Determination of flow patterns in rocks: an introduction and overview
Description of homogeneous flow
2.5PT
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 used to determine the velocity field at a particular time.
c. Marker points in the flow pattern can be connected by lines.
d. For each line stretching rate (𝑆̇; 𝑆̇=l1/l0) and angular velocity (ω) are defined.
e. 𝑆̇ and ω are plotted in curves again line orientation (0-360°; note: dextral rotation is positive!).
Special directions are:
ISA = instantaneous stretching axes: two lines (in 2D), along which the stretching rate (𝑆̇) has its
maximum and minimum values; they are always orthogonal.
Irrotational material lines: can have any orientation.
𝑆̇k: the amplitude of the 𝑆̇-curve is a measure of the strain rate.
Vorticity: the elevation of the symmetry line of the ω-curve.
f. Orientations of ISA and irrotational lines can be read from graphs.
1
Types of flow
2.6PT
Isochoric flow: if the stretching rate (𝑆̇) is symmetrically arranged with respect to the zero stretchingrate axis, no area change is involved in the flow (area in/decrease 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 of lines that is irrotational is parallel to the ISA. This pure shear flow has
orthorhombic shape symmetry.
General non-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.PT5) 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.
Vorticity: 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 exists.
Flow descriptions (see parameter definitions in Fig. 2.6 PT)
αk: angle between one of the ISA and the side of the shear box (shear zone boundary)
𝑆̇k = 𝑆̇1-𝑆̇2 = ω1-ω2; a measure of strain rate (amplitude of the stretching/rotation rate curve)
Wk = (ω1+ω2)/ 𝑆̇k; kinematic vorticity number
Ak = (𝑆̇1+𝑆̇2)/ 𝑆̇k; a measure of areal change with time
E.g.: simple shear: Wk = 1, Ak = 0. (mit ω: Wk = 1+0/1-0 = 1; ω1 = x, ω2 = 0; all material lines rotate
e.g. dextrally; with 𝑆̇k: take scales, e.g. ω1 = 2, ω2 = 0, 𝑆̇k = (1-(-1 )= 2)
pure shear: Wk = 0, Ak = 0
2
Concept of vorticity and spin
Box 2.4PT
a. If the velocity of a river is fastest in the middle, paddle wheels in the river will rotate in opposite
direction at the sides, but will not rotate in the middle; they reflect the vorticity of flow in the river at
three different sites.
b. Vorticity is defined as the sum of the angular velocity with respect to ISA of any pair of orthogonal
material lines (such as p and q); additional rotation of ISA (and all the other lines and vectors) in an
external reference frame is known as spin.
3
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 staged of a deformation sequence.
b. The deformation pattern.
c. Sets of marker points can be connected by material lines and the rotation (r) and stretch (s) of each
line is monitored.
d. These can be plotted against initial orientation of the lines. In the curves, principal strains can be
distinguished.
e. Finite deformation as deduced from these curves contains elements of strain and rotation (ρk). βk
defines the orientation of a material line in the undeformed state that is to become parallel to the
long axis of the strain ellipse in the deformed state.
4
Progressive and finite deformation
2.8PT
Homogeneous 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 some 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 effect of deformation history
a. Two identical squares of material with two marker lines (red and green lines) are deformed up to
the same finite strain value in simple shear and pure shear progressive deformation, respectively.
The initial orientation of the squares is chosen such that the shape and orientation of deformed
squares is identical.
b. The finite stretch and relative orientation of both marker lines is identical in both cases, but the
history of stretch and rotation of each line is different.
c. Circular diagrams show the distribution of all material lines in the squares of a. Ornamentation
shows where lines are shortened (s), extended (e) or first shortened, then extended (se) for each
step of progressive deformation. The orientation of ISA is indicated.
5
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.
