Characterization of the moduli of elasticity of an anisotropic rock using dynamic and static methods

J. Rock Mech.Min.Sci, & Geomech.Abstr.Vol.30, No. 5, pp. 527-535,t993 Printed in GreatBritain.All rightsreserved Int.

0148-9062/93$6.00+ 0.00 Copyright© 1993PergamonPressLtd

Characterization of the Moduli of Elasticity of an Anisotropic Rock Using Dynamic and Static Methods F. H O M A N D t E. M O R E L ~ J.-P. H E N R Y ~ P. CUXAC~f E. H A M M A D E ~ The tests reported in this paper were intended to determine the mechanical characteristics of a slate. This rock has a very definite discontinuous planar anisotropy superimposed on mineral lineation. As a first step, ultrasonic measurements were carried out to determine the main structural axes. The dynamic moduli of elasticity can be calculated if the velocities of the compressional and shear waves are known. This in turn allows a complete stiffness matrix to be calculated. As a second step, taking account of the ultrasonic measurements, the moduli of elasticity were determined from uniaxial and triaxial loading tests. The confining pressure is shown to have a definite influence on Young's modulus, measured at right-angles to the schistosity. Calculating the shear modulus is a fairly complex operation; clinotropic tests with loading-unloading cycles are necessary for an understanding of the large difference in observed values between dynamic and static shear moduli. The laws of evolution of the moduli as a function of confinement levels are validated by compressibility tests.

INTRODUCTION

Most rocks are anisotropic, i.e. their properties vary with direction. The structure of a rock is characterized by a number of factors, which may have a particular orientation: bedding, stratification, schistosity planes, foliation, cracking, joints. Anisotropy, which is a characteristic of metamorphic rocks such as schist, slate or gneiss, is due to the existence of mineral foliation. Sedimentary rocks such as sandstone, shale and limestone may be anisotropic, as a result of stratification. On a larger scale, any rock mass may be crosscut by one or more families of discontinuities. In this paper, we focus on a type of rock with a particular anisotropy which is interesting in that it combines mineral orientation and cracking. This rock is a slate of a type known to structurologists as discontinuous anisotropic. The symmetries most frequently encountered are those of a hexagonal (transverse isotropy material) or an orthorhombic (orthotropy) system.

The objective of this paper is to suggest a methodology that could be used to determine the type or degree of anisotropy in given types of anisotropic rock. This method is based on the propagation velocity, Vp and Vs, of the ultrasonic waves in various directions, the analysis of the polarization measurements of the S-waves being conducted without making any assumptions as concerns symmetry. This provides the matrix of elastic coefficients. As the degree of symmetry is known, uniaxial and triaxial loading strain tests can be carried out in the directions that are necessary for the determination of a constitutive law of behaviour (elastic constitutive law). Changes in the moduli of elasticity as a function of confinement have to be evaluated, and these changes are then validated using a test that is easy to perform; we suggest a compressibility test. In the case of rock with discontinuous anisotropy, the most difficult parameter to determine is the shear modulus. A method of achieving this is suggested. MATERIALS AND PRELIMINARY MEASUREMENTS

tLaboratoire de C~omreanique, ENSG, Nancy, P.O. Box 40, 54501 Samples under study, and their preparation Vandoeuvre les Nancy, France. :l:Laboratoire de Mreanique de Lille, URA CNRS, EUDIL, Lille, The rock under study is a slate with a very fine France. granolepidoblastic texture (grain size of the order of 527

528

HOMAND el al.: ANISOTROP1CROCK MODULI

10-20/zm), only slightly metamorphised (epizone). Its main mineralogical components are chlorite, muscovite and quartz. Pyrite, calcite and chloritoids may also be present. The pyrite may appear in various forms (isolated crystals, amygdalae, veins) and can either create zones of weakness or give increased strength. Schistosity is strongly defined, and merges with the stratification, at least in the experimental sample blocks. Three types of sample were used: --for determining anisotropy and dynamic moduli, 18-face polyhedric samples. --for compression tests, samples of 38 mm dia. with an aspect ratio of 2. --for compressibility tests, cubes with a 5 cm edge. All the faces of the samples were very carefully trued, in order to ensure the most rigorous flatness and parallelism.

