Determination of flow patterns in rocks: an introduction and overview

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Nov 14, 2013 (3 years and 9 months ago)

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1


Determination of flow

patterns
in rocks
: an introduction and overview


Description of homogeneous flow




2.5
PT



a
.

Stages of progressive
homogeneous deformation.
Reference frame is attached to the lower
boundary of the
experiment (deformation
-
zone

boundary
)
.

b
.

Two subsequent
stages
are us
ed
to determine the velocity fi
eld at a particular time
.

c
.

Marker points in the flow pattern can be connected by
lines.

d
.

Fo
r each line stretching rate (

̇
;

̇
=l
1
/l
0
) and angular velocity
(ω)
are defined
.

e
.


̇

and ω are plotted in curves again line
orientation

(
0
-
360°;
note: dextral rotation is positive!)
.

S
pecial
di
rections are:

ISA = instantaneous stretching axes
: two lines

(in 2D)
,

along which the stretching rate (

̇
) has its
maximum
and

minimum values; they are always orthogonal.

Irrotational

material lines
:
can have any orientation
.


̇
k
: t
he amplitude of the

̇
-
curve
is

a mea
s
ure of the
st
rain rate
.

Vorticity: the

elevati
on

of the s
ymmetry line

of the
ω
-
curve
.

f
.

Orientations of ISA and irrotational lines can be read from graphs
.


2


Types of flow


2.6PT



Isochoric flow: if the stretching rate

(

̇


is symmetrically arranged with

respect to the zero stretching
-
rate
axis, no

area change is involved

in the flow

(area in
/de
crease involves extra stretch in all
directions: the curve is shifted upwards
/downwards
)

Coaxial flow: if in a reference frame fixed to the ISA
,

the angular velocity curve is symmetrically
arranged with respect to the angular velocity axis, no ‘bulk rotation’ is involved in the flow, and
lines of zero angular velocity (irrotational lines) are orthogonal.

Flow is said to be coaxial
because a pair o
f lines that is irrotational is parallel to the ISA. This pure shear flow has
orthorhombic

shape symmetry.

General n
on
-
coaxial flow: if all material lines are given an identical extra angular velocity, the
angular velocity curve is shifted upwards (dextral

rotation; see definition of + and


in Fig.
2.
PT
5) or downwards (sinistral rotation). In both

(sinistral, dextral)
cases, flow is non
-
coaxial
since irrotational lines are no longer parallel to the ISA.

All non
-
coaxial

flows

have a monoclinic
symmetry.

Vor
ticity: the deviation of the angular velocity curve from
zero angular velocity

is a measure of the
irrotational character of the flow.

Simple shear: when the angular velocity curve touches the zero angular velocity axis and only one
irrotational line exist
s
.


Flow descriptions

(see parameter definitions in Fig. 2.6

PT)


α
k
: angle between one of the ISA and the side of the shear box (shear zone boundary)


̇
k

=

̇
1
-

̇
2
= ω
1
-
ω
2
; a measure of strain rate (amplitude of the stretching
/rotation

rate curve)

W
k

= (ω
1

2
)/


̇
k
; kinematic vorticity number

A
k

= (

̇
1
+

̇
2
)/


̇
k
;
a measure of areal change with time

E.g.: s
imple shear: W
k

= 1, A
k

= 0.

(
mit
ω
:
W
k

= 1+
0/1
-
0

= 1; ω
1

= x, ω
2

= 0; all material lines rotate
e.g. dextrally
; with

̇
k
: take scales, e.g.
ω
1

=
2
, ω
2

= 0
,

̇
k

= (1
-
(
-
1 )= 2
)



pure shear: W
k

= 0, A
k

= 0


3


Concept of vorticity and spin


Box 2.4PT




a
.

If

the velocity

of

a river
is fastest in the

middle
,
paddle whee
ls in the r
iver will ro
t
a
t
e in

opposite

direc
t
ion at
the

si
de
s
, but will not rotate in
t
he middle; th
ey reflect

the vor
t
icit
y

of
flow in

the river at
t
hree di
f
fe
rent sites
.

b
.

Vorticity is defined as the s
um of the angular

v
elocit
y

with re
spect

to ISA of a
n
y pair of

or
t
hogonal
ma
t
erial

line
s

(
s
uch a
s

p
and
q)
; additional ro
t
a
t
ion of ISA

(and all
t
he other line
s

and vector
s)

in
an
external ref
e
rence fra
me
is known as s
pin
.




4


Types of deformation (not flow)


2.7PT



Homogeneous
deformation

(instead of flow)
can be envisaged by the distribution patterns of stretch
and rotation
.
Note that deformation is normally composed of strain (which describes a
change in
shape) and a rotation; thus deformation ≠ strain.


a
.

Two
st
age
d of a
de
f
o
rmat
ion
sequence.

b
.

T
he
def
ormation p
attern
.

c
.

Sets of marker

points c
an be
connec
ted b
y

materia
l
l
ines and the rotation (
r
) and stretch (s) of
each
l
ine
is
monitor
ed.

d
.

The
s
e
can be p
l
otted
agai
ns
t
initia
l or
ientation of

the
li
n
e
s.
In the
curves,
p
r
incipa
l

s
trai
ns can

be
distingu
ished
.

e
.

Finit
e def
ormation as deduced from
the
s
e
curves
contai
ns el
e
ments

of

st
r
ain and

rotation
(
ρ
k
)
.

β
k

def
in
es the orienta
t
ion of a material
l
ine i
n

the
undeform
ed state that i
s

to
becom
e paralle
l
to the
l
ong

a
xi
s

of the
s
train
ellipse
i
n

the

defo
rm
ed
s
tate
.



5


Progressive and finite deformation


2.8PT



H
omogeneous
finite

deformation carries no information on the deformation path or on progressive
deformation. However, the stretch and rotation
history

of material lines does depend on the flow type
by which it accumulated. In the case of
inhomogeneous

deformation on s
ome scales, as is common in
deformed rocks, pure shear and simple shear progressive deformation can produce
distinct, different
structures.
The difference is best expressed in the symmetry of fabric structures.


The ef
fect of

def
or
mation
hi
s
to
r
y

a
.

Two
identical squa
r
es of material with two
m
a
r
ker
line
s (
red a
n
d gree
n

l
ine
s)

are defo
rm
ed
u
p
to

the same

fi
nite

str
ai
n
va
lue in simpl
e shea
r

and pure

s
hear progre
ssi
ve deformation
, res
pe
c
tive
ly
.
The

initia
l

orientation of the
squares
is
chosen
such
th
at the

s
hape and orie
nt
ation of defo
rmed
s
quare
s

is
identic
al
.

b
.

The
finite

stretch and
r
elative o
rientation of both marker

l
ine
s is

ide
ntica
l

in both
cases,

but

the
h
istory
of

st
retch and rotation of each l
ine

i
s

different.

c
.

C
ir
cul
a
r

diagra
m
s
show

the
di
s
tribu
ti
on

of all mate
ri
al
l
ines in

the

s
quare
s of a
. O
rn
amenta
tion

shows whe
re
li
ne
s
are

s
h
o
rte
n
ed
(s),

exte
nd
ed
(e) or
f
i
rst
shortened,

the
n

extended
(se)

for each

s
tep

of

progre
ss
ive

deformation. The orie
n
tation of

I
SA
is

i
ndi
cated
.




6


Concept of the
fabric attractor


If flow (patterns like 2.6PT) works on a material for some time, material lines (e.g. long axis of finite
strain) rotate toward an axis, which coincides with the extending irrotational material line; this axis
‘attracts’ material lines in

progressive deformation.


2.9PT



In
both pure shear and simple shear deformation, material lines rotate
towards
and concentrate near
an attractor direction. The line is

the fabric attractor
(FA).
Both foliation and lineation rotate
permanently
toward
the attractor.



7


Concept of fabric attractors: flow apophyses or flow eigenvectors
(e.g. Passchier 1987)


In a body deforming by homogeneous isochoric plane
-
strain flow, the rate of displacement (‘X
i
) of
particles at X
i

in a fixed Cartesian co
-
ordinate system
is

described by the velocity gradient tensor L'
(Malvern 1969) as:


[









]


[





















]



[






]



where
W

is the vorticity of the flow and
S

is a scalar defining the stretching rate of the fl
ow. L' can be
expressed as the sum of a symmetric tensor D' and an anti
-
symmetric tensor W':


[





















]





[














]






[














]






The eigenvectors d
i

of D' are the orthogonal instantaneous

stretching axes of the
flow and the
eigenvalues d
1
, d
2

and d
3

are instantaneous stretching rates with values 0,
S/2
and
-
S/2.
The vorticity
vector of the flow is parallel to d
1
(Fig. 1).