6
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 (‘Xi) of
particles at Xi in a fixed Cartesian co-ordinate system is described by the velocity gradient tensor L'
(Malvern 1969) as:
0
0
0
′𝑋1
𝑋1
0
(𝑆 + 𝑊)/2] ∙ [𝑋2 ]
[′𝑋2 ] = [0
𝑋3
′𝑋3
0 (𝑆 − 𝑊)/2
0
where W is the vorticity of the flow and S is a scalar defining the stretching rate of the flow. L' can be
expressed as the sum of a symmetric tensor D' and an anti-symmetric tensor W':
0
0
0
[0
0 (𝑆 − 𝑊)/2
0
0
(𝑆 + 𝑊)/2] = [ 0
0
0
(𝐿′ )
0
0
𝑆/2
0
0
𝑆/2]
+ [0
0 (𝐷′ )
0
0
0
0
𝑊/2]
−𝑊/2
0 (𝑊′ )
The eigenvectors di of D' are the orthogonal instantaneous stretching axes of the flow and the
eigenvalues d1, d2 and d3 are instantaneous stretching rates with values 0, S/2 and -S/2. The vorticity
vector of the flow is parallel to d1 (Fig. 1).
Fig. 1 (Fossen; Passchier ’87). Left: definition of the vorticity vector. Right: schematic representation
of a homogeneous plane-strain non-coaxial flow with vorticity number Wk = 0.76 described by L'. X’i
— coordinate system, di — eigenvectors of D', instantaneous stretching axes. li — eigenvectors of L', =
flow apophyses. Ornamented surfaces are traced by material lines parallel to X’i. Arrow around X’i
indicates sense of shear.
7
Eigenvectors li of L' do not usually coincide with those of D', except for pure shear. l 1 is parallel to d1,
and l2 and l3 lie in the X’2 – X’3 plane, symmetrically arranged with respect to d2 and d3 (Fig. 1)
(Bobyarchick 1986, Passchier 1986). The instantaneous non-coaxiality of the flow can be expressed
by a vorticity number:
𝑊𝑘 = |𝑑
𝑊
,
2 −𝑑3 |
which for isochoric plane-strain flow equals the kinematic vorticity number of Truesdell (1954):
‘𝑊
𝑊
.
√2(𝑑12 +𝑑22 +𝑑32 )
The planar surfaces through l1, l2 and l1, l3, 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 Wk ≤ 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 l2 and 13 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 X1-direction)).
[Explanations: (i) the term asymptotic means approaching a value 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]
hyperbola definition
eigenvector plane
Flow apophyses are theoretical 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 apophyses.
8
Poles of rotating passive lines move along trajectories (great circles) in a stereoplot during simple
shear (Fig. cf. Fossen). 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 orthogonal flow apophyses.
9
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).
10
Relationship between Wk and pure and simple shear.
Fig. 1FB
calculation
below
Scale relations between the Wk-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 Wk and components of pure and simple shear for
instantaneous 2D flow; pure and simple shear components make equal contributions to flow at Wk =
0.71 (arrow). Wk = 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: Wk = cosα, with α as the angle between the AP is the most common definition of vorticity; actually Wk = 0.707
= cos45)
11
Deformation in various Wk fields. Simple 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 Wk-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)
12
Movement of axially-symmetric rigid objects (e.g. Passchier 1987)
General theory—isochoric plane-strain flow. Equations 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 along 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=
𝑎 2 −𝑏2
.
𝑎 2 +𝑏2
B can represent a material line (B = 1), a prolate ellipsoid (0 < B < 1), a sphere (B = 0), an oblate
ellipsoid (-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 Xi is chosen with X1 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. The 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 Xi: 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 ω1, the angular velocity components of OSA around
X1. 𝜃,̇ 𝜑,̇ 𝜓̇ ∶ rate of change of Eulerian angles (which define the object orientation), v:
instantaneous displacement rate of OSA on a sphere around center 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'RT
and
W = RW'RT.
13
The instantaneous angular velocities of the object around the Xi axes are (Freeman 1985):
ω1 = -W32 = ((S·Wk)/2)·cosφ·cosψ
ω2 = -W13 - B·D32 = ((B·S - Wk·S)/2))·cosφ·sinψ
ω3 = -W21 + B·D21 = (S/2)·sinφ·(Wk + B·cos2ψ).
ωl 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.
The 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 positions, is plotted as a vector
v in stereograms (Figs. 2, 3P). 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. Wk = 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).
14
OSA-trajectories for S = 10 are shown for a complete range of B and Wk values in Fig. 4P.
