIP), dual phase (DP), complex phase (CP), twinning in-
duced plasticity (TWIP) and martensite steels. Phase transformation and additional
strengthening by deformation mechanism are characteristics, due to their multiphase
microstructure. Due to this, they possess a combination of high strength and high
ductility allowing for good formability resulting in wide applications in automotive in-
dustry.
Figure 1.1 Example of different steel types used in a car body 74% DP and 3% TRIP
[1].
DP steels as the name says have two phases, normally ferrite and martensite. The
soft ferrite has a body centered cubic (BCC) crystal structure, which normally pro-
vides 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 mech-
anism and increase in volume of martensitic fraction. This induces mobile disloca-
tions at ferrite-martensite interfaces to compensate for the volume change, also bet-
ter known as geometrically necessary dislocations (GNDs).

8. 2
Figure 1.2 Schematic representation of the microstructure of a dual phase steel.
[1].
For a certain material, the microstructural features determine its macroscopic me-
chanical properties. Therefore for any material application, correlating between its
macroscopic properties and microstructure is significant.
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. Many of the research works incorporate an
empirical approach based on local chemical composition to approximate the flow
curve of ferrite and martensite phase. The effort to calibrate these parameters se-
verely hinders the application of it to a general or industrial scale. These empirical
approaches include Ludwik- Hollomon equation, Rodriguez Equation [2].
However, this quite simplistic approach gives very often significant deviations from
experiment and is not able to describe plasticity. Another main disadvantage of the
Rodriguez model is that it gives only a rough estimation of a certain phase. In particu-
lar for ferrite martensite steels, the effect of the strengthening on the ferrite produced
by the formation of the martensite is not considered [3]. In particular for martensite
phase, the flow behaviour is dependent on lot of microstructure features like the lath
distance, the lath orientation and the prior austenite grain size which the Rodriguez
approach does not consider. In the present study the flow behaviour of ferrite is
based on CPFEM which uses the phenomenological model, whereas the flow curve
of martensite is based on nanoindentation test which uses the J2 – plasticity model
respectively. In particularly with reference to martensite flow curve, the response from
the nanoindentation test are accurate. The mechanical behaviour is completely
based on the response of the single martensite phase.
Hard Martensite

9. 3
Crystal plasticity finite element method (CPFEM) is applied, in order to describe the
mechanical behavior. Taking into account the orientation information and applying
appropriate boundary conditions, CPFEM is able to map the elastic to plastic defor-
mation with the various types of deformation mechanisms which includes dislocation
slip, twinning, transformation-induced plasticity and so on. Particular application of
CPFEM to a certain material, requires good calibration of parameters used in the
crystal plasticity model. This suggests that the numerical investigations should be
accompanied with well-designed experiments. One of these experiments includes
nanoindentation. It is a powerful tool to characterize the mechanical behavior of a
single grain within a poly grain material. Load-displacement and pile-up curve are
acquired from the experiment. From the comparison of the experimental data and the
calculated curve from the simulation with CPFEM, the material parameters are then
be calibrated. Tensile test is performed to investigate the mechanical properties.
The aim or novelty of this thesis was to study the effect of CPFEM parameters on
nanoindentation simulation for ferrite grain and the effect of swift law parameters for
martensite grain. Especially for the CPFEM, nanoindentation simulations were per-
formed at three different strain rates. It was studied to acquire a better fitting of the
results.
In the present study RVE was constructed from an artificial microstructure using RSA
[4, 5] and MW-Voronoi algorithm [6]. It was then applied to simulate the experimental
process and get a good agreement with the experimental results.
Due to the similarity of the effect of different parameters, in both RVE and nanoinden-
tation simulation, it was possible to solve different sets of parameters which produce
the same results. To constrain the range of parameters, both the RVE and
Nanoindentation simulations were applied together to calibrate the crystal plasticity
parameter. In the end, one unique set of parameter within a certain range was
solved.

10. 4
2 Theoretical background
2.1 Definition of DP steels
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) [7]. As per Liedl [8] these materials show an excel-
lent 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 [9]. From Leslie [10] the strength of martensite shows a linear depend-
ence 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 (YS) from 1000
to 1265 MPa. Foresaid by Speich and Miller [11] 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 [10]. DP steels display high ultimate tensile
strength (UTS) 800 – 1000 MPa and high ductility (15 – 20%). The strength of dual
phase steels is a function of percentage of martensite in the structure. Figure 2.1
[12], illustrates the elongation vs. strength curve and relative strength of DP steels
along with other categories.
Figure 2.1 Illustration of Dual phase steels with other categories
[12].

11. 5
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, Fig-
ure 2.2 [13]. In case of cold rolled steel the specimen is heated to intercritical tem-
perature between A1 & A3 where austenite is partially formed. The austenite trans-
forms to martensite after quenching.
Figure 2.2 Production of dual phase steel by Hot Rolling and Cold Rolling
[13].
Percentage of martensite in DP steel depends on its carbon content, annealing tem-
perature and hardenability of austenitic region. Higher martensitic fraction results in
higher 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 [14] by tempering the DP steel
up to 200°C, YS increases slightly. This increase is due to volume contraction of fer-
rite grains accompanied by tempering and rearrangement of dislocations in ferrite.
Strengthening is further enhanced by pinning effect created by diffusing carbon at-
oms 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 mar-
tensite finally transforms to BCC and carbon concentration of tempered martensite
approaches to that of ferrite. Hence, the strength difference between ferrite and tem-
pered martensite is reduced.

12. 6
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 [14].
2.2 Damage mechanisms of 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.
Ahmed et. al [15] 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 [16] 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 [17] 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

13. 7
phase and the shape of the martensite phase. As per Shen et. al [18], 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 [19], 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
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 [20, 21, 22] given by:
𝜎𝑡 = 𝜖 𝑡
𝑛
𝑘 (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 de-
creases approximately linear with increasing percent martensite up to 50% marten-
site. Davies further applied the composite theory [23] (change in uniform elongation
and tensile strength in composites of two ductile phases) to calculate change in duc-
tility 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 sec-
ond 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 me-
chanical properties of two phases and composite is given by [23]:
𝑉𝑚 =
1
1+𝛽
𝜖 𝑐−𝜖 𝑚
𝜖 𝐹−𝜖 𝑐
×𝜖 𝑐
𝜖 𝑚−𝜖 𝐹
(2)
Where, 𝛽 =
𝜎 𝑚
𝜎 𝐹
×
𝜖 𝐹
𝜖 𝐹
𝜖 𝑚
𝜖 𝑚
×
𝑒 𝜖 𝑚
𝑒 𝜖 𝐹
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 respec-
tively.

14. 8
2.3 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 rep-
resentation 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 microstruc-
ture is an important method for computational mechanics simulation of heterogene-
ous 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 mul-
ti phases with differing strength. The stresses and strain show a distribution and parti-
tion 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 [24] 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 [25] used a meshing program OOF (Object Oriented Fi-
nite Element analysis software [26]) to generate a 2D RVE from real high resolution
micrographs. Sun et.al [27] first processed the microstructure image in photo pro-
cessing 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 tri-
angular elements. Figure 2.4 [27] illustrates the method followed by Sun et.al [27].
Paul [27a] used Hypermesh to mesh the 2D RVE.

15. 9
Figure 2.4 2D RVE from a LOM image [27].
2.3.1 Mechanical properties of single phase
In order to predict the overall deformation of DP steel, these constituent properties
and the partitioning of stress and strain between two phases during deformation have
to be known. During the mechanical modelling, the flow behaviors of ferrite and mar-
tensite are input as the material properties [28].
2.3.1.1 Ferrite Phase
Ferrite is the softest phase of steel, which has a BCC crystal structure. It contains a
maximum of 0.02% carbon at 723℃ and less at the room temperature. The primary
phase in the low carbon steel is ferrite and the matrix of DP steel is ferrite. The good
ductility of ferrite is the main reason of good ductility of DP steel. But the strength of
ferrite is too low, that it have to be strengthen through different strengthen mecha-
nisms. It is important to evaluate these mechanisms.
2.3.1.2 Martensite Phase
Martensite is a non-equilibrium phase that develops when austenite is rapidly
quenched down to room temperature. Due to the high cooling rate, the carbon atoms
in the austenite have no time to diffuse, so martensite can be regarded as a supersat-
urated solid solution of carbon in ferrite with a generally body- centered tetragonal
structure (BCT).
Martensite exhibit very high hardness and strength resulting from several strengthen-
ing mechanisms existing in the martensite. The high dislocation density which results
from the transformation leads to work hardening. The supersaturated carbon acting as
solid solute atoms will increase the stress, which will prevent the dislocation move-
ment. The volume change during transformation from austenite to martensite will in-

16. 10
duce an elastic stress field to martensite which, strengthens the martensite. In general
terms the hardness of martensite mainly depends on the carbon content, thus increas-
ing carbon content in turn increases the strength of martensite.
According to Leslie [29], the yield strength of martensite increases linearly with its
carbon content. It was also seen that increasing the carbon content of martensite
from 0.2 to 0.3% the yield stress increases linearly from 1000 to 1265 Mpa. The sub-
stitutional alloy elements also do increase the yield strength of martensite, but their
effect is secondary compared to carbon. Due to the high strength of martensite, its
main role in DP steel is to carry significant applied load. As previously stated, in the
second stage of work hardening of DP steels, the ferrite deforms plastically while
martensite deform elastically, the ferrite transfers the most applied stress to marten-
site. At the same time, martensite exhibit brittleness, which leads to a possibility of
cleavage fracture during deformation and initiate the failure of DP steel.
2.3.2 Homogenization and Boundary Conditions
The RVE is strained in FE based software Abaqus [30] under different loading condi-
tions. This requires the RVE to be constrained as per the loading condition which is
termed as Boundary Conditions. The boundary conditions to be applied on the RVE
model simulate the stress and strain evolution and distribution or the failure behav-
iors. Two modes of boundary conditions are usually employed for calculations: Ho-
mogeneous and periodic boundary conditions. In this study the latter is applied.
2.3.1.3 Homogeneous boundary condition (HBC’s)
The HBC’s 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
𝑋 𝑇
⃗⃗⃗⃗ + 𝑋4
⃗⃗⃗⃗ − 𝑋3
⃗⃗⃗⃗ = 0
𝑋 𝑅
⃗⃗⃗⃗ + 𝑋2
⃗⃗⃗⃗ − 𝑋3
⃗⃗⃗⃗ = 0 (3)

