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Journal of Structural Geology,Vol. 13, No. 1, pp. 101 to 104, 1991 0191-8141/91 $03.00+0.00Printed in Great Britain (~ 1991 Pergamon Press plc

T h e c l a s s i fi c a t i o n o f d i l a t a n t f lo w t y p e s

C. W. PASSCHIER

Instituut voor Aardwetensch appen, Utrecht University, Budapestlaan 4 , P . O . Box 80.021, 3508 TA Ut recht ,The Netherlands

(Received26 February1990; accepted in revised orm 4 July 1990)

A b s t r a c t The kinematic vorticity number $V has been used to classify the geometry of two-dimensional,non-dilat ant flow as a function o f flow vorticity. A similar kinematic dilatancy numbe r M can be defin ed, whichexpresses flow geometr y as a function of dilatancy rate. General two-dimensio nal low geometry can be descri bedin a simple, non-am biguous way by just W" and M. Mohr circles for flow can b e us ed to illustrate this classificationmethod.

INTRODUCTION

THE fabric geometry in a deform ed material depends in

part on the geometry of the parent flow, i.e. on the shape

of the velocity field during de formation. Definitions of

flow geometry are the refore an essential part of studies

on deformat ion in rocks (cf. Ramberg 1975, McKenzie

1979, Free man 1985, Bobyarchick 1986, Passchier 1986,

1988a). Two-dimensional and 'plane-strain' flow types

are most popu lar in geological modelling. In the absence

of dilatation (here used to mean area-change in the

plane o f the flow), they can be satisfactorily described in

terms of pure shear and simple shear components, or

more exactly by a vorticity number (Truesdell 1954,

Ramberg 1975, Means e t a l . 1980). A classification

problem may appear with the introduction of dilatation

into models of flow and progressive deforma tion. Figure

l(a) illustrates progressive non-dilatant deformation by

steady-state simple shear flow. Dila tant flow types such

as those responsible for progressive defo rmation in Figs.

l(b) & (c) are less easy to define. Clearly, some para-

meters must be chosen to describe such flow geometries

in an unambiguous way. The four-component velocity

gradient tensor L gives a complete mathematical de-

(a )

( b )

i a ) r e a l s p a c e

L = [ ~ q ]

(b ) Mohr circlefOr flow

,q)

~ 1 o

£

Ico

( c ) t b (d) ~ simplet~raer shear

Fig. 2. (a) The angular velocity ~b and stret ching rate k of materiallines in real space subject to non-dilatant two-dimensional flow asdescribed by the velocity gradient tensor L can be plotted to produce(b) a Mohr circle for flow in Cartesian & - k space. &l and d~ aredirections of maximum and minimum angular velocity; kl and k2 aredirections of maximum and mi nimum stretching rate; clo sed circles arepoints representing co-ordinate axes in real space, constructed byplotting coefficientsof L in pairs, as indicated. (c) Mohr circle for pure

shear flow; (d) M ohr circle for simple shear flow.

scription of flow and could serve this purpose. L, how-

ever, contains information on the orientation of flow

eigenvectors and on strain rate besides flow geometry.

Two parameters are sufficient to describe flow geometry

completely, as explained below.

( c )

Fig. 1. Three sequences of progressive deformation produced bydiffere nt steady-s tate flow types: (a) no n-dilatan t simple shear; (b) &

(c) dilatant flow types.

DESCRIPTION OF FLOW

Any material line in a plane subject to homogeneo us

non-dila tant flow has a specific angular velocity (~b) and

stretching rate (~), depending on its orientation (Fig.2a). When values for an infinite number of differently

oriented material lines are plotted as points in a Carte-

sian tb - k space, they define a circle known as the Mohr

101

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10 2 C . W . P A S S C H I E R

c i rc le fo r f low (F ig . 2b ) (Lis te r & Wi l l i ams 1983 , Pass -

ch ier 1986 , 1988b) . The ang le be tween mater ia l l ines

m e a s u re d a ro u n d t h is c ir c le ( a s in a n y o t h e r M o h r c i r c l e )

is t w i c e t h e a n g l e m e a s u re d i n r e a l s p a c e . M a x i m u m a n d

min im um s t re tch ing ra tes , ka and e2 , occ ur on o r tho -

gonal mater ia l l ines , def ined by a ho r izon ta l d iagonal

t h ro u g h t h e c i r c l e (F i g . 2 b ) ; m a x i m u m a n d m i n i m u mangu lar ve loc i t i es , &a and &2 , o f mater ia l l ines occur on

o r t h o g o n a l l in e s d e f i n e d b y a v er t i c a l d i a m e t e r t h ro u g h

the c i rc le (F ig . 2b ) .

