# Dislocation correlations and the continuum dynamics of the weak line bundle
ensemble

Joseph Pierre Anderson  School of Materials Engineering, Purdue University,
West Lafayette, IN 47907, USA.
(Currently with Department of Physics and Astronomy, Carthage College,
Kenosha, WI 53140, USA.)  Anter El-Azab  School of Materials Engineering,
Purdue University, West Lafayette, IN 47907, USA.

###### Abstract

Progress toward a first-principles theory of plasticity and work-hardening is
currently impeded by an insufficient description of dislocation kinetics, that
is, of the dynamic effect of driving forces in a given dislocation theory.
Current continuum theories of dislocation kinetics are often incapable of
treating the short-range interaction of dislocations. This work presents a
kinetic theory of continuum dislocation dynamics in a vector density framework
which takes into account the short-range interactions by means of suitably
defined correlation functions. The weak line bundle ensemble of dislocations
is defined, whereby the treatment of dislocations by a vector density is
justified. It is then found by direct averaging of the dislocation transport
equation that additional driving forces arise which are dependent on the
dislocation correlations. A combination of spatial coarse-graining and
statistical averaging of discrete dislocation systems is then used to evaluate
the various classes of tensorial dislocation correlations which arise in the
line bundle kinetic theory. A novel, chiral classification of slip system
interactions in FCC crystals is introduced in order to define proper and
improper rotations by which correlation functions corresponding to six
interaction classifications can be evaluated. The full set of these six
dislocation correlations are evaluated from discrete data. Only the self-
correlations (for densities of like slip system) are found to be highly
anisotropic. All six classes of correlation functions are found to be of
moderate range, decaying within 2-4 times the coarse-graining distance. The
correlations corresponding to the coplanar interactions are found to be
negligible. Implications of the evaluated correlations for the implementation
of vector density continuum dislocation dynamics are discussed, especially in
terms of an additional correlation component of the driving force and a
gesture toward a coarse-grained dislocation mobility.

##  I Introduction

Since the discovery of dislocations as the mediators of plastic deformation
almost a century ago [1, 2, 3], dislocation theory has been an important topic
of interest in physics, mechanics, and materials science. Due to the role of
dislocations in determining the plastic strength of metals and alloys, a major
part of the theory of dislocations was quickly developed, focusing on the
mechanics [4], interactions [5], and reactions of individual dislocations [6]
in various crystal structures. However, the initial question that dislocations
were posited to answer remains unsolved even today: precisely _how_ do
dislocations give rise to work hardening in metals? The development of a
first-principles theory of work hardening is still underway because it is a
problem of the collective motion of dislocations. Collective motion always
involves a certain set of complexities, but dislocation theory comes with its
own peculiar difficulties.

To begin to sort out the types of difficulties faced in treating the
collective dynamics of dislocations, let us first distinguish between
dislocation _kinematics_ and _kinetics_ [7, 8]. In treating a collection of
dislocations, one often describes the collection by means of a reduced
representation. This commonly involves some set of density measures.
Constructing a theory of dislocation _kinematics_ involves defining these
density measures and their evolution under a prescribed velocity field. There
are several such theories which are appropriate for different purposes, as we
will discuss below. However, within a given kinematic framework, one will have
to define driving forces [5] and mobilities [9] whereby the velocity field can
be related to the density state. This closure is the problem of dislocation
_kinetics_. The fundamental question of both problems is one of scale.
Dislocations as line objects, as well as their interactions, are associated
with a fine scale near the lattice spacing (on the order of a few Burgers
vectors, or 1 nm). Descriptors of a collection of dislocations are necessarily
associated with a coarser scale, be it as small as a few tens of nm or as
large as several μm. The problem of dislocation kinematics is associated with
finding appropriate dislocation density measures which capture the relevant
dislocation information on the desired coarse scale [10]. The problem of
dislocation kinetics, which is the focus of the present work, is in defining
the short-range dislocation interactions—which naturally occur at the fine
scale—in terms in terms of the density measures at the coarse scale. After
situating our approach to the kinematic problem, we will then proceed to treat
dislocation kinetics for this particular kinematic framework.

Dislocations are topological defects in crystals [11]. A volume containing a
dislocation will be recognizable by means of the dislocation’s topological
invariant, analogous to a ‘charge’ of the dislocation. This topological
invariant is the dyad formed by its line tangent and Burgers vector. When this
is summed over all dislocations in a given volume, we obtain the Kröner-Nye
tensor [12, 13]. The long-range mechanical fields (and thus dislocation
interactions) are produced by the geometric incompatibility which this tensor
represents [10, 13]. However, the signedness of this measure is the
fundamental stumbling block of collective theories of dislocation motion: in
summing the topological invariant over a collection of dislocations multiple
kinds of cancellation can occur, essentially losing track of some portion of
the dislocation population in the volume [14]. The portion contributing to the
Kröner-Nye tensor is commonly referred to as the geometrically necessary
dislocation content (GND). Various kinematic theories define additional
density measures in order to account for the cancelled dislocation
content—commonly referred to as the statistically stored dislocation content
(SSD)—with various levels of precision.111While in this work we will refer to
geometrically necessary or statistically stored dislocations in the plural, we
remind the reader that these are net effects of a collection, and a
geometrically necessary or statistically stored character is not attributable
to a single dislocation.

We will discuss two different kinematic theories which treat SSDs in different
ways. While there are kinematic theories which model only the evolution of
GNDs [15, 16]., it is difficult to interpret the velocity associated with this
evolution in terms of the velocities of individual dislocations. As a result,
kinetic closure of such theories is performed by free-energy or
phenomenological arguments rather than coarse-graining arguments [17]. When
considering the problem of statistical storage of dislocations, however, a
certain distinction immediately arises corresponding to two modes by which the
topological invariant might cancel. Firstly, the presence of dislocations of
different slip system in the volume leads to cancellation by means of the
Burgers vector and the representative plane of possible dislocation tangents
[18]. We will refer to this SSD content as multi-slip SSDs. However,
cancellation can also occur due to dislocations on a single slip system by
cancellation of the tangent vector [7]. We will refer to this SSD content as
single-slip SSDs. This can arise either due to nontrivial curvature of the
dislocations on the scale of the representative volume or due to multiple non-
parallel dislocations passing through the volume. In a curved segment, the
component of the length perpendicular to the end-to-end distance is lost. In
the extreme case, a full dislocation loop would be entirely stored in this
mode. To create a theory that resolves multi-slip SSDs but not single-slip
SSDs, one must consider the density of dislocations on each slip system and
their average direction (either in the form of an angular parameter [19] or in
the form of a vector density [20, 21, 22, 23]), resulting in two kinematic
variables per slip system. We will refer to this as the _vector density
framework_. If one wishes to resolve even single-slip SSDs, it is necessary to
treat the dislocations as having some distribution over angular space as well
as to treat a curvature density [7]. Alternatively, this treatment may be
closed at lower order by considering only the first several coefficients in a
series expansion of this distribution [24]. We will refer to this as the
_higher order kinematic framework_. Closure at second order (already requiring
approximation [25, 26]) requires the treatment of the total line length in the
volume, the average curvature of those lines, the average line tangent, and
the average of the dyad formed by the line tangent with itself. These are
described by one, one, two, and two variables, respectively, for a total of
six kinematic variables per slip system when all vector quantities are
confined to the glide plane of dislocations.

Regardless of the kinematic framework chosen, kinetic closure, as will be the
focus of the present work, amounts to determining relations between the
structure at two scales. The relative arrangement of the dislocations at the
discrete scale results in all the interaction fields, forces, and, eventually,
velocities of interest in the dynamics. Some of this relative arrangement
information is preserved in the continuum density field; this is why the
coarse-grained Kröner-Nye tensor gives rise to the long-range stress field.
This tradeoff between short- and long-range portions of the stress field is at
the heart of the fast Fourier methods which have gained recent popularity in
the discrete dislocation dynamics community [27, 28, 29, 10, 30]. In such a
case, one solves the long-range interactions using the Kröner-Nye tensor while
retaining recourse to the discrete dislocation positions to compute the short-
range interactions overlooked by the Kröner-Nye tensor [31]. In the continuum
case, however, this short-range stress contribution must be considered in an
average sense. Consideration of the average short-range dislocation structure
is performed by analysis of the correlation functions. These allow the
relative arrangement information of the coarse-grained density field to be
corrected in order to approximate more closely the relative arrangement
information in the discrete system. For the higher order theory, where the
local dislocation arrangement is more general, this correction is a complex
object involving multiple contributions from the connectedness of the lines
and even something like a local microstructure [32]. For the vector density
theory, the form is simplified and the correlation largely encodes line
connectivity information and the average structure of parallel line bundles
[33].

The kinetic analysis and correlation functions in the present work will be
entirely situated in the vector density framework. However, if there is a more
complete framework (the higher order framework), why would we bother with the
vector density framework? The short answer is parsimony: the vector density
framework requires two variables per slip system (24 variables for FCC metals
with 12 slip systems), while the higher order framework requires thrice as
many variables (72 variables for FCC). This lack of parsimony becomes more of
an issue when kinetics begin to be considered. Kinetic closure, as we will
describe in more detail shortly, will require a description of average
quantities which are quadratic in the density variables (e.g. total elastic
energy [8]). These pair densities will have to bridge the two relevant length
scales: they relate the relative arrangement information on the order of the
discrete dislocations to the relative arrangement stored in the coarse density
field. The difference between these relative arrangements, in the form of
correlation functions, must be evaluated for every combination of density
field variable [8, 32]. A threefold increase in variables becomes a ninefold
increase in kinetically relevant variables. Of course, if that is the price
for accurately modelling the physical system, so be it. However, what if this
increase in variables gained no new information regarding the dislocation
configuration? That is, what if you are considering a scale on which physical
dislocation arrangements do not contain single-slip SSDs? We know that below
some critical distance, considered to be on the order of 50-100 nm [21, 34,
35], opposite signed dislocations tend to annihilate. At or below this scale,
the additional variables become irrelevant, as collections of dislocations on
like slip system would bear more resemblance to bundles of parallel
lines.222This assumes the absence of significant dislocation dipole
populations, which would represent non-annihilating single-slip SSDs. We will
more rigorously define this weak line bundle regime, where the line bundle
assumption holds in a specific sense, in more detail below.

Work on the dislocation correlation functions has only just begun in recent
years. While several works have attempted to outline kinetic frameworks for
continuum dislocation dynamics by appealing to dislocation correlations [8,
36, 37], only just recently have the correlation functions themselves been
studied in three dimensions [33, 32]. Here we approach the kinetic portion of
a vector density theory in a more direct manner, demonstrating how the
correlation functions arise naturally by directly averaging the dislocation
transport equation. In so doing, we will find that an additional driving force
for dislocation motion arises. This additional non-local force, which we call
the correlation force, bears a similarity to the correlation-based effective
stresses found in two-dimensional models of dislocation motion [38, 39]. This
previous work in two-dimensions, however, only considers a local density
approximation to this force [8], considered here in a more general non-local
form. Also, whereas the earlier investigations of correlation functions only
treat a small subset of the relevant correlations, here we will evaluate all
correlations which can in principle arise in the vector density framework.

In the remainder of this work, we present a kinetic theory of dislocation
motion in the vector density framework. In section II, we outline the
transport relationship for discrete dislocations and how this is coarse-
grained in the line bundle regime. In this way, we show how correlation
functions naturally arise and close the kinetic equations. In section III, we
present a means by which these correlation functions might be evaluated from
discrete dislocation data. This is followed in section IV by a symmetry
analysis of slip systems in FCC crystals which reduces the number of unique
correlation functions which must be considered between pairs of slip systems.
In sections V and VI we present the calculations and their results, followed
in section VII by a discussion of the implications of these findings for
continuum theories of dislocation motion.

##  II Discrete and Coarse-grained Kinetics of Dislocation Transport

Beginning with a treatment of discrete dislocation dynamics and the coarse-
graining process, we will show how dislocation correlation functions arise in
the kinetics of coarse-grained dislocation dynamics. Upon application of the
(weak) line bundle assumption, we show how the correlation functions enter
into the dynamics in a straightforward manner.