17. 11
2.3.1.4 Periodic boundary conditions (PBC’s)
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 ten-
sile test. The equations for Periodic boundary conditions can be expressed as
𝑋 𝑇
⃗⃗⃗⃗ = 𝑋 𝐵
⃗⃗⃗⃗ + 𝑋4
⃗⃗⃗⃗ − 𝑋1
⃗⃗⃗⃗
𝑋 𝑅
⃗⃗⃗⃗ = 𝑋 𝐿
⃗⃗⃗⃗ + 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 2.5 [30].
Figure 2.5 Periodic boundary condition schematic diagram [30].
2.3.1.5 Homogenization strategy
The homogenization phase aims to combine the micro and macros scales and in this
phase of solution, the averaged stress and consistent tangent stiffness matrix are
calculated for the multi scale constitutive relations using the computational homoge-
nization method outline as per Kouznetsova [31, 32]. With this method a RVE for a
certain macroscopic material point is chosen; then a deformation or stress is applied
on it and the results would give feed back to the macroscopic model.
2.4 Crystal Orientation and Texture
Materials like minerals, ceramics and metals are crystalline. Crystalline structure in
physical sense means, periodic arrangement of atoms. These crystallites are charac-
terized by size, shape but most importantly by crystallographic orientation. The rela-
tionship between crystal and sample coordinate system, is defined as crystal orienta-

18. 12
tion and can be seen in Figure 2.6. Miller Indices and Euler angles, describe the crys-
tal orientation.
Figure 2.6 represents the schematic diagram of definition of crystal orientation [33].
2.4.1 Rotation matrix, g
The crystallographic orientation is represented as a rotation matrix g, for transforming
the specimen coordinate into crystal coordinate. Figure 2.7 represents the relation-
ship between specimen and crystal coordinate system.
Figure 2.7 The rotation matrix: relationship between specimen and crystal coordinate
system [33].
g, can be mathematically expressed as
𝑔 = (
𝑔11 𝑔12 𝑔13
𝑔21 𝑔22 𝑔23
𝑔31 𝑔32 𝑔33
) = (
𝑐𝑜𝑠𝛼1 𝑐𝑜𝑠𝛽1 𝑐𝑜𝑠𝛾1
𝑐𝑜𝑠𝛼2 𝑐𝑜𝑠𝛽2 𝑐𝑜𝑠𝛾2
𝑐𝑜𝑠𝛼3 𝑐𝑜𝑠𝛽3 𝑐𝑜𝑠𝛾3
) (5)
Here 𝛼1, 𝛽1, 𝛾1 are the angles between the first crystal axis [100] and the three sam-
ple axes 𝛼2, 𝛽2, 𝛾2 are the angles between the second crystal axis [010] and the
three sample axes 𝛼3, 𝛽3, 𝛾3 are the angle between the third crystal axis [001] and
the three samples.

19. 13
2.4.2 Miller indices
For a quantitative characterization of crystallographic planes and directions, miller
indices are used. It is denoted as (hkl). The three elements in Miller indices are de-
fined by the inverses of the interception points of the chosen plane and the coordi-
nate axis. In a cubic lattice, the symmetry leads the atomic arrangement of some
planes and directions to be indistinguishable. In this case, all the crystallographically
equivalent planes and directions are defined by { } for planes and by < > for direc-
tions. For specific plane and direction, ( ) and [ ] are used respectively [34].
2.4.3 Euler angles
Euler angles are another efficient way to express the crystal orientation. It is defined
by three angles of rotation which transform the specimen coordinate system into the
crystal coordinate system when performed in the correct order.
Figure 2.8 represents the rotation through the Euler angles [33].
Bunge definition [35] is one of the most widely used for expressing Euler angles. As
stated above in the Figure 2.8, the rotation through Euler angles go as follows:
(i) Rotation about the Normal Direction (ND) by 𝜑1 to transform the rolling direction
(RD) to RD’
(ii) Rotation about 𝜑 about the axis RD’, to transform ND direction to ND’ (i.e [001])
(iii) Rotation of 𝜑2 about the ND’
The above steps can be analytically expressed as:

20. 14
𝑔𝜑1 = (
𝑐𝑜𝑠𝜑1 𝑐𝑜𝑠𝜑1 0
−𝑠𝑖𝑛𝜑1 𝑐𝑜𝑠𝜑2 0
0 0 1
)
𝑔𝜑 = (
1 0 0
0 𝑐𝑜𝑠𝜑 𝑠𝑖𝑛𝜑
0 − 𝑠𝑖𝑛𝜑 𝑐𝑜𝑠𝜑
)
𝑔𝜑2 = (
𝑐𝑜𝑠𝜑2 𝑠𝑖𝑛𝜑2 0
−𝑠𝑖𝑛𝜑2 𝑐𝑜𝑠𝜑2 0
0 0 1
) (6)
2.4.4 Texture
The distribution of orientation is not necessarily random for polycrystals. Certain ori-
entations are preferred, due to the material processing like forming or heat treatment.
Texture is defined as the distribution of orientation. Many material properties are tex-
ture related, therefore texture is important for materials. According to Engler et al.
[33], properties of materials are influenced by texture includes Young’s modulus,
Poission’s ratio, strength, ductility, toughness, magnetic permeability, electrical con-
ductivity and thermal expansion. These make the study of texture meaningful for un-
derstanding the material properties and guiding the industrial productive process.
Experimentally, textures are acquired from X-ray pole figures or Electron backscatter
diffraction. Many metallic materials have orientation distributions where certain orien-
tations are preferred, due to the materials processing like heat treatment or forming.
Table 2.1 represents the typical texture fibers in BCC materials [33].
Fiber Fiber axis Euler angles
A <011>/RD 0º,0º,45º-0º,90º,45º
Γ <111>/RD 60º,54.7º,45º-
90º,54.7º,45º
H <001>/RD 0º,0º,0º -0º, 45º, 0º
Z <011>/ND 0º ,45º, 0º-90º ,45º, 0º
E <110>/TD 90º,0º ,45º, 0º-90º , 90º
,45º
B 0º,35º,45º-90º,54.7º,45º

21. 15
Figure 2.9 Schematic representation of the most important textures in BCC materials
in 𝜑2 = 450
section [33].
2.5 Deformation of single crystal
2.5.1 Slip systems in FCC and BCC crystals
To move a dislocation on the slip plane, the dislocation must pass through a high en-
ergy configuration, corresponding to shear stress (Peierls stress) on its slip planes.
According to Gottstein [34] the Peierls stress increases with increasing distance be-
tween lattice planes and decreasing burgers vector. The densest packed planes
have the smallest plane distance, and the densest packed directions have the largest
burgers vector. The most dense packed planes and directions for FCC crystals are
{111} planes and <110> directions, that makes the {111} <110> as the primary slip
systems in FCC crystals. There are four {111} planes with three <110> directions
each, therefore FCC crystals have twelve different slip systems. <111> directions are
the densest packed directions for BCC crystals. Due to the slightly different packing
density of {110}, {112} and {123} planes, there is a defined slip direction but no de-
fined slip plane. Therefore, in BCC crystals, slip systems of {110} <111>, {112}
<111>, {123} <111> are observed. According to Gottstein [34], in this case, defor-
mation like an axial displacement of a stack of pencils can be visualized, which is
referred to as “pencil glide” or {hkl} <111>

22. 16
Crystal
structure
Slip plane Slip direction Number of
non-parallel
planes
Number of
slip systems
FCC {111} <110> 4 12
BCC
{110} <111> 6 12
{112} <111> 12 12
{123} <111> 24 24
HCP
{1000} <112̅0> 1 3
{101̅0} <112̅0> 3 3
{101̅1} <112̅0> 6 6
Table 2.2 represents the slip systems of basic lattice types.
2.5.2 Schmid’s law
Dislocation sets into motion, if the force on the dislocation nod and the corresponding
resolved shear stress exceeds a critical value 𝜏0. Figure 2.10 represents the relation-
ship between the resolved shear stress 𝜏 and the tensile stress 𝜎.
𝜏 = 𝜎 𝑐𝑜𝑠𝜅. 𝑐𝑜𝑠𝜆 = 𝑚 (7)
𝜅 Is the angle between tensile direction and slip plane normal
𝜆 Is the angle between tensile direction and slip direction
m is the Schmid factor
Schmid Law, states that the resolved critical shear stress is equal on all slip systems.
Figure 2.10 represents determining the Schmid factor [34].

23. 17
To determine the activated slip system, Schmid factor is important. According to
Gottstein [34] the system with the highest m will be activated first and carry the plas-
tic deformation. To explain it, during tensile loading, as the load increases, the re-
solved shear stress on each slip system increases until 𝜏c reaches on a system with
the largest Schmid factor. Plastic deformation of the crystal begins with dislocation
slip on this system, which is referred to as primary slip system. The required stress to
cause slip on the primary slip system is the yield stress of the single crystal. With fur-
ther increase in load, 𝜏c can be reached on other slip systems. These secondary slip
systems then begin to operate, without deactivating the primary slip system.
2.5.3 Strain hardening effect
The driving force for strain hardening effect is due to the dislocation multiplication and
their interactions. One can assume that, only the primary slip system in single crystal
is activated at the beginning (𝛾 < 0.4). According to Gottstein [34] under this condi-
tion, the shear stress and the shear strain of the primary slip system is given by
𝜏 = 𝜎𝑡 .
𝑐𝑜𝑠𝑘0
(1+𝜀)2 √(1 + 𝜀)2 − 𝑠𝑖𝑛2 𝜆 𝜊 (8)
𝛾 =
1
𝑐𝑜𝑠𝑘 𝜊
[√(1 + 𝜀)2 − 𝑠𝑖𝑛2 𝜆 𝜊 − 𝑐𝑜𝑠𝜆 𝜊] (9)
𝜆 𝜊 is the angle between tensile axis and the slip direction
𝑘 𝜊 is the angle between tensile axis and slip plane normal
Figure 2.11 represents the typical hardening curve, by single crystals deforming by
single slip.
Figure 2.11 represents typical strain hardening curve of FCC single crystal [34].

