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Methods for determining deformation history for chocolate tablet boudinage with fibrous crystals
Tecronophysics,
211
92 (1983) 21 l-239
Elsevier Scientific
Publishing
Company,
Amsterdam
- Printed
in The Netherlands
METHODS FOR DETERMINING DEFORMATION HISTORY FOR CHOCOLATE TABLET BOUDINAGE WITH FIBROUS CRYSTALS
M. CASEY,
D. DIETRICH
and J.G. RAMSAY
Geologisches Institut der ETH, CH 8092, Zurich (Switzerland) (Received
September
1, 1982)
ABSTRACT
Casey,
M., Dietrich,
D. and
chocolate
tablet boudinage
Processes
in Tectonics.
Chocolate localities
tablet
were studied.
a succession
Britain,
with fibrous
In one, from Leytron,
of plane strain and fiite
J.G.,
1983. Methods
increments
direction
strains
crystal
strain increments.
determining
with the shortening
showing
were evaluated.
deformation
history
and S. Cox (Editors),
between
a progressive
the boudinaged
the deformation
direction change
for
Deformation
The other specimen,
strains
which allowed the simulation
of chocolate
was applied
specimen
results.
from
two
to the boudinaged within
from Parys Mountain,
with diachronous
gave inconsistent
plates
history was found to be
perpendicular
in orientation
the sheet. The Anglesey
break up of the competent
It was found that under these circumstances
finite and incremental to the Anglesey
growths
Valais, Switzerland,
was found to have a more complex history
~atte~ng
for determining
In: M. Etheridge
Tectonophysics, 92: 21 l-239.
boudinage
sheet and the extension incremental
Ramsay,
with fibrous crystals.
the direct graphical
A numerical
model
tablet structure
with a complex
deformation
and three possible
strain histories
for this structure
Great
layer and methods
of
was developed
history.
The model
were tried.
INTRODUCTION
Boudinage of a competent layer in various directions may lead to “tablette de chocolat” structure (Wegmann, 1932). This structure arises during finite deformations of flattening type (e, Z e2 > 1 > e3, where e,, e2 and e3 are the principal finite elongations). The spaces between the boudins are often filled with fibrous growths of quartz or calcite and the fibres are frequently curved and usually show no evidence of internal deformation. The effect of deformation on the competent layer is a breaking up of the sheet into small plates and the separation of these plates. The motion of the plates is rigid body translation and rotation which is reflected in the geometry of the fibres. The object of the work is to deduce a deformation history for the rock above and below the boudinaged layer which could have given rise to the observed geometry of the 0040- 195 l/83/~-~/$03.~
0 1983 Elsevier Scientific
Publishing
Company
212
structure, and to develop methods tectonic deformation sequences. In the following examples
sections geometrical
of chocolate
Parys Mountain,
tablet
Anglesey,
lie close to the X-Y overall
shortening
applicable
structure
analyses
direction
of the finite
of regional
are carried out on particularly
from Leytron,
Great Britain.
plane
to the determination
Valais, Switzerland,
good
and from
The planes of the fibres of both examples strain
ellipsoid
in the rock. Incremental
and perpendicular
and finite strains
to the
are worked
out for both examples. For the Anglesey structure the problems of using the fibres as displacement paths of the separating plates are shown. A numerical model of chocolate tablet structure is developed order to simulate the fibre geometry.
and applied
to the Anglesey
structure,
in
There was considerable interaction among the authors during the course of the work, but each author is responsible for a particular part of the paper: The analysis of the Leytron specimen
by
GEOMETRIC
specimen
was carried
DD, and MC performed ANALYSIS
out by JGR and that of the Parys Mountain the computer
OF VEIN SYSTEMS
ARISING
modelling.
FROM SUCCESSIVE
PLANE
STRAIN
INCREMENTS
The example is illustrated in Fig. 1 and was taken from deformed Liassic slates at Leytron (Badoux, 1963, 1972). This locality is situated in the root zone of the Morcles nappe, one of the tectonic units forming the Helvetic nappes of the Swiss Alps. The surface shown in the diagram illustrates the appearance of a stretched
Fig. I. Quartz-calcite
vein system from Leytron,
VaIais, Switzerland.
unstippled. The area shown measures IS0 by 200 mm.
Quartz is stippled,
and calcite
213
sheet of relatively coarse-grained limestone in a matrix of calcareous shale. The limestone layer was clearly more competent than its host and now consists of a number of detached microboudinaged “islands”, separated by fibrous vein crystals. The veins are of composite type (Dumey and Ramsay, 1973), consisting of a central zone of fibrous quartz and edges of fibrous calcite. The long axes of the quartz fibres are sub-perpendicular to the vein walls and they grow away from a centrally located median line. The long axes of the calcite fibres at the contact of the vein and the boudinaged host are also sub-perpendicular to the vein walls. Along the quartz-calcite interface the fibres of both crystals are oblique to the wall and both crystal species increase slightly in size towards their common interface. There is a smooth linkage of long axis orientations across the vein and the fibres trace out a double sigmoid form. This geometry is consistent with a simultaneous progressive growth of quartz and calcite, with new material being added along the quartz-calcite interface. The incremental strains for a particular orientation (or more exactly for a 10” range of orientation) were determined by measuring the sum of the fibre lengths in this orientation, and finding the proportion of this incremental extension to the length of wall rock and earlier deposited fibre in the same direction. In order to make the calculations isogons were drawn on the fibre directions, measurements made in several parallel traverses across the specimen and the results averaged. The data are plotted in Fig. 2. There is a systematic change in the amount of incremental strain with direction. The geometry of these changes is consistent with the progressive superposition of strain ellipses with differing maximum extension values, but with no change in length along their minimum extension axes. For example, there are no fold-like instabilities developed in directions normal to the fibres as would be expected in the case of increments of constrictive type, and there are no geometric data supporting simultaneous growth of fibres in several directions as would be expected if the increments were of flattening type. In three dimensions the total strain appears to be built up by progressive superposition of plane strain ellipsoids
Fig. 2. Plot of ~crement~
strain against orientation for the vein system of Fig. 1.
214
with the maximum incremental superposition
axis (e,)
aligned
becomes
and the intermediate
in the plane
of these incremental
the finite strain determine
strain
ellipsoid
plane
a true flattening
compute
strain
ellipsoids
deformation.
graphical
restorations
the successive
strains
directly
of each
in a non-coaxial Two methods
manner
were used to
and also determine
stages in the strain history.
of the boudinaged
axis (e,)
vein. As a result of the
the value of the total finite strains on this surface,
values of the strain state at different to make
strain
of the stretched
fragments,
from the strain increment
the
The first method the second
is
is to
measurements.
Method I, progressive finite strain There appears and it is possible
to be very little internal deformation in the limestone fragments to find the original form of the present boudinaged aggregate by
reassembling them in a jigsaw puzzle manner. By matching their shapes, and by restoring the fragments utilising the fact that any curved fibre links two points that were once in contact, it is possible to make a convincing reassembly (Fig. 3a). On
Fig. 3. Sequential reconstruction the deformation. deposited
of the vein system. a. Before deformation.
b and c. Intermediate stages of
d. Final deformed state. The unstippled areas in b, c and d represent the fibre material
in the preceding increment,
and the areas indicated with vertical ornament
material added during the incremental extension.
show new vein
215
this reassembly two circles were drawn. By placing the particles in their final positions, or by restoring them to an intermediate displacement position, it is possible to obtain the average shape of the finite strain ellipse or any intermediate strain ellipse respectively (Fig. 3b, c, d). In this example the total finite strain values are e, = 1.7 to 2.2, e2 = 0.1 and the orientation of the long axis is 8’ = - 13”. By studying a succession of reconstructions at differing stages during the deformation (two intermediate steps are shown in Figs. 3b and 3c), the complete evolution of the vein system becomes very clear. Method 2, incremental strains The second technique for determining the total strains consists of determining the incremental deformation gradient matrix for each of the incremental meas~ements, multiplying the successive matrices together to obtain the successive finite deformation gradient matrices, and from this to compute the principal total strains. For an incremental strain with values e, = ei, ez = 0 and orientation 8’ the incremental deformation gradient matrix in the reference coordinate system is given by: ai
bi
(1 + ei ) cost@’ + sin2B’
ci
d, = H
ei cos 8 sin 8’
ei COS 8’ sin 8’
cos26)’ + (1 + ei) sin28’ 1 Successive matrix multiplications are then made (Ramsay, 1967, p. 324) and the total strain computed for the final matrix product (Ramsay and Graham, 1970, p. 790). In the example given here this gives principal strain values e, = 2.37, e, = 0.15 and orientation 8’ = - 9.6’, fairly close to those derived by the somewhat crude graphical construction method. The values of the increments, incremental and total deformation gradient matrices, and progressive and finite strains are given in Table I. i
Fig. 4. Plot of the proportion of quartz to calcite against orientation of strain increment for the Leytron vein.
1.13
1.11
1.03
1.01
1.01
-20
- 30
-40
-50
-60
-70
1.15
1.15
1.05
1.03
1.02
I .05
1.01
1.17
-10
1.20
0
1.18
I .78
10
a,
I .8
8:
- 0.02
-0.01
-0.01
- 0.02
- 0.07
- 0.05
- 0.03
0.00
0.14
data
total strains
bi
strain and deformation
progressive
1.2
ei
Incremental
Data for determining
TABLE I
1.00
- 0.02
-0.01
-0.01
- 0.02
- 0.07
- 0.05 3.25
3.34
3.31
3.29
3.15
2.83
2.50
2.13
1.76
a
I .02 1.02 I .02 1.04
data
0.01
0.03
0.04
0.06
0.08
0.13
0.16
0.16
0.14
b
Finite deformation
strain
1.04
1.02
1.01
1.02
0.14 0.00 -0.03
c,
from incremental
- 0.47
-0.40
-0.37
- 0.32
- 0.23
1.17
1.12
I.10
1.09
1.07
1.03
- 0.05
1
3.31
3.34
3.31
3.26
3.16
2.83
2.5
1.80 2.15
1.02
1-r e,
1.15
1.11
1.10
1.09
I .07
1.04
1.02
1.00
1.00
1 i e,
Finite strain data
I .02 I .02
0.14
0.14
d
0.07
c
data
- 9.08
- 1.60
- 6.92
- 5.86
-4.21
0.03
3.7%
7.35
10
8’
._._
217
One especially for different changes greatly
interesting
incremental
(Durney,
feature stretch
1971; Ramsay,
in excess of calcite,
progressively
decreases
until,
of the Leytron directions,
composite
the proportion
vein systems is that, of quartz
1981). At the start of vein formation
but as vein growth in the last veins,
continues quartz
to calcite quartz
the quartz-calcite is absent
is
ratio
(Fig. 4). Fluid
inclusion geothermometry by Durney has shown that there is a systematic decrease in homogenization temperature during the deformation suggesting a change of solution
chemistry
THE DEFORMATION
with temperature HISTORY
and with time (Durney,
INFERRED
1971).