###  II.1 Discrete Dislocation Dynamics

Let us begin with a treatment of the underlying discrete dislocation lines
before introducing the coarse-graining treatment. The dislocation system in an
FCC crystal can be described as a collection of twelve line objects
(1-manifolds) $\mathcal{L}^{[\alpha]}$ embedded in a crystal space.
$\mathcal{M}$ which is roughly equivalent to $\mathbb{R}^{3}$.333Although
deformation per se contradicts the equivalence between the crystal manifold
$\mathcal{M}$ and $\mathbb{R}^{3}$, we note that for short-range effects (as
will be examined in this work), the crystal manifold is approximately flat.
These embedded line objects also define two measures on the crystal manifold,
namely a scalar line length $\Lambda^{[\alpha]}$ and the geometric dislocation
content $\bm{\kappa}^{[\alpha]}$ in a given region
$\Omega\subseteq\mathcal{M}$:

$\displaystyle\Lambda^{[\alpha]}(\Omega\subseteq\mathcal{M})$
$\displaystyle:=\int_{\mathcal{L}^{[\alpha]}\cap\Omega}dl,$ (1)
$\displaystyle:=\int_{\Omega}\varrho^{[\alpha]}(\bm{r})d^{3}\bm{r},$ (2)
$\displaystyle\bm{\kappa}^{[\alpha]}(\Omega\subseteq\mathcal{M})$
$\displaystyle:=\int_{\mathcal{L}^{[\alpha]}\cap\Omega}\bm{\xi}^{[\alpha]}(\bm{r}_{l})dl,$
(3) $\displaystyle:=\int_{\Omega}\bm{\varrho}^{[\alpha]}(\bm{r})d^{3}\bm{r},$
(4)

where we have denoted by $\bm{\xi}^{[\alpha]}(\bm{r})$ the unit direction of
the differential line element $dl$. While the scalar line content
$\Lambda^{[\alpha]}$ will still be of some use in the analysis of discrete
dislocation data, the theoretical formalism is concerned much more with the
geometrically necessary dislocation content. These measures imply the
existence of a singular distribution or density $\bm{\varrho}^{[\alpha]}$.
This singular density is analogous to a Dirac distribution concentrated on the
line object $\mathcal{L}^{[\alpha]}$. This distribution allows an
identification of the discrete line object with a singular density field, or,
in the language of differentiable manifolds, with a two-form. The nature of
this density as a two-form allows us to leverage a result from the theory of
differentiable manifolds, namely that the evolution of the lines is given by a
Lie derivative with respect to some velocity field $\bm{v}^{[\alpha]}$:

$\displaystyle\dot{\bm{\varrho}}^{[\alpha]}$
$\displaystyle=\nabla\times\left(\bm{v}^{[\alpha]}\times\bm{\varrho}^{[\alpha]}\right)-\bm{v}^{[\alpha]}\left(\nabla\cdot\bm{\varrho}^{[\alpha]}\right),$
(5)
$\displaystyle=\nabla\times\left(\bm{v}^{[\alpha]}\times\bm{\varrho}^{[\alpha]}\right).$
(6)

We have chosen not to burden the reader with the interior product and exterior
derivative notation. Rather we will opt for the above vector calculus
notation, albeit with the caveat that the particular action of the gradient
operators on the singular density is to be understood as an exterior
derivative.444As a matter of fact, dislocations are not most perfectly
represented by singular densities. Rather, they are better represented by
well-defined continuum densities resolved on the order of the lattice spacing
[40, 41]. In such a case, the gradient operators used in Eqs. (5,6) require no
qualification. The transport equation for open lines [Eq. (5)] has three
terms: the first describes transport due to dislocation glide, while the
second describes the motion of the endpoints of the lines. In the case where
the dislocation line object $\mathcal{L}^{[\alpha]}$ is a collection of closed
curves, the divergence (exterior derivative) of $\bm{\varrho}^{[\alpha]}$ is
null, corresponding to the physical fact that dislocation lines cannot
terminate in the crystal. For a treatment of systems where assumption is
relaxed to treat the closure failure created in connected dislocation networks
(e.g. by way of cross-slip, junction reactions, etc.), see [42]. In the
present work we wish only to treat the kinetics of dislocation glide, and so
ignore the endpoint transport, resulting in Eq. (6).

The above treatment is a kinematic description of the evolution of discrete
line objects in space. In order to describe a kinetic theory of dislocation
motion, we must include a physically motivated prescription of the velocity
field $\bm{v}^{[\alpha]}$. In discrete models of dislocation dynamics, the
motion of the dislocation lines is assumed to be overdamped and therefore
linearly related to the stress field evaluated on the dislocation line [9, 43,
44]. The form of the velocity law is given by the following simplified
functional notation:

$B\bm{v}^{[\alpha]}(\bm{r}):=\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\bm{\xi}^{[\alpha]}(\bm{r})\otimes\bm{\varrho}^{[\beta]}(\bm{r}^{\prime})\right].$
(7)

Here $B$ is a material specific drag parameter, and the Peach-Koehler
functional $\bm{F}_{\mathrm{PK}}^{[\alpha]}$ is a linear, vector-valued
functional. As a result, the brackets should be understood to denote an
integration over $\bm{r}^{\prime}$. From the details of how this Peach-Koehler
functional is derived, cf. appendix 1. Its form is given by:

$\left(\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\bm{\xi}^{[\alpha]}(\bm{r})\otimes\bm{\varrho}^{[\beta]}(\bm{r}^{\prime})\right]\right)_{l}:=\epsilon_{jkl}b_{i}^{[\alpha]}\xi_{k}^{[\alpha]}(\bm{r})\left(\sigma^{0}_{ij}+\sum_{\beta=1}^{N_{S}}\int_{\mathcal{M}}\hat{\sigma}_{ijm}^{[\beta]}(\bm{r}-\bm{r}^{\prime})\varrho_{m}^{[\beta]}(\bm{r}^{\prime})d^{3}\bm{r}^{\prime}\right),$
(8)

where $b^{[\alpha]}_{i}$ is the Burgers vector of slip system $[\alpha]$,
$\sigma^{0}_{ij}$ the externally applied stress,
$\hat{\sigma}_{ijm}^{[\alpha]}(\bm{\Delta r})dl_{m}$ is the stress field
surrounding a differential dislocation segment $dl_{m}$, and $\epsilon_{jkl}$
is the standard Levi-Civita tensor. The summation runs over all $N_{S}$ slip
systems considered in the crystal.

As simple as they may seem, Eqs. (6-8) amount to a complete theory of the
glide of discrete dislocations. Discrete frameworks consist primarily in
evaluating interaction forces [Eq. (8)], and updating segment positions, which
due to the line expansion inherent in Eq. (6) is not a trivial operation [44].
We now turn our attention for the remainder of this section to the evolution
of continuum densities of dislocations. To do so, we will outline a class of
averaging operators which allow the discrete dislocation kinetics to be
coarse-grained rather straightforwardly. We will then show how correlations
naturally enter into such a kinetic theory.

###  II.2 Continuum Dislocation Dynamics
in Line Bundle Ensembles

Before showing how the discrete transport kinetics are averaged, we first must
define what we mean by an averaging operator. We will denote this average with
angle brackets $\left\langle\cdot\right\rangle$. Its main quality of interest
is that it takes discrete densities to a smooth continuum density field:

$\bm{\rho}^{[\alpha]}(\bm{r}):=\left\langle\bm{\varrho}^{[\alpha]}(\bm{r})\right\rangle.$
(9)

Additionally, the average is a linear projection operator. That is, it has the
following properties for constants $c,d$ and linear functionals of the line
configuration $A,B$:

$\displaystyle\left\langle cA(\mathcal{L})+dB(\mathcal{L})\right\rangle$
$\displaystyle=c\left\langle A\right\rangle+d\left\langle B\right\rangle,$
(10) $\displaystyle\left\langle\left\langle A\right\rangle\right\rangle$
$\displaystyle=\left\langle A\right\rangle.$ (11)

Using only these properties, it is possible to derive a kinematic theory of
dislocation motion in a continuum sense [7, 24]. However, deriving a kinetic
theory becomes difficult without making use of further assumptions [8]. These
are chosen by considering what type of dislocation states are considered in
the average, analogously to the operation in statistical mechanics of
selecting an ensemble. In the present case, we choose to utilize the so-called
line-bundle assumption [33, 23]. This corresponds to an assumption that the
dislocation system is highly polarized on the representative distances on
which the density field is modelled. That is to say, this assumption
corresponds to the case of few single slip SSDs. For clarity, we will define
two variations of this assumption: a strong line bundle assumption and a weak
line bundle assumption.

In our previous work with correlations in line bundle ensembles [33], we
utilized a strong form of the line bundle assumption, which states that for
any constant vector $\bm{a}$,

$\left\langle\lvert\bm{a}\cdot\bm{\varrho}^{[\alpha]}(\bm{r})\rvert\right\rangle=\lvert\bm{a}\cdot\bm{\rho}^{[\alpha]}(\bm{r})\rvert.$
(12)

An equivalent statement would be to say that fluctuations in the line
direction,

$\delta\bm{\xi}^{[\alpha]}(\bm{r}):=\bm{\xi}^{[\alpha]}(\bm{r})-\hat{\bm{\rho}}^{[\alpha]}(\bm{r})$
(13)

–where $\hat{\bm{\rho}}^{[\alpha]}$ represents the unit direction of
$\bm{\rho}^{[\alpha]}$–are by definition null. This strong line bundle
assumption has the benefit that it produces convenient properties as
corollaries of this assumption, especially in the form of the correlation
functions. However, this is a rather stringent condition which in practice is
rarely fulfilled in physical dislocation systems.

Nonetheless, appropriate averaging processes can be defined where the line
bundle assumption holds approximately. As a result, we introduce a weak line
bundle assumption. Rather than defining fluctuations in line direction as
null, we assume they are simply small to second order:
$\delta\xi_{i}^{2}\approx 0$. In this case, any quantities second order or
higher in the line direction can factor out the average line direction
once:555Technically in the weak line bundle approximation [Eq. (14)], the
right-hand-side should be fully symmetrized. This is omit ted for brevity.

$\left\langle\bm{\xi}^{\otimes
n}A(\mathcal{L})\right\rangle\approx\hat{\bm{\rho}}\left\langle\bm{\xi}^{\otimes(n-1)}A(\mathcal{L})\right\rangle.$
(14)

Whereas the result of the strong line bundle assumption is that the
dislocation system is perfectly polarized
($|\bm{\rho}|=\left\langle\varrho\right\rangle$), the weak line bundle
assumption simply operates in the high polarization regime
$|\bm{\rho}|/\left\langle\varrho\right\rangle\approx 1$. In the following
section, we will examine a particular averaging procedure which satisfies this
property, but at this time we will simply examine the effects of a weak line
bundle averaging operator on the kinetics of discrete dislocation transport.

Because the averaging operator is linear, it commutes with derivatives and
other linear operators. As a result, averaging both sides of Eq. (6) results
in the following coarse-grained transport equation:

$\dot{\bm{\rho}}^{[\alpha]}=\nabla\times\left\langle\bm{v}^{[\alpha]}\times\bm{\varrho}^{[\alpha]}\right\rangle.$
(15)

As is evident from this equation, a solution of the dislocation dynamics
amounts to finding a closed-form expression for the cross product of the
velocity and density vector fields. We do this by introducing the relation for
the velocity field to obtain:

$\displaystyle
B\left\langle\bm{v}^{[\alpha]}\times\bm{\varrho}^{[\alpha]}\right\rangle$
$\displaystyle=\left\langle\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\bm{\xi}^{[\alpha]}(\bm{r})\otimes\bm{\varrho}^{[\beta]}(\bm{r}^{\prime})\right]\times\bm{\varrho}^{[\alpha]}(\bm{r})\right\rangle$
$\displaystyle=\left\langle\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\bm{\varrho}^{[\alpha]}(\bm{r})\otimes\bm{\varrho}^{[\beta]}(\bm{r}^{\prime})\right]\times\bm{\xi}^{[\alpha]}(\bm{r})\right\rangle$
$\displaystyle\approx\left\langle\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\bm{\varrho}^{[\alpha]}(\bm{r})\otimes\bm{\varrho}^{[\beta]}(\bm{r}^{\prime})\right]\right\rangle\times\hat{\bm{\rho}}^{[\alpha]}(\bm{r})$
$\displaystyle=\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\left\langle\bm{\varrho}^{[\alpha]}(\bm{r})\otimes\bm{\varrho}^{[\beta]}(\bm{r}^{\prime})\right\rangle\right]\times\hat{\bm{\rho}}^{[\alpha]}(\bm{r})$
(16)

by the application of, respectively, the definition of the unit direction, the
weak line bundle assumption, and the linearity of the average and Peach-
Koehler functional.

In the final relation above, a new quantity of interest has arisen: the two-
point density $\rho^{[\alpha,\beta]}(\bm{r},\bm{r^{\prime}})$ which is given
by,

$\bm{\rho}^{[\alpha,\beta]}(\bm{r},\bm{r}^{\prime}):=\left\langle\bm{\varrho}^{[\alpha]}(\bm{r})\otimes\bm{\varrho}^{[\beta]}(\bm{r}^{\prime})\right\rangle.$
(17)

As is common in the analysis of density hierarchies, we consider this average
to be expressed without loss of generality by the product of the single-point
densities and a correlation function
$d^{(i,j)[\alpha,\beta]}(\bm{r},\bm{r}^{\prime})$. This correlation function
expresses the scalar correlation between each respective tensor component of
the product density for each combination of slip systems. That is,

$\rho^{[\alpha,\beta]}_{ij}(\bm{r},\bm{r}^{\prime}):=\rho^{[\alpha]}_{i}(\bm{r})\rho^{[\beta]}_{j}(\bm{r}^{\prime})\left(1+d^{(i,j)[\alpha,\beta]}(\bm{r},\bm{r}^{\prime})\right).$
(18)

The scalar operation of the correlation function is one way of denoting the
diagonal nature of an in general tensorial correlation:

$d^{[\alpha,\beta]}_{kijl}(\bm{r},\bm{r}^{\prime}):=d^{(i,j)[\alpha,\beta]}(\bm{r},\bm{r}^{\prime})\delta_{ik}\delta_{jl},$
(19)

where $\delta_{ij}$ here represents the Kronecker delta. This diagonality
property was shown to be a corollary of the strong line bundle assumption
[33]; we presume that it holds, at least approximately, in the case of the
weak line-bundle approximation as well.