Fig.
1 (Foss
en; Passchier ’87).
Left: d
efinition of the vorticity vector. Right:
s
chematic representation
of a homogeneous plane
-
strain non
-
coaxial flow with vorticity number W
k

= 0.76 described by L'.

X’
i



co
ordinate system, d
i



eigenvectors of D', inst
antaneous stretching axes. l
i



eigenvectors
of

L',
=
flow apophyses. Ornamented surfaces

are traced by material lines parallel to X

i
. Arrow around X

i

indicates

sense of shear.




8


Eigenvectors l
i

of L' do not usually coincide with those of D', except for pure shear. l
1

is parallel to d
1
,
and l
2

and l
3

lie in the X’
2



X’
3

plane
,

symmetrically arranged with respect to d
2

and d
3

(Fig. 1)
(Bobyarchick 1986, Passchier 1986). The
instantaneous

non
-
coax
iality of the flow can be expressed
by a vorticity number:






|





|
,


which for isochoric plane
-
strain flow equals the kinematic

vorticity number of Truesdell (1954):







(











)
.


The planar surfaces through
l
1
,

l
2

and
l
1
,

l
3
, defined as

eigenvector planes (Passchier 1986)
,

have
special properties

for all types of instantaneous steady flow and for

progressive steady flows following

integration of L'.


For

W
k



1 all particle paths in the flow defined by L' follow

hyperboloid

curves (Fig
. 1)
,

which
approach the eigenvector

planes asymptotically; particles within the eigenvector

planes approach or
depart from the X

1

axis along

paths within the planes. For these reasons, eigenvectors

l
2

and 1
3

have
been named flow apophyses by Ramberg

(1975a, b). For plane
-
strain flow, particle paths within

the
eigenvector planes are the only straight orbits in the

flow (Fig. 1

(no flow in X
1
-
direction)
).

[
Explanations
: (i)
the term asymptotic means approaching a val
ue or curve arbitrarily closely; (ii)

a

hyperbola is defined as
the loci of all points within the drawing plane, for which the difference of the distances to the given points F1 and F2 is
constant]






Flow apophyses are theoretic
al

planes that compartmentalize the flow pattern. The number of possible
apophyses in a flow system ranges from 1
-
3. For planar deformations (2D) the maximum number is 2.
Particles cannot cross
flow

ap
ophys
e
s.

hyperbola definition

eigenvector plane

9



Poles of rotating passive lines move along trajectories (great circles) in
a

stereoplot during simple
shear

(
Fig. cf.
Foss
en)
. There is only one flow apophysis (AP), which parallels the x
-
z plane.

(
Explanation
: here only ω is visualized, the
lines does not stretch or shorten; just rotate a line and trace its end points)



Stereographic projection of pole trajectories of passive lines during progressive pure shear. AP =
flow
apophyses
. The particles move in four quadrants separates by two orth
ogonal flow apophyses.

10



Stereographic projection of the pole trajectories of passive lines during
sub
-
simple shear

(=
general
shear
). The particles move in four compartments separated by two flow apophyses at an angle
between 0
°

(3D simple shear) and 90
°

(3D pure shear
)
.




11


Relationship between W
k

and pure and simple shear
.


Fig. 1FB









Scale relations between the W
k
-
value and percent simple shear. Zones of pure shear dominated,
general shear, and simple
shear dominated deformations (from Forte and Bailey, 2007).


Relationship between kinematic vorticity number W
k

and

components of pure and simple shear for
instantaneous 2D flow; pure

and simple shear componen
ts make equal contributions to flow at

W
k

=

0.
71 (arrow). W
k
=

cos
α
, where
α

is the angle between flow

apophyses (Bobyarchick 1986), and
varies from 0
°

(simple shear) to 90
°
(pure shear). Relative proportion of pure shear to simple shear is
given

by
α/
90
°
; at
α

=

45
°

there should be an equal contribution of pure and

simple shear (45°/
90
°

=

0
.
5); for simple shear there is no pure shear

component (0
°/
90
°

=

0).

(
Explanation
: W
k

= cosα, with α as the angle between the AP is the most common definition of
vorticity
; actually W
k

= 0.7
07

= cos45
)


calculation
below

12


D
eformation in various W
k

fields
. S
imple shearing produces displacement with

little
shortening/thinning, general shear is a combination of both

thinning and displacement, and pure shear
zones experience

a great deal

of thinning with modest displacement relative to

overall strain

(Fig.
2FB)
.


Fig. 2FB


a. The initial condition before high
-
strain zone formation. Dikes, points A and B, and the strain ellipse
are provided as reference.

b. Pure shear deformation with two eigenvectors orthogonal to each other. White arrows indicate zone
parallel stretching.

c. General shear deformation with a W
k
-
value of 0.9. The acute angle between the two eigenvectors is
correspondingly 26°

(cos26 = ~0.
9)
.

d. Simple shear deformation with one eigenvector.

(
Explanation
:
thickening/
thinning

(e.g. of dikes)

and lateral flow of material is not shown)






13



Movement of axially
-
symmetric rigid objects

(e.g. Passchier 1987)


General theory

i
s
ochoric
plane
-
strain flow
.
E
quations of the movement of rigid objects

in
homogeneous flow (e.g. Jeffery 1922) are least complex for axially symmetric ellipsoidal objects. The
axial ratio of such objects can be expressed by a parameter B, where ‘a’ is the length al
ong the

symmetry axis and ‘b’ the radius in the circular section (Fig. 2
; in this Fig. a is larger than b, a prolate
object
):














.



B can represent a material line (B = 1), a prolate ellipsoid (0 < B < 1), a sphere (B = 0), an oblate
ellips
oid (
-
1 < B < 0
; a flat disc with the short axis along the
object symmetry axis
(
OSA
) or a material
plane (B =
-
1).


If an

internal reference frame X
i

is chosen with X
1

fixed to the

OSA
, the rotation of the object

with
respect to the external reference frame X

i

is given

by the Eulerian angles
θ, φ, ψ, described by a
rotation tensor R.
T
he orientation of an axially symmetric object
is

defined by two angles only
,
which

reduces
φ

to the azimuth and
ψ

to the plunge of
the OSA

(Fig. 2a).


Fig. 2P




a. Reference frame for rotation of axially symmetric rigid ellipsoids. X’
i
and X
i
: external and internal
co
-
ordinate systems. φ and ψ: azimuth and plunge of object symmetry
axis (OSA; this is the axis
that contains the vorticity
-
vector component ω
1
, the angular velocity components of OSA around
X
1
.


̇



̇


̇


rate of change of Eulerian angles (which define the object orientation),

v:
instantaneous displacement rate of OSA
on a sphere around cent
er

of co
-
ordinate systems.

b. Stereographic projection of OSA and v.


D' and W' can be expressed in terms of the

internal reference frame by:

D

= RD'R
T

and

W = RW'R
T
.



14


The instantaneous angular velocities of the object around the
X
i

axes are (Freeman 1985):

ω
1

=
-
W
32

= ((S∙W
k
)/2)

cosφ

cosψ

ω
2
=
-
W
13

-

B∙D
32

= ((B∙S
-

W
k
∙S)/2))

cosφ

sinψ

ω
3

=
-
W
21

+ B∙D
21

= (S/2)·sinφ·(W
k

+ B·cos2ψ).


ω
l

describes the axial rotational velocity of the

object around its symmetry axis, zero if the OSA
lies in

the X

2

X

3

plane

(=flow plane)

and a maximum if it lies parallel to X

1
.


T
he instantaneous

displacement rate of the OSA along a sphere around the

center

of the co
-
ordinate
systems
,

calculated from

the ω
equations (
above
) for a number of OSA posi
tions
, is

plotted

as a
vector
v

in stereograms (Fig
s
.
2,
3
P
). S appears as

a
constant and its magnitude does not

influence the shape
of the flow patterns: in
Fig. 3P,
S = 10. A half stereogram is

sufficient for presentation of the
movement pattern

because
of its bilateral symmetry.


Fig. 3P



Wulff projection of displacement rate vectors of OSA at 343 positions. 10° intervals of φ and 5°
intervals
of ψ
. W
k

= 0.4, B = 1.0

(material line)
. The vectors in Fig. 3 describe the
instantaneous
movement

of the OSA.

(
Explanation
:
there two stable positions, somewhere intermediate between X’
3

and X’
2
).




15


OSA
-
trajectories for S = 10 are shown for a complete range of B and W
k

values in Fig. 4P.


