Computers and Geotechnics 11 (1991 ) 173-208

MODELLING OF ANISOTROPIC SAND BEHAVIOUR

M. Pastor Sector Ingenleria Computacional

Centro de Estudios y Experimentacl6n de Obras P~blicas Alfonso XII 3

28014 Madrid (Spain)

ABSTRACT

This paper presents an extension of a previous model for isotropic granular soils, to account for both initial and stress-induced anisotropy. The proposed approach is based on Generalized Plasticity Theory, and introduces material micro-structure through a second order tensor, which is used to define a new set of "modified invariants". LoadinE-unloadin E direction n and plastic modulus H are then related to micro-structure.

Model predictions are checked against experimental data obtained on sand exhibiting structural anisotropy which is caused by a preferred orientation of Erains alon E horizontal planes. This initial anisotropy is subsequently modified by loading.

INTRODUCTION

Samples of sand from natural deposits exhibit anisotropic behaviour

caused by the arrangement of their major axes on the bedding planes. When

tested, it is usually found that material response is much stiffer

alone deposition direction and that behaviour depends not only on the

intensity of the applied stress but on orientation of the sample. This

initial or fabric anisotropy varies through the test as grains are

rearranged.

These effects can cause important errors when trying to determine 173

Computers and Geotechnics 0266-352X/91/$03.50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

174

liquefaction or cyclic mobility strength of a particular sand from

standard triaxial tests as orientation of principal stress axes relative

to bedding planes is not kept. In addition to that, rotation of principal

stresses can cause an increase in pore water pressure leading to failure

under undrained conditions.

Extensive testing has been carried out during the last decade to

investigate basic features of anisotropic granular materials [I-5].

Modelling of anisotropic materials has also attracted the effort of

many researchers and has become a focal point of interest [6-22]

The purpose of this work is to show how Generalized Plasticity

Theory [23] can be extended to model the behaviour of granular soils

with a initial or stress induced anisotropic fabric. The model was

previously applied to sands under earthquake loading conditions and

showed good agreement with experimental results [24-2?]

and centrifuge model tests. [28]

GENERALIZED PLASTICITY

B a s i c Theory

If material response does not depend on the velocity at which stress

vary, i.e. behaviour is elastoplastic, increments of effective stress

and s t r a i n are r e l a t e d by

o r

d¢' = D ep : dc

d~ = C ap : d~' (I)

where the fourth-order constitutive tensors C ep and D ep depend not only

on the stress and strain state and material history but on direction of

s t r e s s increment as well.

It Is to be noted that Increment of co-rotational stress is of the

Jaumann-Zaremba type, which can be related to Cauchy stress increment d~

by the relation

where

175

d c = d& + q dw ÷ o" dw tJ tJ iX kJ Jk ki

1 ( v i , j vj tldt (2) dwi j = ~ - ,

v I = l-th component of particle veloclty

In what follows, all stress increments will be assumed co-rotational

and the superscrlpt wlll be dropped.

Next, deformation will be assumed to be caused by M mechanlsms of

deformation, all of them subjected to the same stresses (Series

mechanisms)

M

dc = Z d~(') (31 I=1

T h e s e c a n b e c h o s e n a s p h y s i c a l m l c r o p l a n e s , [ 17 ] a s p l a n e - s t r a i n

m e c h a n i s m s [ 1 8 ] , o r a s r e p r e s e n t i n g d i f f e r e n t p h y s i c a l p h e n o m e n a s u c h a s

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

A dlrectlon n (=) is postulated for each mechanlsm such that It

separates all poslble stress increments In two classes, wlch will be

referred to as loading and unloadlng

d ~ ' : n (a) > 0 l o a d l n g

d ~ ' : n (~) < 0 u n l o a d i n g ( 4 a )

Neutral loadln8 corresponds to the limit case for whlch

d ~ ' : n (m) = 0 ( 4 b )

Now, dependence of material behavlour on dlrection of stress

Increment can be formulated In a simple way by Introducing two dlfferent

constltutlve tensors for loadlng and unloadlng

176

dE (m) C (m) = : d~' loading ~ -L

de (m) C (m) = : d ~ ' u n l o a d i n g - u -

C5)

Taking into account all M different mechanisms it can be seen that

2 M tensorial zones [20] have been introduced. For every zone, a different

constitutive tensor is defined and therefore material behaviour will be

dependent on direction of the stress increment.

The condition of continuity of material response for a given

mechanism (m) leads to the conclusion that constitutive tensors have the

form: [23]

1 (m) (m) C (m) = C e(m) + ~ n ® n ~L ~ ~gL ~

L

C(m) = ce(m) + I (m) (m) U ~ ~-- ng u ® n (6)

U

(m) (m) I n a b o v e , n a n d n a r e a r b i t r a r y t e n s o r s o f u n i t no rm a n d H

gL gU L/U

two scalar functions defined as loading and unloading plastic moduli.

Tensor C e(m) characterizes reversible material behavlour, and it can be

easily checked that any infinitesimal cycle of stress { d~ , (-d~)} ,

where d~ corresponds to neutral loading conditions, causes zero

accumulated strain.