FROM PYRITE BOUDINAGE
Pyrite layers boudinaged during a deformation episode may be found in deformed sedimentary rocks of low metamorphic grade. Excellent examples of chocolate tablet structure have been found in the Ordovician slates of Parys Mountain, Anglesey, Great Britain. The pyrite had an original sheet-like form, and was probably developed during the period of sulphide mineralization that gave rise to widespread ore deposits in the area (Greenly, 1919). The example investigated is shown in Fig. 5. The
Fig. 5. Micrograph
of the pyrite chocolate
tablet boudinage
Britain. The area shown measures 22 by 15 mm.
pyrite
sheets
from Parys Mountain,
are
Anglesey,
a few
Great
218
millimetres thick. They have been broken into a number of separate flat plates parallel to the slaty cleavage of the rock and the plates are separated by fibrous veins consisting
entirely
from the median pyrite-quartz lattice they
of quartz. surface
interface.
The mode of growth of the fibres is antitaxial.
towards This
misfit. The individual are curved.
quartz-pyrite
They
contacts.
the pyrite plate, new material
is an interface
of major
fibres have a constant
often Certain
show
a progressive
fibres
mechanical
optical
weakness
orientation,
increase
are more suitably
that is,
being added
and
even when
in size towards
oriented
at the
for growth
the and
widen at the expense of less suitably oriented neighbours. The median surface is not always parallel to the faces of the pyrite plates on each side but shows. overall, a relative clockwise Strain induced
rotation. features such as the formation
of deformation
lamellae,
undulose
extinction or recrystallization of early formed fibres are not seen in these fibrous quartz crystals. It is therefore concluded that their curved shape was acquired by growth during progressive deformation. The fibres between the pyrite plates are considered
to represent
the displacement
paths of the spreading pyrite plates, to a first approximation, as were the fibres of the Leytron veins. The following strain analyses were undertaken to show some of the basic properties
of the deformation
of pyrite chocolate
tablets.
Method I, total strain from geometric reconstruction It is easily seen that a geometrical best fit of the pyrite plates, as was used for the Leytron vein system, allows a reconstruction of the undeformed state of the pyrite sheet, Fig. 6. A circle drawn on the best fit initial configuration becomes fragmented to the overall shape of the finite strain ellipse. The half axes of this ellipse are 1 + e, = 1.63 and 1 -t e, = 1.17. For this method see also Durney (1971). This direct,
(a)
Fig. 6. Geometrical a. Undeformed b. Deformed
reconstruction
of the Parys Mountain
state plus circle. A-B state.
connection
boudinage.
of two adjacent
gravity
centres.
219
but somewhat approximate, method of determining finite strain is used as an independent comparison for the other methods applied below. Method 2, finite strain from elongation of lines
The centres of gravity of the individual pyrite plates were determined. The centres of gravity of adjacent plates were then connected with lines in the best fit configuration of Fig. 6, and their lengths (I,) before deformation measured. Their lengths after deformation (1) were then measured and the inverse quadratic elongations calculated as: A’ = (f&)2 The total finite strain in the components of the reciprocal coordinate system having the reciprocal quadratic extension transformation:
plane of the chocolate tablet is defined by the three strain tensor Xi, A; and yi_,,, which is expressed in a x axis parallel to an arbitrary reference direction. The A’ in a direction IY’to the x axis is given by the tensor
A’ = A!!cos2ff’ + 2~4, cos (Y’sin (Y’+ hY sin’& The values of A:, A; and y;,, which give the least deviation of h’ values calculated using the above equation and the measured valu& can be found by a standard least squares procedure. When the conditions for a minimum of the sum of the squared deviations are applied three simultaneous equations for the unknowns are obtained. Once A>, Xi and yiY are determined, A;, A; and 6’ can be found. A; and A; are the principal reciprocal quadratic elongations describing the total finite strain state of the specimen, and 8’ is the angle between the reference line and the direction of maximum extension after deformation. For the technique of determining the principal values and their orientations from a tensor see, for example, Ramsay (1967, pp. 81-82). We have measured A’ and CX’of 38 lines connecting gravity centres of adjacent pyrites. The reference direction was the principal extension direction of the best fit ellipse previously established by the r~onstruction technique of method 1. The resulting total strain is 1 + e, = 1.75, 1 + e2 = 1.22, and 8’ = 0.86”. Figure 7 gives a plot of the results of this method allowing their critical discussion. This graph is based on the relationship (Ramsay, 1967, eq. 3.31): 2 A'=
!y i
=A;
cos'tl'+X;
sin2B’
i
and, consequently: h’=(h;-A;)cos%‘+X;
(11
In a homogeneous strain state plots of A’ against co&P should give a straight line. The data points in Fig. 7 refer to the measured A’ values, which are plotted against
A’ 0.7
0.5
0.3
0.1
l
I
0 Fig. 7. Results measurements
0.2 of method
0.6
0.4
2. Plot of A’ against
of the specimen
08
wsz
8’
cos28’, derived
shown in Fig. 5. Line o represents
b the total finite strain calculated
from the data of method
from 38 gravity
centre
to gravity
the finite strain result of method
centre 1, line
2.
the angle made with the calculated extension direction of the total strain. The two straight lines in Fig. 7 are plots of the resulting total strain of methods 1 and 2, using eq. 1, Line a was constructed from the results of method 1 and line b from the results of method 2.8’ is in this case the angle from the calculated 1 -t e, direction of the total strain. The results of the two methods correspond well. In particular the extension directions of the total strain of the two methods are quite similar. Although both lines are very close to the data points, there is a wide dispersion of the points which shows that it is inaccurate to consider that the centres of gravity of the spreading pyrite plates moved as independent points displaced by a homogeneous strain. The best interpretation of the data spread is that the break up of the pyrite plates probably occurred progressively during the deformation, so that not ail the distances between adjacent pyrite plates will be simply related to the geometry of a homogeneous strain. Another explanation of the variable data dist~bution could be that the total strain was heterogeneous. We did not observe any differences in overall data trends from one side of the specimen to the other and therefore conclude that the assumption of homogeneous strain is justified. Method 3, progressive finite strain
This method (termed the pyrite method by Durney and Ramsay, 1973) assumes that the strain state in the X-Y section was achieved by a succession of infinitesimal increments of plane strain and that the orientation of the fibres is parallel to the incremental extension directions. Individual curving fibres were divided into a
221
number of straight finite increments. The strain for each increment was evaluated by dividing its length by the sum of the lengths of previously formed increments resolved into the direction of the current increment plus the pyrite radius. The finite strain was calculated by superposition of the incremental strains. Justifications for the application of this method to the example of Fig. 5 are: (a) the absence of deformation features in the fibres; (b) adjacent fibres, as well as fibres from differently oriented faces, are more or less parallel, hence the assumption of a deformation history consisting of plane strain increments seems acceptable. Sixty fibres distributed over the whole specimen were measured, the arbitrarily chosen reference direction being that of the long axis of the ellipse of method 1. The progressive strain for each fibre was calculated in the relative direction. These strain data were then used to establish the overall strain state of the specimen, using the least squares technique outlined above. The results are: 1 + e, = 1.60, 1 + e2 = 1.01, 8’ = 0.46’. The directions of the principal strains correspond well for the three methods, but the total finite strain value of method 3 is lower than the values of methods 1 and 2. On Fig. 8 the fibre strains are plotted against their angle with the calculated total finite extension direction. Line c was ~onst~cted from the calculated total finite strain, applying eq. 1. The fibre strain data show considerable spread. There are several factors responsible for this spread: the pyrite method assumes that fibres which have grown during the same interval of incremental strain have the same direction. This means that an isogon plot of fibre directions corresponds to a map of isochronous lines. Figure 9 shows an isogon plot of the specimen and Fig. 10 shows
t 0
1..
.
02
0.4
*
*
OS
.
f
Qd
.
-
caG8'
Fig. 8. Results of method 3. Plot of h; and hi against cos*B’, derived from 60 fibres of the specimen shown in Fig. 5. Line c represents the total finite strain, calculated from the data of method 3.
Fig. 9. Isogon plot of the Parys Mountain numbered
Fig. IO. Stages in the spreading constructed formed
specimen.
The orientations
of the fibres corresponding
to the
regions in the figure are given by the key in the top right of the diagram.
from the isogons
during
of three of the pyrite plates of Fig. 5. The eight increments
of Fig. 9, according
the same time interval.
to the assumption
shown are
that fibres of the same orientation
223
the individual steps of spreading using the isogons as marker lines for the increments. A comparison of Fig. 10 with the corresponding part of Fig. 5 shows that the model which describes the deformation history as sequence of plane strain increments gives an acceptable result for the three pyrite plates considered. Between the first and the last increment of Fig. 10 there is an anticlockwise rotation of the three pyrite plates, as well as a separation of plates 1 and 3 after the first four increments only and a breaking apart of plates I and 2 after the first six increments. Using the fibre pattern between the plates it is possible to construct a more or less symmetric, if simplified, isogon plot. If, however, a more complicated part of the specimen is considered as, for example, the borders of Fig. 9, it is evident that the isogons do not correspond in every case to previous positions of the plates: isogons are not always isoehrons. A consequence of this observation is that during the same time interval the fibres can grow in different directions. This feature arises when the incremental strains are of a truly flattening type instead of a series of non-coaxially superimposed plane strain increments. The dispersion of points in Fig. 8 seems best explained by a flattening incremental history and, in parts of the sample, by the diachronous breaking apart of the pyrite plates. Discussion
Methods 1 and 2 give a very similar total finite strain, but method 3 gives appreciably lower values. The low 1 -t- e2 value of method 3 is probably a further indication that the deformation history is progressive flattening and not a sequence of non-coaxially superimposed plane strain increments. Method 3 also gives a slightly lower 1 + e, value than those calculated from methods 1 and 2, notwithstanding that method 3 and method 2 give theoretically the same result for 1 + e,, if one of the straight lines of method 2, which connects two gravity centres, is divided in increments and the incremental strains are accumulated. This shows that the strain calculation from an individual fibre can give only a fair estimate for the total 1 + e, in the corresponding direction, and that the total 1 + e2 from an individual fibre cannot account for true flattening deformation. Line c in Fig. 8 could be drawn considering only the Xi values and would then lie closer to lines a and b of Fig. 7. In all the preceding analyses it has been assumed that the fibres define the displacement paths of the separating vein walls or pyrite plates. This assumption is not strictly correct for the pyrite chocolate tablets. Figure 11 shows that when the pyrite plates are reassembled on the basis of their shapes, the ends of the two fibres which grew from the same point on the median surface do not exactly coincide. One possible explanation for this is that there is often a tendency for the fibres to grow perpendicular to the pyrite surface as in the case of pyrite cubes (Pabst, 1931; Durney and Ramsay, 1973). Another explanation is that wrench-fault-like shear motions took place along the pyrite-quartz interface during deformation to allow
224
for compatibility
between
the fibre geometry, In conclusion
Fig. 11. Geometrical
the rigid plates. These displacements
and they are responsible
it is possible
reconstruction
to formulate
are not registered
in
for the lower total finite strain obtained. a list of properties
of the deformation
of three pyrite plates. The points lettered A- I represent
ends of fibres which grew from the same point on the median
of
points at the
surface.
pyrite chocolate tablets which must be incorporated reproduce the fibre geometry:
in any model which attempts
to
(1) The pyrite plates can be considered as rigid bodies moving in a plane. Their final position partially reflects the total strain history, but it is also a function of the diachronous break-up history of the rigid sheet into its component plates. (2) A total flattening strain can be built up in an infinite number of ways ranging from the superposition flattening increments. histories
of successive plane strains to the superposition of successive The model should explore the consequences of different strain
on the resulting
fibre geometry.