Fig. 4P


Movement patterns of the object symmetry axis (OSA) for the range of possible W
k

and B values.
Upper hemisphere Wulff projection.

Representative flow types shown along the top, representative
object shapes on the left. Symbols in stereograms: open circles: sources of OSA; open squares: sinks
of OSA; solid circles: transient stationary positions of OSA; bold lines at W
k

=
|

|
:
planes
of transient
stable positions, d
i
:
eigenvectors
D’
(instantaneous stretching axes).

l
i
:
eigenvectors of
L'.


(
Explanations
:
e.g,
spheres are irrotational in 3D pure shear and rotate infinitely in all other deformations; there
are
one
to

three

stable positions)


Stable positions according to B and W
k
:

a. W
k

<

|

|


3 stable positions: a transient position parallel to X’
1

and a source

sink pair in the X’
2
-
X’
3

plane. Source and sink are symmetrically arranged with respect to d
i
, and their actual po
sitions depend
on the sense of vorticity, B, and W
k
. At W
k

= 0 source and sink coincide with eigenvectors of D' and L'
at 45° to X’
2

and X’
3
. For any W
k
, material lines (B = 1) follow a trajectory

towards a sink parallel to

l
2
;

the normal to material

planes (B =
-
1) approaches a sink normal to l
2
. The

OSA of objects with
other B
-
values have a source and

sink at an angle
±β/2
from X

3

defined by:

cosβ =
W
k
/B.

(
E
xplanation:
β gets smaller with higher W
k
; β gets smaller with higher B)
.

b
.

W
k

=
|

|
. A 'stab
le plane' exists through X

1

and

X’
3

(B > 0) or X

2

(B < 0). All OSA within this
plane are

in transient equilibrium. Outside the stable plane
,

OSA

rotate along
straight planar paths
towards X’
3

or X’
2
. If

W
k

= 1, the stable plane coincides with the shearing plane of simple shear flow.

c
.

W
k

>
|

|

(these are objects with small axial ratios
).
Only a transient position exists along

X
’1
. In all other
positions the OSA rotate continuously
.

For general flow and axi
ally non
-
symmetric objects
,

the theory predicts that sink positions exist.

coordinates

X3

X2

X1

rotation

16



Application.

W
hich objects had reached a stable position, and

which
we
re still
rotating as deformation
stopped?

Rigid

porphyroclasts in a ductilely deforming matrix often

recrystallize along their margins
and produce tails of recrystallized material
that

stretch
es

out into the matrix.

Immobile objects with a
symmetry axis at a sink in the

X

2

X

3

plane

(normal to the flow plane)

of the flow show

straight
σ
-
type
tails with '
stair
-
stepping'. E
ll
ipsoidal objects rotate by periodic

accelerations and decelerations
,

which
also influence

recrystallization rates. Tail development will be

significant during the period of slow
rotation when the

long axes of the object are near the X

1

X

3

(
shear)

plane, and

these tails will
become distorted to
δ
-
types during the

subsequent fast rotation when the long axes are near the

X

1

X

2

(normal to the shear)
plane.
δ
-
type tails can also develop around

spherical rotating objects if
recrystallizati
on is slow.

Thus, complex and

-
type clast
-
tail systems are considered

to be indicative of
permanently rotating objects,

while objects with straight
σ
-
type tails are probably at

stable positions.

R
ecrystallized tails will reach parallelism with
l
2

at high

finite strains

(the outer tails do not rotate
anymore)
.







X3

X2

X1

17


Tails of recrystallized material will tend to rotate

towards the extensional eigenvector
l
2

of L'
throughout

the deformation. The angle
η
between the M
x
(long)
-
axis of
an

irrotational rigid object,
which lies at a sink

in the

X

2

X’
3

plane, and

l
2

or the straight domain of the tail

away from the object
is a function of W
k

and B
*

only:



























}
, with



















and M
x

and M
n

are the object’s long
and short symmetry axes in the
X’
2

X’
3

plane
.


Fig
.

8
P
-
a

shows η

for the entire range of

W
k

and B
*

values
.
η
increases with decreasing

B
*

up to the
value B
*

= W
k
, the 'cut
-
off point'. At still

lower

B
*

values, the objects are rotatin
g permanently
.
W
k

can be derived from these graphs in two ways (Fig.

8
P
-
b
): (a) from the value

B
*
c
ri
t

= W
k
,

which
separates the
σ
-
type immobilized part of the clast

population in a rock from the complex and
δ
-
type
rotational

part, and (b) theoretically from
η

and B
*

for

individual objects if sub
-
parallelism

of
recrystallized

tails with
l
2

during the last stages of the flow can be

proven.


Fig. 8P


a. Curves for the orientation of stable sink
positions of rigid objects in the X’
2

X’
3

plane of flow for a range of W
k

values, η

angle between the long axis of an object
cross
-
section and l
2
, marked by
recrystallized tails at high finite strain.
B
*

shape

factor. If W
k

> B
*

no stable
sink positions exist.

b. Example of the expected geometries of rigid
-
object recrystallized tail systems in cross
-
section parallel to X’
2

X’
3

at W
k

= 0.5. At
low B
*
values, objects rotate permanently
and generate δ
-
type and co
mplex tails. At
high B
*

values, to the right of the 'cutoff
point', objects have their long axis at a
stable sink positi
on and generate σ
-
type
tails. η
decreases with increasing B
*
.



freely rotating clasts

B*
crit.

for W
k
=0.1

X3

X2

X1

flow apophyses

18


T
he

following requirements should be met

for an application
. (1)

The fabric and general setting of the
samples should

indicate that deformation was reasonably homogeneous

on the scale of the sample
.
Samples from the

limb of a major fold are unsuitable, but
samples from a

straight, regular shear zone
may be useful. (2) Grain size

in the matrix should be significantly smaller than the size

of the objects
in order to make reasonable the assumption

of homogeneous flow. (3) High finite strains

accumulated
by homo
geneous flow are required to

rotate sufficient objects towards sink positions. (4)

Object shape
should be regular and closely approach

orthorhombic shape symmetry. Deviations of object

shape
from an ellipsoid are not expected to influence the

position of s
inks (Bretherton 1962). (5) A sample
should contain a large number of spatially well dispersed objects

with variable B
*

values.


Fig. 10P shows an application: All porphyroclasts

in a mylonite

with

approximately orthorhombic
shape symmetry and two

symmetry

axes in the plane of the section were analyzed: B* values in the
inferred

X

2

X

3

plane and

η
,

the angle between the long axis and the trace of the tail

away from the
clast, were plotted. Because of the high inferred finite strain values, the

tails are as
sumed to parallel
l
2
,
at least during the last

stages of deformation. Nearly all complex and
δ
-
type clast
-
tail systems (open

circles) plot left of the B* = 0.6 line. A dense cluster of
σ
-
type systems (dots), which dip in the
opposite

direction to the stair

stepping, plot to the right of this

line. The solid curves represent
theoretical
η
values for W
k

= 0.6 and 0.7 from Fig. 8
P
.



Data plot of K
-
feldspar clast
-
tail systems in
quartzite
mylonite
,

St. Barth
e
1emy Massif,
France, from
thin sections normal to the

inferred vorticity vector of the flow.
Orientation of object long axes

with respect to
trace of recrystallized tails, in degrees, plotted
against

B*, as in Fig. 8
P
. Open circles, clast
-
tail systems with complex or
δ
-
type

geometr
y
indicative of permanent rotation; dots,
σ
-
type
clast
-
tail

systems.





19


M
ethod of rotating porphyroclasts

elaborations


General considerations
.
The behavior of porphyroclasts depends on the flow type (W
k
; i.e. orientation
of the eigenvectors) and the axial ratio of the
porphyroclasts

(B
*
). During deformation, rigid
porphyroclasts with an aspect ratio greater than a critical value rotate towards material attractors
nearly coincident with the flow eigenvect
ors. Porphyroclasts with axial ratios smaller than this critical
value rotate independent of the bulk flow attractors. In the case of general shear, porphyroclasts either
rotate backwards or forwards towards the flow attractors. Porphyroclasts that

forward

rotate reach a
stable end position within the obtuse angle

field

(β)

between the two eigenvectors
.


Porphyroclasts that have reached a stable position can be identified by sigma tails
.
Sigma tails form
during

slow rotation
, which occurs as
porphyroclasts

approach

their stable posi
tions. Backward rotated
clasts

are identified by long axes orientations antithetic to

the overall sense of shear and the presence
of synthetic sigma

tails on antithetic porphyroclasts. Synthetic si
gma tails on

antithetically oriented
porphyroclasts must be produced through

back rotation of the porphyroclast, because forward rotation
of

an antithetic porphyroclast would inhibit tail formation

(Fig. 4).


