Tectonophysics,
Elsevier
136 (1987) 159-163
Science Publishers
B.V., Amsterdam
- Printed
1Letter Section 1
in The Netherlands
Efficient use of the velocity gradients tensor in flow modelling
CEES
Insriruut uoor Aurdwetenschappen,
W. PASSCHIER
Rijksuniuersitert
(Received
December
Utrecht, Postbus 80.021, 3508 TA Utrecht (The Netherlands)
9, 1986; accepted
January
30, 1987)
gradients
in flow modelling.
Abstract
Passchier,
C.W., 1987. Efficient
For models
reference
frame
elements
which
of fabric
development
for instantaneous
develop
fibres,
crystallographic
orient
themselves
mica preferred
use of the velocity
in rocks,
flow can
in deforming
preferred
orientation
orientation
ment of such fabric
elements
clear
parallelism
benefit
as a variable
presentation
steady
to instantaneous
with the extensional
from a choice of reference
the choice
of most
of instantaneous
flow: syntaxial
and immobilised
axes (di),
apophysis
frame attached
such as occur
nate
In a homogeneously
coming increasingly
important
(Ghosh and Ramberg, 1976; McKenzie
and Jackson,
1983; Passchier,
1986;
commonly
deforming
Bobyarchick,
1986).
The
velocity
ii = L,,
treat fabric development
in a matrix
by homogeneous
flow, described by the
seem to
elements
deforming
of particles
and
for develop-
body,
the in-
at X in a coordi-
x can be described
rate of displacement
models
fabric
to either d, or I,.
stantaneous
system
fabric
(I,) of the flow. Models
In modern
structural
geology, mathematical
modelling
of the development
of fabric elements
rocks is be-
rigid objects
while shape
Velocity gradients tensor
deformed
of an unsuitable
direction
Introduction
in naturally
136:159-163.
The orientation
state foliations
stretching
parameter,
Tectonophysm,
of results.
to some principal
patterns,
relative
approach
with vorticity
hamper
rocks is attached
in a fixed position
tensor
by the Eulerian
equations:
x,
where L is the velocity gradients tensor. L can be
decomposed
into a symmetric rate of deformation
velocity gradients tensor L (Malvern, 1969). Studies which analyse the effect of changing vorticity
on fabric development
(such as the rotational
be-
according to Malvern
al. (1980):
haviour of rigid bodies in a ductile medium) can
be presented by simple equations and diagrams if
the principal directions of L are kept fixed in the
external reference frame. This fact is not generally
The column vectors of L can be used to construct
a Mohr circle of the first kind (De Paor and
realised and is therefore explained in this paper.
The treatment
is restricted
here to two-dimensional or plane strain flow, but can easily be
extended to more general flow types.
0040.1951/87/$03.50
0 1987 Elsevier
Science Publishers
B.V.
tensor
D and
antisymmetric
vorticity
tensor
W
(1969, p. 146) and Means et
Means, 1984) representing
the tensor. This is done
using a convention
first given by Means (1983) to
plot the angular velocity (w) and stretching rate
(i) of material
lines instantaneously
coinciding
with the axes of the coordinate system as points in
160
i-w
L,,,
space following:
-L,,
L 227
= (t,
L,,=
(i,
a)
w, = w. (2df
In this paper only isochoric
which case d, = -d,
top
An off-axis Mohr circle can now be drawn through
both points and centred on the connecting
line
W, = W/2d,
where
diameter
and
and
w values
of a material
two points
angle between
material
1983). The coordinates
are ( V, W/2),
change
line
the angle
is double
the
of the centre
representing
of the circle
instantaneous
volume
on the
opposite points d, and d, represent the orthogonal eigenvectors
of D; they are the instantaneous
stretching axes of the flow along which maximum
shear
simple
shear
flow
with
1; Lister
the
and
flow
W, = 1
W, = 0 (Means
et al.,
1980).
Reference frames for L
As a tensor, L can be presented in a reference
frame of any orientation.
