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Title: PII: 0040-1951(78)90004-5

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Tectonophysics, 45 (1978) 107-158
0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

107

THE SIMULATION OF FABRIC DEVELOPMENT
IN PLASTIC
DEFORMATION
AND ITS APPLICATION TO QUARTZITE:
THE
MODEL

G.S. LISTER *, M.S. PATERSON and B.E. HOBBS **
Research School of Earth Sciences, Australian National University, Canberra, A. C. T.
(Australia)
(Submitted July 20, 1976; revised version accepted March 24, 1977)

ABSTRACT
Lister, G.S., Paterson, M.S. and Hobbs, B.E., 1978. The simulation of fabric development
in plastic deformation and its application to quartzite: the model. Tectonophysics,
45: 107-158.
The Taylor-Bishop-Hill model for polycrystalline deformation has been applied as the
basis of a computer program for simulating the development of preferred crystallographic
orientations in deforming rocks in which the predominant mode of deformation is
dislocation glide within the grains. The model assumes homogeneous deformation on
the scale of the grains and rigid-plastic flow obeying Schmid’s critical resolved shear stress
law for the glide systems. The input data for the simulation are the initial orientation
distribution, the set of possible glide systems and their relative critical shear stresses, and
the details of the deformation including its path, which can involve non-coaxial deformation histories in the general case.
The analysis is applied to model quartzites in which four possible choices of glide
systems and their critical resolved shear stresses are considered, for three different
deformations. Some of the simulated fabrics bear close similarity to observed fabrics,
suggesting that fabric development as a result of rotation of crystal axes during dislocation glide is potentially an important geological process, and that the Taylor-Bishop-Hill
model is suitable for analyzing it. From the profound influence of deformation history
on the simulated fabric and the sensitivity of the fabric to the choice of glide systems and
their relative critical shear stresses, important possibilities are suggested for gaining information about geological environment and deformation history from the analysis of natural
deformation fabrics.

INTRODUCTION

A deformation
fabric may hold important clues to the geological history
of a given rock mass. In particular, information
may be stored about the
* Present address:

Department of Tectonics, Geologisch Instituut, Rijksuniversiteit
Leiden, Leiden, The Netherlands.
** Present address: Department of Earth Sciences, Monash University, Melbourne,
Australia.

108

nature of the total deformation
and the path by which it was achieved, as
well as about the deformation
mechanisms and the environmental
conditions. It is therefore
desirable to model theoretically
the development
of
fabrics during deformation,
taking into account
the influence of these
various factors, as an aid in interpreting observed fabrics.
Preferred orientation
of the crystallographic
axes of the grains of a polycrystalline mass constitutes
one of the most important
types of fabric. In
metallurgical
and structural geological research, the origin of such fabrics
(“textures”
in metallurgical parlance) has been much studied for over half a
century and considerable
sophistication
developed for their measurement.
Experimental
work has shown that, at least at relatively low temperatures
(“cold-work”
regime where recrystallization
or recovery processes are subordinate),
the particular preferred
crystallographic
orientations
developed
during plastic deformation
of a single-phase polycrystalline
mass arise from
the reorientation
of the crystal axes of the grains as they undergo intragranular plastic deformation
(slip or twinning), involving the glide motion of
dislocations
(e.g. Tullis et al., 1973). There is also evidence that such
reorientation
is important in naturally deformed rocks such as quartzites, at
least under some geological conditions; this is indicated, for example, by the
observation within the quartz grains of appreciable densities of dislocations
of types known from laboratory
studies to be glissile. There are other ways
in which preferred
crystallographic
orientation
can develop in naturally
deforming rocks but here we shall focus exclusively on preferred orientations
resulting from intragranular
plastic deformation
as a step in establishing a
framework to aid in the interpretation
of real situations.
Various attempts have been made in metallurgy to set up a theoretical
model of this process for the development
of a deformation
fabric, but the
model introduced
by Taylor (1938a,b)
and proposed again in somewhat
different terms by Bishop and Hill (1951) has received most attention and
it appears to be the most promising
for application
to rocks. Modern
methods of computing now make it possible to exploit this model fully. It
is therefore the purpose of this work to apply the Taylor-Bishop-Hill
model
as a basis for a computer
program to simulate fabric development
in
quartzite masses plastically deformed by means of intragranular dislocation
glide. The influences of various types of deformation,
of the deformation
path itself and of changes in preferred glide systems have been explored.
THE

PHYSICAL

BASIS

FOR

LATTICE

ROTATION

DURING

CONSTRAINED

DE-

FORMATION

Models such as the Taylor-Bishop-Hill
analysis offer no more than a
solution to the problem of predicting the change in lattice orientation
that
occurs when a crystalline
volume is constrained
to undergo a specific
deformation
increment. Before going on to consider the model itself, we will
first look at this process of crystallographic
reorientation
during constrained

109

:::::::::::::::::::::::::
:::::::::::::::::::::::::
......... ...............
::::::::::::::;piiiiiii
;~;i;i!~;Bs!