Fig. 4FB


Tail formation on rotating porphyro
clasts.

a. Sigma tails growing on a back
-
rotating porphyroclast.

b. Growth of sigma tails is inhibited by forward rotation of a porphyroclast with a long axis oriented
antithetic to the sense of shear. Presence of sigma tails on antithetically oriented
porphyroclasts is
evidence of back rotation.


Porphyroclasts will only reach stable end positions

if sufficient amounts of strain have accumulated.




obtuse angle field

20


Orientations of vorticity vectors and fabric asymmetries
.
Orthorhombic deformation s
ymmetries are
charact
erized by
parallelism between the finite strain elements (foliation and lineation) and the high
-
strain zone boundary and an abundance of symmetric structures (W
k

= 0). Monoclinic deformation
produces an angular discordance between the foliation and shear

zone boundaries as well as
asymmetric structures normal to the foliation and parallel to the elongation lineation (Fig. 3FB).
Triclinic deformation is characterized by asymmetric structures on sections both normal and parallel to
elongation lineations. In

zones of heterogeneous triclinic deformation, elongation lineations may vary
between strike
-
parallel and dip
-
parallel orientations.


The vorticity vector is referenced relative to the plane orthogonal to the vector. Maximum rotation
within the flow occurs

within this vorticity profile plane (VPP) and the plane contains the shear
direction. Relations between the VPP, lineation, and foliation depend upon the geometry of the shear
zone (Fig. 5FB). General shear zones should have maximum (fabric) asymmetry in
the VPP.


Fig. 3FB


a. Block diagram of monoclinic shear. The
transport direction and correspondingly
the vorticity profile plane (VPP) are
parallel to the lineation. Maximum
symmetry is expected in the lineation
-
normal plane with a

zero W
k
-
value.

b. In triclinic shear, the VPP and transport
direction are not parallel to the lineation.
Therefore, both lineation
-
parallel and
lineation
-
normal planes should have non
-
zero W
k
-
values, but neither are the VPP.



In

monoclinic general shear zones, the maximum asymmetry

plane (and VPP) should be the lineation
-
parallel foliation
-
normal

plane.
T
he plane normal to both foliation

and lineation is expected to have
maximum symmetry, because

material will not rotate in this
plane and will record only the pure

shear
component of the general shear deformation (Fig. 3
FB
).



Triclinic shear is analogous to multiple instantaneous

monoclinic deformations superimposed on the
previous deformation,

but between each incremental
monoclinic deformation,

the shear direction is
changed, and the end result yields a single

triclinic deformation. In triclinic shear zones, the lineation
is

not expected to be parallel to the shear direction, but rather

oriented between the ISA and the fin
ite
strain axes. Orientation of lineations would also be

expected to change

with respect to the vorticity
vector throughout

the shear zone. For a triclinic deformation
,

the VPP is no longer parallel to fabric
elements in the rock (Fig. 3
FB
).
In the

field,
the identification of triclinic deformation relie
s

on

the
presence of a wide variation of lineation orientations within

a shear zone and a noticeable
porphyroclast asymmetry in both

lineation
-
normal and lineation
-
parallel planes.


21


Fig. 5FB


Relations
between the location of the VPP, lineation, and foliation in monoclinic shear zones. Dotted
lines are an aid to the visualization of the three
-
dimensional shapes of the figures. White arrows
indicated shear directions and directions of shortening and exten
sion.

a. Transtension, the VPP is parallel to both foliation and lineation; the zone widens with increasing
deformation.

b. Monoclinic general shear, the VPP is parallel to lineation and orthogonal to foliation; the zone can
widen or shorten.

c. Transpr
ession, the VPP is orthogonal to both lineation and foliation; the zone shortens with
increasing deformation.




22


The porphyroclast hyperbolic distribution

(PHD)

method

(Simpson and De Paor,
1993, 1997)
.
It

is
based on the premise that

the orientation of th
e long axes of backward rotated grains

within the acute
angle field between the flow eigenvectors delineates the orientation of the unstable eigenvector. The
stable

eigenvector is assumed

to be parallel with foliation.
Porphyroclasts in a given plane of a
sample

are identified as either forward or backward rotated

based on the orientation of long axes relative to
the overall sense

of shear. The angle

between the long axis of the grain and the normal to foliation

is
the phi (
φ
) angle, with positive
φ

values
indicating forward

rotated grains and negative
φ
values
indicating back
-
rotated

grains.



Axial ratios (R) of the porphyroclasts are also measured.

Both the
φ

and R
-
values are plotted on a
hyperbolic stereonet

(De Paor, 1988). A hyperbola is drawn to inc
lude all of the

back
-
rotated grains,
and the angle between the two limbs of

the hyperbola represents the acute angle between the two
eigenvectors,

such that the cosine of this angle (
ν
) yields
W
k
.




Simplification:
Plotting the data on a hyperbolic net
is not necessary because the porphyroclast with
the lowest φ angle always defines the kinematic vorticity number. W
k

is given by: W
k
= cos (90
-
φ),
where the φ is the smallest angle made with the normal to foliation by back
-
rotated grains. Grain
orientation
s are more easily visualized with a radial distribution plot than a hyperbolic

net (Fig. 6
FB
).
Porphyroclasts with small axial ratios (
<
1.4) were removed from consideration because sub
-
spherical

grains are not actually back
-
rotated, but rather are continuo
usly

forward rotated
.
Although an axial
ratio

of 1.4 is an arbitrary cutoff, clasts below this ratio are sub
-
spherical,

commonly difficult to
measure, plot close to the

origin on the hyperbolic net, and do not affect the determined

opening angle
of the hyp
erbola.

















23


Fig. 6FB

a. Hyperbolic stereonet plot of axial ratios and long axis orientations of both forward and backward
rotated porphyroclasts. Solid circles are backward rotated and hollow diamonds are forward
rotated.

b. Radial distribution plot of backward rotated porphy
roclasts with maximum opening angle defined
by the solid black line.







24


Additional comments.

The rotational behavior of rigid elliptical porphyroclasts is controlled by the
bulk kinematic vorticity (W
k
), the axial ratio of the mineral grains (R), and
the orientation of their long
axes with respect to a fixed reference frame (φ). Three reference frames are used to evaluation
vorticity: (1) the finite strain axes; (2) the infinitesimal strain axes; (3) the shear zone boundary. The
shear zone boundary and

its normal are employed as the most reliable frame of reference. Axially
asymmetric porphyroclasts whose long axes are inclined ‘‘downstream’’ (a downstream dip
-
direction)
of the bulk transport direction at an orientation that falls within the acute angle

between the two
eigenvectors of the non
-
coaxial flow field will rotate opposite to the bulk shear sense within a
mylonitic shear zone (Fig. 2KN).


Fig. 2KN


Schematic diagram of mantled

porphyroclasts
that are inclined ‘‘upstream’
’ or ‘‘down
-
stream’’ relative to the bulk

sense of transport.



































25


Backward
-
rotated porphyroclasts

(see e.g. x in Fig. 3KN)

are inclined ‘‘downstream’’ relative to the
bulk sense

of shear and exhibit
σ
-
type asymmetric tails of recrystallized

material attached to the broad
or long sides of the elongate grain

(Figs. 2
,
3
KN)
. Forward
-
rotated

porphyroclasts
a
re distinguished
by
:

(1)
Approximately

equate

or

spherical
δ
-
grains indicating continuous forward
rotation

and

(2)
σ
-
grains that are inclined ‘‘upstre
am’’
that exhibit recrystallized material attached to their

narrow
ends.


Klepeis et al. (1999) described two variations of backward
-
rotated

grains based on the following
criteria:

(1) ‘‘upstream’’ or


‘downstream’’ inclined porphyroclasts exhibiting a sense of shear

contrary to the
bulk direction of transport with
σ
-
type tails of

recrystallized material attached to either the narrow or
broad sides

of the grain (
β
1

grains; Fig. 3
KN
); and

(2)
σ
-
type
porphyroclasts

inclined ‘‘downstream’’ exhibiting asymmetric tails attached to the

broad
sides of the grain and a rotational direction concurrent with

the bulk flow field (
β
2

grains; Fig. 3
KN).


Fig. 3KN

x


Schematic diagram illustrating fields of forward and backward rotation in a
dextral general shear re
gime, as well as, various micro
structures used in PHD analyses. Dashed lines
represent directions of maximum angular shear strain rate.
Those grains incline
d downstream with
sinistral tails on their narrow ends are probably near their stable orientations.




