~ec~~~o~~ysjes, 149 (1988) 323-338

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

323

The use of Mohr circles to describe non-coaxial progressive deformation

C.W. PASSCHIER

Instituut vow Aardwetenschuppen, R$.wniversiteit Utrecht, Budapestkan 4, Utrecht (The Netherlands)

(Received June 4,1987; revised version accepted January 1,1988)

Abstract

Passchier, C.W., 1988. The use of Mohr circles to describe non-coaxial progressive defo~ation. Tectonop~~s~es, 149:

323-338.

Mohr circles are useful in strain analysis but are also potentially powerful tools to translate geological fabric data

into tensors describing the deformation and to reconstruct flow parameters. This paper shows how Mohr circles can be

used to visualize the effects of variable flow parameters such as stretching rate, volume change rate and vorticity

number on parameters of cumulative deformation as described by the position gradient tensor. A “mean” vorticity

number is introduced as a deformation parameter besides finite (cumulative) strain and volume change.

The fabric of ductile-deformed rocks contains information on finite (cumulative) strain and volume change, and also on vorticity of the parent flow type (Lister and Williams, 1983; Passchier, 1986). In many cases, it should be possible to reconstruct flow parameters such as the ratio of vorticity to stretching rate from the final fabric, but in practice it proves difficult to determine which data could be used and to translate mea- surements such as the stretch and rotation of material lines into tensors which fully describe the deformation and the parent flow type. One par- ticularly promising approach to bridge this gap is the application of Mohr circles to represent tensors (Means, 1983; DePaor and Means, 1984); they can easily be constructed by plotting individual m~urements as points in Mohr space; matrix notations for the tensor can be read from the Mohr circle, and problems at hand can further be solved numerically. The range of uncertainty in data derived from naturally deformed rocks can

be incorporated in such Mohr ~nst~ctions to find uncertainties in derived parameters of cumu- lative deformation or flow (Passchier and Urai, in press). This paper demonstrates the use of Mohr circles to describe the effect of flow parameters on the accumulation of deformation and on the final deformation state.

A list of symbols is given in Notation 1.

Flow in rocks

Homogeneous flow in a volume of rock can be described in a purely geometrical way by a veloc- ity gradient tensor L of nine components, regard- less of the presence of anisotropies or the rheology of the material involved (Malvem, 1969; Spencer, 1980). L is composed of the coefficients of Eulerian rate of displacement equations:

(1)

in an orthogonal Cartesian reference frame x (Fig. la). L can be decomposed additively into a

0040-1951,‘88/$03.50 6 1988 Hsevier Science Publishers B.V.

124

NOTATION 1

List of symbols

Xl APa ISA

Tensors

L

D

W

F

4

Ff

Fl

F, H

HI C

R,

reference axes

flow apophyses; eigenvectors of L

instantaneous stretching axes; eigenvectors of D

velocity gradient tensor; L = D + W

strain rate tensor (symmetric)

vorticity tensor (antisymmettic)

position gradient tensor (Lagrangian)

incremental position gradient tensor

cumulative or finite position gradient tensor

left position gradient tensor

right position gradient tensor

position gradient tensor (Eulerian)

cumulative position gradient tensor (Eulerian)

Cauchy’s deformation tensor

orthogonal rotation tensor

Flow parameters

i stretching rate of a material line

3, stretching rate along ISA eigenvalues of D

s mean stretching rate in X,-X, plane s = (st - s2)/2

w angular velocity of a material line

W vorticity

w, vorticity number W, = W/23

wd vorticity number W, = 1 - W, J

wk Truesdell’s kinematic vorticity number

a rate of area change in X, - X, plane

” volume change rate

(Y angle between flow apophyses

Deformation parameters

e, principal elongation along finite strain axes

R, finite strain ratio in X, - X, plane

Km “mean” vorticity number Wmrn = Q/R

AA area change in X, - X, plane: AA = e, e2 + e, + ez

AV volume change: BY= (1 + e,)(l+ e,)(l + e3)- 1

Y shear strain on flow plane of simple shear

6 angle of rigid body rotation around X axis

Mohr circle parameters for Ff

R circle radius

T distance of circle centre to origin

Q vertical elevation of circle centre

symmetrical tensor D, which contains the instan-

taneous stretching rates, and an antisymmetrical

tensor W describing the vorticity of the flow

(Malvern, 1969; Lister and Williams, 1983). ‘The

description can be simplified if the vorticity vector

of the flow remains parallel to one of the instanta-

neous stretching axes (ISA), i.e. to one of the

eigenvectors of D. In that case, L takes the form:

(2)

with only five non-zero components. Throughout

this paper, a two-dimensional cross section of a

flowing material in the X, - X, plane of the exter-

nal reference frame is considered in order to re-

strict the complexity of the model (Fig. la). Flow

in this cross section can be described by a matrix:

L,, L,2 L=L L L 1 21 22

and a component L,, which represents the

stretching rate parallel to the vorticity vector in

the direction of the X, axis.