::::::::::::.

::::::::::::::::::m:::x:::
::;
::::::
........ ...... .......... .......*.::.::.::::!:
.... ... .. ..
t::::u..I”t

::::::l:::::::::::~:i:::::

. . . .

/

.

:::::::::::::::ja;r:;:;:i::
iiQ~~iiili%:?~iiii8I:il

~:X:iii::x:::::::::~::::

:::::

::-:::::-:::::::-:::~:~~a~~~~
;l::;

:::::::

;;::~::;:::

::::::ii::::i:::::::::~::::::
liiiiialiililPBli8~~iiiiit

(A) fine

slip

:I::::U:J::::::“lf:.:..“...
::::::::::^I:::::::::l::::z:::::
::::::::~::::p:::“:‘:“‘:“‘:“:
iiPlfiii8ii:SS~S~~~~S~~~~~

(8) coarse

slop

.

Ea

CC) mechanical

twinning

CD)

Fig. 1. Simple shear resulting from the operation of one glide system. a. Fine slip. b.
Coarse slip. c. The effect of mechanical twinning. In d is shown the spatial finite-strain
ellipsoid developing from a material sphere. The overall deformation can be decomposed
into two stretches at right angles parallel to the axes shown, plus a rotation.

deformation. The discussion is limited to deformation taking place solely
by the conservative motion of dislocations (dislocation glide) but it is not
restricted to any particular model. It will be shown how the lattice rotations
arise out of certain aspects of the mechanics of dislocation glide.
The glide motion of dislocations is essentially a discrete or heterogeneous
process and even their collective movement is commonly heterogeneous, as
revealed by discrete surface steps or striae (e.g., “slip lines”). However, on
the scale of the whole grain, or subgrain, the deformation resulting from the
distributed movement of dislocations of a given type on parallel glide planes
can be approximated as a continuous simple shear, ignoring the finer details
of the distribution (Fig. la and lb).
The direction of shear or slip is that of the Burgers vector, and the glide
plane is the invariant plane of no distortion for the simple shear. Together
the glide plane and slip direction comprise a slip system, or glide system, and

110

this continuum description of the deformation
caused by the operation of a
single glide system is fundamental
to models such as the Taylor-Bishop-Hill
analysis.
It is assumed that all actual glide motions of different types of dislocations can be resolved into a finite number of glide systems and that the
activities of each glide system do not mutually interfere so that activity on
one system is not directly coupled to that on others. Further it is assumed
that the simple shears resulting can be simply superimposed.
In this connection cross-slip is also to be reckoned as a separate glide system if its contribution
to the total strain is not directly coupled with the amount of
activity on the primary slip plane.
Since the crystal structure
within the grains remains recognizable
and
relatively
undistorted
during the operation
of a given glide system, the
crystallographic
axes can be used as an internal reference frame in describing
the geometric changes within a given grain during deformation.
In particular
the simple shearing in the continuum
description
of the gliding can be
resolved into a pure strain and a rigid body rotation relative to these axes.
In Fig. ld the rotation of the axis system marked becomes ~ymptotic~ly
closer to 45 degrees as the simple shear becomes infinite. The rate of rotation of the finite-strain axes decreases as the shear becomes larger.
It is more instructive
to examine this situation kinematically.
At any
instant during the progressive simple shear above the crystalline material
has developed
vorticity
(Malvem, 1969, p. 145) with respect to its own
c~st~lo~aphic
axes. As in Fig. 2 all radial lines have angular velocity with
respect to their reference
frame (outer circle). By subtracting
a constant
angular velocity (middle circle) equal and opposite to the vorticity, angular
velocities appropriate
to a pure shear are obtained (inner circle). The angular
velocity of a material line coinciding with a stretching axis is then zero. In
the progressive simple shear above: (a) the stretching
axes (infini~sim~
straining axes) are fixed in orientation
throughout
the deformation;
and
(b) as long as the rate of shear is constant,
material lines continuously
migrate past and through the position of any reference vector (e.g., normal
to the basal plane, or parallel to a stretching axis) at a constant rate.
Now we can consider what happens when a crystalline grain is constrained
so that it must undergo a pa~icular (imposed) defo~ation.
An external
reference
frame is chosen so that the stretching axes are held stationary.
At any instant the vorticity of the material in any crystalline grain with
respect to these axes must be the vorticity of the imposed deformation.
Because the crystal can only accomplish this deformation
by combinations
of simple shears, the vorticity of the deforming material in the grain with
respect to crystul axes need not be identicui with the imposed vorticity, atA
in general these vorticities will not be the same. The geometrical constraints
posed by this situation require that the crystallographic
axes rotate at a
compensating
rate with respect to the external axes, so that the imposed
vorticity is maintained.