= delta clast

key clast

26


Application example

(Law et al. 2004).
Th
e orientation and

aspect ratio of porphyroclasts that have
either forward rotated or

back rotated
are

r
ecord
ed

o
n a hyperbolic net
,

the porphyroblasts are

cod
ed

w
ith respect to the type of recrystallization tail

(
σ

and
δ
). The

hyperbola that encloses all back
-
rotated
sigma
-
type porphyroclasts,

and separates them from all other types, is chosen. One

limb of this
hyperbola is asymptotic to the foliation, and the

mean kinematic vorticity number W
m

is given by the
cosine of

the acute angle between the two limbs of the hyperbola.
T
he
choice of the
most acute
hyperbola

available on the hyperbolic net
usually
reduc
es

th
e pure shear

component to a minimum
; t
he
estimated W
m

value is therefore regarded as

a maximum value.


Fig. 12L


Porphyroclast hyperbolic

distribution polar plot. Estimated orientation of flow

apophyses is given by
the hyperbola that

encloses all
back
-
rotated sigma
-
type

porphyroclasts, and separates them from all

other types. One limb of this hyperbola is

asymptotic to the foliation, and the mean

kinematic vorticity
number W
m

is given by

the cosine of the acute angle between the

two limbs of the hy
perbola. Average

orientation of shear bands in this sample

approximately bisects

acute angle between flow apophyses
(limbs

of hyperbola), and central segment of

leading edge of quartz c
-
axis fabric is

orthogonal to
average shear band

orientation


(
Comment
:

the plot is somewhat ridicules

but good
, as a 360° representation is given and data are mirrored)
.


For an explanation of the hyperbolic net see De Paor (1988)



27



Rigid grain net method

and updating of other rigid grain methods


Nomenclature

W
m

mean kinematic vorticity number

R porphyroclast aspect ratio (long axis/short axis)

B* shape factor of Bretherton (1962)

M
n

short axis of the porphyroclast

M
x

long axis of the porphyroclast

θ angle between long axis and the foliation
(or shear zone bounda
ry)

X’
2

X’
3

plane normal to the rotational axis X’
1

β angle between the stable
-
sink and source
-
sink in the X’
2

X’
3

plane

R
c

critical threshold between grains that rotate infinitely and those that reach a stable
-
sink position

R
cmin

minimum R
c

as defined by

Law et al. (2004)

R
cmax

maximum R
c

as defined by Law et al. (2004)


Models for the rotation of
rigid
elliptical objects in a fluid demonstrate

that during simple shear (mean
kinematic vorticity

number W
m

=

1) rigid objects will rotate infinitely, regardle
ss

of their aspect ratio
(R). With increasing contributions of pure

shear (0 <

Wm < 1), porphyroclasts will either rotate with
the

simple shear component (forward) or against it (backward)

until they reach a stable
-
sink
orientation that is unique to R

and
W
m

(Fig. 1
J).


Fig. 1J



Rotation of two simplified elliptical porphyroclasts within a regime of general shear.

Porphyroclast on
the left has an aspect ratio of 2 (B* = 0.6) and is in the stable
-
sink orientation of θ = 27° and
represents one of many possible original orientations that rotated forward to the stable
-
sink position.
The porphyroclasts on the right is b
ack rotated, due to the pure shear component, and has a long axis
at a negative angle (
θ
) to the foliation.




28


T
he

Passchier (1987)

(“Passchier plot”)

and Wallis (1995)
(“Wallis plot”)
methods
and the


porphyroclast hyperbolic distribution

(PHD) plot
(Simpson and De Paor,
1993, 1997)

are used for
practical applications (Fig. 3J)
.


Fig. 3J


Examples

of tailless porphyroclast data.
The Passchier plot uses the shape factor






















(where M
n

=

short axis

and M
x

=

long axis of the porphyroclast) vs. angle between
porphyroclast long axis and foliation (
θ
) to define the critical threshold used to estimate W
m
. The
Wallis plot uses the aspect ratio (R
=

long axis/short axis) and angle from macroscopic foliation (
θ
) to
locate the critical threshold (R
c
). W
m

is calculated

using R
c

where





















. Upper
and lower R
c

values are used to estimate a range in likely W
m

estimates. The PHD plot uses the

hyperbolic net to plot aspect ratio (R) and
θ
.
T
he cosine of
the opening angle (
β
) of the best
-
fit
enveloping

hyperbola yields
the

W
m
.


Review of techniques
.
During general shear, rigid grain analysis assumes that the

orientation of
porphyroclasts within a flowing matrix record

a critical threshold (R
c
) between
porphyroclasts that
rotate indefinitely

(low aspect ratio), and therefore do not develop

a preferred orientation, and those
that reach a stable
-
sink orientation

(higher aspect ratio). This unique combination

of W
m
, R or B* and
θ

define the value of R
c

betw
een these two

groups of rigid grains. W
m

to B* and
θ

are related by (see
also earlier):















{
























}
























(where M
n

=

short axis

and M
x

=

long axis of the porphyroclast)


Passchier
-
plot.

The θ
-
equation
generate
s

a hyperbolic

curve in
θ

vs. B* space that represents the ideal
distribution of

grains for a particular W
m
. The vertices of this hyperbola mark

the unique R
c

value
where W
m

=

B*. Assuming high

strain, a natural distribution of porph
yroclasts should define

a limb of
this hyperbola for a range of B* values that is greater

than B* at R
c
. With low strains, a misleading
distribution

of porphyroclasts has the potential to overestimate

the simple shear component because
high aspect ratio po
rphyroclasts

have yet to rotate into their stable
-
sink orientation. Porphyroclasts

with a B* < B* at R
c

will rotate infinitely and

should define a broad distribution with
θ

=

±

90
°
. In
contrast, porphyroclasts

with a B* > B* at R
c

are predicted to reach stable

sink

orientations with a
limited range in
θ

values (Fig. 3
J
A). Whether a porphyroclast will rotate forward or backward to

a
stable
-
sink position depends on the initial
θ

at a specific B* and W
m
. R
c

should be defined by the
ei
ther B* or R

and

an

abrupt change in range of
θ

values (Fig. 3
J
A
,

B).


Although the distribution of porphyroclasts on the Passchier plot can be informative

for the high

quality data sets (Fig. 3
J
A), without a reference frame for comparing

complex natural d
ata

with the
theoretical values established by

the θ
-
equation
, defining R
c

will remain ambiguous.


B* = W
m

at R
c

29


Wallis
-
plot.
The Wallis plot still uses
θ

on the Y
-
axis,

but replaces B* with the more intuitive
porphyroclast aspect

ratio (R
=

long axis/short axis) on the

X
-
axis (Fig. 3
J
B; Wallis,

1992, 1995). W
m

is calculated from the R
c

values separating

porphyroclasts that reach a stable
-
sink orientation (
θ

<
θ

at
R
c
)

from those that rotate continuously (
θ

>
θ

at R
c
). W
m

is
calculated
from

(Wallis et al., 1993):






















.


The distribution of porphyroclasts often

defines a gradual transition between
continuously rotating
(random orientation) porphyroclasts and stable
-

to semi
-
stable porphyroclasts that define R
c
.
T
he
original Wallis plot

is improved

by
drawing

an enveloping surface to better
-
define the grain
distribution, and use a range in possible W
m

values (R
cmin

and

R
cmax

; Fig. 3
J
B; Law et al., 2004; Jessup
et al., 2006).


PHD
-
plot.

The porphyroclast hyperbolic distribution (PHD) method

estimates W
m

by using R and the
angle between the pole to

foliation and long axis of tailed porphyroclasts (Fig. 3
J
C);

plotted

using the
hyperbolic net (HN). Each hyperbola of the HN represents

the theoretically predicted orientation of
porphyroclasts for

a particula
r R and W
m

as plotted in
polar

coordinates

(see sign convention)
.




R
c

is defined as the vertices of the hyperbola. One limb of the hyperbola represents the stable

sink

orientation for porphyroclasts while the other is the metastable

position

(Fig. 3JC).
At
θ

> the
metastable orientation, porphyroclasts

will rotate forward until they define another semi
-
hyperbolic

cluster on the concave side of the same hyperbola. Assuming

significant shear, back
-
rotated clasts
with variable aspect

ratios, plo
tted on the HN, should define a semi
-
hyperbolic

cluster representing the
stable
-
sink orientation. The linear

cluster is then rotated to find the best
-
fit hyperbola whose limbs

represent the two eigenvectors of flow, one of which is asymptotic

to the foliat
ion (i.e., the source
-
sink
and stable
-
sink positions). The vertex of this hyperbola

separates the low aspect ratio porphyroclasts
with random

orientat
ion (i.e., infinitely rotating)

from higher aspect ratio

porphyroclasts with a narrow
range of orientation
s (
Fig. 3
J
C).