A M o h r c i r c le c e n t r e d o n t h e o r i gi n o f t h e r e f e r e n c e

f r a m e r e p re s e n t s p u re s h e a r f l o w (p u re s h e a r in g ) (F ig .

2c) ; a c i rc le cen t red on the ver t i ca l ax i s and t ransec t ing

t h e o r i g i n r e p re s e n t s s i m p l e s h e a r f l o w ( s i m p l e s h e a r in g )

(F ig . 2d ) . A c lass i fi ca t ion o f f low in the range be tw ee n

p u re a n d s i m p l e s h e a r s h o u ld t h e r e fo re b e b a s e d o n t h e

ra t io o f the ver t i ca l pos i t ion (cb) o f the c i rc le cen t re to the

s ize o f the c i rc le rad ius . Th is ra t io can be desc r ibed as W,

a k i n e m a t i c v o r t i c it y n u m b e r o f f l ow (Tru e s d e l l 1 95 4,

M e a n s e t a l . 1980, Passchier 1986):

e[~ ( ] ' )1 q" ( ] )2 _ d) l "+- O)2- - - - c o s a , ( 1 )

0)1 - & 2 2 r

wh ere (&a + &z) i s the f low v or t i c i t y a n d

E 1 - - k 2 _ _ d ~ l - d )2r - ~ 2 (2)

i s the c i rc le rad ius , a i s the ang le m eas ured in geo me t r ic

( r e a l) s p a c e b e t w e e n A a a n d A 2 , t h e d i r e c t io n s o f z e ro

a n g u l a r v e l o c i t y (F i g . 3 ) (R a m b e rg 1 9 7 5 , P a s s c h i e r

1 9 8 8 a ,b ). F o r p u re s h e a r f l o w W = 0 , a n d fo r d e x t r a l

s imple shear f low W = 1 .

I f d i l a tancy occu rs , f low can st il l be p rese n ted as a

M ohr c i rc le , bu t i t s cen t re wi l l no t l i e on the ver t i ca l (d) )

ax i s (p r oo f in Appe nd ix ) (F ig . 2 ) . I f the cen t re l i es to the

l e f t o f th e o r i g i n , t h e a r e a o f a n o b j e c t d e fo rm i n g b y

s tead y-s ta te f low is p rogress ive ly dec reas ing ; i f i t li es to

the r igh t , i t is p rogress ive ly increas ing .

B y a n a l o g y w i t h W , i t i s p o s s i b l e t o q u a n t i fy t h e

h o r i z o n t a l d e v i a t i o n o f t h e M o h r c i r c l e b y a k i n e m a t i c

d i l a ta n c y n u m b e r sg, the ra t io o f the ho r izon ta l dev ia t ion

of the c i rc le cen t re and th e c i rc le rad ius ;

E 1 - - k 2 2 r

w h e re f l i s t h e a n g l e m e a s u re d i n g e o m e t r i c ( r e a l ) s p a c e

b e t w e e n B a a n d B 2 , t h e d i r e c t io n s o f z e ro i n s t a n t a n e o u s

s t r e t c h in g r a t e ( a l s o k n o w n a s ' li n e s o f n o i n s t a n t a n e o u s

long i tud ina l s t ra in ' ) (F ig . 3 ) . A and B axes a re no t

n e c e s s a r i ly o r t h o g o n a l .

C L A S S I F I C A T I O N O F D I L A T A N T F L O W

~ 0

W =cos 13

A2X,, A ! "~

F i g . 3 . T h e g e o m e t r y o f f lo w , o r t h e o r i e n t a t i o n o f t h e M o h r c i r c l e

w i t h r e s p e c t t o i t s r e f e r e n c e f r a m e , c a n b e e x p r e s s e d i n t e r m s o f ° W , t h e

k i n e m a t i c v o r t i c i t y n u m b e r , a n d s ~ , t h e k i n e m a t i c d i l a t a n c y n u m b e r .

