N M P FCC

N M P FCC
NUMERICAL MODELING OF PLASTICITY IN
FCC CRYSTALLINE MATERIALS
USING
DISCRETE DISLOCATION DYNAMICS
Arash Hosseinzadeh Delandar
Stockholm, Sweden
2015
Licentiate Thesis
Division of Materials Technology
Department of Materials Science and Engineering
School of Industrial Engineering and Management
KTH Royal Institute of Technology
SE-100 44 Stockholm, Sweden
ISBN 978-91-7595-705-0
Materialvetesnkap
KTH
SE-100 Stockholm
Sweden
Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan
framlägges till offentlig granskning för avläggande av Teknologie Licentiatexamen,
i materialveteskap torsdagen den 22 oktober 2015 kl 15:30 i N111 (KUBEN),
Binellvägen 23, Materialvetesnkap, Kungliga Tekniska Högskolan.
 Arash Hosseinzadeh Delandar, 2015
Tryck: Universitetsservice US AB
Abstract
Plasticity in crystalline solids is controlled by the microscopic line
defects known as “dislocations”. Decisive role of dislocations in
crystal plasticity in addition to fundamentals of plastic deformation
are presented in the current thesis work. Moreover, major features of
numerical modeling method “Discrete Dislocation Dynamics (DDD)”
technique are described to elucidate a powerful computational method
used in simulation of crystal plasticity.
First part of the work is focused on the investigation of strain rate
effect on the dynamic deformation of crystalline solids. Single crystal
copper is chosen as a model crystal and discrete dislocation dynamics
method is used to perform numerical uniaxial tensile test on the single
crystal at various high strain rates. Twenty four straight dislocations of
mixed character are randomly distributed inside a model crystal with
an edge length of 1 µm subjected to periodic boundary conditions.
Loading of the model crystal with the considered initial dislocation
microstructure at constant strain rates ranging from 103 to 105s1 leads
to a significant strain rate sensitivity of the plastic flow. In addition to
the flow stress, microstructure evolution of the sample crystal
demonstrates a considerable strain rate dependency. Furthermore,
strain rate affects the strain induce microstructure heterogeneity such
that more heterogeneous microstructure emerges as strain rate
increases.
Anisotropic characteristic of plasticity in single crystals is investigated
in the second part of the study. Copper single crystal is selected to
perform numerical tensile tests on the model crystal along two
different loading directions of [001] and [111] at two high strain rates.
Effect of loading orientation on the macroscopic behavior along with
microstructure evolution of the model crystal is examined using DDD
method. Investigation of dynamic response of single crystal to the
mechanical loading demonstrates a substantial effect of loading
orientation on the flow stress. Furthermore, plastic anisotropy is
observed in dislocation density evolution such that more dislocations
are generated as straining direction of single crystal is changed from
[001] to [111] axis. Likewise, strain induced microstructure
heterogeneity displays the effect of loading direction such that more
i
heterogeneous microstructure evolve as single crystal is loaded along
[111] direction. Formation of slip bands and consequently localization
of plastic deformation are detected as model crystal is loaded along
both directions.
Keywords:
Dislocations, crystal plasticity, discrete dislocation dynamics, Cu
single crystal, high strain rate deformation, strain rate sensitivity,
plastic anisotropy, slip band formation.
ii
Acknowledgements
First of all, I am deeply grateful to my main supervisor Assoc. Prof.
Pavel Korzhavyi for his sincere support, valuable time and thorough
supervision during this thesis work. His priceless scientific guidelines
enabled me to finish the present work.
I would also like to thank my co-supervisor Prof. Rolf Sandström for
his guidance and help through my work.
I am grateful to Dr. Masood Hafez Haghighat at Max Planck Institute
für Eisenforschung, for his valuable assistance regarding various
features of dislocation dynamics modeling method and in particular
ParaDis code.
Svensk Kärnbränslehantering (SKB), the Swedish Nuclear Fuel and
Waste Management Company is gratefully acknowledged for
providing financial support of the present thesis work. In addition,
SKB’s expert Christina Lilja is especially thanked.
Finally, Swedish National Infrastructure for Computing (SNIC) is
acknowledged for providing computational resources for this thesis
work at PDC, HPC2N and Triolith High Performance Computing
Centers.
iii
List of Appended Papers
Paper A
“Three-dimensional dislocation dynamics simulation of plastic
deformation in copper single crystal”
A. Hosseinzadeh Delandar, S. M. Hafez Haghighat, P. A. Korzhavyi, R. Sandström
Submitted to "Technische Mechanik" for publication in the proceedings of the
4th International Conference on Material Modeling.
Paper B
“Investigation of loading orientation effect on dynamic deformation of
single crystal copper at high strain rates: Discrete dislocation
dynamics study”
A. Hosseinzadeh Delandar, P. A. Korzhavyi, R. Sandström
Submitted to the journal "Computational Materials Science".
Comment on my own contribution
All numerical calculations in the two supplements (Papers A and B) in
addition to writing of the two manuscripts were performed by the
author.
iv
List of Figures
Figure 1. Illustration of the waste package and its subsequent disposal in the final
repository.. .................................................................................................................. 1
Figure 2. (a) A perfect crystal lattice. (b) An edge dislocation created by inserting an
extra half plane of atoms. (c) A screw dislocation with a Burgers vector b parallel to
the dislocation line. (d) A mixed dislocation which is shown by a curve line inside
the lattice…. ............................................................................................................... 5
Figure 3. (a) Dislocation microstructure in pure bcc molybdenum single crystal
deformed at temperature 278 K. (b) Formation of bundles in the microstructure of
copper single crystal due to deformation at 77 K. (c) Dislocation structure formed in
single crystal bcc molybdenum deformed at temperature 500 K .. ............................. 6
Figure 4. Three Burgers circuits drawn on atomic plane perpendicular to an edge
dislocation in a crystal. The start and end points of the circuits are 𝑆𝑖 and 𝐸𝑖 ,
respectively. Circuit 1 does not enclose dislocation whereas circuits 2 and 3 do. The
sense vector 𝜉 is defined to point out of the paper so that all three circuits flow in the
counterclockwise direction ........................................................................................ 7
Figure 5. (a) Illustration of periodic energy function 𝑬𝒃 and its reduction with local
stress. (b) Mechanism of kink formation and dislocation motion in thermal
fluctuation regime. .................................................................................................... 10
Figure 6. A cylindrical single crystal subjected to the uniaxial tensile load 𝑭 and
deformed along slip direction on the slip plane ....................................................... 13
Figure 7. In the fcc crystal structure (a) motion of dislocations on the parallel planes
in the easy glide stage (b) dislocations glide on the intersecting planes resulting in
relatively strong interaction between dislocations ................................................... 15
Figure 8. Resolved shear stress as a function of shear strain in 99.98% copper single
crystal along various orientations. Note the beginning and the end of stage II marked
in each curve . ........................................................................................................... 16
Figure 9. Shear stress as a function of shear strain for 99.999% copper single crystal
with orientation near [100] at different low temperatures. Note the existence of stage
III at the larger shear strains. .................................................................................... 16
v
Figure 10. TEM micrographs illustrating microstructure development and pattern
formation in copper single crystal at flow stress (a) 28 and (b) 69MPa. At both
deformation, single crystal was oriented along [100] direction (multislip condition)
at room temperature . ................................................................................................ 17
Figure 11. TEM image and a sketch of a microstructure in a grain of 10% cold–
rolled specimen of pure aluminum (99.996%) in longitudinal plane view. One set of
extented noncrystallographic dislocation boundaries is marked A, B, C, etc., and
their misorientations are shown. .............................................................................. 19
Figure 12. Illustration of macroscopic grain subdivision and subsequently formation
of (a) deformation bands (b) shear bands in pure Aluminum during deformation .. 19
Figure 13. An arbitrary dislocation network represented by a set of nodes connected
by straight segments. 𝑏𝑖𝑗 is a Burgers vector associated with a dislocation line
connecting node 𝑖 to 𝑗. .............................................................................................. 22
Figure 14. Conservation of Burgers vector at both types of nodes, i.e., physical and
discretization nodes. ................................................................................................. 23
Figure 15. Delete and add operators, node E is added between nodes A and B and
node D located between nodes B and C is deleted ................................................... 32
Figure 16. (a) Two colliding dislocation segments (1-2, 3-4). Nodes 𝑃 and 𝑄 are
added on the segments and they are in contact distance from each other. (b) Nodes 𝑃
and 𝑄 are merged into new single node, 𝑃’. (c) Node 𝑃’ is splited into two nodes, 𝑃’
and 𝑄’ ........................................................................................................................ 33
Figure 17. Time distribution in parallel computing. Three distinct regions shown by
the dark blue bars, the white area and the light blue bar correspond to the time spent
for computing 𝑡𝑐 , the time spent by each processor in waiting 𝑡𝑤 for last calculation
to be finished and the time spent for inter-processors communication 𝑡𝑚 ,
respectively. .............................................................................................................. 35
Figure 18. Simulation volume is divided into 3 ⨉ 3 ⨉ 2 domains along three axes
and each domain is assigned to its own processor.. .................................................. 36
Figure 19. Stress as a function of plastic strain for tensile deformation of copper
single crystal along [001] orientation at three sets of strain rates. ............................ 38
Figure 20. Total dislocation density evolution as a function of plastic strain at
various strain rates. ................................................................................................... 38
vi
Figure 21. Dislocation density distribution of individual slip systems for different
strain rates (a) 103 (b) 104 and (c) 105 s-1. ................................................................. 40
Figure 22. Distribution of the dislocation density on x-y plane at 1.4 percent plastic
deformation at strain rates (a) 104 s-1 and (b) 105 s-1. Lengths on x-y plane are in
Burgers vector unit. .................................................................................................. 41
Figure 23. Mechanical response of the copper single crystal to the uniaxial tensile
loading along [001] and [111] directions at two imposed strain rates. ..................... 43
Figure 24. Illustration of dislocation density evolution as a function of plastic strain
at two imposed strain rates of 105 and 106 s-1 for loading of single crystal along [001]
and [111] orientations. .............................................................................................. 44
Figure 25. Dislocation density distribution of slip systems for loading of model
crystal along [001] and [111] orientations at (a) 10 5 s-1 and (b) 106 s-1 strain rates. .. 45
Figure 26. Microstructure development resulting from straining of the model crystal
along [001] and [111] directions. (a) and (b) at strain rate of 10 5 s-1, (c) and (d) at
strain rate of 106s-1. ................................................................................................... 46
vii
List of Tables
Table 1. Calculation of the parameter Ω at three imposed strain rates. .................... 42
Table 2. Initial dislocation density and input parameters for current DD simulations
.................................................................................................................................. 42
viii
TABLE OF CONTENTS
1
2
3
INTRODUCTION ................................................................................................. 1
1.1
MANAGEMENT OF SPENT NUCLEAR FUEL IN SWEDEN ................................ 1
1.2
RESEARCH OBJECTIVES ............................................................................. 2
1.3
STRUCTURE OF THE THESIS ........................................................................ 3
FUNDAMENTALS OF CRYSTAL DISLOCATIONS ................................................ 4
2.1
TYPES OF CRYSTAL DISLOCATIONS ............................................................ 4
2.2
OBSERVATION OF DISLOCATIONS ............................................................... 6
2.3
THE CONCEPT OF BURGERS VECTOR ........................................................... 6
2.4
DISLOCATIONS MOTION ............................................................................. 8
2.5
LATTICE RESISTANCE TO DISLOCATION MOTION ........................................ 9
CRYSTAL PLASTICITY ..................................................................................... 11
3.1
3.1.1
RESOLVED SHEAR STRESS ON THE SLIP PLANE ................................. 13
3.1.2
STAGES OF STRAIN HARDENING....................................................... 14
3.1.3
MICROSTRUCTURE EVOLUTION ....................................................... 17
3.2
4
5
MECHANISM OF PLASTIC DEFORMATION IN SINGLE CRYSTALS ................. 11
MECHANISM OF PLASTIC DEFORMATION IN POLYCRYSTALS .................... 18
METHODOLOGY .............................................................................................. 20
4.1
INTRODUCTION ........................................................................................ 20
4.2
DISCRETE DISLOCATION DYNAMICS ......................................................... 22
4.2.1
CALCULATION OF NODAL FORCES ................................................... 23
4.2.2
EQUATION OF MOTION .................................................................... 28
4.2.3
TIME INTEGRATORS ......................................................................... 30
4.2.4
TOPOLOGICAL CHANGES ................................................................. 31
4.2.5
PARALLEL COMPUTATION ............................................................... 34
RESULTS AND DISCUSSION .............................................................................. 37
5.1
SUMMARY OF APPENDED PAPER A............................................................ 37
5.1.1
DETAILS OF DD MODELING ............................................................. 37
5.1.2
MACROSCOPIC BEHAVIOR............................................................... 37
ix
5.1.3
EFFECT OF STRAIN RATE ON SLIP ACTIVITY ..................................... 39
5.1.4
HETEROGENEOUS MICROSTRUCTURE EVOLUTION ........................... 40
5.2
6
SUMMARY OF APPENDED PAPER B ............................................................ 42
5.2.1
SIMULATION DETAILS ..................................................................... 42
5.2.2
MECHANICAL PROPERTIES AND DISLOCATION DENSITY EVOLUTION 43
5.2.3
SLIP ACTIVITY ................................................................................. 44
5.2.4
LOCALIZATION OF PLASTIC DEFORMATION ..................................... 45
CONCLUDING REMARKS AND FUTURE WORK ............................................... 47
6.1
CONCLUSION ........................................................................................... 47
6.2
FUTURE WORK ........................................................................................ 48
References ............................................................................................................... 49
x
1
1.1
INTRODUCTION
MANAGEMENT OF SPENT NUCLEAR FUEL IN SWEDEN
In Sweden radioactive waste produced by operating power plants is
managed by Swedish Nuclear Fuel and Waste Management
Company (SKB). This company is the main responsible for
environmentally safe treatment and disposal of highly radioactive
spent nuclear fuel.