24. 18
Three stages would be distinguished without considering the elastic regime.
Stage I: Easy gliding stage, hardening coefficient
𝑑𝜏
𝑑𝑦
is very small.
Stage II: Large linear increase of strength
Stage III: Decrease in hardening rate (dynamic recovery)
During stage I, very few primary dislocations are activated. The dislocation density
are very low, therefore dislocations can move long distances without meeting disloca-
tions (other dislocations). Stage II occurs due to the interaction of primary disloca-
tions with dislocations on the secondary slip systems, that generates a network of
immobile dislocations (e.g. Lomer – Cottrell locks). An increase in internal stress oc-
curs due to the reason, that successive dislocations get stuck at these network of
immobile dislocations in the crystal. Thus, secondary slip systems are activated more
easily. To maintain the imposed external strain rate, for each immobilized dislocation,
another mobile dislocation has to be generated. This causes a rapid increase in the
dislocation density in stage II. Apart from this, a lot of dislocations would be generat-
ed from the internal dislocation source referred as Frank Read source. In stage III,
the hardening rate decreases, and hence is the longest stage. The driving force is
mainly due to the dominating cross slip of screw dislocations. Under high stresses,
cross slip enables screw dislocations to circumvent obstacles. It is more likely, that a
cross slip dislocation meets an antiparallel dislocation on the new glide plane so both
the dislocations are annihilated. Hence, the dislocation density decreases, which
equals to the slip length of a dislocation. Whereas, in BCC crystal structures, stage I
is not prominent as there are 48 slip systems. A number of dislocations may be acti-
vated simultaneously on different slip systems at the beginning. The interaction be-
tween these dislocations leads to the initiation of stage II. In stage III, both the dislo-
cation climb and cross slip of screw dislocations contribute to the lower hardening
rate.
2.5.4 Influence of strain rate on strain hardening
According to Miyakusu [64] for strain dependent materials:
𝜎 = 𝐾. 𝜀 𝑚
(10)
𝐾, 𝑚 are material constants
The measure of material’s hardness sensitivity to strain rate is referred as strain rate
sensitivity. In a Nanoindentation test, generally the strain hardening effect increases
with larger strain rate. As the material undergoes severe deformation, the stored

25. 19
elastic distortion energy and the number of defect sites for dislocation nucleation is
therefore larger. Thus, more dislocations nucleate and interact with each other, re-
sulting in an increased dislocation density. As there is not enough time for dislocation
annihilation and formation of stable dislocation network structures, e.g. cross slip of
screw dislocations, the material presents a ‘harder’ behaviour with increase in strain
rates. Accordingly, the recorded load – displacement curves shift left with higher load
values, causing the same penetration depths.
2.6 Crystal plasticity finite element method
CPFEM is a well-developed tool for describing the mechanical response of single
crystals and polycrystals, where driving force for the main deformation mode is as-
sumed as crystallographic slip. The method yields a stress-strain response and orien-
tation evolution for a given initial texture information and material parameters. Ac-
cording to Roters et.al [37] the advantage of CPFEM lies in its efficiency, in dealing
with crystal mechanical problems under complicated external and internal boundary
conditions. Apart from these, it provides great flexibility for underlying constitutive
formulations of the elasto-plastic anisotropy. In numerical simulation, it serves a plat-
form for multi-mechanism and multi-physics. It also enables the user to define differ-
ent deforming mechanisms as martensite formation and mechanical twinning. Ac-
cording to [38 - 40], the further advantage of CPFEM is that, they can be compared
with experimental results in a variety of properties and in a very detailed way. Some
of the studies include shape changes, forces, strains, strain paths, texture evolutions,
local stresses and so on. CPFEM simulation is widely used in both microscopic and
macroscopic scales, owing to its capability in dealing with complicated crystalline
matter [37]. For small-scale application, CPFEM can be used for simulating damage
initiation, inter – grain mechanics, micromechanical experiments (e.g. beam bending,
indentation , pillar compression) and the prediction of local lattice curvatures and me-
chanical size effects. Whereas, macro applications include large scale forming and
texture simulations. These studies would require, an appropriate method of homoge-
nization for the reason that a large number of crystals or phases are taken into con-
sideration in each volume element. According to [41 - 44], for CPFEM in macro appli-
cations, prediction of material failure, forming limits, texture evolution and mechanical
properties of the formed parts are the primary objectives. Tool deign, press layout
and surface properties are further related applications [37]. As a finite element meth-

26. 20
od (FEM), CPFEM can be regarded as a class of constitutive materials models.
Therefore, it can be integrated into FE code or implemented as user-defined subrou-
tines into commercially available solvers at reasonable computational costs.
2.6.1 Kinematics
The kinematics is the study of displacements and motions of the material object with-
out consideration of the forces that cause them. According to Roters et al. [45], in the
CPFE framework, the kinematics of isothermal finite deformation is used to describe
the process where a body originally in a reference configuration is deformed to the
current state by a combination of externally applied forces and displacements over a
period of time.
Assuming a material body with an infinite number of points occupy the original region,
𝑩 𝟎and deformed into region 𝑩 , the regions 𝑩 𝟎 and 𝑩 are referred to as undeformed
(or reference) configuration and deformed (or current) configuration, respectively.
The locations of arbitrary material points in the reference configuration are given by
vector 𝐱, whereas those in the current configuration are denoted by vector 𝐲. Thus,
the displacement between two configurations is given by 𝐮 = 𝐲 − 𝐱 in Figure 2.12.
Besides, as shown in Figure 2.13, the positions of infinitesimal neighborhood of arbi-
trary material points in both reference configuration and current configuration are re-
spectively represented by 𝑑𝒙 and 𝑑𝒚, which are related by deformation gradient F
that is a second-rank tensor given by the partial differential of the material point coor-
dinates in the current configuration with respect to the reference configuration (Eq.
11, Eq. 12).
𝑑𝒚 = (
𝜕𝒚
𝜕𝒙
) 𝑑𝒙 = 𝐅𝑑𝒙 (11)
𝐅 =
𝜕𝒚
𝜕𝒙
(12)
Figure 2.12 Deformable body occupies region 𝑩 𝟎 in the reference configuration and
region 𝑩 in the current configuration. The positions of material points are denoted by
𝐱 and 𝐲, respectively. The spatial displacement 𝐮 entails a deformation [37].

27. 21
Figure 2.13 Infinitesimal neighborhood around a material point in the reference con-
figuration and in the current configuration [46].
According to deformation gradient, the volume change of material body J (or Jacobi-
an determinant), the Green-Lagrange and Almansi finite strain tensors 𝐄 and 𝐄∗
, as-
sociated with a deformation, respectively, are defined as:
J = det(𝐅) (13)
𝐄 =
1
2
(𝐂 − 𝐈) =
1
2
(𝐅 𝐓
𝐅 − 𝐈) (14)
𝐄∗
=
1
2
(𝐈 − 𝐁−𝟏) =
1
2
(𝐈 − 𝐅−𝐓
𝐅−𝟏
) (15)
where 𝐈 is the second rank identity tensor, 𝐂 and 𝐁 are termed right and left Cauchy-
Green deformation tensors, respectively, and the superscript (T and −1) indicates the
transpose and inverse of the tensor.
For any deformation gradient, it can be expressed by a pure rotation tensor 𝐑 and a
symmetric tensor that is a measure of pure stretching as follows:
𝐅=𝐑𝐔=𝐕𝐑 (16)
where the symmetric tensors 𝐔 and 𝐕 are, respectively, the right and left stretch ten-
sors. This formula is referred to as polar decomposition theorem.
With the development of kinematics of finite deformations, a time-dependent dis-
placement of material body entails a non-zero velocity field, measured in current con-
figuration, given by the time derivative of the corresponding displacement field:
𝐯 =
𝑑
𝑑𝑥
𝐮 = 𝐮̇ (17)
Based on the deformation gradient, the spatial gradient of the total velocity, 𝐋, is de-
fined as:
𝐋 =
𝜕𝐯
𝜕𝐲
= 𝐅̇ 𝐅−𝟏
(18)

28. 22
Which is also termed velocity gradient and can be decomposed into a symmetric and
a skew-symmetric part:
𝐋 =
1
2
(𝐋 + 𝐋 𝐓) +
1
2
(𝐋 − 𝐋 𝐓) = 𝐃 + 𝐖 (19)
Where 𝐃 is the stretch rate tensor describing the instantaneous rate of pure stretch-
ing and 𝐖 is the spin tensor quantifying the rate of rigid-body rotation.
Considering the physical elasto-plastic deformation, the total deformation gradient
can be decomposed into two components:
𝐅 = 𝐅e 𝐅p (20)
Where 𝐅e is the elastic deformation gradient, due to the reversible response of the
lattice to external loads and displacements, and 𝐅p is the plastic deformation gradient
that is an irreversible permanent deformation under external forces and displace-
ments. It is based on the transformation from reference state to current state going
through an intermediate configuration which is only deformed plastically with a certain
rotation (to match both coordinate systems) but maintains the constant lattice frame.
Subsequently, the intermediate state transforms into current state corresponding to
elastic stretching of the lattice (plus potential rotation) as shown in Figure 2.12.
Figure 2.14 Multiplicative decomposition of total deformation gradient into plastic de-
formation gradient and elastic deformation gradient [46].
For finite deformation, the Cauchy stress tensor 𝝈 relates the stress in the current
configuration, but deformation gradient and strain tensors are described by relating
the motion to the reference configuration. Thus, in CPFE method the Piola-Kirchhoff
stress tensor is utilized to describe the situation of stress, strain and deformation ei-
ther in the reference or current configuration.