(3) Compatibility problems between the spreading fibre mass would be revealed in a model. (4) Fibre growth is continuous and therefore
plates and the already the
strain
history
formed
should
be
considered as being continuous rather than a series of finite increments. (5) Rotational components of the deformation can be evaluated from the relative positions of the pyrite plates and from the curvature of the fibres. Rotation sense is defined relative to a particular reference frame. The pyrite method (method 3) uses the fibres as fixed reference directions and in the case of the analysis carried out here an overall anticlockwise rotation sense of the pyrite plates results (Fig. 10). When the reference direction was fixed in either of plates 1 or 2 (Fig. 12) the other two show a relative anticlockwise rotation sense (cases a and c in Fig. 12) whereas if it was fixed in plate 3, the other two show a relative clockwise rotation.
Fig. 12. Superposition of three pyrites from the undeformed and deformed state of the specimen to show differential rotation. In a plate 1 is used as the reference, in b plate 3 is the reference and in c plate 2 is the reference.
SIMULATION
OF FIBRE GEOMETRY
The basic geometric features of the naturally deformed chocolate tablet structure set out in the last section were used to control a numerical simulation study. In the following sub-sections the mathematical basis of the model is set up, details of computer based solutions are given and some examples of its use are described. Specification of the deformation history
Deformation histories may be described in two ways. One way is to consider the deformation as being made up of a sequence of small, but finite, increments and combining these by matrix multiplication (Ramsay, 1967). An example is given in eq. 2:
(E~j=(~
Y+ej”=(il
I)l+e)“)
226
gradient matrix ( :’ i; ) has been y+, ) which corresponds to uniaxial works well for simple increments such as the one given.
where the finite strain, described formed extension The calculate
by the deformation
from n steps of the small increment in y. This method alternative
approach
is to specify
its effect by integrating
berg, 1975). The velocity
(d
gradient
the velocity
over the time interval
gradient
matrix
of the deformation
and
to
(Ram-
matrix vk,, is given by eq. 3:
\
*k
(3)
.I
The velocity gradient matrix can be decomposed into a symmetric and antisymmetric part. The symmetric part is the deformation rate tensor, d,,:
an
au
_;z
ax d,,=
(4)
which describes the rate of change of shape of the material. The antisymmetric part is the spin tensor wk,:
which describes the rate of rigid rotation. The simultan~us effect of several velocity them. No approximation is involved. In general the velocity gradient matrix
I Fig. 13. The reference
gradient
is a function
matrices
is given by adding
of time. One simple way in
X system
given in x’, y’ is constant
x’. y’ rotating
with time.
with respect
to system
x, y. The velocity
gradient
matrix
227
which
this variation
in time, t, can occur
system, Fig. 13, the strain rate, represented the x’, y’ coordinate according e(t)
is as follows.
In the x’, y’ coordinate
by the velocity gradients,
system is made to rotate
relative
to the x, y reference
but
system
to eq. 6:
= e, + LYi
(6)
The value of uk,,(t)
in the reference cos
I
is constant,
au,(t)
a+) ax
ay
6y1)
=
I
system x, y is given by eq. 7: \ -sin f?(r) case(t)
sine(t)
I /
cos
e(t)
sin e(t) (7)
-sin
e(t)
COST
where vi ,I is the velocity gradient matrix of the x’, y’ coordinate system. The deformation at any instant in the surrounding rock is defined by the velocity gradient matrix the deformation Eringen
(eq. 3). The relationship between the velocity gradient matrix and gradient matrix (F f;) is given by eqs. 8, which were adapted from
(1965, p. 70, eq. 2.4.1):
da=av,a+av,c dt ax
dc 3% z=zu+ayc
ay
au,
This is a set of simultaneous ordinary differential equations which can be solved numerically by the fourth order Runge-Kutta technique (Ralston and Wilf, 1960, ch. 9). The solution will give the values of the deformation gradient matrix at equally spaced intervals in time. From these the usual parameters of finite strain X,, X, etc. can be calculated (Ramsay from natural examples.
and Graham,
1970) for comparison
with measurements
As will be shown below, the motion of a rigid plate embedded in a deforming medium can be related to the velocity gradient matrix. The use of the velocity gradient matrix brings time into consideration. In numerical simulation this does not present any problem. In natural examples the length along a reference fibre can be
228
used to represent necessarily
time (Elliott,
represent
1972) but equal lengths
of the reference
fibre do not
equal times.
The motion of a rigid plate embedded in a deforming
material
To simplify the determination of the motion of a rigid plate in a deforming some assumptions are made. The first is that the plates interact mechanically
rock with
the surrounding material across the surface only and that the effects of the deformation at the edges are negligible. This assumption is reasonable because the surrounding rock does not flow around the ends of the plates in the case of chocolate tablet boudinage. The second is that the matrix exerts a force on the plate which is proportional adjacent matrix. This
to the relative velocity of points in the plate and in the assumption, though impossible to justify without further
analysis, is adopted because the force will certainly increase with increasing velocity and a linear relation is easiest to treat mathematically. The velocities /
of points
in the surrounding
relative
rock are given by eq. 9:
\ f-)x i=
i
*Y
(9) -ax
where the right-hand
3.Y
,Y
side contains
rock. The rigid plate can respond axis at x0, y,. The velocity
the velocity
gradient
to the deformation
matrix
of the surrounding
only by a rotation,
o, about
an
of a point in the rigid plate is given by:
Thus the parameters describing the motion of the plate are x0, y, and w and the problem is to find values of these parameters which give the minimum work rate of the forces exerted by the surrounding rock on the plate. The work rate is equal to the force times the velocity and as the force is assumed to be proportional to the velocity the work rate will be proportional
to the square of the relative
velocities,
u:. 0::
where the integral is carried out over the surface of the plate. Subtracting eq. 9 from eq. 10 to give an expression for the relative velocity and substituting this expression in eq. 11 gives:
229
To find the minimum of this function with respect to x0, y0 and w the function is first differentiated with respect to each of the three parameters and the resulting expressions are set equal to zero.
(13) (14)
-a ah
(15)
w can be cancelled from eqs. 14 and 15:
(16)
ds=O $-w)(y-yO)-
(~~+~y)}
(17)
ds=O
Expand eq. 16 and remove constant terms from the integrals: w xds-ox, I
I
ds=z
(18)
jxds+%/yds
Expand eq. 17 in the same way: -w
/
yds+wyO
/
ds=%/xds+au”jyds
(19)
ay
Expand and reorganize eq. 13: --w
/
(y2+x2)ds+2wy0
I
yds-
oy,2
ds - z/x*
Rearrange eq. 18 with x0 on the left:
X0=
a0 yax / xds+- 2 --w
j yds -wfxds
I
ds
ds+2wx,
J
xds-ox,2
I
ds
ds}
z/xds+%/y
}-x0{ jy’
/
ds
(20)
230
=-
A-wB
(21)
--WC i3V
A=’
3X
s
8V
xds+-I:
Rearrange
aY s
yds,
B=
s
xds,
C=
s
ds
(22)
eq. 19 with Y, on the left:
-wjyds-
[2jxds+%jyds)
Yo = --w
J
ds
-wD-E
=
(23)
--UC E=zjxds+au*jyds,
D= 3Y
/
yds
Substitute for x0 and y0 in eq. 20, multiply to eqs. 22 and 23:
J(y2 + x2)
cw2
(24) by WC and make substitutions
according
ds - 2( w2D + wE) D -t- {w2D2 + 2wDE -i- E2}
-2( Bo2 -Aw)B+{A’-2wAB+02B}-(uD+E)E-(A-Bti)A
Gather
terms in w and substitute
(y2+x2)ds-D*-B2
-(
for the integrals:
1--w i
DE-BA-C[($+)/,,,,
(25)
+~/y2ds-~/x2dslj_0
The roots are:
DE-BA-C[[~-~)/xyds+$y2ds-~/x2dj w=O
or
w= q(Y’+
x2) ds - D2 - B2
(26) Ignoring
the w = 0 root and resubstituting
for A, B, C, D and E:
231
-jds[(%-!$)j X {j
dsj(y2+x2)
Substituting gradient
ds-(jyds)‘-
deformation
matrix
(jxds)2}j’
rate and spin terms
using eqs. 4 and 5 and rearranging
for the elements
of the velocity
gives:
w= (jyds(d,,jxds+d,,jyds)-jxds(d,,,jxds+d,jyds) -
ds
(d,,
j
- d,,) jxy
ds + d,, j( y2 - x2) ds
i
I)
x{jdsj(y2+x2)ds-(jds)2-(jxds)2}j’+~2,
(27)
The values of x0 and y, can be obtained by back-substitution in eqs. 21 and 23. If both solutions are zero, then the motion of the plate is just a rigid translation of velocity u,, uY. The relative velocities are given by:
au
*,‘=*Y-
i
a~
dx+Ly ax
The following
au,
1
ay
equations
give the conditions
(28) for zero net force on the object:
au,
ux,,x+,,y
ds=O )
au,
au,
uY--x+-y
ay
j( Rearranging:
a*, xds+- a*, yds
ax
u, =
J
ay j
(29) / au, ax
VY=
j
xds+-
ds an, ay j
yds (30)
J
ds
Under special circumstances only will it be necessary to use eqs. 29 and 30, that is, if the instantaneous strain is a &axial extension and the rigid object is of equant shape.
232
Rotational
deformation
and rigid objects
Equation 27 shows that the rotation rate of the rigid plate is determined by a factor representing the interaction of the shape of the object and the deformation rate of the imposed
deformation
effect of the spin of the imposed
and the spin of the imposed deformation
is independent
deformation.
The
of the shape of the
plate. If the motion of a disc is considered it can be shown that the first term on the right-hand side of eq. 27 is zero. The disc will therefore rotate at an angular velocity equal to the spin of the matrix. To explore some of the consequences of this result consider the effects of a shearing deformation. The reference systems and the deformation are shown in Fig. 14. The spatial
reference
system x, y serves to define
the imposed
deformation.