remains identifiable (Table 1). These parameters were calculated as follows [2]: AM%=100

Am%:

1

V2+I"3 ,

100r2(~---/--~-3)-]. L 2+ 3 3

From this first approach, it is possible to conclude that the rock under study has an orthotropic symmetry (similar to the orthorhombic system in crystallography) that can be described by means of a stiffness matrix combining 9 independent components. DETERMINATION OF THE DYNAMIC MODULI ASSUMING SYMMETRY

Theoretical r e m a r k s

Hooke's law can be given for any elastic medium: a O = Cijktckt,

Preliminary measurements

Ultrasonic measurements were carried out to check that the samples were homogeneous. The rock's own axes were also defined, and the anisotropy quantified. A rigorous experimental protocol was devised, so as to obtain a signal of high quality [1]. Measurements were made on 18-face polyhedric samples, with the schistosity plane as a reference plane. The results were plotted on Wulff's diagram (lower hemisphere) as shown in Fig. 1 (with the centre of the circle representing the axis at right-angles to the schistosity plane). The lowest velocity is characteristic of transmission along a direction at right-angles to the schistosity plane, while the highest velocity is associated with wave propagation parallel to the cleavage plane. In this plane, wave velocity varies little, which indicates the existence of a major anisotropy AM (the schistosity plane) and a minor anisotropy Am (the lineation); the latter is weak, but

a

(1)

where Cijk~ respresents the elastic stiffnesses. These elements conform to the following relations: C,jk~ = Cj, k~ = G~,j = C,jtk.

Thus, in the most general case, only 21 of the 84 components of the C~kl tensor are independent. The equation of motion is as follows: ~2Ui

O2Ul

P - ~ T = C,jk, ~xj 3 x k '

One solution of the equation can be written:

Then, by carrying over (3) into (2) we find that: CijktnknjAj - p V 2 A i = O.

b

Fig. 1. (a) Velocitiesmeasured in a polyhedron.(b) Representation of the structural axes of the rock sample.

(2)

HOMAND et al.: ANISOTROPICROCK MODULI Table I. Major and minor anisotropies measured in three different blocks Block A Block B Block C AM(°/,) 28.5 30.4 27.1 Am(%) 5.4 1 4.2 If we assume that Fij

=

Cijklnkny,

it follows that:

529

These measurements can be obtained from the polyhedric samples, and our result is given in Table 2 as an example. The minimum value of Young's modulus is measured perpendicular to the schistosity plane; its maximum value is situated in that plane. The elastic anisotropy can be determined from the following two parameters [4, 5]:

F q A j - p V2Ai = 0

K = E3 El'

[Christoffel's equation (4)]. • Fij is called Christoffel's tensor (or the acoustic tensor), and is symmetrical; • n k and nj are the components of a unit vector n which defines the direction of propagation of the wave; • A; is the polarization of the wave. We show that Ai is the eigenvector of the tensor r u, and that the latter can have three eigenvalues p V~, confirming: det(F u - p V2k6U) = O.

y = G23. GI3

(4)

- - I f K > 1 the structure is planar. - - I f K < 1 the structure is linear. High Y-values indicate easy sliding between the schistosity planes. Our measurements give the following values: K = 0.48; Y = 3.38. DETERMINATION OF THE DYNAMIC MODULI IN THE ABSENCE OF HYPOTHESES RELATING TO THE SYMMETRY OF THE MATERIAL Theoretical aspects

Thus, in the most general case, at one point on a wavefront there are three orthogonal polarization directions (as F u is symmetrical), each associated with one velocity [3]. For certain symmetries of the material being studied, the lowest number of propagation and polarization directions necessary to determine all the elastic parameters can be determined. The moduli thus calculated are said to be dynamic, as opposed to the static moduli obtained during compression tests.

