ELSEVIER

Computers and Geotechnics 17 (I 995) 177-201 0 1995 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0266-352W958.9.50

A NEW INCREMENTALLY NON LINEAR CONSTITUTIVE LAW FOR FINITE ELEMENT APPLICATIONS IN

GEOMECHANICS

M. Boulon and A. Alachaher Institut de Mkcanique de Grenoble

Laboratoire 3s Sols, Solides, Structures Domaine Universitaire

B.P 53X, 38041 Grenoble Cedex, Prance

ABSTRACT

The authors present, in the first part of this paper, a new incrementally non linear constitutive model for non-viscous materials. This model allows for avoiding constitutive iterations, in the case where the loading path is purely defined in terms of a strain history. This situation is always encountered in the finite element analyses using the displacement approach. This model is obtained by a generalization of the soil-structure interface laws (Boulon and al [5], Boulon [4]), using a three-dimensional rheological interpolation and a part of the formulations proposed by Darve [ll]. An other advantage of this model is its ability to incorporate an unlimited number of identifi- cation paths. The detail of this constitutive model is exposed by Alachaher [l]. The second part of the paper shows an identification and a validation of this new model for global loading paths. The performances of this model have been shown for several loading paths on sand like undrained, oedo- metric and isotropic paths. These tests seems to be very favourable to the passage to finite element applications.

INTRODUCTION

In the following, stress means intergranular stress. The general non linear

incrementally formulation of constitutive laws for non-viscous materials can

be written (Darve [ll]) :

dei = itdij(~k) daj (1) where dei and daj respectively represent the components of the strain in-

crement and of the stress increment tensors; iWij(uk) is the “tangent” cons-

titutive matrix and Uk the vector of incremental stress directions. This ex-

pression involves a non linearity linked to the directional dependency of 177

178

the constitutive matrix IIfij. For being used in the classical finite elements

(displacements j analysis, this formulation requires a directional research

using an iterative process for the set of pat#hs defined by strain rat es. The

interpolation-type incremental models (Robinet 1241, Chambon [S], Darve

[12], Di Benedetto [15] and Royis (251) 11 a are subjected to this limitation.

The models involving non linearities explicitly depending on strain rates

are free of this restriction. That is the case of new incremental models like

those by Kolymbas ([19], [20]), Gudehus [16], Chambon ([7], [9])> Hammad

[17], the endochronic models (Valanis [26], Bazant [3], Cuellar [lo], Lanier

[21]) and the elasto-plastic models (Prevost [23], Aubry and Al [2], Hujeux

[18], Modaressi [22]). The following notations will be used in this paper : the incremental st’ress

tensor & and the incremental strain tensor & will be represented in the

framework of the pseudo-vect,ors :

and &=

Assuming that the material is initially isotropic, we will limit ourselves to

the case of non rotational loadings, or in other words we will exclude a

rotation of the principal stress and strain directions. Eliminating the null

components of these pseudo-vect,ors, they can be simplified and conventio-

nally written as below :

or (3)

or (4)

Taking these notations int,o account, The Darve’s law [ll] defined by the

hereabove relationship (1) and further on called “D” does not allow to di-

rectly have the response if the loading path is entirely defined in terms of

st,rain rate. Direct,ional iterations often lead to high CPU time consuming

for realizing the convergence of the incrementma response (Boulon and al

179

[6]). This is the current case in the displacement finit,e elements. In order to

overcome this drawback, we propose a new constitutive interpolation allo-

wing to obtain the incremental response without directional searching, the

loading path being completely defined by st,rain rates.

This scheme takes into account elementary basic paths which are defined

in the incremental strain space &. In this paper, this new law wilI be called

” ABD” .

The law “ABD”, based on a rheological interpolation between basic paths

(not regulary distributed in the incremental strain space) will be first pre-

sented and the calculation of the weight functions related to these paths

will be described afterward.

After developing the general algorithm of integration of this constitutive

model, global results for undrained, oedometric and isotropic tests invol-

ving various initial stress levels will be compared with the results given by

the law ‘ID”. It should be noticed that the laws “ABD” and “D” use the

same analytical formulations of the basic paths and the same tangent ma-

trices characterizing these paths.