A
reference system x’, y’ is attached to the rigid object in such a way that it rotates with the object and the coordinate directions are always parallel to the same material lines. The deformation
The rotation
in the X, y frame is:
of the object relative
3%
@=
I/*%
to the x, y reference
system is given by: (32)
The imposed deformation (31) is constant in the X, y reference frame so that the principal extension directions are also constant in this system. The rigid object rotates o relative to X, y so that the orientations of the principal extensions rotate in the x’, y’ coordinate
system.
Fig. 14. A round rigid object in a rotating matrix deforming by sinistral simple shearing. The coordinate system x’, y’ is fixed to the rigid object.
233
Calculation
of the finite fibre geometry
Once the parameters plates,
x,,, y,, and w, which determine
have been obtained
mass can be determined this information
the velocity
of separation
the movement
from eqs. 10 or 29 and 30. To calculate
several
further
techniques
of the rigid
of pyrite plate and rigid fibre
are needed.
These
fibre forms from are:
a means
of
representing the fibres numerically, means of evaluating the integrals in eqs. 21, 23, 27, 29 and 30, and a means of projecting the deformation over a time interval. The basic element in the simulation of the development of chocolate tablet structure
is to analyse
mass between. The pyrite geometry,
plates
although
the relative motions can finite
be represented element
analysis
of two pyrite plates and the growing fibre by finite
element
meshes
is not used to obtain
of constant
a solution
of the
deformation. The fibre mass can be represented by a finite element mesh to which new rows are added on each side at each time step. At the start of the deformation an embryonic fibre mass is inserted as a single row of elements between the pyrite plates. On each side of the initial row of elements a row of elements of zero width is set up, one set of nodes being attached to the pyrite. During the time step the zero width elements are allowed to widen to take up the displacements between the fibre mass and the pyrite. At the end of the time step another row of zero width elements is added ready for the next time step. The integrals can be performed over each element and the contributions of all elements summed to give the totals. The method of calculating the integrals for each element was taken from Zienkiewicz (197 1, pp. 144- 149). Gauss quadrature was used with four integration points in each element. This allows the accurate integration of second-order functions such as xy and x2. To step through
time the fourth-order
and Wilf, 1960, ch. 9). This involves step. In the present coordinates velocities.
case the variables
Runge-Kutta
four evaluations of the ordinary
of the nodes of the finite element
method
was used (Ralston
of the gradient differential
in each time
equations
mesh and the gradients
are the
are the nodal
Results of simulations Simulations were performed to assess the properties of the model and to attempt to reproduce the fibre geometry of a portion of the chocolate tablet boudinage structure from Parys Mountain, Anglesey, Great Britain, shown in Fig. 5. The portion chosen for study comprised three pyrite plates with their intervening fibre masses. The undeformed geometry of the model, derived from the best fit reconstruction of the pyrite boudins (Fig. 11) is shown in Fig. 15. In all the simulations pyrite plates I and 2 were held .together for the first quarter of the deformation, the fibre masses between plates 1 and 3 and 2 and 3 were
234
connected histories principal amounts
and that between
plates
chosen there was a clockwise strain
rates
of flattening
were varied
the relative
from
run
to run
to investigate
strain rate. The results and strain histories
16, 17 and 18. There is considerable although
f and 2 was mechanically separate. In all the rotational component. The ratios between the
positions
variation
the different
are shown in Figs.
in fibre forms produced
by the models
of the pyrite plates are quite close in all cases. One
feature of the natural example of Fig. 5 is the frequent parallelism of fibres on opposite sides of the median surface, shown particularly clearly in the isogon plot of Fig. 9. This feature is also observed in the simulations, especially when the deformation is an extension in x with little change in J’ (Fig. 16). The strongest flattening deformation gives the greatest departure from parallelism of fibres on opposite sides of the median surface (Fig. 18). The general motion of a rigid plate is a rotation about a pole, rather than a simple translation. The rotation motion results in non-parallel fibres forming at the same time on the same pyrite which is opposite
surface.
This is shown
in Fig. 17b on the face of plate 2
plate 3 and in the isogon plot of the natural
example
of the plates in the right-centre of Fig. 9. Differing growth rates of the fibres on each side of the median Fig. 16d. This is related
on the sides
surface is shown in
to the size of the pyrite plates and can be seen also in the
natural example (Fig. 5). A major aim of the simulations was to reproduce the observed pyrite plate positions and geometric features of the fibres of the natural example in order to estimate the deformation history in the plane of the sheet. The median surface between plates 1 and 3 in the natural example does not connect with a median surface between plates I and 2. From this it was thought that plates I and 2 separated later than I and 3 and 2 and 3 and the models were set up accordingly. Broadly comparable results could be obtained especially in the model with very
Fig. 15. The undeformed
configuration
Compare with Figs. 10, 11 and 12.
of the model with reference
numbers
for the pyrite plates.
235
little extension in y (Fig. 16d). The positions of the pyrite plates are comparable as are the curvatures of the fibres. The position of plate 2 in the model is not in good agreement with its position in the natural example and hence the assumption of diachronous breaking of the pyrite may not be correct. Qualitative comparison of the results of simulations (Figs. 16 18) and the natural sample (Fig. 9) indicates that the best agreement would occur for a time just later than that of stage c. Reference to Table II gives a value for 1 + e, of around 1.8 and
Fig. 16. Computer model A. Equally spaced stages in the development
of the model. Defo~ation
history:
Velocity gradients of: o.l~lo-’ ( -o.l.10-7
0.1. IO--’ o.l*lo-‘”
for eight steps of 0.1.10’
1 set:
d,,=0.1~10-', dyy= 0.1. IO-lo,
d,,= 0.. w2,= -0.1.
lo-’
The circle and ellipse above the diagrams represent the increment of strain from the previous diagram. The circle and ellipse below each diagram represents the finite strain. The median surfaces are defined by the jogs on the fibres.
236
Fig. 17. Computer model B. Equally spaced stages in the development of the model. Deformation history: Velocity gradients of: o.1.10-7 i -0.5.10-s
0.5. 1o-8 o.1~10-8 1
for eight steps of 0.1 IO8 set: d,,=O.I.lO-‘,dyy=O.l.lO-s,dxy=O.,W*,=
-0.5.10-s
Circles and ellipses are the same as for Fig. 16.
varying values for 1 + e2 (1.0178- 1.2119). The results of strain analysis method 2 (Fig. 7) indicate that the highest value (1.2119) is the most likely. The value for 1 + e, is in good agreement with the direct methods of strain estimation. In none of the models was it possible to obtain the correct rotation of the fibre masses as a whole relative to the pyrite plates. This may be the result of not having the fibre masses joined correctly or that the deformation history was one in which the extension direction changes in the spatial reference frame (x, y). The computer program cannot calculate the effects of this kind of deformation history at present.
231
Fig. 18. Computer model C. Equally spaced stages in the development of the structure. Deformation history: Velocity gradients of: 0.1*10-7 0.1.10-7 0.3.10-* 1 ( -0.1.10-7 for eight steps of 0.1 f 10’ see: d,,=O.l~lO-‘,
d,=0.3.10-*, d,,=O., y,= -O.l.lO-'
Circles and ellipses are the same as for Fig. 16.
The rotational component of the imposed deformation causes the three pyrite plates to rotate relative to the reference coordinate system as can be seen from Figs. 16-18. Relative rotations of the pyrite plates do occur but these are much smaller
238
TABLE
II
Finite strain data for the steps of the three computer
simuIations
Model A
step 1.22
1.0008
- 5.73
- 11.47
2
1.48
1.0156
- 11.19
- 22.63
4
1.79
1.0178
- 16.66
- 33.44
6
2.14
1.0407
-21.87
- 43.80
8
1.22
1.0203
- 2.56
- 5.73
2
1.49
- 5.73
- 11.35
4
-8.51
- 16.78
6
2.21
I .042 1 1.0660 1.0932
- 10.89
- 22.01
8
I .22 l.49
- 5.73
- 11.42
2
I.1316
- 11.48
- 22.78
4
1.80
1.2119
- 16.86
- 33.90
6
2.16
1.3076
- 22.33
-44.80
8
1.81
1.0623
stage
than the general rotation. This is a consequence of eq. 32 which shows that an equant plate rotates at the vorticity of the matrix which, in the cases presented here, was greater than the rotations produced by the non-equant shapes of the pyrite plates. CONCLUSIONS
(1) When the boudinaged plates move apart by a simple translation, without rotation, as in the case of the Leytron vein, the finite strain at the end of the deformation and at stages in the development of the structure can be estimated from the fibre geometry and reassembly of the plates with acceptable accuracy. (2) When the separation of the plates and the intervening fibre masses is accompanied by relative rotations, as in the Parys Mountain specimen, the calculation of the incremental and finite strains from the fibre geometry under the assumption that the strain history consists of plane strain increments is not strictly accurate. Fibre increments which were deposited during the same time interval do not always have the same orientation and pyrite plates may break apart during the deformation. (3) A model for the simulation of chocolate tablet structure can be set up if certain simplifying assumptions are made about the mechanical interaction of the rigid plates and the surrounding rock. The model is capable of reproducing many features of the natural example from Parys Mountain. The model also shows that the rigid rotation rate of an equant plate is equal to the vorticity of the deformation in the surrounding rock.
239
ACKNOWLEDGMENTS
The authors are grateful to D.W. Durney for extensive discussion of one of his computer programs and to SM. Schmid for valuable text. Financial 5.521.330.785/l,
and for the use criticism of the
support is acknowledged as follows: MC Nationalfonds DD ETH project 0.330.079.27/8, JGR Nationalfonds
project project
no. no.
Sci. Nat.,
68:
5.521.330.622/8. REFERENCES
Badoux,
H., 1963. Les belemnites
tron9onCes
de Leytron
(Valais).
Bull. Sot. Vaudoise
233-239. Badoux,
H., 1972. Tectonique
de la nappe
de Morcles
entre Rhone
et Lizerne.
Mat. Carte Geol. Suisse,
NS 43: l-78. Durney,
D.W.,
1971. Deformation
Thesis, Univ. London, Durney,
history
London,
D.W. and Ramsay,
J.G.,
K.A. De Jong and R. Scholten Elliott,
D., 1972. Deformation
Eringen, Greenly, Pabst,
Helvetic
Nappes,
Valais,
1973. Incremental
strains
(Editors),
and Tectonics.
Gravity
paths in structural
geology.
measured
by syntectonic
Ph.D.
Wiley, New York, N.Y., 502 pp.
E., 1919. The Geology
of Anglesey.
Mem. Geol. Surv., 2 vols., 980 pp.
shadows
crystal
growth.
In:
Geol. Sot. Am. Bull., 83: 2621-2638.
of Continua.
Mineral.,
Switzerland.
Wiley, New York, N.Y., pp. 67-96.