Substituting this definition into the transport relation [Eqs. (15,16)], we
obtain:

$\displaystyle\dot{\bm{\rho}}^{[\alpha]}$
$\displaystyle=B^{-1}\nabla\times\left\\{\left(\bm{F}_{\mathrm{MF}}^{[\alpha]}+\bm{F}_{\mathrm{C}}^{[\alpha]}\right)\times\bm{\rho}^{[\alpha]}\right\\},$
(20) $\displaystyle\bm{F}_{\mathrm{MF}}^{[\alpha]}(\bm{r})$
$\displaystyle=\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\hat{\bm{\rho}}^{[\alpha]}(\bm{r})\otimes\bm{\rho}^{[\beta]}(\bm{r}^{\prime})\right],$
(21) $\displaystyle\bm{F}_{\mathrm{C}}^{[\alpha]}(\bm{r})$
$\displaystyle=\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\hat{\bm{\rho}}^{[\alpha]}(\bm{r})\cdot\bm{d}^{[\alpha,\beta]}(\bm{r},\bm{r}^{\prime})\cdot\bm{\rho}^{[\beta]}(\bm{r}^{\prime})\right].$
(22)

The mean-field force term, $\bm{F}_{\mathrm{MF}}^{[\alpha]}$, captures the
driving force arising due to the eigenstress field generated by the continuum
densities $\bm{\rho}^{[\beta]}$. Assuming the dislocation correlation
functions to be non-trivial, however, we can see that the mean-field solution
of the driving force actually misrepresents the dislocation transport. The
correlation-dependent force term, $\bm{F}_{\mathrm{C}}^{[\alpha]}$, then
serves to correct the dynamics, reincorporating into the theory the short-
range dislocation interactions lost in the coarse-graining process.

We have already arrived at an important result: a kinetic theory for the
evolution of the dislocation density vector in highly polarized dislocation
systems. If the correlation function can be evaluated from discrete data, the
kinetic theory would be complete. In the remainder of this work, we present a
method for averaging discrete dislocation data in a way consistent with the
line bundle assumption so that the correlation functions can be evaluated from
discrete dislocation data.

##  III Averaging of Discrete Dislocation Systems

We have shown that for certain averaging operators—i.e. those for which the
weak line bundle assumption [Eq. (14)] holds—the correlation functions enter
straightforwardly into the evaluation of the dislocation velocity field. In
the section that follows, we present a specific averaging operator which
fulfills the line bundle assumption and outline an empirical method for
evaluating the correlation function from discrete dislocation dynamics. This
averaging process consists of a spatial convolution of the dislocation line
with a weight function of finite range as well as a statistical homogeneity
argument that treats all points in the crystal as equivalent.

###  III.1 Spatial coarse-graining of discrete dislocations

For some discrete dislocation line configuration $\mathcal{L}_{0}$ with
corresponding discrete density field $\bm{\varrho}_{0}$, define the continuum
density $\bm{\rho}(\bm{r};L)$ as follows:

$\bm{\rho}(\bm{r};L):=\int_{\Omega_{L}}w_{L}(\bm{r}-\bm{r}^{\prime})\bm{\varrho}_{0}(\bm{r}^{\prime})d^{3}\bm{r}^{\prime}.$
(23)

where $w_{L}$ is some weight function with compact support $\Omega_{L}$
characterized by some characteristic coarse-graining length $L$. We note that
for this spatially averaged density field, there will obviously be a
qualitative difference between the density field as this length tends to 0 and
the density field as this length tends to infinity. This qualitative
difference is closely related to the emergence of what we have discussed in
the introduction as single-slip SSDs. For small coarse-graining lengths, we
note that there is actually a small length $L^{*}$ which is used in discrete
dislocation dynamics to remedy the non-physical singularity in the elastic
field of dislocations ([38], [39]). That is to say, physical, discrete
dislocations are better modelled as density fields of the type
$\bm{\rho}(\bm{r};L^{*})$, where $L^{*}$ is on the order of 5-10 times the
Burgers vector magnitude, or about 2-3 nm in copper. Here single-slip SSDs are
absent by definition. On the other extreme, it is obvious that at some
sufficiently great $L$, at least on the order of 1 µm, the volume $\Omega_{L}$
surrounding any point could contain dislocation loops, or at least
dislocations sufficiently curved so as to make the vector cancellation in the
spatial average non-negligible, resulting in significant populations of
single-slip SSDs. However, we expect to find intermediate lengths, on the
order of 20-100 nm, where there is a de facto absence of single-slip SSDs.
This corresponds to the physical requirement that these single-slip SSDs
(opposite signed dislocations in $\Omega_{L}$) actually annihilate within some
critical distance. This critical distance is often estimated to be on the
order of 50 nm [45].

A consequence of the above considerations is that the definition of the
correlation function in Eq. (18) now supports a more precise interpretation.
Namely, the correlation encodes some relationship between density products at
different scales $L^{*}$ and $L$. It attempts to recover the fine product
densities by means of an appropriate linear relationship with the coarser
product densities.

The result of applying this density field calculation to a discrete
dislocation configuration is that for some collection of points
$\\{\bm{r}_{i}\\}_{i=1}^{N_{r}}$, we now have in hand the density field at
those points $\bm{\rho}^{[\alpha]}(\bm{r}_{i};L)$ for various convolution
lengths $L$. However, in terms of measuring the product density, this only
amounts to a single measurement of the product density. This is because the
product density

$\bm{\rho}^{[\alpha,\beta]}(\bm{r}_{i},\bm{r}_{j};L)=\bm{\rho}^{[\alpha]}(\bm{r}_{i};L)\otimes\bm{\rho}^{[\beta]}(\bm{r}_{j};L).$
(24)

has no sense of equivalence between some classes of point pairs. In order to
get some sense of the representative behavior of classes of these point pairs,
we must define a further averaging process.

###  III.2 Ensemble averaging of dislocation product densities

If we hope to build some picture of the average relationships between product
densities at different scales we will need to reduce the space of dependencies
of the correlation function. In its form in Eq. (18), it depends on a six-
dimensional space, a tuple of two positions $(\bm{r},\bm{r}^{\prime})$.
However, it is evident that in bulk crystals there is some degree of
arbitrariness to any specific choice of origin. If the only local information
that the correlation can be dependent on is the dislocation density, then it
follows from such a statistical homogeneity argument that the correlation
should only be dependent on the separation distance $\bm{\Delta
r}:=\bm{r}^{\prime}-\bm{r}$ [33]. Additionally, we should expect the long
range behavior of the dislocation state $\bm{\Delta r}\to\infty$ to be
uncorrelated. That is, for some function of the system state at two points
$f(\bm{r},\bm{r}^{\prime}):=f_{1}(\bm{r})f_{2}(\bm{r}^{\prime})$, we expect

$\lim_{\bm{r}^{\prime}-\bm{r}\to\infty}\left\langle
f(\bm{r},\bm{r}^{\prime})\right\rangle=\left\langle
f_{1}(\bm{r})\right\rangle\left\langle f_{2}(\bm{r}^{\prime})\right\rangle.$
(25)

In any case, this statistical homogeneity argument raises further questions.
Namely, which points do we treat as equivalent and how can we use this
equivalence to reason to a correlation function? As should be evident from our
kinetic derivation, we are interested in points where there is a dislocation
at both $\bm{r}$ and $\bm{r}^{\prime}$. Otherwise, there is no transport nor
interaction to be spoken of. As a result, we are interested in the atmosphere
of discrete density surrounding a point with non-zero discrete density.
Considering a sample set $S^{[\alpha]}$ of points where the discrete density
is nonzero,

$S^{[\alpha]}:=\left\\{\bm{r}\in\mathcal{M}\;:\;\rho^{[\alpha]}(\bm{r},L^{*})>0\right\\},$
(26)

we may then partition the direct product of the sample space into sets of like
separation vector in order to find pairs of interest to us:

$\Pi^{[\alpha,\beta]}(\bm{\Delta r}):=\left\\{(\bm{r},\bm{r}^{\prime})\in
S^{[\alpha]}\times S^{[\beta]}\;:\;\bm{r}^{\prime}-\bm{r}=\bm{\Delta
r}\right\\}.$ (27)

We will then simply examine the density product data at two scales, $L^{*}$
and $L$, for position pairs in $\Pi^{[\alpha,\beta]}(\bm{\Delta r})$. That is,
we will examine the scattered product density data,

$\displaystyle\Big{\\{}\Big{(}\rho^{[\alpha,\beta]}_{ij}(\bm{r},\bm{r}^{\prime};L^{*}),\;$
$\displaystyle\rho^{[\alpha,\beta]}_{ij}(\bm{r},\bm{r}^{\prime};L)\Big{)}$
$\displaystyle\forall\;(\bm{r},\bm{r}^{\prime})\in\Pi^{[\alpha,\beta]}(\bm{\Delta
r})\Big{\\}},$ (28)

and fit various correlation functions relating the two.

##  IV Interaction Symmetries in FCC Crystals

While it certainly could be of some utility to fit correlation functions for
all unique pairs of slip systems [$\alpha,\beta$], it would not be
particularly parsimonious. If all 12 slip systems are considered, there are a
total of 144 slip system interactions (78 if these are considered symmetric
under interchange of $\alpha,\beta$). Add to that the 4 tensorial components
of the correlation for each and things quickly get out of hand. However, it is
expected that some of these correlation functions are measuring qualitatively
similar behavior. More specifically, it is expected that the spatial features
of the correlation functions will be somehow relative to a special coordinate
system dependent on the combination of slip systems being considered. As a
result, we may wish to revise the collection of equivalent position pairs [Eq.
(27)] to also treat certain classes of slip system interactions as equivalent.

With this motivation in mind, we take a brief digression from the main
investigation of correlation functions to examine relationships between slip
systems, and from those relationships to form certain equivalence classes.
These equivalence classes will allow us to define distinct classes of
correlation functions which can be averaged together in the sense outlined in
the previous section. For the present work, we consider only the case of FCC
crystals, for which slip systems are commonly defined in terms of the faces
and edges of the so-called Thompson tetrahedron [6]. In this section, we will
examine the algebraic structure of the Thompson tetrahedron and their
resulting equivalence classes. Some of the results will be standard–the
relationships of various classes of slip systems are well known in the theory
of dislocation reactions in FCC crystals [21, 46]–but we will also make a
distinction which to our knowledge is novel, namely that certain interactions
have an associated handedness or chirality. In doing so we reduce the number
of slip system interactions which must be considered from 144 (or 78) to 6.

This classification problem to which this section is devoted can be defined as
follows. Given two slip systems $\alpha,\beta$, let us find:

  1. 1.

all other systems $\alpha^{\prime},\beta^{\prime}$ such that the resulting
dislocation reaction is of the same type,

  2. 2.

a (possibly improper) coordinate transformation such that the dihedral planes
and Burgers directors of the two slip systems are mapped onto constant
positions in space, and

  3. 3.

signs according to which the Burgers vectors—and accordingly, the dislocation
line sense—should be flipped in order to map the Burgers vectors onto constant
positions.

In order to devise such a scheme, we will first outline the algebraic
structure of the 12 slip systems in FCC crystals, augmenting the well-known
algebra of the twelve perfect dislocations in FCC crystals described by the
Thompson tetrahedron [6]. This algebra will then allow us to enumerate six
general classifications of slip system interactions and the transformations to
which they naturally give rise. These classifications correspond to the
traditional classification according to the reaction process which takes place
between them, i.e. self, cross-slip, coplanar, glissile junction, Lomer-type,
and Hirth-type interactions. We will also find that the latter four classes
can be further divided into left- and right-handed subclasses.

###  IV.1 Algebraic operations on the Thompson tetrahedron

####  IV.1.1 Vectors and directors on the Thompson tetrahedron

To begin, we must first define two sets of quantities, namely a set of slip
plane normal vectors $\bm{n}_{i}$ and Burgers vectors $\bm{b}_{i}$. These are
the faces and edges, respectively, of a tetrahedron. As a result there are
four {111}-type normal vectors and six ⟨110⟩-type Burgers vectors, listed in
Table 1666Note that these vectors are all defined rationally, and thus do not
have unit magnitude.. The senses of the normals can be well-defined as the
outward-facing normal vectors of the tetrahedral faces. However, it is not
possible to adequately define a sense of the Burgers vectors, and as a result
the sense of these in Table 1 is arbitrarily chosen following [47].

Table 1: Normal vectors and Burgers vectors defined. Normal vectors | Burgers vectors
---|---
Identifier | Plane | Identifier | Direction
| $(\bar{1}\bar{1}\bar{1})$ | a | $[\bar{1}01]$
| $(1\bar{1}1)$ | b | $[011]$
| $(11\bar{1})$ | c | $[\bar{1}10]$
| $(\bar{1}11)$ | d | $[110]$
|  | e | $[01\bar{1}]$
|  | f | $[101]$

One can see that the sign of the Burgers vector is irrelevant by considering
that the topological invariant of a dislocation with line sense $\bm{\xi}$ and
Burgers vector $\bm{b}$ is the outer product $\bm{\xi}\otimes\bm{b}$.
Therefore the dislocation is preserved upon reversal of the sign of the
Burgers vector so long as the line sense is also reversed. Thus it is more
useful to our present purposes to consider Burgers directors, which are non-
polar vectors commonly used in the theory of liquid crystals [48]. To consider
vectors as directors, we need an operation which can eliminate the sign of the
Burgers vectors. We will denote this sign product as a dot product $b_{i}\cdot
b_{j}$:

$b_{i}\cdot
b_{j}:=\frac{\bm{b}_{i}\cdot\bm{b}_{j}}{|\bm{b}_{i}\cdot\bm{b}_{j}|}\in\\{\pm
1\\}$ (29)

This operation returns the ‘alignedness’ of one Burgers vector with respect to
another, and as such is commutative. The multiplication table of this sign
product operator is shown in table II. This operation also has the expected
property that the resulting sign is sensitive to the sign reversal of either
input vector:

$b_{i}\cdot(-b_{j})=(-b_{i})\cdot b_{j}=-(b_{i}\cdot b_{j})$ (30)

and so is well-defined for the flipped Burgers vectors as well.