GENERAL EXPRESSION OF THE LAW “ABD”

General:

The interpolation can be based on any number of basic paths (six in this

case). The response is constituted by the incrementa stress pseudo-vector

da and the loading, by the incremental strain pseudo-vector &. The res-

ponse equals the weighted sum :

where :

i is the number of the basic incremental path (here, i = 1 to 6).

2 is the constitutive matrix associated to the i-th incremental basic

path.

and W’i is the weight of the i-th basic path.

The new incrementaIly non linear constitutive law is expressed in the cur-

rent orthotropic axes relat,ed to the geomaterial. The constitutive relation

(5) assumes that this law is orthotropic (i.e. admits structural symmetries

180

versus three orthogonal planes called the orthotropic planes). Therefore, the

constitutive law is invariant by a symmetry versus each orthotropic plane.

Essential Notions

Basic Paths:

For the sake of comparison, the basic paths are normalized by the norm

of the incremental strain pseudo-vector.

Then, the Gth incremental basic path is defined by the pair of vectors as :

incremental loading

and incremental response

After normalization, these vectors become :

normalized incremental

loading

and normalized incremental

response

Now, all the possible incremental loadings are located on a unit sphere

Fi 1 1 ‘li =

in the (X , p , v) space. Then, we can speak about loading directions.

The basic paths are the generalized triaxial paths. Table 1 points out the

parameters related to these pat,hs.

Features of the Basic Paths: As previously indicated, due to their nor-

malization, the basic points 1 to 6 are moving on the unit sphere of the

incremental loading space during the global loading (when the state of the

material changes).

On the other hand, in the incremental response space, the basic points move

181

TABLE 1

The Set of Parameters of the Six Basic Paths ((1, 72, (3, &, 53, (6 > 0)

basic incremental incremental path loading response remarks nr. xpv (11C

t1 0 0 compression in the direction 1 0 00 -r/20

-c3

extension extension in in the the direction direction 3 2

extension in the direction 1 ie750 0” compression in the direction 2 0 0 cs compression in the direction 3

on their own half-axis during the global loading (see figure 1). Using the

notations adopted by Darve [ll] concerning the tangent Poisson ratios(‘),

the value of Xi, pi and pi can be obtained exactly for the basic paths 1 to 6

(see table 2).

Angular Distance Between the Actual Path and the Basic Paths: Let us

take an arbitrary loading direction, say d (whose response is unknown), of

the incremental loading space, as actual loading path. d is different from

the basic directions d(‘) (see figure 2). The angular distance oi between the

actual direction d and every basic direction d(“) is defined as (arithmetic

value) :

(6)

where the operator ]I 112 refers to the norm L’.

Moreover, the definition of CK~ is completed by the condition :

0 5 Qi 5 +a (7)

Constitutive Interpolation: The construction of the constitutive matrix e

corresponding to the direction (X , p , v) requires to know the basic ma-

trices 2 related to each basic direction and their weight functions Wi-

182

TABLE 2

Summary of the Basic Directions in the (X p , Space

r

t

basic path director r director 12% parameters cosines

Xl = +1/q 1 r1 = [l + ($+)” + (v:+)~]~”

3

4 x4 = -l/q

r4 = [l + ($-)” + (~:-)~]l’~

5 r5 = [If (d’)” + (G+)‘y2

x5 = -v;+/rg

115 = +1/T5 V5 = _~~‘,T5

6

183

unit sphere

3

surface of in&mental responses passing through the points 1 to 6

incremental loading space incremental response space

FIGURE 1 Incremental loading and response spaces and basic paths

Basic Constitutive Matrices & The matrices & may be calculated by

explicit inversion of the matrices i& of rank 3 of the law “D” (Darve ill]):

Q. = Ad;’ = { &+ + g-) + ;(g+ - g-j &}-I

where nCi) =da represents the incremental stress directions tensor of the basic

pat’hs.

The matrices x+ and g- depend upon the state parameters through the

triaxial tangent moduli Ei and the tangent Poisson ratios z/j, which are

defined in compression and in extension. As the state of the material and the

loading parameters change from one increment to the next, these matrices

(Darve [ll]) change too. Their expression is :

&+= -q+/E;

[

1 Is,+ 4 -vz’+/E; 1 Ez+ $ -v;+ jE,+ -vz+/E$

and in the same way for g-, substituting the

-v;+/E; -v;+fE; 1 (9) l/E,+ sign (-) for the sign (+).