A.C., 1965. Mechanics A., 1931. Pressure
Ralston,
of the Western
327 pp. (unpublished).
and the measurement
of the orientation
of minerals
in rocks.
Am.
16: 55-70.
A. and Wilf, H.S.,
1960. Mathematical
Methods
for Digital
Computers,
Vol. 1. Wiley, New
York, N.Y., 293 pp. Ramberg,
H.,
1975.
Tectonophysics,
Particle
Ramsay,
J.G., 1967. Folding
Ramsay,
J.G., 1981. Tectonics
and Nappe Ramsay,
Zienkiewicz, PP.
Tectonics.
J.G. and Graham,
Wegmann,
paths,
displacements
and
progressive
strain
applicable
to
rocks.
28: l-37. and Fracturing of the Helvetic
Geol. Sot. London, R.H.,
of Rocks. McGraw-Hill, Nappes.
O.C., 1971. The Finite Element
(Editors),
Thrust
pp. 293-309.
1970. Strain variation
A., 1932. Note sur le boudinage.
New York, N.Y., 568 pp.
In: N.J. Price and K.R. McClay
in shear belts. Can. J. Earth Sci., 7: 786-813.
Bull. Sot. Geol. France, Method
in Engineering
Ser. 5, 2: 477. Science. McGraw-Hill,
London,
521
211
92 (1983) 21 l-239
Elsevier Scientific
Publishing
Company,
Amsterdam
- Printed
in The Netherlands
METHODS FOR DETERMINING DEFORMATION HISTORY FOR CHOCOLATE TABLET BOUDINAGE WITH FIBROUS CRYSTALS
M. CASEY,
D. DIETRICH
and J.G. RAMSAY
Geologisches Institut der ETH, CH 8092, Zurich (Switzerland) (Received
September
1, 1982)
ABSTRACT
Casey,
M., Dietrich,
D. and
chocolate
tablet boudinage
Processes
in Tectonics.
Chocolate localities
tablet
were studied.
a succession
Britain,
with fibrous
In one, from Leytron,
of plane strain and fiite
J.G.,
1983. Methods
increments
direction
strains
crystal
strain increments.
determining
with the shortening
showing
were evaluated.
deformation
history
and S. Cox (Editors),
between
a progressive
the boudinaged
the deformation
direction change
for
Deformation
The other specimen,
strains
which allowed the simulation
of chocolate
was applied
specimen
results.
from
two
to the boudinaged within
from Parys Mountain,
with diachronous
gave inconsistent
plates
history was found to be
perpendicular
in orientation
the sheet. The Anglesey
break up of the competent
It was found that under these circumstances
finite and incremental to the Anglesey
growths
Valais, Switzerland,
was found to have a more complex history
~atte~ng
for determining
In: M. Etheridge
Tectonophysics, 92: 21 l-239.
boudinage
sheet and the extension incremental
Ramsay,
with fibrous crystals.
the direct graphical
A numerical
model
tablet structure
with a complex
deformation
and three possible
strain histories
for this structure
Great
layer and methods
of
was developed
history.
The model
were tried.
INTRODUCTION
Boudinage of a competent layer in various directions may lead to “tablette de chocolat” structure (Wegmann, 1932). This structure arises during finite deformations of flattening type (e, Z e2 > 1 > e3, where e,, e2 and e3 are the principal finite elongations). The spaces between the boudins are often filled with fibrous growths of quartz or calcite and the fibres are frequently curved and usually show no evidence of internal deformation. The effect of deformation on the competent layer is a breaking up of the sheet into small plates and the separation of these plates. The motion of the plates is rigid body translation and rotation which is reflected in the geometry of the fibres. The object of the work is to deduce a deformation history for the rock above and below the boudinaged layer which could have given rise to the observed geometry of the 0040- 195 l/83/~-~/$03.~
0 1983 Elsevier Scientific
Publishing
Company
212
structure, and to develop methods tectonic deformation sequences. In the following examples
sections geometrical
of chocolate
Parys Mountain,
tablet
Anglesey,
lie close to the X-Y overall
shortening
applicable
structure
analyses
direction
of the finite
of regional
are carried out on particularly
from Leytron,
Great Britain.
plane
to the determination
Valais, Switzerland,
good
and from
The planes of the fibres of both examples strain
ellipsoid
in the rock. Incremental
and perpendicular
and finite strains
to the
are worked
out for both examples. For the Anglesey structure the problems of using the fibres as displacement paths of the separating plates are shown. A numerical model of chocolate tablet structure is developed order to simulate the fibre geometry.
and applied
to the Anglesey
structure,
in
There was considerable interaction among the authors during the course of the work, but each author is responsible for a particular part of the paper: The analysis of the Leytron specimen
by
GEOMETRIC
specimen
was carried
DD, and MC performed ANALYSIS
out by JGR and that of the Parys Mountain the computer
OF VEIN SYSTEMS
ARISING
modelling.
FROM SUCCESSIVE
PLANE
STRAIN
INCREMENTS
The example is illustrated in Fig. 1 and was taken from deformed Liassic slates at Leytron (Badoux, 1963, 1972). This locality is situated in the root zone of the Morcles nappe, one of the tectonic units forming the Helvetic nappes of the Swiss Alps. The surface shown in the diagram illustrates the appearance of a stretched
Fig. I. Quartz-calcite
vein system from Leytron,
VaIais, Switzerland.
unstippled. The area shown measures IS0 by 200 mm.
Quartz is stippled,
and calcite
213
sheet of relatively coarse-grained limestone in a matrix of calcareous shale. The limestone layer was clearly more competent than its host and now consists of a number of detached microboudinaged “islands”, separated by fibrous vein crystals. The veins are of composite type (Dumey and Ramsay, 1973), consisting of a central zone of fibrous quartz and edges of fibrous calcite. The long axes of the quartz fibres are sub-perpendicular to the vein walls and they grow away from a centrally located median line. The long axes of the calcite fibres at the contact of the vein and the boudinaged host are also sub-perpendicular to the vein walls. Along the quartz-calcite interface the fibres of both crystals are oblique to the wall and both crystal species increase slightly in size towards their common interface. There is a smooth linkage of long axis orientations across the vein and the fibres trace out a double sigmoid form. This geometry is consistent with a simultaneous progressive growth of quartz and calcite, with new material being added along the quartz-calcite interface. The incremental strains for a particular orientation (or more exactly for a 10” range of orientation) were determined by measuring the sum of the fibre lengths in this orientation, and finding the proportion of this incremental extension to the length of wall rock and earlier deposited fibre in the same direction. In order to make the calculations isogons were drawn on the fibre directions, measurements made in several parallel traverses across the specimen and the results averaged. The data are plotted in Fig. 2. There is a systematic change in the amount of incremental strain with direction. The geometry of these changes is consistent with the progressive superposition of strain ellipses with differing maximum extension values, but with no change in length along their minimum extension axes. For example, there are no fold-like instabilities developed in directions normal to the fibres as would be expected in the case of increments of constrictive type, and there are no geometric data supporting simultaneous growth of fibres in several directions as would be expected if the increments were of flattening type. In three dimensions the total strain appears to be built up by progressive superposition of plane strain ellipsoids
Fig. 2. Plot of ~crement~
strain against orientation for the vein system of Fig. 1.
214
with the maximum incremental superposition
axis (e,)
aligned
becomes
and the intermediate
in the plane
of these incremental
the finite strain determine
strain
ellipsoid
plane
a true flattening
compute
strain
ellipsoids
deformation.
graphical
restorations
the successive
strains
directly
of each
in a non-coaxial Two methods
manner
were used to
and also determine
stages in the strain history.
of the boudinaged
axis (e,)
vein. As a result of the
the value of the total finite strains on this surface,
values of the strain state at different to make
strain
of the stretched
fragments,
from the strain increment
the
The first method the second
is
is to
measurements.
Method I, progressive finite strain There appears and it is possible
to be very little internal deformation in the limestone fragments to find the original form of the present boudinaged aggregate by
reassembling them in a jigsaw puzzle manner. By matching their shapes, and by restoring the fragments utilising the fact that any curved fibre links two points that were once in contact, it is possible to make a convincing reassembly (Fig. 3a). On
Fig. 3. Sequential reconstruction the deformation. deposited
of the vein system. a. Before deformation.
b and c. Intermediate stages of
d. Final deformed state. The unstippled areas in b, c and d represent the fibre material
in the preceding increment,
and the areas indicated with vertical ornament
material added during the incremental extension.
show new vein
215
this reassembly two circles were drawn. By placing the particles in their final positions, or by restoring them to an intermediate displacement position, it is possible to obtain the average shape of the finite strain ellipse or any intermediate strain ellipse respectively (Fig. 3b, c, d). In this example the total finite strain values are e, = 1.7 to 2.2, e2 = 0.1 and the orientation of the long axis is 8’ = - 13”. By studying a succession of reconstructions at differing stages during the deformation (two intermediate steps are shown in Figs. 3b and 3c), the complete evolution of the vein system becomes very clear. Method 2, incremental strains The second technique for determining the total strains consists of determining the incremental deformation gradient matrix for each of the incremental meas~ements, multiplying the successive matrices together to obtain the successive finite deformation gradient matrices, and from this to compute the principal total strains. For an incremental strain with values e, = ei, ez = 0 and orientation 8’ the incremental deformation gradient matrix in the reference coordinate system is given by: ai
bi
(1 + ei ) cost@’ + sin2B’
ci
d, = H
ei cos 8 sin 8’
ei COS 8’ sin 8’
cos26)’ + (1 + ei) sin28’ 1 Successive matrix multiplications are then made (Ramsay, 1967, p. 324) and the total strain computed for the final matrix product (Ramsay and Graham, 1970, p. 790). In the example given here this gives principal strain values e, = 2.37, e, = 0.15 and orientation 8’ = - 9.6’, fairly close to those derived by the somewhat crude graphical construction method. The values of the increments, incremental and total deformation gradient matrices, and progressive and finite strains are given in Table I. i
Fig. 4. Plot of the proportion of quartz to calcite against orientation of strain increment for the Leytron vein.