This suggests that the strain increment can be decomposed into

elastic a n d plastic p a r t s :

de (m) = de e(m) + de p(m) (7)

where

dE e(m) = C e(m) : dcr'

d~P(m) __ 1 In (m} ® n Cm)) : dcr' (8 ) ~ H ( m ) [ ~ g L / U ~ ~

l , , / U

177

Plastic strain may also develop under unloading conditions, its

importance depending on type of material, history and mobilized stress

ratio from wich unloading takes place [25,2?]

(m) From above expression the unit tensor n can be seen to give

~gL/U direction of plastic flow corresponding to mechanism (m).

Total increment of strain caused by an increment of stress d@ can be ~

found by adding contributions of all mechanisms:

o r

" " i [n,.i ) d<_ = I] _°'<" : <'=" + . = , . = , I] ~ t -o~ ,o" - n<'' L/U

:d¢' (9)

dc=Ce: d@, +~ I In(m) ® n(m)) ~ ~ ~ m:l H (m) t "g~ ~

L/U

: dcr'

d~ = C ep : d@' (10)

Material behavlour can be fully characterized by providing following

items for every mechanism:

(I) The elastic constitutive tensor Ce

(ii) Loading/Unloading directions n (m)

(iii) Directions of plastic flow n (m) ~gL/U

(iv) Plastic modulus H (m) L/U

I n v e r s i o n o f t h e c o n s t i t u t i v e t e n s o r C ep

Implementatlon in FE codes of a contltutlve model requlres in many

occaslons to invert relatlon (I0) which is only Immedlate if all H (m)

are not zero. Should it not be the case, inverslon would have to be

carrled out accordlr~ to the procedure described below.

178

First, we introduce a set of scalars

(m) n : do" .,. (m) _ ~ ~

A Him)

which, after being replaced in (I0), results in

(li)

M

dc = C e : do" + Z A(m)n(m) . . . . gL/U

m=!

If we now premultiply by n(m): D, e we arrive at

(12)

(k) H(k)A(k) n(k) I ~ A(m) (m) ~ n : De: dc = +_ : D': ngwu j m=1

where D e is the inverse of C e

(13 )

Expression (13) may be wrltten in a more compact form as

N T { D e } X = b . A : D e : de = a + N : : NgL/u (14)

where N and N are given by ~gL/U

(k) N = n klJ |J

. ( k )

(NgL/U)lJ = (ngL/U)11

(k) A =A k

and a is a diagonal matrix with

(15)

a = H (1) ( 1 6 ) i i

179

From equation (14) it is possible to obtain

A = c N : D e : dc . . ~ ~ ~

a s ~

(17)

with

cb = I ~ .

Finally, substituting (17) into (14) we obtain

d~' = D ep : dc

DeP = [ De - De : N _ - ~ -g ~c N : D e]- (18)

whlch in the case of a single mechanlsm reduces to : [23]

D D" : n. n :D } DeP = e . ~ ~g ~ ~e ~ H +n : D:n

-- ~ e ~g

(19)

S o f t e n i n g

A positive plastic modulus accounts for hardening effects, while

negative values correspond to softening phenomena.

Loadlng and unloadln E deflnltlons have to be modlfled to include

softening behavlour. If d~ e(m) denotes the Increment of stress produced

by dc If the material behaved elastically In mechanism (m)

(m) de(m) n : > 0 loading (20)

(m) de(m) n : < 0 unloadlng

Care should be taken when analyzlng experlmental data exhibiting

softening. First of all, they may be influenced by the development of

180

one or several shear bands in the specimen, with a lose of homoEeneity.

Shear bands may appear before a peak in deviatoric stress is reached

for materials with non-associated flow rules. Wether the softening

exhibited by many geomaterials is or is not a consequence of

localization phenomena has not been clearly established yet, but it is

logical to believe that in some cases it should exist (although not so

remarked as experiments show) This could be the case of granular

materials in the dilatant regime.

A second point to remark is that not all peaks in deviatoric stress

strain curves correspond to softeninE. For instance, liquefaction of very

loose sand under undrained conditions shows the existence of a peak in

the deviatoric stress althouEh the mobilized friction angle or the

stress ratio are continuously increasinE (Fig. la)

On the other hand, overconsolidated cohesive soils exhibit no peak

at all in many occasions, as far as the deviatoric stress is concerned

but examination of mobilized stress ratio shows such a peak which can

occur without deformation localizin E in a shear band.(FiE. Ib)

I so tropic mater ia l s

Many EeoloEical materials exhibit either initial or stress induced

anisotropy, resultin E on a behaviour which is dependent upon orientation

of principal axes of stress increment tensor.

However, accurate results can be obtained assumln E that this

dependence is not of paramount importance. Such models have been

successfully applied to reproduce soil behaviour under monotonic and

cyclic loads [24,27].