The calculations given above are based on the knowledge of the degree of symmetry of the material. It is, nevertheless, possible to determine a complete elastic matrix (21 coefficients) without making any assumption as to the type of symmetry. If we consider A as a unit vector, Christoffel's equation can be written as follows:

A p p l i c a t i o n to the r o c k under s t u d y

with

The analysis of the wave velocities showed that this rock could be said to possess hexagonal symmetry. To find the eigenvalues and eigenvectors of Christoffel's tensor, 4 of the 5 components of the stiffness matrix have to be obtained, which entails knowing the wave propagation velocities in the main directions. The velocity of the P-wave at 45 ° of schistosity gives the fifth component (Fig. 2).

~

lvpl(C33)

CuklnjnkAiA i = p V2,

or else as X~ Ck = M'

X I = njnkAtAi,

where the exponent l corresponds to the lth measurement. In the present case, the number of measurements, and thus of equations, is higher than the number of unknowns. The matrix [X] is first transformed by using an orthogonal factorization: [X] = [Q] x [R]. With the type of sample used, [X] is a 21 x 27 matrix; [Q] is a square orthogonal matrix and [R] is a higher triangular matrix. We then have a solution, written:

[C] = {[Q]T

/

~ ((211)

~'2t = ~'3t

dence with the elements of the stiffnessmatrix.

x

[M]}

x

[R]-'.

Table 2. Dynamic moduli of elasticity (VC = variation coefficient) Moduli VC (%) EI(MPa) 58,870 4 E2(MPa) 121,300 1.3 vt2= vl3 0. I 1 2 V23 = V32

Fig. 2. Wave propagation and polarizationdirections,and correspon-

(6)

M t = p V2,

Vp45(C13)

:

(5)

G21(MPa) G23(MPa)

0.19 0.24

15,100 51,000

6.6 3.4

7.3 1.3

(7)

530

HOMAND et al.: ANISOTROPICROCK MODULI

This equation is solved by first taking into account the estimated polarization directions only. The values of Cijk~ thus calculated are then introduced into Christoffel's equation in order to recalculate the eigenvectors of F,/. The operation is then repeated until stabilization occurs [6].

Application to the rock under study Experimentally, these calculations came from measurements made by using the principle of S-wave birefringence. A transversely polarized transducer was situated on either side of the sample. A control regulates two stepped motors which turn the transducers, either simultaneously or with respect to each other, with a jack which can apply a force reaching 15 kN to the sample. The samples used are 18-face polyhedrons and can be used to measure 27 velocities; the system (6) is thus overdetermined. After determination of the velocities of the P-wave and the two S-waves, propagating in the 9 measurement directions, the following matrix can be calculated: 1.257

transducers, and then applied with a servo-controlled machine in such a way that the chosen strain velocity of i.4 × 10-5/sec remained constant throughout the test. As we did not want sudden impacts to damage the ultrasonic transducers, the tests were stopped as soon as the stress-strain curve appeared to go beyond the elastic domain. The experiments were first carried out in a state of isotropic stress, and the deviator stress was then increased.

One appraoch to the dynamic moduli of elasticity Assuming the symmetry of the rock, it is possible to determine the moduli of elasticity by carrying out ultrasonic measurements in three directions. During triaxiai cell tests, however, the signal can be sent in only one direction, which means that only some of the elements of the stiffness matrix can be calculated: - - a transverse schistosity sample will give the C33 and C44 parameters; - - a n axial schistosity sample will give the Cj~ and C66 (or C44) parameters.

0.212

0.297

0

- E

0

1.135

0.272

0

0

0.01

0.582

-0.08

0

0

C(10 t~ MPa) =

0.159 -0.03

0.044

0.134 0.011 0.487 where E is a term close to zero. The values of the 9 coefficients corresponding to the orthotropy appear to be much higher than those of the other coefficients. It is thus possible to attribute orthotropic symmetry to the rock. However, lineation, as measured by ultrasonic waves (Dm ~ 5%, and the variation coefficient of measured velocities ~ 2%), had very little influence, as was confirmed by preliminary compression tests carried out in the three first directions determined, and the problem was thus treated in rotational orthotropic symmetry (hexagonal system).

