Numerical Implementation of Continuum Dislocation Theory Shengxu Xia

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2011 Numerical Implementation of Continuum Dislocation Theory Shengxu Xia

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COLLEGE OF ARTS AND SCIENCES

NUMERICAL IMPLEMENTATION OF CONTINUUM DISLOCATION THEORY

SHENGXU XIA

A Thesis submitted to the Department of Scientiﬁc Computing in partial fulﬁllment of the requirements for the degree of Master of Science

Degree Awarded: Fall Semester, 2011 Shengxu Xia defended this thesis on November 7, 2011. The members of the supervisory committee were:

Anter El-Azab Professor Directing Thesis

Tomasz Plewa Committee Member

Xiaoqiang Wang Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the thesis has been approved in accordance with the university requirements.

ii To my parents and friends

iii ACKNOWLEDGEMENTS

It has been a great honour to spend two years in Florida State University, where I constantly felt inspired by the intelligence and humanity surrounding me. The completion of my thesis has been a joyful journey and I could not have succeeded without the invaluable assistance of many individuals, so I want to express my deepest appreciation to them. First of all, I would like to give special thanks to my advisor, Professor Anter El-Azab, for his patient, friendly and encouraging support over the past two years, and for his knowledge and enthusiasm which made this thesis possible. I am very grateful to all members of my thesis committee as well as faculty and staff at scientific computing Department for their input, valuable discussions and accessibility. In particular, I would like to thank Professor Max Gunzburger, Professor Xiaoqiang Wang, Professor Tomasz Plewa for their support. The discussion with professionals in this research area is always interesting and cheerful. In particular, the communication with Dr. Thomas Hochrainer, Dr. Jie Deng, Dr. Stefan Sandfeld, Dr. Giacomo Po, and Dr. Mamdouh Mohamed benefits me a lot. I will take this chance to thank them for all the help they offer. In addition to the academic assistance above, I received equally important assistance from my friends and family. I am grateful to my parents for their love, support and trust throughout not only the course of my Master study, but throughout my entire life.

iv TABLE OF CONTENTS

List of Figures ...... vi Abstract ...... viii

1 Introduction 1

2 Discrete Dislocation Dynamics Model 4 2.1 Basic Concepts of Dislocations ...... 4 2.2 Burgers Vector ...... 6 2.3 Slip System and Dislocation Motion ...... 6 2.3.1 Slip Systems ...... 6 2.3.2 Dislocation Motion ...... 7

3 Continuum Theory of Dislocation 8 3.1 Crystal Plasticity ...... 8 3.2 Dislocation Plasticity Theory ...... 10 3.3 Dislocation Evolution Equation ...... 12

4 Numerical Scheme for Simulation 14 4.1 Numerical Scheme for Eigenstrain Problem ...... 14 4.2 Numerical Scheme for Curl Equation ...... 15 4.2.1 Least Square Finite Element Method(LSFEM) ...... 15 4.2.2 Finite Diﬀerence Method ...... 17 4.3 Coupled Scheme for Stress-Dislocation Simulation ...... 18

5 Numerical Results 20 5.1 Test Problem for Equilibrium Equation ...... 20 5.2 Test Problem for Curl Equation ...... 24 5.2.1 Simulation Results Obtained by LSFEM ...... 24 5.2.2 Comparison between Diﬀerent Numerical Schemes for Curl Equation 27 5.3 Test Problem for Stress-Dislocation Coupled Scheme ...... 29

6 Summary and outlook 34 References ...... 36 Biographical Sketch ...... 40

v LIST OF FIGURES

2.1 edge dislocation and screw dislocation ...... 5

2.2 mixed dislocation pattern formation ...... 5

2.3 determination of Burger’s vector ...... 6

3.1 kinematics of crystal plasticity ...... 9

4.1 coupled scheme for the simulation of dislocation evolution ...... 19

5.1 formation of a single edge dislocation ...... 20

5.2 γ ﬁeld distribution ...... 21

5.3 distorted shape due to an edge dialocation ...... 22

5.4 σxx component ﬁeld ...... 23

5.5 σzz component ﬁeld ...... 23

5.6 σxz component ﬁeld ...... 24

5.7 the movement of a straight dislocation line ...... 25

5.8 movement of a tilt dislocation line ...... 26

5.9 loop expansion ...... 26

5.10 pileup process of a loop ...... 27

5.11 curl equation solved by four distinct numerical methods ...... 28

5.12 stress ﬁeld σ13 in 3D crystal and 2D slip plane ...... 29

5.13 dislocations absorbed into the boundary ...... 30

5.14 dislocations absorbed into the boundary(given by upwind scheme) ...... 31

5.15 Plastic distortion component β31 recedes...... 32

vi 5.16 A distorted crystal partly recovers to its original state where the cloud denotes the magnitude of displacement...... 32

vii ABSTRACT

This thesis aims at theoretical and computational modelling of the continuum dislocation theory coupled with its internal elastic field. In this continuum description, the space-time evolution of the dislocation density is governed by a set of hyperbolic partial differential equations. These PDEs must be complemented by elastic equilibrium equations in order to obtain the velocity field that drives dislocation motion on slip planes. Simultaneously, the plastic eigenstrain tensor that serves as a known field in equilibrium equations should be updated by the motion of dislocations according to Orowan’s law. Therefore, a stress- dislocation coupled process is involved when a crystal undergoes elastoplastic deformation. The solutions of equilibrium equation and dislocation density evolution equation are tested by a few examples in order to make sure appropriate computational schemes are selected for each. A coupled numerical scheme is proposed, where resolved shear stress and Orowan’s law are two passages that connect these two sets of PDEs. The numerical implementation of this scheme is illustrated by an example that simulates the recovery process of a dislocated cubic crystal. The simulated result demonstrates the possibility to couple macroscopic(stress) and microscopic(dislocation density tensor) physical quantity to obtain crystal mechanical response.

