# Symmetry-adapted yield criterion for non-Schmid materials

Roman Gröger<EMAIL_ADDRESS>Institute of Physics of Materials and CEITEC IPM,
Czech Academy of Sciences, Žižkova 22, 61600 Brno, Czech Republic

###### Abstract

The current form of the effective yield criterion that is used to describe
non-Schmid behavior of dislocations in crystalline materials is not invariant
under the simultaneous change of the slip direction and the slip plane normal
as dictated by crystallography. Here, we explore the consequences of this
symmetry on the derivation of the effective stress. This is defined by a
linear combination of six stress functions that are represented by their
Maclaurin series. Only the lowest order terms that obey this symmetry are
retained, which results in a homogeneous effective stress of order two. It is
shown that the quadratic terms can be further eliminated by resolving the
respective stresses into another coordinate system in the zone of the slip
direction. The application of the ensuing linear yield criterion to body-
centered cubic and hexagonal close-packed metals shows under which conditions
the non-Schmid stress terms become significant in predicting the onset of
yielding. In the special case, where the contributions of all non-Schmid
stresses vanish, this model reduces to the maximum shear stress theory of
Tresca.

###### keywords:

non-Schmid stresses , yield criterion , body-centered cubic , hexagonal close-
packed , symmetry , crystal plasticity

##  1 Introduction

Yield criteria play indispensable roles in identifying the loading conditions
that cause the initiation of elastic-plastic deformation (20). At the single-
crystal level, the onset of plastic deformation is associated with the stress
state that activates dislocation glide on the most highly stressed slip system
compatible with the underlying crystallographic space group. A detailed
understanding of the effect of each stress component is thus a prerequisite to
predicting the activation of individual slip systems under the applied load.
Schmid and Boas (31) postulated that slip on a particular crystallographic
system commences when the shear stress parallel to the slip direction acting
in the slip plane reaches its critical value. This idea constitutes a central
point in the maximum shear stress theory developed by Tresca (37). Within this
theory, one tacitly assumes that the microscopic mechanism of yielding remains
unaltered by coarse-graining through some six to eight orders of magnitude of
the length scale.

Almost a century ago, Taylor and Elam (36, 35) pointed out that single
crystals of $\alpha$-Fe and $\beta$-CuZn do not behave according to the rules
established by the Schmid law. Since then, the pool of materials for which the
Schmid law is not capable of correctly predicting the initiation of plastic
flow grew steadily to include body-centered cubic (bcc) metals (9, 13, 21), B2
and L12 intermetallics (28), CuZnAl (1) and NiTi shape memory alloys (2). It
is now widely accepted that the main distinguishing factor between the
yielding of close-packed and non-close-packed metals are non-planar cores of
screw dislocations in the latter structures (21, 6, 38). Moreover, the
widespread observation of the breakdown of the Schmid law implies that
Tresca’s model of yielding is a special case of a more general yield
criterion.

The importance of non-Schmid stresses was first recognized by Asaro and Rice
(4) on the basis of their finite deformation studies of strain localization in
ductile crystals. They observed the formation of shear bands that formed due
to stress bifurcations, which were attributed to micromechanical processes
such as cross-slip of screw dislocations. If the crystal deforms by multiple
slip, lattice rotation plays an important role in the localization process
(11). In order to quantify the effect of non-Schmid stresses on the initiation
of elastic-plastic deformation, Qin and Bassani (30) proposed a simple yield
criterion

$\tau^{*\alpha}(\bm{\sigma},\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha})\leq\tau^{*}_{cr}\
,$ (1)

where the effective stress $\tau^{*\alpha}$, corresponding to the slip system
$\alpha$, was written as a linear combination of several components of the
stress tensor $\bm{\sigma}$ with coefficients representing the relative
importance of each stress. The work of Duesbery (13), Ito and Vitek (21), as
well as our more recent contribution (14, 18), have demonstrated that the
relevant stress components are primarily shear stresses acting both parallel
and perpendicular to the slip direction. To incorporate these effects, the
effective stress assigned to the slip system $\alpha$ was expressed using
stress components resolved in two orthogonal systems rotated by $60^{\circ}$
in the zone of the slip direction $\hat{\bm{m}}^{\alpha}$. One has the slip
plane normal $\hat{\bm{n}}^{\alpha}$, whereas the other (auxiliary) plane has
normal $\hat{\bm{n}}^{\prime\alpha}$. In terms of the stresses resolved in
these systems, the effective stress was defined as

$\tau^{*\alpha}=\sigma_{nm}+a_{1}\sigma_{n^{\prime}m}+a_{2}\sigma_{tn}+a_{3}\sigma_{t^{\prime}n^{\prime}},$
(2)

where $\sigma_{nm}$ and $\sigma_{n^{\prime}m}$ are shear stresses parallel to
the slip direction resolved in the two systems, and $\sigma_{tn}$,
$\sigma_{t^{\prime}n^{\prime}}$ the shear stresses perpendicular to the slip
direction resolved in the same two systems. In the latter two stress
components, the vectors
$\hat{\bm{t}}^{\alpha}=\hat{\bm{n}}^{\alpha}\times\hat{\bm{m}}^{\alpha}$ and
$\hat{\bm{t}}^{\prime\alpha}=\hat{\bm{n}}^{\prime\alpha}\times\hat{\bm{m}}^{\alpha}$.

Detailed atomistic simulations using empirical potentials or the Density
Functional Theory (DFT) must be employed to obtain the coefficients $a_{1}$,
$a_{2}$, $a_{3}$ in (2) and the effective yield stress $\tau^{*}_{cr}$ in (1)
to parameterize the yield criterion for a particular material. So far, these
developments have been made for the bcc metals Mo (15, 16, 17), W (15, 16, 17,
7, 24), Ta (3), ferromagnetic $\alpha$-Fe (8), non-magnetic phase of Cr (19),
as well as for the shape memory alloys NiTi (2), and CuZnAl (1). The
orientational dependence of the critical resolved shear stress for all bcc
metals was characterized also in DFT simulations by Dezerald et al. (12).
Similar formulations of the yield criteria that go beyond the Schmid law were
proposed independently by other authors (40, 23, 25). Although non-Schmid
effects in bcc metals originate from non-planar cores of $1/2\langle
111\rangle$ screw dislocations, their effect was also observed in interacting
dislocation networks studied using discrete dislocation dynamics models (33,
39). The yield criteria involving non-Schmid stress terms form the basis of
constitutive equations used in a number of crystal plasticity models (22, 29,
7), and in kinetic Monte Carlo calculations (34). The mathematical basis of
these models in the framework of finite deformations was investigated by
Cleja-Ţigoiu and Paşcan (10), and Soare (32).

The effective yield criterion should not only capture the results of atomistic
simulations and experiments, but the symmetry of the effective stress must be
compatible with crystallography. To be more specific, consider an arbitrary
slip system $\alpha$ that is characterized by two unit vectors:
$\hat{\bm{n}}^{\alpha}$ representing the normal of the slip plane, and
$\hat{\bm{m}}^{\alpha}$ that is parallel to the slip direction. The two
possible choices of the slip plane normal, i.e. $\pm\hat{\bm{n}}^{\alpha}$,
suggest that the Burgers circuit around the dislocation can be made in two
different directions. While these choices are equivalent, they lead to
opposite slip directions, $\pm\hat{\bm{m}}^{\alpha}$ and thus to opposite
directions of the Burgers vector of the dislocation,
$\pm\hat{\bm{b}}^{\alpha}=\pm\hat{\bm{m}}^{\alpha}b$, where $b$ is the
magnitude of the Burgers vector. However, the Burgers vector is a topological
quantity and does not depend on the coordinate frame. In the spirit of
Neumann’s principle (26), we thus demand that any physical quantity involving
the slip system $\alpha$ must be invariant under a _simultaneous_ change of
sign of the vectors $\hat{\bm{m}}^{\alpha}$ and $\hat{\bm{n}}^{\alpha}$.
Therefore, the effective stresses involved in all single crystal yield
criteria must obey the inversion symmetry

$\tau^{*\alpha}(\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha}|\bm{\sigma})=\tau^{*\alpha}(-\hat{\bm{m}}^{\alpha},-\hat{\bm{n}}^{\alpha}|\bm{\sigma})\
.$ (3)

It can be easily shown that Eq. (3) is the statement of invariance of
$\tau^{*\alpha}$ under rigid-body rotation about the vector
$\hat{\bm{t}}^{\alpha}$ defined above. An arbitrary lattice vector $\bm{u}$ is
expressed relative to the fixed coordinate frame spanned by the unit vectors
$\hat{\bm{X}}$, $\hat{\bm{Y}}$, $\hat{\bm{Z}}$ as
$\bm{u}=i\hat{\bm{X}}+j\hat{\bm{Y}}+k\hat{\bm{Z}}$ or as a set of three Miller
indices $\bm{u}=[ijk]$. In Fig. 1, we define the orientation of an arbitrary
slip system $\alpha$ by three lattice vectors $\hat{\bm{t}}^{\alpha}$,
$\hat{\bm{n}}^{\alpha}$, $\hat{\bm{m}}^{\alpha}$, all written relative to the
fixed frame. Assume that the effective stress for this system is
$\tau^{*\alpha}(\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha}|\bm{\sigma})$.
Now, carry out a rigid-body rotation of the crystal by 180∘ around the axis
$\hat{\bm{t}}^{\alpha}$, which brings
$\hat{\bm{m}}^{\alpha}\rightarrow-\hat{\bm{m}}^{\alpha}$ and
$\hat{\bm{n}}^{\alpha}\rightarrow-\hat{\bm{n}}^{\alpha}$. For this new
orientation, the effective stress is
$\tau^{*\alpha}(-\hat{\bm{m}}^{\alpha},-\hat{\bm{n}}^{\alpha}|\bm{\sigma})$.
Obviously, in non-polar crystal structures, rigid-body rotations by
$180^{\circ}$ are symmetry operations and all physical quantities must be
invariant under this transformation111In polar structures the rotation by
$180^{\circ}$ is not a symmetry operation.. This proves the validity of Eq.
(3).

Figure 1: Schematic illustation of the validity of (3) under rigid-body
rotation about the axis $\hat{\bm{t}}^{\alpha}$. The vectors used to define
this condition are highlighted in green.

It is evident from (2) that only the Schmid stress
$\sigma_{nm}=\bm{\sigma}:(\hat{\bm{n}}^{\alpha}\otimes\hat{\bm{m}}^{\alpha})$
in the first term obeys the symmetry (3), where : is the double-dot product
(yielding a scalar), and $\otimes$ the tensor (element-wise) product. However,
this symmetry is broken by the shear stress perpendicular to the slip
direction,
$\sigma_{tn}=\bm{\sigma}:[(\hat{\bm{n}}^{\alpha}\times\hat{\bm{m}}^{\alpha})\otimes\hat{\bm{n}}^{\alpha}]$,
which changes sign under the symmetry operation
$(\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha})\rightarrow(-\hat{\bm{m}}^{\alpha},-\hat{\bm{n}}^{\alpha})$.
The symmetry (3) is broken also for the remaining two stress terms, but this
matter requires a deeper analysis that will be made in the next Section.

Despite the fact that the yield criterion (1) with the effective stress
defined by (2) incorporates correctly the effect of non-Schmid stresses, it
does not obey the fundamental inversion symmetry (3) that is dictated by
crystallography. To remove this drawback, we first derive a general form of
the effective stress as a power series expansion in terms of all components of
the stress tensor and investigate which stress terms are allowed by the
symmetry (3). The obtained yield criterion is then applied to materials of
cubic and hexagonal symmetry to predict the primary and secondary slip systems
for a few combinations of their parameters.