EVOLUTION OF MODULI OF ELASTICITY AS A FUNCTION OF CONFINEMENT AND

There are two families of coefficients (Fig. 3): --those that increase with stress (C33, C44); --those that remain stable (C~t, C66). At first, C33 and C44 increase fairly rapidly. This stage ends when confinement reaches 20-24 MPa, after which C44 becomes stable, while C33 continues to increase slowly. The impossibility of determining the evolution, especially that of the Young's moduli, is the weak point of this method. However, considering that dynamic moduli are often determined in situ using the traditional formulas for isotropic media, it seemed a promising idea to 30

DEVIATOR STRESS

An analysis of the influence of confinement on the moduli of elasticity is necessary to the completeness of the study. Measurements of ultrasonic propagation in a triaxial cell should be give useful results for the modelling stage.

20

Test methodology Measurements of ultrasonic propagation were undertaken during triaxial compression tests. The axial stress was transmitted to the sample by steel blocks containing

-10

I0

/I.._+__

-

j,c,

0 Stress (MPa) ii,

0

20

40

60

80

100

120

140

160

Fig. 3. Variationof the elastic parameters as a functionof the triaxial stress (confinementof 40 MPa).

HOMAND et al.: ANISOTROPICROCK MODULI 55000

531

C33 and C,~ by making use of the following relations (Fig. 5):

(MW)

C44 = 15,000 x (1 + 0"3)°°s5, 50000

C33 = 65,000 x (1 + 0"3)0"027. " ". ~ 45 000 0

"

03 (MPa) i

.

,

,

-'

I

~

5

I0

15

20

25

30

35

GENERAL ASPECTS OF TRIAXIAL TESTS 40

Fig. 4. Variationof the Young's pseudo-modulias a functionof axial stress. use the method to calculate the moduli of elasticity and the Poisson's ratios. The following expressions were used in this case:

V~(I + v)(l - 2v) (l -v)

E =p

0 . 5 - (vs/vp) 2 1 -(vs/vo ~

The pseudo-modulus E~ increases very rapidly with confining pressure, becoming stable at around 50,000 MPa (Fig. 4). In the direction parallel to the schistosity, E2 varies only slightly. The Poisson's pseudo-ratios do not show any clear evolution. Certain authors [7, 8] suggest relations linking the values of the moduli to the confining pressure. In the present case, it is possible to express coefficients

19000

C44 (MPa)

~

®

The triaxial tests were carried out on core samples, at various orientations 0 to the schistosity plane. Each test was assigned three numbers, indicating the orientation of the schistosity plane, the confining pressure and the test number, respectively. Figure 6 shows the positions of the gauges for the 0 and 90 ° samples. The number of gauges was then doubled and the mean signal values were processed. As concerns the other orientations, a 45 ° dial was placed at fight-angles to the schistosity, with one gauge following the orientation of the schistosity (Er), while another was placed on the schistosity plane (e3//) in a transverse direction. As this arrangement was not entirely satisfactory, we later placed a gauge at fight-angles to the schistosity planes. INTERPRETATION OF THE RESULTS Preliminary remarks

For the 0 and 90 ° orientations, failure took place with an extension mechanism, inside either the rock matrix or the schistosity; for the other orientations, the failure mechanism consists of one plane sliding on another.

74 000' C33 (MPa)

18000

72 000 s C44 = 15 000 (I+c3)0"085

17000

70 000

16000

=

"

68 000

15000

66 000 o 3 (MPa)

14 000

64000 I0

15

20

25

30

35

40

0

: 5

: 10

: 15

: 20

Fig. 5. Variation of the C33 and C~ parameters as a function of pressure.

Fig. 6. Positions of the gauges for 0 = 0°, 90°, 45°.

: 25

: 30

o 3 (MPa)

I 35

I 40

HOMAND et al.: ANISOTROPIC ROCK MODULI

532

Table 3. Value of EI Tests E~(MPa) 90/0/1 31,200 90/0/2 34,300 90/2.5/I 37,400 90/5/1 37,400 90/7.5/1 41,200 90/10/1 38,100 90/20/1 41,350 90/30/1 42,200 90/40/1 42,700

Kulhawy [7] gives an expression of the modulus as a function of pressure, in the form of a power of a3. Santarelli [8] suggested that this law be modified by making use of a power of 1 + a3. In the present case, the following approximation can be assumed to apply:

Table 4. Values of E2 Tests E2 (MPa) 0/0/1 117,500 0/5/1 123,000 o/lo/I 181,000 0/10/2 114,000 0/20/1 153,000 0/20/2 120,000 0/30/1 120,000 0/4011 118,000

Poisson's ratios

El = 33,190(1 + 0"3)0"072, R = 0.93. The curve is shown in Fig. 7.