Fig. 4P
X3
X2
rotation
coordinates
X1
Movement patterns of the object symmetry axis (OSA) for the range of possible Wk 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 Wk = |𝐵|: planes of transient
stable positions, di: eigenvectors D’ (instantaneous stretching axes). li: 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 Wk:
a. Wk < |𝐵|. 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 di, and their actual positions depend
on the sense of vorticity, B, and Wk. At Wk = 0 source and sink coincide with eigenvectors of D' and L'
at 45° to X’2 and X’3. For any Wk, material lines (B = 1) follow a trajectory towards a sink parallel to
l2; the normal to material planes (B = -1) approaches a sink normal to l2. The OSA of objects with
other B-values have a source and sink at an angle ±β/2 from X’3 defined by: cosβ =Wk/B.
(Explanation: β gets smaller with higher Wk; β gets smaller with higher B).
b. Wk = |𝐵|. A 'stable 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 Wk = 1, the stable plane coincides with the shearing plane of simple shear flow.
c. Wk > |𝐵| (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 axially non-symmetric objects, the theory predicts that sink positions exist.
15
Application. Which objects had reached a stable position, and which were still rotating as deformation
stopped? Rigid porphyroclasts in a ductilely deforming matrix often recrystallize along their margins
and produce tails of recrystallized material that stretches 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'. Ellipsoidal 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
recrystallization is slow. Thus, complex and 𝛿-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.
Recrystallized tails will reach parallelism with l2 at high finite strains (the outer tails do not rotate
anymore).
X2
X3
X1
16
Tails of recrystallized material will tend to rotate towards the extensional eigenvector l2 of L'
throughout the deformation. The angle η between the Mx(long)-axis of an irrotational rigid object,
which lies at a sink in the X’2–X’3 plane, and l2 or the straight domain of the tail away from the object
is a function of Wk and B* only:
1
𝜂 = 2 𝑠𝑖𝑛−1
𝑊𝑘
√1 −
𝐵∗
𝑊𝑘2 − √𝐵∗ 2 − 𝑊𝑘2 }, with 𝐵∗ =
𝑀𝑥 2 −𝑀𝑛2
𝑀𝑥 2 +𝑀𝑛2
and Mx and Mn are the object’s long
and short symmetry axes in the X’2–X’3 plane.
Fig. 8P-a shows η for the entire range of Wk and B* values. η increases with decreasing B* up to the
value B* = Wk, the 'cut-off point'. At still lower B* values, the objects are rotating permanently. Wk
can be derived from these graphs in two ways (Fig. 8P-b): (a) from the value B*crit = Wk, 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 l2 during the last stages of the flow can be proven.
Fig. 8P
B*crit. for Wk=0.1
freely rotating clasts
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 Wk values, η—
angle between the long axis of an object
cross-section and l2, marked by
recrystallized tails at high finite strain.
B*—shape factor. If Wk > B* no stable
sink positions exist.
b. Example of the expected geometries of rigidobject recrystallized tail systems in crosssection parallel to X’2–X’3 at Wk = 0.5. At
low B* values, objects rotate permanently
and generate δ-type and complex tails. At
high B* values, to the right of the 'cutoff
point', objects have their long axis at a
stable sink position and generate σ-type
tails. η decreases with increasing B*.
X2
flow apophyses
X3
X1
17
The 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 homogeneous 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 sinks (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 assumed to parallel l2,
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 Wk = 0.6 and 0.7 from Fig. 8P.
Data plot of K-feldspar clast-tail systems in
quartzite mylonite, St. Barthe1emy 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. 8P. Open circles, clasttail systems with complex or δ-type geometry
indicative of permanent rotation; dots, σ-type
clast-tail systems.
18
Method of rotating porphyroclasts—elaborations
General considerations. The behavior of porphyroclasts depends on the flow type (Wk; 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 eigenvectors. 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.
obtuse angle field
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 positions. 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 sigma 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 porphyroclasts.
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.
19
Orientations of vorticity vectors and fabric asymmetries. Orthorhombic deformation symmetries are
characterized by parallelism between the finite strain elements (foliation and lineation) and the highstrain zone boundary and an abundance of symmetric structures (Wk = 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 lineationnormal plane with a zero Wk-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 nonzero Wk-values, but neither are the VPP.
In monoclinic general shear zones, the maximum asymmetry plane (and VPP) should be the lineationparallel foliation-normal plane. The 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. 3FB).
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 finite
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. 3FB). In the field, the identification of triclinic deformation relies 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.
20
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 extension.
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. Transpression, the VPP is orthogonal to both lineation and foliation; the zone shortens with
increasing deformation.