Supplementary information.

Table 1J:
Critical threshold values

R


W
m


θ at Rc


B*


β


cos(β)

1.1


0.1


42



0.1


84


0.1

1.21

0.2


39



0.2


78


0.2

1.3


0.3


36



0.3


73


0.3

1.5


0.4


33



0.4


66


0.4

1.7


0.5


30



0.5


60


0.5

2


0.6


27



0.6


53


0.6

2.4


0.7


23



0.7


46



0.7

3


0.8


18



0.8


37


0.8

4.4


0.9


13



0.9


26


0.9

R = aspect ratio (long axis/short axis).

W
m

= mean kinematic vorticity number.

θ = angle from foliation
(Fig. 1J).

R
c

= critical threshold.

B* = shape factor.

β = opening angle of hyperbola.




30


Fig. 2J


Plot showing the relationship between mean kinematic
vorticity number (W
m
), shape factor (B*), and aspect
ratio (R) at critical value
s.



31


The Rigid Grain Net (RGN)
.

Eq.















{
























}

is

used to
calculate

semi
-
hyperbolas for a range of W
m
-
values that express

the relationship between
θ

and B*
(
Fig. 4
J
, location A). The

second set of curves represent the possible R
c

(vertices curves)

values for
when W
m

=

B* (
Fig. 4
J
, location B). Each

semi
-
hyperbola was calculated for a particular W
m

and a
series

of B* values.
T
he shape factor (B*
)
enables W
m

values to be obtained directl
y

from the RGN.

P
ositive and negative semi
-
hyperbolas are

plotted at 0.025 increments

(of B*)

for a range in W
m

(0.1
-
1.0) by

solving for
θ

using
the θ
-
equation.
For

a particular shape factor, when B*
=
W
m

and
θ

>
θ

at
R
c
, the

semi
-
hyperbolas transition
into vertical lines to define the

maximum B* value below which
grains rotate freely

(
Fig. 4
, location C).
To highlight the continuity

in R
c

values for the range in W
m

values represented by the

RGN, a second curve (vertices curve) links the R
c

values on

eac
h hyperbola
(
Fig. 4
, location B).


Fig. 4J



The Rigid Grain Net (RGN) using semi
-
hyperbolas. Location A is an example of a semi
-
hyperbola;
location B highlights the vertices curve; location C is an

example of a R
c

value when W
m

=

B*;
location D points to one of a series of aspect ratio (R) values included on the RGN to demonstrate its
relationship with the

less intuitive shape factor (B*); location E is a W
m

value for a semi
-
hyperbola.


To relate hyperbolas on the HN and the RGN,
full hyperbolas

are plotted

on the RGN
;
highlight
ed

are
the critical curves
that

define W
m

=

0.6;
plotted
are
also

hypothetical distribution
s

of

porphyroclasts
(
Fig. 5
J
). The two hyperbolas that are included

on the simplified HN are highlighted in black on

the
RGN

(W
m

=

0.6) for positive and negative
θ

(
Fig. 5
J
A
,

B).
Fields of the HN and RGN plots that
represent the maximum

aspect ratio for porphyroclasts that are predicted to rotate

forward
infinitely

and thereby have a complete range in
θ

between

±90°
, ar
e represented by a circle of constant R for all

orientations that is tangential to the vertex of each hyperbola

and plotted on the center of the HN (i.e.,
when B* <W
m
;

Fig
.
5
J
A
,
C). On the RGN, this area includes all of

the potential range in B* between 0
and B* at R
c

(i.e., to the

left of the apex of the hyperbolas;
Fig. 5
J
B
,

D). An additional

section of the
RGN is highlighted on the HN that defines

porphyroclasts that will rotate to the vertices curve for R
c

values


th
e ‘‘true’’ R
c

for the sample. On the HN, this curve

is defined by linking the R
c

value from
each potential hyperbola

greater than the ‘‘true’’ R
c

for this sample to generate

a small section of the
vertices curve as shown on the RGN

(
Fig. 5
J
A
,

C). This impo
rtant clarification shows that these

porphyroclasts must be considered as stable
-
sink orientations

when c
hoosing the best
-
fit hyperbola.


The RGN is available as
an Excel
-
worksheet. This enables to

monitor how the distribution of
porphyroclasts

is
developing during data acquisition.

32



Fig.5J


a.

Half of the HN simplified to graphically demonstrate the relationship between one hyperbola (W
m

=

0.60) and the vertices curve for

that hyperbola. The vertices curve is drawn using the
vertices of
several hyperbolas (dashed) for a range in W
m

values greater than the W
m

for the sample (0.60).

Gray circles with letters (a
-
i) on the hyperbola for W
m

=

0.60 are shown to compare how these
define t
he hyperbolas on the HN and RGN.
The circle th
at

defines the highest aspect ratio (R
=

2)
below which porphyroclasts are predicted to rotate infinitely is also included.

b.

The RGN with complete hyperbolas.

Highlighted in black are the positive and negative hyperbolas
that correspond to a W
m

=

0.60, as well as the section of the vertices curve for B* > B* at R
c
. A

series of gray circles represent equivalent points on the RGN and HN.

c.

The same plot as
a.

with an overlay of different types of hypothetical porphyroclasts

in their
predicted dist
ribution; gray squares are infinitely rotating, black crosses are limited rotation, gray
circles are stable
-

to metastable
-
sink positions.

d.

The

same plot as
b.

with hypothetical porphyroclasts distributed in various sections of the RGN.



vertices
-
curve

33


Application

of

the
RGN
.

The choice of rigid porphyroclasts
i
s highly selective, ignoring

all but the
most appropriate grains for estimating W
m

(Fig. 7J).


Fig. 7J





All R
c
-
based methods may tend to underestimate the vorticity number if clasts of large aspect ratio
are
not present, and therefore within individual samples the upper bound of this W
m

range is probably
closest to the true value.




34


SC’
-
type shear
-
band

method



Fig. 4KN


Rose diagrams generated from orientation data for synthetic and antithetic SC’
-
type shear bands and
other related structures such as synthetically and antithetically imbricated mineral grains are
combined with PHD diagrams. This combination is meant to ill
ustrate geometric relationships and
frequency distributions of shear band orientations relative to the inferred flow field determined from
PHD analyses. Porphyroclast axial ratio data relates to the scale along the base of each diagram.
Annular dashed half
-
circles measure shear band frequency data associated with rose diagrams.
Dotted lines indicate the maximum inclination of synthetic SC’
-
type shear band within two standard
deviations of the mean and are used to estimate kinematic vorticity
. Black X’s repr
esent plotted
backward
-
rotated porphyroclasts

while gray
-
filled circles depict plotted forward
-
rotated
porphyroclasts.




35


Synthetic shear bands are oriented either parallel to, or at an angle less than the acute bisector (AB;
Fig. 4KN). Antithetic shear ba
nds populations show a range of inclination with a mean inclination
lying near the obtuse bisector (OB). These data suggest that extensional SC’
-
type shear bands initially
form parallel to AB and OB.

Shear bands that are inclined at an angle less than AB a
nd OB are

the
result of either: (1) rotation towards the shear zone boundary

during progressive non
-
coaxial flow; (2)
formed under heterogeneous

non
-
steady
-
state conditions and/or varying bulk vorticity;

or (3) formed
during separate episodes of
deformation. Assuming

a steady
-
state general flow regime, synthetic and
antithetic

extensional shear bands are expected to rotate towards the stable

eigenvector and away from
AB and OB throughout progressive non
-
coaxial

deformation. Assuming steady
-
state

v
orticity during
progressive deformation, the most steeply inclined

shear bands may provide the best direct estimate of
AB and OB in

that shear bands at this orientation may not have been significantly

rotated.

Estimated

values for bulk vorticity
are

determ
ined

by utilizing the most steeply inclined shear band
orientation within

±

two standard deviations of the mean in order to eliminate outliers

(Fig.
4KN)
.


Samples may exhibit moderately to well
-
defined SC’
-
type shear bands that form an anastomosing
networ
k of quartz dominated

micro
-
shear zones and enclose large microlithons of feldspar

(Fig. 7
KN).