The advantage of a reduction of flow analysis

to 2D lies in the possibility to use Mohr circles for

the presentation of data. Such a description is not

academic since many deformed rocks show evi-

dence that the vorticity vector remained subparal-

lel to the intermediate instantaneous stretching

axis during progressive deformation; e.g. in shear

zones the intersection lineation of various folia-

tions usually parallels the symmetry axis of micro-

structures with monoclinic shape symmetry (Pas-

schier and Simpson, 1986; Passchier and Urai, in

prep.). The vorticity component of L can be composed

of an “internal” vorticity, describing the angular

velocity of material lines with respect to instanta-

neous stretching axes (ISA; Fig. la), and a spin

component describing the angular velocity of ISA

in the external reference frame (Lister and Wil-

liams, 1983; Passchier, 1986). If flow in two or

more connected domains is described in one study,

it is useful to fix ISA in one of the domains to the

external reference frame, and to describe spin of

ISA in the others. In this paper, however, only

single domains of homogeneous flow are consid-

ered which can always be described fixing ISA in

the external reference frame. The spin component

can therefore be neglected in the model.

325

lsoplanar flow

If ISA are fixed in such a way that one parallels the X, axis and the other two lie at 45” to Xi and X,, L can be represented by a matrix:

[

0 L=

s + w,s s- W”S 0 1

and the component L,, =x3. If si, s2 and sj are

instantaneous stretching rates along ISA, s is the mean instantaneous stretching rate s = (s, - s,)/2 in the Xi - X, plane and W, a “neutral” vorticity number W, = W/2s. W is the flow vorticity (Malvern, 1969; Spencer, 1980). This description of L involves no instantaneous change in area in the X, - X, plane i.e. si + s2 = 0 and volume change rate u = sg_ In this case of “isoplanar” flow, W, equals the kinematic vorticity number of Truesdell (Truesdell, 1954; Means et al., 1980).

Mohr-circle presentation of L

A four-component matrix description of L, as presented above can be plotted as a circle in Mohr space (Fig. lb; Means, 1983; Lister and Williams, 1983; Passchier, 1986, 1987). A point on the Mohr circle represents a material line in the Xi - X, plane of true space, whose angular velocity (w)

(a>

and instantaneous stretching rate (i) can be read as Cartesian coordinates on the vertical and hori-

zontal coordinate axes of the Mohr space (Fig. lb). Tensor components are plotted as opposite points on the circle and represent the orientation of material lines instantaneously coinciding with Xi and X, (Fig. lb). s is the circle radius and W,/2 is the elevation of the circle centre above the horizontal axis (Fig. lb). Material lines which instantaneously coincide with ISA plot opposite each other on a horizontal diameter of the circle.

In this case of isoplanar flow, si = -sz. and the circle is centred on the vertical axis. The only two material lines in the flow which are instanta- neously irrotational with respect to ISA are called flow apophyses (AP, and AP,) after Ramberg (1975a, 1975b). All material points on flow apophyses move instantaneously towards (AP,) or away from (AP,) the origin of the coordinate system in real space (Fig. la; Passchier, 1986, 1987). Flow apophyses are eigenvectors of L. s is a stretching rate independent of the flow vorticity W and can be used to describe the rate of strain accumulation in the Xi - X, plane for any flow type. Decomposition of L to f) by subtraction of W is represented in Mohr space by a shift of the circle centre towards the origin of the coordinate system without a change in diameter.

Fig. 1. Representation of the velocity gradient tensor L for an isoplanar flow field in real space (a) and Mohr space (b);

X,-X,-X,--extemaI reference frame; ISA-~st~~~~ stretching axes (internal reference frame); AP-flow apophyses;

W-vorticity; s-mean instantaneous stretching rate; u-angle between flow apophyscs. Each point on the Mohr circle represents

the angular velocity ( w ) and the instantaneous stretching rate i of a material line in real space. Components of L can be used to plot

position of X, and X, in Mohr space as indicated.

lsoplanar deformation 1979; Passchier, in press):

Just as flow can be effectively described by L,

incremental and cumulative (finite) deformation

can be described by Lagrangian or Eulerian posi-

tion gradient tensors F or H which relate the

position of material particles in undeformed and

deformed state:

X; = FpqXq and X, = HP, X, (5)

H can serve for geological applications using data

on cumulative deformation to reconstruct the un-

deformed state, but F is more useful to illustrate

the way in which flow parameters influence the

final deformation fabric. F is therefore used

throughout most of this paper. H is simply the

inverse of F.