21
The porphyroclast hyperbolic distribution (PHD) method (Simpson and De Paor, 1993, 1997). It is
based on the premise that the orientation of the 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 include 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 Wk.
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. Wk is given by: Wk = cos (90-φ),
where the φ is the smallest angle made with the normal to foliation by back-rotated grains. Grain
orientations are more easily visualized with a radial distribution plot than a hyperbolic net (Fig. 6FB).
Porphyroclasts with small axial ratios (<1.4) were removed from consideration because sub-spherical
grains are not actually back-rotated, but rather are continuously 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 hyperbola.
22
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 porphyroclasts with maximum opening angle defined
by the solid black line.
23
Additional comments. The rotational behavior of rigid elliptical porphyroclasts is controlled by the
bulk kinematic vorticity (Wk), 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 ‘‘downstream’’ relative to the bulk sense of transport.
24
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, 3KN). Forward-rotated porphyroclasts are distinguished
by:
(1) Approximately equate or spherical δ-grains indicating continuous forward rotation and
(2) σ-grains that are inclined ‘‘upstream’’ 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 (β1 grains; Fig. 3KN); 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 (β2 grains; Fig. 3KN).
Fig. 3KN
x
= delta clast
key clast
Schematic diagram illustrating fields of forward and backward rotation in a
dextral general shear regime, as well as, various microstructures used in PHD analyses. Dashed lines
represent directions of maximum angular shear strain rate. Those grains inclined downstream with
sinistral tails on their narrow ends are probably near their stable orientations.
25
Application example (Law et al. 2004). The orientation and aspect ratio of porphyroclasts that have
either forward rotated or back rotated are recorded on a hyperbolic net, the porphyroblasts are coded
with 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. The choice of the most acute
hyperbola available on the hyperbolic net usually reduces the pure shear component to a minimum; the
estimated Wm 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 Wm is given by the cosine of the acute angle between the two limbs of the hyperbola. 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)
26
Rigid grain net method and updating of other rigid grain methods
Nomenclature
Wm mean kinematic vorticity number
R porphyroclast aspect ratio (long axis/short axis)
B* shape factor of Bretherton (1962)
Mn short axis of the porphyroclast
Mx long axis of the porphyroclast
θ angle between long axis and the foliation (or shear zone boundary)
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
Rc critical threshold between grains that rotate infinitely and those that reach a stable-sink position
Rcmin minimum Rc as defined by Law et al. (2004)
Rcmax maximum Rc 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 Wm = 1) rigid objects will rotate infinitely, regardless 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 Wm (Fig. 1J).
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 back rotated, due to the pure shear component, and has a long axis
at a negative angle (θ) to the foliation.
27
The 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
B* = Wm at Rc
Examples of tailless porphyroclast data. The Passchier plot uses the shape factor 𝐵 = (𝑀𝑥2 − 𝑀𝑛2 )/
(𝑀𝑥2 + 𝑀𝑛2 ) (where Mn = short axis and Mx = long axis of the porphyroclast) vs. angle between
porphyroclast long axis and foliation (θ) to define the critical threshold used to estimate Wm. The
Wallis plot uses the aspect ratio (R = long axis/short axis) and angle from macroscopic foliation (θ) to
locate the critical threshold (Rc). Wm is calculated using Rc where 𝑊𝑚 = (𝑅𝑐2 − 1)/(𝑅𝑐2 + 1). Upper
and lower Rc values are used to estimate a range in likely Wm estimates. The PHD plot uses the
hyperbolic net to plot aspect ratio (R) and θ. The cosine of the opening angle (β) of the best-fit
enveloping hyperbola yields the Wm.