by the rotation of
small paddle whC?el
at the centre

Fig. 2. At any instant during the progressive simple shear, radiaf lines have the angular
velocities shown on the outer circle with respect to the external axes. By subtracting a
constant angular velocity, equal and opposite to the vorticity (see middle circle) angular
velocities are obtained that are appropriate to a coaxial deformation (see inner circle).
The stretching axes are fixed in orientation, and material lines pass through these directions with a constant angular velocity, as long as the rate of shear is constant.

This principle is illustrated in Fig. 3. A crystal block (Fig. 3a) is constrained so that it undergoes a pure shear, with extension axis inclined at
45 degrees to horizontal (the outline box - Fig. 3b). The crystal is obliged
to accommodate #is deformation by the operation of a single glide system,
glide plane initially horizontal. For the small increment of shear shown the
shear angle is of magnitude y in unit time, and the vorticity of the material
with respect to crystal axes is thereby T/2 clockwise about a horizontal axis.
The imposed deformation has zero vorticity, and the crystal axes therefore
rotate at a compensating rate j/2 (coun~r~lock~se).
A more general situation, but still only two-dimensional, is illustrated in
Figs. 3c, d, e. The deformation imposed on the crystal results in material

112

crystal block

the otilitne box

t

,al
...............

.............
.............
.............
.............

.....

.....

...

.............
.......

.......

......
.....

.

...

......

.....

.............
.............
.............

(a)

.............

...............

_~

this material ltne
held stattonary

L.----r
crystal axes

same material line
“?B

(c) angular velocity ‘A’ of crystat
axes with respect to a
particular material line

externa

(d) angular velocity ‘5’ of material
ltne with respect to the stretching
axe2

(e) angular velocity ‘C’ of crystal axes
with respect to the stretching axes

Fig. 3. A crystal block (a) is constrained
to undergo a specific deformation.
In (b) a pure
shear (the outline box) with zero vorticity
is accommodated
by a simple shear with vorticity y/2 clockwise
(with respect to crystal axes). The crystal axes rotate at a rate 9/2
(counter-clockwise)
to compensate.
The vorticity
of the imposed deformation
must be
maintained.
This is shown in (c), (d) and (ef. The imposed deformation
(d) results in the
angular velocity
“B” of the material line coinciding
with the stretching
axes at a particular instant,
Because
glide systems
operate
the crystal
axes have angular velocity
“A”
with respect to these lines (c.) and the crystal axes rotate at rate “C = A + B” as a result,
compensating
for the difference
between
the imposed vorticity,
and the vorticity
of the
deforming
crystalline
material with respect to the crystal axes.

113

lines rotating past and throu~ a particular reference direction at a specific
angular velocity. Figure 3d shows the relative angular velocity of the material
line coinciding with a stretching axis at a particular instant, with respect to
that axis. Because the crystal is undergoing combinations of simple shears
(with respect to crystal axes) this same material line, at the same instant in
general has a d~~~e~en~angular velocity with respect to the crystal axes (Fig.
3c), so that the crystal axes rotate at a compensating rate with respect to the
stretching axes (Fig. 3e).
The fundamental assumption of the Taylor and Bishop-Hill analyses is
that specific deformation increments are imposed on a crystalline volume.
The stress state in the volume is calculated as a result of the analysis, and it is
therefore the strain ~~cre~e~t imposed, rather than the strtss state that
caused the deformation, that is the independent variable. (This deformation
is imposed in total disregard to the strength of each grain in relation to the
strength of adjacent grains, unfortunately leading to the consequence that
strong grains are deformed at the same rate as soft grains.)
During the small deformation increment the vorticity with respect to
crystal axes results in a rotation given by the infini~sim~ rigid body rotation tensor:
nc = [f(M,

