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. 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