##  2 Derivation of the effective stress

The stress state at a point of an elastic body is characterized by six
components of a symmetric stress tensor. An infinite number of possibilities
exist to combine these stress components to arrive at a scalar quantity
(hereafter called as an effective stress), which can be used in the yield
criteria to determine whether the material is locally in the state of purely
elastic or elastic-plastic deformation. The simplest representation of this
effective stress, due to Tresca, is based on the validity of the Schmid law.
In the following, our objective is to generalize this formulation to all
crystalline materials in which the onset of plastic deformation depends also
on other (non-Schmid) stresses.

###  2.1 Nonlinear effective stress

We begin by writing the effective stress ($\tau^{*}$) as a weighted linear
combination of six stress functions $f_{ij}(\sigma_{ij})$,

$\tau^{*}=\sum_{i=1}^{3}\sum_{j=i}^{3}c_{ij}f_{ij}(\sigma_{ij})\ .$ (4)

Each function $f_{ij}(\sigma_{ij})$ will be represented by its Maclaurin
series as
$f_{ij}(\sigma_{ij})=\sum_{n=0}^{\infty}(1/n!)\sigma_{ij}^{n}f_{ij}^{(n)}(0)$,
where the superscript $(n)$ represents the $n$-th derivative with respect to
$\sigma_{ij}$. To simplify the notation, we introduce a new set of
coefficients $c_{ijn}=c_{ij}(1/n!)f_{ij}^{(n)}(0)$. Incorporating the last two
expressions into (4) then yields an equivalent representation of the effective
stress,

$\tau^{*}=\sum_{i=1}^{3}\sum_{j=i}^{3}(c_{ij0}+c_{ij1}\sigma_{ij}+c_{ij2}\sigma_{ij}^{2}+\dots)\
.$ (5)

We first require that $\tau^{*}=0$ for zero applied stress, i.e. when all
$\sigma_{ij}=0$. This can be satisfied easily by setting all $c_{ij0}$ to
zero.

In order to write (5) for a particular slip system $\alpha$, we first resolve
all stress components $\sigma_{ij}^{\alpha}$ using the slip plane normal
$\hat{\bm{n}}^{\alpha}$ and the slip direction $\hat{\bm{m}}^{\alpha}$
corresponding to the system $\alpha$. The vector orthogonal to both is then
$\hat{\bm{t}}^{\alpha}=\hat{\bm{n}}^{\alpha}\times\hat{\bm{m}}^{\alpha}$. Only
the components of $\sigma_{ij}^{\alpha}$ that obey the inversion symmetry
defined by Eq. (3) will be retained. All three normal stresses, i.e.
$\sigma_{tt}^{\alpha}=\bm{\sigma}:(\hat{\bm{t}}^{\alpha}\otimes\hat{\bm{t}}^{\alpha})$,
$\sigma_{nn}^{\alpha}=\bm{\sigma}:(\hat{\bm{n}}^{\alpha}\otimes\hat{\bm{n}}^{\alpha})$,
and
$\sigma_{mm}^{\alpha}=\bm{\sigma}:(\hat{\bm{m}}^{\alpha}\otimes\hat{\bm{m}}^{\alpha})$,
and also the shear stress
$\sigma_{nm}^{\alpha}=\bm{\sigma}:(\hat{\bm{n}}^{\alpha}\otimes\hat{\bm{m}}^{\alpha})$
are clearly invariant under this symmetry. To obtain the simplest possible
form of the effective stress for the system $\alpha$, we retain in (5) only
the lowest-order (i.e., linear) terms containing these stresses. The remaining
two shear stresses perpendicular to the slip direction are defined as
$\sigma_{tm}^{\alpha}=\bm{\sigma}:(\hat{\bm{t}}^{\alpha}\otimes\hat{\bm{m}}^{\alpha})$
and
$\sigma_{tn}^{\alpha}=\bm{\sigma}:(\hat{\bm{t}}^{\alpha}\otimes\hat{\bm{n}}^{\alpha})$.
Both these stresses violate the aforementioned inversion symmetry in that they
change sign as
$(\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha})\rightarrow(-\hat{\bm{m}}^{\alpha},-\hat{\bm{n}}^{\alpha})$.
Therefore, in order to comply with (3), only even powers of these stresses are
allowed in the yield criterion with the lowest order being quadratic. Owing to
these arguments, it follows from Eq. (5) that the effective stress exerted on
any slip system $\alpha$ that satisfies the condition (3) has the following
form:

$\tau^{*\alpha}=c_{tt1}\sigma_{tt}^{\alpha}+c_{tn2}(\sigma_{tn}^{\alpha})^{2}+c_{tm2}(\sigma_{tm}^{\alpha})^{2}+c_{nn1}\sigma_{nn}^{\alpha}+c_{nm1}\sigma_{nm}^{\alpha}+c_{mm1}\sigma_{mm}^{\alpha}\
.$ (6)

If all non-Schmid contributions to $\tau^{*\alpha}$ vanish (i.e., when
$c_{tt1}=c_{tn2}=c_{tm2}=c_{nn1}=c_{mm1}=0$), $\tau^{*\alpha}$ must equal to
the Schmid stress $\sigma_{nm}^{\alpha}$. One possibility222Another
possibility is to divide both sides of (6) by $c_{nm1}$. This would
renormalize all remaining coefficients in the right-hand side of (6) as well
as the effective yield stress $\tau^{*}_{cr}$. to satisfy this condition is to
take $c_{nm1}=1$. In the other extreme, i.e. when all non-Schmid stresses play
role in the initiation of plastic deformation, it is customary to express
$\tau^{*\alpha}$ in terms of the hydrostatic stress $\sigma_{h}^{\alpha}$. The
presence of $\sigma_{h}^{\alpha}$ and $\sigma_{mm}^{\alpha}$ causes that the
two remaining normal stresses ($\sigma_{tt}^{\alpha}$ and
$\sigma_{nn}^{\alpha}$) are not independent. Starting with (6), it is possible
to rewrite the first, the fourth, and the sixth term using
$\sigma_{nn}^{\alpha}-\sigma_{tt}^{\alpha}$, $\sigma_{h}^{\alpha}$ and
$\sigma_{mm}^{\alpha}$. This results in an equivalent representation of the
effective stress:

$\tau^{*\alpha}=\sigma_{nm}^{\alpha}+c_{tm2}(\sigma_{tm}^{\alpha})^{2}+a_{2}(\sigma_{nn}^{\alpha}-\sigma_{tt}^{\alpha})+c_{tn2}(\sigma_{tn}^{\alpha})^{2}+a_{4}\sigma_{mm}^{\alpha}+a_{5}\sigma_{h}^{\alpha}\
,$ (7)

where we have introduced new coefficients to replace the linear combinations
of $c_{ijn}$. The first term in (7) is the resolved Schmid stress for the
system $\alpha$. The second term is the shear stress parallel to the slip
direction acting in the plane with the normal $\hat{\bm{t}}^{\alpha}$ (this is
perpendicular to the slip direction and lies in the slip plane. The third term
is a combination of two normal stresses, which generates a _shear_ stress
parallel to the slip direction in another coordinate system, where the axes
$\hat{\bm{t}}^{\alpha}$ and $\hat{\bm{n}}^{\alpha}$ are rotated by
$45^{\circ}$ in the zone of $\hat{\bm{m}}^{\alpha}$. The fourth term is
directly the shear stress perpendicular to the slip direction. The fifth and
the sixth terms incorporate the effects of the normal stress parallel to the
dislocation line and of the hydrostatic stress, respectively. A notable
feature of (7) is that all stress components entering $\tau^{*\alpha}$ are
resolved in a _single_ coordinate system, which is spanned by the vectors
$\hat{\bm{t}}^{\alpha}$, $\hat{\bm{n}}^{\alpha}$ and $\hat{\bm{m}}^{\alpha}$.

###  2.2 Linear effective stress using an auxiliary system

Although the form of the effective stress (7) obeys the inversion symmetry
(3), the presence of quadratic terms may lead to a number of complications in
homogenization and developments of plastic flow rules for random and textured
polycrystals (11, 5). In the following, we demonstrate that a form equivalent
to (7) can be obtained if the Maclaurin series is terminated _before_ the
quadratic terms $(\sigma_{tm}^{\alpha})^{2}$ and $(\sigma_{tn}^{\alpha})^{2}$.
Since the linear terms of these stresses are absent by symmetry, the effective
stress reduces to a simple form

$\tau^{*\alpha}\approx\sigma_{nm}^{\alpha}+a_{2}(\sigma_{nn}^{\alpha}-\sigma_{tt}^{\alpha})+a_{4}\sigma_{mm}^{\alpha}+a_{5}\sigma_{h}^{\alpha}\
.$ (8)

All stresses in this expression are resolved in a single orientation defined
by the vectors $\hat{\bm{t}}^{\alpha}$, $\hat{\bm{n}}^{\alpha}$ and
$\hat{\bm{m}}^{\alpha}$. Therefore, this form cannot capture the variation of
the Peierls stress with the orientation of the plane in which it is applied
and on the orientation of the shear stress perpendicular to the slip
direction, as explained in the Introduction. The simplest way to re-introduce
these effects is to write $\tau^{*\alpha}$ as a linear superposition of the
stresses $\sigma_{nm}^{\alpha}$ and
$\sigma_{nn}^{\alpha}-\sigma_{tt}^{\alpha}$ resolved in two different
orientations in the zone of the slip direction, which is the same step as
taken previously by Qin and Bassani (30) to define their effective stress (2).
This superposition extends (8) to

$\tau^{*\alpha}=\sigma_{nm}^{\alpha}+a_{1}\sigma_{n^{\prime}m}^{\alpha}+a_{2}(\sigma_{nn}^{\alpha}-\sigma_{tt}^{\alpha})+a_{3}(\sigma_{n^{\prime}n^{\prime}}^{\alpha}-\sigma_{t^{\prime}t^{\prime}}^{\alpha})+a_{4}\sigma_{mm}^{\alpha}+a_{5}\sigma_{h}^{\alpha}\
,$ (9)

where we have added the terms with coefficients $a_{1}$ and $a_{3}$, in which
$\hat{\bm{n}}^{\prime\alpha}$ is the normal of an auxiliary plane in the zone
of $\hat{\bm{m}}^{\alpha}$, and
$\hat{\bm{t}}^{\prime\alpha}=\hat{\bm{n}}^{\prime\alpha}\times\hat{\bm{m}}^{\alpha}$.
The presence of $\hat{\bm{n}}^{\prime\alpha}$ and
$\hat{\bm{t}}^{\prime\alpha}$ requires a generalization of the symmetry
condition (3). We can see from Fig. 2 that a rigid-body rotation of the
crystal by $180^{\circ}$ about the axis $\hat{\bm{t}}^{\alpha}$ results in the
transformations $\hat{\bm{m}}^{\alpha}\rightarrow-\hat{\bm{m}}^{\alpha}$,
$\hat{\bm{n}}^{\alpha}\rightarrow-\hat{\bm{n}}^{\alpha}$ and
$\hat{\bm{n}}^{\prime\alpha}\rightarrow-\hat{\bm{n}}^{\prime\prime\alpha}$.
The last transformation involves normals of two different planes, which is a
natural consequence of introducing the auxiliary plane. Because the effective
stress must be invariant under this transformation, it follows that