In order to dispose spent nuclear fuel in a highly efficient and safe
manner, a proposed plan based on KBS-3 concept is followed.
According to KBS-3 concept, initial short-term storage of the nuclear
waste in the Central Interim Storage Facility or CLAB will be
proceeded by the final placement of the spent nuclear fuel inside the
designated waste package. The waste package which will be located at
500 m down in the bedrock consists of two parts. The inner part is
nodular cast iron insert which acts as load bearing part and is
composed of quadratic channels where the radioactive nuclear waste
are positioned. The outer part is known as copper canister which is
corrosion resistance part of the waste package and cast iron insert is
placed inside it. Figure 1 demonstrates the waste package and its
subsequent disposal in the final repository.
Figure 1. Illustration of the waste package and its subsequent disposal in the final
repository.
1
The cylindrical copper canister with an approximately 1 m diameter,
5 m length and a wall thickness of 50 mm will be initially exposed to
a temperature close to 100°C and an external pressure of around
15 MPa from the surrounding bentonite clay and the groundwater
(R. Sandström and J. Hallgren, 2011) [1].
Because of the applied external pressure on the copper canister in
addition to the relatively high temperature, the canister will deform
plastically. While the rate of the deformation is considerably low,
however, it will lead to the gradual closure of the existing gap
(1-2 mm) between the copper canister and cast iron insert.
Due to the fact that integrity of the copper canister should be
maintained over 100,000 years, a thorough investigation of
mechanism of plastic deformation and microstructure evolution in
copper is extremely important to describe the deformation behavior of
the canister. The relevant scientific data can be obtained by means of
computer simulation of plasticity in crystalline solids using state of the
art numerical modeling techniques such as discrete dislocation
dynamics method.
1.2
RESEARCH OBJECTIVES
The main objectives of this thesis work are as follows:
1. To study the main concepts of plastic deformation in face
centered cubic (fcc) crystals and to develop an understanding
of fundamentals of crystal plasticity.
2. To implement numerical modeling “Discrete Dislocation
Dynamics (DDD)” method for simulation of plastic
deformation in copper single crystal as a material of interest.
3. To investigate effect of various factors such as strain rate and
loading orientation on the deformation behavior of copper
single crystal using DDD method.
4. To examine formation of dislocation microstructures and to
establish relations between microstructure evolution and
macroscopic behavior of the copper single crystal.
2
1.3
STRUCTURE OF THE THESIS
The present thesis is organized into 6 chapters. Chapter 1 is the
introduction chapter where an overall view of the thesis is presented.
A brief description of fundamentals of dislocations is provided in
Chapter 2. In Chapter 3 mechanism of plastic deformation in single
crystals and polycrystals is discussed. The main focus of this chapter
is to present a theoretical background for single crystal plasticity.
Chapter 4 is devoted to the discrete dislocation dynamics method and
the main ingredients of this numerical modeling technique are
explained. Description of discrete dislocation dynamics method is
followed by a summary of the obtained modeling results along with a
brief discussion regarding these results in Chapter 5. Finally, the
conclusions of the current thesis work and a brief outline of the
intended future work are provided in Chapter 6.
3
2
FUNDAMENTALS OF CRYSTAL DISLOCATIONS
The concept of dislocations initially appeared as an abstract
mathematical notion. Italian mathematician Vito Volterra was a
pioneer mathematician who created mathematical foundation for the
dislocations in the late 19th century. For a few decades, the concept of
dislocations was purely mathematical; however, in 1926 Frenkel’s
first attempt to calculate the strength of perfect crystal led to a major
scientific breakthrough in the field of dislocations. The existing
discrepancy by many orders of magnitude between the theoretical
calculations and experimental observations resulted in the conclusion
that crystal line defects, i.e., dislocations should be responsible for
observed deformation behavior in metals. After Frenkel, fundamental
role of dislocations in crystal plasticity was introduced by three
scientists, Taylor, Polany and Orowan independently in 1934. These
scientists proposed that crystals should deform plastically by means of
dislocations. In spite of all efforts in understanding the crystal
plasticity and introducing the decisive role of dislocations in plastic
deformation of metals, the existence of dislocations as line defects in
crystals was entirely theoretical until the late 1950s. However,
observation of dislocations by transmission electron microscopy
(TEM), in the late 1950s, ascertained the actual existence of
dislocations in crystalline materials. Since then, the ubiquity and
importance of dislocations for crystal plasticity and numerous other
aspects of material behavior have been regarded as firmly established
as, say, the role of DNA in promulgating life (Bulatov and Cai, 2006)
[2].
In this chapter a brief introduction to the fundamentals of dislocations
is presented. For further information about the crystal dislocations the
reader is referred to the first chapter of Bulatov and Cai, 2006 [2].
2.1
TYPES OF CRYSTAL DISLOCATIONS
Dislocations are line defects inside an otherwise perfect crystal.
Formation of these linear defects in the crystals leads to the distortion
of the crystal lattice. In order to grasp the concept of dislocations an
imaginary course of action is followed.
4
Figure 2(a) shows a perfect crystal which does not contain any lattice
defect. By inserting an extra half plane of atoms inside this perfect
crystal from above, a linear defect, i.e., dislocation is introduced
inside the host lattice, see Figure 2(b). Apparently, removing a half
plane of atoms from below, will also lead to the same situation. Due
to the location of the dislocation on the edge of an extra half plane of
atoms, the created dislocation in Figure 2(b) is called an edge
dislocation. Lattice distortion generated by this type of dislocation is
perpendicular to the dislocation line and is proportional to the
magnitude of dislocation’s Burgers vector, b. Figure 2(c) illustrates a
screw dislocation embedded in the host lattice. In a screw dislocation,
atoms around the dislocation line are located in such a way that
resembles a spiral as it can be seen by the white arrows in Figure 2(c).
Lattice distortion generated by a screw dislocation is parallel to the
dislocation line. Figure 2(d) represents a mixed or general dislocation
with a curve line inside the lattice. General dislocation has a mixed
character (edge and screw) with a Burgers vector which has
components both perpendicular and parallel to the dislocation line
(Jonsson, 2010) [3].
(a)
(b)
(d)
(c)
Figure 2. (a) A perfect crystal lattice. (b) An edge dislocation created by inserting an
extra half plane of atoms. (c) A screw dislocation with a Burgers vector b parallel to
the dislocation line. (d) A mixed dislocation which is shown by a curve line inside
the lattice (Bulatov and Cai, 2006) [2].
5
2.2
OBSERVATION OF DISLOCATIONS
In order to examine the properties and microstructure of dislocations
in crystalline materials, a number of experimental techniques have
been used during the last few decades. These techniques such as
surface methods, decoration methods, X-ray diffraction, transmission
electron microscopy, field ion microscopy and atom probe
tomography have shown a remarkable potential to provide useful
information about dislocations. However, Transmission Electron
Microscopy (TEM) has been the dominant method among the other
experimental techniques. In this method, either an individual
dislocation or a large number of dislocations can be examined at
relatively high magnification. Figure 3 shows TEM pictures of
dislocation microstructures in single crystals of bcc molybdenum and
fcc copper.
(a)
(b)
(c)
Figure 3. (a) Dislocation microstructure in pure bcc molybdenum single crystal
deformed at temperature 278 K. (b) Formation of bundles in the microstructure of
copper single crystal due to deformation at 77 K. (c) Dislocation structure formed in
single crystal bcc molybdenum deformed at temperature 500 K . The dark regions
contain a high density of entangled dislocation lines that can no longer be
distinguished individually (Bulatov and Cai, 2006) [2] .
2.3
THE CONCEPT OF BURGERS VECTOR
In addition to TEM technique to examine the existence of dislocations
in crystalline materials, an efficient theoretical approach exists to
elaborate the concept of dislocations. In order to demonstrate the
presence of a dislocation inside the host lattice, a well-known Burgers
circuit test is used. Figure 4 shows a plane of atoms along with an
edge dislocation which is perpendicular to this two-dimensional plane.