Fig. 7KN


Composite image of sample CC
-
6
-
30
-
1,
example of weakly recrystallized coarse to
medium
-
grained quartz
-
feldspar aggregates.
An

anastomosing network of quartz
-
rich,
micro
-
scale shear zones have accommodated
the majority of the strain within the rock mass.


Because these coarse to medium
-
grained quartz
-
feldspar

aggregates appear to have undergone
comparatively

smaller amounts of strain, the arithmetic mean of shear band

orientations has been used
to estimate bulk vorticity.

It is assuming that this average orientation roughly bisects the angle ν (Fig.
8KN).


Fig. 8KN


Rose diagrams illustrating
SC’
-
type shear band orientations. Dashed line represents the estimated
orientation of AB.

36




Quartz

c
-
axis fabric
s and oblique grain shape
vorticity method
s (review in Xypolias 2009)


These methods are based on two

assumptions:

(1) Under progressive simple,

pure
,

and general shear, the central

girdle segment of quartz c
-
axis
-
fabrics develops nearly

orthogonal to the flow/shear plane (A
1
)
.
Therefore, the angle,
β
, between the

perpendicular to the central girdle segment of quartz c
-
axis

fabric

and the foliatio
n (S
A
) is equal to the
angle between the

flow plane (A
1
) and the flattening plane of finite strain

(
Fig. 1
a,c;
Wallis, 1992
).

(2) The long axes of quartz neoblasts within an oblique
-
grain

shape

foliation

align nearly

parallel to
the extensional
instantaneous stretching axis (ISA
1
)

(
Fig. 1
a,b;
Wallis, 1995
). This is interpreted to be
the result of a complex process of

continuous nucleation, passive deformation
,

and rotation of the

recrystallized grains composing the S
B
.

Therefore, the maximum obser
ved angle,
δ
, between the S
B

and

the main (S
A
) foliation is equal to the angle between the ISA
1

and the largest principal axis, X, of
the finite strain (
Fig. 1
a,b;

Wallis, 1995
).


Fig. 1X





According to the oblique
-
grain
-
shape/quartz c
-
axis
-
fabric

method (referred to hereinafter as
δ
/
β
-
method), if both angles
δ

and

β

are known then an estimate of vorticity number can be obtained

using
the equation (
Wallis, 1995
):


W
m

= sin 2 (
δ

+
β
)


This e
q. only

holds for two
-
dimensional flow. The
δ
/
β
-
method
record
s

the last increments of plastic
deformation (
Wallis, 1995
).




Fig. 1. (a) Simplified sketch showing the
relative orientation o
f instantaneous flow
elements and their angular relationships in
real space for a dextral general shear flow.

A
1

and A
2



flow apophyses; ISA
1

and ISA
3



instantaneous stretching axes; X and Z


principal strain axes. The vorticity vector lies
perpendicular to the page. (b) The angle, δ,
between the oblique
-
grain
-
shape fabric (S
B
)
and the main foliation (S
A
) is inferred to be
equal to the angle between t
he ISA
1

and the
principal finite strain axis X. (c) The angle, β,
between the perpendicular to the central girdle
segment of quartz c
-
axis fabric and the main
foliation (S
A
) is inferred to be equal to the
angle between the flow apophysis A
1

(flow
plane) an
d the principal finite strain axis X.


37


R
XZ

-
method
:
The strain
-
ratio/quartz c
-
axis
-
fabric method
estimates

W
m

(finite deformation)
as:


Fig. 13L


V
orticity analysis after Wallis (1995). Flow plane is inferred to be orthogonal to central segment of
the leading edge of quartz c
-
axis fabric (or a
-
axis maximum), measured in section oriented

perpendicular to foliation and parallel to lineation. Acute angle between foliation and inferred flow
plane (i.e. normal to central segment of fabric) defines β; complementary angle ψ between foliation
and central segment of fabric is a measure of externa
l fabric asymmetry. β is used in combination with
the ratio of principal

stretches, R
xz

=(1
-

ε
1
)/(1
-

ε
3
), to estimate mean kinematic vorticity number W
m
.


This method assumes plane strain deformation.
Plane strain conditions are indicated by the cross
-
gir
dle pattern of quartz c
-
axis fabrics, and the orthogonal relationship between the cross
-
girdle fabric
and sample Y direction (within foliation and perpendicular to lineation) also argues for monoclinic
rather than triclinic flow.
Tikoff and Fossen (1995)

i
nvestigated the effect of the third stretching
direction (Y
-
axis) on

vorticity
-
number estimates and demonstrated that the
2D
vorticity analysis can
overestimate the actual,
3D
-
W
m

by only a small amount (<0.05).
The overestimation will be greatest
for
nearly equal components of coaxial and non
-
coaxial deformation and will decline to zero as W
m

goes to zero or one.
V
orticity number estimates using R
XZ
/
β
-
method are very sensitive

to small
changes in the evaluated angle
β
.

I
n most quartz c
-
axis

fabric

diag
rams the angle
β

can be evaluated
with an error
±
2
°;
the method

becomes unreliable in high
-
strain samples (R
XZ

>10

15) with small

β

(<5
°).


R
XZ
/δ method
:
T
he maximum observed angle
δ

between S
B

and S
A

grain shape foliation is primarily
related to both

the

degree of non
-
coaxiality and the shape of the strain ellipse just

prior to the end of
deformation.


W
n

(instantaneous deformation) =
sin
(
2
δ) [(
R
XZ

+

1
)/(R
XZ



1
)]


which shows that W
n

is related to R
XZ

and
δ
.



38


This e
q. can be solved for
δ

and the result can be plotted for different W
n

values
.


δ = 1/2
sin
-
1
[
Wn
(R
XZ
-
1)/(
R
XZ

+

1
).


T
his plot (RXZ versus
δ
) is illustrated in the synthetic diagram of

Fig. 3
, which also includes an R
XZ
versus
β

plot
.


Fig. 3X




Fig. 3. Synthetic diagram
showing plots of the finite strain
-
ratio R
XZ

versus
β

and R
XZ

versus
δ
.


This diagram indicates

that during progressive steady
-
state deformation the angle

δ

attains a high
value at low strain (R
XZ

<

4) and then is subjected

to small changes. This observat
ion appears to be in
accordance with

experimental work
that indicates

that the oblique
-
grain
-
shape fabric forms early
during non
-
coaxial

shear attaining a stable orientation at low imposed shear

strain (
γ

<1.5). The p
lot of
R
XZ

versus
δ

also indicates that

the

W
n

curves

get broader spaced for increasing strain values (
Fig. 3
).
This implies

that, in contrast to the R
XZ
/
β
-
method,
W
n

estimates are

relatively insensitive to small
changes in the values of R
XZ

and
δ
.



The

R
XZ
/
δ method
describes instantaneous
deformation and, therefore, does

not require the assumption
of steady
-
state deformation.


Comparison to the particle methods
.
D
ifference could

be due to: (1) analytical problems

associated
with individual methods (e.g.
the particle methods tend

to

give a
minimum estimate of W
m

if rigid
grains of sufficiently

high aspect ratio are not present); (2) the two methods record

different parts of
the deformation history (i.e. they have different

lengths of ‘strain memory’); (3) the two methods
reflect a

contrast i
n synchronous flow behavior between rigid particle

rotation of the porphyroclasts
and plastic deformation of the

surrounding matrix quartz grains.




39


A vorticity nomogram

incorporates

all the parameters (
β
,
δ
, R
XZ

and

W
n
/W
m
) involved in the three

vorticity

analysis methods. The advantages of such nomographic

approach are
:

(1)

it instantly provides the vorticity number for all methods;

(2) it

allows to check, for all methods, the sensitivity of estimated

vorticity numbers values in changes
of input
parameters (
β
,
δ
, R
XZ
);

(3) it provides a rapid means of evaluating the consistency of values

estimated by the various methods;
(4) it permits the presentation

of uncertainties in vorticity number estimates as error bars;

(5) it

enables the rapid evaluati
on of mean strain level for a suite of

samples, where only
β
-

and
δ
-
angle data are available.


Fig. 4X




Fig. 4. Suggested nomogram for estimating W
m

(or W
n
) and R
XZ

in naturally deformed quartzites and
an application to five representative samples. Circ
les indicate estimates obtained using the best
assigned input data. Error bars were constructed using uncertainty in the estimations of the β
-
angle as
well as error in R
XZ

values.






40


Stretch across a plane
-
strain shear zone

(Law et al. 2004, Wallis et al. 1993)



Calculation of shortening value (S) measured perpendicular to flow plane, taking into account both
strain magnitude and vorticity of

flow (adapted from Wallis et al. 1993). Assuming plane strain

deformation, stretc
h measured parallel to the flow plane in the transport

direction is given by S
-
1
.