$\tau^{*\alpha}(\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha},\hat{\bm{n}}^{\prime\alpha}|\bm{\sigma})=\tau^{*\alpha}(-\hat{\bm{m}}^{\alpha},-\hat{\bm{n}}^{\alpha},-\hat{\bm{n}}^{\prime\prime\alpha}|\bm{\sigma})\
.$ (10)

Similarly as (3) represents the symmetry of $\tau^{*\alpha}$ in (7) under
rigid-body rotation about the axis $\hat{\bm{t}}^{\alpha}$, (10) is a
statement of the symmetry of $\tau^{*\alpha}$ in (9) under rigid-body rotation
about the same axis $\hat{\bm{t}}^{\alpha}$. The latter is further
demonstrated in Fig. 2. The effective yield criterion,
$\tau^{*}(\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha},\hat{\bm{n}}^{\prime\alpha}|\bm{\sigma})$,
is initially described in terms of the lattice vectors
$\hat{\bm{m}}^{\alpha}$, $\hat{\bm{n}}^{\alpha}$ and
$\hat{\bm{n}}^{\prime\alpha}$ which are highlighted in Fig. 2 in green. After
rotation of the crystal by 180∘ about the axis $\hat{\bm{t}}^{\alpha}$, the
same green motif refers to the lattice vectors $-\hat{\bm{m}}^{\alpha}$,
$-\hat{\bm{n}}^{\alpha}$, $-\hat{\bm{n}}^{\prime\prime\alpha}$ and the
corresponding effective stress is
$\tau^{*\alpha}(-\hat{\bm{m}}^{\alpha},-\hat{\bm{n}}^{\alpha},-\hat{\bm{n}}^{\prime\prime\alpha}|\bm{\sigma})$.
Both values of $\tau^{*\alpha}$ must be the same, which proves the validity of
the condition (10).

Figure 2: Schematic illustation of the validity of (10) under rigid-body
rotation about the axis $\hat{\bm{t}}^{\alpha}$. The auxiliary plane as
defined in the text is drawn in blue. The vectors used to define this
condition are highlighted in green.

In the previously used effective stress (2), the effects of shear stresses
perpendicular to the slip direction are incorporated by directly including
_shear stresses_ $\sigma_{tn}$ and $\sigma_{t^{\prime}n^{\prime}}$ in two
different orientations. We have shown that these stress terms violate the
symmetry represented by (3) and (10). On the contrary, in the effective stress
(9), the effects of these shear stresses are incorporated via _differences of
normal stresses_ $\sigma_{nn}-\sigma_{tt}$ and
$\sigma_{n^{\prime}n^{\prime}}-\sigma_{t^{\prime}t^{\prime}}$, again expressed
in two different orientations. These stresses do not violate the
aforementioned symmetries and thus the form (9) is consistent with
crystallography.

The effective stress (9) together with the yield criterion (1) constitute the
primary result of this paper. The linear dependence of $\tau^{*\alpha}$ on all
stress components is the consequence of avoiding the quadratic stresses
$\sigma_{tm}^{\alpha}$ and $\sigma_{tn}^{\alpha}$ in (7) and, instead, writing
$\tau^{*\alpha}$ as a linear superposition of the shear stresses parallel and
perpendicular to the slip direction resolved in two coordinate systems. All
stress components in (9) are invariant under the generalized inversion
symmetry defined by (10). The presence of different auxiliary systems for
$(\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha})$ (with the normal
$\hat{\bm{n}}^{\prime\alpha}$), and
$(-\hat{\bm{m}}^{\alpha},-\hat{\bm{n}}^{\alpha})$ (with the normal
$-\hat{\bm{n}}^{\prime\prime\alpha}$) is the consequence of choosing the angle
of the auxiliary plane different from $90^{\circ}$. This projection cannot be
used, because $\sigma_{n^{\prime}m}^{\alpha}$ would equal
$-\sigma_{tm}^{\alpha}$, which is the stress that is not allowed by the
symmetry conditions (3) and (10). The coefficients $a_{k}$ $(k=1\dots 5)$ in
(9) are usually determined from atomistic simulations on isolated dislocations
that move in low-index slip planes. However, they may be also obtained from
molecular dynamics simulations of interacting dislocation networks or directly
from low-temperature experiments in tension and compression. Since no details
have been given so far about the Miller indices representing the vectors
$\hat{\bm{m}}^{\alpha}$ and $\hat{\bm{n}}^{\alpha}$, the effective stress (9)
should apply to any crystal structure. In the following, we demonstrate its
applicability to bcc and hcp crystals.

##  3 Application to cubic and hexagonal crystals

###  3.1 Body-centered cubic crystals

The plastic deformation of bcc metals is governed by the glide of $1/2\langle
111\rangle$ screw dislocations. We will be concerned with the usual case,
where the initiation of plastic deformation is not affected by the hydrostatic
stress and by the tension/compression acting parallel to the dislocation line.
Under these assumptions, the effective stress (9) reduces to

$\tau^{*\alpha}=\sigma_{nm}^{\alpha}+a_{1}\sigma_{n^{\prime}m}^{\alpha}+a_{2}(\sigma_{nn}^{\alpha}-\sigma_{tt}^{\alpha})+a_{3}(\sigma_{n^{\prime}n^{\prime}}^{\alpha}-\sigma_{t^{\prime}t^{\prime}}^{\alpha})\
.$ (11)

The coefficients $a_{1}$, $a_{2}$ and $a_{3}$ in (11) as well as the value of
$\tau^{*}_{cr}$ in (1) need to be determined using atomistic simulations or
from experiments. In the former case, the parameterization may follow the
procedure described in (15). This is based on the observation that only the
shear stresses perpendicular and parallel to the slip direction affect the
Peierls stress to move $1/2\langle 111\rangle$ screw dislocations in bcc
metals (13, 21, 14). The twinning-antitwinning asymmetry in bcc metals
dictates that the coefficient $a_{1}$ must be positive so that the Peierls
stress to move the dislocation is higher when shearing the crystal in the
antitwinning sense than when shearing it in the twinning sense. The magnitudes
and signs of the coefficients $a_{2}$ and $a_{3}$ are determined by the
dependence of the Peierls stress on shear stresses perpendicular to the slip
direction.

Table 1: The 24 slip systems in bcc crystals. The crystallographic vectors
$\bm{m}^{\alpha}$, $\bm{n}^{\alpha}$, $\bm{n}^{\prime\alpha}$ have to be
normalized before their use in (11).

$\alpha$ | system | $\bm{m}^{\alpha}$ | $\bm{n}^{\alpha}$ | $\bm{n}^{\prime\alpha}$ |  | $\alpha$ | system | $\bm{m}^{\alpha}$ | $\bm{n}^{\alpha}$ | $\bm{n}^{\prime\alpha}$
---|---|---|---|---|---|---|---|---|---|---
1 |  ($01\bar{1}$)[$111$] |  [$111$] |  [$01\bar{1}$] |  [$\bar{1}10$] |  | $1^{*}$ |  ($01\bar{1}$)[$\bar{1}\bar{1}\bar{1}$] |  [$\bar{1}\bar{1}\bar{1}$] |  [$01\bar{1}$] |  [$10\bar{1}$]
2 |  ($\bar{1}01$)[$111$] |  [$111$] |  [$\bar{1}01$] |  [$0\bar{1}1$] |  | $2^{*}$ |  ($\bar{1}01$)[$\bar{1}\bar{1}\bar{1}$] |  [$\bar{1}\bar{1}\bar{1}$] |  [$\bar{1}01$] |  [$\bar{1}10$]
3 |  ($1\bar{1}0$)[$111$] |  [$111$] |  [$1\bar{1}0$] |  [$10\bar{1}$] |  | $3^{*}$ |  ($1\bar{1}0$)[$\bar{1}\bar{1}\bar{1}$] |  [$\bar{1}\bar{1}\bar{1}$] |  [$1\bar{1}0$] |  [$0\bar{1}1$]
4 |  ($\bar{1}0\bar{1}$)[$\bar{1}11$] |  [$\bar{1}11$] |  [$\bar{1}0\bar{1}$] |  [$\bar{1}\bar{1}0$] |  | $4^{*}$ |  ($\bar{1}0\bar{1}$)[$1\bar{1}\bar{1}$] |  [$1\bar{1}\bar{1}$] |  [$\bar{1}0\bar{1}$] |  [$01\bar{1}$]
5 |  ($0\bar{1}1$)[$\bar{1}11$] |  [$\bar{1}11$] |  [$0\bar{1}1$] |  [$101$] |  | $5^{*}$ |  ($0\bar{1}1$)[$1\bar{1}\bar{1}$] |  [$1\bar{1}\bar{1}$] |  [$0\bar{1}1$] |  [$\bar{1}\bar{1}0$]
6 |  ($110$)[$\bar{1}11$] |  [$\bar{1}11$] |  [$110$] |  [$01\bar{1}$] |  | $6^{*}$ |  ($110$)[$1\bar{1}\bar{1}$] |  [$1\bar{1}\bar{1}$] |  [$110$] |  [$101$]
7 |  ($0\bar{1}\bar{1}$)[$\bar{1}\bar{1}1$] |  [$\bar{1}\bar{1}1$] |  [$0\bar{1}\bar{1}$] |  [$1\bar{1}0$] |  | $7^{*}$ |  ($0\bar{1}\bar{1}$)[$11\bar{1}$] |  [$11\bar{1}$] |  [$0\bar{1}\bar{1}$] |  [$\bar{1}0\bar{1}$]
8 |  ($101$)[$\bar{1}\bar{1}1$] |  [$\bar{1}\bar{1}1$] |  [$101$] |  [$011$] |  | $8^{*}$ |  ($101$)[$11\bar{1}$] |  [$11\bar{1}$] |  [$101$] |  [$1\bar{1}0$]
9 |  ($\bar{1}10$)[$\bar{1}\bar{1}1$] |  [$\bar{1}\bar{1}1$] |  [$\bar{1}10$] |  [$\bar{1}0\bar{1}$] |  | $9^{*}$ |  ($\bar{1}10$)[$11\bar{1}$] |  [$11\bar{1}$] |  [$\bar{1}10$] |  [$011$]
10 |  ($10\bar{1}$)[$1\bar{1}1$] |  [$1\bar{1}1$] |  [$10\bar{1}$] |  [$110$] |  | $10^{*}$ |  ($10\bar{1}$)[$\bar{1}1\bar{1}$] |  [$\bar{1}1\bar{1}$] |  [$10\bar{1}$] |  [$0\bar{1}\bar{1}$]
11 |  ($011$)[$1\bar{1}1$] |  [$1\bar{1}1$] |  [$011$] |  [$\bar{1}01$] |  | $11^{*}$ |  ($011$)[$\bar{1}1\bar{1}$] |  [$\bar{1}1\bar{1}$] |  [$011$] |  [$110$]
12 |  ($\bar{1}\bar{1}0$)[$1\bar{1}1$] |  [$1\bar{1}1$] |  [$\bar{1}\bar{1}0$] |  [$0\bar{1}\bar{1}$] |  | $12^{*}$ |  ($\bar{1}\bar{1}0$)[$\bar{1}1\bar{1}$] |  [$\bar{1}1\bar{1}$] |  [$\bar{1}\bar{1}0$] |  [$\bar{1}01$]

Figure 3: Predictions of the symmetry-adapted yield criterion for bcc metals
(11) for different choices of the parameters $a_{1}$, $a_{2}$ and $a_{3}$
shown on the left (the remaining parameters are zero). The numbers refer to
the slip systems in Tab. 1.