181

(A- l )

%

( B - t )

(A-2)

m,

p ,

(8 -2)

L

p'

j (B-3)

FIGURE I: Very loose sand llquefactlon (A) and oversonsolldated clay

behavlour under undrained condltlons (B)

182

Under this assumption, constitutive relations may be derived in

terms of stress and strain invariants. To characterize the stress

tensor, following invariants are defined:

I' = t r ( o " ) 1 ~

1 J2 = 2 t r (S 2) (21)

1 J~ = ~ t r (S 3)

where S is the deviatoric stress tensor given by

1

S = ¢ ' - ± I' I (22) ~ 3 1 -

and I the identity tensor 6 ~ lj

Alternatively, other sets of invariant-based functions may be

defined for convenience. Such is the case of (p', q, 8) which are

defined as

p , 1 , :~ 11

q = / 3 J ' (23) 2

1 ~ J3 8 = g s i n - * . - 2-- j , 3/---------2--"

2

181 ~ ~ / 6

Strain Invarlants have to be chosen in such a way that plastic

work is conserved.

183

Thls representatlon of the stress and strain tensors now reduced to

three components, has the advantage of simplifying the process of

derlvatlon of constltutlve laws from laboratory tests. In fact, they

are arranged as stress and strain "vectors" ~' and c for whlch all

relations presented above hold. Therefore, we can write

d~P- 1 ngn/u[ nT . d~ '') (24)

HL/U

where a single mechanism of deformation has been considered.

Once the isotropic model has been developed, it can be extended to

general three-dimensional conditions [29]

.a : f i a- ~ , _

~g ~g aO"

n : ~ L/U L/U

(25)

ANISOTROPY MODELLING

Introductory remarks

Several theories have been proposed within the framework of

plasticity to descrlbe both inltlal and stress induced anlsotropy.

Basically, anlsotropy has been approached most of the times by changing

the posltlon, orlentatlon and shape of Isotroplc yield, loading or

plastlc potentlal surfaces, in such a way that those changes were

dependent on tensors such as stress or strain and not only on their

Invarlants.

Inltlal or fabric anlsotropy could be reproduced as well by

introducing Inltlal movements and dlstorslons on the surfaces.

184

Combination of kinematic and anisotropic hardening laws proposed by

Hroz [6], have provided a suitable way to model anisotropic behaviour

of soils [7-10]. Surfaces can be allowed to expand following isotropic

hardening rules and to translate and rotate under anisotropic hardening

laws.

If only initial or fabric anisotropy is to be considered, a simple

way to introduce anisotropic surfaces is to define a modified second

invariant of deviatoric stress tensor:

- i A S ( 2 6 ) J; : 2 Sij i]kl kl

where A is a fourth order tensor. This method was initially lJkl suggested by Hill [11] and extended by Nova [12,13] to soils and soft

rocks. If J' is substituted by J' in any isotropic plasticity model 2 2

(such as the Cam-Clay, for instance), one finds yield surfaces which

have been distorded and rotated.

This effect can be introduced also by directly formulating the

surfaces on the stress space [14] or by deriving them from modified

anisotropic flow rules, as proposed by Anandarajah and Dafalias [15]

Baker and Desal [16] suggested to include the effect of stress

induced anisotropy via joint stress and stress invariants.

Surfaces were made dependent on

=tF { ° <) K 1 ~

= t r ( ¢ . cp2) K 2 ~

K3 = t r { 2 . cp)

K4 = t r { ~ 2 . c p2)

(27)

185

in addition to stress and strain Invarlants I' , J' and J'. 1 2 3

This approach is based on the representation theorem of scalar

functions depending upon two symmetric second order tensors (v and c p in

this case)

All theories mentioned above are able to introduce the anlsotroplc

response of geomaterlals even when there Is a single mechanlsm of

deformation.

Multimechanism theories can also describe anisotropic behaviour,

provided that they are not formulated in terms of the three stress

invariants only.

Multi-lamlnate models as introduced by Pande [17] consider that

deformation is caused by dilation and slip taking place at all possible

contact planes within the material. Of all possible active planes only a

reduced number of sampling planes is considered. The overall response is

obtained by a process of numerical integration extended to sampling

planes.

If attention is focused only on planes normal to XY, Y'Z and ZX, and

their responses are grouped together, one finally arrives at a three

mechanisms model [18,19].Each plane mechanism response may be assumed to

depend only on plane stress invariants defined as

where

p, (k) 0" + (7"

i j

q : 2 + .2 (28) ~JJ

k = • ( 1 , 2 , 3 }

( l , J ) = • { ( 2 , 3 ) , ( 3 , 1 ) , ( 1 , 2 ) }

186

Both multllaminate and multimechanism models of the type described

above can produce plastic strain under pure rotation of principal stress

axes [17,27].

Alternatively, behaviour of material can be assumed to be caused by

i superposition of responses to variations in ~, ~ and ~3 ' and then

generalized to more general stress conditions. This has been proposed by

Darve [20] and applied to complex stress paths, including anisotropy

effects.

Finally, it should be mentioned that material fabric plays a

paramount role on geomaterials anisotropic response and it is in turn

modified by the deformation process. The fabric may be approximated by a

second order tensor, which can be incorporated into the constitutive

equations. An interesting way has been recently proposed by Pietruszczak

and Krucinski [21], and consists on adding two components to obtain the

increment of plastic strain. First one corresponds to an isotropic

hardening mechanism, and the second accounts for deviations of isotropy,

which are made dependent on fabric tensor.