A 1 a n d A 2 a r e d i r e c t i o n s o f z e r o a n g u l a r v e l o c i t y ; B 1 a n d B 2 a r e

d i r e c t i o n s o f z e r o s t r e t c h i n g r a t e .

- 1 < W < 1 ; B a x e s e x is t o n l y i f - 1 < ~ < 1. T h r e e

b a s i c t y p e s o f fl o w g e o m e t ry c a n b e d i s t i n g u i s h e d , d e -

p e n d i n g o n t h e r e l a t i v e p o s i t io n o f A a n d B a x e s .

T y p e I : W 2 + g/2 < 1

The se f low types p lo t in s ide the c i rc le o f F ig . 4 and a r e

c h a ra c t e r i z e d b y t w o A a n d t w o B a x e s o f o p p o s i t e si gn .

F o u r t y p e s o f i n s t a n t a n e o u s d e fo rm a t i o n o f m a t e r i a l

l ines a re poss ib le ; dex t ra l ly ro ta t ing and ex tend ing ;

dex t ra l ly ro ta t ing and shor ten ing ; s in i s t ra l ly ro ta t ing

and ex tend ing ; and s in i s t ra l ly ro ta t ing and shor ten ing

l in e s . M o s t o f t h e f l o w t y p e s r e s p o n s i b l e fo r n a t u r a l

d e fo rm a t i o n w i t h s m a l l v o r t i c i t y n u m b e r s a n d l i m i t e dd i la tancy p lo t in th i s ca tegory . Pure shear f low i s a wel l -

k n o w n e x a m p l e (F i g . 4 d ) .

T y p e H : W 2 + ~/2 = 1

Th e s e f l o w t y p e s p l o t o n t h e c i rc l e in F i g . 4 . Tw o o r

t h r e e t y p e s o f r o t a ti n g l i n e s e x is t , a n d o n e A a n d B a x is

co inc ide ; i . e . there i s one l ine wh ich i s ne i ther ro ta t ing

n o r s t r e t c h in g . F l o w t y p e s i n n a t u ra l s h e a r z o n e s w i t h

r ig id wal l rocks be long to th i s ca tegory . S imple shear

f low wi th ou t d i l a ta t ion (F ig . 4a) is a we l l -know n

e x a m p l e .

T y p e I I I : W 2 + s ~2 > 1

The se f low types p lo t ou t s ide the c i rc le in F ig . 4 . A o r

B a x e s h a v e t h e s a m e s i g n a n d t h r e e o r l e s s t y p e s o f

ro ta t ing l ines ex i s t (F ig . 4e) . S om e f low types wh ich

e i t h e r l a c k A a x e s ( su p e r - s i m p l e s h e a r ; D e P a o r 1 9 8 3) ,

o r B a x e s (p u re d i l a t a t i o n ) b e l o n g t o t h i s c a t e g o ry .

T h e s e f lo w ty p e s a r e p r o b a b l y u n c o m m o n i n n a tu r a l

d e fo rm a t i o n .

I t w i l l b e c l e a r f ro m F i g . 3 t h a t t h e g e o m e t ry o f t h e

M o h r d i a g ra m fo r fl o w , a n d t h e r e fo re o f t w o -d i m e n s i o n a l f l o w i ts e l f, c a n b e c o m p l e t e l y d e s c r i b e d b y

t w o p a ra m e t e r s , W a n d ~ . F i g u re 4 i l lu s t r a te s t h e ful l

r a n g e o f p o s s i b le f l o w g e o m e t r i e s . A a x e s e x is t o n l y i f

D I S C U S S I O N

In p ro g re s s i v e d e fo rm a t i o n b y a d e x t r a l s i m p l e s h e a r

f low wi th ou t d i l a ta t ion (F ig . l a ) , one m ater ia l l ine i s

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The classification of dila tant flow types 103