6
In order to begin the Burgers circuit test, first, a line sense, 𝝃, should
be defined for the dislocation line. In our present test, the line sense is
selected to point out of the paper. Based on the direction of the line
sense, the Burgers circuit can be constructed following the right-hand
rule. Since the line sense of the edge dislocation has been selected to
point out of the plane, therefore, the Burgers circuit will flow
counterclockwise. The present Burger circuit around an edge
dislocation consists of a sequence of jumps starting from point 𝑆𝑖 and
ending to the point 𝐸𝑖 . Construction of Burgers circuit in the perfect
lattice creates a complete loop where both starting point, 𝑆𝑖 , and
ending point, 𝐸𝑖 , coincide, see Figure 4. However, when the same
circuit is built inside the defective lattice which incorporates an edge
dislocation the starting point 𝑆𝑖 and ending point 𝐸𝑖 will not coincide
and a gap will be present between the two points. As a result, a vector
can be drawn to connect the starting point to the ending point which is
regarded as the Burgers vector of the edge dislocation. This
constructed vector, i.e., Burgers vector is associated with the
distortion of the crystal lattice which encloses the line defect, i.e.,
edge dislocation.
Figure 4. Three Burgers circuits drawn on atomic plane perpendicular to an edge
dislocation in a crystal. The start and end points of the circuits are 𝑆𝑖 and 𝐸𝑖 ,
respectively. Circuit 1 does not enclose dislocation whereas circuits 2 and 3 do. The
sense vector 𝜉 is defined to point out of the paper so that all three circuits flow in the
counterclockwise direction (Bulatov and Cai, 2006) [2].
7
2.4
DISLOCATIONS MOTION
Dislocations respond to the applied stress through glide on the slip
planes. Therefore, dislocation motion takes place when a sufficient
stress acts on a dislocation. It is assumed that dislocations are
embedded in the linear elastic continuum and driving force on the
dislocation lines can be calculated using linear elasticity theory.
However, the way that dislocations respond to the applied driving
force is governed by the atomistic mechanisms and therefore is
beyond the scope of continuum elasticity theory. Hence, only
calculation of applied force on the dislocation line is presented in this
section.
Suppose that a force per unit length of a dislocation 𝒇 is applied at the
arbitrary point (𝑃) on the dislocation line. The applied force is
calculated using local stress at point 𝑃 such that:
𝒇 = (𝝈 · 𝒃) × 𝝃 ,
(2.1)
where 𝝈 is the local stress, 𝒃 is the Burgers vector and 𝝃 is the local
line tangent at point 𝑃. Equation (2.1) is known as Peach-Koehler
formula and relates the force applied on the dislocation line to the
local stress acting on it regardless of the origin of this stress.
Dislocation motion is categorized into the two main mechanisms so
called conservative and non-conrservative motions. An edge
dislocation moves on its slip plane by conservative motion or glide
and out of its slip plane by non-conservative motion or climb.
However, unlike an edge dislocation, motion of a screw dislocation is
not confined to the uniquely defined glide plane and it can move in
other available planes by only glide mechanism. In a real crystalline
solids dislocations maily have mixed character and they move by
both glide and climb mechanisms. Temperature in addition to
mechanical stress acting on dislocations are the main factors
determining the extent of each mechanism. For example, at high
temperatures due to the higher possibility for atomic diffusion climb
mechanism dominates whereas at low temperature glide is the
dominant mechanism.
8
Calculation of generated plastic strain by motion of a dislocation
When a dislocation glides on the slip plane, plastic strain is produced
in the crystal which is proportional to the magnitude of dislocations
Burgers vector along with the swept area by gliding dislocation.
Suppose dislocation with the Burgers vector 𝒃=[𝑏𝑖 𝑏𝑗 𝑏𝑘 ] glides on the
slip plane with normal vector 𝒏 = [𝑛𝑖 𝑛𝑗 𝑛𝑘 ] and sweeps out an area
of ∆𝐴 on the respective slip plane. As a result of dislocation glide,
𝑝
plastic strain 𝜀𝑖𝑗
will be introduced into the host lattice according to
𝑝
𝜀𝑖𝑗
=
(𝑏𝑖 𝑛𝑗 + 𝑏𝑗 𝑛𝑖 )∆A
,
2Ω
(2.2)
where Ω is the volume of the crystal.
In the case of collective motion of a number of dislocations, if 𝐿
denotes the total length of all dislocations, the total area swept out by
movement of all dislocations during a period ∆𝑡 will be ∆𝐴 = 𝑣𝐿∆𝑡,
where 𝑣 represents the average velocity of the moving dislocations.
Calculation of plastic strain rate resulting from collective motion of
dislocations leads to the well known Orowan’s equation which
demonstrates the existing relation between plastic strain rate 𝜀̇ 𝑝 ,
dislocation density 𝜌, and the average velocity of dislocations 𝑣 such
that
𝜀̇𝑝 = 𝜌𝑏𝑣
(2.3)
where 𝑏 represents the magnitude of Burgers vector.
2.5
LATTICE RESISTANCE TO DISLOCATION MOTION
When a dislocation moves inside a crystal it experiences resistance
against its motion resulting from the crystal lattice. This intrinsic
lattice resistance can be defined by two parameters: the Peierls barrier
and the Peierls stress.
Imagine a straight dislocation moves in its glide plane, a periodic
energy function of the dislocation position can describe the effect of
crystal lattice on dislocation motion, see Figure 5a. When the local
stress acting on the dislocation line is negligible, a dislocation inside
the crsytal lattice must overcome an energy barrier 𝐸𝑝 to move from
9
one Peierls valley to the adjacent one. This energy barrier is referred
to as Peierls barrier. However, as local stress increases actual energy
barrier 𝐸𝑏 experienced by a dislocation decreases. Finally, when the
applied stress on the dislocation line reaches the critical value, i.e.,
Peierls stress ( 𝜏𝑝 ), the energy barrier 𝐸𝑏 vanishes entirely as it is
illustrated in Figure 5a.
(a)
(b)
Figure 5. (a) Illustration of periodic energy function 𝑬𝒃 and its reduction with local
stress. (b) Mechanism of kink formation and dislocation motion in thermal
fluctuation regime (Bulatov and Cai, 2006) [2].
The local stress acting on the dislocation line can be compared to the
magnitude of the Peierls stress to determine the respective regime for
dislocation motion. When the local stress is less than Peierls stress,
dislocation motion takes palce by means of thermal fluctations. Figure
5b illustrates the mechanism of dislocation motion in the thermal
fluctuation regime. Creating a pair of kinks enables dislocation to
move from one Peierls valley to the next one without moving the
entire disloction at once. When dislocations move in thermal
fluctuation regime, dislocation mobility increases with temperature
due to the higher possibility for kink pair formation at high
temperatures.
On the other hand, when the local stress on the dislocation is higher
than the Peierls stress, a dislocation can move without the help of
thermal fluctuations. In this so-called “viscous drag” regime,
dislocation velocity becomes a linear function of stress and is usually
limited by the viscosity due to dislocation interaction with lattice
vibrations, i.e. sound waves (Bulatov and Cai, 2006) [2]. When a
dislocation moves in the viscouse drag regime, increase in temperature
leads to decrease in mobility due to the higher interaction between the
dislocation and phonons.
10
3
CRYSTAL PLASTICITY
When a crystalline solid is subjected to a mechanical loading
geometrical shape of the sample may change. However, if the initial
shape of the material can be retrieved when applied external stress is
relieved; the material has been deformed elastically. Elastic
deformation of a crystalline material usually manifests itself by a
linear relation between applied external stress 𝝈 and resultant elastic
strain 𝜀𝑒 such that
𝝈 = 𝐸𝜀𝑒
(3.1)
In equation (3.1) which is referred to as Hooke’s law, 𝐸 represents the
material specific elastic constant known as Young’s modulus.
As external applied stress exceeds a critical level of 𝜎𝑦 (yield stress), a
permanent change in the geometrical shape of the sample takes place.
This permanent shape change which mainly results from linear defects
i.e., dislocations in materials is known as plastic deformation. Hence,
dislocations are considered as main carriers of plasticity in crystalline
solids and collective motion of a large number of these line defects
leads to the plastic deformation of the crystals.
Furthermore, numerous dislocation-based mechanisms involved on
mesoscale govern the macroscopic behavior of crystalline materials.
Therefore, an essential relation exists between microstructure
evolution and macroscopic mechanical properties.
In the current chapter, fundamental aspects of plastic deformation of
crystalline solids are discussed with a particular emphasis on the
plasticity of single crystals.
3.1
MECHANISM OF PLASTIC DEFORMATION IN SINGLE CRYSTALS
Motion of dislocations in close-packed directions on the close-packed
crystallographic planes leads to the slip of these planes over each
other and consequently generation of plastic strain. The required stress
for plastic deformation is reduced by several orders of magnitude
when the simultaneous motion of the entire lattice plane is replaced by
successive motions of embedded dislocations in the plane. Each single
11
crystal has a limited number of planes with highest density of atoms,
these crystallographic planes are known as slip planes. Additionally,
restoration of the crystal structure after slip indicates that slip should
be limited to the particular crystallographic directions known as slip
directions. Combination of slip planes and slip directions form the slip
systems for a crystal structure. Each crystal structure, i.e., fcc, bcc,
hcp has a number of slip systems which indicates the possible planes
and directions of slip.
For face centered cubic (fcc) structure slip occurs on the closedpacked {111} crystalographic planes. Therefore, fcc crystal structure
has four unique slip planes with three possible slip direction of
<110> type on each plane which results in 4 × 3 = 12 slip systems
for this structure. This large number of fully closed-packed slip
systems allows fcc materials to exibit high ductility at all temperatures
and under all loading conditions (Hansen and Barlow, 2014) [4]. In
the case of body centered cubic (bcc) structure, slip can occur on two
slip planes of {110} and {112} types. There are six unique {110} slip
planes and each slip plane contains two <111> slip directions. Hence,
12 slip systems of type {11̅0}<111> enable bcc crystal structure to
deform plastically. In addition, there are 12 {112} slip planes in bcc
structure and on each plane there is only one <111> slip direction.
Thus, 12 slip systems of type {112̅ }<111> exist for body centered
cubic strucure. Hence, bcc structure has 24 distinguishable slip
systems to contribute to the total plastic strain.
The hexagonal close packed (hcp) structure has a relatively complex
deformation mechanism in comparison with bcc and fcc structures. In
the hcp crystal structure a number of slip systems exist which are
rather difficult to activate. Therefore, in some loading conditions,
plastic deformation by dislocations slip is relatively restricted and as a
result the imposed deformation is accomodated by an alternative
mechanism so called twinning. However, there are two relatively easy
to activate slip systems in the hcp structure. These two slip systems
are known as basal-<a> slip and prismatic-<a> slip. One basal plane
of (0001) type with three Burger vectors (slip directions) of type
<112̅0> leads to the three (0001)<112̅0> slip systems in hcp crystal
structure. Furthermore, three slip planes of {11̅00} type, where each
of them contains one Burgers vector of type <1120>, result in three
{11̅00}<1120> slip systems in hcp structure.