Proposed approach

It has been mentioned above that material structure or fabric has to

be incorporated in the constitutive equations to account for both

initial and induced anisotrpy. Here, it will be assumed that fabric

can be described by a second order structure tensor A, which will N

determine its type of symmetry. If Q is a rotation or reflection tensor,

the class of symmetry will be defined by the set of operators Q wich

fulfill

A = QT AQ (29)

For instance, transversely isotroplc materials wlll be described by

A invarlant under Q given by

187

1 0 O]

Q = 0 cose - sene (30)

0 sen{) cos8

where it has been assumed that the plane of isotropy is XY

The structure tensor will have the form

A I t 0 0

A = 0 A22 0 (31)

0 0 A33

with

A =A 22 33

and it can be easily seen that relation (29) is verified.

If the initial "structure" of the material is described by A °, A

will vary along the loading process, according to

A = A 0 + dA (32)

where dA w i l l d e p e n d on p l a s t i c s t r a i n

dA = dA(de p) (33)

Now, the structure tensor can be used to define a fourth order

anisotropy tensor B ° from which a modified second invariant J' can be - 2

derived as suggested by Hill [11] and Nova [12,13]. Following Cowin [22]

B ° can be expressed as a combination of terms listed below:

(i) ~ ®

(ii) ~ ® A , A ®

(iii) 6 ® A 2 A 2 ® P

(lv) A ® A

(v) A ® A ~ A 2 ® A

( v i ) A 2 ® A 2

(34a)

1 88

where compact notation has been used. Full tensorlal products have to be

expanded as follows:

M ® N = {MijNkl, MlkNjl, MiiNjk } (34b)

If a transverse isotroplc material is considered, the tensor B

referred to principal axes is given by

11

22

33

12

21

13

31

23

32

11 22 33 12 21 13 31 23 32

B B B 1 2 3

B B B 3 2 4

B B B 3 4 2

B 5

B 5

B 5

n B 5

B 6

B 6

35)

with 2B = B - B 6 2 4

It can be seen that resulting anisotropy tensor depends only on five

constants, and that the form proposed by Nova [12] is a particular case

of (35) in which B and B have been made one and zero respectively. 1 4

So far,only J' has been extended to account for anlsotropy. However, 2

geomaterials behaviour is also dependent on first and third invariants,

and anisotropy should be also reflected or them. New invariants ~

and J' can be introduced in a similar manner, by defining B I and BIII

which are tensors of orders two and six. The first anisotropy tensor B I

would be dependent on

(i)

(Ii) (36)

189

and B HI on double tensorlal products of ~ ,A and A 2 such as 6 ® ~ e (~,

A ® ~ ® ~ , A ® ~ ® ~ , e t c .

Therefore, the extended set of invarlants is given by

[ ' = B I ¢ ' % ij IJ

] ' = c ' B I I c ' (37) 2 iJ iJkl kl

J' = 0" Or' cr' BllI 3 lJ kl mn I Jklmn

Finally, constitutive laws derived for is®tropic materials in terms

of It' ' J'2 and J'3 can be generalized to anisotropic situations by

substituting them by modified forms given above.

An interesting particular case is obtained when A is taken as

P A = C (3S)

Then, the constitutive law can be seen to be dependent on joint

stress-strain invariants, as proposed by Baker and Desai[16]

A SIMPLE MODEL FOR GRANULAR SOILS

Sand deposits exhibit anisotropic response caused by the alignment

of sand grains on horizontal planes. This initial or inherent anisotropy

may be modified by subsequent strains developed as the material is loaded

If an specimen of such material is brought to failure, grains will

be reorganized as deformation increases, changing the initial structure.

It has been shown above how materlal response can be described by

providing suitable expressions for tensors ngL/U, and plastic modulus

HL/U. Following this approach, slmple models have been derived for

Isotroplc materials [24,27] in terms of Invarlants I~, J' and J' or p' 2 3 '

190

q and 8 We will concentrate now on how to obtain these elements

for anisotropic sands.

Flow rule

An important feature of granular soils is the existence of a surface

in the space of stress invariants on which plastic flow is such that no

plastic volumetric strains are produced. The trace of this surface with

the triaxial plane (~2 = ~3 ) consists on two lines which have often

being referred to as "characteristic state" [30] or "phase

transformation" lines [31] and which coincide with the projection of the

Critical State Line on the ~ = ~ plane. 2 3

If experimental data obtained on granular soils with initial

anisotropy such as given in Ref.[2] are analyzed, it can be found that

the zero volumetric incremental strain surface may be described by a

simple relation

W = ~ = M (8) (39) P g

where M depends only on Lode's angle and n is the stress ratio. g

The tests performed by Yamada and Ishihara [2] consisted on

proportional, radial paths performed at constant p' and 8 . A detailed

description of both the testing procedures and the obtained results is

given in Ref. [2]. The samples exhibited a strong anisotropy as grains

were arranged such their long axes were horozintal. Therefore,

different behaviour was observed along paths such as ZC and YC which

have the same value of Lode's angle.