184

d(i) : b asic direction no i 2 : current direction

di

FIGURE 2 Definition of the angular distance cy; in the incremental loading space

Weight Functions Wi: They represent the weight of each constitutive ma-

trix .Q and therefore of each basic path in the matrix D. -

Definition of the Functions Wi: The present approach has to be able to

model all material behaviour from the most simple to the most complex

oue. The most simple case is the isotropic linear elasticity corresponding to:

& = 4 (V i = 1 to 6) (16)

In this situation, the relations (5) and (10) imply :

185

ggK=l (11)

The physical sense requests that the weight of the basic path which is

the nearest from the current has to be dominant. This condition is realized

by :

?Ji Wi ((Yi)” = Ca (12)

with :

x : Shape exponent of the interpolation functions.

yi : Adjustment functions - formula (18).

Analytical Expression of the Functions Wi: Using (11) and (12), it follows:

WI w2 5Wi 1

i=l -=-=_*.r 1 1 (13)

Yl ffl Y2 a2

k’=k’

kl Yi (wf kl yi Q$

and finally :

One can verify that for cyj = 0 with j # i, Wi will be zero.

Hence, the interpolation functions Wi verify all the classical relations well

known for the finite elements. Here, the basic paths play the role of the

nodes in the spatial finite elements.

Formulation of the Functions yi: In three-dimensional cases, the functions

Wi obviously should be able to vanish not only for crj = 0 (j # i). For

that reason, the yi functions are choosen more complicated that the sign

functions.

186

Let be :

h=izCVi (15)

I----

&: a= z _ ($2

6 (16)

6 is the algebric mean value of the angular distances between the cur-

rent path and the basic paths , and 5 is a mean value of the gaps between

the distances cq and the previous algebric mean. The functions yi are the

sum of two functions depending of the reduiced variable ,f3i :

pi = “” (17)

l a power function, preponderant for the low values of \/3il.

l an hyperbolic tangent function, preponderant for the high values of ]/3il.

So, the final expression of IJ~ is :

yi = pp - tad (X (Pi + k)) (18)

Where n and X are real coefficients and k is also a real value which adjust

the number of positive and negative functions y/;. A complete parametric

study has been made in order to determine the optimal values of 2, X,

n and k. Several sets of these parameters have been tested successfully to

get the best incremental responses of geomaterials. Among them, we have

selected the following set :

z = 8., X = 1.5, n = 1.5 and k = 1.

Rem.: An incremental response is the locus of responses obtained by ap-

plying all the possible incremental loadings (directions) from a same initial

state.

Figures 5, 6, 7 and 8 respectively show : the incremental response to the

normalized loading located in the bisector plane (p = v), the functions oi,

the functions yi and the weight functions Wi, from an undrained path.

187

zo-

IO-

-l

o-

-lO-

II d&Ii 2 = 0.0001 initial state: E loose state (&=O.~.y % initial state of consolidation: IOOkPa

-20 / , , , , ( , , / ,,I”,_

-20 -10 0 10 ? 0

da,JZ (kPa)

1

FIGURE 3 Incremental response space from an undrained state

260 lIdclIp = 0.0001

initial state: E, = 0.015 % loose state (e. = 0.73) initial state of consolidation: 1 OOkPa

160

60

10 and

e a.‘ A-M-b+. a~ and as

-40 ,,,,,r,,,,r,,,,,,,,,,,,,,,,,,,,,,,,,

0 90 180

0 (deg.)

270 3i

FIGURE 4 Evolution of the angular distance (Remark: (3~2, ~23, ~5 and (Y6 never vanish)

ai, undrained loading path

15000 7

i 1

188

lId&ll~ = 0.0001 initial state: E

initial state of consolidation: 1 OOkPa

5000 -

I r\+ - - -

1 I+---

-15000 f,,,,,,,,,,,,,,,,,,,,,,,,,~~,,,

0 90 180 270 :

0 (deg.)

FIGURE 5 Evolution of the functions yi, undrained loading path

,OOkPa

0 90 I80 270 360

0 (deg.)

FIGURE 6 Evolution of the weight funct,ions IV;, undrained loading path

189

General Flow Chart of Treatment: The general flow chart of the numerical

integration of the law is shown in figure 7, for the general loading case (loa-

ding not given as a strain history) assuming that there is no rotation of

principal directions. In classical non linear finite element analysis (displace-

ment method), the local paths are more particular because they are known

in terms of strain history. Hence, the directional iterations shown in figure

7 are suppressed. In that special case, obtaining the incremental response

only needs to describe a set of values of d, for each increment (loop 2).