12
3.1.1
RESOLVED SHEAR STRESS ON THE SLIP PLANE
When a mechanical load is applied on a single crystal, deformation
occurs by activation of possible slip system/systems in the crystal.
Which slip systems are activated is determined by the orientation of
the applied stress and resultant resolved shear stress acting on the slip
planes along the slip directions. Figure 6 illustrates a uniaxial tensile
loading of a single crystal along the cylindrical axis.
Figure 6. A cylindrical single crystal subjected to the uniaxial tensile load 𝑭 and
deformed along slip direction on the slip plane (Hull and Bacon, 2011) [5].
Straining of sample crystal by the force 𝑭 leads to generation of
tensile stress 𝝈 along this load such that
𝝈=
𝑭
,
𝐴
(3.2)
where 𝐴 is the cross sectional area of the cylinder. The respective
component of load 𝑭 along slip direction shown in the Figure 6 is
𝑭𝑐𝑜𝑠𝜆, where λ is the angle between load 𝑭 and slip direction.
Similarly, the area of the slip plane where load 𝑭𝑐𝑜𝑠𝜆 is applied can
be obtained by dividing 𝐴 with cos 𝜑, such that 𝐴/ cos 𝜑, where 𝜑
denotes the angle between load 𝑭 and normal vector of the slip plane.
Hence, the resoved shear stress acting on the slip plane along the slip
direction is calculated as following
𝝉=
𝑭𝑐𝑜𝑠𝜆
= 𝝈𝑐𝑜𝑠𝜆 cos 𝜑
𝐴/ cos 𝜑
(3.3)
13
The equation (3.3) is referred to as Schmid’s formula and the quantity
𝑐𝑜𝑠𝜆 cos 𝜑 is known as Schmid factor. Plastic deformation of a single
crystal takes place as resolved shear stress on the slip plane exceeds a
threshold value known as Critically Resolved Shear Stress (CRSS)
such that
𝝈𝑐𝑜𝑠𝜆 cos 𝜑 ≥ 𝝉𝑐 ,
(3.4)
where 𝝉𝑐 corresponds to the critical shear stress required for the onset
of plastic deformation on a specific slip system.
Although equation (3.4) is used to predict the necessary shear stress
to generate shear strain, however, this expression can be simply
rearranged to define the required normal stress to introduce shear
strain on a particular slip system such that
𝝈(𝜆, 𝜑) =
1
𝝉
cos 𝜆 cos 𝜑 𝑐
(3.5)
1
In the equation (3.5) the ratio cos 𝜆 cos 𝜑 is the inverted Schmid factor
and is referred to as the Taylor factor represented by 𝑚. This equation
plays a significant role in determining the activation of slip systems.
When yielding of a single crystal takes place the stress 𝝈(𝜆, 𝜑) can be
calculated and compared for each slip system to detect the minimum
value of 𝝈(𝜆, 𝜑) . The slip system which becomes active with the
lowest level of stress 𝝈(𝜆, 𝜑) is called primary slip system and plastic
strain is accommodated in this system as deformation starts.
3.1.2
STAGES OF STRAIN HARDENING
Onset of plastic deformation on the primary slip system in a single
crystal leads to the motion of dislocations on the parallel slip planes
due to the single slip condition. In this situation, a considerably weak
interaction exists between dislocations, thus, they can move freely
through the material and contribute to the plastic deformation
significantly. This easy glide stage is referred to as the first stage of
the strain hardening of a single crystal and manifests itself by a very
low deformation hardening rate. Stage I has a strong dependency on
the orientation of the crystal and if yielding of the crystal starts with
the multiple slip this stage will be suppressed.
14
Figure 7(a) demonstrates a single slip situation as dislocations move
on the parallel planes without creating noticeable obstacles against
each other’s motion. However, as straining of the single crystal
advances, activation of successive slip systems leads to the stage II of
work hardening. At this stage, due to the multiple slip, dislocations
start to move on the various non-parallel slip planes and they may
intersect each other as it is shown in Figure 7(b). In the multislip
condition, substantial increase in interaction between dislocations
results in the dislocation storage (forest dislocations) and formation of
subgrains or dislocation cells within the crystal. These dislocation
cells consist of cell walls, i.e., regions with relatively high dislocation
density where dislocations are immobile along with cell interiors, i.e.,
dislocation-poor regions with mobile dislocations.
(a)
(b)
Figure 7. In the fcc crystal structure (a) motion of dislocations on the parallel planes
in the easy glide stage. (b) dislocations glide on the intersecting planes resulting in
relatively strong interaction between dislocations (Jonsson, 2010) [3].
When dislocation cells form inside the crystal the immobile stored
dislocations on the cell walls known as Statistically Stored
Dislocations (SSDs) block the further movement of dislocations.
SSDs do not contribute to the plastic deformation of single crystal by
slip, however, they act as a source of dislocation generation.
Obstruction of the dislocations movement by SSDs leads to reduction
of dislocations glide distance and subsequently increase in dislocation
generation and work hardening rate. The steepest rate of work
hardening observed in the deformation of single crystal corresponds to
the stage II of strain hardening. Figure 8 illustrates the shear stress
versus shear strain curves for copper single crystal with different
15
orientations. Both stage I and stage II deformation behaviors are
displayed in the figure.
Figure 8. Resolved shear stress as a function of shear strain in 99.98% copper
single crystal along various orientations. Note the beginning and the end of stage II
marked in each curve (Kocks and Mecking, 2002) [6].
.
Further increase in plastic strain leads to the stage III of deformation
hardening where a considerable decrease in work hardening rate is
observed. At stage III, due to the sufficiently high stress level
immobile dislocations locked in the crystal become mobile again.
Therefore, dislocation generation is reduced and subsequently work
hardening rate decreases, see Figure 9.
Figure 9. Shear stress as a function of shear strain for 99.999% copper single
crystal with orientation near [100] at different low temperatures. Note the existence
of stage III at the larger shear strains (Kocks and Mecking, 2002) [6].
16
3.1.3
MICROSTRUCTURE EVOLUTION
When a single crystal deforms plastically dislocations tend to organize
themselves into the heterogeneous microstructures in the form of
patterns where a rather regular structure composed of alternating
dislocation-rich and dislocation-poor regions emerges spontaneously.
In the wavy glide crystalline materials such as Copper (Cu), Nickel
(Ni) and Aluminum (Al) with relatively high stacking fault energy,
formation of three-dimensional cell structures is the commonly
observed dislocation pattern. Multiple slip condition where slip occurs
on several active slip systems and dislocations storage takes place as a
result of strong interaction between gliding dislocations with forest of
immobile dislocations is crucial for dislocation pattern formation.
Therefore, observation of dislocation patterns in stage I of work
hardening of single crystal is unfeasible. In addition to multislip
condition, cross slip mechanism which enables screw dislocations to
change slip plane plays a major role in dislocation pattern formation.
Figure 10 displays dislocation patterns as a copper single crystal is
strained along [100] orientation (multislip condition). The regions
with dense dislocation entanglements, i.e., cell walls along with less
dense regions, i.e., cell interiors can be clearly identified.
(a)
(b)
Figure 10. TEM micrographs illustrating microstructure development and pattern
formation in copper single crystal at flow stress (a) 28 and (b) 69MPa. At both
deformation, single crystal was oriented along [100] direction (multislip condition)
at room temperature (Kocks and Mecking, 2002) [6].
The underlying reason for formation of heterogeneous structures by
dislocations is attributed to the internal stress field which is positive
(tensile) at cell walls and negative (compressive) in the cell interiors.
During plastic deformation, dislocations tend to reduce the internal
stress and consequently reach to the the minimum energy level of the
17
system. Formation of a heterogenous microstructure where
dislocation-rich regions are seperatad by dislocation-poor regions
leads to a reduction in internal energy of the dislocation network.
3.2
MECHANISM OF PLASTIC DEFORMATION IN POLYCRYSTALS
Due to the existance of a large number of grains with different
orienations in the texture of polycrystalline materials, mechanism of
plastic deformation in polycrystals is relatively different from that in
the single crystals. Grain boundaries embedded in a polycrystalline
solid act as impenetrable obstacles for dislocations motion. Therefore,
unlike the single crystal, plastic deformation of polycrystals require
activation of various slip systems from the beginning of deformation.
Accommodation of deformation on the various slip systems enables
grains to deform individually while maintaining continuous grain
boundaries.
Subdivision of grains
Plastic deformation of a polycrystal takes place by the subdivision of
grains on different length scales. Grain subdivision leads to the
formation of different but compatible deformation regions. On the
grain scale, grain subdivision can be categorized into the macroscopic
and microscopic scales.
On the microscopic scale, the grain is subdivided into the cells where
storage of dense immobile dislocations form the cell walls with
misorientation of approximately 1-2°. In contrast to the cell walls,
within the cells, i.e, cell interiors dislocation density is relatively
lower and dislocations are mobile.
Dislocation cells tend to organize themselves into the bands, which in
turn results in formation of dislocation boundaries with thinkness of a
few cell-blocks. These dislocation boundaries which are known as
Dense Dislocation Walls (DDWs) coexist with the cell boundaries,
however, they have a completely different morphology from cell
boundaries. Unlike the random orienation of cells, dislocation
boundaries form extended and rather planar boundaries. In addition,
DDWs show higher misorientation than individual cell blocks inside
the bands. Figure 11 illustrates a grain subdivision in the microscale
where cell blocks and dislocation boundaries can be clearly identified.
18
Figure 11. TEM image and a sketch of a microstructure in a grain of 10% cold–
rolled specimen of pure aluminum (99.996%) in longitudinal plane view. One set of
extented noncrystallographic dislocation boundaries is marked A, B, C, etc., and
their misorientations are shown (Hansen and Barlow, 2014) [4].
Combination of the two structures on the microscale leads to the
formation of deformation bands on the macroscopic scale. Inside each
deformation band the same principle orientation exists and each band
is separated from neighboring bands by the borders referred to as
transition bands.
Finally, on the highest macroscopic level, formation of shear band
may be observed. These shear bands which are formed by plastic
instability are the regions where strain localization and consequently
major concentration of plasic deformation takes place, see Figure 12.
(a)
(b)
Figure 12. Illustration of macroscopic grain subdivision and subsequently formation
of (a) deformation bands (b) shear bands in pure Aluminum during deformation
(Jonsson, 2010) [3].
19
4
4.1
METHODOLOGY
INTRODUCTION
After experimental observation of dislocations in the late 1950s
extensive studies have been performed over the past several decades
to understand the significant role of dislocations in plasticity of
crystalline materials. While experimental studies was dominant
approach for elaboration of plasticity in the beginning, however,
advent of computational modeling techniques paved an alternative
road to profound investigation of plastic deformation in various
materials. A number of different numerical methods from atomistic to
continuum models have been used in the recent years to answer the
fundamental questions in crystal plasticity. Molecular dynamics (MD)
method is an example of atomistic modeling techniques which have
been used by various researchers to study the mechanism of plasticity
in crystals. For example, Horstemeyer et al., 2001 [7] performed a
MD simulation to study the strain rate, simulation volume size and
crystal orientation effect on the dynamic deformation of fcc single
crystals. Guo et al., 2007 [8] used MD simulation with single crystal
copper blocks under simple shear to investigate the size and strain rate
effects on the mechanical response of face-centered metals.