In Fig. 3, we demonstrate how different choices of the coefficients $a_{1}$,
$a_{2}$ and $a_{3}$, and thus the three non-Schmid stress components in (11),
affect the prediction of the primary and secondary slip systems. These are the
first two systems $\alpha$ for which the value of $\tau^{*\alpha}$ reaches
$\tau^{*}_{cr}$ for a given orientation of the applied load. These
calculations were made for uniaxial tension and compression in all directions
covering the area of the stereographic triangle $[001]-[011]-[\bar{1}11]$.
Preliminary parameterizations of the yield criterion (11) for existing
atomistic studies on bcc metals suggests that both coefficients $a_{2}$ and
$a_{3}$ may be negative, whereas $a_{1}$ must always be non-negative to obtain
correct trend of the twinning-antitwinning asymmetry. In the following, we
thus consider $a_{1}=\\{0,0.3\\}$, and $a_{2},a_{3}=\pm 0.3$. The first row of
images gives the predictions of the Schmid law that were obtained by setting
$a_{1}=a_{2}=a_{3}=0$. The primary and secondary slip systems predicted for
tension and compression are obviously identical, as dictated by the symmetry
between tension and compression embodied in the Schmid law. The remaining maps
demonstrate how individual non-Schmid stress components (i.e., nonzero values
of $a_{1}$, $a_{2}$, or $a_{3}$) cause deviations of the slip activity from
the prediction of the Schmid law. The effect of the twinning-antitwinning
asymmetry is shown in the second row for $a_{1}=0.3$ (the remaining two
parameters are zero). For most orientations, the primary slip system is still
number 2 in tension (number 2* in compression). This generally agrees with the
atomistic simulations, where the application of pure shear stress parallel to
the slip direction results in slip on the most highly stresses slip system 2
(2*). The effect of the shear stress perpendicular to the slip direction is
demonstrated in the maps for nonzero values of the coefficients $a_{2}$ and
$a_{3}$. The presence of these shear stresses in the yield criterion makes the
predictions of the slip activity more orientational-dependent. For $a_{2}>0$
or $a_{3}<0$, loading in tension in the center-triangle orientation gives a
similar prediction as the Schmid law, i.e. primary slip on the system 2 (2*).
The same is true for compression when $a_{2}<0$ or $a_{3}>0$. However, very
different predictions are obtained for $a_{2}<0$ or $a_{3}>0$ when loading in
tension, as well as for $a_{2}>0$ or $a_{3}<0$ if the load is applied in
compression. In these four cases, the primary slip is predicted to occur on
the slip system 4 (4*), 3 (3*), or 8 (8*) for most orientations of the loading
axis.

###  3.2 Hexagonal crystals

In hexagonal crystallography, it is customary to characterize directions and
normals of planes using four-index notations. The first three indices
constitute projections onto the axes $\bm{a}_{1}$, $\bm{a}_{2}$, $\bm{a}_{3}$
in the basal plane, whereas the fourth is the projection onto the $\bm{c}$
axis, as shown in Fig. 4. The addition of the third axis, i.e.
$\bm{a_{3}}=-(\bm{a}_{1}+\bm{a}_{2})$, ensures that crystallographically
equivalent directions have similar Miller indices in the four-index notation.
In order to simplify mathematical operations in the hexagonal system, Otte and
Crocker (27) proposed to convert the four-index representations of slip
directions and slip plane normals in the hexagonal system into an ortho-
hexagonal system spanned by mutually orthogonal lattice vectors $\bm{a}$,
$\bm{b}$, $\bm{c}$ shown in Fig. 4. We will use these projection rules in the
following to derive expressions for the six stress components $\sigma_{ij}$ in
(9) directly in terms of the slip directions and slip plane normals defined in
the four-index hexagonal notation.

Figure 4: Schematic illustration of the primitive hexagonal cell (black) and
the ortho-hexagonal cell (green) with the lattice vectors shown by the same
color. Several low-index directions in the hexagonal lattice are shown in
gray.

####  3.2.1 Slip directions in the ortho-hexagonal basis

Consider an arbitrary direction $\bm{m}$ that is represented by the Miller
indices $[uvtw]$ in the four-index hexagonal system. The vector $\bm{m}$ can
be recovered from these Miller indices using the basis vectors of the
hexagonal system, i.e.
$\bm{m}=u\bm{a}_{1}+v\bm{a}_{2}+(u+v)(\bm{a}_{1}+\bm{a}_{2})+w\bm{c}$, where
we have used well-known identities $t=-(u+v)$ and
$\bm{a}_{3}=-(\bm{a}_{1}+\bm{a}_{2})$. In the ortho-hexagonal system, the same
direction $\bm{m}$ is represented by the Miller indices $[pqr]$, i.e.
$\bm{m}=p\bm{a}+q\bm{b}+r\bm{c}$. It was shown by Otte and Crocker (27) that
the relations between the lattice vectors in the two systems are
$\bm{a}=\bm{a}_{1}$, $\bm{b}=\bm{a}_{1}+2\bm{a}_{2}$, whereas the third axis
remains the same in both systems. Equating the right-hand sides of the two
representations of the vector $\bm{m}$ above and implementing these
transformation rules provides the indices of directions in the ortho-hexagonal
system in terms of those in the hexagonal system: $p=3u/2$, $q=u/2+v$, $r=w$.
It is clear that the third index in $[uvtw]$ is redundant and can be omitted
(it is replaced in the following by a dot). Therefore, any direction
represented by the vector $\bm{M}=[uv.w]$ defined in the hexagonal system can
be expressed by the vector $\bm{m}=[pqr]$ in the ortho-hexagonal system as

$\bm{m}=\bm{M}\bm{Q}\ ,$ (12)

where both $\bm{m}$ and $\bm{M}$ are row vectors with three values. The
transformation matrix $\bm{Q}$ has the form

$\bm{Q}=\left[\begin{array}[]{ccc}3/2&1/2&0\\\ 0&1&0\\\
0&0&1\end{array}\right]\ .$ (13)

It is important to emphasize that the indices $[pqr]$ representing $\bm{m}$ in
the ortho-hexagonal basis refer to the basis vectors $\bm{a}=a\hat{\bm{a}}$,
$\bm{b}=a\sqrt{3}\hat{\bm{b}}$ and $\bm{c}=c\hat{\bm{c}}$, where
$\hat{\bm{a}}$, $\hat{\bm{b}}$, $\hat{\bm{c}}$ are unit vectors along the
three directions. In terms of these basis vectors, $\bm{m}=[pqr]$ can be
expressed as

$\bm{m}=pa\hat{\bm{a}}+qa\sqrt{3}\hat{\bm{b}}+rc\hat{\bm{c}}\ .$ (14)

We want to find the unit vector $\hat{\bm{m}}=[\hat{p}\hat{q}\hat{r}]$
parallel to $\bm{m}$, which we define as

$\hat{\bm{m}}=\hat{p}\hat{\bm{a}}+\hat{q}\hat{\bm{b}}+\hat{r}\hat{\bm{c}}\ .$
(15)

The two vectors are related by $\hat{\bm{m}}=\bm{m}/\lambda_{m}$, where
$\lambda_{m}$ is a normalization factor. This condition gives three equations
that define the components of the unit vector $\hat{\bm{m}}$:
$\hat{p}=pa/\lambda_{m}$, $\hat{q}=qa\sqrt{3}/\lambda_{m}$,
$\hat{r}=rc/\lambda_{m}$. Substituting these equations into the normalization
condition $\hat{p}^{2}+\hat{q}^{2}+\hat{r}^{2}=1$ provides the normalization
factor $\lambda_{m}=a\sqrt{p^{2}+3q^{2}+(c/a)^{2}r^{2}}$. It is evident that
substituting $\lambda_{m}$ back into the expressions for $\hat{p}$, $\hat{q}$,
$\hat{r}$ eliminates the lattice parameter $a$. For the sake of simplicity, we
thus redefine the components of the unit vector
$\hat{\bm{m}}=[\hat{p}\hat{q}\hat{r}]$ and the normalization factor as

$\displaystyle\hat{p}=p/\lambda_{m}\ ,\quad\hat{q}=q\sqrt{3}/\lambda_{m}\
,\quad\hat{r}=r(c/a)/\lambda_{m}$
$\displaystyle\lambda_{m}=\sqrt{p^{2}+3q^{2}+(c/a)^{2}r^{2}}$ $\displaystyle\
.$

Employing (12) in the definition of $\hat{\bm{m}}$ allows to express the unit
vector parallel to the slip direction in the (orthonormal) ortho-hexagonal
basis simply as a division by the factor $\lambda_{m}$:

$\hat{\bm{m}}=\frac{\bm{M}\bm{Q}}{\lambda_{m}}\ .$ (17)

####  3.2.2 Slip plane normals in the ortho-hexagonal basis

Different transformation rules apply for directions of plane normals as
explained in detail in Otte and Crocker (27). To demonstrate this, consider an
arbitrary plane defined by the indices $(hkil)$ in the four-index hexagonal
system, where $i=-(h+k)$ as explained above. The vector normal to this plane
is then $\bm{n}=h\bm{a}_{1}^{*}+k\bm{a}_{2}^{*}+l\bm{c}^{*}$, where
$\bm{a}_{1}^{*},\bm{a}_{2}^{*},\bm{c}^{*}$ are reciprocal lattice vectors. In
the ortho-hexagonal system, this plane is represented by the Miller index
$(efg)$, which defines the same vector
$\bm{n}=e\bm{a}^{*}+f\bm{b}^{*}+g\bm{c}^{*}$. The transformation rules between
the two reciprocal bases are $\bm{a}^{*}=\bm{a}_{1}^{*}-\bm{a}_{2}^{*}/2$,
$\bm{b}^{*}=\bm{a}_{2}^{*}/2$ with $\bm{c}^{*}$ being parallel in both bases.
Equating the factors in front of $\bm{a}_{1}^{*}$, $\bm{a}_{2}^{*}$ and
$\bm{c}^{*}$ provides the relations between the indices in the three-index
ortho-hexagonal system in terms of the indices in the four-index hexagonal
system: $e=k$, $f=h+2k$, $g=l$. Similarly as above, the third index in
$(hkil)$ is redundant and does not enter the transformation (it is again
replaced by a dot). Therefore, the vector normal to the plane $(hk.l)$ in the
hexagonal system, represented by the vector $\bm{N}=[hk.l]$, can be expressed
as the vector $\bm{n}=[efg]$ in the ortho-hexagonal system as

$\bm{n}=\bm{N}\bm{Q}^{*}\ ,$ (18)

where both $\bm{n}$ and $\bm{N}$ are again row vectors with three values. The
transformation matrix $\bm{Q}^{*}$ has the form

$\bm{Q}^{*}=\left[\begin{array}[]{ccc}1&1&0\\\ 0&2&0\\\
0&0&1\end{array}\right]\ .$ (19)

Similar procedure as above can be applied to derive the normalization factors
for plane normals. In the ortho-hexagonal basis, the basis vectors for plane
normals are $\bm{a}^{*}=\bm{a}/a^{2}$, $\bm{b}^{*}=\bm{b}/3a^{2}$,
$\bm{c}^{*}=\bm{c}/c^{2}$. Using these basis vectors, the plane normals
$\bm{n}=[efg]$ can be expressed as