However, no such dependence was found for M .Therefore, g

assumed that M varies according to a law of the type g

it may be

6M M (e) = qc (40)

g 6 + f M (1- sin 38) g gc

191

where M is the value obtained for 8 = 30 ° and f a material parameter q¢ g

wlch corrects observed devlatlons from Mohr-Coulomb type behavlour.

XE

Z¢

¥C

ZE

FIGURE 2. Trace of characteristic state line on the ~ plane

• experiments from Yamada and Ishihara, 1979

- Predicted fg = 0.5

- - Predlcted fg = 1.0

FIg. 2 shows both the experimental results of Ref.[2] and the predlctlons

of expression (40). The paramater f has been taken as 0.5. g

192

So far, it has not been necessary to introduce the modified

invariants I' , J' and J' described above,and expression (40) holds for i 2 3

both isotropic and anisotropic materials.

A similar analysis may be carried out to study how soil dilatancy,

defined as

de p d~ d - v = ~ (41)

g de p dc s s

is affected by soil anisotropy. A simple expression relating dilatancy

to stress ratio W was given in Ref. [25]

d = (I + ~) (M - n) (42) q g

Miura and Toki [32] analyzed experiments of Ref.[2] and concluded

that soil anlsotropy did not greatly affect the dllatancy behaviour of

sand. It was found that parameter ~ depended on e , and that it was

related to M by g

(I + ~(e)] M (e) = (I + ~ )M (43) g c g c

from which ~( 8 ) can be obtained as

M =(e) = (I + ~ ) ~c I (44)

c M (e} g

where ~ and M are the values obtained for e = + 30 ° . ¢ gc

There fore , sand d l l a t a n c y may be approximated by e q u a t i o n s (40-44),

where no reference to the anlsotroplc fabric has been made. Of course

this is only strictly valid under the assumptions made, i.e., virgin

loading and stress ratlos n higher than 0,2 M (e). g

193

Once a theoretical law for sand dllatancy has been selected, It ls

possible to derive the components of tensor n in triaxial coodinates ~gL

{p', q, e } and then to generalize it to 3D conditions applying the

transformation equations given In (25)

from

Components of n In the space of stress Invarlants a r e obtalned ~gL

9gL,p = (1 + C()[Mg - pq-~ --)

9 =I g i , q

and

1 VgL, e = - ~ fg Mg q cos 38 (45)

_ qL

-gL IVgL I

It can be easlly seen that the proposed flow rule corresponds to a

plastlc potentlal surface glven by

[ g ( o . ' ) = n - 1 - L p ' g ]

If ~ dependence on Lode's angle ls neglected.

(46)

In above,

and M i s glven by (40) g

[ '} ng = Mg 1 +

-- q ' / p

194

Loading-unloading d i r e c t i o n n

In the case of isotropic granular soils, it was proposed in [19]

that n could be defined in a similar manner by

and

where M is given by f

9 = (I + ~) (M - q/p') ,p f

9 = I ,q

I = - - f M q cos 38 (47a) 9,8 2 r f

6M H C8) = fc (47b) £

6 + f M (I- sin 38) £ £c

It is important to notice that associated flow rules can be obtained

as a particular case of above expressions by choosing f = f and f g

= M . However, non-associated rules are needed to account for very Mfc qc

loose sand liquefaction under monotonic or cyclic loading [25].

To account for initial and induced anlsotropy effects it is now

necessary to include the microstructure tensor A in above definitions.

This can be easily done by substituting p', g and 8 by

p' : I ' 1 3 1

2

1 = s i n -1 6

195

where ~' I

(37).

, 3' and 3' are the modified Invarlants 2 3

defined in equations

The three anlsotropy tensors B*, B *x and B HI may be taken as

BI =A

B I! = (6 ® 6 - A ® A) (49)

B n I = ": 6 ® 6 ® 6 - ~ : ( A ® 6 ® 6 + 6 ® A ® 6 + 6 ® 6 ® A) - - 6 ~ - - 9 - - - - . . . .

1 1 + - (6 ® A ® A + A ® 6 ® A + A ® A ® 6) - tr(A3). (A ® A ® A)

It can be demonstrated that, for Classical Plasticity models, this

choice corresponds to a rotation of the yield surface.

Plastic Modulus H L

A suitable expression for the plastic modulus H h

= - -- H v + H H o ~f DM

where

H, = - ~I exp(-~o ~}

and HDm iS a discrete memory function defined as

i s

(5o)

(Sl)

196

HD. (52)

The mobilized stress function ~ characterizes the intensity of the

stress conditions, and it is given by

- ( 5 3 1

Memory of past loading events affects material behaviour, and can be

taken into account by keeping ~max' the maximum value reached by the

mobilized stress function.

Therefore, plastic modulus depends on material microstructure

through p', q and

In above expressions, four additional nondimensional parameters H o,

8o, 81 and ~ have been introduced.