Review of techniques. During general shear, rigid grain analysis assumes that the orientation of
porphyroclasts within a flowing matrix record a critical threshold (Rc) 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 Wm, R or B* and
θ define the value of Rc between these two groups of rigid grains. Wm to B* and θ are related by (see
also earlier):
1/2
1/2
𝜃 = 1/2𝑠𝑖𝑛−1 𝑊𝑚 /𝐵∗ {(1 − 𝑊𝑚2 ) − ((𝐵∗2 − 𝑊𝑚2 }
𝐵∗ = (𝑀𝑥2 − 𝑀𝑛2 )/(𝑀𝑥2 + 𝑀𝑛2 ) (where Mn = short axis and Mx = long axis of the porphyroclast)
Passchier-plot. The θ-equation generates a hyperbolic curve in θ vs. B* space that represents the ideal
distribution of grains for a particular Wm. The vertices of this hyperbola mark the unique Rc value
where Wm = B*. Assuming high strain, a natural distribution of porphyroclasts should define a limb of
this hyperbola for a range of B* values that is greater than B* at Rc. With low strains, a misleading
distribution of porphyroclasts has the potential to overestimate the simple shear component because
high aspect ratio porphyroclasts have yet to rotate into their stable-sink orientation. Porphyroclasts
with a B* < B* at Rc will rotate infinitely and should define a broad distribution with θ = ± 90°. In
contrast, porphyroclasts with a B* > B* at Rc are predicted to reach stable sink orientations with a
limited range in θ values (Fig. 3JA). Whether a porphyroclast will rotate forward or backward to a
stable-sink position depends on the initial θ at a specific B* and Wm. Rc should be defined by the
either B* or R and an abrupt change in range of θ values (Fig. 3JA, B).
Although the distribution of porphyroclasts on the Passchier plot can be informative for the high
quality data sets (Fig. 3JA), without a reference frame for comparing complex natural data with the
theoretical values established by the θ-equation, defining Rc will remain ambiguous.
28
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. 3JB; Wallis, 1992, 1995). Wm
is calculated from the Rc values separating porphyroclasts that reach a stable-sink orientation (θ < θ at
Rc) from those that rotate continuously (θ > θ at Rc). Wm is calculated from (Wallis et al., 1993):
𝑊𝑚 = (𝑅𝑐2 − 1)/(𝑅𝑐2 + 1).
The distribution of porphyroclasts often defines a gradual transition between continuously rotating
(random orientation) porphyroclasts and stable- to semi-stable porphyroclasts that define Rc. The
original Wallis plot is improved by drawing an enveloping surface to better-define the grain
distribution, and use a range in possible Wm values (Rcmin and Rcmax ; Fig. 3JB; Law et al., 2004; Jessup
et al., 2006).
PHD-plot. The porphyroclast hyperbolic distribution (PHD) method estimates Wm by using R and the
angle between the pole to foliation and long axis of tailed porphyroclasts (Fig. 3JC); plotted using the
hyperbolic net (HN). Each hyperbola of the HN represents the theoretically predicted orientation of
porphyroclasts for a particular R and Wm as plotted in polar coordinates (see sign convention).
Rc 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, plotted 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 foliation (i.e., the source-sink
and stable-sink positions). The vertex of this hyperbola separates the low aspect ratio porphyroclasts
with random orientation (i.e., infinitely rotating) from higher aspect ratio porphyroclasts with a narrow
range of orientations (Fig. 3JC).
Supplementary information.
Table 1J: Critical threshold values
R
Wm θ at Rc
B*
β
1.1
0.1
42
0.1
1.21
0.2
39
0.2
1.3
0.3
36
0.3
1.5
0.4
33
0.4
1.7
0.5
30
0.5
2
0.6
27
0.6
2.4
0.7
23
0.7
3
0.8
18
0.8
4.4
0.9
13
0.9
R = aspect ratio (long axis/short axis).
Wm = mean kinematic vorticity number.
θ = angle from foliation (Fig. 1J).
Rc = critical threshold.
B* = shape factor.
β = opening angle of hyperbola.
84
78
73
66
60
53
46
37
26
cos(β)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
29
Fig. 2J
Plot showing the relationship between mean kinematic
vorticity number (Wm), shape factor (B*), and aspect
ratio (R) at critical values.
30
1/2
1/2
The Rigid Grain Net (RGN). Eq. 𝜃 = 1/2𝑠𝑖𝑛−1 𝑊𝑚 /𝐵∗ {(1 − 𝑊𝑚2 ) − ((𝐵∗2 − 𝑊𝑚2 } is used to
calculate semi-hyperbolas for a range of Wm-values that express the relationship between θ and B*
(Fig. 4J, location A). The second set of curves represent the possible Rc (vertices curves) values for
when Wm = B* (Fig. 4J, location B). Each semi-hyperbola was calculated for a particular Wm and a
series of B* values. The shape factor (B*) enables Wm values to be obtained directly from the RGN.