29. 23
The 1st Piola-Kirchhoff stress tensor 𝐏 relates forces in the current configuration with
areas in the reference configuration, which is defined as:
𝐏 = det(𝐅) . 𝛔. 𝐅−𝐓
(21)
This asymmetric stress is energy-conjugate to the deformation gradient.
The 2nd Piola-Kirchhoff stress tensor 𝐒 relates forces in the reference configuration
to areas in the reference configuration. The force in the reference configuration is
obtained via a mapping that preserves the relative relationship between the force di-
rection and the area normal in the current configuration. The expression of the 2nd
Piola-Kirchhoff stress tensor is:
𝐒 = det(𝐅) . 𝐅−𝟏
. 𝛔. 𝐅−𝐓
(22)
This tensor is symmetric and energy-conjugate to the Green-Lagrange finite strain
tensor. If the material is under a rigid rotation, the components of the 2nd Piola-
Kirchhoff stress tensor remain constant.
2.6.2 Constitutive models
With the multiplicative decomposition of the deformation gradient and the velocity
gradient:
𝐅 = 𝐅e 𝐅p (23)
𝐋 = 𝐅̇ 𝐅−𝟏
(24)
the plastic deformation evolves as:
𝐅𝐩
̇ = 𝐋 𝐩 𝐅𝐩 (25)
In the case of dislocation slip as the only deformation process according to Rice et. al
[47], 𝐋p can read:
𝐋 𝑝 = ∑ 𝛾̇ 𝛼𝑛
𝛼=1 𝐦 𝛼
⊗ 𝐧 𝛼
(26)
Where the vectors 𝐦 𝛼 and 𝐧 𝛼 are unit vectors describing the slip direction and the
normal to the slip plane of the slip system 𝛼, respectively, and ⊗ is the dyadic prod-
uct of two vectors; 𝑛 is the number of the active slip systems; 𝛾̇ 𝛼 is the shear strain
rate for that same slip system.
Such constitutive equations connect the external stress with the microstructural state
of the material. Different from the kinematic formalism, the constitutive equations cap-
ture the physics of the material behavior, in particular of the dynamics of those lattice
defects that act as the elementary carriers of plastic shear. There are two classes of
constitutive models, namely phenomenological models and physics-based models. In
this thesis, the phenomenological constitutive models are utilized.

30. 24
The phenomenological constitutive models mostly use a critical resolved shear stress
𝜏 𝑐
𝛼
as state variable for each slip system. Therefore, the shear rate is a function of
the resolved shear stress and the critical resolved shear stress:
𝛾̇ 𝛼
= 𝑓(𝜏 𝛼
, 𝜏 𝑐
𝛼) (27)
Herein the resolved shear stress is defined as:
𝜏 𝛼
= 𝐅𝐞
𝐓
𝐅𝐞 𝐒. (𝐦 𝛼
⊗ 𝐧 𝛼) (28)
and for metallic materials the elastic deformation in small, Eq. 2.24 is usually approx-
imated as:
𝜏 𝛼
= 𝐒. (𝐦 𝛼
⊗ 𝐧 𝛼) (29)
Besides, the evolution of the material state is formulated as function of total shear 𝛾,
and the shear rate 𝛾̇ 𝛼:
𝜏 𝑐
𝛼
= g(𝛾, 𝛾̇ 𝛼) (30)
In the framework the kinetic law on the slip system is:
𝛾̇ 𝛼
= 𝛾0̇ |
𝜏 𝛼
𝜏 𝑐
𝛼|
1
𝑚
𝑠𝑔𝑛(𝜏 𝛼) (31)
Where 𝛾̇ 𝛼 is the shear rate for slip system 𝛼 subjected to the resolved shear stress 𝜏 𝛼
at a slip resistance 𝜏 𝑐
𝛼; 𝛾0̇ and 𝑚 are material parameters that quantify the reference
shear rate and the strain rate sensitivity of slip, respectively. The influence of any slip
system 𝛽 on the hardening behavior of slip system 𝛼 is given by:
𝜏̇ 𝑐
𝛼̇ = ∑ ℎ 𝛼𝛽
𝑛
𝛽=1 |𝛾̇ 𝛽
| (32)
Where ℎ 𝛼𝛽 is referred to as the hardening matrix:
ℎ 𝛼𝛽 = 𝑞 𝛼𝛽 [ℎ0 (1 −
𝜏 𝑐
𝛽
𝜏 𝑐
𝑠 )
𝑎
] (33)
Which indicates the micromechanical interaction among different slip systems empiri-
cally. Herein 𝜏 𝑐
𝑠, ℎ0 and 𝑎 are slip hardening parameters. They are assumed to be
identical for all slip systems owing to the underlying characteristic dislocation reac-
tions. The parameter 𝑞 𝛼𝛽 is a measure for latent hardening. Its value is taken as 1.0
for coplanar slip systems and 1.4 otherwise, which renders the hardening model ani-
sotropic.
In principle, the phenomenological formulations mentioned above are used in the
FCC materials [37, 47, 48] but they can also be used in BCC materials. However,
due to the intricacy of the atomic scale in BCC materials, there exists the nonplanar

31. 25
spreading of screw dislocation cores, which result in more complicated plasticity
mechanism in bcc materials [49, 50]. To take these effects into account, the expres-
sion of slip resistance for bcc materials can be modified to [51, 52]:
𝜏 𝑐,𝑏𝑐𝑐
𝛼
= 𝜏 𝑐
𝛼
− 𝑎 𝛼
𝜏 𝑛𝑔
𝛼
(34)
where 𝑎 𝛼
is a coefficient that gives the net effect of the nonglide stress on the effec-
tive resistance, and 𝜏 𝑛𝑔
𝛼
is the resolved shear stress on the non-glide plane with
normal 𝐧 𝛂̃ , which is given by [53, 54]:
𝜏 𝑛𝑔
𝛼
= 𝐒. (𝐦 𝛂
⊗ 𝐧 𝛂̃) (35)
Therefore, the kinetic law is in this case constructed by inserting the modified critical
resolved shear stress instead of the classical slip resistance into the power-law ex-
pression for the plastic slip rate (Eq. 31).
2.6.3 Numerical model
As stated previously, CPFEM is able to be integrated into FE code or implemented as
user-defined subroutines into commercially available solvers like Abaqus for simula-
tion.
The overall simulation task can be conceptually split to four essential levels as illus-
trated in Figure 2.15 from top to bottom: To arrive (under given boundary conditions)
at a solution for equilibrium and compatibility in a finite strain formalism one requires
the connection between the deformation gradient 𝐅 and the (first Piola–Kirchhoff)
stress 𝐏 at each discrete material point. Provided the material point scale consists of
multiple grains, a partitioning of deformation 𝐅 and stress 𝐏 among these constituents
has to be found at level two. At the third level, a numerically efficient and robust solu-
tion to the elastoplastic straining, i.e. 𝐅e and 𝐅p is calculated.
This would depend on the actual elastic and plastic constitutive laws. The former
links the elastic deformation 𝐅e to the (second Piola–Kirchhoff) stress 𝐒. The latter
keeps track of the grain microstructure on the basis of internal variables and consid-
ers any relevant de-formation mechanism to provide the plastic velocity gradient 𝐋p
driven by 𝐒. Both are incorporated as the forth level in the hierarchy [55, 56].

32. 26
Figure 2.15 Schematic diagram of the hierarchy at a material point.
Based on the theory presented in the previous section, and concerning the commer-
cial FE codes, the CPFE constitutive laws can be implemented in the form of a user
subroutine e.g. UMAT in ABAQUS as a material model. The purpose of this model is
to calculate the stress required to reach the final deformation gradient in both implicit
and explicit scheme and to determine the material Jacobian 𝐽 =
𝜕Δ𝜎
𝜕Δ𝜀
, only in implicit
code fpr the iterative procedure by perturbation methods.
Figure 2.16 represents the visualized clockwise loop of calculations, where the stress
is calculated by using a predictor-corrector scheme.
Figure 2.16 Clockwise loop of calculations during stress determination (𝐒 is second
Piola-Kirchhoff stress, 𝛾̇ 𝛼
is shear rate, 𝐋p is plastic velocity gradient, 𝐦 𝛼 is slip direc-
tion, 𝐧 𝛼 is slip plane normal, 𝐅p is plastic deformation gradient, 𝐅e is elastic defor-
mation gradient, I is identity matrix, C is elastic tensor) [57].

33. 27
According to [45], theoretically, the prediction can start from any of the quantities in-
volved, and one can follow the circle to compare the resulting quantities with the pre-
dicted one. Subsequently, the prediction would be updated using, for instance, a
Newton-Raphson scheme. Although various implementations using different quanti-
ties as a starting point lead to the same results, there are two numerical aspects to
consider: firstly, the inversion of the Jacobian matrix occurring in the Newton-
Raphson algorithm (the dimension of the Jacobian matrix is equal to the number of
independent variables of the quantity that is used as the predictor); secondly, the
evaluation of the character of equations (the numerical convergence of the overall
system).
2.7 Nanoindentation test
2.7.1 Nanoindentation test definition
Nanoindentation simply refers to the indentation in which the length scale of the pen-
etration is measured in nanometers. Indentation test is a widely used method for
measuring mechanical properties of materials. It can be used to calculate properties
like hardness, elastic modulus, strain hardening exponent, fracture toughness (brittle
materials) and so on. With high resolution equipment, this measurement can be done
at micrometer and nanometer scales.
2.7.2 Strain rate definition in nanoindentation tests
Normally a nanoindentation test contains an elastic-plastic loading segment and an
unloading segment. Figure 2.17 reveals the contact between a rigid sphere and a flat
surface. The related variables are listed in the following.
Figure. 2.17 Schematic diagram of contact between a rigid spherical indenter and a
flat surface [58].