Table 2: Dot product of Burgers vectors | a | b | c | d | e | f
---|---|---|---|---|---|---
a | + | + | + | - | - | 0
b | + | + | + | + | 0 | +
c | + | + | + | 0 | + | -
d | - | + | 0 | + | + | +
e | - | 0 | + | + | + | -
f | 0 | + | - | + | - | +

####  IV.1.2 Wedge products of normal vectors and Burger directors

The two algebraic operations which will define the equivalence classes of slip
systems are as follows. There are natural operations which arise in the
discussion of planes and vectors. First, an operation that takes two normal
vectors and returns the dihedral direction: the oriented line direction along
which the planes intersect. Secondly, an operation that takes two Burgers
directors and returns the plane to which they both belong. We will define both
of these as a wedge product. The wedge product of two normal vectors
$n_{i}\wedge n_{j}$ is defined as

$n_{i}\wedge n_{j}:=\frac{1}{2}(\bm{n}_{i}\times\bm{n}_{j})\in\\{\pm b_{k}\\}$
(31)

and returns a direction which is a valid Burgers vector. The multiplication
table for this operation is shown in Table 3; one can see that it inherits
anticommutativity from the cross product. While this product will be of use
for defining a preferred polarity of the Burgers vectors, it does not lead to
interesting classifications of slip system interactions.

Table 3: Normal vector wedge product |  |  |  |
---|---|---|---|---
| 0 | a | -c | e
| -a | 0 | b | -d
| c | -b | 0 | f
| -e | d | -f | 0

Additionally, the wedge product $b_{i}\wedge b_{j}$ is defined as

$b_{i}\wedge b_{j}:=(b_{i}\cdot b_{j})\bm{b}_{i}\times\bm{b}_{j}\in\\{\pm
n_{k}\\}$ (32)

and returns a directed slip plane normal vector. The multiplication table for
this operation is shown in table IV. The result of this product gives the slip
plane to which the aligned Burgers vectors belong in a right-handed sense.
Consistently with this interpretation, we see that this product is also anti-
commutative, but it is insensitive to the sign of either Burgers vector:

$\displaystyle b_{i}\wedge b_{j}$ $\displaystyle=-(b_{j}\wedge b_{i})$ (33)
$\displaystyle=(-b_{i})\wedge b_{j}$ (34) $\displaystyle=b_{i}\wedge(-b_{j})$
(35) $\displaystyle=(-b_{i})\wedge(-b_{j})$ (36)

The Burgers wedge product gives two interacting slip systems a well-defined
chirality with respect to each other. As we will see below, this chirality
actually represents an important distinction within the traditional FCC slip
system interaction classes, and will be essential to defining equivalent
coordinate systems as well.

Table 4: Burgers director wedge product | a | b | c | d | e | f
---|---|---|---|---|---|---
a | 0 | \-  |  |  | \-  | 0
b |  | 0 | \-  | \-  | 0 |
c | \-  |  | 0 | 0 |  | \-
d | \-  |  | 0 | 0 |  | \-
e |  | 0 | \-  | \-  | 0 |
f | 0 | \-  |  |  | \-  | 0

###  IV.2 Classification of slip system interactions

Having defined these algebraic operations between slip planes and Burgers
directors, we may now define our twelve slip systems. Each allowable
combination of slip plane and Burgers vector is considered (i.e. those for
which the Burgers and normal vectors are orthogonal), and these arbitrarily
numbered combinations are shown in Table 5.

Table 5: Slip system definitions.  Slip system | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---
Burgers vector | a | a | b | b | c | c | d | d | e | e | f | f
Normal vector |  |  |  |  |  |  |  |  |  |  |  |

These slip systems form nodes of a graph structure shown in Fig. 1. We note
that some of the edges of this graph are directed, and some undirected. Two
slip systems are connected by a thin, undirected line if they share a Burgers
vector. Two edges are connected by a thick directed line if they share a
normal vector. The direction of these edges is specified such that for two
slip systems connected $1\to 2$, the wedge product of their Burgers directors
is right-handed on their shared normal plane; i.e.

$b^{[1]}\wedge b^{[2]}=n^{[1]}=n^{[2]}$ (37)

This graph structure is also shown superposed on the traditional net
representation of the Thompson tetrahedron to demonstrate the correspondence.

Figure 1: The slip system graph structure. Nodes represent slip systems; nodes
connected by an undirected edge share a Burgers vector, while nodes connected
by a directed edge share a slip plane normal vector. The direction of the edge
between two nodes $1\to 2$ is chosen such that $b^{[1]}\wedge
b^{[2]}=n^{[1]}=n^{[2]}$ . The Thompson tetrahedron is shown faintly
superimposed on the graph for reference.

####  IV.2.1 Classification conditions

The classification scheme is as follows. Self-interactions are those for which
$\alpha=\beta$, or

$\mathrm{self}:=\Big{\\{}[\alpha,\beta]\;:\;n^{[\alpha]}=n^{[\beta]}\;\mathrm{and}\;b^{[\alpha]}=b^{[\beta]}\Big{\\}}.$
(38)

Cross-slip interactions are those slip systems which share a Burgers vector
but glide on different slip planes, or:

$\mathrm{xslip}:=\Big{\\{}[\alpha,\beta]\;:\;n^{[\alpha]}\neq
n^{[\beta]}\;\mathrm{and}\;b^{[\alpha]}=b^{[\beta]}\Big{\\}}.$ (39)

Coplanar reactions occur when two dislocations on the same slip plane react to
form a dislocation segment with a new Burgers vector which can still glide on
the original slip plane. This interaction class is given by:

$\displaystyle\mathrm{coplanar}(\chi):=\Big{\\{}[\alpha,\beta]\;:\;$
$\displaystyle b^{[\alpha]}\wedge b^{[\beta]}=\chi n^{[\alpha]}$
$\displaystyle\mathrm{and}\;$ $\displaystyle
n^{[\alpha]}=n^{[\beta]}\Big{\\}}.$ (40)

where we have introduced a chirality $\chi=-1,+1$ for left- and right-handed
interactions, respectively.

Glissile junction reactions occur when a similar process occurs between
dislocations on different slip systems, resulting in a third glissile segment
on one of the two original slip planes. We consider only the interactions
where the resulting segment would be glissile on the plane of the second
reacting segment.777As a result of this asymmetric constraint, the glissile
junction class is the only class which is not symmetric under interchange of
the two slip systems considered. This corresponds to the case in which the
first Burgers vector is along the dihedral direction, i.e.

$\displaystyle\mathrm{glissile}(\chi):=\Big{\\{}[\alpha,\beta]\;:\;$
$\displaystyle(b^{[\alpha]}\wedge b^{[\beta]})=\chi n^{[\beta]}$
$\displaystyle\mathrm{and}\;$ $\displaystyle n^{[\alpha]}\neq
n^{[\beta]}\Big{\\}}.$ (41)

Lomer locks are a sessile junction segment formed when two dislocations react
to form a valid ⟨110⟩-type Burgers vector, but there does not exist a
tetrahedral face which contains both the dihedral direction (the line
direction of the resulting segment) and the resulting Burgers vector [49, 50].
For a right-handed Lomer-type interaction, the dihedral direction
$n^{[\alpha]}\wedge n^{[\beta]}$ is right-handed w.r.t. $b^{[\alpha]}$ and
left-handed with $b^{[\beta]}$, i.e.

$\displaystyle\mathrm{lomer}(\chi):=\Big{\\{}[\alpha,\beta]\;:\;$
$\displaystyle(n^{[\alpha]}\wedge n^{[\beta]})\wedge b^{[\alpha]}=\chi
n^{[\alpha]}$ $\displaystyle\mathrm{and}\;$ $\displaystyle
b^{[\beta]}\wedge(n^{[\alpha]}\wedge n^{[\beta]})=\chi n^{[\alpha]}\Big{\\}}.$
(42)

Lastly, Hirth locks are a sessile junction formed when dislocations with
orthogonal Burgers vectors react, forming a dislocation segment with a
$\langle$100$\rangle$-type Burgers vector [51]. These segments are sessile
because this is not a close-packed direction and thus the energy barrier for
slip along this direction is prohibitively high [46]. Although it is possible
to use the requirement $b_{i}\cdot b_{j}=0$ to identify Hirth-type
interactions, this definition would ignore the chiral nature of the
interaction. Rather, we define it similarly to the Lomer-type interactions,
with the distinction that for a right-handed Hirth-type interaction, the
dihedral direction $n^{[\alpha]}\wedge n^{[\beta]}$ is right-handed w.r.t.
both $b^{[\alpha]}$ and $b^{[\beta]}$:

$\displaystyle\mathrm{hirth}(\chi):=\Big{\\{}[\alpha,\beta]\;:\;$
$\displaystyle(n^{[\alpha]}\wedge n^{[\beta]})\wedge b^{[\alpha]}=\chi
n^{[\alpha]}$ $\displaystyle\mathrm{and}\;$ $\displaystyle(n^{[\alpha]}\wedge
n^{[\beta]})\wedge b^{[\beta]}\wedge=\chi n^{[\alpha]}\Big{\\}}.$ (43)

For a more intuitive representation of these conditions, please refer to Fig.
2, which shows these conditions represented as the various unique classes of
paths one can take on the slip system graph structure shown in Fig. 1. The
classes and their right- and left-handed variants are shown as well. From
these graphical representations, it is clear that the classes are not only
defined by the types of connections traversed between two slip systems, but
also the direction of the edges traversed. The former results in the
traditional classification scheme, while it is the directionality of the graph
edges which results in the handedness discussed above. For completeness, a
table of the interactions of all 144 combinations of slip system is shown in
Table 6, with left-handed interactions highlighted.

Table 6: Enumeration of all interaction types.

####  IV.2.2 Equivalence Spaces

The handedness is central to the classification problem that presents itself
to us in the analysis of correlation functions because it allows us to define
coordinate transformations which map the Burgers vectors and dihedral planes
onto constant directions. There are two such coordinate transformations
$\bm{R}^{[\alpha,\beta]}$. In the case where the two interacting slip systems
share a slip plane normal, the rotation is given by

$\bm{R}^{[\alpha,\beta]}:=\left(\;\bm{b}^{[\alpha]}\;|\;\chi(\bm{n}^{[\alpha]}\times\bm{n}^{[\beta]})\;|\;\bm{n}^{[\alpha]}\;\right)^{T}.$
(44)

In the case where the two slip systems have different slip plane normal
vectors, the rotation is given by

$\displaystyle\bm{R}^{[\alpha,\beta]}:=$
$\displaystyle\left(\;(\bm{n}^{[\alpha]}\wedge\bm{n}^{[\beta]})\times\bm{n}^{[\beta]}\;|\;\chi(\bm{n}^{[\alpha]}\wedge\bm{n}^{[\beta]})\;|\;\bm{n}^{[\beta]}\;\right)^{T}.$
(45)

The result of these transformations is that for self (assumed right-handed)
and coplanar interactions, the Burgers vector of the first (or only)
interacting slip system is aligned to the x-axis and the shared normal
direction to the z-axis. The y-axis is then defined in a right- or left-handed
sense depending on the chirality of the interaction. For the remaining
interactions—cross-slip (assumed right-handed), glissile junction, Lomer-type,
and Hirth-type—the dihedral direction of the two planes is aligned to the
y-axis and the second slip plane normal to the z-axis. The x-axis is then
defined in a right- or left-handed sense according to the chirality of the
interaction.

Figure 2: Classes of interactions on the Thompson tetrahedron. Different
classes of paths along the graph denote different slip system interactions.
The first slip system of a given interaction is marked by a diamond, the
second by a square. Shown are a) self, b) cross-slip, c)coplanar, d) glissile
junction, e) Lomer-type, and f) Hirth-type interactions. Right- and left-
handed interactions are marked.

These all represent simple coordinate rotations, although according to the
interaction handedness these will be proper or improper rotations. Proper or
improper, they map the dihedral planes and Burgers vectors onto constant
directions in what will for the remainder of the work be referred to as the
equivalence space of the respective class of interactions.

####  IV.2.3 Burgers reversals

Lastly, we also define a sign by which the Burgers vector and line direction
of the two slip systems ought to be reversed in order to make use of the above
spatial rotations. Like the rotation tensors, the behavior is dependent on the
two slip systems sharing or not their slip plane normal. In the case where the
slip plane normal is shared, the second sign ought to be reversed according to
the sign product of the two Burgers vectors:

$\displaystyle s^{[\alpha]}$ $\displaystyle=1$ $\displaystyle s^{[\beta]}$
$\displaystyle=b^{[\alpha]}\cdot b^{[\beta]}$ (46)

In the case where the slip plane normal is not shared, both signs are reversed
according to the sign product of the respective Burgers vector with the
dihedral direction:

$\displaystyle s^{[\alpha]}$
$\displaystyle=b^{[\alpha]}\cdot\left(n^{[\alpha]}\wedge n^{[\beta]}\right)$
(47) $\displaystyle s^{[\beta]}$
$\displaystyle=b^{[\beta]}\cdot\left(n^{[\alpha]}\wedge n^{[\beta]}\right)$
(48)

###  IV.3 Final remarks regarding classification scheme

In closing this section of the analysis, we show how these equivalence classes
will be leveraged in the analysis of the two-point density data. Whereas
previously we were considering the behavior of the product density,

$\bm{\rho}^{[\alpha,\beta]}(\bm{r},\bm{r}+\bm{\Delta r};L),$ (49)

we will now be considering the qualitatively equivalent behavior of distinct
groups of interactions. That is, for $[\alpha,\beta]\in\mathrm{type}$, where
$\mathrm{type}$ is some class of interactions as outlined above, we now
consider the behavior of the following product density:

$\displaystyle\bm{\rho}^{[\mathrm{type}]}(\tilde{\bm{r}},$
$\displaystyle\tilde{\bm{r}}+\bm{\Delta}\tilde{\bm{r}};L)$ $\displaystyle\sim
s^{[\alpha]}s^{[\beta]}\bm{\rho}^{[\alpha,\beta]}\left(\bm{r},\bm{r}+(\bm{R}^{[\alpha,\beta]})^{-1}\bm{\Delta}\tilde{\bm{r}};L\right),$
(50)

where $\tilde{\bm{r}}$ here denotes a vector in the equivalence space and we
have flipped both dislocation vectors
$\bm{\rho}^{[\alpha]},\bm{\rho}^{[\alpha]}$ by the Burgers signs
$s^{[\alpha]},s^{[\beta]}$.