This algorithm is essentially the same algorithm introduced by Darve [ll].

But the treatment is simplified for the finite elements by the displacement

method.

GLOBAL VALIDATION OF THE LAW “ABD” ON

CLASSICAL PATHS

Identification: As the law “ABD” is obtained by interpolation between the

basic paths which are generalized drained triaxial paths, it seems to be ne-

cessary to verify at first the prediction of this law for these loading paths.

Figure 8 shows the results of a simulated triaxial axisymetric test in com-

pression for several initial lateral stresses (a~ = 100, 200 and 400 kPa). The

responses of the laws “D” and “ABD” are exactly the same, which is not

surprising because the formulations are the same.

Axisymetric Constant Volume Path (Undrained Loading Path) : Figure 9

(q - p and q - ~1 plane) shows the comparison between the laws “ABD”

and “D” for an axisymetric undrained triaxial path, from two initial lateral

stresses (100 kPa and 200 kPa) on Hostun R-F sand, with a loose initial

state (eo = 0.73). Th ese curves show that the integration of the law “ABD”

can be performed to the origin point of the (q , p) plane while it is impos-

sible to inverse the law “D”, because the determinant of the constitutive

matrix becomes zero as the mean pressure fall down to 130 kPa for the

high initial lateral stress (200 kPa), and to 55 kPa for the low initial lateral

stress (100 kPa). This difference may come from the interpolation (loss of

stability of the law “D”) since the basic paths of both laws are strictly the

190

START r Mechanical initialization : isotropic state , local path . . .

Num. initialization : integr. step, max. number of incr. (IRc,,, ) , interpolation parameters . . .

$ inc = I

r

LOOP 1 * * Beginning of the computations for each increment

Hypothesis on the incremental loading direction 4 or value evaluated by “continuity ”

Storage of the history of material at the previous increment

Calculation of the 6 tangent moduli E, and the 6 Poisson tangent ratios V; corresponding to each basic path

Construction of the matrices fl+and g- LOOP 2

4_

Computing the angular distan;y) pi and the 6 basic directions d *

Construction of the 6 constitutive basic matrices JIi 2

Calculation of & $

Calculation of the 6 coefficients y; ‘3

Calculation of the 6 weight functions W; $ .z ‘il

Next increment

Calculation of the matrix Q = 2 W, pi

.z 9

.z i=l t;

Evaluating the stress increment 0”

v

&=L3&

Directional convergence test

dst = 11~2 - rz$‘ll 5 lob4 radians No

YES

Updating void ratio and

stress and strain state

a rnc _ --cr ix-1 + &

f’“C = f inc-1 + &

c YES

NO

1 ENDI

FIGURE 7 General flow chart of numerical integration of the law “ABD” (explicit method, order one in time)

191

6.0

5.0 - DENSE STATE (eo = 0.55)

_t+t++ Law "ABD',-..D., (lOOkPa) _M Law "ABD..--D" (ZOOkPa)

s .m Law -ABD"-"D- (4OOkPa)

4.0 -

c

2

E 3.0 - v)

2

$

2.0

: 1.0 -

I

0.0 1

-1.0 ! ,,I,/ !,I,,18 11, I! I 0.0 2.5 5.0 7.5 10.0

axial strain c, (X)

0.0 2.5 5.0 7.5 18

axial strain e, (X)

FIGURE 8 Response path from a compression triaxial drained path

same (see figure 8).

Oedometric Path: The results of this loading path which is characteri-

zed by zero axisymetric lateral strain and then defined by strain rates are

shown in figure 10. The initial conditions are :

El = E2 = E3 = 0 %

01 = 02 = (~3 = 300 kPa

Figure 10 leads to conclude that there are more differences between the

INL (non-linear) and octolinear laws of Darve ([ll], [13]), than between the

INL law (“D”) and the law “ABD”, for this loading path.