34. 28
𝑎: radius of contact circle
𝑅𝑖: indenter radius
ℎ 𝑝: penetration depth, distance between bottoms of the contact to the contact circle
ℎ 𝑎: distance between contact circle and sample free surface
ℎ𝑡: total displacement
Hertz equation combines the radius of contact circle 𝑎 with reduced modulus of in-
denting system 𝐸 𝑟:
𝑎3
=
3𝑃𝑅 𝑖
4𝐸 𝑟
(36)
Where
1
𝐸 𝑟
=
1−𝑣2
𝐸 𝑖
+
1− 𝑣 𝑠
2
𝐸𝑠
𝑃: applied force
𝐸 𝑟: reduced modulus
𝜈𝑖, 𝜈 𝑠: Poisson’s ratio of indenter and sample
𝐸𝑖, 𝐸𝑠: Young’s modulus of indenter and sample
The mean contact pressure is simplified as:
𝑝 𝑚 =
𝑃
𝐴
=
𝑃
𝜋𝑎2
(37)
𝐴: contact area
Combining Eq. 36 and Eq. 37, we obtain:
𝑝 𝑚 = (
4𝐸 𝑟
3𝜋
)
𝑎
𝑅
(38)
𝑝 𝑚 is indentation stress
𝑎
𝑅
is indentation strain
In a uniaxial tensile or compression test, the strain (ε) varies linearly with stress (σ)
within the elastic deformation range:
𝜎 = 𝐸𝜀 (39)
Where E is the Young’s modulus.
Even though there exists a similar linear relation between indentation stress and in-
dentation strain in elastic condition, an indentation stress-strain relationship yields
valuable information about the elastic-plastic properties of the material that is not
generally available from uniaxial tensile and compression tests [58].
Figure 2.18 represents the situation of contact between a rigid conical indenter and a
flat specimen surface.

35. 29
Figure 2.18 Diagram of a contact between a rigid conical indenter and flat surface, 𝛼
is cone semi-angle [58].
For a conical indenter:
𝑃 =
𝜋𝑎
2
𝐸𝑟ℎ 𝑝 (40)
Where 𝑎= ℎ 𝑝∙𝑡𝑎𝑛𝛼, 𝐴= 𝜋ℎ 𝑝
2
𝑡𝑎𝑛2
𝛼, so
𝑝 𝑚 =
𝑃
𝐴
=
𝜋ℎ 𝑝 𝑡𝑎𝑛𝛼
2
𝐸 𝑟ℎ 𝑝
𝜋ℎ 𝑝
2 𝑡𝑎𝑛2 𝛼
=
𝐸 𝑟
2
𝑐𝑜𝑡𝛼 (41)
Because 𝛼 is always constant during loading and unloading, it seems like the inden-
tation strain does not change throughout the test, which is called “geometrical similar-
ity” [59]
Figure 2.19, the indentation strain 𝑐𝑜𝑡𝛼 =
𝛿1
𝑎1
=
𝛿2
𝑎2
for a conical or pyramidal in-
denter is unchanged, which can be explained by the fact that the strain within the
sample is too small compared to the whole sample size. Therefore, an external refer-
ence is necessary to evaluate the strain in the sample. The only length scale availa-
ble for nanoindentation is the penetration depth of the indenter ℎ 𝑝, so the strain in a
nanoindentation test would be evaluated by the penetration depth ℎ 𝑝, the strain rate
𝜀̇ =
𝑑𝜀
𝑑𝑡
is then defined as 𝜀 =
𝑑ℎ 𝑝
ℎ 𝑝
1
𝑑𝑡
.This results in an exponential load function
for the constant strain rate nanoindentation. The indentation strain
𝑎
𝑅
for a spherical
indenter increases with increasing load as the contact circle radius 𝑎 increases while
indenter radius 𝑅 remains constant. Hence indentations with a spherical indenter are
not geometrically similar.

36. 30
Figure 2.19 represents the geometrical similarity for a conical or pyramidal indenter
[58].
2.7.3 Load displacement curves and pile up
Load displacement curves are very useful to calculate elastic modulus and hardness
of indented materials. Generally the specimen goes through a deformation process
elastically and then plastically. After unloading an indent impression left on the sur-
face while a release of elastic strain occur at the initial unloading segment. Through
fitting the initial unloading part the contact stiffness of material is generated as shown
in Figure 2.20:
𝑆 =
𝑑𝑃
𝑑ℎ
(42)
Figure 2.20 Typical load displacement curve of a nanoindentation experiment: 𝑃𝑡 is
the maximum applied load; ℎ𝑡 is the maximum indent displacement; ℎ 𝑟 is the depth of
residual impression; ℎ 𝑒 is released elastic displacement [58].

37. 31
The contact depth ℎ𝑐 (equal to ℎ 𝑝) is calculated with:
ℎ 𝑐 = ℎ 𝑡 − 0.72 ∗
𝑃𝑡
𝑆
(43)
ℎ𝑡: maximum indent displacement
𝑃𝑡: maximum load
0.72: geometry parameter for conical indenter
Hardness is calculated via equation:
𝐻 =
𝑃𝑡
𝐴(ℎ 𝑐)
(44)
(ℎ𝑐): contact area function
The reduced modulus is finally deduced:
𝐸𝑟 =
√ 𝜋
2√𝐴(ℎ 𝑐)
∗ 𝑆 (45)
The reduced modulus combines the Young’s modulus of the indenter and the sam-
ple, the hardness reflects the strength of the material.
When performing indentation test with elastic material, the area near the indenter on
the surface of the specimen are often pushed inwards and downwards and sinking-in
occurs. In another case, if the contact includes plastic deformation, the material can
either sink in or pile up. Research shows that in the fully plastic regime, the ratio of
E/Y and the strain hardening properties of the material decide the behavior of sinking
in or piling up, where E stands for the Young’s modulus and Y is the yield stress.
σ = Eε ε ≤ Y/E (46)
σ =Kεx
ε ≥ Y/E (47)
The degree of piling up or sinking in depends on the ration E/Y of the specimen ma-
terial and the strain hardening exponent x. Piling up or sinking in can be quantified by
a pile up parameter given by the ratio of the contact depth hc over the total depth hmax
as shown in Figure 2.21

38. 32
Figure 2.21 The pile up parameter given by hc/hmax [60].
Both piling up and sinking in have a large effect on the actual contact area of the in-
denter with the specimen. When sinking in occurs, the actual contact area in smaller
than the indenter cross-sectional area while piling up causes larger actual contact
area.

39. 33
3 Methodology
The aim of this study is to analyze the effect of the material parameters on simulation
results, to calibrate the material parameters for each single phase for DP600 steel.
The methodology of investigations is illustrated in Figure 3.1.
Figure 3.1 Methodology of Investigation.
The first step is the fundamental study of the chemical composition, microstructure
and mechanical properties of DP600 steel. Second step includes generation of an
artificial microstructure model based on the statistical parameters of grain size distri-
bution. Third step includes a series of numerical simulations performed by the com-
mercial FE program ABAQUS/Standard using the generated artificial microstructure
model. The fourth step is altering the crystal plasticity parameters to study the rela-
tionship between these parameters and nanoindentation simulation results, and cali-
bration of the parameters interactively based on the comparison of the experimental
and numerical results. The fifth step is generation of microstructure model. 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.
Characterization of chemical and mechanical properties of DP600 steel
Generation of stastically based microstructure model
Virtual laboratory on the model
Parameter study, adjustment and calibration
Generation of microstructure model
Unixial loading condition on the microstrcuture model

40. 34
4 Material
4.1 Chemical composition
The Material for our study is DP600 steel, which is categorised to low carbon content
steel for its carbon content is lower than 0.2%. Besides carbon, many other alloying
elements are remained due to form the stable martensite island and improve the duc-
tility of the material. The particular chemical composition of DP600 is shown in Table
4.1, mass content in %. N2 was calculated as 40 ppm.
Table 4.1 Chemical composition of the DP600 steel, mass contents in percentage.
C Si Mn P S Mo N
0.11 0.39 1.38 0.017 <0.001 0.05 0.004
The effect of carbon content on DP steel was discussed in details in numerous stud-
ies. Results showed that the strength of DP steel strongly depends on the martensite
content, with little influence of the carbon content in martensite or in ferrite.
Manganese as the highest alloying content (1.5-2.0 wt%) in DP steel plays an im-
portant role on improvement the tensile strength and wear resistant of the steel. The
similar functions of alloying elements to DP steel are chromium and molybdenum.
0.8-1.0 wt% chromium and molybdenum is added to DP steel to improve the me-
chanical strength properties also contribute to the chemical resistance of the steel.
Furthermore, Mo, Cr, Ni and V are applied to delay the formation of pearlite and re-
duce the martensite transformation start temperature so that the critical cooling rate
of production process is reduced.
Silicon is added to DP steel as 0.15-0.30 wt% to improve the electrical resistivity
when it is necessary and also hardens the DP steel slightly as a common alloying
element. But high amount of Si will lead a brittle behaviour.
At last 0.01 wt% sulphur and 0.05 wt% phosphorous are not expected in DP steel
because of their contribution of improvement of brittleness. But obviously completely
removal of these elements is difficult in process and not economically worthy.
4.2 Microstructure
The microstructure of our material DP600 is studied by light optical microscope (Fig-
ure 4.1) and by scanning electron microscope (Figure 4.2). In Figure 4.1, the metal-

41. 35
lographic graph of DP600 observed through light optical microscope shows the rep-
resentative phase distribution of the material. Martensite islands (dark phase), which
mainly contribute to the high strength, formed along grain boundaries of ferrite matrix
(white phase), especially in double and triple connection point. The phase volume
fraction is therefore gained, as the martensite and ferrite are respectively 13% and
87% of total volume.
Figure 4.1 Light optical microscopic graph of DP600 microstructure.
Figure 4.2 Scanning electron microscopic graph of DP600 microstructure.
4.3 Tensile properties
The tensile properties of the material are investigated by the tensile test of flat speci-
men whose geometry is mentioned in Figure 4.3 and described in Table 4.2.