M o b r P i n g . m i r ed i . ~ m m s

for f l ow deformati on

F i g. 4 . F ~ w g e ~ m e t r y d i a g r a m i ~ u s t r a t i n g a ~ p ~ s s i b ~ e t w ~ - d i m e n s i ~ n a ~ w t y p e s a s a f u n c t i ~ n ~ f W a n d s ~ . I , I l a n d l I I a r e

main f l ow ca t egor i es . I n se t a t r i gh t shows M o hr d i ag r ams and p r ogr ess ive de f o r m at ion seq uences f o r som e spec i fi c f l owtypes ( a - e ) .

neither stretching nor rota ting, while all other lines are

rotating dextrally. If we take the presence of a non-

rotating, non-stretching material line as a characteristic

of simple shear deformation, the flow type responsible

for progressive deformat ion in Fig. l(b) could be classi-

fied as a simple shear flow with area-increase (McCaig

1987, fig. 8). In this flow type, however, one group of

material lines is rotating sinistrally. This is a character-

istic of non-dilatant flow types between pure and simple

shear; apparently, the addition of a dilatation com-ponen t to simple shear changes the vorticity number as

used here. The flow type responsible for progressive

deformation in Fig. l(c) contains a non-rotating, non-

stretching material line as for the flow types of Figs. l(a)

& (b). Nevertheless, it would be classified by most

geologists as a pure dila tation, or a pure shear flow with

area-increase, rather than a flow type related to simple

shear.

Apparently, the flow types responsible for progress-

ive deforma tion in Fig. 1 cannot satisfactory be classified

with the available terminology. In the classification

suggested here, they all belong to flow Type II (Figs. 4a-

c), but have d ifferent vorticity and dilatancy numbers. In

order to avoid confusion as outlined above, it may be

advantageous to reserve the terms 'simple shear' and

'pure shear' to non-d ilatant flow types (Fig. 4; ~ = 0

axis), and to use W and ~ values to give an exact, non-

ambiguous description of flow geometry. Mohr dia-

grams for flow can easily be construc ted from W and ,~

values, and can be used to visualize the different flow

types. Obviously, many alternative pairs of parameters

could be used to classify two-dimensional flow geom-

etry, but the method suggested here has the advantage

that it combines the already familiar kinematic vorticity

number with a similar paramete r, and allows an easy linkto Mohr constructions.

A cknowledgemen t s - - - Chr i s T a l b o t p r o p o s e d u s e o f t h e t e r m ' k i n e m a -

t i c d i la t a n c y n u m b e r ' f o r t h e p a r a m e t e r ~ . A r e v i e w b y A n d y B o b y a r -ch i ck he lped to impr ove the manuscr ip t .

REFERENCES

Bobya r ch i ck , A . R . 1986. T he e igenva lues o f s t eady s t a te f l ow in M ohrspace . Tec tonophys i cs 122, 35-51.

De P aor , D . G . 1983 . Or thogr aph ic ana lys is o f geo log ica l s t r uc tu r es . I .D e f o r m a t i o n t h e o r y . J. Struct. Geol. 5 , 2 5 5 - 2 7 8 .

F r eem an , B . 1985 . T he mot ion o f r i g id e l l i p so ida l pa r t i c l es i n slowflows. Tectonophysics 113, 163-183.

L i s t e r , G . S . & Wi l l i ams , P . F . 1983 . T he pa r t i t i on ing o f de f o r mat ionin f l owing rock masses . Tec tonophys i cs 92, 1-33.

M a l v e r n , L . E . 1 9 6 9 . Introduction to the Mechanics of a Con t inuousM e d i u m . P r e n t i c e - H a l l , E n g l e w o o d C l i ff s , N e w J e r s e y .

M c C a i g , A . M . 1 9 8 7 . D e f o r m a t i o n a n d f l u i d - ro c k i n t e r a c ti o n i nmetasomat i c d i l a t an t shea r bands . Tec tonophys i cs 135 ,121- 132 .

M cK enz ie , D . 1979. F in i t e de f o r mat ion du r ing f l u id fl ow. Geophys . J .R . astr. Soc. 5 8 , 6 8 9 - 7 1 5 .