Isotropic Path: Figure 11 shows a comparison between the law “ABD”

and the two versions (octolinear and non linear) of the law “D” for an iso-

tropic path on Hostun RF sand from a very loose initial state (eo = 0.92)

UNDRAINED TEST

loose state (e. = 0.73)

192

250.0

200.0 -

150.0 -

100.0 -

50.0 ~

tw++ Law .‘D,, (lOOkPa) x++w+,c Law =D,, (2OOkPa) 08880 Law “ABD” (lOOkPa) m Law .‘ABD- (200kP.z)

mean stress p (kPu)

60.0

n

loose state (e. = 0.73)

M Law ‘ED,, (lOOkPa) = Law “D“ (ZOOkPa) 0888t) Law “ABD., (1 OOkPa) &++&e Law “ABD.. (ZOOkPa)

axial strain E, (%)

FIGURE 9 Response path for an axisymetric undrained path. (Remark: the integration of the law plane)

“ABD” is possible up to the origin of the (p,q)

193

6000 I

b

FIGURE 10 Response path for an axisymetric oedometric loading path

5000 -

4000 -

3000 -

eooo -

lOOO-

OEDOMETRIC TEST mg = 300 kPa dense state (e, = 0.55)

// I A?

INL Darve’s law (“D’,) Otto-linear Darwzs law PD,,)

- Law ‘.ADD,, 0

1.0

0.8

axial strain El (X)

axial strain cl (X)

194

1400c 1200 ISOTROPIC TEST

very loose state (eo=0.92)

a uO= 300 kPa

a, 1000

5

b' 600

s

L t; 600

5

8 400

200 e Law .‘ABD’, ~-h INL Dawex law (.‘D,‘) s Otto-linear Darue~s law c.D..)

0.2 0.4 0.6 0.8 1.0

axial strain E, (%)

FIGURE 11 Response path for an isotropic path

and initial lateral stress of 300 kPa. The different curves are globally very

close between them.

DETAILED ANALYSIS OF THE UNDRAINED PATH

Evolution of the Response: Figure 12 consecutively shows the evolution of

the weight functions W; and of the angular distances Qi characterizing the

basic paths for an undrained test. So, we can say that the basic path no 1

remains the nearest path of the current path during the whole evolution.

Consequently, its weight has a value close to unity. Globally, the basic paths

no 1, 2 and 3 (compression paths) are closer to the undrained compression

than the basic paths no 4, 5 and 6 (extension paths).

195

Analysis of Global Loss of Unicity for the Undrained Loading: We represent

on the same chart in figure 13, the evolution of the determinant AM of the

rheological matrix M (where; & = g da) for both the laws “ABD” and - “D”. It is clear that the origin of the non convergence of the law “D” re-

sults from the abrupt change in the determinant sign during the directional

iterations. Then, the rheological matrix M becomes non inversible causing - a loss of unicity of the solution (Darve and al [14]). On the other hand, we

can say that the determinant of the rheological matrix of the law “ABD”

never vanishes, due to the choosen interpolation.

CPU TIME NEEDED BY THE TWO LAWS

Figures 14 and 15 show a comparison of the CPU times needed for the in-

tegration of the laws “D” and “ABD” for an undrained test on Hostun R-F

sand from a loose state and for two lateral stresses, 100 and 200 kPa (re-

fering to figure 9 for the corresponding responses). The two computations

have been done on the same computer (DECstation 5000/200) installed in

laboratory 3s.

So, we confirm through these figures, some advantages of the model “ABD”

to cover this class of paths with a small CPU time (see figure 15).

The gain in CPU time is between 20 to 40 %. When we know that the

computation time is one of the main handicaps of the incremental laws,

this gain, cumulated with others (use of an adaptive step, . ..) is not so ne-

gligible.

196

UNDRAINED TEST CT,, = 100 kPa Loose state

zo- inc=50 (0.50 X)

0 + ,,,,‘,,,,,,,,,,,,,,‘,,,,,,,,,,,,,,,,,,,,,,,,,,,,p,,,,,,,,~

0 20 40 60 a0 100 I

mean stress p (kPa)

‘0

2.0 -i- I

m basic path TLO 1 UNDRAINED TEST ++ttt basic path ~LO 2 and 3

1.5 m+++- basic path 7~0 4

o0 = 100 kPa

- basic path no 5 and 6 Loose state

-0.5 ~,,,,,,,,,,,rm,,,,,,,,,,I,,,,,,,,,,,~,,,,,,,, 0.0 1.0 2.0 3.0 4.0 5.0

axial strain E, (%)