Deformation with constant flow parameters

The incremental position gradient tensor Fi can

be derived from L by integration as:

F,= J

L dt

cosh( W,st ) 1+w n sinh( W’,,st )

F;- = I Yi

1-W w sinh( W,st ) cosh( W,st )

d

(ifOg W,<l) (9)

The vorticity number W, is a useful derivation

from W,, and both vorticity numbers can be ex-

pressed in terms of the angle between flow

apophyses as (Fig. 1; Bobyarchick, 1986; Pas-

schier, 1986):

W, = sin (Y = \il - W,’

W” = cos (Y = 7 1- w,

If W, = 1 (simple shear), Ff reduces to:

(10)

4 (W,=l>

=I+Lt= :, 2ff [ 1

(11)

and for W, > 1 because of complex eigenvalues of

Fi to:

1 COS(iWdH) 1+w -&sin(iW,St)

-I 1 ”

s A41 + w,> + c I (6) cos( iW,st)

0

The integration constant reduces to the unit ma-

trix since at At = 0, F, = I, and therefore:

(if W, > 1; i = J-1) (12)

1 F,=

s At(1 + W,)

s At(1 - W,) 1 1 (7)

For a deformation history with constant flow

parameters s and W, the position gradient tensor

Ff for cumulative deformation can be derived

from Fi by the eigenvector method @eider et al.,

1972) or by sequential multiplication of n tensors

Fi (Elliott, 1972). The latter produces a binomial

series of powers of L which can be solved using

the Cayley-Hamilton theorem (e.g. Spencer, 1980)

and rewritten as:

E; = AymO (Fi)’ = exp(Lt)

(“.A;=,)

(8)

If 0 Q W, < 1 this can be written as (McKenzie, Ff can be illustrated in Mohr diagrams using

Representation of F, in Mohr circles

The well known Mohr circle for finite (cumula-

tive) strain (e.g. Choi and Hsu, 1971; Ramsay and

Huber, 1983; p. 93), illustrates Cauchy’s deforma-

tion tensor C=H,' . H, and is useful for the re-

construction of the finite strain ellipse from the

elongation of material lines and their rotation with

respect to each other. Its component H, or its

Lagrangian counterpart Ff as used in this paper

can also be illustrated by Mohr circles (Means,

1982). Contrary to the symmetric Cauchy’s defor-

mation tensor, H, or Ff contain information on

the bulk rotation of material lines between the

deformed and the undeformed state in the exter-

nal reference frame, and are therefore presented

by asymmetric matrices.

327

(a)

“& d

slnh(b&s.t) =R=&(q-s) f

W,.R =Q= W,m.R

Fig. 2. Presentation of cumulative deformation in real space (a) and in Mohr space (b) for the position gradient tensor 4. Points on

Mohr circle represent stretch and rotation of material lines read in polar coordinates. q-angle between material lines before

deformation; $-after deformation; $-rotation of material line k in external reference frame. Components of Fr represent

material lines coinciding with the external reference frame before deformation and plot as opposite points in Mohr diagram. Other

parameters explained in text.

the matrix components as Cartesian coordinates

(F,,? -4) and (k F12) to plot two points in Mohr space (Fig. 2b; Means, 1982, 1983). These points lie 180 o apart on the Mohr circle since they represent material lines which coincided with X1 and X, before deformation. Since a non-coaxial deformation history leads to asymmetric I;; tensors, such Mohr circles do not generally plot on the horizontal axis of the Mohr space coordinate system (Fig. 2b). Means (1982, 1983) has shown that, though tensor components are plotted and read as Cartesian coordinates in Mohr space, polar coordinates of a point on the I;rMohr circle repre- sent the stretch and rotation of a material line in true space resulting from deformation (Fig. 2). This rotation is measured with respect to the external reference frame in true space. The origi- nal angle between two material lines before defor-

mation is read as a double angle in the Mohr circle (Fig. 2). The angle between two deformed lines in true space cannot be directly read from the Ff Mohr circle, but must be calculated from the individual rotation of each line due to defor- mation and the original angle between the lines (Means, 1982). Material lines coinciding with finite strain axes plot along the line from the origin through the Mohr circle centre. Lines of no finite longitudinal strain plot on a circle arc with radius 1 around the origin of Mohr space (Fig. 2). The two lines of no finite rotation, which plot on the horizontal axis of the Mohr space, are eigenvec- tors of Ff. In the case of constant flow parameters, these lines coincide with flow apophyses throughout the deformation history. Eigenvalues of I;; are exp( IV&) and exp( - IQt).