42. 36
Figure 4.3 Geometric illustration of the smooth dog-bone specimen in tensile tests.
Table 4.2 Dimensions of the smooth dog-bone tensile specimen, all units in mm.
Standard Sample description a b L0 Lv Lt B R
EN 10002-1
flat specimen 20 x
80
1.5 20 80 120 250 30 20
Figure 4.4 represents the engineering stress-strain curve, using the data from tensile
test, and the characterization values selected from the curve are illustrated in Table
4.3
Figure 4.4 Engineering stress{strain curves of six samples of DP600 steel, three
loading along the rolling direction (indicated as 0°) and three loading perpendicular to
the rolling direction (indicated as 90°).

43. 37
Table 4.3 Mechanical properties of DP600 steel sheet.
Young’s
modulus
𝑬
Yield
strength
𝑹 𝐩𝟎.𝟐
Ultimate tensile
strength
𝑹 𝐦
Uniform
elongation
𝑨 𝐮
Fracture
elongation
𝑨 𝟖𝟎
214 GPa 390 MPa 704 MPa 16.5 % 25.4 %
The Young’s modulus and yield strength from tensile test shows DP600 steel per-
forms similar as normal low carbon content steel during elastic deformation. And the
UTS indicates a much better mechanical strength of DP600 than ferrite-pearlite steel
and conventional high strength steel. Moreover, compared to martensitic steel, the
uniform elongation and fracture elongation of DP600 is much larger even though its
mechanical strength is a little lower than martensitic steel.
Figure 4.5 Electron Back-Scattered Diffraction (EBSD) micrograph of DP600 steel
microstructure; the color map corresponds to the grain orientation of ferrite grains
and the martensite phase is represented by black color.

44. 38
5 Experimental and numerical investigation
5.1 Grain size characterization by EBSD
As shown in Figure 5.1, the microstructure of DP steel sample was investigated by
EBSD. In the micrograph, the color areas are ferrite phase (different colors indicate
different orientations), while the black areas are regarded as martensite grains. Ac-
cording to the EBSD image in Figure 5.1, the martensite volume fraction of DP steel
can be determined: Vmartensite=9.6%, Vferrite=90.4%.
Because generation of microstructure model based on the statistical parameters of
grain size distribution, the grain size data for both martensite and ferrite phase were
required as input parameters. Therefore the grain diameters of both ferrite phase and
martensite phase from the micrograph obtained by EBSD analysis were measured
respectively. Because of the irregularity of grain shape, the maximum and minimum
grain diameters were measured and the average was used as grain size data (Figure
5.2).
Figure 5.1 EBSD micrograph of DP600 steel microstructure.

45. 39
Figure 5.2 The grain diameter evaluation by horizontal and vertical measurement
[25].
The histograms of ferrite and martensite grain size data are illustrated in Figure 5.3
and Figure 5.4 respectively.
Figure 5.3 The histogram of ferrite grain size.

46. 40
Figure 5.4 The histogram of martensite grain size.
Log-normal distribution function was chosen to be fitted with the characterized histo-
gram of grain size of both phases. The probability density function is described as:
𝑓𝑙𝑜𝑔−𝑛𝑜𝑟𝑚𝑎𝑙(𝑥) =
1
𝑠√2𝜋
𝑒
(ln𝑥−𝑀)2
2𝑆2
, 𝑥 > 0 (48)
Where 𝑀 and 𝑠 can be calculated from 𝜇𝑙𝑜𝑔−𝑛𝑜𝑟𝑚𝑎𝑙 and 𝜎𝑙𝑜𝑔−𝑛𝑜𝑟𝑚𝑎𝑙 , which are mean
and standard deviation for the log-normal distribution function, respectively:
𝑀 = ln (𝜇𝑙𝑜𝑔−𝑛𝑜𝑟𝑚𝑎𝑙
2
/√𝜎𝑙𝑜𝑔−𝑛𝑜𝑟𝑚𝑎𝑙
2
+ 𝜇𝑙𝑜𝑔−𝑛𝑜𝑟𝑚𝑎𝑙
2
) (49)
𝑆 = √ln((𝜎𝑙𝑜𝑔−𝑛𝑜𝑟𝑚𝑎𝑙/𝜇𝑙𝑜𝑔−𝑛𝑜𝑟𝑚𝑎𝑙)2 + 1 (50)
Table 5.1 Represents the summary of the estimated statistical parameters for ferrite
and martensite.
𝑴 (µm2
) 𝑺 (-) 𝝁 (µm) 𝝈(-)
Ferrite 1.5056 0.5988 6.307 4.309
Martensite 0.095 0.69 1.586 1.214

47. 41
The statistical parameters based on the study from Vajragupta et al. [61] are shown
in Table 5.1 and used as input parameters for calculating diameters for each grain.
The fitted log – normal curves are shown in Figure 5.5 and Figure 5.6 respectively.
Figure 5.5 Comparison of grain size histogram and log-normal distribution functions
calculated by estimated statistical parameters for ferrite.
Figure 5.6 Comparison of grain size histogram and log – normal distribution functions
calculated by estimated statistical parameters for martensite.

48. 42
5.2 RVE model construction
In this study, RVE was used to describe different multiphase microstructures, their
morphologies, distribution, as well as failure behavior at the micro scale. Figure 5.7,
illustrates the steps followed for RVE generation.
Figure 5.7 Illustrates steps performed for RVE generation.
The RVE model in this study, was constructed based on the artificial microstructure
model which was generated from the statistical parameters of grain size distribution,
i.e. log – normal calculated by estimated statistical parameters in Table in 5.1.
All the seeds and their corresponding circles were placed into the defined RVE area
by applying the random sequential algorithm RSA [4, 5]. Positions and diameters cal-
culated before were used as input parameters to the MW-Voronoi algorithm [6] to
generate the artificial microstructure geometry model illustrated in Figure 5.8. The
solid black grains are martensite, and the other grains with different colors represent
the ferrite grains with different Euler angles. Besides, there are also several input pa-
rameters to decide the ellipticity of each phase to change the shape of grains.
Fitting of the experiment grain size histogram with log-normal distribution
Applying RSA into defined RVE to place the seeds
Inputting positions and diameters to MW - Voronoi
Generation of the artificial microstructure geometry model
Reading coordinates to create the microstructure model in ABAQUS

49. 43
Figure 5.8 Artificial microstructure model.
The linearized stored coordinates were then read by the Python script in order to pro-
vide the necessary data to create the microstructure model in ABAQUS. At this
stage, the random distribution of grain orientation was also assigned to the individual
grains. The colors represented different orientation of ferrite grains, while the black
grains represented the martensite phase, since no orientation was assigned for the
martensite phase. As it could be observed from the artificial microstructure models,
the generated grains of both phases were randomly dispersed throughout the micro-
structure.
The mechanical deformation of martensite phase was assumed to be homogeneous,
i.e. isotropic elasticity and J2 plasticity laws were applied to martensite. On the con-
trary, the crystal –level inhomogeneity of ferrite phase was taken into the account by
using CPFEM. As there was only minor anisotropic behavior found for the DP600
steel sheet, a random texture distribution in terms of Euler angles 𝜑1, 𝜑, 𝜑2 [34] was
assigned to the ferrite grains for the current study.

50. 44
Based on Yang’s study [62], in which the phase fraction of DP600 steel sample es-
timated by EBSD measurement was: Vmartensite = 9.6 %, Vferrite = 90.4%, 90% ferrite
and 10% martensite were used for this present study. The 3D RVE model for DP600
steel microstructure is illustrated in Figure 5.9. The dimensions of the work piece in
this model are 75 x 75 x 1 µm3
, which represents a periodic repeating cell as 1/8
symmetric dimensions of a complete model. It is noted that periodic boundary condi-
tions were applied to minimize constraint effect. The periodic boundary condition was
applied in X, Y and Z directions.
On the basis of previous theory, the crystal plasticity constitutive model is implement-
ed in a commercial finite element (FE) code (ABAQUS/Standard) using a user-
defined UMAT subroutine which is Fortran coded in this thesis.
Figure 5.9 3D FEM model generated in ABAQUS.

51. 45
5.3 Nanoindentation simulations
The crystal plasticity finite element method (CPFEM) simulations on a single grain
are conducted to investigate the constitutive behavior under the Nanoindentation.
The 3D model setup is shown in Figure 5.1. Dimensions of the workpiece in this
model are 10 × 30 × 30 μm, which represents a single grain for the material. The in-
denter is modeled as a rigid body with the sphere tip of 1 μm radius which is the
same as that in the experiment. The contact interaction between indenter and speci-
men is implemented by the standard Abaqus contact algorithms (frictionless). Due to
the highest value of the stress occurring underneath the indenter, a fine mesh (finest
as 0.056μm) is applied near the contact area of the indenter and the sample while a
coarse mesh is applied for other regions. The total number of elements is 13830 and
the element type is defined as 8-node linear brick (C3D8). Besides, the crystal plas-
ticity model served as a user-defined subroutine UMAT is combined with the previous
FEM model, in which material parameters and the crystal orientation are concerned.
Fig 5.10 Model for Nanoindentation simulation.

52. 46
5.4 Parametric study of crystal plasticity parameters
Before performing simulation for calibration, the parametric effect on the nano inden-
tation simulation is also carried out. As stated before, according to the constitutive
law of CPFEM and the subroutine of Abaqus, there are 6 parameters in all to be stud-
ied. 𝜏0 is the resolved shear stress on slip system. 𝛾̇0 and 𝑚 are material parameters
that quantify the reference shear rate and the strain rate sensitivity of slip. 𝜏 𝑐
𝑠
, ℎ0 and
𝑎 are slip hardening parameters. Based on the study of Vajragupta [61], in which the
parameters of DP600 steel were set as shown in Table 5.2.
Table 5.2 Set of parameters used by Vajragupta [61].
𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
55 2.25 2.7 0.001 180 148
And as per the study of Choi [63], in which the parameters of DP980 steel were set
as shown in Table 5.3.
Table 5.3 Set of parameters used by Choi [63].
𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
170 4 2.7 0.001 250 170
The parametric set (Table 5.4) for CPFE was determined, and is shown in Table 5.4
Table 5.4 Reference set of parameters for CPFE simulations.
𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
105 1.5 19 0.0007 850 470
Change of 𝑚 can affect other parameters’ influence on flow curve, since it has close
connection or relation to the material. 𝜏0, is also strongly depending on the material.
So an appropriate 𝜏0 was determined at first during the effect of crystal plasticity (CP)
parameters on the results of nanoindentation simulation being studied. The strategy
of parametric study is shown in the Table 5.5 to Table 5.10.