Rotation
of the reference frame by coordinate
transformation
causes a
change in actual values of matrix coefficients of L
and in the position
of L (Bobyarcl~ck,
1986; Passchier, 1986) sometimes called “flow apophyses”
(Ramberg,
1975;
Passchier, 1986): they are the lines of no instantaneous angular velocity, which are not usually orthogonal and only exist if eigenvalues
of L are
real. con-coa~ality
of the flow, i.e. non-coinciof the centre of the Mohr circle with the
can be expressed by the kinematic vorticity
of Truesdell
for pure
For
(Fig.
of points
representing
nate axes on the Mohr circle (Fig. 1; Means,
(d,) and minimum (d,) stretching rates of material
lines are realised. Intersections
I, and I, of the
Mohr circle with the Z-axis represent eigenvectors
number
1983).
coinciding
circle
( W) of the flow.
flow types plot as circles centred
w-axis, coaxial flow types (pure shear) as circles
centred on the i-axis. In Fig. 1 the diagonally
dence
i-axis,
factor
lines in real space (Means,
(I’) and half the vorticity
Jsochoric
and
on the circle
in
= W/S
of the Mohr
Williams,
flow is considered,
and:
S is a stretch
(Fig. 1; cf. Lister and Williams,
1983; Means,
1983). Each point on the Mohr circle represents i
between
+ 2d:)-l’?
Xl
(1954):
coordi1983)
but leaves tensor invariants
unmodified.
This
principle is illustrated in Fig. 2 where three different tensors
L are each presented
in four orienta-
tions by a Mohr circle, in spatial orientation
of
eigenvectors
and in matrix form. Presentation
I is
a random orientation,
II is often used to present
simple shear flow (IIc), III for pure shear flow
(IIIa) and IV for general non-coaxial
flows (IVb),
sometimes called sub-simple shear (De Paor, 1983;
Bobyarchick,
1986). II and III are related by a 45 a
rigid body rotation.
In presentations
I, II and III, a change in W,
does not affect the orientation
of d, and d, in the
external
reference
between
I, and I, according
frame, but changes
the angle
(Y
to:
cos cy = w,
sin cy = J-1 - W, - 21,/S
Fig. 1. Mohr circle presentation
Points
on Mohr
(i) and angular
-volume
Mohr
open
d,,
and closed
coordinate
system.
rate;
S-stretching
tensor L.
stretching
lines. W-vorticity;
rate factor,
d2 and I,, I,--eigenvectors
circles-orientation
gradients
instantaneous
velocity (w) of material
change
circle;
of velocity
circle represent
rate
V
diameter
of
of R and
L;
of X, and
X2 axes of
This is illustrated
in E’--0 space by a shift of the
Mohr circle along the o-axis (Figs. 1 and 2).
Presentation
IV is rather special in that one of
the eigenvectors
of L (I, in Fig. 2-N) is permanently fixed in the external reference frame. This
implies that d, and d, rotate in the external
frame by an angle a/2 upon change of Wk. The
angle between d, or d, in the first three presentations and in IV is therefore a function
of W,
(Fig. 2).
161
+w
‘(a)
+w
(b)
II
PURE
SHEAR
SIMPLE
0
_- s
2
L
Fig. 2. Presentation
I-IV
of velocity
show four differently
anticlockwise;
S-stretching
< and
J
gradients
oriented
w-instantaneous
rate factor,
!d&
2
diameter
_I
_$
J
L
stretching
of .C in the coordinate
rate and angular
system
velocity
2
5
2
-l
l-
tensor L in Mohr space, real space and in matrix
presentations
SHEAR
X,-X,.
of material
form for three values of Wk. Diagrams
P-angle
lines;
between
Wk-kinematic
Xt and dt. measured
vorticity
number;
of Mohr circle.
In many recent models of fabric development,
the effect of a flow type with variable W, is
studied instead of pure or simple shear only (e.g.,
Ramberg, 1975; Ghosh and Ramberg, 1976; Mc-
Kenzie and Jackson, 1983; Hanmer, 1984; Freeman, 1985; Bobyarchick, 1986). For this purpose,
presentation IV of L is commonly used (ibid.)
although it is difficult to see to which advantage.