The stress-strain curves indicate linear elastic behaviour. It may nevertheless happen that a sudden failure (45 ° tests) entirely masks this elastic phase, in which case it is difficult to interpret the results.

Young' s moduli El and E2 The longitudinal gauges (Et) of the 0 and 90 ° tests give the values for E2 and El, respectively. The results are shown in Table 3 and 4. The E: modulus is independent of the confining pressure. The high value (tests 0/05/1 and 0/20/1) may be explained by heterogeneity within the samples. Setting aside these values, the mean value is approx. 119,000 MPa. El is strongly influenced by the pressure. For the 90 ° tests, the deviator was applied at right-angles to the schistosity planes, which resulted in a compaction of these planes as a function of the confinement; this increased the apparent stiffness of the material.

44000

The 90 ° tests give the values of vl2 (E3 gauge). The 0 ° tests give the values of 1'21 ((~3p) and v23 (E3//). The results are indicated in Tables 5 and 6. As concerns v~2 and v2l, the values appear to decrease slightly with confinement, while, for 1'53, they appear to increase. Such marked dispersion rules out any chance of finding a significant correlation between variation and confinement; it was thus decided that, in our calculations, we would use the mean values of the coefficients, namely: 1'12=0.14, v23 = 0.16; v21 = 0.34.

Validity of the elastic hypothesis In this paper, our hypothesis was that of elastic behaviour, which implies that v~2/El = v:l/E2. Table 7 gives the values of these ratios for the different pressures. The vj2/E 1 and vl:/E2 ratios were calculated from the following values: vl2 = 0.14; v21 = 0.31; E2 = 119,000 MPa. The Young's modulus is given by the following relation: El = 33,190(1 + 0"3)0"072. The elastic hypothesis lapsed because of differences, which were large at low pressures but decreased at higher pressure. The calculated values of vl2/E I correspond to the law of variation of this mean value, 0.5 (vl2/Ei + v21/E2) = 3.536 x 10-6(1 +tr3) -°°4j. They are shown in Fig. 8.

El 0riP,)

42 000 40000

38 000 36000 34000 32000 a3 (MPa)

30000

I

I

I

I

I

I

I

I

5

10

15

20

25

30

35

40

Fig. 7. Variation of E~ as a function of confining pressure.

HOMAND et al.:

ANISOTROPIC ROCK MODULI

Table 5. Values of vt2 (90° tests) Tests

v'12

90/0/1 90/0/2 90/2.5/1 9015/I 90/7.5/1 90/10/1

0.17

9o/2o/1 9o/3o/1

O.ll o.15

90/40/1

0.093

the slopes of the linear sections were almost identical (Fig. ll). In theory, the gauges should give the following readings:

0.13 O. 14 0.16 0.15

dEF = dec = d P I ~

dEA = dee = d E D = dEE =

Tests

v2~

v23

0.39 0.42 0.31 0.43

0.12 0.175 0.12 0.16 0.19 0.165 0.13 0.2

0.27 0.23 0.34

Table 7. v J E j ratios (10-6/MPa) P (MPa)

Vl2/E !

•21/E2

0 5 10 20 30 40

7.05 3.74 3.34 2.66 3.55 2.18

3.32 3.41 3.77 2.25 1.92 2.88

],

dpF-V12

L El +

It is theoretically possible to measure a shear modulus during a compression test. There are, however, quite a number of practical difficulties to this, as the strain gauges do not provide sufficient data, and may produce aberrant results. Let us consider the tests carried out at 0 and 90 °. The E(O) moduli are determined by using the longitudinal gauges (E,). The values obtained are shown in Table 8. For linear elasticity, the equation is as follows:

This test was carried out to validate the laws suggested for the different moduli. It used a cube with a 5 em edge with the gauges placed as shown in Fig. 9. The sheathed sample was placed in a confining cell. Two loading cycles were applied, between 0 and 55 MPa. The responses of the C and F gauges were similar (Fig. 10). The important non-linear stage indicates a principal crack in the schistosity plane (crack porosity of the order of 400 x 10-6). As concerns the other gauges,

1

sin40

E(0) =

(1

vl2

E--T-+ \~z

v21"~

E, E:J cos4 0 x sin: 0 cos: 0 + - -

E:

We use approximations for E,, E2 and the mean value

l_ (v,: v2, 2 \ E +~--J '

Vl2/E1 calculated 3.5 3.4 3.3 3.2 3.1 3.0

5

V23"]

EVALUATION OF THE SHEAR MODULUS G12

COMPRESSIBILITY TEST

0

1 --

We carried out a simulation of the compressibility test by using approximations of the moduli. The comparisons are shown in Figs 12 and 13, for the F and E gauges respectively. The simulation does not take into account compaction due to open planes during sample preparation (F gauge, Fig. 12); the variation of E L as a function of confinement is expressed by a continuous media behaviour law that does not simulate the opening of the planes. However, the slopes of the curves show a high degree of similarity.

Table 6. Values of v2t and v23 (0 ° tests) 0/0/1 0/5/1 0/10/1 0/10/2 0/20/l 0/20/2 0/30/1 0/40/1

533

10

15

20

25

30

35

Fig. 8. Evolution of calculated v,2/E , as a function of confinement.

40

HOMAND et al.: ANISOTROPIC ROCK MODULI

534

60 03 (MPa)

50

$1nmla~perin~n

t

40 30 20 10 I

0

I

50

I

100

I

150

I

200

250

Fig. 12. Experiment-simulation comparison for the F gauges. 60'

Fig. 9. Positions of gauges during the compressibility test. 60

50. 40.

03 (MPa)

30 20

40 30 20 10 0

l0 -6) 0 200 400 600 800 1000 1200 1400 Fig. 10. Responses of the C and E gauges.

and suggest the following method of determining Gi2: - - T h e theoretical E(O) curves are plotted for each confining pressure by varying G~2. The maximum value is assumed to be 15,000MPa (taking into account the values determined by ultrasonic waves). Gl2 was then made to vary from 12,000 to 2000 MPa, with intervals of 2000 MPa. - - T h e experimental values of E(O) are superimposed on these curves (Table 8). This is done for each pressure value. An example is given in Fig. 14. The values of G~2 can be deduced from these curves. It has been estimated that, for the 15c orientation, Gl2 is situated between 2000 and 3000 MPa; at 30 °, the value is between 6000 and 7000 MPa. These values increase with the slope, although there is no indication of a clear correlation with confinement; in any case, they remain lower than 14,500 MPa. On the whole, the other gauges did not provide much information, either because the values were too widely scattered or because the gauges

00

200

400

600

80O

111110 1200 1400

Fig. 13. Experiment-simulation comparison for the E gauges. were strongly influenced by the sliding of one plane over another. It is nevertheless possible to provide an interpretation of the signals obtained from the ( % ) gauge. In theory, this gauge should give: =

I

1 . ~ vl2

xsin 2 0 c o s 20

hE El

The method previously described for E(O) was used to evaluate the values of G,2, which were found to remain within approximately the same value range. The variation in the values of Gi2 as a function of the anisotropy orientation led us to question the validity of the shear coefficient thus calculated. Moreover, the static value of Gt2 ( ~ 2000 MPa) is extremely low when compared to the value found by ultrasound ( ~ 15,000 MPa). There is no certainity that the strain characteristic recorded during a uniaxial or triaxial loading test is an elastic quantity. We then proceeded to carry out ad120 000

60'

100 000

50

80 000

4O

60000

30 40000

2O

20 000

10 0

.' 0

.'

:;

I.

!.

~ (1E'6)1

50 100 150 200 250 300 350 Fig. 11. Responses of the A, B, D and E gauges.

0

I

0

I

I

I

I

I

I

10 20 30 40 50 60 70 Fig. 14. E(O) carrier for p = 5 MPa.