In spite of numerous promising results of studies obtained using
atomistic modeling methods, however, due to the time and length
scale limitations of these methods, an alternative simulation technique
so called Dislocation Dynamics (DD) method, has been developed to
overcome the limitations of atomistic methods regarding modeling of
plasticity. DD method is a powerful numerical modeling technique
which allows us to directly simulate dislocation aggregates motion
and subsequently to investigate the microstructure evolution in the
deformed materials. DD simulations can offer important insights that
help answer the fundamental questions in crystal plasticity, such as the
origin of the complex dislocations patterns that emerge during plastic
deformation and the relationship between microstructure, loading
conditions and the mechanical strength of the crystals (Bulatov and
Cai, 2006) [2].
Two dimensional dislocation dynamics models were the dominant
technique at the onset of DD modeling. Numerous researchers such as,
20
Lepinoux and Kubin, 1987 [9], Gullouglu et al., 1989 [10] and
Ghoniem and Amodeo, 1990 [11] used two dimensional DD models
to shed light into the dislocations behavior under different conditions.
However, due to the severe limitations of two dimensional simulations,
a number of major features such as line tension effects, slip geometry
and multiplication could not be treated properly using two
dimensional techniques.
The first three dimensional DD simulation was implemented by
Kubin et al., 1992 [12]. In this pioneer work, dislocation loops of
arbitrary shapes were discretized into a series of edge and screw
dislocation segments of rudimentary length. After this first attempt
several three dimensional dislocation dynamics models have been
presented by various scientists such as Devincre and Kubin, 1997 [13],
Zbib et al., 1998 [14], Schwartz, 1999 [15], Ghoneim and Sun, 1999
[16], Shenoy et al., 2000 [17], Monnet et al., 2004 [18], Wang et al.,
2006 [19] and Arsenlis et al., 2007 [20].
In addition to pure dislocation dynamics method, a hybrid modeling
technique has been developed to couple two dimensional dislocation
dynamics with Finite Element Method (FEM) by Van der Giessen and
Needleman, 1995 [21] and three dimensional dislocation dynamics
with FEM by Yasin et al., 2001 [22]. Numerous numerical modeling
studies were performed using these hybrid techniques for example,
Liu et al., 2008 [23] carried out a combined finite element and discrete
dislocation dynamics studies to investigate strain rate effect on
dynamic deformation of single crystal copper at strain rates ranging
from 102 to 105 s-1. Shehadeh et al., 2005 [24] studied the deformation
process in copper and aluminum single crystal under shock loading
(high strain rates) using a coupled dislocation dynamics and finite
element analysis.
In the present chapter the main ingredients of discrete dislocation
dynamics method are elaborated.
21
4.2
DISCRETE DISLOCATION DYNAMICS
In the line dislocation dynamics technique, dislocation lines are
discretized into the straight segments connected by the discretization
nodes. Nodal representation of dislocation network is shown in Figure
13. For every individual segment, a non-zero Burgers vector exists
and the line sense direction determines the possible sign of the
Burgers vector for each segment. Here, 𝑏𝑖𝑗 is defined as Burgers
vector of a segment with a line sense pointing from node 𝑖 to node 𝑗.
Similary, 𝑏𝑗𝑖 can be defined as Burgers vector of the same segment
with a line sense pointing from node 𝑗 to node 𝑖. Furthermore, at each
segment 𝑏𝑖𝑗 + 𝑏𝑗𝑖 = 0.
Figure 13. An arbitrary dislocation network represented by a set of nodes connected
by straight segments. 𝒃𝒊𝒋 is a Burgers vector associated with a dislocation line
connecting node 𝒊 to 𝒋.
In addition, at both discretization nodes (connecting only two
segments) and physical nodes (connecting arbitrary number of
segments), the conservation of Burgers vector should be enforced, see
Figure 14. Therefore, at every single node inside the dislocation
network the following criteria should be satisfied
∑ 𝑏𝑖𝑗 = 0.
𝑘
Sum is taken over the all nodes 𝑘 connected to node 𝑖.
22
Figure 14. Conservation of Burgers vector at both types of nodes, i.e., physical and
discretization nodes.
Since dislocation lines cannot terminate inside the crystal therefore
each node should be connected to the at least two other nodes.
Additionally, in order to avoid redundancy, no two nodes can be
directly connected by more than one segment (Bulatov and Cai, 2006)
[2].
4.2.1
CALCULATION OF NODAL FORCES
The force 𝑭𝑖 is defined as the force applied on the node 𝑖 in the
dislocation network and can be described by considering the total
energy, 𝜀 , of the system. Thus, the nodal force, 𝑭𝑖 , is derived by
taking negative derivitive of the total energy of the system with
respect to the nodal position, 𝑿𝑖 , such that
𝑭𝒊 ≡ −
𝜕𝜀 ( 𝑿𝑗 , 𝒃𝑖𝑗 , 𝑻𝑠 )
,
𝜕𝑿𝑖
(4.1)
where the total energy, 𝜀, is a function of all nodal positions along
with their connectivity defined by the Burgers vector and the
externally applied surface traction 𝑻𝑠 .
The total energy of the dislocation network can be partitioned into the
two different parts; elastic part, 𝜀 𝑒𝑙 , which is attributed to the long
23
range interactions among dislocations and is predicted using
continuum elasticity theory. Core part, 𝜀 𝑐 , which is associated with
dislocation core energies and results from the atomic configurations of
the dislcation cores. Therefore, 𝜀 = 𝜀 𝑒𝑙 + 𝜀 𝑐 .
Similarly, the force on node 𝑖 can also be partitioned into two parts,
first part corresponds to the elastic force and consequently is derived
as minus derivative of 𝜀 𝑒𝑙 with respect to the nodal position and the
second part is associated with the core force resulting from core
𝑐
energy 𝜀 𝑐 . Hence, we will have: 𝑭𝑖 = 𝑭𝑒𝑙
𝑖 + 𝑭𝑖 .
Calculation of the elastic force acting on a dislocation node
While it is possible to describe the applied elastic force on node 𝑖 by
differentiating the total elastic energy with respect to the nodal
position, however, a more convenient approach, i.e., virtual work
argument is presented here to describe elastic force.
The elastic force on node 𝑖 inside the dislocation network can be
defined by considering the contributions of the segments which are
connected to node 𝑖. Therefore, in order to calculate the nodal
force, 𝑭𝑖 , it suffices to sum up the elastic forces acting on the
segments which are connected to node 𝑖 such that
𝑒𝑙
𝑭𝑒𝑙
𝑖 = ∑ 𝒇𝑖𝑗 ,
(4.2)
𝑗
where 𝒇𝑒𝑙
𝑖𝑗 is the elastic force acting on segment 𝑖𝑗 and sum is taken
over the all segments connected to node 𝑖.
The elastic force on segment 𝑖𝑗, 𝒇𝑒𝑙
𝑖𝑗 , is defined using Peach-Koehler
𝑝𝑘
force, 𝒇 , applied on the point 𝒙𝑖𝑗 on the dislocation segment such
that
𝑙
2
𝑝𝑘
𝒇𝑒𝑙
(𝒙𝑖𝑗 (𝑙)) 𝑑𝑙,
𝑖𝑗 ≡ |𝒍𝑖𝑗 | ∫ 𝑁(−𝑙) 𝒇
(4.3)
𝒇𝑝𝑘 (𝒙𝑖𝑗 ) = [ 𝝈 (𝒙𝑖𝑗 ) · 𝒃𝑖𝑗 ] × 𝒕𝑖𝑗 ,
(4.4)
−
𝑙
2
24
where 𝑁(𝑙) represents a linear shape function to describe the position
of each point 𝒙𝑖𝑗 on the segment 𝑙𝑖𝑗 such that
𝒙𝑖𝑗 (𝑙) = 𝑁(−𝑙)𝑿𝑖 + 𝑁(𝑙)𝑿𝑗 ,
and 𝑁(𝑙) =
1
1
(4.5)
1
+ 𝑙 for ( - 2 ≤ 𝑙 ≤ 2 )
2
(4.6)
𝒇𝑝𝑘 (𝒙𝑖𝑗 ) is the Peach-Koehler force at point 𝒙𝑖𝑗 on dislocation line
and is described by local stress 𝝈, Burgers vector 𝒃𝑖𝑗 and line tangent
𝒕𝑖𝑗 at point 𝒙𝑖𝑗 . Local stress, 𝝈, in the Peach-Keohler expression may
be defined by superposition principle and subsequently summation of
all stress fields resulting from applied external load (the imposed
surface traction), long range elastic interactions among dislocation
segments and dislocations own line tension. Similarly, it is possible to
describe the elastic force applied on the dislocation segment using
superposition principle as following
𝑛−1
𝒇𝑒𝑙
𝑖𝑗
=
𝒇𝑒𝑥𝑡
𝑖𝑗
+
𝑠
𝒇𝑖𝑗
𝑛
+ ∑ ∑ 𝒇𝑘𝑙
𝑖𝑗
(4.7)
𝑘=1 𝑙=𝑘+1
and [𝑘, 𝑙] ≠ [𝑖, 𝑗] 𝑜𝑟 [𝑗, 𝑖],
𝑠
where 𝒇𝑒𝑥𝑡
𝑖𝑗 represents the applied external force, 𝒇𝑖𝑗 corresponds to
the segment’s own stress field and 𝒇𝑘𝑙
𝑖𝑗 is attributed to the long range
elastic interaction between segments 𝑖𝑗 and 𝑘𝑙.
In dislocation dynamics simulation with periodic boundary conditions
typically we deal with a situation where the applied external force
results in a uniform stress field 𝝈𝑒𝑥𝑡 inside the crystal, in this case,
𝒇𝑒𝑥𝑡
𝑖𝑗 can be easily defined according to
𝒇𝑒𝑥𝑡
𝑖𝑗 =
1
{[ 𝝈𝑒𝑥𝑡 · 𝒃𝑖𝑗 ] × 𝒍𝑖𝑗 },
2
(4.8)
𝑒𝑥𝑡
and 𝒇𝑒𝑥𝑡
𝑖𝑗 = 𝒇𝑗𝑖 ,
𝑠
Both 𝒇𝑖𝑗
and 𝒇𝑘𝑙
𝑖𝑗 can be predicted using analytical solutions of the
non-singular continuum theory of dislocations. In this new theory, the
core singularity is removed by replacing a single distribution of
25
Burgers vector around dislocation line with a spherically symmetric
Burgers vector distribution at every point on dislocation line. Using
this approach leads to the smooth distribution of the Burgers vector at
every point on dislocation segment which results in simple analytical
expressions to describe the internal forces applied on the dislocation
segments. Details of these analytical solutions can be found elsewhere
(Arsenlis et al., 2007) [20].