Functions H and H may be related to volumetric and deviatoric v

strain hardening functions.

It can be seen that residual conditions are reached at the Critical

State Line, where

H = 0 v

llm H ~-) m s

= 0 (54)

M i c r o - s t r u c t u r e t e n s o r A

Mlcrostructure tensor A was assumed to have an initial value A °

corresponding to initial fabric. Therefore, it should reflect material

symmetries. Naturally deposited sands exhibiting transverse isotropy

will have

197

A I

A ° = A 3 (55)

A 3

with axis X coinciding with direction of deposition, while isotropic !

materials will be characterized by

[AI 1 A ° = A 1 (56)

A t

As the material is loaded, deformation will produce rotation of

grains and rearrangement of mlcrostructure. Therefore, A ° will change

according to (32), and dA will be a function of the increment of strain.

General expressions for this functions have been suggested by

Pletruszczak and Kruclnskl [21]. Here we will assume that dA may be

expressed as

dA = de p . {AI ex p I- Ao csl} (57)

where A and A are two materlal parameters. If A and A are taken as o 1 o 1

zero a n d one respectively, a n d no Initial anlsotropy exists, A will

coincide wlth c p, and the modlfled Invarlants I' , ]' and ]' wlll be ~ 1 2 3

functions of both the stress and mlxed stress-straln Invarlants.

C y c l i c l o a d i n g

A fundamental feature of granular soils under cycllc loadlng is that

plastic strains may develop durlng unloading. A slmple flow rule for

unloading of granular soils was suggested in [25]

n = abs (n ] gO, p gL, p

n = - n ( 5 8 ) gup q g L , q

n ~ - n qu, 8 gu, 8

198

It can be seen that volumetric component is always positive to

produce densification according to observed behaviour of sand under

unloading.

Plastic modulus is taken as

(59)

which accounts for the fact that higher plastic strains are produced

when unloading from high stress ratios.

H UO

In above, ~u is the stress ratio from which unloading takes place,

and ~u being two additional parameters.

MODEL PERFORMANCE

Many well documented tests showing the effects of initial and

induced anisotropy on the response of granular soils have been presented

in the literature [I-5] Among them, we have chosen a set of tests carried

out on Fuji River sand by Yamada and Ishihara [l,2].Basically, sand

specimens were constructed by pluviation of sand through water, to

simulate natural deposition processes. This resulted in a highly

anisotropic structure which was modified by subsequent loading.

Cubic samples of sand were tested on a true triaxial apparatus along

proportional stress paths for which p'and 8 and were kept constant. The

tests wich will be considered here correspond to paths ZC, ZE, XE and

YC (Fig. 2) where OZ axis has been taken along the direction of

deposition. As expected, sand exhibited isotropy on the plane XY

caused by the random orientation of particles on it. Therefore, responses

along paths XC and YC, or XE and YE were found to be identical.

First of all we will consider undrained behaviour along YC, ZC, XE

and ZE. Material anisotropy of Fuji River sand was characterized by

199

0.6767 0 0

J AO = 0 0.5205 0

0 0 0.5205

(60)

from which anlsotropy tensors B I, B xx and B xlx were derived.

Fig. 3 shows experimental and predicted behaviour for paths YC and

ZC, which have the same value of Lode's Angle (8 = + 30"). It can be

seen how material behaviour is much stiffer if it is compressed

along the direction Of deposition, ln fact, deviatoric stresses are more

than 50Z higher in this case.

This tendence is reversed when load is applied along directions ZE

and XE (Fi E . 4). This effect may play a paramount role in the dynamic

response of anisotropic deposits o f saturated sand and therefore

simplified analysis methods based on symmetric triaxial tests should not

be used if accurate predictions are desired.

Drained test results are shown in Figs 5 and 6 together with model

predictions corresponding to the same set of parameters obtained for the

undrained tests.

AEaln, samples tested alone directions ZC and YC repeat the same

pattern found in the undrained case, i.e., material bein E stiffer when

tested alone ZC. It can be observed that maximum contraction takes place

in both cases at the same value of mobilized stress ratio ~ , the peak

being higher for the YC specimen. This situation corresponds to

= M (e) (61) g

with M (6) is given in (40). As explained above, this value of ~ at g

which the contraction is maximum depends only on Lode's angle.

200

E

o <~ t~i Q:

u o ({ > w

I0

0.5

I I

ZC / / / / / . . . . ~ o , ~ o

oi5 ll-o MEAN EFFECTIVE STRESS p' (kg/cm 2 )

A

Z

oi

~J

{n

3

2 Y ~ Z C

//~,',

, //,,,/ . . . . . ,.;i

STRESS RATIO 7

FIGURE 3. Undrained compression of Fuji River sand (Exp. from Yamada and

Ishlhara, 1981].