Positive and negative semi-hyperbolas are plotted at 0.025 increments (of B*) for a range in Wm (0.11.0) by solving for θ using the θ-equation. For a particular shape factor, when B* = Wm and θ > θ at
Rc, 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 Rc values for the range in Wm
values represented by the RGN, a second curve (vertices curve) links the Rc values on each 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 Rc value when Wm = 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 Wm value for a semi-hyperbola.
To relate hyperbolas on the HN and the RGN, full hyperbolas are plotted on the RGN; highlighted are
the critical curves that define Wm = 0.6; plotted are also hypothetical distributions of porphyroclasts
(Fig. 5J). The two hyperbolas that are included on the simplified HN are highlighted in black on the
RGN (Wm = 0.6) for positive and negative θ (Fig. 5JA, 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°, are 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* <Wm; Fig. 5JA, C). On the RGN, this area includes all of the potential range in B* between 0
and B* at Rc (i.e., to the left of the apex of the hyperbolas; Fig. 5JB, D). An additional section of the
RGN is highlighted on the HN that defines porphyroclasts that will rotate to the vertices curve for Rc
values ≥ the ‘‘true’’ Rc for the sample. On the HN, this curve is defined by linking the Rc value from
each potential hyperbola greater than the ‘‘true’’ Rc for this sample to generate a small section of the
vertices curve as shown on the RGN (Fig. 5JA, C). This important clarification shows that these
porphyroclasts must be considered as stable-sink orientations when choosing 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.
31
Fig.5J
vertices-curve
a. Half of the HN simplified to graphically demonstrate the relationship between one hyperbola (Wm =
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 Wm values greater than the Wm for the sample (0.60).
Gray circles with letters (a-i) on the hyperbola for Wm = 0.60 are shown to compare how these
define the hyperbolas on the HN and RGN. The circle that 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 Wm = 0.60, as well as the section of the vertices curve for B* > B* at Rc. 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 distribution; 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.
32
Application of the RGN. “The choice of rigid porphyroclasts is highly selective, ignoring all but the
most appropriate grains for estimating Wm“ (Fig. 7J).
Fig. 7J
All Rc-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 Wm range is probably
closest to the true value.
33
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 illustrate 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 represent plotted
backward-rotated porphyroclasts while gray-filled circles depict plotted forward-rotated
porphyroclasts.
34
Synthetic shear bands are oriented either parallel to, or at an angle less than the acute bisector (AB;
Fig. 4KN). Antithetic shear bands 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 and 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 vorticity 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 determined 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
network of quartz dominated micro-shear zones and enclose large microlithons of feldspar (Fig. 7KN).
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.
35
Quartz c-axis fabrics and oblique grain shape vorticity methods (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-axisfabrics develops nearly orthogonal to the flow/shear plane (A1). Therefore, the angle, β, between the
perpendicular to the central girdle segment of quartz c-axis fabric and the foliation (SA) is equal to the
angle between the flow plane (A1) and the flattening plane of finite strain (Fig. 1a,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 (ISA1) (Fig. 1a,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 SB. Therefore, the maximum observed angle, δ, between the SB
and the main (SA) foliation is equal to the angle between the ISA1 and the largest principal axis, X, of
the finite strain (Fig. 1a,b; Wallis, 1995).
Fig. 1X
Fig. 1. (a) Simplified sketch showing the
relative orientation of instantaneous flow
elements and their angular relationships in
real space for a dextral general shear flow.
A1 and A2 – flow apophyses; ISA1 and ISA3 –
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 (SB)
and the main foliation (SA) is inferred to be
equal to the angle between the ISA1 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 (SA) is inferred to be equal to the
angle between the flow apophysis A1 (flow
plane) and the principal finite strain axis X.
According to the oblique-grain-shape/quartz c-axis-fabric method (referred to hereinafter as δ/βmethod), if both angles δ and β are known then an estimate of vorticity number can be obtained using
the equation (Wallis, 1995):
Wm = sin 2 (δ + β)
This eq. only holds for two-dimensional flow. The δ/β-method records the last increments of plastic
deformation (Wallis, 1995).
36
RXZ/β-method: The strain-ratio/quartz c-axis-fabric method estimates Wm (finite deformation) as:
Fig. 13L
Vorticity 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 external fabric asymmetry. β is used in combination with
the ratio of principal stretches, Rxz =(1 - ε1)/(1- ε3), to estimate mean kinematic vorticity number Wm.