In this manner, a given interaction class now can be treated in toto by
revising our definition of the equivalent pairs [Eq. (27])] to include all
pairs which share a separation vector in the equivalence space:

$\Pi^{[\mathrm{type}]}(\bm{\Delta}\tilde{\bm{r}}):=\bigcup_{[\alpha,\beta]\in\mathrm{type}}\Pi^{[\alpha,\beta]}\left((\bm{R}^{[\alpha,\beta]})^{-1}\bm{\Delta}\tilde{\bm{r}}\right).$
(51)

With this classification scheme in hand, we have reduced the number of unique
correlation tensors which need to be calculated from 144 to 6.

##  V Calculation of Correlation Functions

The current correlation calculations are performed using the same library of
discrete dislocation dynamics simulations used in [33]. In summary, this
represents 45 unique simulations of dislocations in copper which were run to
0.3% plastic strain in uniaxial ([100] loading axis) strain-controlled
simulations using the microMegas code [47]. Because this code is lattice-
based, the lines were resampled to the midpoints of the discrete segments in
order to fully populate the angular space [52]. The dislocation densities in
these simulations evolved from an initial value of 2 µm-2 to an average final
value of 6 µm-2 and the simulation domain measured 4.40 x 4.87 x 5.74 µm.

We have made use of the cloud-in-cell function [53] as our choice of weight
function for averaging the dislocations:

$w_{L}(\bm{r}):=\begin{cases}\prod_{i=1}^{3}\left(1-\frac{|r_{i}|}{L}\right)&\text{for
}|r_{i}|<L\\\ 0&\text{otherwise}\end{cases}$ (52)

For the details of the analytical line integral of this weight function over
the straight segments used in discrete dislocation dynamics simulations,
please refer to appendix 2. The main result is that for a collection of line
segments $\lambda_{i}$ with line tangents $\hat{\bm{t}}_{i}$ passing through
the cube $\Omega_{L}$ of side length $2L$ surrounding the point $\bm{r}$, the
line integral of each can be evaluated as a weight $w_{L}(\lambda_{i})$. This
allows us to calculate both a vector-valued dislocation density and a scalar
line density as [cf. Eq. (23)]:

$\displaystyle\bm{\rho}(\bm{r};L)$
$\displaystyle=\sum_{\lambda_{i}}\hat{\bm{t}}_{i}w_{L}(\lambda_{i}),$ (53)
$\displaystyle\rho(\bm{r};L)$
$\displaystyle=\sum_{\lambda_{i}}w_{L}(\lambda_{i}).$ (54)

These may not be of equivalent magnitude for volumes containing more than one
dislocation line. The line bundle approximation only requires that the points
where $|\bm{\rho}|$ is not approximately equal to $\rho$ are of little
interest.

The majority of the density calculations performed utilized a ‘discrete’ mesh
of 720 cubic voxels along the longest box dimension. The fundamental length
$L^{*}$ was taken as equivalent to this mesh spacing, 8.1 nm. The continuum
density field calculations were performed using coarse-graining lengths L that
were multiples of the mesh spacing, from 5 times the mesh spacing (40.5 nm) to
10 times this spacing (81 nm). A small minority of the calculations were
performed comparing the continuum coarse-graining length of 81 nm and varying
the fundamental discrete length $L^{*}$ from 16.2 nm down to 8.1 nm by
increasing the number of mesh elements along the longest box dimension.

Having density fields in hand, we now turn our attention to how these are used
in the scheme discussed above to calculate correlation functions. As outlined
above, once grouped into spatial equivalence classes, for every separation
distance $\bm{\Delta}\tilde{\bm{r}}$, we have the scattered data:

$\displaystyle\Big{\\{}\Big{(}D^{[\mathrm{type}]}_{ij}(\bm{r},\bm{r}^{\prime}),\;$
$\displaystyle C^{[\mathrm{type}]}_{ij}(\bm{r},\bm{r}^{\prime})\Big{)}$
$\displaystyle\forall\;(\bm{r},\bm{r}^{\prime})\in\Pi^{[\mathrm{type}]}(\bm{\Delta
r})\Big{\\}},$ (55)

where we have relabeled the discrete product density
$D^{[\mathrm{type}]}_{ij}(\bm{r},\bm{r}^{\prime}):=\rho^{[\mathrm{type}]}_{ij}(\bm{r},\bm{r}^{\prime};L^{*})$
and continuum product density
$C^{[\mathrm{type}]}_{ij}(\bm{r},\bm{r}^{\prime}):=\rho^{[\mathrm{type}]}_{ij}(\bm{r},\bm{r}^{\prime};L)$
for notational clarity. An investigation of the correlation amounts to, for
every separation distance, fitting the following linear relationship between
the two:

$D^{[\mathrm{type}]}_{ij}(\bm{\Delta}\tilde{\bm{r}}):=C^{[\mathrm{type}]}_{ij}(\bm{\Delta}\tilde{\bm{r}})g^{[\mathrm{type}](i,j)}(\bm{\Delta}\tilde{\bm{r}}),$
(56)

where
$g^{[\mathrm{type}](i,j)}(\bm{\Delta}\tilde{\bm{r}}):=1+d^{[\mathrm{type}](i,j)}(\bm{\Delta}\tilde{\bm{r}})$
is a more compact relationship for this slope. Notice, this is simply a
compact notation for Eqs. (17,18), the definition of the correlation. This
compact notation represents the data-driven context in which we will evaluate
the correlation functions.

For this reason, we outline a series of fitting methods for
$g^{[\mathrm{type}](i,j)}(\bm{\Delta}\tilde{\bm{r}})$. These include: ordinary
least squares fitting with discrete (OLSD) and continuum residuals (OLSC), and
a principal component analysis of the covariance matrix (PC). All three
methods make use of the covariance matrix:

$\mathrm{cov}(D,C,\bm{\Delta}\tilde{\bm{r}}):=\left(\begin{matrix}\frac{\left\langle
D^{2}\right\rangle(\bm{\Delta}\tilde{\bm{r}})}{D^{2}_{\infty}}&\frac{\left\langle
CD\right\rangle(\bm{\Delta}\tilde{\bm{r}})}{CD_{\infty}}\\\ \frac{\left\langle
CD\right\rangle(\bm{\Delta}\tilde{\bm{r}})}{CD_{\infty}}&\frac{\left\langle
C^{2}\right\rangle(\bm{\Delta}\tilde{\bm{r}})}{C^{2}_{\infty}}\end{matrix}\right)$
(57)

where the averages are now understood as empirical averages over the set
$\Pi^{[\mathrm{type}]}(\bm{\Delta}\tilde{\bm{r}})$:

$\left\langle
f\right\rangle(\bm{\Delta}\tilde{\bm{r}})=\frac{1}{|\Pi^{[\mathrm{type}]}(\bm{\Delta}\tilde{\bm{r}})|}\sum_{(\bm{r},\bm{r}^{\prime})\in\Pi^{[\mathrm{type}]}(\bm{\Delta}\tilde{\bm{r}})}f(\bm{r},\bm{r}^{\prime})$
(58)

Here, $|\Pi^{[\mathrm{type}]}(\bm{\Delta}\tilde{\bm{r}})|$ represents the
number of pairs in the set $\Pi^{[\mathrm{type}]}(\bm{\Delta}\tilde{\bm{r}})$.
Additionally, each average is measured relative to the uncorrelated [in the
sense of Eq. (25)] average. That is, for some function
$h(C,D):=h_{1}(C)h_{2}(D)$ satisfying the requirements of Eq. (25), we may
express the uncorrelated average as

$\displaystyle h_{\infty}(C^{[\mathrm{type}]}_{ij},D^{[\mathrm{type}]}_{ij})=$
$\displaystyle\left[\frac{1}{\sum_{(\alpha,\beta)\in\mathrm{type}}|S^{[\alpha]}|}\sum_{(\alpha,\beta)\in\mathrm{type}}\sum_{\bm{r}\in
S^{[\alpha]}}h_{1}\big{(}\rho^{[\alpha]}_{i}(\bm{r};L)\big{)}h_{2}\big{(}\rho^{[\alpha]}_{i}(\bm{r};L^{*})\big{)}\right]$
$\displaystyle\times\left[\frac{1}{\sum_{(\alpha,\beta)\in\mathrm{type}}|S^{[\beta]}|}\sum_{(\alpha,\beta)\in\mathrm{type}}\sum_{\bm{r}\in
S^{[\beta]}}h_{1}\big{(}\rho^{[\beta]}_{j}(\bm{r};L)\big{)}h_{2}\big{(}\rho^{[\beta]}_{j}(\bm{r};L^{*})\big{)}\right].$
(59)

Each of the four fitting schemes outlined above utilize the covariance
averages, as follows:

$\displaystyle g_{\mathrm{OLSC}}(\bm{\Delta}\tilde{\bm{r}})$
$\displaystyle:=\frac{\left\langle
D^{2}\right\rangle(\bm{\Delta}\tilde{\bm{r}})}{\left\langle
CD\right\rangle(\bm{\Delta}\tilde{\bm{r}})}\frac{CD_{\infty}}{D^{2}_{\infty}}$
(60) $\displaystyle g_{\mathrm{OLSD}}(\bm{\Delta}\tilde{\bm{r}})$
$\displaystyle:=\frac{\left\langle
CD\right\rangle(\bm{\Delta}\tilde{\bm{r}})}{\left\langle
C^{2}\right\rangle(\bm{\Delta}\tilde{\bm{r}})}\frac{C^{2}_{\infty}}{CD_{\infty}}$
(61) $\displaystyle g_{\mathrm{PC}}(\bm{\Delta}\tilde{\bm{r}})$
$\displaystyle:=\frac{\beta_{D}^{(1)}}{\beta_{C}^{(1)}}$ (62)

where $\bm{\beta}^{(1)}:=(\beta_{D}^{(1)},\beta_{C}^{(1)})$ represents the
eigenvector of the covariance matrix corresponding to the highest eigenvalue.
This expansion is performed as follows:

$\mathrm{cov}(D,C):=(\bm{\beta}^{(1)}|\bm{\beta}^{(2)})\cdot\left(\begin{matrix}\sigma_{(1)}^{2}&0\\\
0&\sigma_{(2)}^{2}\end{matrix}\right)\cdot(\bm{\beta}^{(1)}|\bm{\beta}^{(2)})^{T}$
(63)

with $\sigma_{(1)}^{2}\geq\sigma_{(2)}^{2}$. That is, the covariance matrix is
diagonalizable into two statistically uncorrelated linear combinations of $C$
and $D$. The eigenvector corresponding to the combination with the larger
variance is chosen as a correlation function [Eq. (62)]. As an additional
measure of the variance in the scatter relationship [Eq. (55)] which is not
captured by the linear correlation relationship, we will also examine the
following relation as the fluctuation variance:

$\sigma_{\mathrm{f}}^{2}=\frac{\sigma_{(2)}^{2}}{\sigma_{(1)}^{2}+\sigma_{(2)}^{2}}.$
(64)

The relative strength of the principal component analysis just described is
that the random components of the data along each principal component (the
correlation fit line and the remaining fluctuation variance) are statistically
uncorrelated. For a more detailed description of principal component analysis,
cf. [54].

##  VI Results

Figure 3: Scatter plots of discrete and continuum density product data. (a)
Volumetric representation of the edge-edge self-correlation function given for
reference of separation space locations. The slip plane (hexagon), Burgers
direction (blue line), and coarse-graining volume (cube) are shown for
reference. In this figure, blue regions are anti-correlated regions ($d<1$),
and red regions are positively correlated ($d>1$). (b-f) Scatter plots of the
underlying discrete and continuum density products, and the linear
relationship which the correlation function describes. In (a), blueregions
correspond to anti-correlated regions ($g<1$) and red to correlated regions
($g>1)$), while transparent regions correspond to uncorrelated regions
($g=1$). Orienting features of (a) include the Burgers vector direction
(blueline), slip plane (hexagon), and the coarse-graining volume (cube). The
points in separation space examined in (b-f) are shown as well. In (b-f), the
dotted line represents the relation that would obtain in the perfectly
uncorrelated case (a slope of unity). The solid lines show the principal
components of the covariance matrix. The slope of the larger component is
taken as the correlation value and reported in each plot. Points in (b-f) are
colored according to the effective polarization [Eq. (65)], with blue denoting
low polarization ($\ll 1$) and red denoting high polarization ($\approx 1$)
according to the colorbar shown on the right.

We begin this section with a figure showing the scattered nature of the
product density data (Fig. 3) as an aid for conceptually understanding the
correlation fitting procedure. The first plot shows a spatial correlation
function with several points labeled. Corresponding to each of these points in
the separation space, the product density at the two scales is represented as
scattered data [cf. Eq. (55)]. The linear relationship passes through the
origin–this is required due to the lack of constant terms in Eq. (18)888The
lack of constant terms in the definition of the correlation relation is no
accident. Rather, it is a corollary of a certain property of the coarse
grained densities in line bundle ensembles, namely that the coarse densities
are absolutely continuous with respect to the discrete densities [33].–but
differs from point to point; this results in a correlation field that is a
function of separation distance (cf. Fig. 3a).