Nodal force calculation using analytical solutions is a non-trivial task
which requires considerable computational expense to describe the
long range interactions between dislocations. No matter how efficient
the nodal force calculation from a pair of segments becomes, the total
amount of calculation scale as 𝛰(𝑁 2 ) for a system of 𝑁 segments, if
the interaction between every segment pair is accounted for
individually (Arsenlis et al., 2007) [20]. Therefore, it is desirable to
introduce a method to reduce the computational costs of nodal fore
calculations for large scale simulations.
Fast Multipole Method (FMM) by Greengard and Rokhlin,1997 [25]
is an efficient technique which has been applied in DD modeling to
treat the long range elastic interaction between dislocations and
consequently to reduce the costs of numerical calculations. In this
method when the distance between interacting dislocations is
sufficiently large, it is assumed that dislocations are lumped into
groups and the respective interactions between them are not accounted
for individually. Therefore, the force between distant dislocations is
approximated to reach higher numerical efficiency.
For further information regarding application of FMM in dislocation
dynamics the reader is referred to Arsenlis et al., 2007 [20].
Dislocations core energy and corresponding force calculation
Dislocation core is a region where interatomic interactions play a key
role and, as a result, linear elasticity theory is incapable of predicting
energy and forces in this region. Therefore, in order to evaluate the
core energy and its associcated force acting on the dislocation line,
atomistic caluclation should be used.
26
Imagine that 𝐶 represents the entire dislocation network, hence, the
core energy can be described using a single integral along the network
such that
𝐸𝑐𝑜𝑟𝑒 (C, 𝑟𝑐 ) = ∮ 𝐸𝑐 (𝒙; 𝑟𝑐 )𝑑𝑳,
(4.9)
where 𝐸𝑐 (𝒙; 𝑟𝑐 ) is the energy per unit length of dislocation line at
point 𝒙 on the dislocation network and the integral is taken over 𝐶, i.e.,
entire dislocation network. Furthermore, 𝑟𝑐 is the cut-off radious, i.e.,
the radius which incorporates the core region and it is usually
approximated to a few Burgers vector lengths.
Since the core energy of a dislocation only depends on its line
direction, thus, for the current discretized framework (straight
dislocation segments) the core energy can be evaluated according to
𝐸𝑐𝑜𝑟𝑒 (C, 𝑟𝑐 ) = ∑ 𝐸𝑐 (𝜃𝑖−𝑗 , 𝜑𝑖−𝑗 ; 𝑟𝑐 )| 𝒓𝑖 − 𝒓𝑗 | ,
(4.10)
(𝑖−𝑗)
where the sum is taken over all segments (𝑖 − 𝑗) and 𝐸𝑐 is a core
energy which is a function of 𝜃𝑖−𝑗 and 𝜑𝑖−𝑗 (the orientation angles of
segement (𝑖 − 𝑗)).
Due to the lack of atomistic data in addition to the fact that
contribution of core energy to the nodal foce is usually small and
short ranged, thus, core energy contribution to the nodal forces
mostly has not been accounted for in dislocation dynamics simulations.
Periodic Boundary Conditions
Because the simulation volume used in the DD method is relatively
small in comparison with the volume of a macroscopic specimen, thus,
in order to model a bulk material and to avoid the surface effects
(resulting from small size of simulation volume), Periodic Boundary
Conditions (PBC) are often applied in DD numerical technique.
Using PBC in dislocation dynamics leads to the situation where a
node inside the dislocation network which is located at position 𝒓
repeats itself at positions 𝒓 + 𝑛1 𝒄1 + 𝑛2 𝒄2 + 𝑛3 𝒄3 , where 𝑛1 , 𝑛2 , 𝑛3
are integers and 𝒄1 , 𝒄2 , 𝒄3 are the repeat vectors of the periodic
27
sepercell. Therefore, the FMM algorithm is applied to predict the nonlocal long ranged interaction between dislocation segments located
inside the simulation volume and all their periodic images.
4.2.2
EQUATION OF MOTION
Dislocations repond to the applied force through motion on the glide
planes. Therefore, calculation of nodal force should be followed by
evaluation of dislocation response. Consequently, deriving an
equation of dislocation motion is necessary. In dislocation dynamics
technique, nodal forces are related to the nodal velocities through a so
called “mobility function”. This mobility function includes most of
material specific aspects of DD method and several factors such as
type of material, pressure, temperature, applied force and geometry of
the dislocation play a role in the construction of this function.
However, in order to avoid uprising complexity and to be able to
obtain a generic mobility function which reproduces the fundamental
aspects of kinetics of dislocations network, relevant simplifications
should be considered. Hence, explicit dependency of mobility function
on different factors is neglected and only applied local forces acting
on dislocation nodes are considered. Furthermore, due to the limited
circumstances where dislocation interia play a major role on kinetics
of dislocations such as very high strain rate state, e.g., shock
propagation, dislocation inertia is neglected and therefore,
accelerations and masses of dislocations are ignored. Thus, it is
assumed that dislocations move in an over-damped regime and their
motion is controlled only by forces acting on them. By considering the
above assumptions mobility funtion will take the following form
𝒗(𝒙) = 𝑀[𝒇(𝒙)] ,
(4.11)
where 𝒗(𝒙) is the velocity of point 𝒙 on dislocation line and 𝒇(𝒙) is
the force per unit length of dislocation line at the same point.
Moreover, 𝑀 represents a generic mobility function and in the present
framework only depends on the applied local force 𝒇(𝒙).
Due to the fact that linear dislocation segments should remain linear
over the entire course of simulation, equation (4.11) cannot be
applied as is in the discretized dislocation dynamics framework. Thus,
it is more convenient to invert the mobility function and define a local
28
drag force per unit length 𝒇𝑑𝑟𝑎𝑔 along the dislocation line as a
function of the velocity at that point, with the form (Arsenlis et al.,
2007) [20]
𝒇𝑑𝑟𝑎𝑔 (𝒙) = −𝑀−1 [𝒗(𝒙)] = −𝓑[𝒗(𝒙)] ,
(4.12)
where 𝓑 represents the drag function. In the present discretized
system, due to to the linear variation of velocity along a segment
equilibrium condition between the driving force on dislocation node,
𝑭𝑖 , and drag force 𝒇𝑑𝑟𝑎𝑔 is satistifed in the weak form i.e., only at the
discritized nodes such that
𝑭𝑖 = −𝑭𝑖 𝑑𝑟𝑎𝑔 ,
(4.13)
where
−𝑭𝑖
𝑑𝑟𝑎𝑔
1
2
≡ ∑|𝒍𝑖𝑗 | ∫ 𝑁(−𝑙)𝓑𝑖𝑗 [𝒗𝑖𝑗 (𝑙)]𝑑𝑙 ,
𝑗
−
1
2
(4.14)
where 𝑁(𝑙) is a shape function and 𝓑𝑖𝑗 is the drag function of the
dislocation segment connecting node 𝑖 to node 𝑗. Thus, with enforced
equilibrium in the weak sense a set of linear equations relating the
nodal velocities to the nodal forces are obtanied.
For linear mobility model, i.e., 𝓑𝑖𝑗 [𝒗𝑖𝑗 ] = 𝑩𝑖𝑗 · 𝒗𝑖𝑗 , the intergral
expression will take a simple algebraic form for all nodes according to
𝑭𝑖 = ∑
𝑖
|𝒍𝑖𝑗 |
𝑩𝑖𝑗 · (2 𝑽𝑖 + 𝑽𝑗 ) ,
6
(4.15)
where 𝑩𝑖𝑗 represents the drag coefficient tensor for segment 𝑖𝑗 .
Summation is over all nodes 𝑗 that are connected to node 𝑖.
If we assume that in the dislocation network velocities of all nodes
connected to node 𝑖 are approximately identical, i.e., 𝑽𝑖 ≈ 𝑽𝑗 and
𝒗𝑖𝑗 (𝑙) ≈ 𝑽𝑖 , then equation (4.15) can be further simplified such that
𝑭𝑖 =
1
∑|𝑙𝑖𝑗 | 𝑩𝑖𝑗 [𝑽𝑖 ] ,
2
(4.16)
𝑗
29
where 𝑙𝑖𝑗 is the length of segment 𝑖 − 𝑗 and the sum is over all nodes 𝑗
connected to node 𝑖. Equation (4.16) is a simplified expression which
relates the velocity of a node 𝑖 to the nodal force applied on it.
4.2.3
TIME INTEGRATORS
Calculation of nodal velocities enables us to advance the nodal
positions within a certain time step of the simulation. Choosing an
appropriate time step is necessary to ensure the convergence of
numerical calculations. Several algorithms are available to move the
nodal positions accordingly; however, we only consider two
algorithms known as explicit Euler forward method and implicit
trapezoidal method.
Equation of motion in over-damp regime is a first-order ordinary
differential equation which can be solved numerically using a simple
numerical integrator known as explicit Euler forward method such
that
𝑿𝑡+∆𝑡
= 𝑿𝑡𝑖 + 𝑽𝑡𝑖 ∆𝑡 ,
𝑖
(4.17)
where 𝑿𝑡𝑖 is the position of node 𝑖 at time 𝑡 and 𝑽𝑡𝑖 is the velocity of
the same node at this time. In addition, ∆𝑡 is the time step which
should be adjusted appropriately to enable an accurate integration of
the equation of motion. Explicit Euler forward method is a simple
algorithm with a relatively low computational cost. However, this
algorithm is fairly inefficient in dislocation dynamics modeling due to
its numerical instability along with too small allowable time steps.
More efficient algorithm which is frequently used in dislocation
dynamics modeling is implicit trapezoidal method. This technique is
the combination of the Euler forward and Euler backward methods
and takes the following form
𝑿𝑡+∆𝑡
= 𝑿𝑡𝑖 +
𝑖
1 𝑡
(𝑽 + 𝑽𝑡+∆𝑡
)∆𝑡.
𝑖
2 𝑖
(4.18)
The implicit trapezoidal method generates a set of equations which
should be solved iteratively. Although this mehtod requires higher
computational costs, however, the allowable time step is relatively
larger in comparison with the implicit Euler forward technique.
30
In order to solve the above equation an iterative update technique is
considered such that
(𝑛 + 1) = 𝑿𝑡𝑖 +
𝑿𝑡+∆𝑡
𝑖
(1) = 𝑿𝑡𝑖 +
𝑿𝑡+∆𝑡
𝑖
1 𝑡+∆𝑡
(𝑽𝑖 (𝑛) + 𝑽𝑡𝑖 )∆𝑡,
2
1 𝑡−∆𝑡
(𝑽
+ 𝑽𝑡𝑖 )∆𝑡,
2 𝑖
(4.19)
(4.20)
(𝑛 + 1) − 𝑿𝑡+∆𝑡
(𝑛) ||< 𝑟𝑡𝑜𝑙 , is
and when the the criterion, ||𝑿𝑡+∆𝑡
𝑖
𝑖
satisfied the iteration stops. 𝑟𝑡𝑜𝑙 is the maximum position error
tolerated in time step integration.
Unlike the explicit methods, implicit algorithms such as trapezoid
technique do not have any a priori known time step and optimum size
of time step is adjusted dynamically over the course of simulation.