300, !z

k? ‘ti ~ 250

. i

UNDRAINED TEST - basic path TW 1 o0 = 100 kPa +++++ basic path rw 2 and 3 Loose state - basic path no 4

a-

M basic path no 5 and 6

f,-, 200 G

: 7

150- I: ;i _ ~3undrained extension path

: w x: 10 v11 :ti

2 : II 8: : II j:

73 < w’ ;; -oedometric extension path

lz 0‘ j loo- =isotropic extension path

3 *isotropic compression path

t 50-L:: *oedometric compression path

? .’ O-&L, I I I, I, I I Y1 I/ I n, / No- n I,/, , I/ 1 I’, I I I -undrained compression path 0.0 1.0 2.0 3.0 4.0 5.0

axial strain E,(W)

FIGURE 12 Evolution of the weight functions and of the angular distances in the global loading path for the law “ABD”. Undrained loading path

197

23.0

20.5 { UNDRAINED TEST u0 = 100 kPa Loose state

Axial strain E, (X)

FIGURE 13 Evolution of the determinant of the rheological matrix of the laws “ABD” and “D”, undrained loading path

? $ 0.5

. ..l”.. 1 I.._.. rn.,““. ’ zos 3

UNDRAINED TEST 0, .

z 2 loose state (e. = 0.73) a0 = 200 kPa

UNlJKAlNdlJ ‘I.&S1

loose state (eO = 0.73) u0 = 100 kPa

0.0 I”;, / I,, I,, , I,, I,, / 1 ,I 0.0 0.1 0.2 0.3 0.4

axial strain .zi (X)

-Law -ABP

lJ"'IJ"~I~'J'I""I""l'~'~ 0.00 0.05 0.10 0.15 0.20 0.25 0. 0

axial strain c, (X)

FIGURE 14 CPU time needed for integration of the laws “D” and “ABD” on an undrained path

198

1.0 *

-7

F$ ;0.9- UNDRAINED TEST

! 3 loose state (e. = 0.73)

2 d 4 ~o.s-

i! 1 c&7-

$b ?

3% 0.6 -

u 2 ; 2

z g o.5 -

-ttm 100 kPa

2 2 w 200 kPa

0.4 ‘,,,,,,,,,,,,,,~,,,,,,,,,,I,,,,,,,,,,,

o.bo 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.4

axial strain c, (%)

0

FIGURE 15 Comparison of the CPU time needed for integration of the laws “D” and “ABD” on an undrained path (ratio of CPU time)

CONCLUSION

The authors have developed for the soils a new incrementally non linear

constitutive law “ABD” derived from the law “D” (Darve [ll]), but expli-

citly defined by a prescribed strain history. This new law uses the genera-

lized triaxial analytical formulations (extension and compression) of Darve

WI. This new law is more adjusted for using in the finite element applications

because constitutive or directional iterations are no more needed. On the

other hand, it could integrate other basic paths (for example, undrained

loading paths), if these paths were analytically formulated, which in fact

could considerably improve the predicted results, for any loading path.

As it was previously discussed, building a three-dimensional interpolation

is an extremely sensitive exercise. At the present time, we can’t say if all

the interpolation problems are definitively solved. The comparisons of this

new incremental non linear law and the Darve’s INL law generally show a

good agreement with a best capacity for this new law to simulate paths,

which are completely defined by strain rate.

199

(1) u;+ = -($) ( corn p ression loading in the direction j # i, no sum-

mation). -

u;- = -($) (extension loading in the direction j # i, no summation).

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NOTATIONS

The following notations have been used in this paper:

: strain increment pseudo-vector.

: stress increment pseudo-vector.

: tangent constitutive matrix.

: new law’s constitutive matrix.

: vector of incremental stress directions.

: vector of incremental strain directions. : number of basic path.

: weight functions.

: tangent basic constitutive matrix.

: new law’s basic constitutive matrix. : compression tangent Poisson ratios.

: extension tangent Poisson ratios.

: compression triaxial tangent moduli.

: extension triaxial tangent mbduli.

: compression state matrices.

: extension state matrices. : angular distance.

: current loading direction.

: basic direction.

: geometric distance between the extremities of the vectors d(“) and d. : sign and correction functions.

: shape coefficient of the interpolation functions.

: algebric mean value of the angular distances oi.

: mean value of the gaps between (Y; and CW.

: reduiced variable. : real parameters of the interpolation.

: parameter of adjustment.

Received 21 September 1993; revised version received 17 January 1994; accepted 18 January 1994