An entirely uncorrelated product density would result in the scattered data
falling perfectly on the line of unit slope passing through the origin. The
principal components of the covariance matrix for the scattered data at each
point are shown in Fig. 3(b-f) as black lines, the length of which denotes the
variance along each component. The direction of the longer component gives the
PC fit for the correlation function [Eq. (62)], which is reported in each
plot. This direction also defines the fluctuation direction (these two
directions are orthogonal), and the length of the shorter component is related
to the fluctuation variance [Eq. (64)]. Thus, Fig. 3 not only shows
qualitatively what the correlation represents, but also the limits of the
information it contains. This limitation, quantified by the fluctuation
variance, will be revisited at the end of the present section and we will
examine its implications in a later section.

One additional feature of the scatter plots in Fig.3(b-f) is that the the
effective polarization of each data point is shown. That is,

$\mathrm{Pol}=\sqrt{\frac{|\bm{\rho}(\bm{r})|}{\rho(\bm{r})}\frac{|\bm{\rho}(\bm{r}^{\prime})|}{\rho(\bm{r}^{\prime})}}.$
(65)

It is worth noting that the linear relationship is especially erroneous for
data points with low effective polarization; such points have high discrete
product density but low continuum product density and cluster along the
vertical line that would represent an infinite value of the correlation. Such
data obviously violate the line bundle assumption; however, we note that they
represent a negligibly small portion of the total data.

Figure 4: Various fits for the self-correlation function in the slip plane.
Self-correlation functions are shown for separation vectors in the slip plane,
where $\Delta r_{b}$ and $\Delta r_{a}$ are the separation in the Burgers
vector direction and the line direction of an edge dislocation, respectively.
The three methods of fitting the correlation are compared, namely ordinary
least-squares with continuum (OLSC) and discrete (OLSD) residuals, and the
prinicpal component (PC, cf. Fig. 3). These are shown for the screw-screw,
screw-edge, and edge-screw components of the self-correlation function. The
morphology of the correlation function is unaffected by the choice of fitting
method, although the magnitude of the respective maxima differs slightly
between the three methods. The faint hexagons represent the boundaries of the
fundamental and coarse-graining volumes (cube of side length $2L^{*}$ and
$2L$, respectively).

Fig. 4 shows a typical self-correlation field (self in the sense defined in
section IV, as pertaining to density products of like slip system). Results
for the various fitting methods outlined above are shown for separation
distances which lie in the slip plane, i.e. those separation vectors which
have no component in the slip plane normal direction. The three unique tensor
components of the self-correlation field (screw-edge and edge-screw are
symmetric under inversion of separation space) are shown for the three fitting
methods described in Eqs. (60-62). The distinctive morphologies of the self-
correlation field can be seen in this figure. All of the fitting methods
demonstrate roughly equivalent morphologies, differing mainly in the magnitude
of their respective extrema. The principal component analysis fitting method,
however, will be utilized in the remainder of the manuscript because of the
property that the error axis is statistically uncorrelated (in the sense of
null covariance) to the correlation axis. This fluctuation axis, briefly
discussed above, will be the subject of later discussion.

Figure 5: Radial plots for all correlation functions. (a) Cylindrical
correlation function [Eq. (67)] for the self-correlation . (b-g) Spherical
correlation function [Eq. (66)] from the correlation functions corresponding
to each interaction class.

Next we move on to the full cross-correlation results for each type of density
field combination enumerated in section IV. Reporting on the cross-correlation
is one of the main objectives of the present work. Firstly, in Fig. 5 we show
the radial convergence of these cross-correlations. The spherical correlation
functions shown in Fig. 5(b-g) are the average correlation value over the
spherical shell of radius r:

$g(r):=\frac{1}{4\pi r^{2}}\int_{0}^{4\pi}r^{2}d\Omega\;g(\bm{r}).$ (66)

Because of the strong anisotropy of the self-correlation function surrounding
the slip plane, we also compute for that correlation function a cylindrical
correlation function. This is computed as the average over the disc of height
$L$ and radius $s$ centered on the origin:

$g(s):=\frac{1}{2\pi
sL}\int_{-L/2}^{L/2}dr_{n}\int_{0}^{2\pi}d\varphi\;g(\bm{r}).$ (67)

In addition to these radial correlation functions, full volumetric contour
plots are also shown in Fig. 6 to display the full spatial morphology of the
correlation functions.

Figure 6: Volumetric plots of all correlation functions. Volumetric
isosurfaces of the correlation functions for all interaction classes are
colored according to the value of the correlation function. The color scale,
denoting the value of the correlation, is constant across all plots. The
transparency scale of each plot ranges from the maximum deviation from unit
correlation (opaque) to uncorrelated (unit correlation, transparent). For
spatial reference, several relevant spatial features are shown. Cubes show the
coarse-graining volume (cube of side length $2L$). Hexagons show the
intersection of the slip planes of the respective slip system with the cube of
side length $6L$. Dotted line shows the intersection of the two slip planes if
applicable. blueand orange lines show the Burgers vector of the first and
second slip systems, respectively.

Of foremost significance to the dislocation system is the range of the
correlation functions, which is displayed by the radial convergence shown in
Fig. 5. As these ranges define the degree of non-locality which must be
considered for corrections to the Peach-Koehler functional, it is reassuring
that these converge to uncorrelated values in most cases by 2-4 times the
coarse-graining length. Of particular prominence are the longer-ranged and
more strongly correlated cross-slip and glissile junction type correlations.
Of least prominence are the coplanar correlations, which densities it would
seem are almost completely uncorrelated in the simulations analyzed.

Additional insights into the features of the correlation functions can be
gained from the volumetric renderings. In Fig. 6, it is clear that at least
some degree of anisotropy is evident in all cross-correlations. The self-
correlation is seen to be strongly anisotropic, with significant variation in
the slip plane (cf. Fig. 4) as well as strong anti-correlations ($g<1$) for
small distances normal to the slip plane. The dominance of these
anticorrelated regions just off the slip plane can be seen in the cylindrical
and radial correlations of the self-interaction type (Fig. 5a-b). While not as
dramatic as in the self-correlations, the cross-correlations do display some
weak anisotropies. Notably, they seem to decay more slowly in the direction of
the dihedral between their two slip planes.

Figure 7: Dependencies of screw-screw self-correlation on various calculation
parameters. (a) Self-correlations calculated with various coarse-graining
lengths and a fixed fundamental length. (b) Self-correlations calculated with
various fundamental lengths and a fixed coarse-graining length. (c) Self-
correlations calculated from configurations with various values of plastic
strain. Note, all separation distances scaled by the coarse-graining length.

Of considerable interest to any implementation of the correlation functions is
the question whether these functions vary as the system evolves, that is, as
the total dislocation line length in the volume increases. Two additional
parameters are the choices of fundamental and coarse-graining lengths, the
length scales associated with the discrete and continuum representations of
the dislocation densities. These remain fixed in a continuum theory, but in
principle they should affect the correlation functions. Screw-screw self-
correlation calculations which independently vary each of these parameters are
shown in Fig. 7. The significant dependencies are on the relationship between
the discrete and continuum coarse-graining lengths. These are seen in the
first two series of plots [Fig. 7(a-b)]. In the former (a), correlation
functions are calculated from density fields with 16 nm fundamental length and
coarse-graining lengths increases from 32 to 81 nm. In the latter (b), the
coarse-graining length is held fixed at 81 nm while the fundamental length is
decreased from 16 to 8 nm. From these plots, it can be seen that the relevant
spatial features of the self-correlation functions are relative to the
discrete and continuum coarse-graining lengths. This strong dependence
demonstrates that the correlations arise as features of the spatial averaging
process itself, rather than some particularly favored dislocation
arrangements. Additionally, the dependence of the correlation function on
plastic strain is shown in Fig. 7(c). The correlation functions are seen to be
stable with respect to strain, showing little evolution over the course of the
simulation. This relationship will be further elaborated in the discussion
section to follow.

Figure 8: Fluctuation variances of the self-correlation calculation. Displayed
are the fluctuation variance field for the self-correlation calculation as a
fraction of the total variance in the product density data at each point [Eq.
(64)] for separation vectors in the slip plane.

Lastly, the portion of the variance at each separation distance which was
unexplained by the correlation function (cf. Fig. 3) is shown in Fig. 8. While
the correlation was defined to capture the error which is introduced in the
discrete product density by naively assuming it to be equal to the continuum
product density, some error remains even after defining a correlation
function. This is due to the closure of the density hierarchy at order two as
well as the suppression of the fully six-dimensional dependence of the
correlation function. Nonetheless, the noticeable spread off the correlation
axis in the scattered data shown in Fig. 3 still only accounts for no more
than 5% of the total variance at any separation distance. Of note, however, is
that the areas with the highest fluctuation variance correspond to the areas
of strong correlation ($g>1$). These regions, occurring near the boundary of
the coarse-graining region, arise due to the small continuum field values
which occur near the boundaries of the coarse-graining volume in the
‘unconnected’ direction.

##  VII Discussion

The calculated correlation functions reported in the previous section, while
changing little with strain, depend strongly on the size of the coarse-
graining volume. This suggests that correlations arise due to the averaging of
a field of planar lines, rather than encoding a contingent arrangement of the
microstructure. This is in contrast to theories at longer length scales, where
the correlation begins to capture the length scale of such a microstructure
[55]. Nonetheless, the correlations measured above still encode important
kinetic information. In this section we discuss our interpretation of what
information that is by examining the impact of the presented results on future
investigations in vector density continuum dislocation dynamics. In
particular, we will examine the implications of these correlation functions
for mean-field continuum dislocation dynamics. However, we also discuss some
implications of our findings which gesture towards the possibility of
stochastic theories of continuum dislocation motion as well.

###  VII.1 Correlation effects in mean-field continuum dislocation dynamics

Current vector-density continuum dislocation dynamics simulations only treat
the mean-field portion of the kinetic equations [Eqs. (20-22)]. That is, they
ignore the correlation force given by Eq. (22). The mean field force term is
generally evaluated by considering the mechanical solution to the eigenstrain
produced by the Kröner-Nye tensor (El-Azab 2018):

$\displaystyle\nabla\cdot\bm{\sigma}(\bm{r})$ $\displaystyle=0,$ (68)
$\displaystyle\bm{\sigma}(\bm{r})$
$\displaystyle:=\mathbb{C}\;:\;\left(\nabla\bm{u}_{\mathrm{tot}}-\bm{\beta}_{\mathrm{p}}(\bm{r})\right)_{\mathrm{sym}},$
(69) $\displaystyle\bm{\beta}_{\mathrm{p}}(\bm{r})$
$\displaystyle=\nabla\bm{z}(\bm{r})+\bm{\chi}(\bm{r}),$ (70)
$\displaystyle\nabla\times\bm{\chi}(\bm{r})$
$\displaystyle=\bm{\alpha}(\bm{r}),$ (71) $\displaystyle\bm{\alpha}(\bm{r})$
$\displaystyle:=\sum_{\eta=1}^{N_{S}}\bm{\rho}^{[\eta]}(\bm{r})\otimes\bm{b}^{[\eta]}.$
(72)

These equations allow the total displacement field–and hence the stress field
$\bm{\sigma}$–to be calculated from the dislocation arrangement. They do so in
the following manner. The total displacement gradient is (additively)
decomposed into an elastic part and a plastic part. These are distinguished by
the constitutive relation [Hooke’s law, Eq. (69)], whereby only the elastic
distortion $\nabla\bm{u}_{\mathrm{tot}}-\bm{\beta}_{\mathrm{p}}$ is used to
compute the stress field. The plastic distortion ($\bm{\beta}_{\mathrm{p}}$),
of course, can be decomposed [Eq. (70)] into a compatible part–i.e. a part
which can be represented by a gradient of some plastic displacement field
$\bm{z}$–and an incompatible part $\bm{\chi}$ which has a nontrivial curl. The
closure fault of this plastic distortion corresponds precisely to the Burgers
circuit definition of dislocations, allowing the curl of the incompatible
distortion to be represented in terms of the Kröner-Nye tensor $\bm{\alpha}$.
The Kröner-Nye tensor, being a measure of the geometrically necessary
dislocation content of the crystal region, is given straightforwardly in terms
of the dislocation densities [Eq. (72)]. In practice, the constitutive
equations [Eqs. (69,68)] give a second order differential equation for the
elastic displacements $\bm{u}_{\mathrm{tot}}-\bm{z}$ in terms of the
dislocation arrangement. Once these displacements are fixed, Hooke’s law [Eq.
(69)] yields the stress field. This solution scheme, known as field
dislocation mechanics (FDM) is commonly solved by finite element [22] or
spectral methods [27].

A well-known but oft-minimized downfall of the FDM approach is that the short-
range stress contribution is ignored, resulting in a dislocation transport
equation which is, a priori, incorrect [31]. One creative approach to solve
this problem is to utilize FDM within DDD simulations to solve for the long-
range stress field while solving the short-range stresses and dislocation
transport in terms of the discrete dislocation lines [10]. This may prove
advantageous to DDD formalisms by reducing computational cost, but continuum
formalisms are necessary to push into geometrically nonlinear regimes of
strain [23, 56]. In continuum dislocation dynamics, a correlation dependent
correction to the transport equation is necessary to capture the effects of
short-range stresses.