Therefore, the simulation will proceed with a distribution of time steps,
and the code will perform sub-optimally, in that the maximum
allowable time step is not always used (Arsenlis et al., 2007) [20].
4.2.4
TOPOLOGICAL CHANGES
In order to represent a realistic dislocation behavior and to obtain a
better numerical efficiency, it is necessary to allow topological
changes, i.e., changes in the connectivity of the nodes in dislocation
dynamics method. During the numerical modeling various topological
changes may be required such as change in the length and/or curvature
of dislocation line accompanied with the number of nodes
representing this line. In addition, when two dislocation lines
approach each other, they may annihilate or recombine and form a
junction which for both cases topological modifications should be
considered. Within the nodal representation adopted in the current
framework, arbitrarily complex topological changes can be produced
by combination of only two elementary topological operators: “merge”
(two nodes merge into one node) and “split” (one node splits into two
nodes) (Bulatov and Cai, 2006) [2]. The conservation of Burgers
vector should still be enforced in every node and segment involved in
the merge or split operation.
31
Remeshing
During the simulation, it may be desirable to add nodes or delete
existing nodes. Adding or deleting nodes enable us to improve
dislocation line representation over the course of simulation. These
operators can be regarded as special case of merge and split operators.
Figure 15 illustarates a geometry where node E is added to the
existing set of nodes and node D is deleted from them.
Figure 15. Delete and add operators, node E is added between nodes A and B and
node D located between nodes B and C is deleted .
Merge and Split Operators
As it was discussed earlier, interaction between dislocations should be
accounted for in dislocation dynamics modeling, thus, it is important
to have appropriate operators such as split and merge to allow
dislocations to interact. These operators act as bookkeepers of nodal
connectivity.
The merge operator acts when two dislocation lines are at a contact
distance from each other. If distance between two dislocation lines, 𝑑
becomes less than a collision distance parameter, 𝑟𝑎 : 𝑑 < 𝑟𝑎 , then two
dislocation lines are considered to be in contact and merge operator
should be called.
Figure 16 illustartes two dislocation segements 1-2 and 3-4 which are
at contact distance from each other. In order to allow these two
dislocation lines to interact the following course of action should be
implemented. First, two nodes at points 𝑃 and 𝑄 will be added on the
segments 1-2 and 3-4. Then, due to the fact that these two new
intorduced nodes are within a contact distance from each other, they
will merge into a new node 𝑃′. Finally, node 𝑃′ will be splited into
32
two nodes such as 𝑃′ and 𝑄’ to complete the intercation bewteen the
two dislocation segments.
(a)
(b)
(c)
Figure 16. (a) Two colliding dislocation segments (1-2, 3-4). Nodes 𝑃 and 𝑄 are
added on the segments and they are in contact distance from each other. (b) Nodes
𝑃 and 𝑄 are merged into new single node, 𝑃’. (c) Node 𝑃’ is splited into two nodes,
𝑃’ and 𝑄’ (Bulatov and Cai, 2006) [2].
There are several topological ways to split node 𝑃′ into two nodes.
One distinct way is (13)(24) which is shown in Figure 16(c). In
addition, it is possible to split node 𝑃′ into two nodes following
different ways such as (12)(34) and (14)(23). In order to choose a
proper way to split a node, the minimum energy principle should be
considered. In the over-damped regime descent of the dislocation
system towards the minimum free energy can be quantified by
introducing the energy reduction rate which is regarded as a rate of
heat production, 𝑄̇ , in the dislocation network. From the various
available ways to split node 𝑃′ in Figure 16(b) the ultimate arrangment
which the system will evolve to is the one with the highest heat
production rate, i.e., (13)(24).
Imagine that node 𝑖 in the dislocation netwok experiences nodal force
𝑭𝑖 and moves in the response to this force with the velocity, 𝑽𝑖 . Node
𝑖 will contribute to the total energy dissipation rate of the system such
that
𝑄̇ 𝑖̇ = 𝑭𝑖 · 𝑽𝑖 ,
(4.21)
where 𝑄̇ 𝑖̇ represents the rate of energy dissipation of the system which
is attributed to the node 𝑖. Now, suppose that node 𝑖 splits into two
nodes 𝑃 and 𝑄 , hence, the contribution of these new nodes to the
energy dissipation rate of the system will be
33
𝑄̇𝑃𝑄 = 𝑭𝑃 · 𝑽𝑃 + 𝑭𝑄 · 𝑽𝑄 ,
(4.22)
where 𝑭𝑃 and 𝑭𝑄 denote the force acting on nodes 𝑃 and 𝑄
respectively, and 𝑽𝑃 and 𝑽𝑄 represent their respective velocities.
If 𝑄̇𝑃𝑄 > 𝑄̇ 𝑖̇ , it will be desirable for node 𝑖 to split into two nodes, 𝑃
and 𝑄 to increase the energy dissipation in the network and
consequently to reach the minimum free energy.
4.2.5
PARALLEL COMPUTATION
As numerical modeling proceeds, initial number of distributed
dislocations inside the simulation volume increases. Therefore, in
every time step, detailed dynamics of a large number of interacting
dislocations should be followed which requires vast calculations and
consequently substantial computational power. At the moment, the
only means to acquire the required computational muscle is through
massively parallel computing, which involves running a single
simulation simultaneously on a large number (~103 ) of processors
(Bulatov and Cai, 2006) [2].
Scalability
In the parallel computing scheme, scalability of a code plays a
decisive role to achieve the computational efficiency. A code is
considered to be scalable if assigning more processors to the code
leads to either less computational time to solve a problem or same
computational time to solve a larger problem. Several factors can limit
scalability of a code such as the time spent on communications
between the processors along with the time wasted due to an uneven
distribution of computational load among CPUs. Thus, we introduce
the parameter 𝜂 to represent the parallel efficiency of a code such that
𝜂=
𝑡𝑐
,
𝑡𝑐 + 𝑡𝑤 + 𝑡𝑚
(4.23)
where, 𝑡𝑐 is the total computation time summed over all the CPUs,
𝑡𝑤 is the total wasted time in waiting due to load imbalance and 𝑡𝑚 is
the total message passing time spent for communication between
processors. Figure 17 demonstrates distribution of all three times in a
simple parallel computation.
34
Figure 17. Time distribution in parallel computing. Three distinct regions shown by
the dark blue bars, the white area and the light blue bar correspond to the time spent
for computing 𝒕𝒄 , the time spent by each processor in waiting 𝒕𝒘 for last calculation
to be finished and the time spent for inter-processors communication, 𝒕𝒎 ,
respectively.
Thus, in order to achieve higher scalability, communication time
between processors (𝑡𝑚 ) should be reduced and computational load
should be uniformly distributed among the processors to minimize the
waiting time 𝑡𝑤 . The following section describes an approach to
increase the efficiency of a parallel code through reducing both
communication time 𝑡𝑚 and waiting time 𝑡𝑤 .
Spatial domain decomposition and dynamic load balance
In parallel computing having a partitioning algorithm which divides
the entire simulation domain into the number of subdomains and
assigns each subdomain to a separate processor is crucial to achieve
the acceptable efficiency. However, increase in the number of
processors due to the partitioning of simulation volume leads to the
significant increase in the communication time, i.e., message passing
time (𝑡𝑚 ) between the CPUs which in turn results in considerable
reduction in scalability of the code. Therefore, in order to achieve a
decent scalability while increasing the number of the CPUs an
35
efficient algorithm, e.g., Fast Multipole Method should be followed.
Implementation of this method in DD modeling leads to decrease in
the communication time between processors 𝑡𝑚 in each time step and
consequently improvement of the scalability in parallel computing
framework.
Formation of heterogeneous microstructure during plastic deformation
results in substantial variations in local dislocation density, thus,
partitioning of simulation domain into the subdomains with equal
sizes may lead to load imbalance and subsequently very poor
scalability. Therefore, in order to distribute the computational
expenses more uniformly among the CPUs and to reach higher
scalability, a partitioning model which divides the simulation volume
into the sub-volumes with different sizes is of great interest. Uniform
distribution of nodes inside the subdomains and subsequently
improvement of load balancing can be achieved by recursive
partitioning of the simulation domain along 𝑥 , 𝑦 , 𝑧 directions, see
Figure 18. First, the simulation volume is sectioned into 𝑁𝑥 slabs
along the 𝑥 direction. Then, each slab is divided into 𝑁𝑦 columns
along the 𝑦 direction. Finally, each column is partitioned into 𝑁𝑧
boxes along the 𝑧 direction. As recursive partitioning model is
followed every subdomain will contain approximately equal number
of nodes.
Moreover, further improvement of computationl load balancing can be
achieved by adjusting the domain boundaries during the modeling.
Dynamic load balancing can lead to significant increase in effeciently
of parallel computing.
Figure 18. Simulation volume is divided into 3 ⨉ 3 ⨉ 2 domains along three axes
and each domain is assigned to its own processor (Bulatov and Cai, 2006) [2].
36
5
RESULTS AND DISCUSSION
5.1
SUMMARY OF APPENDED PAPER A
In paper A, numerical tensile test was performed on the copper single
crystal to investigate plastic deformation in fcc crystalline materials.
In this work, model crystal was uniaxially loaded along [001]
crystallographic direction at high strain rates ranging from 103 to
105s-1 and macroscopic response of the crystal was examined.
Moreover, microstructure development during plastic deformation of
the crystal was studied. All numerical modelings were carried out
using Parallel Dislocation Simulator (ParaDis) (Arsenlis et al., 2007)
[20] code which implements dislocation dynamics method for
numerical calculations.
5.1.1
DETAILS OF DD MODELING
In the present numerical modeling, simulation volume is set as 1×1×1
µm and three dimensional periodic boundary conditions are applied.
Random distribution of twenty four straight dislocations inside the
cubic cell leads to the 𝜌0 = 2.7 × 1013 m−2 initial dislocation
density over the simulation volume. The material parameters for
copper are set as follows
 Shear modulus, µ = 42 GPa
 Poisson’s ratio, 𝜈 = 0.31
 Burgers vector, 𝑏 = 0.256 nm
In addition, dislocation drag coefficient is set as 𝐵 = 10−4 Pa·s,
which is the highest acceptable value for copper at room temperature
(Edington, 1969) [26]. The simulations are performed to 1.4 percent
plastic strain and stress-plastic strain curve, dislocation density in
addition to microstructure evolution are analyzed.
5.1.2
MACROSCOPIC BEHAVIOR
Figure 19 illustrates stress as a function of plastic strain for tensile
deformation of copper single crystal at three sets of imposed strain
rates. As the copper single crystal is strained at high strain rates
>> 103 s-1, flow stress demonstrates significant strain rate sensitivity;
37
thus, a considerable increase in plastic flow is observed when strain
rate increases from 103 s-1 to 105 s-1, see Figure 19.
Figure 19. Stress as a function of plastic strain for tensile deformation of copper
single crystal along [001] orientation at three sets of strain rates.