This method assumes plane strain deformation. Plane strain conditions are indicated by the crossgirdle 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) investigated 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-Wm 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 Wm
goes to zero or one. Vorticity number estimates using RXZ/β-method are very sensitive to small
changes in the evaluated angle β. In most quartz c-axis fabric diagrams the angle β can be evaluated
with an error ±2°; the method becomes unreliable in high-strain samples (RXZ >10–15) with small β
(<5°).
RXZ/δ method: The maximum observed angle δ between SB and SA 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.
Wn (instantaneous deformation) = sin(2δ) [(RXZ + 1)/(RXZ – 1)]
which shows that Wn is related to RXZ and δ.
37
This eq. can be solved for δ and the result can be plotted for different Wn values.
δ = 1/2 sin-1[Wn(RXZ-1)/(RXZ + 1).
This plot (RXZ versus δ) is illustrated in the synthetic diagram of Fig. 3, which also includes an RXZ
versus β plot.
Fig. 3X
Fig. 3. Synthetic diagram showing plots of the finite strain-ratio RXZ versus β and RXZ versus δ.
This diagram indicates that during progressive steady-state deformation the angle δ attains a high
value at low strain (RXZ < 4) and then is subjected to small changes. This observation 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 plot of
RXZ versus δ also indicates that the Wn curves get broader spaced for increasing strain values (Fig. 3).
This implies that, in contrast to the RXZ/β-method, Wn estimates are relatively insensitive to small
changes in the values of RXZ and δ.
The RXZ/δ method describes instantaneous deformation and, therefore, does not require the assumption
of steady-state deformation.
Comparison to the particle methods. Difference could be due to: (1) analytical problems associated
with individual methods (e.g. the particle methods tend to give a minimum estimate of Wm 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 in synchronous flow behavior between rigid particle rotation of the porphyroclasts
and plastic deformation of the surrounding matrix quartz grains.
38
A vorticity nomogram incorporates all the parameters (β, δ, RXZ and Wn/Wm) 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, RXZ);
(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 evaluation of mean strain level for a suite of samples, where only β- and δangle data are available.
Fig. 4X
Fig. 4. Suggested nomogram for estimating Wm (or Wn) and RXZ in naturally deformed quartzites and
an application to five representative samples. Circles 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 RXZ values.
39
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, stretch measured parallel to the flow plane in the transport direction is given by S-1.
40
Non-perfectly matrix bounded rigid inclusions (Marques 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) Jeffery’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 identification of the material property contrast between
inclusion and matrix, µi/µm, the aspect ratio, Ar, 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 Ar, ratio Wr 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 (µi),
rim (µr ) and matrix (µm), and finally the relative strength of pure to simple shear Sr. Note that the
angle and quadrant convention used (see 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 clasts as isolated perfectly bonded rigid inclusions, therefore allowing for
application of Jeffrey’s solution. Factors that can change the rotation behavior of rigid inclusions are:
(i) the addition of a pure shear component to simple 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.
(iii) slip 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 is produced.
(iv) flow confinement. Depending on the Wr 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.
41
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 equivalent void derived
by Schmid and Podladchikov (2004), and the
transtension (Sr = -0.3) case of combined pure
and simple shear (Ghosh and Ramberg, 1976;
Marques and Coelho, 2003).
Site 1: Given the φ-Ar 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 equivalent
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 Sr case is opposite to the field data as illustrated with Sr = -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.
42
Fig. 5M et al., Fig. 7M et al.
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 rectangle
(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.
43
‘‘Site 2 Data Fit’’, ‘‘ice experiments’’, and ‘‘equivalent void’’ models. Transpression is plotted for Sr
= 1 according to Ghosh and Ramberg’s model (1976). ‘‘FEM Slip Sr = 1’’ represents finite element
calculations for slipping inclusions in pure and simple shear
.
Site 2. Two 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.
The clasts with negative inclinations can be fitted to Ghosh and Ramberg’s model with a shear zone
flattening component that equals the simple shear component, Sr = 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 finite 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. Varying 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: thin rims of mica
and/or quartz and/or fine-grained feldspar commonly surround 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 confined 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.
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last modified: June 16, 2011
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