1.13
1.11
1.03
1.01
1.01
-20
- 30
-40
-50
-60
-70
1.15
1.15
1.05
1.03
1.02
I .05
1.01
1.17
-10
1.20
0
1.18
I .78
10
a,
I .8
8:
- 0.02
-0.01
-0.01
- 0.02
- 0.07
- 0.05
- 0.03
0.00
0.14
data
total strains
bi
strain and deformation
progressive
1.2
ei
Incremental
Data for determining
TABLE I
1.00
- 0.02
-0.01
-0.01
- 0.02
- 0.07
- 0.05 3.25
3.34
3.31
3.29
3.15
2.83
2.50
2.13
1.76
a
I .02 1.02 I .02 1.04
data
0.01
0.03
0.04
0.06
0.08
0.13
0.16
0.16
0.14
b
Finite deformation
strain
1.04
1.02
1.01
1.02
0.14 0.00 -0.03
c,
from incremental
- 0.47
-0.40
-0.37
- 0.32
- 0.23
1.17
1.12
I.10
1.09
1.07
1.03
- 0.05
1
3.31
3.34
3.31
3.26
3.16
2.83
2.5
1.80 2.15
1.02
1-r e,
1.15
1.11
1.10
1.09
I .07
1.04
1.02
1.00
1.00
1 i e,
Finite strain data
I .02 I .02
0.14
0.14
d
0.07
c
data
- 9.08
- 1.60
- 6.92
- 5.86
-4.21
0.03
3.7%
7.35
10
8’
._._
217
One especially for different changes greatly
interesting
incremental
(Durney,
feature stretch
1971; Ramsay,
in excess of calcite,
progressively
decreases
until,
of the Leytron directions,
composite
the proportion
vein systems is that, of quartz
1981). At the start of vein formation
but as vein growth in the last veins,
continues quartz
to calcite quartz
the quartz-calcite is absent
is
ratio
(Fig. 4). Fluid
inclusion geothermometry by Durney has shown that there is a systematic decrease in homogenization temperature during the deformation suggesting a change of solution
chemistry
THE DEFORMATION
with temperature HISTORY
and with time (Durney,
INFERRED
1971).
FROM PYRITE BOUDINAGE
Pyrite layers boudinaged during a deformation episode may be found in deformed sedimentary rocks of low metamorphic grade. Excellent examples of chocolate tablet structure have been found in the Ordovician slates of Parys Mountain, Anglesey, Great Britain. The pyrite had an original sheet-like form, and was probably developed during the period of sulphide mineralization that gave rise to widespread ore deposits in the area (Greenly, 1919). The example investigated is shown in Fig. 5. The
Fig. 5. Micrograph
of the pyrite chocolate
tablet boudinage
Britain. The area shown measures 22 by 15 mm.
pyrite
sheets
from Parys Mountain,
are
Anglesey,
a few
Great
218
millimetres thick. They have been broken into a number of separate flat plates parallel to the slaty cleavage of the rock and the plates are separated by fibrous veins consisting
entirely
from the median pyrite-quartz lattice they
of quartz. surface
interface.
The mode of growth of the fibres is antitaxial.
towards This
misfit. The individual are curved.
quartz-pyrite
They
contacts.
the pyrite plate, new material
is an interface
of major
fibres have a constant
often Certain
show
a progressive
fibres
mechanical
optical
weakness
orientation,
increase
are more suitably
that is,
being added
and
even when
in size towards
oriented
at the
for growth
the and
widen at the expense of less suitably oriented neighbours. The median surface is not always parallel to the faces of the pyrite plates on each side but shows. overall, a relative clockwise Strain induced
rotation. features such as the formation
of deformation
lamellae,
undulose
extinction or recrystallization of early formed fibres are not seen in these fibrous quartz crystals. It is therefore concluded that their curved shape was acquired by growth during progressive deformation. The fibres between the pyrite plates are considered
to represent
the displacement
paths of the spreading pyrite plates, to a first approximation, as were the fibres of the Leytron veins. The following strain analyses were undertaken to show some of the basic properties
of the deformation
of pyrite chocolate
tablets.
Method I, total strain from geometric reconstruction It is easily seen that a geometrical best fit of the pyrite plates, as was used for the Leytron vein system, allows a reconstruction of the undeformed state of the pyrite sheet, Fig. 6. A circle drawn on the best fit initial configuration becomes fragmented to the overall shape of the finite strain ellipse. The half axes of this ellipse are 1 + e, = 1.63 and 1 -t e, = 1.17. For this method see also Durney (1971). This direct,
(a)
Fig. 6. Geometrical a. Undeformed b. Deformed
reconstruction
of the Parys Mountain
state plus circle. A-B state.
connection
boudinage.
of two adjacent
gravity
centres.
219
but somewhat approximate, method of determining finite strain is used as an independent comparison for the other methods applied below. Method 2, finite strain from elongation of lines
The centres of gravity of the individual pyrite plates were determined. The centres of gravity of adjacent plates were then connected with lines in the best fit configuration of Fig. 6, and their lengths (I,) before deformation measured. Their lengths after deformation (1) were then measured and the inverse quadratic elongations calculated as: A’ = (f&)2 The total finite strain in the components of the reciprocal coordinate system having the reciprocal quadratic extension transformation:
plane of the chocolate tablet is defined by the three strain tensor Xi, A; and yi_,,, which is expressed in a x axis parallel to an arbitrary reference direction. The A’ in a direction IY’to the x axis is given by the tensor
A’ = A!!cos2ff’ + 2~4, cos (Y’sin (Y’+ hY sin’& The values of A:, A; and y;,, which give the least deviation of h’ values calculated using the above equation and the measured valu& can be found by a standard least squares procedure. When the conditions for a minimum of the sum of the squared deviations are applied three simultaneous equations for the unknowns are obtained. Once A>, Xi and yiY are determined, A;, A; and 6’ can be found. A; and A; are the principal reciprocal quadratic elongations describing the total finite strain state of the specimen, and 8’ is the angle between the reference line and the direction of maximum extension after deformation. For the technique of determining the principal values and their orientations from a tensor see, for example, Ramsay (1967, pp. 81-82). We have measured A’ and CX’of 38 lines connecting gravity centres of adjacent pyrites. The reference direction was the principal extension direction of the best fit ellipse previously established by the r~onstruction technique of method 1. The resulting total strain is 1 + e, = 1.75, 1 + e2 = 1.22, and 8’ = 0.86”. Figure 7 gives a plot of the results of this method allowing their critical discussion. This graph is based on the relationship (Ramsay, 1967, eq. 3.31): 2 A'=
!y i
=A;
cos'tl'+X;
sin2B’
i
and, consequently: h’=(h;-A;)cos%‘+X;
(11
In a homogeneous strain state plots of A’ against co&P should give a straight line. The data points in Fig. 7 refer to the measured A’ values, which are plotted against
A’ 0.7
0.5
0.3
0.1
l
I
0 Fig. 7. Results measurements
0.2 of method
0.6
0.4
2. Plot of A’ against
of the specimen
08
wsz
8’
cos28’, derived
shown in Fig. 5. Line o represents
b the total finite strain calculated
from the data of method
from 38 gravity
centre
to gravity
the finite strain result of method
centre 1, line
2.
the angle made with the calculated extension direction of the total strain. The two straight lines in Fig. 7 are plots of the resulting total strain of methods 1 and 2, using eq. 1, Line a was constructed from the results of method 1 and line b from the results of method 2.8’ is in this case the angle from the calculated 1 -t e, direction of the total strain. The results of the two methods correspond well. In particular the extension directions of the total strain of the two methods are quite similar. Although both lines are very close to the data points, there is a wide dispersion of the points which shows that it is inaccurate to consider that the centres of gravity of the spreading pyrite plates moved as independent points displaced by a homogeneous strain. The best interpretation of the data spread is that the break up of the pyrite plates probably occurred progressively during the deformation, so that not ail the distances between adjacent pyrite plates will be simply related to the geometry of a homogeneous strain. Another explanation of the variable data dist~bution could be that the total strain was heterogeneous. We did not observe any differences in overall data trends from one side of the specimen to the other and therefore conclude that the assumption of homogeneous strain is justified. Method 3, progressive finite strain
This method (termed the pyrite method by Durney and Ramsay, 1973) assumes that the strain state in the X-Y section was achieved by a succession of infinitesimal increments of plane strain and that the orientation of the fibres is parallel to the incremental extension directions. Individual curving fibres were divided into a
221
number of straight finite increments. The strain for each increment was evaluated by dividing its length by the sum of the lengths of previously formed increments resolved into the direction of the current increment plus the pyrite radius. The finite strain was calculated by superposition of the incremental strains. Justifications for the application of this method to the example of Fig. 5 are: (a) the absence of deformation features in the fibres; (b) adjacent fibres, as well as fibres from differently oriented faces, are more or less parallel, hence the assumption of a deformation history consisting of plane strain increments seems acceptable. Sixty fibres distributed over the whole specimen were measured, the arbitrarily chosen reference direction being that of the long axis of the ellipse of method 1. The progressive strain for each fibre was calculated in the relative direction. These strain data were then used to establish the overall strain state of the specimen, using the least squares technique outlined above. The results are: 1 + e, = 1.60, 1 + e2 = 1.01, 8’ = 0.46’. The directions of the principal strains correspond well for the three methods, but the total finite strain value of method 3 is lower than the values of methods 1 and 2. On Fig. 8 the fibre strains are plotted against their angle with the calculated total finite extension direction. Line c was ~onst~cted from the calculated total finite strain, applying eq. 1. The fibre strain data show considerable spread. There are several factors responsible for this spread: the pyrite method assumes that fibres which have grown during the same interval of incremental strain have the same direction. This means that an isogon plot of fibre directions corresponds to a map of isochronous lines. Figure 9 shows an isogon plot of the specimen and Fig. 10 shows
t 0
1..
.
02
0.4
*
*
OS
.
f
Qd
.
-
caG8'
Fig. 8. Results of method 3. Plot of h; and hi against cos*B’, derived from 60 fibres of the specimen shown in Fig. 5. Line c represents the total finite strain, calculated from the data of method 3.
Fig. 9. Isogon plot of the Parys Mountain numbered
Fig. IO. Stages in the spreading constructed formed
specimen.