162
In
some
papers
Bobyarchick,
(Ghosh
1986)
velocity
by addition
gradients
the standard
tensors.
tion of L results.
the mathematical
sentation
of results.
rigid objects
Rotating
of eigenvectors
is treated
in
a type IV presentastudies,
uncritical
can unnecessarily
treatment
shear
IIc and IIIa of Fig. 2,
frame,
For many
of this presentation
W, can be
Since this is done using
presentations
reference
1976;
seems to be
of pure and simple
which differ by a 45” rotation
the external
Ramberg,
of L with variable
that a presentation
constructed
and
the only reason
of rigid objects
in the flow
arrows)
II in Fig. 2). Further
angular
explanation
I,
from 0
frame (presentation
reference
I,,
velocity
(p,, p2) with W’, increasing
reference
Fig. 2). b. d, and d, fixed in external
tation
of d,, d,,
and minimum
IV in
frame (presen-
in text.
clear pre-
behaviour
of
the permanently
symmetrical
p2 and other critical
and d, is immediately
as an example.
positions
obvious.
arrangement
of pi,
with respect
to d,
rigid objects
Ghosh and Ramberg (1976) and Freeman
have investigated
the rotational
behaviour
(1985)
of el-
lipsoidal rigid objects in a homogeneously
flowing
viscous matrix using presentation
IV for L. Ghosh
and Ramberg
ties of object
object axial
(1976) find that the angular
veloci-
symmetry axes are a function of the
ratio,
W, and initial
orientation.
are given which define
the critical
posi-
tions where symmetry axes have maximum
(pl)
and minimum
(p2) angular velocities or become
irrotational
(curved
of maximum
to 1. a. I, fixed in external
Orientation
Equations
in orientation
complicate
and hamper
The rotational
use
Fig. 3. Change
and of positions
for some W, and axial ratio. Careful
investigation
of these equations
shows that the
critical positions are symmetrically
arranged with
respect to d, and d,, and remain so upon changes
in W, or axial ratio of the object (Fig. 3; Passchier, in press). p1 and p2 are orthogonal
and
fixed at 45’ to d, and d,, irrespective of W, and
object axial ratio. These important
results are not
immediately
obvious
in the equations
and diagrams of Ghosh and Ramberg (1976) due to their
use of presentation
IV for L; all critical positions
of object symmetry axes, d, and d, are subject to
an extra rotation over ~y/2 in the external reference frame upon change in W, (Fig. 3a). The
diagrams
in Freeman
(1985) which demonstrate
actual orbits of object symmetry axes for non-plane
strain flows, show the same rotation over a/2 for
diagrams with different Wk. In contrast, Passchier
(in press) uses presentation
II for L in which d,
and d, do not rotate in the external reference
frame upon change of W, (Fig. 3b): this leads to
relatively simple equations and diagrams in which
of fabric elements
An obvious question
is whether presentation
in
any
situation.
Many
arising from the foregoing
IV of L is advantageous
simple
fabric
elements
which develop in response to ductile deformation
in rocks seem to have a fixed orientation
in the
kinematic
frame of the flow. Some fabric elements
such as syntaxial fibres (Ramsay and Huber, 1983)
crystallographic
preferred
orientation
patterns
(Lister and Hobbs, 1980) steady-state
foliations
(Means, 1981) and stable positions of ellipsoidal
rigid objects (Ghosh and Ramberg,
1976; Passchier, in press) seem to be fixed to eigenvectors of
D. They
ciently
can obviously
using presentations
be modelled
most
I, II or III for L.
effi-
Others, such as mica preferred
orientations,
planar and linear shape fabric elements (Ramsay
and
Huber,
around
1983)
tails of recrystallized
porphyroclasts
(Passchier
and
material
Simpson,
1986) and pressure shadows (Malavieille
et al.,
1982) trend towards parallelism
with the extensional eigenvector
of L (I, in this paper) with
progressive
deformation.
Studies of such fabric
elements may benefit from the use of presentation
IV.