I

I

80

90

HOMAND et al.: ANISOTROPIC ROCK MODULI Table 8. Values of E(0) as a function ofthe confining pressure (MPa) Pressure (MPa) Orientation (degrees)

0

5

I0

20

30

40

15 30 45 60 75

40,600 45,800 15,480 44,160 40,000

28,000 33,000 11,890 36,750 21,000

17,340 15,390 58,000 20,090 61,100

21,940 15,400 40,500 12,860 47,560

16,560 13,500 48,300 12,500 32,650

37,700 54,000 12,300 39,690 26,810

40 30

(O t "O 3) MPa

25 20

15 10 5

535

values of the Poisson's ratio are widely scattered; and furthermore, the calculation of GI2 is difficult and can only be achieved indirectly. Clinotropic tests with loading-unloading cycles indicated that the moduli obtained from the cycles had a higher value. The scattering encountered is due, it is thought, on the one hand, to the discontinuous anisotropic aspect of this material, and, on the other hand, to its inhomogeneity. There is obviously no question of comparing the dynamic moduli to the static moduli; the mechanisms brought into play by the two types of stress lead to different results, and the discontinuous nature of anisotropy only accentuates these differences. The evolution of the different parameters is identical, however, in static and dynamic conditions, which leads us to put forward a somewhat simplified model. This model should be introduced in a numerical modelling code, and will be validated through triaxial tests which are to be carried out in the large-capacity true triaxial machine at the University of Lille.

0 200

400

600

800 1000 1200 14001600

18002000

Fig. 15. Deviator axial strain curve for the sample with 30° schistosity.

ditional triaxial compression tests, with load-unloading cycles, on clinotropic samples. These cycles immediately indicated that the moduli calculated for one cycle are far higher than the moduli calculated for the envelope (Fig. 15). CONCLUSIONS

An analysis making use of ultrasonic wave velocities can, as a first step, be used to determine the type of symmetry under consideration, and thus to define the main axes which may be used for a static analysis. A second step is to determine a dynamic stiffness matrix for the material concerned. However, an analysis using wave velocities alone would be incomplete, since the determination of different moduli as a function of confinement is lacking. This is due to the impossibility, in a fairly non-homogeneous anisotropic medium, of calculating all the elastic parameters from the measurements provided by our experimental arrangement. There are also certain experimental problems linked to the extensometric measurements. For example, the

RMMS 30/5--E

Acknowledgements--The research described in this paper was funded by the French National Agency for the Management of Radioactive Wastes (ANDRA). The authors acknowledge the valuable discussions with P. Lcbon. Accepted for publication 9 March 1993.

REFERENCES

!. Cuxac P. Etude de la propagation d'ultrasons dans des roches fissurres et anisotropes. Thrse de Doctorat, INPL, Nancy (1991). 2. Guyader J. and Denis A. Propagation des ondes dans les roches anisotropes sous contrainte. Evaluation de la qualit6 des schistes ardoisiers. Bull. Assoc. Int. Gdol. Ing. 3, 49-55 (1986). 3. Dieulesaint E. and Royer D. Ondes 61astiques darts les solides. Application au Traitement du Signal. Ed. Masson, Paris (1974). 4. Sirieys P. Anisotropie mrcanique des roches. Comportement mrcanique des solides anisotropes, Coll. du CNRS, No. 295, pp. 481-532 (1982). 5. Laqueche H. Drtermination de rrnergie de fracturation des roches anisotropes et sismogrnrse. Thrse de Doctorat, Universit6 Bordeaux I (1985). 6. Arts R. J., Helbig K. and Rasolofosaon P. N. J. General anisotropic elastic tensor in rocks. Approximation, invariants and particular directions. 61st Society of Exploration and Geophysicists Meetings, Extend Abstract, Houston, Texas, Paper ST 2.4, pp. 1534-1537 (1991). 7. Kulhawy F. H. Stress deformation properties of rock and rock discontinuities. Engng Geol. 9, 327-350 (1975). 8. Santarelli F. Theoretical and experimental investigation of the axisymmetrie wellbore. Ph.D. Thesis Imperial College, London (1987).