53. 47
1. Effect of 𝜏0
Table 5.5 Strategy of study for 𝜏0.
Variation 𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
1 50 1.5 19 0.0007 850 470
2 100 1.5 19 0.0007 850 470
3 150 1.5 19 0.0007 850 470
2. Effect of 𝑎
Table 5.6 Strategy of study for 𝑎.
Variation 𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
1 105 1.3 19 0.0007 850 470
2 105 3.0 19 0.0007 850 470
3 105 5.0 19 0.0007 850 470
3. Effect of 1/𝑚
Table 5.7 Strategy of study for 1/𝑚 .
Variation 𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
1 105 1.5 5 0.0007 850 470
2 105 1.5 25 0.0007 850 470
3 105 1.5 35 0.0007 850 470
4. Effect of 𝛾̇ 0
Table 5.8 Strategy of study for 𝛾̇ 0 .
Variation 𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
1 105 1.5 19 0.0001 850 470
2 105 1.5 19 0.001 850 470
3 105 1.5 19 0.01 850 470

54. 48
5. Effect of ℎ0
Table 5.9 Strategy of study for ℎ0.
Variation 𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
1 105 1.5 19 0.0007 700 470
2 105 1.5 19 0.0007 1000 470
3 105 1.5 19 0.0007 1200 470
6. Effect of 𝜏 𝑐
𝑠
Table 5.10 Strategy of study for 𝜏 𝑐
𝑠
.
Variation 𝝉 𝟎(MPa) 𝒂 (-) 𝟏/𝒎 (-) 𝜸̇ 𝟎 (s-1
) 𝒉 𝟎 (MPa) 𝝉 𝒄
𝒔
(MPa)
1 105 1.5 19 0.0007 850 320
2 105 1.5 19 0.0007 850 620
3 105 1.5 19 0.0007 850 1000
5.5 Parametric study of Swift law parameters
To model strain dependence on flow stress for plasticity, several empirical formula-
tions were suggested [64]. In cold working processes, in which strain hardening is
prevalent, the mechanical behaviour is usually described by parabolic equations and
one such equation is by Swift Law [65].
𝜎 = 𝑘(𝜀0 + 𝜀) 𝑛
(51)
Where 𝜎 and 𝜀 represent the equivalent stress and strain, respectively, and 𝑘,𝑛 and
𝜀0 are constants for a particular material, determined usually in uniaxial tension tests.
Satisfactory correlations between theory and experimental were found when carefully
determining the adjustable parameters (𝑘,𝑛 and 𝜀0). Similar to section 5.4, parametric
study of Swift law parameters namely 𝑘,𝑛 and 𝜀0 are studied in this thesis. Based on
studies of Vajragupta [61] for DP600, Sharma [66] for XA980 and Kadkhodapour et
al. [67] for DP800 steels’, the Peierl’s stress, the local chemical composition of Mar-
tensite were determined. Equation (51) was then fit to this resulting flow curve from
[66] and were used for studying parametric effect of 𝑘,𝑛 and 𝜀0 on nanoindentation
simulations. The strategies of parametric study are shown in the Table 5.12 – 5.14.
The reference set of parameters for J2-plasticity are shown in Table 5.11.

55. 49
Table 5.11 Reference set of parameters for J2-plasticity simulations.
𝒌 𝜺 𝟎 𝒏
1020 0.001 0.2
1. Effect of 𝑘
Table 5.12 Strategy of study of 𝑘.
Variation 𝒌 𝜺 𝟎 𝒏
1 800 0.001 0.2
2 1200 0.001 0.2
2. Effect of 𝜀0
Table 5.13 Strategy of study of 𝜀0.
Variation 𝒌 𝜺 𝟎 𝒏
1 1020 0.05 0.2
2 1020 0.1 0.2
3. Effect of 𝑛
Table 5.14 Strategy of study of 𝑛.
Variation 𝒌 𝜺 𝟎 𝒏
1 1020 0.001 0.4
2 1020 0.001 0.005

56. 50
6 Results and discussion
6.1 Results of CP parameters on nanoindentation
In this section, the effect of variation in each CP parameter at different strain rates
has listed and discussed in the following.
6.1.1 Effect of strain rate parameter, 𝒎
From the figures below it can be observed that at very low strain rates the higher the
value of
1
𝑚
, the more harder the material is and vice versa. It is also worth to mention
that at higher values of
1
𝑚
, the increase in hardness with increase in 1/𝑚 value is al-
most stagnant. In this case, it can be observed that there is a significant change in
hardness from the
1
𝑚
= 5 to
1
𝑚
= 25 and35. But if observed closely, the hardness for
the
1
𝑚
= 25 and 35 is more or less the same. From the slope vs displacement curve,
it can be observed that for
1
𝑚
= 5, the slope is increasing with increase in displace-
ment, whereas for
1
𝑚
= 25 𝑎𝑛𝑑 35, the slope begins initially with higher value but after
certain displacement, the change in slope is almost constant. This observation
strengthens the point that at slow strain rate, hardening becomes stagnant even with
increasing
1
𝑚
values. From the trendline of the load displacement curves, it can be
observed that the elastic-plastic loading curve for both
1
𝑚
= 5 𝑎𝑛𝑑
1
𝑚
= 25, 35 is pretty
divergent, suggesting that hardening occurs right from the initial loading. From the
load-displacement graphs it can also be observed that at lower values of
1
𝑚
, the depth
of the residual impression ℎ 𝑟is lower, indicating the effect of strain rate on elastic un-
loading of nanoindentation.

57. 51
Figure 6.1 The effect of 1/𝑚 or n on nanoindentation simulation with a strain rate
0.01 s-1
.
(a) (b) (c)
Figure 6.2 Pile up contours of nanoindentation simulation results for (a) n = 5, (b) n =
25 and (c) n = 35 respectively, for a strain rate of 0.01 s-1
.
Unlike slow strain rate, strain rate with 0.1 s-1
shows interesting results. It can be ob-
served from figures below that, with increase in
1
𝑚
values there is decrease in harden-
ing. From the trendlines of load-displacement curves, one can observe at the onset of
load, till certain displacement the elastic – plastic curve for all the
1
𝑚
values is more or
less the same. But with increase in load, lower
1
𝑚
values gets hardened at much
higher rate than that with higher
1
𝑚
values. This point can be further agreed from the

58. 52
slope - displacement graph, where it can be seen that the slope for
1
𝑚
= 5 is increas-
ing exponentially as the load increases. Whereas, for
1
𝑚
= 25, 35, the slope is initially
higher indicating that there is increase in hardness till certain load.
Figure 6.3 The effect of 1/𝑚 or n on nanoindentation simulation with a strain rate 0.1
s-1
.
(a) (b) (c)
Figure 6.4 Pile up contours the nanoindentation simulation results for (a) n = 5, (b) n
= 25 and (c) n = 35 respectively, for a strain rate of 0.1 s-1
.

59. 53
Figure 6.5 The effect of 1/𝑚 or n on nanoindentation simulation with a strain rate 1.0
s-1
.
(a) (b) (c)
Figure 6.6 Pile up contours of nanoindentation simulation results for (a) n = 5, (b) n =
25 and (c) n = 35 respectively, for a strain rate of 1.0 s-1
.
The effect of strain rate on 𝑚 can be very well observed in the load displacement
curves. It can also be inferred that the strain sensitivity is high, for higher values of 𝑚
at high strain rates. This statement can be further strengthened from the slope-
displacement graph, where the slope with higher value of 𝑚 is increasing polynomi-
ally, with increase in load. It can be inferred that at high strain rates, higher 𝑚 values,
are harder than lower 𝑚 values.

60. 54
6.1.2 Effect of shear rate parameter, 𝜸̇ 𝟎
At lower strain rates, the change in hardness is inversely proportional to the value of
𝛾̇0 . In other words, the higher the 𝛾̇0 value, the lesser the hardness. From the trend-
line of load-displacement curves, it can be observed that at the onset of loading, the
elastic-plastic curve for all different values of 𝛾̇0 remains the same until certain load. It
can be also observed from the slope displacement graph that the slope is pretty high
for lesser values of 𝛾̇0 and as the load increases, the slope gradually increases..
From load-displacement graph it can be observed that low strain rates, lower 𝛾̇0 val-
ues has lower depth of residual impression ℎ 𝑟 than the higher values of 𝛾̇0 . This indi-
cates the effect of strain rate on 𝛾̇0 .
Figure 6.7 The effect of 𝛾̇0 on nanoindentation simulation with a strain rate 0.01 s-1
.

61. 55
(a) (b) (c)
Figure 6.8 Pile up contours of nanoindentation simulation results for (a) 𝛾̇0 = 0.0001,
(b) 𝛾̇0 = 0.001 and (c) 𝛾̇0 = 0.1 respectively, for a strain rate of 0.01 s-1
.
Figure 6.9 The effect of 𝛾̇0 on nanoindentation simulation with a strain rate 0.1 s-1
.
(a) (b) (c)
Figure 6.10 Pile up contours of nanoindentation simulation results for (a) 𝛾̇0 =
0.0001, (b) 𝛾̇0 = 0.001 and (c) 𝛾̇0 = 0.1 respectively, for a strain rate of 0.1 s-1
.