The orientations
of the fibres corresponding
to the
regions in the figure are given by the key in the top right of the diagram.
from the isogons
during
of three of the pyrite plates of Fig. 5. The eight increments
of Fig. 9, according
the same time interval.
to the assumption
shown are
that fibres of the same orientation
223
the individual steps of spreading using the isogons as marker lines for the increments. A comparison of Fig. 10 with the corresponding part of Fig. 5 shows that the model which describes the deformation history as sequence of plane strain increments gives an acceptable result for the three pyrite plates considered. Between the first and the last increment of Fig. 10 there is an anticlockwise rotation of the three pyrite plates, as well as a separation of plates 1 and 3 after the first four increments only and a breaking apart of plates I and 2 after the first six increments. Using the fibre pattern between the plates it is possible to construct a more or less symmetric, if simplified, isogon plot. If, however, a more complicated part of the specimen is considered as, for example, the borders of Fig. 9, it is evident that the isogons do not correspond in every case to previous positions of the plates: isogons are not always isoehrons. A consequence of this observation is that during the same time interval the fibres can grow in different directions. This feature arises when the incremental strains are of a truly flattening type instead of a series of non-coaxially superimposed plane strain increments. The dispersion of points in Fig. 8 seems best explained by a flattening incremental history and, in parts of the sample, by the diachronous breaking apart of the pyrite plates. Discussion
Methods 1 and 2 give a very similar total finite strain, but method 3 gives appreciably lower values. The low 1 -t- e2 value of method 3 is probably a further indication that the deformation history is progressive flattening and not a sequence of non-coaxially superimposed plane strain increments. Method 3 also gives a slightly lower 1 + e, value than those calculated from methods 1 and 2, notwithstanding that method 3 and method 2 give theoretically the same result for 1 + e,, if one of the straight lines of method 2, which connects two gravity centres, is divided in increments and the incremental strains are accumulated. This shows that the strain calculation from an individual fibre can give only a fair estimate for the total 1 + e, in the corresponding direction, and that the total 1 + e2 from an individual fibre cannot account for true flattening deformation. Line c in Fig. 8 could be drawn considering only the Xi values and would then lie closer to lines a and b of Fig. 7. In all the preceding analyses it has been assumed that the fibres define the displacement paths of the separating vein walls or pyrite plates. This assumption is not strictly correct for the pyrite chocolate tablets. Figure 11 shows that when the pyrite plates are reassembled on the basis of their shapes, the ends of the two fibres which grew from the same point on the median surface do not exactly coincide. One possible explanation for this is that there is often a tendency for the fibres to grow perpendicular to the pyrite surface as in the case of pyrite cubes (Pabst, 1931; Durney and Ramsay, 1973). Another explanation is that wrench-fault-like shear motions took place along the pyrite-quartz interface during deformation to allow
224
for compatibility
between
the fibre geometry, In conclusion
Fig. 11. Geometrical
the rigid plates. These displacements
and they are responsible
it is possible
reconstruction
to formulate
are not registered
in
for the lower total finite strain obtained. a list of properties
of the deformation
of three pyrite plates. The points lettered A- I represent
ends of fibres which grew from the same point on the median
of
points at the
surface.
pyrite chocolate tablets which must be incorporated reproduce the fibre geometry:
in any model which attempts
to
(1) The pyrite plates can be considered as rigid bodies moving in a plane. Their final position partially reflects the total strain history, but it is also a function of the diachronous break-up history of the rigid sheet into its component plates. (2) A total flattening strain can be built up in an infinite number of ways ranging from the superposition flattening increments. histories
of successive plane strains to the superposition of successive The model should explore the consequences of different strain
on the resulting
fibre geometry.
(3) Compatibility problems between the spreading fibre mass would be revealed in a model. (4) Fibre growth is continuous and therefore
plates and the already the
strain
history
formed
should
be
considered as being continuous rather than a series of finite increments. (5) Rotational components of the deformation can be evaluated from the relative positions of the pyrite plates and from the curvature of the fibres. Rotation sense is defined relative to a particular reference frame. The pyrite method (method 3) uses the fibres as fixed reference directions and in the case of the analysis carried out here an overall anticlockwise rotation sense of the pyrite plates results (Fig. 10). When the reference direction was fixed in either of plates 1 or 2 (Fig. 12) the other two show a relative anticlockwise rotation sense (cases a and c in Fig. 12) whereas if it was fixed in plate 3, the other two show a relative clockwise rotation.
Fig. 12. Superposition of three pyrites from the undeformed and deformed state of the specimen to show differential rotation. In a plate 1 is used as the reference, in b plate 3 is the reference and in c plate 2 is the reference.
SIMULATION
OF FIBRE GEOMETRY
The basic geometric features of the naturally deformed chocolate tablet structure set out in the last section were used to control a numerical simulation study. In the following sub-sections the mathematical basis of the model is set up, details of computer based solutions are given and some examples of its use are described. Specification of the deformation history
Deformation histories may be described in two ways. One way is to consider the deformation as being made up of a sequence of small, but finite, increments and combining these by matrix multiplication (Ramsay, 1967). An example is given in eq. 2:
(E~j=(~
Y+ej”=(il
I)l+e)“)
226
gradient matrix ( :’ i; ) has been y+, ) which corresponds to uniaxial works well for simple increments such as the one given.
where the finite strain, described formed extension The calculate
by the deformation
from n steps of the small increment in y. This method alternative
approach
is to specify
its effect by integrating
berg, 1975). The velocity
(d
gradient
the velocity
over the time interval
gradient
matrix
of the deformation
and
to
(Ram-
matrix vk,, is given by eq. 3:
\
*k
(3)
.I
The velocity gradient matrix can be decomposed into a symmetric and antisymmetric part. The symmetric part is the deformation rate tensor, d,,:
an
au
_;z
ax d,,=
(4)
which describes the rate of change of shape of the material. The antisymmetric part is the spin tensor wk,:
which describes the rate of rigid rotation. The simultan~us effect of several velocity them. No approximation is involved. In general the velocity gradient matrix
I Fig. 13. The reference
gradient
is a function
matrices
is given by adding
of time. One simple way in
X system
given in x’, y’ is constant
x’. y’ rotating
with time.
with respect
to system
x, y. The velocity
gradient
matrix
227
which
this variation
in time, t, can occur
system, Fig. 13, the strain rate, represented the x’, y’ coordinate according e(t)
is as follows.
In the x’, y’ coordinate
by the velocity gradients,
system is made to rotate
relative
to the x, y reference
but
system
to eq. 6:
= e, + LYi
(6)
The value of uk,,(t)
in the reference cos
I
is constant,
au,(t)
a+) ax
ay
6y1)
=
I
system x, y is given by eq. 7: \ -sin f?(r) case(t)
sine(t)
I /
cos
e(t)
sin e(t) (7)
-sin
e(t)
COST
where vi ,I is the velocity gradient matrix of the x’, y’ coordinate system. The deformation at any instant in the surrounding rock is defined by the velocity gradient matrix the deformation Eringen
(eq. 3). The relationship between the velocity gradient matrix and gradient matrix (F f;) is given by eqs. 8, which were adapted from
(1965, p. 70, eq. 2.4.1):
da=av,a+av,c dt ax
dc 3% z=zu+ayc
ay
au,
This is a set of simultaneous ordinary differential equations which can be solved numerically by the fourth order Runge-Kutta technique (Ralston and Wilf, 1960, ch. 9). The solution will give the values of the deformation gradient matrix at equally spaced intervals in time. From these the usual parameters of finite strain X,, X, etc. can be calculated (Ramsay from natural examples.
and Graham,
1970) for comparison
with measurements
As will be shown below, the motion of a rigid plate embedded in a deforming medium can be related to the velocity gradient matrix. The use of the velocity gradient matrix brings time into consideration. In numerical simulation this does not present any problem. In natural examples the length along a reference fibre can be
228
used to represent necessarily
time (Elliott,
represent
1972) but equal lengths
of the reference
fibre do not
equal times.
The motion of a rigid plate embedded in a deforming
material
To simplify the determination of the motion of a rigid plate in a deforming some assumptions are made. The first is that the plates interact mechanically
rock with
the surrounding material across the surface only and that the effects of the deformation at the edges are negligible. This assumption is reasonable because the surrounding rock does not flow around the ends of the plates in the case of chocolate tablet boudinage. The second is that the matrix exerts a force on the plate which is proportional adjacent matrix. This
to the relative velocity of points in the plate and in the assumption, though impossible to justify without further
analysis, is adopted because the force will certainly increase with increasing velocity and a linear relation is easiest to treat mathematically. The velocities /
of points
in the surrounding
relative
rock are given by eq. 9:
\ f-)x i=
i
*Y
(9) -ax
where the right-hand
3.Y
,Y
side contains
rock. The rigid plate can respond axis at x0, y,. The velocity
the velocity
gradient
to the deformation
matrix
of the surrounding
only by a rotation,
o, about
an
of a point in the rigid plate is given by:
Thus the parameters describing the motion of the plate are x0, y, and w and the problem is to find values of these parameters which give the minimum work rate of the forces exerted by the surrounding rock on the plate. The work rate is equal to the force times the velocity and as the force is assumed to be proportional to the velocity the work rate will be proportional
to the square of the relative
velocities,
u:. 0::
where the integral is carried out over the surface of the plate. Subtracting eq. 9 from eq. 10 to give an expression for the relative velocity and substituting this expression in eq. 11 gives:
229
To find the minimum of this function with respect to x0, y0 and w the function is first differentiated with respect to each of the three parameters and the resulting expressions are set equal to zero.
(13) (14)
-a ah
(15)
w can be cancelled from eqs. 14 and 15:
(16)
ds=O $-w)(y-yO)-
(~~+~y)}
(17)
ds=O
Expand eq. 16 and remove constant terms from the integrals: w xds-ox, I
I
ds=z
(18)
jxds+%/yds
Expand eq. 17 in the same way: -w
/
yds+wyO
/
ds=%/xds+au”jyds
(19)
ay
Expand and reorganize eq. 13: --w
/
(y2+x2)ds+2wy0
I
yds-
oy,2
ds - z/x*
Rearrange eq. 18 with x0 on the left:
X0=
a0 yax / xds+- 2 --w
j yds -wfxds
I
ds
ds+2wx,
J
xds-ox,2
I
ds
ds}
z/xds+%/y
}-x0{ jy’
/
ds
(20)
230
=-
A-wB
(21)
--WC i3V
A=’
3X
s
8V
xds+-I:
Rearrange
aY s
yds,
B=
s
xds,
C=
s
ds
(22)
eq. 19 with Y, on the left:
-wjyds-
[2jxds+%jyds)
Yo = --w
J
ds
-wD-E
=
(23)
--UC E=zjxds+au*jyds,
D= 3Y
/
yds
Substitute for x0 and y0 in eq. 20, multiply to eqs. 22 and 23:
J(y2 + x2)
cw2
(24) by WC and make substitutions
according
ds - 2( w2D + wE) D -t- {w2D2 + 2wDE -i- E2}
-2( Bo2 -Aw)B+{A’-2wAB+02B}-(uD+E)E-(A-Bti)A
Gather
terms in w and substitute
(y2+x2)ds-D*-B2
-(
for the integrals:
1--w i
DE-BA-C[($+)/,,,,
(25)
+~/y2ds-~/x2dslj_0
The roots are:
DE-BA-C[[~-~)/xyds+$y2ds-~/x2dj w=O
or
w= q(Y’+
x2) ds - D2 - B2
(26) Ignoring
the w = 0 root and resubstituting
for A, B, C, D and E:
231
-jds[(%-!$)j X {j
dsj(y2+x2)
Substituting gradient
ds-(jyds)‘-
deformation
matrix
(jxds)2}j’
rate and spin terms
using eqs. 4 and 5 and rearranging
for the elements
of the velocity
gives:
w= (jyds(d,,jxds+d,,jyds)-jxds(d,,,jxds+d,jyds) -
ds
(d,,
j
- d,,) jxy
ds + d,, j( y2 - x2) ds
i
I)
x{jdsj(y2+x2)ds-(jds)2-(jxds)2}j’+~2,
(27)
The values of x0 and y, can be obtained by back-substitution in eqs. 21 and 23. If both solutions are zero, then the motion of the plate is just a rigid translation of velocity u,, uY. The relative velocities are given by:
au
*,‘=*Y-
i
a~
dx+Ly ax
The following
au,
1
ay
equations
give the conditions
(28) for zero net force on the object:
au,
ux,,x+,,y
ds=O )
au,
au,
uY--x+-y
ay
j( Rearranging:
a*, xds+- a*, yds
ax
u, =
J
ay j
(29) / au, ax
VY=
j
xds+-
ds an, ay j
yds (30)
J
ds
Under special circumstances only will it be necessary to use eqs. 29 and 30, that is, if the instantaneous strain is a &axial extension and the rigid object is of equant shape.