viii CHAPTER 1

INTRODUCTION

Foundation of various crystal plasticity theories is mainly motivated by the technological need to understand and improve the strength of metallurgic crystals. Although research efforts that first approached this problem dates back to early nineteens, our understanding of the various aspects of crystal plasticity is still far from completion. There are two approaches to investigate into crystal plastic behaviour: phenomenological level and physical level. On the phenomenological level and within the continuum framework, Tresca[54], de St. Venant[44] et al. extended the elasticity theory by introducing phenomenological laws to account for the observed plastic behaviours of metals, such as yielding and work hardening. These phenomenological continuum plasticity theories have been very successful in a wide rang of engineering applications. They operate on length scales where the properties of materials and systems are scale invariant. However, experimental results showed that this scale-invariance breaks down at very small dimensions, say e.g. a few microns. In torsion tests of copper wire[16], strong scale-dependent response was observed in the range of 170µm to 12µm of wire diameter. In corresponding uni-axial tension tests, where deformation is macroscopically uniform, no comparable scale-dependence was recognized for the same range of copper wire diameter. Clear evidence for scale-dependent plasticity also appears in micro-indentation tests[53, 50, 39] where the hardness of metal crystals increases as the indenter decreases in size. In a word, a deeper understanding of crystal plasticity is required to interpret such phenomena. On the physical level, attempts to explain the microscopic mechanism underlying the phenomenological laws lead to the concept of the ’dislocation’ - a line-like defect inside a perfect crystal - which was firstly created as a mathematical concept by Volterra[56] and developed independently by Orowan[37], Polanyi[38] and Taylor[52]. While successfully explaining most of the puzzling phenomenology of crystal plasticity, dislocations continued as a beautiful hypothesis until the late 1950s when first sightings of them were reported in transmission electron microscopy (TEM) experiments[23]. Since then, it was recognized that modeling of the strain hardening behaviour of metallic crystals should be based on the collective behaviour of the underlying dislocation system. Kondo[26], Nye[36], Bilby et al.[4] and Kroner[29] formulated the classical continuum theory of dislocations independently from the viewpoint of plastic distortion. The common characteristic residing in their theories is a second-rank dislocation density tensor(also referred to as Kroner-Nye’s tensor) which is a continuous physical field to bridge the gap between the atomistic discontinuity and the

1 macroscopic continuity. With the development of computer technology, many discrete dislocation models were also proposed by Kubin[30], Zbib[40], and Ghoniem[18] et al. to simulate plastic deformation of a crystal. In these models,the dislocation lines were discretized into sequences of segments connected by nodes. However, all these models are computationally very expensive due to the huge abundance of dislocations and long range interaction between them. As a result of this computational difficulty, the continuum models of dislocations are coming back and gradually attracting more notice. Acharya et al.[1, 2, 3, 55] and Sedlancek et al.[48, 47] derived dislocation evolution equation and applied it to simulate the movement of only several dislocations, which still gives the hope to represent a huge number of dislocations as a continuous physical field. Since the late 1990s, Groma and co-workers developed a statistical approach towards deriving not only the kinematic evolution of dislocation systems, but also their internal interactions and stress driven dynamics , from systematic averages over ensembles of discrete dislocation systems[20, 58, 21, 28]. In an average representation, these systems can be described in terms of densities of charged particles. From this point of view, Yefi- mov and co-workers[57] simulated the response of a model composite material with elastic reinforcements in a hexagonal pattern under a simple shear load. Anter El-azab and co- workers[13, 12, 9] furtherly utilized the statistical mechanics theory and represented the movement of dislocation lines in a higher dimension phase space. However, The mathematical foundation for dealing with curved dislocations is not s- tudied thoroughly until Hochrainer and co-workers[45] who proposed Continuum Dislo- cation Dynamics Theory(also referred to as CDD) which also adds one more dimension to denote the tangent of a dislocation line. The expansion of a dislocation loop is successfully simulated by solving dislocation evolution equation in a higher dimension space, but the solenoidality of dislocation density tensor is corrupted in this way. Acharya and co-workers[55] proposed Galerkin/Least Square Finite Element Method(also referred to as Galerkin/LSFEM) to discretize the evolution equation in space and the numerical result in such way conserves the solenoidality of dislocation distribution field. Although Both achievements of the aforementioned are pioneering attempts to simulate the dislocation evolution process, they are incomplete and thus only serve as test examples because the velocity field they used is prescribed rather than obtained from stress field. One way to drive dislocation density distribution by a real stress field, which is illustrated in this thesis, is to employ the macroscopic elasticity equilibrium equations and couple the microscopic dislocation evolution equation with it. The present work is motivated by the need to simulate crystal response driven by continuum dislocation dynamics, thus enabling the development of statistical approach in the future. This thesis is organized in the following way: Chapter 2 is dedicated to an overview of the concept of dislocation theory, where we briefly introduce fundamental definitions,conventions, and physical terminologies. Chapter 3 covers the continuum dislocation model on which this thesis relies on mostly. This chapter can be reviewed as a rewriting of chapter 2 in continuum language. The Elastoplasticity theory is briefly presented and then the dislocation density tensor is introduced by incompatibility of plastic distortion tensor. This chapter is concluded by the derivation of dislocation evolution equation.

2 The numerical scheme to solve the coupling process is described in chapter 4 followed by numerical results and analysis in chapter 5, where the recovery process of a plastically distorted cubic crystal is simulated. In the end, this thesis ends up with a summary of current progress and an outlook of future work in chapter 6.

3 CHAPTER 2

DISCRETE DISLOCATION DYNAMICS MODEL

Dislocations ﬁrst appeared as an abstract mathematical concept rather as an experimental observation although they are physically one type of defects in crystal materials. The existence of dislocations was proved by transmission electron microscopy experiments after about 20 years since the proposition of the concept. Starting from the early of 20th century an overwhelming number of observations have been made, which in summary provide un- equivocal evidence of the existence of dislocations in crystals. There is such an abundance of these observations in literatures that readers are referred to reference [24] and [15] to see photos of dislocation bundles. This chapter provides a brief introduction to the basics of dislocations from the discrete point of view.

2.1 Basic Concepts of Dislocations

Crystal lattice defects are usually classiﬁed according to their dimensions: 0-dimension defects, 1-dimensional defects, 2-dimensional defects and 3-dimensional defects. 1-dimensional defect is often referred to as dislocation, on which this thesis will mostly focused. Two types of dislocations that have been well studied are the edge and screw dislocations. The sim- plest model of an edge dislocation can be constructed by inserting an extra half plane into the crystal, as shown in Figure 2.1a. An edge dislocation is denoted by or depending ⊥ ⊤ on where the extra half plane is. An equivalent way of creating the edge dislocation is to cut the crystal across the plane A and slip the upper half of the crystal uniformly relative to the lower half. This process leads to crystal distortion, and as a result, the atoms above and below the plane will feel compressive and tensile stresses respectively. This distortion, however, is signiﬁcant only in the neighbourhood of dislocation lines. Several lattice vectors away from the dislocation line, the crystal can be considered as perfect with minor level elastic deformation.