32x

Mohr-circle parameters

Ff can be described in terms of s, W, and t

(eqns. 9, 11 and 12) but also by three parameters

which define the shape and position of its Mohr

circle (Fig. 2b); Q, the vertical elevation of the

circle centre; R, its radius and T, the distance of

the circle centre to the origin of Mohr space. If I;;

results from progressive defo~ation by isoplanar

constant fiow at 0 < W,, < 1, its Mohr circle

parameters Q, R and T are defined as:

Fn - F,, Q= 2

= $sinh( WdSt) d

F,, + 41 R= 2 = -$- sinh( W,sf ) (13)

d

and for W, = 1 (isoplanar simple shear flow) as:

Q=R=st

T==h-GF Q/R= W,=1

04)

Poku decomposition

A general description of cumulative deforma-

tion such as Ff consists of components of rigid

(b)

body rotation and finite strain. This is illustrated

in Mohr space by the off-axis position of the

Mohr circle. The two deformation elements can be

separated by polar decomposition according to:

I;,=F,.R,=R;F, (15)

where Fi and F, are left and right symmetric

stretch tensors which describe the strain compo-

nent and R, is an orthogonal rotation tensor:

R,= cos 6 sin 6 -sin S cos 6

(16)

where S is the angle of rigid body rotation. Polar

decomposition can be elegantly presented in Mohr

diagrams. Figure 3a illustrates how F, and F, can

be derived from Ff by multiplication with the

transpose of R, as:

Ft = Ff ’ R; and 1;; = R; . Ff (17)

Operation of Ry on Ff rotates the Mohr circle

over an angle S into a symmetric position on the

horizontal axis (DePaor and Means, 1984) where

S can be expressed as:

cosS=JT2-Q2/T

sin 8 = Q/T Cl@

The Mohr diagram illustrates clearly that F, and

Fr represent the same tensor for two different

orientations in the external reference frame. R

Fig. 3. a. Two possible ways of polar decomposition of Ff illustrated in Mohr diagrams. Small open circles represent material lines

coinciding with the external reference frame before deformation; &-angle of rotation imposed by RT. b. Mohr circle presentation of

the incremental position gradient tensor F, for isochoric deformation. Polar decomposition would shift the circle centre to 1,O ’ along

the solid arrow

329

and T have the same value in both tensors I;, and Fr or Fr:,; they contain the same finite strain com- ponent expressed by the ratio:

1 + e, T-I-R R,=_=_ 1 + e2 T-R (19)

Fi in Mohr circle presentation

Figure 3b shows how the incremental position gradient tensor Fi, which is derived from L by integration, and which is used to compose F,, can be presented in Mohr space in a way analogous to F,. Since the radius of the Fi Mohr circle is very small, for isoplanar flows it is symmetrically arranged around a vertical line through the point 1 on the horizontal axis of the Mohr diagram (1, 0 “), i.e. the point representing zero stretch and rotation. It is easy to see that because of the very small value of 6, either type of polar decomposi- tion of Fi has an effect identical to that of additive decomposition into a symmetrical and an antisym- metrical tensor:

1 sAt(1 + W,)

sAt(1 - W,) 1 1 This illustrates why Fi is decomposed additively, and Ff by polar decomposition; the results of additive and polar decomposition approach simi- lar values at infinitesimally small strains. Figure 3b also illustrates why it is possible to derive a correct description of Fi for non-coaxial incre- mental deformation by multiplication of a rota- tion tensor and a symmetric incremental position gradient tensor (e.g. Ramsay and Huber, 1983; session 12): this “lifts” the Fi Mohr circle from the horizontal axis by an amount WAt/2 propor- tional to the amount of incremental rotation.