201

E .,,0 v cr

o

0~ o

>

o

MEAN EFFECTIVE STRESS p'(ko/cm 2) _ o' .s ~/,.o

-05 ~ I I/

\(, Y E

. . . . . . PREDICTED EXPERIM

I I

, i >=

z =<

x

--3 ZE

--2 ~ Y E

--I ~ ! E

~ " ~ L O I

STRESS RATIO 1.5

I

FIGURE 4. Undralned extenslon of FuJl Rlver sand (Exp. from Yamada and

Ishlhara, 1981)

202

A

3 o

s / " YC 1.0 / / ~

-o.s / /

-0.6 / ~

- O, 4 / ~ , , Z C

. / / / -°~ i S - " ~ ' " Oj8 I li2

STRESS RATIO "~,

Z

bJ X:

_____ ;:~,I~ED /.c

/ / / / I

7 //

S a~ I

STRESS RATIO 'Z

FIGURE 5. Drained compression of Fuji River sand. (Exp. from Yamada and

Ishihara, 1979)

203

o,,~I I'0

z I-- ¢n

o

=, o >

-0.8 / Z£

-0.6 /' .0.4 XE

-0.2 --.

S T R E S S R A T I O "~

o~ o

0¢

2

S T R E S S R A T I O

FIGURE 6. Drained extension of Fuji River sand (Exp. from Yamada and

Ishlhara, 1979)

204

CONCLUDING REMARKS

The model introduced in this paper within the framework of

Generalized Plasticity provides a suitable approach to the modelling of

anisotropic behaviour of granular soils.

This is accomplished by introducing suitable "modified invariants"

I~, 3~ and J~ wich are dependent on stress and on a tensor A which is a

measure of material microstructure. These invariants are used in

definitions of loading-unloading direction n and of plastic modulus H,

and reduce to the classical set of invariants for isotropic materials

for which A is taken as the indentity tensor. In this case, the model

obtained is the same proposed in [26] for the behaviour of isotropic

sands under monotonic and cyclic loading.

Model predictions have been checked against experimental data

obtained on Fuji River sand which exhibits a very strong initial

anisotropy caused by the natural process of deposition through water,

and found to reproduce observed hehaviour satisfactorily.

205

TABLE I NODEL P ~

Elastic n n H Anlsotropy Ng

K = 566 Kg/cm 2 N = 0.9 A e v fc 1

K = 424 Kg/cm z emo

Mgc= I. 40

f = 0 . 5 g

~ = 0 . 4 5

F =i.0 F

H ° = 150

80= 4

81 = 0.5

A o

= 10

= 100

ACKNONLEDGEMENTS

This work have been sponsored by the EEC trhough SCIENCE program

"EUROGRECO : Rheologle des geomaterlaux".

The author would like to express his gratitude to them. Technical

help provided by Mrs. M.D. Azc~rraga is also gratefully acknowledged.

REFERENCES

1.

2.

3.

4.

Yamada, Y. and Islhara, K. "Anlsotroplc deformatlon characteristics of sand under three- dimensional stress conditions" Soils and Foundations, Vol. 19, n ~ 2, pp 79-94 (1979)

Yamada, Y. and Ishlrara, K. "Undrained deformation characterlstlcs of loose sands under three-dimensional stress conditions"

Soils and Foundations, Vol. 21, N I, pp 97-107 (1981)

Yamada, Y. and Ishlrara, K. "Yielding of loose sand in three-dlmenslonal stress conditions" Solls and Foundatlons, Vol. 22, N 3, pp 15-31 (1982)

Ochlal, H. and Lade, P.V. "Three dlmenslonal behavlour of sand with anlsotroplc fabric" J. Geotech. En~. ASCE. Vol. 109, n 10, pp 1313-1328 {1983)

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

206

Symes, M. J .P .R . , Gens, A. and Hight , D.W.

"Undrained a n i s o t r o p y and p r i n c i p a l s t r e s s r o t a t i o n in s a t u r a t e d sand"

G~otechnique v o i . 3 4 , N 1, pp 11-27 (1984)

Mroz, Z "On the d e s c r i p t i o n of an i so t r op i c work-hardening" J.Hech. Phys. So i lds , 15, 163-175 (1967)

Hroz, Z , Norr is , V.A., and Z lenk iewicz, O.C. " A p p l i c a t i o n o f an an i so t r op i c hardening model in the ana lys i s of e l a s t i c - p l a s t i c deformat ion o f s o i l s "

Geotechnique, 29, 1-34 (1979)

Hashiguchi, K. " C o n s t i t u t i v e equat ions of e l a s t o p l a s t i c ma te r i a l s w i th e l a s t i c - p l a s t i c t r a n s i t i o n " .