4 (a) edge dislocation pattern formation (b) screw dislocation pattern formation

Figure 2.1: edge dislocation and screw dislocation

A screw dislocation can be imagined by cutting a crystal along a plane and slipping one half across the other by a lattice vector which is perpendicular to cutting direction. The boundary of the cut is a screw dislocation(see Figure 2.1b). As a matter of fact, most dislocations are neither pure edge nor pure screw, but rather have mixed type between them. Pure edge and pure screw parts only appear at special positions of a general dislocation line. A mixed dislocation loop is shown in Figure 2.2, where a dashed line quarter loop represents a dislocation line that appears to be pure screw at A and pure edge at B.

Figure 2.2: mixed dislocation pattern formation

5 Since the dislocation line can also be thought of as the region of localized lattice disturbance separating the slipped and unslipped regions of a crystal[11], there should be a quantity that measures the extent of such disturbance. That quantity is called Burgers vector which is discussed in the following section.

2.2 Burgers Vector

Figure 2.3 shows Burgers circuit[10] that is illustrated for edge(on left side) and screw(on right side) dislocation, respectively. The circuit is drawn ﬁrst in a dislocated crystal as a closed atom-to-atom path. The same path is then reconstructed in a perfect crystal, where the closure failure of the circuit is deﬁned as Burgers vector(vector −−→QM in Figure 2.3)

Figure 2.3: determination of Burger’s vector

It is necessary to point out here that the Burgers vector remains constant along the same dislocation line. Thus in Figure 2.2 the Burgers vector b is a constant along the quarter loop.

2.3 Slip System and Dislocation Motion 2.3.1 Slip Systems Dislocations usually prefer to glide on a set of specific slip planes(denoted by ~n), which are defined according to the crystal structure as those highly closed-packed by atoms. With- in each slip plane, atoms prefer to move in the most closed packed directions(denoted by ~s), which defines the most possible Burgers vector directions for each slip plane. So there are 12 slip systems for face centred crystal(FCC, e.g. copper) and 48 slip systems for body centred crystal(BCC, e.g. α-Fe)[24]. It is to be noted here that none of the BCC slip planes

6 are truly close-packed, which means the slip systems need additional energy(i.e. heat) to operate. Namely, they are highly temperature-dependent. The theory of crystal plasticity, which is introduced in the next chapter, is built up upon the movement of dislocations on these slip systems.

2.3.2 Dislocation Motion Dislocation motion on its slip plane results in relative slip between the crystal regions separated by the slip plane, which ultimately yields plastic distortion of the crystal. Re- solved shear stress component on the slip plane is the only component that contributes to the movement of dislocations. The normal components of the the stress in the slip plane system act perpendicular to the glide plane and thus will not contribute to the dislocation motion. The resolved shear stress is a scalar evaluated as:

τRSS = σ ~n ~b (2.1) · · where, ~n denotes the normal vector of glide plane and ~b represents the unit Burgers vector. Many models are proposed to determinate the velocity from resolved shear stress τRSS, among which a linear relationship is most commonly used:

v = BτRSS where, velocity is assumed to be linearly dependent on τRSS and the coeﬃcient B is called drag coeﬃcient[57]. The resolved shear stress causes a dislocation line to move perpendicular to its line direction ~l: ~v = ~n ~l × where, ~v is velocity direction of the oriented segment ~l. So, the glide velocity vector is expressed as: v = B σ ~n ~b ~n ~l · · ×

7 CHAPTER 3

CONTINUUM THEORY OF DISLOCATION

In this chapter, dislocations are reviewed from continuum theory of mechanics which char- acterizes them as a result of plastic distortion incompatibility, where Kroner’s mathematical convention is adopted[29]. Crystal plasticity theory and dislocation plasticity theory are introduced accordingly, where the former is used to produce initial plastic distortion and the later is used to construct the evolution equation of dislocation density tensor. In another word, this chapter is based on ﬁeld theory and forms a continuum rewriting of the previous chapter.

3.1 Crystal Plasticity

Crystal Plasticity is initially proposed by Kratochvil[27], Green and Naghdi[19], Casey and Naghdi[7], Loret[33], Dafalias[8], Boyce[6], among others, which begins with the kinematics of a deformed crystal. The kinematic representation of elastoplastic deformation of single crystals, in which crystallographic slip is assumed to be the only mechanism of plastic deformation , is shown in Figure 3.1. The crystal material experiences plastic shear via dislocation motion(stage A to stage B in Figure 3.1), and then undergoes elastic deformation via extension and rotation of the lattice(stage B to stage C in Figure 3.1). The plastic deformation is considered to occur in the form of smooth shearing on the slip planes and in the slip directions. In the following statement, the configuration that has only experienced plastic deformation caused by dislocation slip is called intermediate configuration(B in Figure 3.1) ; the configuration that has both experienced plastic and elastic deformation is called current configuration(C in Figure 3.1); the very beginning configuration is called undeformed configuration.

8 Figure 3.1: kinematics of crystal plasticity

The total deformation gradient F is decomposed as:

F = F e F p (3.1) · where F p is the part of F due to slip only, while F e is due to lattice elastic stretching and rotation. Denote the unit vector normal to the slip plane and the unit vector in the α α slip direction in the undeformed configuration by ~n0 and ~s0 , where α designates the slip system index(in the case of FCC, α = 1, 2,..., 12). These two original unit vectors do not change at the intermediate configuration because the slip of dislocations does not introduce α any extension or rotation. However, at the current configuration, the vector ~s0 is changed α e α to ~s = F ~s0 . The normal to the slip plane in the current configuration should still · α α α e 1 be orthogonal to slip direction vector ~s , so it is given by ~n = ~n F − . Generally, 0 · ~sα and ~nαare not unit vectors. According to equation (3.1), the velocity gradient tensor 1 L = F˙ F − can be expressed as ·

e e p p 1 e 1 L = L + F F˙ F − F − (3.2) · · · where Le is the lattice elastic velocity gradient tensor:

e e e 1 L = F˙ F − · and the second term in equation (3.2) is denoted as Lp:

p e p p 1 e 1 L = F F˙ F − F − (3.3) · · ·

9 It is easy to prove that the velocity gradient in the intermediate conﬁguration is produced by the slip ratesγ ˙ α on n active slip systems, such that: n p p 1 α α α F˙ F − = γ˙ ~n ~s (3.4) · 0 ⊗ 0 α=1 X substituting with equation (3.4), equation (3.2) can be written as:

n L = Le + γ˙ α~nα ~sα (3.5) ⊗ α=1 X p p p 1 The velocity gradient tensor at current configuration L is different from F˙ F − · which is the velocity gradient tensor at intermediate configuration. The former tensor is α α α α related to ~n and ~s while the later one is related to ~n0 and ~s0 . Equation (3.5) is an important conclusion in crystal plasticity, which connects the microscopic slip quantityγ ˙ with macroscopic rate of deformation and will be employed in the next section.