The specific form of the correlation force term
$\bm{F}_{\mathrm{C}}^{[\alpha]}$ leverages the weak line bundle assumption to
allow a straightforward calculation of the effective stress field needed to
recapture the true dislocation dynamics. While the mean-field contribution to
the mechanics is best calculated by FDM (with which it is possible to treat
the nuances of geometric nonlinearity), the correlation portion can be treated
using the infinitesimal stress kernel due to the short-range nature of the
integral: the stress kernel converges as $O(r^{-1})$ and the correlations
presented here are of finite range.

The degree of non-locality which needs to be considered in a model that
evaluates the correlation effects can be gleaned from the radial convergence
of the correlation functions (Fig. 5). As previously noted, these decay to an
uncorrelated state over 2-4 coarse-graining lengths. We note that, by
construction, the coarse-graining length represents the typical variation
length of the density field. As a result, a local density approximation to the
correlation force functional would be inappropriate. While it is possible that
gradient theories may be considered to deal with this non-locality, further
investigation is needed to understand what mathematical tools would be needed
to evaluate the non-local correlation force integral.

Having briefly touched upon possible implementations of the correlation force,
we now ask: what is gained by incorporation of the short-range stress effects?
Energetically, the correlation introduces an alternative mechanism to store
strain energy in local neighborhoods (i.e. at small separations). Kinetically,
this should result in the system preferring the coagulation of dislocation
densities to minimize the more costly long-range strain energies. This
shielding effect is thought to be the key to the onset of dislocation pattern
formation in deformed metals. It is well known that the inclusion of a
posteriori phenomenological local energy storage terms (i.e. Taylor-type
hardening terms) in CDD models leads to rudimentary pattern formation,
although at low strains compared to experiment [57, 58, 25]. Our expectation
is that inclusion of the short-range stress effects in the form of correlation
dependent transport terms will naturally give rise to dislocation patterns
without the need to resort to ad hoc terms designed to force pattern formation
[39].

###  VII.2 Stochastic effects

A closure of the density hierarchy at any order will at some level mean
projecting away the fluctuations of the higher order densities (i.e., higher
than 2-point). These fluctuations are commonly ignored, but they do play a
role in the true dynamics of the system, especially when the six-dimensional
pair space is reduced to a three-dimensional separation space. As was
immediately clear in the present investigation of the correlation functions
(cf. Fig. 3), the linear relationship between the discrete and continuum
density products $D=gC$ does not account for all of the variance in the value
of D. In fact, as is seen from Figs. 3 and 8, a significant amount of
fluctuation variance remains in the sense of Eq. (51). This variance could be
incorporated into the model by means of a suitable stochastic process. In such
a case, the stochastic process would be superadded to the correlation
relationship as follows:

$\displaystyle D^{[\mathrm{type}]}_{ij}(\bm{r},\bm{r}^{\prime})=$
$\displaystyle
C^{[\mathrm{type}]}_{ij}(\bm{r},\bm{r}^{\prime})g^{[\mathrm{type}](i,j)}(\bm{r}-\bm{r}^{\prime})$
$\displaystyle\;\;+X^{[\mathrm{type}](i,j)}_{s}(\bm{r}-\bm{r}^{\prime}).$ (73)

The exact stochastic process that would best suit this relationship is beyond
the scope of this work. However, for heuristic purposes, we consider the
following:

$X^{[\mathrm{type}](i,j)}_{s}(\bm{\Delta
r}):=\frac{1}{g^{[\mathrm{type}](i,j)}(\bm{\Delta r})}\varsigma(\bm{\Delta
r})B_{s},$ (74)

where $B_{s}$ is some elementary stochastic process, say a Brownian motion. By
the dependence on $g^{-1}$, this process is naturally uncorrelated to the
correlation effects (via the principal component analysis). The standard
deviation of this process $\varsigma$ can then be given a suitable functional
form in terms of the distance along the correlation axis, as it appears from
Fig. 3 that there are scaling effects at play.

The above is not meant, by any means, to be a firmly held hypothesis. Rather
it is simply a gesture at one possible direction of development one might
pursue in order to consider the effects of higher than second order density
fluctuations. If pursued, this line of reasoning would be well-suited to
investigations in terms of established projection operator methods such as the
GENERIC framework [59]. This projection operator approach to the stochastic
effects would differ in method but be similar in results to two-dimensional
investigations into the density-based mobility of dislocations [60]. That is,
the higher-order effects could alter the drag law defining the relationship
between effective Peach-Koehler force and the time derivative of the density
field.

##  VIII Concluding Remarks

In this work, we have given a full treatment of the coarse-grained kinetics
proper to a vector-density based theory of continuum dislocation dynamics. The
kinetic theory which we developed naturally gave rise to dislocation
correlation functions, which were defined in a straightforward manner.
Utilizing a novel, chiral classification of the FCC slip systems, we were able
to show that the full number of correlation functions needed to describe the
kinetics of the vector-densities is 24, corresponding to 6 interaction classes
and interactions between 4 various combinations of vector-density components.
All 24 of these correlation functions were then computed from discrete
dislocation dynamics simulations. The same were seen to depend strongly on the
coarse-graining length used to define the continuum vector-density, but only
weakly on the cumulative plastic strain and total dislocation density. This
implies that the correlations in vector-density schemes primarily encode line
connectivity information rather than some favorable microstructural
arrangement of dislocations.

The evaluation of the correlation functions unveiled several key features
thereof. First, the dislocation correlations were found to be of moderate
range on the order of 2-4 times the coarse-graining lengths. The self-
correlations, corresponding to the interaction of dislocations of like slip
system, were seen to be more strongly anisotropic than the correlation of
dislocations of unlike slip systems. Of these cross-correlations, the cross-
slip type interactions were seen to have the strongest correlations, while the
coplanar type interactions showed negligible correlation.

The implications of the present work for future progress in vector-density
continuum dislocation dynamics are profound. In discussion of these
implications we especially note the effect of the correlation functions on
driving forces in the transport equations, where it is expected that these
correlations can open a mechanism for dislocation pattern formation in
deformed metals. Additionally, we discussed the stochastic fluctuations not
captured in the correlation function which the present investigation brought
to light and how these might lead in future investigations to a coarse-grained
mobility function.

In summary, we have presented here the complete correlation information
necessary to compute the average evolution of the dislocation density field.
The full set of dislocation correlation functions now lies at our fingertips
and the short-range interactions of dislocations can now be recovered in
vector-density continuum dislocation dynamics. With this new tool, we have
scaled one more barrier in our understanding of dislocations systems; what new
plastic phenomena may now lie within our sight?

## Appendix

###  VIII.1 1\. Form of the Peach-Koehler functional

In discrete dislocation dynamics, the velocity law used is normally one of
overdamped motion. That is, there is some force field $\bm{f}$ which acts on a
dislocation, and the velocity simply linearly related to that force:

$B\bm{v}(\bm{r}):=\bm{f}(\bm{r}),$ (75)

where $B$ is a viscous drag coefficient. In face-centered cubic crystals, a
tensorial factor of $(\mathbb{I}-\bm{n}^{[\alpha]}\otimes\bm{n}^{[\alpha]})$
is applied to project the force onto the slip plane [44], but this will be
omitted for brevity.

Very early in the development of dislocation theory, it was realized that the
configurational force which acts on the dislocation line–the negative of the
work done by moving a dislocation some small distance–is given by the famous
Peach-Koehler force [7]:

$\displaystyle\bm{f}(\bm{r})$
$\displaystyle:=\left(\bm{b}^{[\alpha]}\cdot\bm{\sigma}(\bm{r})\right)\times\bm{\xi}(\bm{r})$
(76)
$\displaystyle=\left[\bm{b}^{[\alpha]}\cdot\left(\bm{\sigma}^{0}+\bm{\sigma}^{\mathrm{(int)}}\right)\right]\times\bm{\xi}^{[\alpha]},$
(77)

where we have introduced the stress field $\bm{\sigma}$ and its decomposition
into a homogenous remote stress $\bm{\sigma}^{0}$ and the internal stress
field $\bm{\sigma}^{\mathrm{(int)}}$ arising from the other dislocations in
the system.

In an isotropic elastic medium, the internal stress field can be expressed as
an integral with respect to the discrete dislocation density [61]. That is,
the internal stress field is a functional of the discrete dislocation density:

$\sigma^{\mathrm{(int)}}_{ij}=\sum_{\beta=1}^{N_{S}}\int_{\mathcal{M}}\hat{\sigma}^{[\beta]}_{ijk}(\bm{r}-\bm{r}^{\prime})^{[\beta]}\varrho^{[\beta]}_{k}(\bm{r}^{\prime})d^{3}\bm{r}^{\prime}$
(78)

where $\hat{\sigma}^{[\beta]}_{ijk}$ represents the third rank Green’s
function for the stress field produced by a dislocation with Burgers vector
$\bm{b}^{[\beta]}$. This Green’s function is given by

$\displaystyle\frac{4\pi}{\mu}\hat{\sigma}_{ijk}^{[\beta]}(\bm{r})$
$\displaystyle=\frac{1}{2}\left(s^{[\beta]}_{i}(\bm{r})\delta_{jk}+s^{[\beta]}_{j}(\bm{r})\delta_{ik}\right)+\frac{1}{1-\nu}S^{[\beta]}_{ijk}(\bm{r})$
(79) $\displaystyle s^{[\beta]}_{i}(\bm{r})$
$\displaystyle:=(\bm{b}^{[\beta]}\times\nabla)_{i}\left(\frac{1}{|\bm{r}|}\right)$
(80) $\displaystyle S^{[\beta]}_{ijk}(\bm{r})$
$\displaystyle:=(\bm{b}^{[\beta]}\times\nabla)_{k}\left(G_{ij}-\frac{2\delta_{ij}}{|\bm{r}|}\right)$
(81) $\displaystyle G_{ij}(\bm{r})$
$\displaystyle:=(\nabla\otimes\nabla)|\bm{r}|=\frac{\delta_{ij}}{|\bm{r}|}-\frac{r_{i}r_{j}}{|\bm{r}|^{3}}$
(82)

It can be seen that the stress kernel itself is independent of any line
information. Thus, it suffices to express the of Peach-Koehler force as a
vector valued functional of the line information as follows:

$\left(\bm{F}_{\mathrm{PK}}^{[\alpha]}\left[\bm{\xi}\otimes\bm{\varrho}^{[\beta]}(\bm{r}^{\prime})\right]\right)_{l}:=\epsilon_{jkl}b^{[\alpha]}_{i}\xi^{[\alpha]}_{k}(\bm{r})\left(\sigma_{ij}^{0}+\sum_{\beta=1}^{N_{S}}\int_{\mathcal{M}}\hat{\sigma}^{[\beta]}_{ijm}(\bm{r}-\bm{r}^{\prime})\varrho^{[\beta]}_{m}(\bm{r}^{\prime})d^{3}\bm{r}^{\prime}\right)$
(83)

As a final comment regarding notation, we note the brackets represent an
integration over $\bm{r}^{\prime}$. Additionally, the quantity in brackets
denotes all dependences on the line information. This notation will be
utilized throughout the main text to refer to the Peach-Koehler functional.

###  VIII.2 2\. Line Integration of the cloud-in-cell weight function

For reference, we give here the line integral of a straight line segment with
respect to the cloud-in-cell weight function. In our previous work [33] we
used a form of this that had a small error (fourth order in a parameter always
less than 1), following a typographical error in [10]. It should be noted that
the correct form is found in the preprint of the same [62]. However, we will
restate the form of this line integral for clarity.

We would like to integrate the dislocation configuration with respect to our
cloud-in-cell weight function $w_{L}$ given by Eq. (52). In general, this
evaluation will be centered at some $\bm{r}_{D}$; for brevity, we consider it
to be centered at the origin. The support of the weight function $\Omega_{L}$
is given by a cube of side length $2L$ centered on the origin. This cube is
composed of eight octants $\Omega_{m}$, cubes with side length $L$ with one
corner on the origin.

Because the discrete dislocation configuration is composed of straight line
segments ($\mathcal{L}=\bigcup_{q}\lambda_{q}$), we represent the dislocations
which contribute to the line integral at this point by $\lambda_{p}^{(L)}$.
Each segment $\lambda$ can be expressed in parametric form in terms of its
origin $\bm{x}$, endpoint $\bm{x}+\bm{t}$, and a dimensionless parameter
$a\in[0,1]$ as:

$\bm{r}^{\lambda}(a):=\bm{x}+a\bm{t}.$ (84)

It then remains to compute the integral of the weight function along this
line.

In order to integrate the absolute value functions present, we use
$\Omega_{m}$ to break the line into segments on which the sign of the three
coordinates remains constant. Letting $a_{m}^{\pm}$ represent the line
parameters at which $\lambda$ enters and leaves $\Omega_{m}$,999The line
parameters of intersection with a rectangular grid can be efficiently computed
using Siddon’s algorithm [63].  we can represent by $s_{i}^{m}=\pm 1$ the sign
of the coordinate $r_{i}(a)$ in the range $a\in[a_{m}^{-},a_{m}^{+}]$.
Utilizing the following expansion of the cloud-in-cell function [Eq. (52)],

$w_{L}(\bm{x},\bm{s}):=\frac{1}{L_{1}L_{2}L_{3}}\left[1-\sum_{i}\frac{s_{i}}{L_{i}}x_{i}+\sum_{i<j}\frac{s_{i}s_{j}}{L_{i}L_{j}}x_{i}x_{j}+\frac{s_{1}s_{2}s_{3}}{L_{1}L_{2}L_{3}}x_{1}x_{2}x_{3}\right],$
(85)

the integral along this portion of the line can be evaluated as follows.