$\bm{n}=\frac{e}{a}\hat{\bm{a}}+\frac{f}{a\sqrt{3}}\hat{\bm{b}}+\frac{g}{c}\hat{\bm{c}}\
.$ (20)

We want to find the unit vector $\hat{\bm{n}}=[\hat{e}\hat{f}\hat{g}]$
parallel to $\bm{n}$, which we define as

$\hat{\bm{n}}=\hat{e}\hat{\bm{a}}+\hat{f}\hat{\bm{b}}+\hat{g}\hat{\bm{c}}\ .$
(21)

The two vectors are related by $\hat{\bm{n}}=\bm{n}/\lambda_{n}$, where
$\lambda_{n}$ is a normalization factor. This condition gives three equations
that define the components of the unit vector $\hat{\bm{n}}$:
$\hat{e}=e/a\lambda_{n}$, $\hat{f}=f/a\sqrt{3}\lambda_{n}$,
$\hat{g}=g/c\lambda_{n}$. Substituting these equations into the normalization
condition $\hat{e}^{2}+\hat{f}^{2}+\hat{g}^{2}=1$ provides the normalization
factor $\lambda_{n}=(1/a)\sqrt{e^{2}+f^{2}/3+g^{2}/(c/a)^{2}}$. It is again
evident that substituting $\lambda_{n}$ back into the expressions for
$\hat{e}$, $\hat{f}$, $\hat{g}$ eliminates the lattice parameter $a$. We thus
again redefine the components of the unit vector
$\hat{\bm{n}}=[\hat{e}\hat{f}\hat{g}]$ and the normalization factor as

$\displaystyle\hat{e}=e/\lambda_{n}\ ,\quad\hat{f}=f/\sqrt{3}\lambda_{n}\
,\quad\hat{g}=g/(c/a)\lambda_{n}$
$\displaystyle\lambda_{n}=\sqrt{e^{2}+f^{2}/3+g^{2}/(c/a)^{2}}$
$\displaystyle\ .$

Employing (18) in the definition of $\hat{\bm{n}}$ allows to express the unit
normal of the slip plane in the (orthonormal) ortho-hexagonal basis simply as
a division by the factor $\lambda_{n}$:

$\hat{\bm{n}}=\frac{\bm{N}\bm{Q}^{*}}{\lambda_{n}}\ .$ (23)

####  3.2.3 Stress components in the yield criterion

Table 2: Hexagonal slip systems given in the four-index (hexagonal) and three-index (ortho-hexagonal) basis (the vectors are not normalized). In the former, the slip direction is $\bm{M}^{\alpha}=[uvtw]$, and the slip plane normal $\bm{N}^{\alpha}=[hkil]$. The indices in the latter systems ($\bm{m}^{\alpha}$, $\bm{n}^{\alpha}$, $\bm{n}^{\prime\alpha}$) were obtained by employing (12) for slip directions and (18) for slip plane normals with the third indices ($t$ and $i$) neglected. Separate numbering is used here for the basal & prismatic, and pyramidal systems. Nomenclature: $\eta=c/a$, $f_{1}=-5/2-2\eta^{2}+3\sqrt{1+\eta^{2}}/(2\eta)+2\eta\sqrt{1+\eta^{2}}$, $f_{2}=-(3+4\eta^{2})/[2\eta(\eta+\sqrt{1+\eta^{2}})]$, $f_{3}=(3+4\eta^{2})(\eta+\sqrt{1+\eta^{2}})/(2\eta)$, $f_{4}=[3+2\eta(\eta+\sqrt{1+\eta^{2}})]/[2\eta(\eta-\sqrt{1+\eta^{2}})]$. type | $\alpha$ | system | $\bm{M}^{\alpha}$ | $\bm{N}^{\alpha}$ | $\bm{N}^{\prime\alpha}$
---|---|---|---|---|---
| 1 | $(0001)[11\bar{2}0]$ | $[11\bar{2}0]$ | $[0001]$ | $[1,-1,0,2\eta/3]$
| 2 | $(0001)[1\bar{2}10]$ | $[1\bar{2}10]$ | $[0001]$ | $[-1,0,1,2\eta/3]$
basal | 3 | $(0001)[\bar{2}110]$ | $[\bar{2}110]$ | $[0001]$ | $[0,1,-1,2\eta/3]$
| $1^{*}$ | $(0001)[\bar{1}\bar{1}20]$ | $[11\bar{2}0]$ | $[0001]$ | $[-1,1,0,2\eta/3]$
| $2^{*}$ | $(0001)[\bar{1}2\bar{1}0]$ | $[1\bar{2}10]$ | $[0001]$ | $[1,0,-1,2\eta/3]$
| $3^{*}$ | $(0001)[2\bar{1}\bar{1}0]$ | $[\bar{2}110]$ | $[0001]$ | $[0,-1,1,2\eta/3]$
| 4 | $(\bar{1}100)[11\bar{2}0]$ | $[11\bar{2}0]$ | $[\bar{1}100]$ | $[-1,1,0,2\eta]$
| 5 | $(\bar{1}010)[1\bar{2}10]$ | $[1\bar{2}10]$ | $[\bar{1}010]$ | $[-1,0,1,-2\eta]$
prismatic | 6 | $(0\bar{1}10)[\bar{2}110]$ | $[\bar{2}110]$ | $[0\bar{1}10]$ | $[0,-1,1,2\eta]$
| $4^{*}$ | $(\bar{1}100)[\bar{1}\bar{1}20]$ | $[11\bar{2}0]$ | $[\bar{1}100]$ | $[-1,1,0,-2\eta]$
| $5^{*}$ | $(\bar{1}010)[\bar{1}2\bar{1}0]$ | $[1\bar{2}10]$ | $[\bar{1}010]$ | $[-1,0,1,2\eta]$
| $6^{*}$ | $(0\bar{1}10)[2\bar{1}\bar{1}0]$ | $[\bar{2}110]$ | $[0\bar{1}10]$ | $[0,-1,1,-2\eta]$
| 1 | $(\bar{1}011)[11\bar{2}3]$ | $[11\bar{2}3]$ | $[\bar{1}011]$ | $[f_{1},f_{2},1,1]$
| 2 | $(0\bar{1}11)[11\bar{2}3]$ | $[11\bar{2}3]$ | $[0\bar{1}11]$ | $[f_{3},f_{4},1,1]$
| 3 | $(\bar{1}101)[1\bar{2}13]$ | $[1\bar{2}13]$ | $[\bar{1}101]$ | $[f_{4},1,f_{3},1]$
| 4 | $(01\bar{1}1)[1\bar{2}13]$ | $[1\bar{2}13]$ | $[01\bar{1}1]$ | $[f_{2},1,f_{1},1]$
| 5 | $(1\bar{1}01)[\bar{2}113]$ | $[\bar{2}113]$ | $[1\bar{1}01]$ | $[1,f_{1},f_{2},1]$
pyramidal | 6 | $(10\bar{1}1)[\bar{2}113]$ | $[\bar{2}113]$ | $[10\bar{1}1]$ | $[1,f_{3},f_{4},1]$
| $1^{*}$ | $(\bar{1}011)[\bar{1}\bar{1}2\bar{3}]$ | $[11\bar{2}3]$ | $[\bar{1}011]$ | $[f_{4},f_{3},1,1]$
| $2^{*}$ | $(0\bar{1}11)[\bar{1}\bar{1}2\bar{3}]$ | $[11\bar{2}3]$ | $[0\bar{1}11]$ | $[f_{2},f_{1},1,1]$
| $3^{*}$ | $(\bar{1}101)[\bar{1}2\bar{1}\bar{3}]$ | $[1\bar{2}13]$ | $[\bar{1}101]$ | $[f_{1},1,f_{2},1]$
| $4^{*}$ | $(01\bar{1}1)[\bar{1}2\bar{1}\bar{3}]$ | $[1\bar{2}13]$ | $[01\bar{1}1]$ | $[f_{3},1,f_{4},1]$
| $5^{*}$ | $(1\bar{1}01)[2\bar{1}\bar{1}\bar{3}]$ | $[\bar{2}113]$ | $[1\bar{1}01]$ | $[1,f_{4},f_{3},1]$
| $6^{*}$ | $(10\bar{1}1)[2\bar{1}\bar{1}\bar{3}]$ | $[\bar{2}113]$ | $[10\bar{1}1]$ | $[1,f_{2},f_{1},1]$

In the previous Section, the stress components in (9) were defined using the
slip directions and slip plane normals expressed in the _orthogonal_
coordinate systems spanned by the vectors $\hat{\bm{m}}^{\alpha}$,
$\hat{\bm{n}}^{\alpha}$ and $\hat{\bm{n}}^{\prime\alpha}$. Using the linear
transformations (17) and (23), we can now express these vectors in terms of
the vectors $\bm{M}^{\alpha}$, $\bm{N}^{\alpha}$, $\bm{N}^{\prime\alpha}$ that
are natural to the hexagonal system. These are shown for the three types of
slip systems in hexagonal crystals in Tab. 2. Within each group, the system
without and with an asterisk is conjugate to the same-numbered system in that
the two have the same slip plane, but opposite slip directions. From (17), the
slip directions transform as
$\hat{\bm{m}}^{\alpha}=(1/\lambda_{m})\bm{M}^{\alpha}\bm{Q}$, whereas from
(23) the two slip plane normals transform as
$\hat{\bm{n}}^{\alpha}=(1/\lambda_{n})\bm{N}^{\alpha}\bm{Q}^{*}$ and
$\hat{\bm{n}}^{\prime\alpha}=(1/\lambda_{n^{\prime}})\bm{N}^{\prime\alpha}\bm{Q}^{*}$.
Using these relations, we can now express the six stress components entering
(9) directly using the Miller indices of slip planes and slip directions given
in Tab. 2. To demonstrate this, consider the Schmid stress that was previously
defined as
$\sigma_{nm}^{\alpha}=\hat{\bm{n}}^{\alpha}\bm{\sigma}(\hat{\bm{m}}^{\alpha})^{T}$.
Employing the relations above provides an equivalent expression,
$\sigma_{nm}^{\alpha}=(1/\lambda_{n}\lambda_{m})\bm{N}^{\alpha}\bm{\Sigma}^{*}(\bm{M}^{\alpha})^{T}$,
where $\bm{\Sigma}^{*}=\bm{Q}^{*}\bm{\sigma}\bm{Q}^{T}$ is a $3\times 3$
matrix that can be regarded as a representation of the stress tensor
$\bm{\sigma}$ in the hexagonal system. It is interesting to observe that the
previous two representations of $\sigma_{nm}^{\alpha}$ have the same
mathematical form and differ only by the expression of the stress tensor in
the two bases and by the normalization factor. The same is true also for the
stress components $\sigma_{n^{\prime}m}^{\alpha}$, $\sigma_{nn}^{\alpha}$,
$\sigma_{n^{\prime}n^{\prime}}^{\alpha}$, and $\sigma_{mm}^{\alpha}$. However,
the expressions for $\sigma_{tt}$ and $\sigma_{t^{\prime}t^{\prime}}$ for the
hexagonal system are quite different from those in cubic systems. These
stresses are defined using cross-products
$\hat{\bm{n}}^{\alpha}\times\hat{\bm{m}}^{\alpha}$ and
$\hat{\bm{n}}^{\prime\alpha}\times\hat{\bm{m}}^{\alpha}$, respectively, which
contain _both_ the slip plane normal _and_ the slip direction. According to
(12) and (18), these two vectors transform differently into the hexagonal
system, which makes the algebra more involved. The derivation of the explicit
forms of these two stress components is given in A.