3.2 Dislocation Plasticity Theory Classical continuum mechanics assumes that the material is continuous such that there is no defects inside the material. Hence, every point has a unique position in three dimensional Euclidean space after deformation. Under this assumption, distortion tensor β is expressed as a gradient of displacement u: β = u (3.6) ∇ where vector calculus enables us to write:

β = 0 (3.7) ∇× The above equation (3.7) is ﬁrst order diﬀerential condition expressing the compatibility of deformation. The integral form of this compatibility condition reads:

du = 0 (3.8) IC which means that the displacement ﬁeld u is a single-valued diﬀerentiable function of coordinates. However, incompatibility occurs inevitably if there are defects within the material. Mathematically speaking, equation (3.7) and equation (3.8) do not hold any longer in the presence of dislocation. Equation (3.8) turns out to be[29]:

b = du (3.9) − IC which is exactly the deﬁnition of Burgers vector in a continuous way. In equation (3.9), du is not a total diﬀerentiation because u is not a single-valued function of the coordinates[31]. Assuming du can be expressed as a function of dx mapped by a second order tensor βp:

du = dx βp (3.10) ·

10 Substituting equation (3.10) into equation (3.9) and applying Green Theorem on the contour integral, it immediately follows that:

b = du − IC = dx βp · IC = dS ( βp) · ∇× ZZSC which gives the deﬁnition of dislocation density tensor α:

α = βp (3.11) −∇ × Accordingly, Burgers vector can be expressed in terms of this tensor as:

b = dS α (3.12) · ZZSC In terms of Nye’s point of view[36], Burgers vector of dislocations on an inﬁnitesimal plane is denoted by ∆b and it is assumed that the dislocation density tensor α maps the oriented inﬁnitesimal plane ∆S to Burgers vector in the way that:

∆bj = αij∆Si (3.13) which is the diﬀerential form of equation (3.12) . The dislocation density tensor αij in equation (3.13) is resulted from a tensor product of ρ and b:

α = ρ b (3.14) ⊗ where, ρ is dislocation density vector[36] whose magnitude is the number of dislocation lines passing through unit area normal to its direction ~l, b is Burgers vector of each dislocation line. The comparison between equation (3.12) and equation (3.13) reveals the fact that Kroner’s and Nye’s dislocation density tensors express the same physical meaning but are deﬁned in diﬀerent ways. With the introduction of plastic distortion tensor βp, the total distortion tensor β can be expressed as a summation of elastic distortion βe and plastic distortion tensorβp:

β = βe + βp where the compatibility condition (3.7) still applies to β but it does not apply to βe or βp. This expresses the basic idea underlying the dislocation plasticity theory: the pure plastic deformation that takes place from undeformed configuration to intermediate configuration is incompatible; the pure elastic deformation that takes place from intermediate configuration to current configuration is also incompatible; the superposition of such two kinds of deformation, however, is compatible. In Kroner’s dislocation plasticity theory, the plastic strain tensor ǫp and plastic distortion tensor βp is assumed to be known. But the way to separate plastic part from the total deformation is not specified in his theory. However, the

11 crystal plasticity theory discussed in the previous section provides an eﬀective approach. Under the small deformation assumption, we appeal to equation (3.5) and eventually have the expression for plastic distortion tensor: n p β˙ = γ˙ α~nα ~sα (3.15) ⊗ α=1 X and: n βp = γα~nα ~sα (3.16) ⊗ α=1 X where, γα is referred to as scalar shear strain[22]. Therefore, Kroner’s dislocation plasticity theory that provides a continuous description of dislocation forms a complete theory accompanied with crystal plasticity.

3.3 Dislocation Evolution Equation Equation (3.11) relates the dislocation density tensor to the plastic distortion βp, but it does not give the law that governs the evolution of α itself. Fortunately, Orowan’s law provides an expression of α in terms of βp: p β˙ = v α (3.17) − × Here, v is the velocity ﬁeld which is perpendicular to the dislocation line direction ~l every- where on the slip plane. Focusing on the small deformation case and taking time derivative of equation (3.11), it gives that: p α˙ = β˙ (3.18) −∇ × At this moment, the equation (3.17) and equation (3.18) can be combined to produce the evolution equation for α: α˙ = (v α) (3.19) ∇× × From the perspective of Nye’s point of view, α can be regarded as a tensor product of ρ and b: αij = ρibj which leads to the drop of b on both sides of equation (3.19):

ρ˙i = eikl∂k (elnmvnρm) , or ρ˙ = (v ρ) (3.20) ∇× × Employing ǫ δ relationship, the above equation can be simpliﬁed to the following form: − ρ˙i = eikl∂k (elnmvnρm)

= elkielnm∂k (vnρm) − = (δkmδin δknδim) ∂k (vnρm) − = δkmδin∂k (vnρm) δknδim∂k (vnρm) − = ∂m (viρm) ∂n (vnρi) −

12 Equation (3.20) is evolution equation for vector ρ, which is referred to as curl equation for simplicity. Under a given velocity ﬁeld, which comes from resolved shear stress, equation (3.20) can be solved, enabling an update of βp through equation (3.17).

13 CHAPTER 4

NUMERICAL SCHEME FOR SIMULATION

There are two essential partial differential equations to be solved in the theory of elastoplasticity based on dislocation dynamics. One of them is equilibrium equation that accounts for plastic strain and another one is dislocation evolution equation that takes velocity field as a parameter which is supposed to be obtained from equilibrium equation. The equilibrium equation is solved by Galerkin finite element method while the dislocation evolution equation is solved by Least Square Finite element method(LSFEM) and first order upwind scheme.