J.Appl. Mech. ASHE, 47, 266-2?2 (1980)

H i r a i , H. "An elastoplastic constitutive model for cyclic behaviour of sands" Int. J. Numer Anal. Methods Geomech, II, 503-520 (1987)

Ghaboussi, J. and Momen, H. "Modelling and analysis of cyclic behavlour of sands" Soll Mechanlcs-Translen~ and Cyclic Lo@~s, pp 313 - 342 G.N. Pande and O.C. Zienkiewicz (Eds),Wiley (1982

Hill, R. "The mathematical theory of plasticity" Oxford at the Clarendon Press (1950)

Nova, R. and Sacchl ,G. "A model of the s t r e s s - s t r a i n r e l a t i o n s h i p o f o r t h o t r o p l c g e o l o g l c a l media"

J. Mec. Theor. ApDI. 1, 6, pp 927-949 (1982)

NOVa, R. "An extended cam c lay model f o r so f t Comp. and Geotech., 2, 69-88 (1986)

a n i s o t r o p i c rocks"

BanerJee, P.K. and Yous l f , N.B. "A p l a s t i c i t y model f o r the mechanica l b e h a v i o u r

a n i s o t r o p i c a l l y c o n s o l i d a t e d c l ay" In t . J.Numer. Anal. Methods Geomech, 10, 521-541 (1986)

AnandaraJah, A. and D a f a l i a s , F. "Bounding s u r f a c e p l a s t i c i t y l l I : A p p l i c a t i o n to a n i s o t r o p i c

c o h e s i v e s o i l s " J .Ene . Mech. ASCE, 112, 12, 1292-1318 (1986)

of

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

207

Baker, R. and Desai, C.S.

"Induced an iso t ropy dur ing p l a s t i c s t r a i n i ng " In t -J . Numer. AnahMethods Geomech., 8, 167-185 (1984)

Pande G.N. and Sharma, E.G. "Multi-laminate model of clays: a numerical evaluation of the influence of principal stress axes"

Int. J. Numer. Anal. Methods Geomech., 7, 397-418 (1983)

Aubry, D.,HuJeux, J.C., Lassoudlere, F., and Melmon, Y. "A double memory model with multiple mechanisms for cyclic soll behavlour" Int. Symp. Num. MOdels l_~n Geomechanlcs, R. Dungar, G.N. Pande and J.A. Studer (Eds} Balkema, Rotterdam (1982)

Matsuoka, H. "Stress-straln relationships of sands based on the mobilized plane"

Soils and Foundations, Vol. 14, N 2, pp 47-61 (1974)

Darve, F and Labanieh, S. "Incremental constltutlve law for sands and clays: simulation of monotonic and cyclic tests" Int. J. Numer. Anal. Methods Geomech, 6, 243-275 (1982)

Pietruszczak, S. Kruclnskl, S. "Description of anlsotroplc response of clays using tensorlal measure of structural dlsorder"

Mech. Materials ,8, 237-249 (1989)

Cowln, S.C. "The relationship between the elasticity tensor and the fabric tensor"

Mech. Materlals, 4, 137-147 (1985}

Zienklewicz,O.C. and Mroz, Z. "Generalized plasticity formulation and appllcations to geomechanlcs"

Mechanics of En~Ineerin~ Materials C.S. Desal and R.H. Gallagher (Eds) Wiley 665-679 (1984)

Zlenkiewicz,O.C.,Leung, K.H. and Pastor, M. "A simple model for transient soll loading in earthquake analysis. I : Basic model and its application" Int. J. Numer. Anal. Methods. Geomech, 9, 953-976 (1985)

Pastor, M., Zlenklewicz, O.C. and Leung, K.H. "A Simple model for transient soil loading in earthquake analysis. II : Non-assoclatlve models for sands" Int. J. Numer. Anal. Methods Geomech, 9, 977-998 (1986)

208

26.

27.

Pastor, M. and Zlenklewlcz, O.C.

"A generalized plasticity hierarchical model for sand under monotonic and cyclic loading"

Proc. 2nd. Int. Conf. on Numerical Models In Geomechanlcs, Ghent G.N. Pande and W.F. Van Impe (Eds) M Jackson and Son Publ., 131-150 (1986)

Pastor, M.,Zlenklelwcz, O.C. and Chan, A.H.C. "Genenerallzed plasticity and the modelling of soil behaviour" Int. J. Numer. Anal.Me%hods G~omech., 14 (1990)

28. Zlenkielwcz,O.C.,Chan, A.H.C.,Pastor, M.,Paul,D.K. and Shioml,T. "Static and Dynamic behavlour of geomaterlals, rational approach to quantitative solutions. I :Fully saturated problems" Proc. Royal Society Series A 429,285-309 (1990)

29. Chan, A.H.C.,Zlenklewlcz,O.C. and Pastor, M. "Transformation of incremental plasticity relation from defining space to general cartesian stress space"

Com~. Appl. Num.Meth., 4, 577-580 (1988).

30. Hablb, P. and Luong, M.P. "Sols pulvurulents sous chargement cyclique" Materlaux e_~£ structures sous char~ement cyclique Ass. Amlcale des Ingenleurs Aclens El~ves de l'Ecole Nationale des Ponts et Chausees, Palaiseau, 28-29 Sept. 1978, pp 47-79

31. Ishlhara,K., Tatsuoka, F. and Yasuda, S. "Undrained deformation and liquefaction of sand under cyclic stress"

Soils and FoundaElons, 15, 29-44 (1975)

32. Mlura, S. and Tokl,S. "Elastoplastlc stress-straln relationship for loose sands with anlsotropic fabric under £hree-dimensional stress conditions"

Soils and Foundations, 24, 43-57 (1984)

Received 1 July 1990; revised version received 30 May 1 991; accepted 31 May 1991