-

Gr)lr

(r, s = 1,2,3)

where y is the amount of shear on the glide system, nl, na, ns are the components of the unit vector n normal to the glide plane, and tl, 22, 1s are the
components of the unit vector t parallel to the glide direction. When several
glide systems are active, the resultant infinitesimal rigid body rotation is the
sum of the rotations associated with the activities of the individual glide
systems.
The vorticity of the imposed deformation results in a rotation with
respect to external axes, described by the rigid body rotation tensor, ax. The
latter imposed rotation, sIzx, will in general differ from the rotation 51’
relative to the crystal axes. The crystal axes must therefore undergo a compensatory rotation nL with respect to the external axes (Fig. 3b), such that:
nL=nx-~c
Drawings ~lust~ting lattice rotation during defo~ation
are often based
on card-deck type models, for a single glide system. Simple lattice rotations
result. For shortening, the pole to the active glide plane rotates at an asymptotic rate, via a great circle, towards the axis of compression. In addition the
process leading to the development of a crystallographic fabric is often
referred to as taking place via grain rotation.
It is important to realize that c~s~o~phic
axes rotate with respect to
the grain shape during glide, and that rotations of the grain shape more or
less as a whole can be considered independently. Neither the crystal axes
nor the incremental strain axes are tied to any particular set of material
points, whereas the grain shape is decided by the overall configuration of

c- ax6

m
...*...*.*
.:..::-.:-.::1:
.**.
a*. .*.

. .,.*..
-.;..*.*:. . . a.. -

.*: .*. .*: .: .*:::

. ...‘..
._.* .*.*.. .*..:

(b)

**:.

5

*.. *

.
* : . : . * . ::: .
. * :*.*:.*:.::
._:. .*: *:: : __..

*_. ’
a**.
* . . ._.*.*..:. . .*:: . *.:
. .*.*. ._. . . . - . * . ..:::
.* .::::

. -

5
3

‘. . .***

.*...*..*.*.**.:*: .*:: ::: :: * .*: . . me-..*:: *. . .* . .*:: * cQ,
.*: ., . .* . . *_. .*. .*. * - z
. .* . .* . . :: - .*: .*:: * *.: *
a.*, . .* . .. . .*..: .**:.**:.*:: ::‘::.::
**

I .. .*.*.*
.: ._..
..~.~~.:::~~~~~.~~~.~~...
**
;
c-axis
4

(d) 10% axially symmetric ;” ln~it~+;;~
shortening
+

‘LI
5

(4
progressive 2-D

strain

Crystal axes rotating
rapidly with
respect to grainshape

symptotic spiralling of

115

these points (Fig. 4a, b, c). The type of behaviour that can be expected is
illustrated
in Fig. 4e, f, g. A crystal block is subjected to a progressively
increasing axially symmetric shortening. The block is able to deform by the
operation of mechanisms as for Model Quartzite II (see page 132), but with
the addition of {prism} (a> glide systems. Note that {prism} (a) glide denotes
glide on the prism planes of dislocations with <a>Burgers vector. Calculations
using the Taylor-Bishop-Hill
analysis (Fig. 4d) show that the c-axis will
rotate from its original position towards the axis of shortening for most
initial orientations.
However, rather than by a direct movement, the c-axis
spirals asymptotically
towards a position about 25 degrees away from this
axis, and continually
swivels as deformation
proceeds.
For illustrative
purposes the rate of travel shown (Fig. 4e, f,g) is about twice that calculated.
The actual shape of the grain has undergone
no rotation
whatsoever,
although the crystallographic
axes are continuously
rotating.
Because of the finite number of available glide systems, and their relative
disposition, the crystallographic
axes tend to rotate towards special orientations relative to the axes of the imposed incremental
strain tensor. It is a
result of this bias in the totality of the lattice rotations in the grains of a
polycrystalline
aggregate that a strong crystallographic
fabric can develop
purely as a result of deformation
by intra-crystalline
slip. Most attempts so
far to model this development
have concentrated
on the prediction
of
rotationally
stable end orientations in imposed coaxial deformation
histories
(Srx = 0), but a more general approach is taken here in order to deal with
geologically more interesting problems.
THE TAYLOR-BISHOP-HILL

MODEL

This model is formulated here from the explicit and implicit assumptions
made in the models presented originally in the papers of Taylor (1938a,b),
Bishop and Hill (1951) and Bishop (1953,1954).
The model incorporates
the following assumptions:
(1) That the deformation
takes place solely through dislocation glide and
that this can be treated in terms of the simultaneous operation of a number
of discrete glide systems, each of which can be approximated
as a continuous simple shear relative to its glide plane, as discussed above.
(2) That the deformation
occurs uniformly throughout the polycrystalline
mass at all stages, that is, in each grain or volume element the imposed

Fig. 4. Glide causes crystal axes to rotate rapidly with respect to the grain shape. The
crystal axes are not tied to any particular set of material points. Card-deck type analogs
usually show the normal to the glide plane rotating asymptotically towards the axis of
compression (sequence a, b, c). In model quartzite II c-axes usually rotate towards a 25
degree girdle, and from then on continuously swivel around the compression axis
(sequence e, f, g drawn at a rate about twice that actually calculated - d - for illustrative purposes).


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