4.1 Numerical Scheme for Eigenstrain Problem The equilibrium equation for linearly elastic solid reads:

σ = 0 ∇· where σ stands for stress tensor ﬁeld which is related to the elastic strain by Hooke’s law[43]:

σ = C : εe (4.1)

Considering εp as a part of total strain, equation (4.1) can be written as:

[C :(ε εP )] = 0 (4.2) ∇· − In equation (4.2), plastic strain εp is treated as a known eigenstrain ﬁeld[35] introduced by scalar shear strain γ: 1 εp = γα (sα nα + nα sα) α 2 ⊗ ⊗ X 1 = (βp)T +(βp) (4.3) 2 The detailed FEM procedures to solve equation (4.1) can be found in reference [25], with only additional term of plastic eigenstrain[35] that appears in equation (4.2). Ultimately, the weak of equation (4.2) can be written as:

T T p [B] [C][B]dΩ[µ]= [N][t]dΓσ + [B] Cε dΩ (4.4) ZΩ ZΓσ ZΩ

14 where, ∂N ∂N ∂N α 0 0 0 α α ∂x ∂z ∂y T  ∂Nα ∂Nα ∂Nα  [B]α = 0 0 0  ∂y ∂z ∂x   ∂N ∂N ∂N   0 0 α α α 0     ∂z ∂y ∂x    and [B]=[B , B , , BN ] . Here, Nf is the total number of nodes and Nα is the shape 1 2 ··· f function for node α. Equation (4.4) reveals that the plastic strain eﬀect on the body is essentially a distributed body force associated with elements that contain plastic distortion domain.

4.2 Numerical Scheme for Curl Equation

The discretization of curl equation (3.20) is divided into two processes: First, discretization in time dimension by appropriate finite difference scheme(implicit Euler or explicit Euler for example); Second, discretization in space using finite element method/finite difference method. In this thesis, only one slip plane is considered, so a 2D version of the curl equation needs to be solved, which is written in index form as:

ρ˙i = eikl∂k (elnmvnρm) (4.5)

The above index form represents three equations:

ρ˙ = ∂ (v ρ ) ∂ (v ρ )+ ∂ (v ρ ) ∂ (v ρ ) 1 2 1 2 − 2 2 1 3 1 3 − 3 3 1 ρ˙ = ∂ (v ρ ) ∂ (v ρ )+ ∂ (v ρ ) ∂ (v ρ ) 2 1 2 1 − 1 1 2 3 2 3 − 3 3 2 ρ˙ = ∂ (v ρ ) ∂ (v ρ )+ ∂ (v ρ ) ∂ (v ρ ) (4.6) 3 1 3 1 − 1 1 3 2 3 2 − 2 2 3 which is simpliﬁed in 2D as:

ρ˙ = ∂ (v ρ ) ∂ (v ρ ) 1 2 1 2 − 2 2 1 ρ˙ = ∂ (v ρ ) ∂ (v ρ ) (4.7) 2 1 2 1 − 1 1 2 Equation (4.7) is essentially a non-linear equation as v is always perpendicular to ρ. To avoid such non-linearity, the velocity vn+1 that couples with ρn+1 is taken to be the same as vn, which means that an implicit time scheme is actually a semi-implicit method. n+1 Furthermore, an implicit time scheme will require a separation of the term [∂i (vjρk)] n+1 n+1 n+1 into ρk ∂ivj + vj∂iρk in order to spawn a set of linear equations about ρi ; an explicit method, on the contrary, just takes ∂i (vjρk) as a whole and uses this space derivative information at time n∆t to update ρ.

4.2.1 Least Square Finite Element Method(LSFEM) Least square ﬁnite element method uses backward Euler method in time and least square approximation in space.

15 Generally speaking, we are considering three PDEs:

ρ˙ = ∂ (v ρ ) ∂ (v ρ ) 1 2 1 2 − 2 2 1 ρ˙ = ∂ (v ρ ) ∂ (v ρ ) 2 1 2 1 − 1 1 2 0 = ∂1ρ1 + ∂2ρ2 (4.8)

The third divergence-free condition in equation (4.8) serves as a complementary gauge condition that simply denotes the divergence of a curl ﬁeld is equal to zero. LSFEM has the ability to handle two unknowns appearing in three equations. For simplicity, equation (4.8) is rewritten in matrix form:

ρ˙1 ∂ ρ1 ∂ ρ1 ρ1 At = Ax + Ay + A (4.9) ρ˙ ∂x ρ ∂y ρ 0 ρ 2 2 2 2 where the coeﬃcient matrix A are all of dimension 3 2, and they are given by: • × 1 0 0 0 v v − 2 1 At = 0 1 ,Ax = v v ,Ay = 0 0    2 − 1    0 0 1 0 0 1       ∂ v ∂ v − 2 2 2 1 A = ∂ v ∂ v 0  1 2 − 1 1  0 0   Equation (4.9) is ﬁrstly discretized in time dimension:

2 n n ∂ n+1 n ∆tA0 At +∆t Ak u = Atu (4.10) − ∂xk − " k # X=1 The abstract form of equation (4.10) can be written as the following expression:

Ln(un+1)= pn (4.11) where,

2 n n+1 n n ∂ n+1 L (u ) = At +∆tA0 +∆t Ak u  − ∂xk  " k=1 #  3 2 3 1 X × × pn = [A ] un  t 3 2 3 1  − × × The least-square weak form of diﬀerential equation (4.11) yields that:

Ln(vn+1),Ln(un+1) = Ln(vn+1),pn

Here,v,u L (Ω), and the inner product of vector function (x,y) is deﬁned as: ∈ 2 (x,y)= xT ydΩ ZΩ

16 It is worth to mention that the counterpart Galerkin weighted residual method is vn+1,Ln(un+1) = vn+1,pn , however it does not work if the number of equations is more than the number of variables. Now, the discretization in space is implemented as following: The vector function un+1 (and vn+1) at any point can be discretized as:

m ρn+1 ρn+1 un+1 = 1 = N 1 ρn+1 i ρn+1 2 i 2 i X=1 where m denotes the total number of nodes and Ni is shape function for note i. Substitute un+1 and vn+1 into the weak form:

Ln(vn+1),Ln(un+1) = Ln(vn+1),pn and it gives: n n+1 n n+1 n n+1 n L (Nivi ),L (Njuj ) = L (Nkvk ),p where vn+1 is an arbitrary function that can be dropped from the above equation, so:

n n n+1 n n (L (Ni),L (Nj)) uj =(L (Nk),p ) which can be written as:

n T n n+1 n T n (L N)2m 3 (L N)3 2m dΩ U 2m 1 = (L N)2m 3 (p )3 2m dΩ ZΩ × × × ZΩ × × which gives the global equation for gauge conditioned curl equation:

KU = P

If velocity ﬁeld changes in the process of simulation, the K matrix should be refreshed and assembled in each time step.