Progressive deformation

The complete stretch and rotation history of all material lines in a two-dimensional deformation

can be illustrated in Mohr space plotting Ff circles for successive steps of the deformation in one diagram (Fig. 4). The result is a series of circles of increasing diameter which shift from the point 1, 0” on the horizontal axis into Mohr space (Bobyarchick, 1986; Passchier, in press). For a flow with constant parameters and ISA fixed in the external reference frame during progressive deformation, a point on the circle at a certain angle from the horizontal always represents the same material line (Fig. 4). It is therefore possible to follow individual material lines in Mohr space with progressive deformation and thus determine their stretch-rotation history. For progressive pure shear (Fig. 4a; W, = 0), the intersections of the circle with the horizontal axis shift from 1, 0” along the horizontal axis towards the origin (1 + e2 principal axis) or in the opposite direction (1 + e, principal axis). For 0 -C W, i 1 (Fig. 4b) lines of no finite rotation do not coincide with finite strain axes; the latter rotate away from the horizontal axis with progressive deformation. The angle be- tween lines of no finite rotation is identical to that between apophyses of the constant flow, and is a simple function of W, (Fig. 1; eqn. 10). For progressive simple shear (Fig. 4c; W, = l), one material line is ii-rotational and remains unde- formed (flow plane). Finite strain axes coincide with different material lines for each stage of the deformation (compare Figs. 2b and 4b, c). The vertical diameter of the circle is the shear strain of the flow plane (y). For W, > 1, i.e. for pulsating progressive deformation (Ramberg, 1975a, 1975b; DePaor, 1983) circles first _ increase in diameter, and then reduce to a point when approaching 1, 180 o (Fig. 4d). After passing this point, they grow again and eventually decrease in size when retum- ing to the point 1, 0 O. These deformations are characterized by a rhythmic strain accumulation and a rigid body rotation with respect to ISA and the external reference frame. Rigid body rotation (Fig. 4e) is represented by a point in circular orbit around the origin; no Mohr circles of finite radius are formed, i.e. all material lines have the same angular velocity and zero elongation.

(a> 1 W,.+J.o PROGRESSIVE PURE SHEAR

(d)w,,=1.5

_,- ~ ' - .

-1

‘t FLOW PLANE

k> y/_/ w,= m

,,-- / /

t

\ ‘*\ \ \ ---

‘r;

Fig. 4. Sets of Mohr circles for the position gradient tensor representing steps of progressive isoplanar deformation at vorticity

numbers W, = 0.0 (a); 0.5 (b); 1.0 (c); 1.5 (d) and infinity (e). Further explanation in text.

Mohr-circle centre curves

In the case of isoplanar flow, R depends on T

according to:

R=JTZ-1 (21)

and progressive deformation can be completely represented by plotting only the trajectory of Mohr

circle centres as defined by T and Q. The shape of this trajectory depends on W, and is therefore a suitable representation of the “deformation path” even if W, is variable during progressive deforma- tion. Figure 5 gives a complete presentation of Mohr-circle centre-paths for different but con- stant W, values. All deformation paths start in 1, 0 O, but only those representing W, > 1 can return

331

Fig. 5. Mohr circle centre paths for isochoric deformation at positive vorticity numbers.

to this starting point. Figure 6 shows two sets of

contours in the same diagram, indicating where

Mohr circle centres will plot for specific R, and

s . r values. Contours of equal R, values are sim-

ply circles around the origin of Mohr space il-

lustrating that Mohr circles at increasing distance

from the origin have an increasing radius and

represent increasing finite strain, independent of

W,. Curves of equal s. t value represent the posi-

tion occupied by a Mohr circle at a specific time

during the deformation for a specific value of s

and W,; doubling of the stretching rate results in

an identical Fr in half of that time. Non-coinci-

dence of curves for R, and s. t means that for a

certain stretching rate S, the strain accumulation

rate decreases with increasing W,; the strain ac-

cumulation is fastest for progressive pure shear

(Pfiffner and Ramsay, 1982).

Non-isoplanar flow history

Where flow instantaneously changes the area in

the X, - X, plane, i.e. if si + sz # 0, L can be

represented by a matrix;

1 a L=

so+ w,) so- w,) a 1

(22)

and a component L,, = s3. a is the rate of change

in area a = s1 + s2 and the instantaneous volume

change rate is defined as u = s1 + sz + s3; a non-

zero value of a does not necessarily imply a

volume change. In Mohr diagrams for L, a non-

I ‘i:

1 I i 5 1 2’ ,+e 3

Fig. 6. Curves in Mohr space giving positions of centres of Mohr circles for the position gradient tensor at specific S. t (heavy lines)

and R f values (thin fines). Isoplanar flow. Mohr circle centre paths for IV,, = 0.0, 1.0, 1.5 and infinity are stippled.