Figure 5: Predictions of the symmetry-adapted yield criterion for hexagonal
metals deforming by basal-prismatic slip for different choices of the
parameters $a_{1}$, $a_{2}$ and $a_{3}$ shown on the left (the remaining
parameters are zero). The numbers refer to the slip systems in Tab. 2.

We have shown that the effective yield criterion (9) is applicable also to
hexagonal crystals after expressing the slip directions and slip plane normals
using the four-indexed slip directions $\bm{M}^{\alpha}$, slip plane normals
$\bm{N}^{\alpha}$, and the normals of the auxiliary planes
$\bm{N}^{\prime\alpha}$ summarized in Tab. 2. For convenience, the explicit
forms of these stresses are given below:

$\displaystyle\sigma_{nm}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\bm{\Sigma}^{*}:(\bm{N}^{\alpha}\otimes\bm{M}^{\alpha})}{\lambda_{n}\lambda_{m}}$
$\displaystyle\sigma_{n^{\prime}m}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\bm{\Sigma}^{*}:(\bm{N}^{\prime\alpha}\otimes\bm{M}^{\alpha})}{\lambda_{n^{\prime}}\lambda_{m}}$
$\displaystyle\sigma_{nn}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\bm{\Sigma}^{**}:(\bm{N}^{\alpha}\otimes\bm{N}^{\alpha})}{\lambda_{n}^{2}}$
$\displaystyle\sigma_{n^{\prime}n^{\prime}}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\bm{\Sigma}^{**}:(\bm{N}^{\prime\alpha}\otimes\bm{N}^{\prime\alpha})}{\lambda_{n^{\prime}}^{2}}$
(24) $\displaystyle\sigma_{mm}$ $\displaystyle=$
$\displaystyle\frac{\bm{\Sigma}:(\bm{M}^{\alpha}\otimes\bm{M}^{\alpha})}{\lambda_{m}^{2}}$
$\displaystyle\sigma_{tt}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{(\bm{N}^{\alpha}\otimes\bm{M}^{\alpha})_{ij}\Sigma_{ijmn}(\bm{N}^{\alpha}\otimes\bm{M}^{\alpha})_{mn}}{\lambda_{n}^{2}\lambda_{m}^{2}}$
$\displaystyle\sigma_{t^{\prime}t^{\prime}}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{(\bm{N}^{\prime\alpha}\otimes\bm{M}^{\alpha})_{ij}\Sigma_{ijmn}(\bm{N}^{\prime\alpha}\otimes\bm{M}^{\alpha})_{mn}}{\lambda_{n^{\prime}}^{2}\lambda_{m}^{2}}\
,$

where the generalized stress tensors are
$\bm{\Sigma}=\bm{Q}\bm{\sigma}\bm{Q}^{T}$,
$\bm{\Sigma}^{*}=\bm{Q}^{*}\bm{\sigma}\bm{Q}^{T}$,
$\bm{\Sigma}^{**}=\bm{Q}^{*}\bm{\sigma}\bm{Q}^{*T}$, and the definition of the
tensor $\Sigma_{ijmn}$ is given in A.

Similarly as in the case of bcc metals, it is instructive to look at the
predictions of the yield criterion for hexagonal metals, which is defined by
(9) with the individual stresses given by (3.2.3). For simplicity, we again
ignore the last two components in (9), because they do not affect the onset of
plastic deformation at small strain rates. We particularly focus on
$1/3\langle 11\bar{2}0\rangle$ dislocations, which may glide on both basal and
prismatic planes. In the following, we thus consider only the twelve systems
1-3, 1*-3*, 4-6, 4*-6* in Tab. 2. The stereographic triangle in which the
analysis of operative slip systems will be made is now
$[0001]-[01\bar{1}0]-[\bar{1}2\bar{1}0]$. Due to lower symmetry of hexagonal
crystals, this triangle contains three elementary stereographic triangles
considered in Fig. 3 for crystals with cubic symmetry. The first row in Fig. 5
again shows the prediction of the Schmid law ($a_{1}=a_{2}=a_{3}=0$), where
the symmetry between tension and compression is evident. Loading in the center
of the stereographic triangle results in primary basal slip on the system 2
(2*) and secondary basal slip on the system 1 (1*). However, deviation of the
loading axis towards the $[01\bar{1}0]-[\bar{1}2\bar{1}0]$ edge of the
triangle activates primary prismatic slip on the system 4 (4*) with the
secondary contribution arising from basal or prismatic slip, depending on the
orientation. Similarly as in bcc metals, the effects of individual non-Schmid
stress components depend on the nature of the particular stress term and on
the character of the applied load. For some combinations of $a_{1}$, $a_{2}$,
$a_{3}$, these non-Schmid stresses result in broadening of the area of basal
slip on the system 2 (2*) and/or 1 (1*), but other combinations promote
prismatic slip on the systems 4 (4*) and/or 5 (5*). The precise values of the
coefficients $a_{1}$, $a_{2}$ and $a_{3}$, and thus predictions of the slip
activity for a particular hexagonal metal, have to be again obtained from
atomistic simulations or from experiments.

##  4 Conclusions

We have shown that the current form of the effective stress that is used to
describe the activity of slip systems in materials that do not obey the Schmid
law is not invariant under the symmetry operation
$(\hat{\bm{m}}^{\alpha},\hat{\bm{n}}^{\alpha})\rightarrow(-\hat{\bm{m}}^{\alpha},-\hat{\bm{n}}^{\alpha})$
dictated by crystallography, where $\hat{\bm{m}}^{\alpha}$ is the slip
direction and $\hat{\bm{n}}^{\alpha}$ the slip plane normal of the slip system
$\alpha$. For example, different values of $\tau^{*\alpha}$ are predicted from
(2) for the $(\bar{1}01)[111]$ system if one makes a choice
$\hat{\bm{m}}^{\alpha}=[111]/\sqrt{3}$,
$\hat{\bm{n}}^{\alpha}=[\bar{1}01]/\sqrt{2}$, or
$-\hat{\bm{m}}^{\alpha}=[\bar{1}\bar{1}\bar{1}]/\sqrt{3}$,
$-\hat{\bm{n}}^{\alpha}=[10\bar{1}]/\sqrt{2}$ although both are equivalent
conventions describing the same slip system. This means that a great care must
be exercised when using these criteria to predict the slip activity.

To remove this drawback, we have explored the possibility to derive a general
form of the effective stress that satisfies this symmetry. The effective
stress is written as a sum of power series for each component of the stress
tensor. For simplicity, we retain only the lowest order stress terms that are
invariant under the symmetry above. This results in a homogeneous effective
stress of order two in which the quadratic stresses are associated with shear
stresses parallel and perpendicular to the slip direction resolved in the two
non-glide planes perpendicular to the slip plane. We show that the presence of
these quadratic terms can be avoided by introducing an auxiliary system in the
zone of the slip direction, as done previously. This transformation provides a
linear yield criterion that is conceptually similar to that proposed by Qin
and Bassani (30), but also satisfies the symmetry above. The presence of the
auxiliary plane leads to generalization of the symmetry condition that
involves also the normal of the auxiliary plane.

The same form of the yield criterion can be used to describe the slip activity
in bcc and hcp crystals. In the case of hexagonal symmetry, we use the
transformation rules between ortho-hexagonal and hexagonal systems to express
all stress terms using four-index representations of slip directions and slip
plane normals. These expressions are very similar to those used for bcc
lattices provided the applied stress tensor is also generalized into the four-
index notation. Somewhat different forms are obtained for the stresses
$\sigma_{tt}^{\alpha}$ and $\sigma_{t^{\prime}t^{\prime}}^{\alpha}$, which
follows from different transformation rules for slip directions and slip plane
normals.

If non-Schmid stresses play no role in the initiation of plastic deformation,
as it is the case in fcc metals and for basal slip in hcp metals, the
effective stress derived here reduces to the Schmid stress. In this case, the
theoretical framework established in this paper reduces to Tresca’s maximum
shear stress theory.

## Acknowledgments

The author acknowledges stimulating discussions on this topic with Vaclav
Vitek and John Bassani. This research was supported by the Czech Science
Foundation, grant no. 19-23411S. It was carried out under the project CEITEC
2020 (LQ1601) with financial support from the Ministry of Education, Youth and
Sports of the Czech Republic under the National Sustainability Programme II.

##  Appendix A Derivation of $\sigma_{tt}^{\alpha}$

Our objective is to write the stress term $\sigma_{tt}^{\alpha}$ in terms of
the applied stress tensor $\bm{\sigma}$, the slip plane normal
$\bm{N}^{\alpha}$, and the slip direction $\bm{M}^{\alpha}$, both defined in
the hexagonal basis. By definition,
$\sigma_{tt}^{\alpha}=(\hat{\bm{n}}^{\alpha}\times\hat{\bm{m}}^{\alpha})\bm{\sigma}(\hat{\bm{n}}^{\alpha}\times\hat{\bm{m}}^{\alpha})^{T}$,
where the slip plane normal $\hat{\bm{n}}^{\alpha}$ and slip direction
$\hat{\bm{m}}^{\alpha}$ are expressed in the orthogonal basis. Using the
transformation rules (12) and (18), these vectors can be expressed as
$\hat{\bm{m}}^{\alpha}=(\bm{M}^{\alpha}\bm{Q})/\lambda_{m}$ and
$\hat{\bm{n}}^{\alpha}=(\bm{N}^{\alpha}\bm{Q}^{*})/\lambda_{n}$. Upon
substituting these vectors above, one obtains an equivalent form of the stress
component above:

$\sigma_{tt}^{\alpha}=\frac{[(\bm{N}^{\alpha}\bm{Q}^{*})\times(\bm{M}^{\alpha}\bm{Q})]\bm{\sigma}[(\bm{N}^{\alpha}\bm{Q}^{*})\times(\bm{M}^{\alpha}\bm{Q})]^{T}}{\lambda_{n}^{2}\lambda_{m}^{2}}$
(25)

that is now written explicitly using the vectors $\bm{N}^{\alpha}$ and
$\bm{M}^{\alpha}$ defined in the hexagonal basis. We want to derive an
equivalent form of this expression in which the vectors $\bm{N}^{\alpha}$ and
$\bm{M}^{\alpha}$ are decoupled from the multiplication with the matrices
$\bm{Q}^{*}$ and $\bm{Q}$, respectively. For that purpose, it is first
convenient to write the cross-products in (25) in tensorial form. The $k$-th
component of the cross-product
$(\bm{N}^{\alpha}\bm{Q}^{*})\times(\bm{M}^{\alpha}\bm{Q})$ is

$\displaystyle[(\bm{N}^{\alpha}\bm{Q}^{*})\times(\bm{M}^{\alpha}\bm{Q})]_{k}=(\epsilon_{i^{\prime}j^{\prime}k}Q^{*}_{ii^{\prime}}Q_{jj^{\prime}})N_{i}M_{j}\
,$ (26)

where $\epsilon_{i^{\prime}j^{\prime}k}$ is the Levi-Civita tensor, and the
Einstein summation is made over the repeated indices. Incorporating (26) in
(25) leads to an equivalent expression of the latter,

$\sigma_{tt}^{\alpha}=\frac{(\bm{N}^{\alpha}\otimes\bm{M}^{\alpha})_{ij}\Sigma_{ijmn}(\bm{N}^{\alpha}\otimes\bm{M}^{\alpha})_{mn}}{\lambda_{n}^{2}\lambda_{m}^{2}}$
(27)

where
$\Sigma_{ijmn}=(\epsilon_{i^{\prime}j^{\prime}k}Q^{*}_{ii^{\prime}}Q_{jj^{\prime}})\sigma_{kl}(\epsilon_{lm^{\prime}n^{\prime}}Q^{*}_{mm^{\prime}}Q_{nn^{\prime}})$
is a representation of the stress tensor in the hexagonal basis. The two terms
in the brackets are cross-products of the second indices of the matrices
$\bm{Q}$ and $\bm{Q}^{*}$. Hence, this generalized stress tensor can be
written in the form

$\Sigma_{ijmn}=(Q^{*}_{i\bullet}\times
Q_{j\bullet})_{ijk}\sigma_{kl}(Q^{*}_{m\bullet}\times Q_{n\bullet})_{mnl}\ ,$
(28)

where $\bullet$ represents the index involved in the calculation of the cross-
product.