62. 56
Figure 6.11 The effect of 𝛾̇0 on nanoindentation simulation with a strain rate 1.0 s-1
.
(a) (b) (c)
Figure 6.12 Pile up contours of nanoindentation simulation results for (a) 𝛾̇0 =
0.0001, (b) 𝛾̇0 = 0.001 and (c) 𝛾̇0 = 0.1 respectively, for a strain rate of 1.0 s-1
.
The effect of strain rate on the shear rate parameter 𝛾̇0 is not that high as compared
to the strain sensitivity on the strain rate parameter 𝑚. It can be seen from the above
figures that with change in strain rate, change in load displacement curves for 𝛾̇0 is
not high, except for the case with same 𝛾̇0 value, there is higher hardness at higher
strain rates, and lower hardness at lower strain rates. This inference can be visually
viewed, when the load-displacement curves from Figures 6.7, 6.9 and 6.11 are com-
pared. The inference made earlier, that the hardness increases with decreasing val-
ues of 𝛾̇0 still holds good, irrespective of the value of the strain rate.

63. 57
6.1.3 Effect of resolved shear stress on slip system, 𝝉 𝟎
Unlike shear rate parameter 𝛾̇0 , the resolved shear stress 𝜏0 has inverse effect of it.
This refers that, hardness increases as the value of 𝜏0 increases. This inference can
be further strengthened from the pile up curves and slop – displacement graphs.
However unlike 𝛾̇0 , the trendlines of load displacement from 𝜏0suggest that, right
from the onset of load, there is a distinguished separation of elastic – plastic region
for different values of 𝜏0. One can observe from the pile up curves that, there is more
pile up with higher 𝜏0 values and lesser pile up with lower 𝜏0values. From pile up
curve and load displacement curve, one can infer that at lower strain rates, as 𝜏0 in-
creases, the residual depth impression ℎ 𝑟 decreases. This indicates, at lower strain
rates it has an effect on 𝜏0.
Figure 6.13 The effect of 𝜏0 on nanoindentation simulation with a strain rate 0.01 s-1
.

64. 58
(a) (b) (c)
Figure 6.14 Pile up contours of nanoindentation simulation results for (a) 𝜏0 = 50, (b)
𝜏0 = 100 and (c) 𝜏0 = 150 respectively, for a strain rate of 0.01 s-1
.
Figure 6.15 The effect of 𝜏0 on nanoindentation simulation with a strain rate 0.1 s-1
.
(a) (b) (c)
Figure 6.16 Pile up contours of nanoindentation simulation results for (a) 𝜏0 = 50, (b)
𝜏0 = 100 and (c) 𝜏0 = 150 respectively, for a strain rate of 0.1 s-1
.

65. 59
Figure 6.17 The effect of 𝜏0 on nanoindentation simulation with a strain rate 1.0 s-1
.
(a) (b) (c)
Figure 6.18 Pile up contours of nanoindentation simulation results for (a) 𝜏0 = 50, (b)
𝜏0 = 100 and (c) 𝜏0 = 150 respectively, for a strain rate of 1.0 s-1
.
From the above figures it can be inferred that, as the strain rate increases the hard-
ness increases. As stated earlier, there is distinguished difference in the load dis-
placement curves right from the onset of the load. From the pile up curves and slope
displacement graphs, it can be inferred that the higher the value of 𝜏0, higher is the
hardness of the material. Furthermore it can be also inferred that at same value of 𝜏0,
the hardness is higher at higher strain rate and is lower at lower strain rate. It can
also be stated that at higher values of 𝜏0 results in a more concentrated pile up and
increases the maximum pile up height.

66. 60
6.1.4 Effect of slip hardening parameter, 𝝉 𝒄
𝒔
The effect of strain rate on one of the slip hardening parameter, 𝜏 𝑐
𝑠
is of an interest. It
can be observed from Figure 6.19, that as the value of 𝜏 𝑐
𝑠
increases not the only the
hardness increases, but the shape of the curve is also varied. From the trendlines of
load displacement curves, it can be observed that after the onset of load, the elastic
–plastic region for all different values of 𝜏 𝑐
𝑠
remains the same, till certain load. It can
be also observed from the pile up and load displacement curves, that as the value of
𝜏 𝑐
𝑠
increases, the value of the residual depth impression ℎ 𝑟 increases. This indicates
that at lower strain rates, it has an effect on 𝜏 𝑐
𝑠
. Also from the pile up curve at low
strain rate it can be observed that, for lower 𝜏 𝑐
𝑠
values, there is more concentrated
pile up and increases the maximum pile up height.
Figure 6.19 The effect of 𝜏 𝑐
𝑠
on nanoindentation simulation with a strain rate 0.01 s-1
.

67. 61
(a) (b) (c)
Figure 6.20 Pile up contours of nanoindentation simulation results for (a) 𝜏 𝑐
𝑠
= 320,
(b) 𝜏 𝑐
𝑠
= 620 and (c) 𝜏 𝑐
𝑠
= 1000 respectively, for a strain rate of 0.01 s-1
.
Figure 6.21 The effect of 𝜏 𝑐
𝑠
on nanoindentation simulation with a strain rate 0.1 s-1
.
(a) (b) (c)
Figure 6.22 Pile up contours of nanoindentation simulation results for (a) 𝜏 𝑐
𝑠
= 320,
(b) 𝜏 𝑐
𝑠
= 620 and (c) 𝜏 𝑐
𝑠
= 1000 respectively, for a strain rate of 0.1 s-1
.

68. 62
Figure 6.23 The effect of 𝜏 𝑐
𝑠
on nanoindentation simulation with a strain rate 1.0 s-1
.
(a) (b) (c)
Figure 6.24 Pile up contours of nanoindentation simulation results for (a) 𝜏 𝑐
𝑠
= 320,
(b) 𝜏 𝑐
𝑠
= 620 and (c) 𝜏 𝑐
𝑠
= 1000 respectively, for a strain rate of 1.0 s-1
.
The observation made on resolved shear stress 𝜏0, can also be implied here on 𝜏 𝑐
𝑠
,
i.e for the same value of 𝜏 𝑐
𝑠
, at higher strain rate has higher hardness and lower
hardness at lower strain rate. Furthermore, it is also observed that with the same val-
ue of 𝜏 𝑐
𝑠
, the shape of the curve changes increasingly (polynomial) with decrease in
strain rate. It is also observed that, irrespective of the strain rate, the initial elastic-
plastic region for all three different values of 𝜏 𝑐
𝑠
at all different strain rates, remains
the same till a particular load. From Figures 6.19, 6.21 and 6.23, it can be concluded
that all strain rates have an effect on 𝜏 𝑐
𝑠
.

69. 63
6.1.5 Effect of slip hardening parameter, 𝒉 𝟎
Like resolved critical shear stress 𝜏 𝑐
𝑠
, similar inferences can be made on one of the
slip hardening parameter ℎ0. As the value of ℎ0 increases, so does the hardness.
From the Figures’ 6.25, 6.27 and 6.29 it can be observed that at the onset of load,
the elastic-plastic curves for different ℎ0 values and at different strain rates, remain
the same until certain load. Furthermore, with increase in the strain rate, there is in-
crease in hardness suggesting that hardness is directly proportional to the strain rate
and ℎ0 values. Higher the strain rate and higher the ℎ0 value, higher is the hardness.
There is evidence of change in the shape of the curve with different ℎ0 values, but
this change is not as prominent as that could be observed in 𝜏 𝑐
𝑠
.
Figure 6.25 The effect of ℎ0 on nanoindentation simulation with a strain rate 0.01 s-1
.

70. 64
(a) (b) (c)
Figure 6.26 Pile up contours of nanoindentation simulation results for (a) ℎ0 = 700,
(b) ℎ0 = 1000 and (c) ℎ0= 1200 respectively, for a strain rate of 0.01 s-1
.
Figure 6.27 The effect of ℎ0 on nanoindentation simulation with a strain rate 0.1 s-1
.

71. 65
(a) (b) (c)
Figure 6.28 Pile up contours of nanoindentation simulation results for (a) ℎ0 = 700,
(b) ℎ0 = 1000 and (c) ℎ0= 1200 respectively, for a strain rate of 0.1 s-1
.
Figure 6.29 The effect of ℎ0 on nanoindentation simulation with a strain rate 1.0 s-1
.
(a) (b) (c)
Figure 6.30 Pile up contours of nanoindentation simulation results for (a) ℎ0 = 700,
(b) ℎ0 = 1000 and (c) ℎ0= 1200 respectively, for a strain rate of 1.0 s-1
.

72. 66
6.1.6 Effect of slip hardening parameter, 𝒂
One other parameter of interest in this thesis is one of the slip hardening parameter 𝑎.
From the Figure 6.31 it can be observed that, the hardness increases with decrease
in values of 𝑎. Also from Figure 6.31 it can be observed that there is change in the
shape of the curve for 𝑎 = 1.3 at very low strain rate, whereas the same cannot be
implied on the other values of 𝑎 at the same strain rate.
Figure 6.31 The effect of 𝑎 on nanoindentation simulation with a strain rate 0.01 s-1
.
(a) (b) (c)
Figure 6.32 Pile up contours of nanoindentation simulation results for (a) 𝑎 = 1.3, (b)
𝑎 = 3 and (c) 𝑎= 5 respectively, for a strain rate of 0.01 s-1
.

73. 67
Figure 6.33 The effect of 𝑎 on nanoindentation simulation with a strain rate 0.1 s-1
.
(a) (b) (c)
Figure 6.34 Pile up contours of nanoindentation simulation results for (a) 𝑎 = 1.3, (b)
𝑎 = 3 and (c) 𝑎= 5 respectively, for a strain rate of 0.1 s-1
.

74. 68
Figure 6.35 The effect of 𝑎 on nanoindentation simulation with a strain rate 1.0 s-1
.
(a) (b) (c)
Figure 6.36 Pile up contours of nanoindentation simulation results for (a) 𝑎 = 1.3, (b)
𝑎 = 3 and (c) 𝑎= 5 respectively, for a strain rate of 1.0 s-1
.
The results from this parametric study can be viewed in similar to the results from the
results of parametric study of 𝑎 with strain rate 0.01s-1
. The increase in strain rate,
results in higher hardness. From the Figure 6.35 for 𝑎 = 1.3, it can be inferred that
there is change in the shape of the curve.