-1.0 . ’ I_ / &’

- 0.5 0.0

/ 3 x3 / / / / /

a 0.5 /

/

Fig. 7. Mohr diagrams for the velocity gradient tensor L and the incremental (Fi) and cumulative (F,) position gradient tensors for

W, = 0.5 and s = 1.0. The range of circles in each diagram illustrates the effect of different values of a as indicated; (c) shows the

effect of different u-values on the shape of Mohr-circle centre-curves for progressive deformation, each at constant flow parameters.

One F, Mohr circle, for s. t = 0.5, is drawn on each curve.

zero value of a leads to a horizontal displacement

of the Mohr circle centre (Fig. 7a). The magnitude

of s3 does not influence the shape and position of

the Mohr circle because W, = W/2s is used to

describe flow vorticity instead of Truesdell’s

kinematic vorticity number w, = w/

J2(S:+.

For non-isoplanar flow, a number of flow types

can be defined which lead to unfamiliar but prob-

ably realistic F,. tensors. If a = s W, or a = -s W,,

then material lines which neither rotate nor stretch

can exist in deformation types other than simple

shear, even in pure shear. Shear zones with an

undeformed wall rock and volume change compo-

nent fall in this range (Ramsay, 1980). If a > s or

a < -s, lines of no finite longitudinal strain do

not exist in the cumulative deformation.

Integration of L for non-isoplanar flow yields

for Fi:

l+uAt Fi =

sAt(1 + W,)

sAt(l- W,) 1 +uAt 1 (23)

which shifts the Mohr circle for Fi by a distance a

parallel to the horizontal axis of the diagram (Fig.

7b). For constant flow parameters s, a and W,,

accumulation of incremental strains result in a

general equation for F,:

F, = exp( Lt ) = exp( at) { exp( Lt)‘“=“‘} (24)

This means that matrix notations for Ff accu-

mulated by non-isoplanar flow are simply derived

from the matrices in eqns. (9) (11) and (12) by

multiplication of the components by the scalar

exp( a. t):

Ff =

exp( at) . cosh( W,st) cl+ VI) exp(at) sinh(W st>

wd d

(0 < w, < 1) I (1 - YJ exd4 sinht w st)

w, d

2st exp( at)

expW 1

333

Ff Mohr circles for a # 0 are related to those for

a = 0, discussed in the first section of this paper

by larger or smaller T and R (Fig. 7~). The

relationship between Mohr circle parameters T, R

and Q, flow parameters a, s and W, and cumula-

tive deformation parameters R, and 1 + AA are

shown in Table 1 and Fig. 8. The most important

results are (s and W, always positive):

(1) T, Q and R are changed in magnitude by a

factor exp( at). This means that the equation Q/R

= W, (13, 14) is equally valid for isoplanar and

non-isoplanar flows, and that the angle between

lines of no finite rotation is a function of W, only.

(2) Mohr circle centres move away from 1, 0’

with progressive deformation, either towards infin-

ity if u/s > 0 and - u/s x W,; towards the origin

if-u/s> W, or if W,>l and u<O; and to-

wards an end position with Cartesian coordinates

(0.5; 0.5 W,/W,) if - u/s = W, (Fig. 8). In the

last case one of the flow apophyses (AP, in Fig. 1)

remains undeformed, which fixes one point of all

the Ff Mohr circles for progressive deformation at

l,OO.

(3) If a < 0 and - u/s < W,, Mohr circle centres

follow a path to infinity but reverse direction

during the first part of the deformation history.

This even applies for pure shear histories.

(4) If W, > 1, Mohr circle centres either move

in spirals to infinity (a > 0), follow a closed orbit

around the origin (a = 0) or follow a spiral to-

wards the origin (a < 0).

exp( at) cosh( Wdst)

Ff =

exp( ut) cos(iwdst) (1 + W,) expt4 sintiWdst)

iWd

(W, > 1) (’ - I%) exp(ut) sin(iWdst) iWd

exp( at) cos( iW,st)

(25)

334

(5) Cumulative area change 1 + AA relates to

cumulative volume change 1 + AV’ by:

1 + AV= (1 + e3)(1 + AA) (26)

With no stretch along the X, axis, 1 + AA repre-

Effect of variable flow parameters

If s, u or W,, change in the course of progres-

sive deformation, F{ can be modelled by sequen-

tial multiplication of different Fi tensors. This sents the volume change. means that an analytical approach to express F, in

5 i

3 I

0.1

or

-0.1

-I.(

-l.I

I

5 11

L! /

CURVE TO Cn

-----b- L_fc -----_-

, k REVERSE0 CURVE TO 0)

/-y

CURVE TO 0

I OUTWARDSPIRAL TO c/)

I

‘b I ’ I INWARD SPIRAL TO 0

/

I

/ /- L 1.0 1.5

1.0 wd 0.5 010 Fig. 8. Scheme showing the effect of different but constant values of W, and = on the shape of F(-Mohr circle centre paths. Mohr

circle centres either move towards infinity (above solid curve), towards the origin of Mohr space (below solid curve) or towards a

fixed pole in Mohr space (on the left part of the solid curve for W, < 1). Closed orbits around the origin occur on the right-hand right

part of the solid curve for W, > 1. Further explanation in text.