Unlike all other stress components, the matrix form of $\sigma_{tt}^{\alpha}$
in hexagonal crystals (27) is different from its expression for cubic systems.
This is due to the cross-product between the slip plane normal and the slip
direction, which transform differently between the ortho-hexagonal and
hexagonal systems as shown in (12) and (18). The same argument applies to the
stress component $\sigma_{t^{\prime}t^{\prime}}^{\alpha}$ that is obtained
from (27) by replacing $\bm{N}^{\alpha}$ with $\bm{N}^{\prime\alpha}$.

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RemIX: A Distributed Internet Exchange
for Remote and Rural Networks

William WaitesJames SweetRoger BaigPeter BunemanMarwan FayedGordon HughesMichael FourmanRichard Simmons

University of Edinburgh
HUBS c.i.c
University of Stirling
Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS><<EMAIL_ADDRESS>
The concept of the IXP, an Ethernet fabric central to the structure of the
global Internet, is largely absent from the development of community-driven
collaborative network infrastructure. The reasons for this are two-fold.
IXP exist in central, typically urban, environments where strong network
infrastructure ensures high levels of connectivity. Between rural and remote
regions, where networks are separated by distance and terrain, no such
infrastructure exists. In this paper we present RemIX a distributed IXP
architecture designed for the community network environment. We examine this
praxis using an implementation in Scotland, with suggestions for future
development and research.

The concept of the IXP is, so central to the
structure of the global Internet has so far been mostly absent
from the world of community-driven collaborative network
infrastructure development. The dual reasons for this are that
IXP typically exist in a central place in an urban area
and community networks are often in rural and remote places,
with long distances, mountains and the sea to contend with.
In this paper we present RemIX a distributed IXP
architecture designed for the community network environment
and examine two instances of this praxis in Scotland and

[inline]Below deprecated for now in discussion with RB
for purposes of abstract submission deadline.
Salient differences between HUBS and Guifi:

* HUBS lacks a centralising structure, a foundation, with an articulated social contract.

* Guifi itself does not provide Internet access

* HUBS is an IP Transit provider

* If anything, Guifi is more like a (larger version of) WHIX

* There is some conflict of interest in HUBS operating WHIX, which Guifi does not suffer from

Rural access networks are designed to bridge the `last-mile' broadband
gap in regions that are under-served by traditional broadband
providers. Their construction is bespoke, driven by their
beneficiaries, and determined by physical landscape, population
distribution, and monetary budget. Irrespective of their differences,
they are joined by one substantial challenge: connecting to the rest
of the Internet is prohibitively expensive. HUBS c.i.c was
created in Scotland to respond to this. It is a co-operative of access
network members that generates the economies of scale required to
afford backhaul and Internet transit. While intermediation at the IP
layer between the members and the Internet is required for reasons of
scale, it is neither necessary nor desirable amongst the member
networks themselves. In urban areas, networks could interconnect with
each other at an Internet Exchange Point (IXP). In Scotland where the
networks are scattered across 80,000km$^2$ of mountain, field, and sea
it is not so easy. We bridge this gap with a design for a distributed
Internet exchange for access networks in remote places. Doing so
allows for bilateral arrangements for mutual support and assistance
between these networks, and increases the resilience of access network
connectivity. We present the relevant components, and describe our
implementation, so that our efforts may be reproduced.

§ INTRODUCTION
In remote and rural regions the last-mile problem has been the subject of much
focus. Deployments in remote regions of the world have shown that it is possible
to build high quality access networks in otherwise under-serviced
regions [8, 6, 9]. Their underlying technologies range in medium (eg. copper
or fibre-optic cabling, licensed or unlicensed wireless), energy (eg. solar or
wind generation or mains supplied), and topology.

Successful deployments, including our own in Scotland,
have two attributes in common:

* Networks designs are bespoke, suggesting
there is no one-size-fits-all solution;

* crucially, communities must be invested and
involved [19, 20].

Though remote access network research is far from complete, the next question is
increasingly clear: What options do remote, isolated networks have for
`backhaul' to interconnect with the rest of the Internet? We define “remote”
as far from urban areas where commodified network infrastructure is available.
For example long-distance circuits, if and where they are available, are both
expensive and difficult to reach. Access networks in remote places serve
populations that are dispersed. The lower population density reduces the size of
their user-base when compared to their urban cousins. With no options for
interconnecting with nearby networks to generate economies of scale,
high-quality backhaul is prohibitively expensive, if it exists at all.

The absence of resource pooling options for remote networks is the focus of this
paper. One such example is operated by the Guifi Foundation [8]. Guifi
operates a regional backbone network as a commons. The abstraction that is
presented to clients is an exchange point implemented over IP. In this type of
network, relationships between end-users are either mediated by Guifi, or
implemented as an overlay.

The IXP is a long-standing structure that plays a pivotal role in
facilitating interconnections between networks [2, 7]. We
are motivated by IXP for two reasons. First, the primary role of an
IXP is economic. Member networks can connect $n$ networks at an IXP with
$n$ circuits, rather than arranging $O(n^2)$ circuits independently.
Second, the IXP model of multilateral public peering leads to high density
interconnections, and traffic across the exchange that can be comparable in
magnitude to the largest global service providers [2]. Together,
they are an indication that such a topology might be used to improve
inter-connectivity between networks in under-serviced regions, and to pool
otherwise expensive backhaul resources.

In this paper we present RemIX, a distributed Internet Exchange for
Remote and rural networks. The RemIX architecture is agnostic to underlying
technologies, embedding the same principles as the successful remote networks it
is designed to serve. It distinguishes itself from IXP by the vast
distances permitted between points of presence, and the lower density of member
networks that connect to them. The trade-off between distance and density gives
rise to the idea of lightweight points of presence. The lightweight
nature is advantageous, in that as few as two member networks are sufficient to
establish a point of presence.

We describe our RemIX implementation in Scotland. In its current form our
deployment services a $\sim 2000\mathrm{km}^2$ region that spans sea and mountainous
mainland. Implementation details are provided, with motivating rationale, so
that others may benefit from our efforts. Functionally, our implementation
appears to its members as a large Ethernet switching fabric. Crucially, RemIX
allows member networks to establish unmediated relationships between themselves.

In the following sections we further motivate IXP as an ideal model. We
then discuss the RemIX architecture in detail. Our deployment is described,
along with lessons learned. Finally, a broader context of the local environment
is presented before concluding remarks.

§ *IXP

As part of the decommissioning of the NSFNET,

four NAP were created. They were operated by large
American telephone companies (MCI, Sprint, PacBell, Ameritech) and
designed to prevent partitioning of the commercial
Internet [2, 7].
The NAP were prohibitively expensive and had arbitrary
technological requirements which created barriers to participation.
Soon IXP emerged as an alternative. IXP appeared in
carrier-neutral facilities allowing dense inter-network connections on
a non-discriminatory basis. Presence at an IXP entails freedom
to make bilateral arrangements with any other network also
present. Worldwide, IXP now number in the hundreds and are a
fundamental feature in the structure of the Internet.

A mirroring of this structure would be useful in joining remote
networks. The increases of interconnection density could then
be used to pool traffic, and make collective use of expensive resources
such as long-distance circuits. However, there are some important
differences between the environment of a typical urban IXP and
the rural regions, as in the West Coast of Scotland:

* There are no data centres, carrier-neutral or otherwise;

* due to geography there is no single facility where all of the
networks could meet.

§ REMIX ARCHITECTURE
In this section we present the RemIX architecture. We compare RemIX with IXP architectures, and relate those
benefits in the context of remote access networks.

§.§ Design Requirements

Our requirements are shaped by three broad goals:

* establish high-quality backhaul to remote regions;

* ensure backhaul affordability for small access networks;

* allow networks to maintain the autonomy that is
fundamental to their sustainability.

Member networks must be able to connect to one or more transit providers.
Members must also be free to arrange and articulate policies among themselves.
These requirements imply that a logical concentration of inter-network
connections is desirable, which suggests a shared switching fabric below the
network layer.

[Traditional IXP]

[Modern urban IXP]

Comparison of exchange point models. Notice density.

Networks that are capable of connecting to the same location can do so with an
Ethernet switch. This is the basis for traditional IXP design (Fig. <ref>) where member networks connect to a central fabric
with their own router that sits inside the IXP facility. Our remote networks
have no such luxury. In response, we take and distribute the contemporary design
of a multi-site IXP (Fig. <ref>). A multi-site IXP
presents a single logical fabric to its members, implemented with switches
that are joined by private circuits.

The RemIX architecture that emerges (Fig. <ref>)
has no large facility nor physical housing. Instead it is
distributed so that lightweight points of presence may be established
where there are as few as two members. Members either colocate their border
routers with the exchange switch, or remotely on the far end of a link, as
circumstances dictate.

These circumstances motivate the lightweight nature of points of presence.
Since the fabric is distributed, fewer networks that will connect from each
site. High port densities are unnecessary. Simultaneously, space and power are
both at a premium. For example, a remote port into RemIX could be housed in a
small cabinet atop a hill, or in space that is donated by a property owner for
this purpose. Equipment is therefore restricted to the small and

§.§ RemIX Components

§.§.§ Switching Fabric

The exchange itself must mimic a distributed Ethernet switch. Multiple
Ethernet-like link options include fibre, 802.11 wireless, licensed wireless, fibre, leased pseudo-wires. The switching fabric may be implemented on
top using BGP-VPLS [12] (as we have in
Section <ref>),
BATMAN [11], or
TRILL [15] protocols. The salient feature
between them is MAC address learning to establish an Ethernet switch
similar to the MEF E-LAN interface
specification [1].

§.§.§ Member AS

Among traditional IXP connected networks are encapsulated into
Autonomous Systems (AS). Among RemIX member networks, the
policies of the small sized member networks are

different from the Internet's DFZ. In particular, member
networks' smaller routers will be neither be capable of storing the
entire Internet routing table, nor are they likely to announce
netblocks large enough to be globally visible.  However, AS
encapsulation enables networks to retain their internal
structures and methodologies, and to interconnect safely  with
neighbours. Due to the likelihood of collisions use of private
ASN is inappropriate for this purpose [14], as are
private IP addresses for the exchange itself [16].