M eans , W. D. 1983 . App l i ca t i on o f t he M oh r - c i r c l e cons t r uc t ion t op r o b l e m s o f i n h o m o g e n e o u s d e f o r m a t i o n . J. Struct. Geol. 5 , 2 7 9 -286,

M eans , W. D. , Hobbs , B . E . , L i s t e r , G . S . & Wi l l i ams , P . F . 1980 .Vor t i c i t y and non- coax ia l i t y i n p r og r ess ive de f o r mat ion . J. Struct.Geo l . 2 , 3 7 1 - 3 7 8 .

Passch i e r , C . W. 1986 . F low in na tu r a l shea r zone s- - th e consequ ences

o f sp inn ing f l ow r eg imes . Earth Planet . Sci . Let t . 77 , 70 - 80 .Passch i e r , C . W. 1988a . Ana lys i s o f de f o r m at ion pa ths i n shea r zones .

Geol . R dsch . 77, 309-318.Passch i e r , C . W . 1988b . T he use o f M ohr c i r c l es t o desc r ibe non-

coax ia l p r og r ess ive de f o r mat ion . Tec tonophys i cs 149 ,323- 338 .Passch i e r , C . W. I n p r ess . Recons t r uc t ion o f de f o r m at ion and f l ow

p a r a m e t e r s f r o m d e f o r m e d v e i n s et s . Tectonophysics.Ram ber g , H . 1975 . Pa r ti c l e pa ths , d i sp l acem en t and p r ogr ess ive s tr a in

app l i cab l e t o r ocks . Tec tonophys ics 28 , 1173-1187.T r uesde l l , C . 1954 . The Kinematics of Vort ici ty . I nd i ana Un iver s i t y

P r e s s, B l o o m i n g t o n .

APPEND IX

Pr oo f o r the statement that a M ohr circle for f low centred o n the verticalreference axis ( d~-axis) represents non-dilatant flo w

Flow in two d imens ions can be r ep r es en t ed by the ve loc i ty g r ad i en tt enso r :

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1 0 4 C . W . P A S S C H IE R

T h e M o h r c i r c l e f o r f l o w ( F i g . 2 b ) c a n b e c o n s t r u c t e d f r o m t e n s o r

c o e f f i c i e n t s b y p l o t t i n g ( p , - u ) a n d ( v , q ) a s C a r t e s i a n c o - o r d i n a t e s i n

t h e r e f e r e n c e f r a m e ( M e a n s 1 9 8 3) ; t h e M o h r c i rc l e p a s s e s t h r o u g h

t h e s e p o i n t s a n d i s c e n t r e d o n t h e d i a g o n a l c o n n e c t i n g t h e m ( M e a n s

1 9 83 ). T h i s i m p l i e s ( F i g . 3 ) , t h a t t h e M o h r c i r c l e i s c e n t r e d o n t h e

ve r t i c a l a x i s i f p = - v .

I n t e g r a t io n o f L g i v es t h e p o s i t i o n g r a d i e n t t e n s o r f o r i n c r e m e n t a l

d e f o r m a t i o n :

f Ip ' A t + I q 'A t ) ( A 2 )F i = L d t = ~ u . A t v . A t + 1 '

w h e r e A t i s t h e t i m e o v e r w h i c h t h e i n c r e m e n t a l d e f o r m a t i o n o p e r a t e s .

I f t h e d e t e r m i n a n t o f F i = 1 , i . e . i f :

( p . A t + 1 ) ( v . A t + 1 ) - - q u . A t 2 =p v . A ~ + p . A t + v . A t + l - - q u . A ~ = l ( A 3 )

t h e r e i s n o i n c r e m e n t a l a r e a c h a n g e . I f At -- -~ 0 , t h e e q u a t i o n ( A 3 )

r e d u c e s t o

p . A t + v . A t = 0 , i . e . p = - v . ( A 4 )

C o n s e q u e n t l y , a M o h r c i r c l e f o r n o n - d i l a t a n t f l o w is a l w a y s c e n t r e d o n

t h e v e r t i c a l r e f e r e n c e a x i s , a n d c i r c l e s w h i c h a r e o f f - a x i s r e p r e s e n t

i n s t a n t a n e o u s a r e a i n c r e as e o r d e c r e a s e .