$\displaystyle I_{m}$
$\displaystyle:=|\bm{t}|\int_{a_{m}^{-}}^{a_{m}^{+}}w_{L}(\bm{r}^{\lambda}(a),\bm{s}^{m})da$
(86)
$\displaystyle=|\bm{t}|(a_{m}^{+}-a_{m}^{-})-|\bm{t}|\sum_{i}\frac{s_{i}}{L_{i}}A_{i}+|\bm{t}|\sum_{i<j}\frac{s_{i}s_{j}}{L_{i}L_{j}}B_{ij}+\frac{s_{1}s_{2}s_{3}}{L_{1}L_{2}L_{3}}C_{123},$
(87)

where we individually compute the various coordinate product integrals:

$\displaystyle A_{i}$
$\displaystyle:=\int_{a_{m}^{-}}^{a_{m}^{+}}(x_{i}+at_{i})da$ (88)
$\displaystyle=\left[ax_{i}+\frac{1}{2}a^{2}t_{i}\right]_{a_{m}^{-}}^{a_{m}^{+}}$
(89) $\displaystyle B_{ij}$
$\displaystyle:=\int_{a_{m}^{-}}^{a_{m}^{+}}(x_{i}+at_{i})(x_{j}+at_{j})da$
(90)
$\displaystyle=\int_{a_{m}^{-}}^{a_{m}^{+}}(x_{i}x_{j}+a(x_{i}t_{j}+x_{j}t_{i})+a^{2}t_{i}t_{j})da$
(91)
$\displaystyle=\left[ax_{i}+\frac{1}{2}a^{2}(x_{i}t_{j}+x_{j}t_{i})+\frac{1}{3}a^{3}t_{i}t_{j}\right]_{a_{m}^{-}}^{a_{m}^{+}}$
(92) $\displaystyle C_{ijk}$
$\displaystyle:=\int_{a_{m}^{-}}^{a_{m}^{+}}(x_{i}+at_{i})(x_{j}+at_{j})(x_{k}+at_{k})da$
(93)
$\displaystyle=\int_{a_{m}^{-}}^{a_{m}^{+}}\left(x_{i}x_{j}x_{k}+a(x_{i}x_{k}t_{j}+x_{j}x_{k}t_{i}+x_{i}x_{j}t_{k})+a^{2}(x_{i}t_{i}t_{j}+x_{i}t_{j}t_{k}+x_{j}t_{i}t_{k})+a^{3}t_{i}t_{j}t_{k}\right)da$
(94)
$\displaystyle=\left[x_{i}x_{j}x_{k}+\frac{1}{2}a^{2}(x_{i}x_{k}t_{j}+x_{j}x_{k}t_{i}+x_{i}x_{j}t_{k})+\frac{1}{3}a^{3}(x_{i}t_{i}t_{j}+x_{i}t_{j}t_{k}+x_{j}t_{i}t_{k})+\frac{1}{4}a^{4}t_{i}t_{j}t_{k}\right]_{a_{m}^{-}}^{a_{m}^{+}}$
(95)

Finally, this allows us to express the total line integral, given by

$\displaystyle I_{\lambda}$
$\displaystyle=|\bm{t}|\int_{0}^{1}w_{L}\left(\bm{r}^{\lambda}(a)\right)da$
(96) $\displaystyle=\sum_{m=1}^{8}I_{m}.$ (97)

The total of the line integral for all segments passing through $\Omega_{L}$
gives a scalar line density field, while incorporating the line direction
gives the vector density field:

$\displaystyle\rho(\bm{r}_{D};L)$
$\displaystyle=\sum_{q}I_{\lambda^{(L)}_{q}},$ (98)
$\displaystyle\bm{\rho}(\bm{r}_{D};L)$
$\displaystyle=\sum_{q}I_{\lambda^{(L)}_{q}}\frac{\bm{t}_{\lambda^{(L)}_{q}}}{|\bm{t}_{\lambda^{(L)}_{q}|}}.$
(99)

If only a single segment passes through the volume, the scalar density and the
norm of the vector density will be equal. In general, however, this is not the
case, and the polarization $|\bm{\rho}|/\rho$ is generally less than unity due
to the presence of two or more straight line segments.

## Acknowledgment

This work was supported by the US Department of Energy, Office of Science,
Division of Materials Sciences and Engineering, through award number DE-
SC0017718 at Purdue University.

## References

  * Orowan [1934] E. Orowan, Zeitschrift für Physik 89, 605 (1934).
  * Taylor [1934] G. I. Taylor, Proceedings of the Royal Society of London, Series A. 145, 362 (1934).
  * Polanyi [1934] M. Polanyi, Zeitschrift für Physik 89, 660 (1934).
  * Burgers [1939] J. Burgers, Proc. Acad. Science 42, 293 (1939).
  * Peach and Koehler [1950] M. Peach and J. S. Koehler, Physical Review 80, 436 (1950).
  * Thompson [1953] N. Thompson, Proceedings of the Physical Society. Section B 66, 481 (1953).
  * Hochrainer [2007] T. Hochrainer,  _Evolving systems of curved dislocations: mathematical foundations of a statistical theory_ , Ph.D. thesis, Karlsruhe Institute of Technology (2007).
  * Zaiser [2015] M. Zaiser, Physical Review B 92, 174120 (2015), arXiv:arXiv:1508.03652v2 .
  * Kubin [2013] L. Kubin,  _Dislocations, mesoscale simulations and plastic flow_ , Vol. 5 (Oxford University Press, 2013).
  * Bertin [2019] N. Bertin, International Journal of Plasticity 122, 268 (2019), arXiv:1804.00803 .
  * Mermin [1979] N. D. Mermin, Reviews of Modern Physics 51, 591 (1979).
  * Nye [1953] J. F. Nye, Acta Metallurgica 1, 153 (1953).
  * DeWit [1973] R. DeWit, Journal of Research of the National Bureau of Standards. Section A, Physics and Chemistry 77, 49 (1973).
  * Kröner [2001] E. Kröner, International Journal of Solids and Structures 38, 1115 (2001).
  * Acharya and Roy [2006] A. Acharya and A. Roy, Journal of the Mechanics and Physics of Solids 54, 1687 (2006).
  * Roy and Acharya [2006] A. Roy and A. Acharya, Journal of the Mechanics and Physics of Solids 54, 1711 (2006).
  * Acharya  _et al._ [2006] A. Acharya, A. Roy, and A. Sawant, Scripta Materialia 54, 705 (2006).
  * Arsenlis and Parks [1999] A. Arsenlis and D. Parks, Acta Materialia 47, 1597 (1999).
  * Sedláček  _et al._ [2003] R. Sedláček, J. Kratochvíl, and E. Werner, Philosophical Magazine 83, 3735 (2003).
  * Mura [1963] T. Mura, Philosophical Magazine 8, 843 (1963).
  * Lin and El-Azab [2020] P. Lin and A. El-Azab, Modelling and Simulation in Materials Science and Engineering 28, 045003 (2020), arXiv:1910.12766 .
  * Lin  _et al._ [2021] P. Lin, V. Vivekanandan, K. Starkey, B. Anglin, C. Geller, and A. El-Azab, International Journal of Plasticity 138, 102943 (2021).
  * Anderson  _et al._ [2022] J. P. Anderson, V. Vivekanandan, P. Lin, K. Starkey, Y. Pachaury, and A. El-Azab, Journal of Engineering Materials and Technology 144, 10.1115/1.4052066 (2022).
  * Hochrainer [2015] T. Hochrainer, Philosophical Magazine 95, 1321 (2015).
  * Sandfeld and Zaiser [2015] S. Sandfeld and M. Zaiser, Modelling and Simulation in Materials Science and Engineering 23, 065005 (2015), arXiv:1501.03527 .
  * Monavari  _et al._ [2016] M. Monavari, S. Sandfeld, and M. Zaiser, Journal of the Mechanics and Physics of Solids 95, 575 (2016), arXiv:1509.05617v5 .
  * Brenner  _et al._ [2014] R. Brenner, A. Beaudoin, P. Suquet, and A. Acharya, Philosophical Magazine 94, 1764 (2014).
  * Bertin  _et al._ [2015] N. Bertin, M. V. Upadhyay, C. Pradalier, and L. Capolungo, Modelling and Simulation in Materials Science and Engineering 23, 065009 (2015).
  * Graham  _et al._ [2016] J. T. Graham, A. D. Rollett, and R. LeSar, Modelling and Simulation in Materials Science and Engineering 24, 085005 (2016).
  * Morin  _et al._ [2019] L. Morin, R. Brenner, and P. Suquet, Modelling and Simulation in Materials Science and Engineering 27, 085012 (2019).
  * Vivekanandan  _et al._ [2022] V. Vivekanandan, J. P. Anderson, Y. Pachaury, M. S. Mohamed, and A. El-Azab, Modelling and Simulation in Materials Science and Engineering 10.1088/1361-651X/AC5DCF (2022).
  * Hochrainer  _et al._ [2022] T. Hochrainer, B. Weger, and S. Gupta, Materials Theory 2022 6:1 6, 1 (2022).
  * Anderson and El-Azab [2021] J. P. Anderson and A. El-Azab, Materials Theory 5, 1 (2021).
  * Essmann and Mughrabi [1979] U. Essmann and H. Mughrabi, Philosophical Magazine A 40, 731 (1979).
  * Xia and El-Azab [2015] S. Xia and A. El-Azab, Modelling and Simulation in Materials Science and Engineering 23, 10.1088/0965-0393/23/5/055009 (2015).
  * Groma  _et al._ [2021] I. Groma, P. D. Ispánovity, and T. Hochrainer, Physical Review B 103, 174101 (2021), arXiv:2012.12560 .
  * Hochrainer [2016] T. Hochrainer, Journal of the Mechanics and Physics of Solids 88, 12 (2016).
  * Groma  _et al._ [2016] I. Groma, M. Zaiser, and P. D. Ispánovity, PHYSICAL REVIEW B 93, 214110 (2016).
  * Ispánovity  _et al._ [2020] P. D. Ispánovity, S. Papanikolaou, and I. Groma, Physical Review B 101, 10.1103/PhysRevB.101.024105 (2020), arXiv:1708.03710v1 .
  * Cai  _et al._ [2006] W. Cai, A. Arsenlis, C. Weinberger, and V. Bulatov, Journal of the Mechanics and Physics of Solids 54, 561 (2006).
  * Po  _et al._ [2018] G. Po, M. Lazar, N. C. Admal, and N. Ghoniem, International Journal of Plasticity 103, 1 (2018), arXiv:1706.00828 .
  * Starkey  _et al._ [2022] K. Starkey, T. Hochrainer, and A. El-Azab, Journal of the Mechanics and Physics of Solids 158, 104685 (2022).
  * Cai and Bulatov [2004] W. Cai and V. V. Bulatov, Materials Science and Engineering: A 387-389, 277 (2004).
  * Sills  _et al._ [2016] R. B. Sills, W. P. Kuykendall, A. Aghaei, and W. Cai, in  _Springer Series in Materials Science_, Vol. 245 (Springer Verlag, 2016) pp. 53–87.
  * Xia [2015] S. Xia,  _Continuum Dislocation Dynamics Modeling of the Deformation of FCC Single Crystals_ , Ph.D. thesis, Purdue University (2015).
  * Hull and Bacon [2011] D. Hull and D. J. Bacon, Introduction to Dislocations 10.1016/C2009-0-64358-0 (2011).
  * Devincre  _et al._ [2011] B. Devincre, R. Madec, G. Monnet, S. Queyreau, R. Gatti, and L. Kubin, in  _Mechanics of Nano-Objects_ (Presses des MINES, 2011) pp. 81–99.
  * Allen and Thomas [1999] S. M. Allen and E. L. Thomas,  _The Structure of Materials_ (John Wiley & Sons, New York, 1999) p. 218.
  * Lomer [1951] W. M. Lomer, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 42, 1327 (1951).
  * Cottrell [1952] A. Cottrell, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 43, 645 (1952).
  * Hirth [1961] J. P. Hirth, Journal of Applied Physics 32, 700 (1961).
  * Anderson [2023] J. P. Anderson,  _The Statistical Foundations of Line Bundle Continuum Dislocation Dynamics_ , Ph.d., Purdue University (2023).
  * Birdsall and Fuss [1969] C. K. Birdsall and D. Fuss, Journal of Computational Physics 3, 494 (1969).
  * Shlens [2014] J. Shlens, arXiv:1404.1100 [cs.LG] (2014).
  * Groma and Balogh [1999] I. Groma and P. Balogh, Acta Materialia 47, 3647 (1999).
  * Starkey  _et al._ [2020] K. Starkey, G. Winther, and A. El-Azab, Journal of the Mechanics and Physics of Solids 139, 103926 (2020).
  * Vivekanandan  _et al._ [2021] V. Vivekanandan, P. Lin, G. Winther, and A. El-Azab, Journal of the Mechanics and Physics of Solids 149, 104327 (2021).
  * Xia  _et al._ [2016] S. Xia, J. Belak, and A. El-Azab, Modelling and Simulation in Materials Science and Engineering 24, 075007 (2016).
  * Öttinger [1998] H. C. Öttinger, Physical Review E 57, 1416 (1998).
  * Kooiman  _et al._ [2015] M. Kooiman, M. Hütter, and M. Geers, Journal of Statistical Mechanics: Theory and Experiment 2015, P06005 (2015).
  * Hirth and Lothe [1968] J. P. Hirth and J. Lothe,  _Theory of Dislocations_ (McGraw-Hill, New York, 1968) p. 99.
  * Bertin [2018] N. Bertin, arXiv:1804.00803 [physics.comp-ph] (2018).
  * Jacobs  _et al._ [1998] F. Jacobs, E. Sundermann, B. De Sutter, M. Christiaens, and I. Lemahieu, Journal of Computing and Information Technology 6, 89 (1998).