§.§.§ Exchange Transit

RemIX members' IP address spaces will be small, and need some entity to
advertise larger netblocks on their behalf. This suggests a specialized transit
provider to mediate between members and the wider Internet. For this reason
RemIX members form a confederation with a transit provider that
presents them collectively to upstream providers and other exchange points. This
is unusual for IXP: Rarely are transit relationships implemented with
exchange points. However, this is normal in RemIX, and likely necessary to
function in the intended environment. We note that transit service should be
optional to members, with no requirement to purchase said provider's transit as
a condition for joining the exchange. Also, nothing prevents other such
providers from participating.

§.§.§ Auxilliary Services

BGP configuratoin can be a complex. For example, upon
connecting to RemIX, member networks need to be configured to peer amongst
themselves. The complexity quickly increases as session numbers grow with the
square of the number of participants. Instead, IXP use
route-servers to repeats announcements from one member to all others. A
route reflector keeps the configuration burden to a minimum. Other useful
services such as NTP clocks and looking glasses for assistance in
debugging may be offered in addition.

The overall RemIX architecture is motivated by our own needs in Scotland. In
the next section we present our first-phase implementation of RemIX,
alongside remarks on usability and directions for the future.

§ REMIX DEPLOYMENT IN SCOTLAND

In this section we describe our first implementation of RemIX in a series of planned deployments across Scotland. In the West Highlands there
is a cluster of

11 small community networks. Their spread across $\sim 2000km^2$

of sea and mountainous islands makes the
construction of an exchange fabric geographically ambitious. Four networks have
a history of interconnecting and sharing network resources, pre-established
relationships that must be respected in our deployment.

Our deployment's location is its namesake, the WHIX. Both logical
and physical layers are described below, with additional lessons and
comments drawn from our experience.

§.§ West Highland IX at Layer 1

The physical WHIX fabric is overlayed onto a stylized map of the region in
Figure <ref>. The map itself preserves critical geographical
features. Red connected nodes are the connection sites. In a traditional
IXP these sites are the Ethernet ports into which subscriber AS
plug-in. WHIX sites are connected by wireless radio links in black, and leased
100Mbps or 1Gbps circuits in orange. The areas enclosed with dotted lines
correspond to the service areas reachable from each site.

[Physical topology of WHIX.]

[Member connections to WHIX]

Physical and logical layout of WHIX. In
Figure <ref> the dark lines correspond to radio links
and the light, curved lines to leased ethernet circuits.
In Figure <ref> the dashed lines
correspond to internal layer-2 circuits forming WHIX
switching fabric and the solid lines to member connections.

We complement WHIX' physical topology in Figure <ref> with
the member network in Figure <ref>. In the latter, unlabeled red
nodes are the WHIX points of presence and correspond with the same set of
red nodes in Figure <ref>. The dashed lines represent the fully
connected virtual topology that implements the exchange E-LAN.

The two places in the region where long-distance ethernet circuits are
available on the mainland are the towns of Mallaig and Kyle of
Lochalsh. Circuits[At the time of writing, the circuit from
Mallaig is in place, and that from Kyle is planned.] from these sites
connect back to the Pulsant datacentre in Edinburgh to facilitate
remote peering — and indeed the provision of Internet access via the
exchange point.

The radio links are implemented with equipment from Ubiquiti Networks,
configured in transparent bridge mode so that they appear as Ethernet from a
functional perspective. The switching fabric itself at each of WHIX points
of presence is implemented with Mikrotik routers. This choice was made because
of their moderate port density, low power consumption, low cost, and adequately
featureful MPLS implementation. We revisit this choice in the next
section. All equipment is configured to pass Ethernet frames of at least 1600
bytes to provide room for the necessary extra protocol headers for implementing
the E-LAN service.

§.§ West Highland IX at Layer 2

We emphasize that layer-2 details are internal to
WHIX, and invisible to members who only see an Ethernet switch. Also, our
implementation decisions are by no means the only possible means of

In WHIX the requirement for functional equivalence to a MAC address
learning Ethernet switch is met using BGP signalled
VPLS [12]. This creates a full set of LSP pseudo-wires
between every pair of WHIX edge routers.

Each WHIX router maintains an OSPF routing protocol
adjacency with its neighbours and distributes reachability information
for its loopback IP address. All addresses used for this purpose are
private IPv4 addresses [16]. This is the basic layer that
ensures reachability throughout the distributed fabric.

Non-IP traffic is carried via LDP [3]

with MPLS labels according to the topology of the underlying OSPF

Routers in WHIX establish BGP peering sessions with routers at
Mallaig and Sabhal Mòr Ostaig that act as route reflectors [5].
Participating routers use route reflectors to exchange reachability information
without requiring a full mesh ($n^2$) of internal peering sessions. The presence
of BGP signalling throughout the WHIX fabric enables the use of
multi-protocol extensions [4]. Routers can use extensions to signal a
desire to form part of the exchange LAN.

The result is a fully meshed VPLS, where each router has a virtual bridge
interface that forms part of the exchange LAN.

Interfaces can be added to virtual bridges, as needed, to
form part of the exchange. Care must be taken to prevent loops in which members
see the traffic that they originate. This is accomplished with a split-horizon
method [13]. Equally, members must be prevented from creating bridge
loops via their own network by employing MAC address filter on relevant

§.§ West Highland IX at Layer 3+

Given logical connectivity between all member ports, it remains to assign IP
addresses to their border routers, as well as public infrastructure such as the
router server. As mentioned above the use of private IP address space for this
purpose is undesireable since it generates risks of collisions with members' own
infrastructure. WHIX, and more generally RemIX, is fortunate in this
regard: The design meets the definition of an IXP [18, 10],
making it possible to acquire IPv4 and IPv6 address allocations from RIPE
NCC [17].

Autonomous System topology. The members of a RemIX form a fully
connected network where each may communicate with another over the
exchange without intermediation.

The full layer-3 WHIX topology is shown in Figure <ref>. At this
stage member network have everything they need. Members can communicate at layer
2. Each has an IP address at layer 3, an autonomous system number for
identification, and their own networks to announce. Bilateral peering
arrangements (an otherwise $n^2$ configuration task) are facilitated by two
route-servers, as before at Mallaig and Sabhal Mòr Ostaig. The route servers
redistribute reachability information, akin to a route-reflector omits its own
ASN from the path.

The transit provider, HUBS (see Section <ref>), is also
present at WHIX as a member. In addition to public multilateral peering, it
establishes bilateral sessions with members wishing to announce either a
default route or full Internet routing tables. HUBS forwards those members'
announcements upstream and to their peers. In this way transit, and hence
connectivity to the global Internet, is provided over the exchange.

§.§ Deployment Discussion

Our experience motivates higher-level comments to further distinguish
RemIX deployments form their larger IXP cousins. Flat networks
consisting of a single layer-2 broadcast domain can be plagued by
problems that are difficult to troubleshoot. By its very design RemIX requires
that members be able to communicate directly without mediation at the
IP layer. Like other IXP RemIX eliminates a large class of
potential problems by allowing only unicast and ARP traffic on
the exchange. Moreover, members must nominate a specific MAC
address for their connections, which reduces the risk of loops and
broadcast storms. We also adopt best practices such as quarantines for
new connections while they are evaluated for correctness.

IP transit in RemIX also deserves to be addressed. Transit
via the exchange, for networks that are not otherwise visible on the Internet,
may evoke notions of conflicting interests that
beset NAP. However the similarity is superficial. Here, the transit
provider and exchange operator HUBS c.i.c., is a cooperative
that exists for the benefit of and is controlled by the members, who
are also members of the IXP. As a consequence all parties'
economic interests are aligned.

Finally, WHIX' implementation using BGP-VPLS
to construct the exchange fabric makes it possible to offer auxiliary
point to point pseudo-wire services to its members. This is
useful for those members that have need for making connections internal to their

§ THE ENVIRONMENT

Scotland contains 1/3 of the area but 10% of the population of Great Britain. It also has 95  inhabited islands with about
100,000 people. The Scottish Highlands and Islands, where this
work is currently focussed, consist of mountainous terrain stretching along a 400km north to south corridor. Islands are scattered on the West together with
deep lakes and glens penetrating the mainland to the East.  The economy was
traditionally maritime, and nearly all habitation is at sea level or in the

Fibre in the region has only recently appeared.  Much of the telephone network
in the region was constructed with microwave links. Infrastructure is improving,
though plans terminate at telephone exchanges. Among them, fibre-based
services are rare. In the medium term future, local wireless distribution is the
only feasible technology for adequate bandwidth and quality of service.

Starting in 2008, the Tegola project [6] started to
experiment with technology that would enable communities to build
their own wireless networks.

The details of Tegola, and its dissemination to nearby communities, are
omitted due to space constraints. Relevant to this project is the technology
that emerged. Figure <ref>, for example, features the type of
robust, inexpensive relay construction that operates in mountainous region, and
that can be constructed by its residents.

A basic relay

Many communities have since constructed their own local distribution networks
with point-to-point wireless links that can span more than 20km. Expertise is
often shared between them, also infrastructure where feasible, yet they operate
independently. Constrained by availability, they acquire backhaul via ADSL lines
nearby to telephone exchanges. Ethernet services have since emerged in two
larger towns, with wholesale pricing that exceeds the budget of any single
community. A resolution has two components: An organizational vehicle that
combines networks to generate economies of scale, and a supporting network

We have learned that solutions are complicated by both terrain and by culture.
In particular we note:
(a) Social aspects and organization of communities can fail to align with the
ideal “electronic” or networked communities, eg. physical landscape constrains
connectivity, while social and economic groups can be determined by vehicles for funding.

(b) Local network infrastructure is non-uniform and varies in complexity.

(c) Communities that share network resources generally do so in a
non-systematic or ad-hoc manner.

§.§ HUBS C.I.C.

In response to the local environment and absence of affordable
backhaul, the Universities of Edinburgh and Stirling launched
HUBS, which is a not-for-profit transit provider whose members are the
community networks that it serves.  HUBS is also a co-operative where
the networks that subscribe also become members. It is the culmination
of collaborations between Universities with communities in the West
Highlands, and later with community networks in the South Scotland.

The need for RemIX-like functionality arose soon after launch. Two of
the subscriber networks took advantage of mutual proximity to
collaborate on an operational basis. Equipment management and
troubleshooting tasks, for example, were shared. Their desire to keep
the details internal was complicated by the fact that their only
interconnection was mediated by HUBS. Circuits were hand-crafted
between them, and demonstrated the benefits of bilateral agreements
between HUBS members. However, while effective, hand-crafted circuits
would fail to scale.

HUBS bridges gaps in backhaul affordability. It has also revealed the
benefits emerge when remote and rural networks are able to act
collectively in the wholesale telecommunications market, and present a
uniform interface to their upstream transit provider. However, a
transit-only solution prohibits autonomous arrangements between
members unmediated at the IP layer. From this need the RemIX
architecture emerges.

§ CONCLUDING REMARKS

The features of RemIX described above will be instantly recognizable to anyone
who has participated in a regular IXP. This is by design. RemIX is
architected to mirror in under-serviced regions, the benefits of IXP in
urban regions. The encapsulation of small community networks in AS means
that they can present a uniform interface to a transit provider, cooperate and
share resources. RemIX provides these benefits to members without sacrificing
their independence, a necessary attribute for longterm sustainability.

§ ACKNOWLEDGEMENTS

This project is supported by the University of Edinburgh, by the
Scottish Government and by local industries including Marine Harvest
Scotland and Benchmark Holdings.  During an incubation period,
WHIX is jointly operated by HUBS and the University of Edinburgh.

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BGP Route Reflection: An Alternative to Full Mesh Internal BGP
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