4.2.2 Finite Difference Method A forward Euler scheme is employed here for numerical time integration of the curl equation. Hence, only values at time n∆t are needed to compute the new values at time (n + 1)∆t. The forward Euler time integration scheme is an unstable scheme which is only first order accurate and suffers from dispersion. Except for its programming simplicity and computing economy, The main reason for using such scheme is: the implicit method requires ∂(vρ) ∂ρ ∂v the separation of into v + ρ while this separation does not necessarily hold in ∂x ∂x ∂x the presence of oscillation of ρ. Because of the advection property of the equation (4.7) spatial derivatives which govern advection are approximated by an upwind scheme. This numerical scheme uses information about the flow direction to determine whether to use a forward or backward difference stencil, such that only information from upstream of the flow is used. The upwind method based on first order forward and backward differences is oscillation-free but at the cost of a very strong numerical diffusion effect while, on the other hand, the second order based upwind scheme is of much better quality than the first

17 order scheme from perspective of numerical diffusion but suffers from slight oscillation[51]. Oscillatory dislocation density vector will cause the oscillation in velocity field which will make the matter worse. We therefore use the first order upwind scheme. The discretization of equation (4.7) is given below as: update ρ : • 1 n+1 n+1 ∆t n n ρ1,ij = ρ1,ij + ∆y (v1ρ2 v2ρ1)i,j (v1ρ2 v2ρ1)i,j 1 if v2 > 0.0 − − − − ρn+1 = ρn+1 + ∆t h(v ρ v ρ )n (v ρ v ρ )n i if v < 0.0  1,ij 1,ij ∆y 1 2 − 2 1 i,j+1 − 1 2 − 2 1 i,j 2 h i update ρ : • 2 n+1 n+1 ∆t n n ρ2,ij = ρ2,ij + ∆x (v2ρ1 v1ρ2)i,j (v2ρ1 v1ρ2)i 1,j if v1 > 0.0 − − − − ρn+1 = ρn+1 + ∆t h(v ρ v ρ )n (v ρ v ρ )n i if v < 0.0  2,ij 2,ij ∆x 2 1 − 1 2 i+1,j − 2 1 − 1 2 i,j 1 h i  4.3 Coupled Scheme for Stress-Dislocation Simulation The movement of dislocations within the crystal will give rise to the change of plastic p distortion tensor field due to Orowan’s law: β˙ = α˙ . As seen from equation (4.3) −∇ × and (4.1). This change in the plastic distortion tensor will contribute to the change in the stress field which in turn governs the motion of dislocations due to the resolved shear stress effect on the slip plane. In other words, the interaction between dislocation density field and stress field forms a coupled process in which the microscopic quantity and macroscopic quantity influence each other. Mathematically, The elastoplastic behaviour of a crystal is a result of this coupled interaction. This coupled process involves two partial differential equations, one ordinary equation and one algebraic equation, all of which are listed below:

σ = 0 (4.12) ∇· ρ˙ = (v ρ) (4.13) ∇× × v = B σ ~n ~b ~n ~l (4.14) · · × p β˙ = α˙ (4.15) −∇ × The equation (4.12) and equation (4.13) govern the macroscopic and microscopic quantity respectively. Equation (4.14) and equation (4.15) serve as bridges that connect the miscro- process and macro-process. Corresponding to the above equations, the numerical scheme is mainly divided into four procedures in each time step: 1. Given a known plastic strain field, equation (4.12) can be solved by Galerkin finite element method which directly gives the solution for displacement. The stress at integral points can be obtained from displacement field. This procedure is done in three dimension cubic crystal and the stress field is computed at grid on slip plane; 2. The velocity field on slip plane is computed from equation (4.14). This velocity field is associated with dislocation density vector and is used as motivation to drive the dislocations to evolute;

18 3. The evolution process is controlled by equation (4.13) which is a hyperbolic equation and has advective characteristics;

4. Orowan’s law expressed in equation (4.15) is used to update plastic distortion tensor βp which in turn contributes to equation (4.12).

Figure 4.1 shows the process of this coupled scheme.

Figure 4.1: coupled scheme for the simulation of dislocation evolution

19 CHAPTER 5

NUMERICAL RESULTS

In this chapter, we firstly investigate two test cases, each of which is solving single PDE without coupling. The first case is the solution of equilibrium equation under a given plastic distortion tensor field. The second case treats the velocity field to be known and solves the curl equation. Finally, the numerical techniques used in two cases are both used in solving coupled process that causes a dislocated crystal to recover partly to its original state.

5.1 Test Problem for Equilibrium Equation The solution of plastic eigenstrain problem[14] described in equation (4.2) yields a de- sired stress ﬁeld that will be used in curl equation (3.20). Similar idea was proposed by Lemarchand[32] and solved by Roy and Acharya[42] using ﬁnite element method, which is going to be resolved here as a preliminary step for coupling. Suppose a cubic crystal is deformed in the way that the body is cut half through and the upper part is extended in the direction AB~ forming an edge dislocation AD in the middle of the cube(shown in Figure 5.1).