Evolution of the total dislocation density with plastic strain is shown
in Figure 20. Dislocation density increases with plastic strain at all
considered strain rates. Similar to the flow stress, dislocation density
evolution is also affected by the strain rate such that the highest
increase in dislocation density takes place when the model crystal is
deformed at the highest strain rate of 105 s-1.
Figure 20. Total dislocation density evolution as a function of plastic strain at
various strain rates.
38
5.1.3
EFFECT OF STRAIN RATE ON SLIP ACTIVITY
In fcc metals there are twelve slip systems of type {111} <11̅0> which
can contribute to the plastic staining process. Loading of the single
crystal along [001] direction may result in activation of eight slip
systems due to the highest symmetry associated with this orientation.
Four (111) slip planes have the same Schmid factor of approximately
0.41 when loading of sample crystal takes place along [001]
orientation. However, identical Schmid factor of slip planes, will not
lead to the similar contribution of the slip systems to the deformation
process. As dislocations generate, accumulate, and organize into low
energy structures, the internal stress state changes. The local internal
forces are inhomogeneous and could favor dislocation glide on some
slip systems while preventing glide on others, regardless of their
Schmid factors (Wang et al., 2009) [27].
In Figure 21 dislocation density distribution among different slip
systems are plotted in order to delineate the contribution of each slip
system to the dynamic deformation of the model crystal. Figure 21
demonstrates that, straining of the same crystal at different strain rates
leads to considerably different contribution of each slip system to the
deformation process. Therefore, imposed strain rate has a remarkable
effect on the contribution of each slip system to the dynamic
deformation.
(a)
39
(b)
(c)
Figure 21. Dislocation density distribution of individual slip systems for different
strain rates (a) 103 (b) 104 and (c) 105 s-1.
5.1.4
HETEROGENEOUS MICROSTRUCTURE EVOLUTION
Emerged microstructure in deformed model crystal shows a prominent
strain induced heterogeneity meaning that dislocation density
distribution is relatively non-uniform across the simulation volume.
Figure 22 demonstrates the distribution of dislocation density on the
thin slice in the x-y plane for imposed strain rates of 104 and 105 s-1.
40
Due to the similar trend observed at lower strain rate, i.e., 103 s-1,
dislocation density distribution at this strain rate is not illustrated here.
(a)
(b)
Figure 22. Distribution of the dislocation density on x-y plane at 1.4 percent plastic
deformation at strain rates (a) 104 s-1 and (b) 105 s-1. Lengths on x-y plane are in
Burgers vector unit.
In order to predict the extent of heterogeneity in microstructure
evolution, the relevant data concerning maximum and minimum
amount of dislocation density distributed on the x-y plane are
extracted for each imposed strain rate and the parameter Ω is
introduced as an indicator of heterogeneity in microstructure evolution
such that:
41
Ω = 𝜌𝑚𝑎𝑥 - 𝜌𝑚𝑖𝑛 .
The calculated values of the parameter Ω at different imposed strain
rates are shown in Table 1.
Table 1. Calculation of the parameter Ω at three imposed strain rates.
Parameter Ω
𝜀̇ = 103 s-1
𝜀̇ = 104 s-1
𝜀̇ = 105 s-1
1.0201⨉1012 𝑚−2
2.4516⨉1012 𝑚−2
3.4292⨉1012 𝑚−2
Our prediction of strain induce heterogeneity demonstrates strain rate
dependency of the microstructure development such that more
inhomogeneous microstructure is evolved as strain rate increases.
5.2
SUMMARY OF APPENDED PAPER B
Anisotropic characteristic of plastic deformation in fcc metals was
investigated in paper B using dislocation dynamics modeling
technique. Copper single crystal was uniaxially loaded along [001]
and [111] directions at two high strain rates of 105 and 106 s-1 and
effect of loading orientation on the flow properties in addition to
microstructure evolution of the model crystal was studied. Parallel
Dislocation Simulator (ParaDis) (Arsenlis et al., 2007) [20] code was
used to perform the numerical modelings.
5.2.1
SIMULATION DETAILS
Cubic simulation volume with an edge length of 2 𝜇m is selected and
three dimensional periodic boundary conditions (PBC) are applied.
Initial configuration of dislocations consists of twenty four straight
dislocations which are randomly distributed inside the simulation box.
Table 2 shows the initial dislocation density along with the input
parameters used in the present work. Simulations are performed to
approximately 0.45 percent plastic strain.
Table 2. Initial dislocation density and input parameters for current DD simulations
Initial dislocation
density (𝜌0 )
Shear
modulus (µ)
Poisson’s
ratio (𝜈)
Burgers
vector (𝑏)
Dislocation
drag
coefficient (𝐵)
7.14 × 10 12 𝑚−2
50 𝐺𝑃𝑎
0.31
0.256 𝑛𝑚
10−4 𝑃𝑎 · 𝑠
42
5.2.2
MECHANICAL PROPERTIES AND DISLOCATION DENSITY
EVOLUTION
Figure 23 illustrates the plastic anisotropy of the copper single crystal
as the model crystal is deformed along two multislip orientations of
[001] and [111] at strain rates of 105 and 106 s-1. The obtained stressplastic strain curves demonstrate the remarkable effect of the loading
orientation on the plastic flow such that at both imposed strain rates
loading of the crystal along [111] axis results in considerably higher
flow stress than loading along [001] direction, see Figure 23.
Figure 23. Mechanical response of the copper single crystal to the uniaxial tensile
loading along [001] and [111] directions at two imposed strain rates of 10 5 and
106 s-1.
Similarly, anisotropic response of the model crystal to the mechanical
loading can be observed in the evolution of total dislocation density.
Figure 24 presents the total dislocation density as a function of plastic
strain for [001] and [111] loading orientations at two imposed strain
rates of 105 and 106 s-1. Dislocation density increases with plastic
strain for all studied cases, however, loading orientation influences the
generation of dislocations such that at both strain rates, higher
dislocation density evolution corresponds to the [111] loading
direction, see Figure 24.
43
Figure 24. Illustration of dislocation density evolution as a function of plastic strain
at two imposed strain rates of 105 and 106 s-1 for loading of single crystal along [001]
and [111] orientations.
5.2.3
SLIP ACTIVITY
In order to demonstrate the contribution of each available slip system
to the total dynamic deformation of the model crystal, distribution of
dislocation density among different slip systems is plotted in
Figure 25 at two imposed strain rates of 105 and 106 s-1. At both strain
rates, two loading orientations, i.e., [001] and [111] are considered.
Figure 25 illustrates that at both considered strain rates loading
orientation affects the contribution of each slip system to the plastic
deformation. Thus, the same slip system contributes differently to the
plastic straining process when loading orientation changes from [001]
to [111] direction. Similarly, strain rate shows a significant effect on
the slip activity. Therefore, straining of the same crystal along similar
orientations (identical Schmid factors) by different strain rates leads to
considerably dissimilar contribution of each slip system to the
deformation process.
44
(a)
(b)
Figure 25. Dislocation density distribution of slip systems for loading of model
crystal along [001] and [111] orientations at (a) 10 5s-1 and (b) 106s-1 strain rates.
5.2.4
LOCALIZATION OF PLASTIC DEFORMATION
The generated microstructure resulted from straining of the single
crystal along [001] and [111] axes demonstrates considerable
localization of plastic strain and consequently formation of slip bands,
see Figure 26. In the observed slip bands dislocation density is
relatively higher than in other regions which indicate the
heterogeneous microstructure development during deformation. While
45
similar morphologies of the dislocation microstructure is observed for
all considered cases, however, strain localization is more prominent at
higher strain rate of 106 s-1 which leads to evolution of well-developed
dense slip bands as deformation of the single crystal progresses at this
strain rate for both loading orientations.
Loading direction: [001]
Loading direction: [111]
(a)
(b)
(c)
(d)
Figure 26. Microstructure development resulting from straining of the model
crystal along [001] and [111] directions. (a) and (b) at strain rate of 105 s-1, (c) and
(d) at strain rate of 106 s-1.
46
6
6.1
CONCLUDING REMARKS AND FUTURE WORK
CONCLUSION
In the present thesis work fundamentals of plastic deformation of
crystalline materials was reviewed. Discrete dislocation dynamics
method was introduced and applied to simulate the plasticity in fcc
metals. Copper single crystal was selected as a material of interest and
numerical tensile tests were performed to study the mechanical
response of the copper single crystal to the different loading
conditions. In addition to macroscopic behavior of the model crystal,
dislocation microstructure evolution and subsequent microscopic
mechanisms were examined.
First part of the study (Paper A) was devoted to the simulation of
dynamic deformation of single crystal copper at high strain rates
ranging from 103 to 105 s-1. The obtained modeling results allowed us
to conclude that strain rate has a significant effect on the mechanical
properties of copper single crystal at high strain rates. Sensitivity of
the plastic flow to the strain rate was clearly observed in all
considered cases. Observed strain rate sensitivity of the flow stress
agrees well with the reported experimental studies concerning
deformation of copper single crystal at strain rates above 103 s-1.
In addition to the macroscopic response of the single crystal to the
mechanical loading, strain rate sensitivity was observed in the
generated microstructure such that strain-induced heterogeneity of the
microstructure was increased with the strain rate. Hence, most
heterogeneous microstructure was developed at the highest strain rate
of 105 s-1.
In spite of the observed heterogeneous microstructure and dislocation
entanglements, due to the relatively small size of the simulation
volume along with the low strain levels reached at the present
simulations, formation of dislocation patterns was not detected in the
present work.
In the second part of the study (Paper B), plastic anisotropy of the
single crystal copper was investigated. Deformation of the model
crystal along two different multislip crystallographic directions was
47
modeled to examine the effect of loading direction on the mechanical
properties of the crystal. At the considered strain rates (105 and 106 s-1)
plastic flow increased significantly when loading orientation changed
from [001] to [111]. In addition to the loading direction, flow stress
demonstrated strain rate dependency such that flow stress increased
when strain rate escalated from 105 and 106 s-1 for both loading
directions.
Furthermore, microstructure development was investigated with the
aim to understand the anisotropic characteristic of plastic deformation
in single crystals. Discrete dislocation dynamics modeling results
showed an emergence of the inhomogeneous structure during plastic
deformation, and highest heterogeneity was associated with the
loading of the model crystal along [111] direction at strain rate of
106 s-1.
For all considered cases in the second work (Paper B), strain
localization and formation of slip bands was detected. Slip band
formation resulting from straining of the sample crystal at high strain
rates was more prominent at the higher strain rate of 106 s-1 for both
loading directions.
6.2
FUTURE WORK
 Analysis of creep deformation in copper canister is the main
objective of the future work. In this framework, dislocation
dynamics method will be used to directly address the
deformation behavior of the canister. Creep deformation of the
copper canister will be modeled using discrete dislocation
dynamics technique.
 Existence of impurities in the host lattice changes the
mechanism of dislocations motion due to the interaction of
dislocation aggregates with impurities. Therefore, analysis of
dislocation-impurity interaction is essential to understand the
effect of impurities on the collective motion of dislocations
and consequently on the macroscopic behavior of the bulk
material. Investigation of behavior of dislocation groups in the
presence of impurities will be performed.
48
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