41


Non
-
perfectly matrix bounded rigid inclusions

(
M
ar
ques et al. 2007)


Overview of inclusion models.

1. Fundamental solutions: Aspect ratio, inclination, and material
property contrast
. If the interest is focused on slow deformation processes, two analytical solutions
address the problem of an inclusion immersed in a matrix of different property:

(1) Je
ffery’s model

for rigid ellipsoids in a deforming viscous matrix (Jeffery, 1922), and

(2) Eshelby’s model for a deformable elastic ellipsoid in a far
-
field loaded matrix with different
properties (Eshelby, 1957, 1959).

Common to these solutions is the id
entification of the material property contrast between
inclusion and matrix, µ
i

m
, the aspect ratio, A
r
, and the inclination φ as the governing
parameters that influence the inclusion behavior (Fig. 1M).


Fig. 1M


Sketch of an inclusion in a shear zone.

The factors that determine inclusion behavior are: inclination
φ (positive counter clockwise), aspect ratio A
r
, ratio W
r

between shear zone width (w) and inclusion
short axis (b), distance to other inclusions (d ), presence of a rim with thickness h that
controls the
degree of welding between inclusion and matrix, shape of the inclusion, viscosities of inclusion (
µ
i
),
rim (
µ
r
) and matrix (
µ
m
), and finally

the relative strength of pure to simple shear S
r
. Note that the
angle and quadrant

convention used (s
ee text) is related to the shear direction, i.e., in top to

the left
shear zones the above sketch has to be flipped horizontally.


The modeling

of inclusion behavior

in

ductile

shear zones used to assume that the matrix acts as a
viscous

fluid and the clas
ts as isolated perfectly bonded rigid inclusions,

therefore allowing for
application of Jeffrey’s solution.

F
actors that can change

the rotation behavior of rigid inclusions

are:
(i) the addition of a pure shear component

to sim
ple shear (variable
vorticity).

(ii) inclusion interaction

in multi
-
inclusion systems.


Inclusion interaction in multi
-
inclusion systems usually leads to tilling effects, which strongly affect
inclusion rotation and bring the inclusion to a stable equilibrium orientation.

(ii
i) sli
p at inclusion/matrix interface.

If the slipping contact is due to the existence of weak mantle phase then the inclination of the clast will
be controlled by the viscosity contrast between mantle and matrix and the rate at which the mantle
material i
s produced.

(iv) flow confinement.
Depending on the W
r

value, elliptical inclusions can rotate backwards from φ =
0° (opposite to Jeffery’s model) and stabilize at shallow positive angles (0≤φ<90°
)

(Marques and
Coelho, 2001).

These factors may furthermore

be affected by the actual shape of the inclusion.




42


First applications.


Fig. 3M et al. and Fig. 4M et al.



Asymmetrical inclusions comprise mainly
disrupted and strongly attenuated quartz
-
rich
material,
which mostly derives from older
quartz veins and segregations within the
phyllonitic gneiss
.






Field data is fitted with a power
-
law and
compared to the ice experiments of Marques
and Bose (2004), the equ
ivalent void derived
by Schmid and Podladchikov (2004), and the
transtension (S
r

=
-
0.3) case of combined pure
and simple shear (Ghosh and Ramberg, 1976;
Marques and Coelho, 2003).

Site 1: Given the φ
-
A
r

data and the fact that the
clasts are found in relative isolation, the explanation
for the SPO is that the clast
-
matrix interface was slipping. This can be seen from the ice in PDMS

(a
viscous polymer)

experiments that correspond very well to the field data and also from the equival
ent
void conjecture. Had the interface been welded,
then

the Ghosh and Ramberg’s (1976) model could be
applicable, which also predicts stable SPOs at positive angles for cases where the pure shear
component is extensional at 90° to the shear plane. However
, the stabilization trend for any such
negative S
r

case is opposite to the field data as illustrated with S
r

=
-
0.3. We can therefore rule out
perfect bonding and, given the good fit between the natural data and the simple shear only lubrication
models, we

can also rule out the necessity of an additional pure shear component to explain the data.




43


Fig. 5M et al., Fig. 7M et a
l.




Features: (1) σ
-
porphyroclast (1), with straight tails rooting with asymmetric development at the
crests, hence showing very high stair stepping and
top to left sense of shear. A dark biotite layer
developed around this clast at opposing faces (contraction

quadrants, marked by small black arrows),
while a white thin rim of quartz (marked by small white arrows) formed at the two other faces
(expansion quadrants).

(ii) Porphyroclasts with shapes that vary from elliptical (3 to 7), to lozenge (2) or skewed re
ctangle
(8).

(iii) Porphyroclasts at negative inclinations (1 and 9) or, more commonly, positive inclinations.

(iv)
Lack of well
-
developed recrystallization tails in most porphyroclasts.

(v) Pressure shadow tails as marked by white bigger arrow on the
left of porphyroclast 10.



44




‘‘Site
2

Data Fit’’, ‘‘ice experiments’’, and ‘‘equivalent void’’ models. Transpression is plotted for S
r

= 1 according to Ghosh and Ramberg’s model (1976).
‘‘FEM Slip S
r

= 1’’ represents finite element
calculations for slipping inclusions in pure and simple shear
.

Site 2
. T
wo distinct distributions

are evident
: (1) a set of positive

inclinations and (2) a set of negative
inclinations. In both

cases, larger aspect ratio clasts plot closer to the shear plane

than smaller ones.
T
he clasts with negative inclinations can be

fitted to Ghosh and Ramberg’s model with a shear zone
flattening

component that equals the simple shear c
omponent,

S
r

=
1. Note though that the
transpression line ceases to exist

below a minimum aspect ratio, i.e. clasts with smaller aspect

ratios
are not stable

(rotate freely)

under the given pure to simple shear ratio

and perfect interface bonding.
The

fini
te element experiments,
with
the addition of

a flattening pure shear component
,

shift the stable
inclinations

of lubricated clasts closer to the shear plane. The actual angles

are too shallow, yet the
trend in the data is reproduced.
V
arying the actual
amount of pure to simple shear

can vary the actual
position of the stable inclination curve.


Evidence for slipping boundaries around part of the clast

population is indicated
by:

t
hin rims of mica
and/or

quartz and/or fine
-
grained feldspar commonly surrou
nd feldspar

porphyroclasts, all typically
weaker phases than the porphyroclasts
.
These weak rims could have worked as effective lubricants

that kept the rigid feldspar in slipping contact with the mylonitic

matrix. Having excluded the
possibility of confin
ed flow,

the straight tails with very high stair stepping are also indicative

that the
inclusion was in slipping contact with the matrix. It is

therefore a reasonable conclusion that the clasts
exhibit slipping and non
-
slipping interfaces with

a far field
flow condition where simple shear and a
flattening

pure shear component are of approximately equal strength.

The total shear strain must be
more than 20.



Literature


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space.
Tectonophysics 12
2,

35
-
51.

Bretherton, F.P., 1962. The motion of rigid particles in shear flow at low

Reynolds number. Journal of Fluid Mechanics 14,
284
-
301.

De Paor, D.G., 1988. R
f
/
φ
f

strain analysis using an orientation net. Journal of Structural Geology 10, 323
-
333.

Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of

the Royal Society of London Series A
-

Mathematical and Physical Sciences 241 (1226), 376
-
396.

Eshelby, J.D., 1959. The elastic fie
ld outside an ellipsoidal inclusion. Proceedings of the Royal Society of London Series A.
Mathematical and Physical
Sciences 252 (1271), 561
-
569.

Forte, A.M., and C.M. Bailey, 2007. Testing the utility of the porphyroclasts hyperbolic distribution method o
f kinematic
vorticity analysis. Journal Struct. Geol., 29, 983
-
1001, 2007.

Fossen, H. Textbook, http://folk.uib.no/nglhe/StructModulesTextbook/Deformation.swf

Freeman, B. 1985. The motion of rigid ellipsoidal particles in slow

flows.
Tectonophysics
113,163
-
183.

Ghosh, S. K. & Ramberg, H. 1976. Reorientation of inclusions by

combination of pure and simple shear.
Tectonophysics
34,
1
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70.

Jeffery, G. B. 1922. The motion of ellipsoidal particles immersed in a

viscous fluid.
Proc. R. Soc. Lond.
A102, 161
-
179.

45


Jessuo, M.J., Law, R.D., and Frassi, Ch., 2007. The rigid grain net (RGN): an alternative method for estimating mean
kinematic vorticity number (W
m
). J. Struct. Geol., 29, 411
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