Figure 5.1: formation of a single edge dislocation

Here, the edge dislocation AD is characterized by dislocation density ﬁeld: ρ = (0,δ(x x )δ(z z ), 0)T (5.1) − 0 − 0

20 where δ(x) is Dirac delta function which is emulated by Gaussian distribution for practical reason. On the other hand, it is seen from Kroner’s definition of dislocation density tensor that plastic distortion is spatial integral of ρ. Hence, it is reasonable to define the scalar shear strain γ in the form of an error function about x: γ = γ Erf(x x ) H(z z ) (5.2) 0 − 0 − 0 where, function H(z) is defined as 0 z >h H(z)= | | (1 z

Figure 5.2: γ ﬁeld distribution

The plastic strain tensor can be obtained by: βp = γ~n ~s and εp = sym(βp) since there ⊗ is only one slip system considered for this problem. The crystal material was chosen to be copper(E = 117MPa,ν = 0.34), where the Burgers vector is taken to be 2.56e 10m. −

21 The entire stiﬀness matrix formed from Galerkin weighted residual method has 20577 × 20577 elements which is stored in element-by-element way. The system is solved by precon- ditioned conjugate gradient method[49]. The shape of distorted crystal is plotted in Figure 5.3 which turns out to be a continuous representation of Figure 5.1.

Figure 5.3: distorted shape due to an edge dialocation

The stress ﬁeld around an edge dislocation in an inﬁnite medium is analytically given by[24], which is singular at point x = y = 0:

Gb z(3x2 + z2) σxx = 2π(1 ν) (x2 + z2)2 − Gb z(x2 z2) σzz = − −2π(1 ν) (x2 + z2)2 − Gb x(x2 z2) σxz = − −2π(1 ν) (x2 + z2)2 − Gbν z σyy = −π(1 ν) x2 + z2 − σxy = 0

σzy = 0 (5.3)

22 Figure 5.4, Figure 5.5, and Figure 5.6 show the stress tensor components: σxx,σzz,σxz compared with analytical solution in equation (5.3), respectively. Due to the singularity of equation (5.3), the stress value at (0, 0) for analytical solution is artiﬁcially set to be zero. The displayed cross-section plane is parallel to x z plane with its position in the middle − of y axis.

(a) numerical solution in ﬁnite medium (b) analytical solution in inﬁnite medium

Figure 5.4: σxx component ﬁeld

(a) Numerical solution in ﬁnite medium (b) analytical solution in inﬁnite medium

Figure 5.5: σzz component ﬁeld

23 (a) numerical solution in ﬁnite medium (b) analytical solution in inﬁnite medium

Figure 5.6: σxz component ﬁeld

The comparison between numerical solution and analytical solution illustrates that the stress ﬁeld distribution caused by plastic eigenstrain in term of equation (5.2) emulates the one that is introduced by a single edge dislocation in an inﬁnite medium.

5.2 Test Problem for Curl Equation

The physical dislocation lines are represented in the form of continuous vector field ρ(x,t) which depends on space and time. The test examples run in this section presents the time evolution process of initially prescribed ρ fields under different circumstances. The velocity orientation is always perpendicular to ρ during the evolution process. Least square finite element method is employed firstly to solve this equation, yielding predictions of dislocation motion. Different numerical methods are also compared with each other, in- cluding first order and second order upwind scheme as well as Lax-Wendroff finite difference method[41].

5.2.1 Simulation Results Obtained by LSFEM The ﬁrst example is simply a straight dislocation moved by an evenly distributed velocity ﬁeld on a slip plane. Assign the initial Gaussian distribution function as:

ρ1(x, 0) = 0 2 1 (x x0) ρ2(x, 0) = 2 exp( − 2 ) ( √2πσ − 2σ The magnitude of velocity is taken to be 5m/s and the ρ ﬁeld moves accordingly as shown in Figure 5.7

24 (a) t =0 (b) t =3.6e − 8s

Figure 5.7: the movement of a straight dislocation line

Figure 5.7 reveals that as the dislocation line moves, the Gaussian distribution is gradually smeared out, which is ascribed to the implicit Euler scheme used here. However, the position of the peak of Gaussian distribution, where there is a bundle of dislocations, is accurately captured and there is no oscillation observed. If the dislocation line is tilted then some appropriate boundary condition should be taken into account. Otherwise, the curl equation does not apply to boundary and thus results in unrealistic value of ρ. For this test case, an advective boundary condition is adopted: the physical quantity moves along the boundary at a speed faster than the velocity evaluated according to body velocity formula so that the quantity remains compatible near the boundary. This boundary velocity is set to be:

v vb = ~t ~v ~t · where, v is the magnitude of body velocity(it is equal to 5m/s in our case), ~v and ~t are direction of body velocity and boundary tangent, respectively. The evolution equation for boundary quantity ρ is given by standard advective equation:

ρ˙ +(vb )ρ = 0 (5.4) ·∇

Equation (5.4) can be numerically taken into account by adding a boundary term to the weak form[5]. Figure 5.8 shows a tilt dislocation line sweeping through the slip plane.

25 (a) t =0 (b) t =1.2e − 8s

Figure 5.8: movement of a tilt dislocation line

The dislocation line seen from Figure 5.8. is naturally elongated by curl equation, which reveals that the equation does not only have advective effect upon a dislocation line but also an extensional effect. This extension effect can be clearly seen in the following example shown in Figure 5.9, where a dislocation loop expands toward the boundary.

(a) t =0 (b) t =1.0e − 8s

Figure 5.9: loop expansion

26 If the magnitude of velocity around the boundary is tuned to be zero, a “velocity wall” is formed. As soon as the dislocation loop hits this wall, it will pile up there and ﬁts its shape to the the boundary, as shown in Figure 5.10.

(a) t =0.85e − 8s (b) t =1.5e − 10

Figure 5.10: pileup process of a loop

All of the above numerical examples show that dislocation curl equation can be solved by LSFEM with backward Euler time scheme. Other schemes(such as Lax-Wendroﬀ scheme, upwind scheme) are going to be compared in the next section.

5.2.2 Comparison between Diﬀerent Numerical Schemes for Curl Equation

In the context of advective equation, first order upwind scheme is known for stability but large diffusion while second order upwind scheme and Lax-Wendroff scheme has less numerical diffusion but defective in oscillation[34, 46]. Hereby, we conduct numerical testing for curl equation to ensure that we choose the appropriate numerical scheme for coupled problem.

27 (a) solution by LSFEM (b) solution by ﬁrst order upwind scheme

Figure 5.11: curl equation solved by four distinct numerical methods

Figure 5.11 shows that LSFEM(a) and first order upwind scheme(b) are free of oscillation while second order upwind scheme(c) and Lax-Wendroff scheme(d) have disturbed dislocation density vector near the Gaussian peak. As discussed before, the scheme that is oscillation free but has numerical diffusion is preferred over the scheme that is better in numerical diffusion but has oscillation problem. Based on this preference, LSFEM and upwind finite element scheme are chosen as candidates to be used in the stress-dislocation coupled example shown in the next section.