335

TABLE 1

Equations for parameters of flow and deformation

Flow parameters Mohr-circle parameters Deformation parameters

l/W,{exp(at) cosh2(Wdst)-W:} =dm= T=m =;{dm

(W”<l)

l/W,{(exp(nt).sinh(W,sr)} =dv = R=m =&I-

(K < 1) -d_}=F

l/W,{(exp(ar). W;sinh(W,sZ)} = Q = Wnm. R

(w,(l)

w,

exp(nr)/Wd(( cosh2( Wdsr) - W,’ +sinh(W,sr)}

(w, < 1)

exp(at)/Wd{( cosh2( Wdsr) - W,” - sinh(W,sr))

(K-=1)

exp(2sr) = (W, = 0)

exp(2nr)

= Q/R = T+R

= T-R

T+R T-R

= T2-R2

= w"m

=l+e,

=l+e,

=R,

=l+AA

terms of flow parameters will be possible in some special cases only. Changes in s result in a change in the accumulation velocity of finite strain, but do not influence the path of the Mohr circle centre. Changes in W,, however, always have a strong effect on the progress of the centre, while a has the greatest effect at high vorticity numbers (Passchier, in press). A different t cannot com- pensate for these effects as it can for changes in s. Thus, if the changes in flow parameters are known, it is always possible to draw a deformation path in Mohr space that does not necessarily coincide with any of the families of curves in Figs. 5 or 8. It is easy to prove numerically that even for such complex deformation paths the relationships given on the right-hand side of Table 1 do still apply. This means, that the Mohr circle radius R for each point on a complex deformation path only depends on T and the cumulative area change, 1 + AA. If flow parameters and their change with time are known, the Mohr circle centre path is an unambiguous and useful presentation of the defor- mation path, carrying all information on the stretching and rotational history of every material line. It should be noted, however, that points on the Mohr circles at equal angle with the horizontal

do not generally represent the same material line in subsequent circles along the deformation path as they do for flow with constant parameters. Since the original angle between two material lines is plotted as a double angle in the Mohr circle and does not change for any flow type, the rotational behaviour of at least one material line, e.g., one which coincided with one of the ISA at the onset of the deformation, must be known.

Discussion

The rotation component of I;;

The position gradient tensor Ff describes the change in position of material particles from the undeformed to the deformed state, and contains no information about the actual path these par- ticles followed to reach their final position. For example, any position of an Fr-Mohr circle centre in Figs. 4, 5 or 8 can be reached along a curve for constant flow parameters, by a pure shear history followed by a rigid body rotation, or any other path. Therefore, Ff can always be split by polar decomposition into a symmetric tensor which de- scribes finite strain and area or volume change,

and an antisymmetric rigid body rotation tensor

R,. The latter can be described by the ratios

Q/T = sin 6 (Fig. 3; eqns. 15 and 16) or by the

more useful ratio Q/R, as outlined below. Unlike

finite strain and volume change, the antisymmetric

tensor component R, of Ff has rarely received

any attention of geologists in the past: it can be

changed by simply altering the orientation of the

external reference frame in real space in which Ff is described. Therefore, R, is usually eliminated

through multiplication of Ff by its transpose to

produce the symmetric Green’s deformation

tensor.

Although different deformation histories can

produce identical position gradient tensors, e.g.,

progressive simple shear and pure shear followed

by rigid body rotation, both deformation histories

can produce distinctly different fabrics in rocks.

This discrepancy arises because, unlike viscous

fluids, the fabric of most naturally deformed rocks

has a “memory” for certain aspects of the defor-

mation history, such as flow vorticity. For geolo-

gists, it is therefore useful to pay attention to the

nature of the usually neglected antisymmetric

tensor component of Ff.