Conclusions
From the example given above, it is obvious
that some care is needed in the choice of a presentation of L in fabric studies. The method of
addition of pure and simple shear tensors to ob-
163
tain
a general
velocity
gradients
tensor
L should
subjects.
l-33.
by use of
Malavieille,
be used with care and only for suitable
In other cases, problems
the method
can be avoided
notation
circle presentation.
ful because
of L directly
This method
of the visual
which the components
rock
J., Etchecopar,
des exemples
from a Mohr
Malvem,
tinuous
over the way in
McKenzie,
rates,
Planet.
References
Means,
A.R.,
1986. The eigenvalues
of steady
122: 35-51.
De Paor, D.G., 1983. Orthographic
analysis
tures.
I. Deformation
theory.
De Paor, D. and Means,
and
second
ations.
kind
J. Struct.
Freeman,
crustal
and
their
S.K. and Ramberg,
by
tectonic
Lister,
use to represent
tensor
oper-
particles
in
pure
flow. Current
G.S. and Hobbs,
and
simple
use of planar
shear.
regime
and elliptical
and kinematics
of
Res., 84: 133-142.
plastic
to quart&e:
the influence
deformation
of fabric
and its application
G.S.
and
Williams,
between
palaeomagnetism,
within a deforming
1981. The concept
finite
zone. Earth
of steady
state
foliations.
of the Mohr-circle
construc-
78: 179-199.
1983. Application
W.D.,
of inhomogeneous
Hobbs,
B.E., Lister,
and non-coaxiality
tions. J. Struct.
Geol., 2: 371-378.
deformation.
J. Struct.
G.S. and Williams,
in progressive
C.W., 1986. Flow in natural
quences
Passchier,
of spinning
flow regimes.
C.W., in press.
non-coaxial
Passchier,
flow-a
Stable
study
of deformation
history.
J. Struct.
1983.
The
partitioning
of
indicators.
Earth
P.F.,
deforma-
shear zones-the
positions
in vorticity
Planet.
conseSci. Lett.,
of rigid objects
in
analysis.
J. Struct.
C., 1986. Porphyroclast
systems
applicable
J.G. and Huber,
em Structural
versity
J. Struct.
H., 1975. Particle
sive strain
Truesdell,
P.F.,
C.W. and Simpson,
as kinematic
Ramsay,
Geol., 2: 355-371.
Lister,
of a con-
Cliffs, N.J.
J., 1983. The relationship
thickening,
1980. Vorticity
Ramberg,
B.E., 1980. The simulation
during
Englewood
Geol.
of strain
development
to the Mechanics
77: 70-80.
of inclu-
34: l-70.
as indicators
a
Geol., 5: 279-286.
Means,
113: 163-183.
of
W.D.,
Passchier,
of rigid ellipsoidal
S.K., 1984. The potential
structures
5: 255-277.
H., 1976. Reorientation
combination
Tectonophysics,
Hanmer,
Geol.,
struc-
Geol., 6: 693-701.
slow flows. Tectonophysics,
sions
of geological
J. Struct.
Means,
de
et application
Sci. Lett., 65: 182-202.
W.D.,
tion to problems
W.D., 1984. Mohr circles of the first
B., 1985. The motion
Ghosh,
flow in
simulation
Prentice-Hall,
Tectonophysics,
Mohr space. Tectonophysics,
92:
C.R. Acad. Sci. Paris, 294: 279-284.
D. and Jackson,
strain
are derived.
naturels.
medium.
Tectonophysics,
A. and Burg, J-P., 1982. Analyse
strain and fault movements
Bobyarchick,
masses.
L.E., 1969. Introduction
seems most use-
control
in flowing
la g&ometrie des zones abritees:
of Means (1983) to derive components
of the matrix
deformation
Geology.
paths,
Geol., 8: 831-843.
displacement
to rocks. Tectonophysics,
MI.,
28: l-37,
1983. The Techniques
Vol. 1. Academic
C., 1954. The kinematics
Press, Bloomington,
and progres-
Ind.
of Mod-
Press, London.
of vorticity.
Indiana
Uni-