232
Rotational
deformation
and rigid objects
Equation 27 shows that the rotation rate of the rigid plate is determined by a factor representing the interaction of the shape of the object and the deformation rate of the imposed
deformation
effect of the spin of the imposed
and the spin of the imposed deformation
is independent
deformation.
The
of the shape of the
plate. If the motion of a disc is considered it can be shown that the first term on the right-hand side of eq. 27 is zero. The disc will therefore rotate at an angular velocity equal to the spin of the matrix. To explore some of the consequences of this result consider the effects of a shearing deformation. The reference systems and the deformation are shown in Fig. 14. The spatial
reference
system x, y serves to define
the imposed
deformation.
A
reference system x’, y’ is attached to the rigid object in such a way that it rotates with the object and the coordinate directions are always parallel to the same material lines. The deformation
The rotation
in the X, y frame is:
of the object relative
3%
@=
I/*%
to the x, y reference
system is given by: (32)
The imposed deformation (31) is constant in the X, y reference frame so that the principal extension directions are also constant in this system. The rigid object rotates o relative to X, y so that the orientations of the principal extensions rotate in the x’, y’ coordinate
system.
Fig. 14. A round rigid object in a rotating matrix deforming by sinistral simple shearing. The coordinate system x’, y’ is fixed to the rigid object.
233
Calculation
of the finite fibre geometry
Once the parameters plates,
x,,, y,, and w, which determine
have been obtained
mass can be determined this information
the velocity
of separation
the movement
from eqs. 10 or 29 and 30. To calculate
several
further
techniques
of the rigid
of pyrite plate and rigid fibre
are needed.
These
fibre forms from are:
a means
of
representing the fibres numerically, means of evaluating the integrals in eqs. 21, 23, 27, 29 and 30, and a means of projecting the deformation over a time interval. The basic element in the simulation of the development of chocolate tablet structure
is to analyse
mass between. The pyrite geometry,
plates
although
the relative motions can finite
be represented element
analysis
of two pyrite plates and the growing fibre by finite
element
meshes
is not used to obtain
of constant
a solution
of the
deformation. The fibre mass can be represented by a finite element mesh to which new rows are added on each side at each time step. At the start of the deformation an embryonic fibre mass is inserted as a single row of elements between the pyrite plates. On each side of the initial row of elements a row of elements of zero width is set up, one set of nodes being attached to the pyrite. During the time step the zero width elements are allowed to widen to take up the displacements between the fibre mass and the pyrite. At the end of the time step another row of zero width elements is added ready for the next time step. The integrals can be performed over each element and the contributions of all elements summed to give the totals. The method of calculating the integrals for each element was taken from Zienkiewicz (197 1, pp. 144- 149). Gauss quadrature was used with four integration points in each element. This allows the accurate integration of second-order functions such as xy and x2. To step through
time the fourth-order
and Wilf, 1960, ch. 9). This involves step. In the present coordinates velocities.
case the variables
Runge-Kutta
four evaluations of the ordinary
of the nodes of the finite element
method
was used (Ralston
of the gradient differential
in each time
equations
mesh and the gradients
are the
are the nodal
Results of simulations Simulations were performed to assess the properties of the model and to attempt to reproduce the fibre geometry of a portion of the chocolate tablet boudinage structure from Parys Mountain, Anglesey, Great Britain, shown in Fig. 5. The portion chosen for study comprised three pyrite plates with their intervening fibre masses. The undeformed geometry of the model, derived from the best fit reconstruction of the pyrite boudins (Fig. 11) is shown in Fig. 15. In all the simulations pyrite plates I and 2 were held .together for the first quarter of the deformation, the fibre masses between plates 1 and 3 and 2 and 3 were
234
connected histories principal amounts
and that between
plates
chosen there was a clockwise strain
rates
of flattening
were varied
the relative
from
run
to run
to investigate
strain rate. The results and strain histories
16, 17 and 18. There is considerable although
f and 2 was mechanically separate. In all the rotational component. The ratios between the
positions
variation
the different
are shown in Figs.
in fibre forms produced
by the models
of the pyrite plates are quite close in all cases. One
feature of the natural example of Fig. 5 is the frequent parallelism of fibres on opposite sides of the median surface, shown particularly clearly in the isogon plot of Fig. 9. This feature is also observed in the simulations, especially when the deformation is an extension in x with little change in J’ (Fig. 16). The strongest flattening deformation gives the greatest departure from parallelism of fibres on opposite sides of the median surface (Fig. 18). The general motion of a rigid plate is a rotation about a pole, rather than a simple translation. The rotation motion results in non-parallel fibres forming at the same time on the same pyrite which is opposite
surface.
This is shown
in Fig. 17b on the face of plate 2
plate 3 and in the isogon plot of the natural
example
of the plates in the right-centre of Fig. 9. Differing growth rates of the fibres on each side of the median Fig. 16d. This is related
on the sides
surface is shown in
to the size of the pyrite plates and can be seen also in the
natural example (Fig. 5). A major aim of the simulations was to reproduce the observed pyrite plate positions and geometric features of the fibres of the natural example in order to estimate the deformation history in the plane of the sheet. The median surface between plates 1 and 3 in the natural example does not connect with a median surface between plates I and 2. From this it was thought that plates I and 2 separated later than I and 3 and 2 and 3 and the models were set up accordingly. Broadly comparable results could be obtained especially in the model with very
Fig. 15. The undeformed
configuration
Compare with Figs. 10, 11 and 12.
of the model with reference
numbers
for the pyrite plates.
235
little extension in y (Fig. 16d). The positions of the pyrite plates are comparable as are the curvatures of the fibres. The position of plate 2 in the model is not in good agreement with its position in the natural example and hence the assumption of diachronous breaking of the pyrite may not be correct. Qualitative comparison of the results of simulations (Figs. 16 18) and the natural sample (Fig. 9) indicates that the best agreement would occur for a time just later than that of stage c. Reference to Table II gives a value for 1 + e, of around 1.8 and
Fig. 16. Computer model A. Equally spaced stages in the development
of the model. Defo~ation
history:
Velocity gradients of: o.l~lo-’ ( -o.l.10-7
0.1. IO--’ o.l*lo-‘”
for eight steps of 0.1.10’
1 set:
d,,=0.1~10-', dyy= 0.1. IO-lo,
d,,= 0.. w2,= -0.1.
lo-’
The circle and ellipse above the diagrams represent the increment of strain from the previous diagram. The circle and ellipse below each diagram represents the finite strain. The median surfaces are defined by the jogs on the fibres.
236
Fig. 17. Computer model B. Equally spaced stages in the development of the model. Deformation history: Velocity gradients of: o.1.10-7 i -0.5.10-s
0.5. 1o-8 o.1~10-8 1
for eight steps of 0.1 IO8 set: d,,=O.I.lO-‘,dyy=O.l.lO-s,dxy=O.,W*,=
-0.5.10-s
Circles and ellipses are the same as for Fig. 16.
varying values for 1 + e2 (1.0178- 1.2119). The results of strain analysis method 2 (Fig. 7) indicate that the highest value (1.2119) is the most likely. The value for 1 + e, is in good agreement with the direct methods of strain estimation. In none of the models was it possible to obtain the correct rotation of the fibre masses as a whole relative to the pyrite plates. This may be the result of not having the fibre masses joined correctly or that the deformation history was one in which the extension direction changes in the spatial reference frame (x, y). The computer program cannot calculate the effects of this kind of deformation history at present.
231
Fig. 18. Computer model C. Equally spaced stages in the development of the structure. Deformation history: Velocity gradients of: 0.1*10-7 0.1.10-7 0.3.10-* 1 ( -0.1.10-7 for eight steps of 0.1 f 10’ see: d,,=O.l~lO-‘,
d,=0.3.10-*, d,,=O., y,= -O.l.lO-'
Circles and ellipses are the same as for Fig. 16.
The rotational component of the imposed deformation causes the three pyrite plates to rotate relative to the reference coordinate system as can be seen from Figs. 16-18. Relative rotations of the pyrite plates do occur but these are much smaller
238
TABLE
II
Finite strain data for the steps of the three computer
simuIations
Model A
step 1.22
1.0008
- 5.73
- 11.47
2
1.48
1.0156
- 11.19
- 22.63
4
1.79
1.0178
- 16.66
- 33.44
6
2.14
1.0407
-21.87
- 43.80
8
1.22
1.0203
- 2.56
- 5.73
2
1.49
- 5.73
- 11.35
4
-8.51
- 16.78
6
2.21
I .042 1 1.0660 1.0932
- 10.89
- 22.01
8
I .22 l.49
- 5.73
- 11.42
2
I.1316
- 11.48
- 22.78
4
1.80
1.2119
- 16.86
- 33.90
6
2.16
1.3076
- 22.33
-44.80
8
1.81
1.0623
stage
than the general rotation. This is a consequence of eq. 32 which shows that an equant plate rotates at the vorticity of the matrix which, in the cases presented here, was greater than the rotations produced by the non-equant shapes of the pyrite plates. CONCLUSIONS
(1) When the boudinaged plates move apart by a simple translation, without rotation, as in the case of the Leytron vein, the finite strain at the end of the deformation and at stages in the development of the structure can be estimated from the fibre geometry and reassembly of the plates with acceptable accuracy. (2) When the separation of the plates and the intervening fibre masses is accompanied by relative rotations, as in the Parys Mountain specimen, the calculation of the incremental and finite strains from the fibre geometry under the assumption that the strain history consists of plane strain increments is not strictly accurate. Fibre increments which were deposited during the same time interval do not always have the same orientation and pyrite plates may break apart during the deformation. (3) A model for the simulation of chocolate tablet structure can be set up if certain simplifying assumptions are made about the mechanical interaction of the rigid plates and the surrounding rock. The model is capable of reproducing many features of the natural example from Parys Mountain. The model also shows that the rigid rotation rate of an equant plate is equal to the vorticity of the deformation in the surrounding rock.
239
ACKNOWLEDGMENTS
The authors are grateful to D.W. Durney for extensive discussion of one of his computer programs and to SM. Schmid for valuable text. Financial 5.521.330.785/l,
and for the use criticism of the
support is acknowledged as follows: MC Nationalfonds DD ETH project 0.330.079.27/8, JGR Nationalfonds
project project
no. no.
Sci. Nat.,
68:
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