28 5.3 Test Problem for Stress-Dislocation Coupled Scheme

The numerical scheme proposed in section 4.3 is implemented to simulate the process that a dislocated cubic crystal attempts to recover to its undeformed state driven by self- produced stress ﬁeld. An edge dislocation line is initially created close to the right-side of the crystal where it will be absorbed to . In this way, the wedge showing up in Figure 5.3. will disappear gradually and the whole crystal will tend to stress free state. Replacing Dirac delta function with Gaussian distribution in equation (5.1), ρ2 ﬁeld can be written as:

2 2 1 (x x0) 1 (z z0) ρ2(x, y, z)= exp − exp − √ 2 2 − 2σ2L2 · √ 2 2 − 2σ2L2 2πσ L 2πσ L where, L is element size. The integral of this initial distribution on x z plane is equal to − one, showing that there is only one dislocation passing through. Figure 5.12 shows the initial stress ﬁeld(a) and its resolved shear stress on the slip plane(b). The arrows plotted in Figure 5.12b denote the velocity caused by the component σ13 of this stress ﬁeld.

(a) isotropic surface of stress ﬁeld σ13 (b) stress ﬁeld component σ13 on slip plane

Figure 5.12: stress ﬁeld σ13 in 3D crystal and 2D slip plane

It can be seen from Figure 5.12b that dislocation line will be moving to the positive x direction by the velocity associated with it. We can also observe that the velocity vector ﬁelds on the two sides of the dislocation line are opposite to each other, which reveals the physical fact that the Gaussian distribution that we use to replace Dirac delta function can only represents a bundle of dislocation lines which have the tendency to repel themselves due to the same Burgers vector they share. Choosing the least square ﬁnite element method as

29 our numerical scheme for curl equation, Figure 5.13 shows the dislocation evolution process on the x y slip plane. −

(a) t =0 (b) t =0.5e − 11s

Figure 5.13: dislocations absorbed into the boundary

As expectation, the dislocations are drawn to the boundary of the crystal by the self- produced stress field. However, there are two problems seen in Figure 5.13: the diffusion of ρ and oscillation of velocity field in the end. Except for numerical diffusion that we noted in section 5.2.1, the repellent velocity field discussed above also tries to smear the dislocation lines out. In another word, some dislocation lines will move towards crystal interior. The

30 oscillatory velocity field at the end of the process originates from slightly oscillating ρ field, which gives rise to a chaos of velocity direction(see Figure 5.13d). Replacing LSFEM with first order upwind scheme, Figure 5.14 shows the same process of an absorbed dislocation line.

(a) t =0 (b) t =0.5e − 11s

Figure 5.14: dislocations absorbed into the boundary(given by upwind scheme)

Comparing each sub-figure in Figure 5.14 and Figure 5.13 respectively, Upwind explicit scheme behaves better over LSFEM in oscillation but worse in numerical diffusion. However, because it is the oscillation that causes the trouble rather than diffusion, upwind explicit scheme is preferred in this coupled scheme. Figure 5.15 shows the initial plastic distortion

31 recedes as the straight dislocation line sweeps though the distorted area, which makes the crystal recover to its original state to some extent(shown in Figure 5.16).

(a) initial plastic distortion β31 (b) plastic distortion β31 in the end

Figure 5.15: Plastic distortion component β31 recedes.

(a) a distorted crystal (b) crystal shape in the end

Figure 5.16: A distorted crystal partly recovers to its original state where the cloud denotes the magnitude of displacement.

32 Figure 5.15 and Figure 5.16 show the recovery process of a distorted crystal but still with some residual plastic distortion. This is partly due to the residual dislocations that live within the crystal in the end. Up to now, we were able to couple stress ﬁeld and dislocation evolution process together to simulate the elastoplastic behaviour of a crystal according to the underlying physical principles rather than the phenomenological laws.

33 CHAPTER 6

SUMMARY AND OUTLOOK

This thesis is based on continuum dislocation theory , which spawns PDEs that describe the microscopic and macroscopic behaviour of a crystal body. The main work done in current thesis is summarized below:

1. The stress field in equilibrium equation is solved by Galerkin finite element method under the assumption that the plastic distortion is already known within the crystal body. This method shows to be very effective by comparing with analytical solution.

2. The 3D evolution equation that governs the movement of dislocation lines is deduced based on the continuum deﬁnition of Burgers vector. These equations can be simpliﬁed to 2D version in the case of only one slip plane considered.

3. Least square finite element method, which proves to be effective in some advective equations, is formulated here for the evolution equation. Several examples are tested to check the movement of dislocations in different circumstances. Other finite difference methods are also considered and compared with LSFEM.

4. Based on the continuum theory of dislocation, a coupling scheme for simulating crystal behaviour is proposed and successfully tested by a simple example. The numerical scheme for curl equation that is oscillation free but has some numerical diffusion(e.g. upwind finite difference method) is more reliable than the one that has better numerical diffusion but is slightly oscillatory(e.g. least square implicit method). Both methods can show macroscopic recovery process of a distorted crystal.

Even though upwind scheme works stably in the current test example, its numerical diffusion is very obvious, which certainly causes computational errors . On the other hand, LSFEM behaves some oscillatory effect in the coupling process but its numerical diffusion effect is better off. Hence, if we want to improve the precision in future, some advanced numerical techniques are required. It is currently assumed here that every dislocation line is geometrically necessary[17] and contributes to the crystalline behaviour. However, a real crystal is always filled with numerous bundles of dislocation lines orienting at various directions rather than just one or several dislocations simulated currently. Statistical model should naturally show up and help here in order to do real simulations in future.

34 In a word, dislocation dynamics described in the continuum way is an interesting, promising and challenging research area which requires many pure and applied mathematical tools, and also provides a bright road to simulate crystal mechanical response instead of phenomenological constitutive relationships.

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39 BIOGRAPHICAL SKETCH

The author was born in Anhui province of China. The educational experience of him is listed below: Florida State University, Tallahasee, United States Master of Science in Scientiﬁc Computing, 2011 Thesis: Numerical Implementation of Continuum Dislocation Theory Hohai University, Nanjing, China Master of Science in Engineering Mechanics, 2009 Hohai University, Nanjing, China Bachelor of Science in Engineering Mechanics, 2007