The “mean” vorticity number

If deformation accumulates by constant flow

with ISA fixed in an external reference frame, the

parameters describing Ff such as Q, R and T or

R, and 1 + AA can be expressed in terms of three

flow parameters s, a and Wn (Table 1). Also, in

this special case the ~tisy~et~c tensor compo-

nent of F, can be expressed as a function of the

flow vorticity number and the principal stretches

following:

sinS=Q/T= W,te, - 4 2+e +e

1 2

or:

Q/R = icy,

In the case of a variable flow vorticity number W, during progressive deformation, it can be written

14:

sin S = Wi”fe, - e2)

2 + e1 i- e, or:

Q/R = Wn”

’ (28)

where Wnrn represents a “mean” value of W, for

the entire deformation history. This “mean”

vorticity number Wnm describes the bulk rotation

of material lines which coincide with principal

strain axes of Ff, with respect to ISA. Wnrn is a

parameter of cumulative deformation which, to-

gether with R, and 1 + AA, can be used to define

F;. IIowever, W,” carries information on the de-

formation history, specifically on the flow vortic-

ity during progressive deformation, while R, and

1 + AA do not.

If ISA are not fixed in the external reference

frame in which progressive deformation is de-

scribed, the angle 6 or the ratio Q/R will contain

a factor Wnrn dependent on flow vorticity, and a

factor describing the bulk rotation of ISA; in

theory, R, or Q/R can be decomposed into these

two factors in a way analogous to d~omposition

of the antisymmetric tensor component of L into

factors describing vorticity and spin of ISA (Lister

and Williams, 1983).

Reco~st~ct~on of 1”; for n~t~r#~ly deformed rocks

In many tektonites, it would be useful to know

not only the finite strain and volume change, but

also the shape of the path in Mohr space along

which deformation accumulated. This would help

to establish the regional tectonic history, and could

also give more information on the distribution

pattern of vorticity in deforming rocks and the

influence of developing anisotropies or other fabric

elements on such patterns.

In practice, accurate reconstruction of the de-

formation path may not be possible, but the fabric

of many naturally deformed rocks contains enough

information for a rough approximation. First of

all the position of the Ff Mohr circle for cumula-

tive deformation should be reconstructed. Two

possible approaches are:

(1) Direct construction of the Mohr circle from

337

fabric data (Passchier and Urai, in press). This can be realized using fibre-filled crack-seal veins, if sigmoidal fibres grew during vein rotation with respect to the instantaneous extension axis; if three veins of initially different orientation can be studied, their rotation as indicated by the curved fibres and their stretch value allow three points to be plotted in Mohr space such that the FrMohr circle can be constructed and I+$“’ can be de- termined from the ratio Q/R. Alternatively, a combination of data on R,, 1 f AV and the stretch and rotation of one vein may also allow construc- tion of the F, Mohr circle (Passchier and Urai, in press).

(2) Calculation of Wnr” from fabric data without reference to ISA orientation. Fabric elements such as the angle of rotation of rigid objects at a specific R,; sets of shortened, extended and short- ened/extended veins; the axial ratio of blocked or permanently rotating ellipsoidal rigid objects, are a function of Wn’” (Passchier, in press). Analysis of such fabric elements in a deformed rock can be done in any external reference frame; they have a “memory” for Warn, but not for bulk rotation of ISA in the external reference frame.

Finally, attempts can be made to find the ac- tual deformation path which led to the final Ff position. If data on the “mean” vorticity number are compared for several stages of the deformation history, or if the Wnm value for the entire deforma- tion history is compared with the W, value for the last deformation stage, it is possible to get a rough impression of the pattern of variation of W, with time (Passchier, in press). Such patterns can be visualized if they are plotted as Mohr circle centre

curves for Ff, which can be compared with the standard curves of Figs. 4 and 8.

Conclusions

Mohr circles help to visualize the effects of flow parameters on progressive deformation. Four parameters; mean stretching rate in a plane nor- mal to the vorticity vector; rate of area change; vorticity, and stretching rate along the vorticity vector are sufficient to describe any type of flow in which the vorticity vector is parallel to one of the instantaneous stretching axes. Incremental and

cumulative deformation can be described as posi- tion gradient tensors in terms of these flow param- eters, and illustrated by off-axis Mohr circles. Progressive deformation histories can be il- lustrated as series of such Mohr circles for succes- sive time intervals or, more elegantly, as curves in Mohr space representing the path of the centre of the Mohr circle with time. Mohr diagrams and Mohr circle centre paths may also serve to recon- struct flow parameters from fabric data in natu- rally deformed rocks, or at least to determine which part of a reconstruction can be attempted with the available data. Besides finite strain and area or volume change, the “mean” vorticity num- ber Wnm can be defined as a third parameter for cumulative deformation. This parameter describes the mean value of flow vorticity along the defor-

mation path. It can be calculated for many natu- rally deformed rocks since several fabric elements have a “memory” for Wnm.

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