Anisotropic strength and deformational behavior of
Himalayan schists
M.H.B. Nasseria,*, K.S. Raob, T. Ramamurthyb
a
Department ofCivil Engineering, University of Toronto, 35 St. George Street, Toronto, ON, Canada M5S 3B1
b
Civil Engineering Department, Indian Institute of Technology Delhi, New Delhi 110016, India
Accepted 11 September 2002
Abstract
Anisotropy, which is characteristic of metamorphic rocks such as schists, is due to a process of metamorphic differentiation.
Preferred orientation of minerals like mica and chlorite in response to tectonic stresses makes schistose rocks foliated. As a result
their engineering properties vary with the direction of loading.
The influence of transverse anisotropy on strength and deformational responses of four schistose rocks obtained from the
foundation of two underground powerhouse sites in the Himalayas has been critically examined. Specimens at different orientation
(b) of the foliations varying from 01 to 901 with respect to the axial stress (s1) in the unconfined state and also in the confined states
up to 100 MPa of confining pressure were tested to evaluate the applicability of the non-linear strength criterion for the prediction of
triaxial compressive strength and modulus. Based on the analysis of large experimental results it has been possible to predict strength
and modulus with minimum pre-evaluation experimental data, i.e. only with three uniaxial compressive strength tests at 01, 301 and
901 and two triaxial compression tests conducted at convenient confining pressures at b = 901orientation. Predicted non-linear
stress-strain curves, using predicted values of strength and modulus have been found to match well with the experimental stressstrain curves even at higher confining pressures.
Keywords: Anisotropy; Strength; Deformation; Failure criteria; Modulus; Schist; Stress-strain
1. Introduction
Out of the three generic categories of rocks, metamorphic rocks exhibit highest degree of anisotropy [1].
Segregation of constituent minerals, in response to high
pressure and temperature gradients, is associated with
tectonic evolution and development of layers of
contrasting mineralogical assemblages. Rocks flow and
recrystallize under new tectonic stresses to form weak
foliation planes. Such planes of weakness (i.e. schistosity) affect the strength and deformational behaviors of
rocks with orientation to the applied stresses. Irrespective of the size of the engineering projects, either dealing
with inherent intact rock anisotropy from an exploratory borehole or induced rock anisotropy due to in situ
fracturing [1], where stability of large rock mass is
concerned, evaluation of intact rock anisotropy in terms
of strength and modulus is inevitable. Prediction of the
anisotropic responses of strength and deformation of
rocks involves study of specimens at different orientation angles, b (the angle between the major principal
stress direction and the foliation plane).
In the past, many investigators have carried out the
measurement of the strength anisotropy for various rock
types e.g. Donath [2], Chenevert and Gatline [3],
McLamore and Gray [4], Hoek [5] Attewell and
Sandford [6] and Brown et al. [7] on shales and slates,
Deklotz et al. [8], Akai et al. [9] McCabe and Koerner
[10] Nasseri et al. [11,12] and Singh and Singh [13] on
gneisses and schists, Ramamurthy et al. [14,15], on
phyllites Horino and Ellicksone [16], Rao et al. [17] and
Al-Harthi [18] on sandstones, Pomeroy et al. [19] on
coal, Allirote and Boehler on diatomite [20] and Tien
and Tsao [21] on artificial material. An overall analysis
and review of their works exhibit that maximum failure
strength is either at b = 01 or 901 and the minimum
value usually is around b = 301, more precisely at
(45 2 f=2) where f is the friction angle along the plane
of weakness, fracture or sliding. The shape of the curve
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
[17], proposed an empirical strength criterion to account
for the non-linear strength response of isotropic intact
rocks in the following form:
Inherent
Induced
30
p, degree
Fig. 1. Possible variation of SC versus b for inherent and induced
anisotropies.
between the uniaxial compressive strength (sc) and the
orientation angle, b; is designated as the 'type of
anisotropy’’ and is found to be generally of three types
namely 'U-shaped", "shoulder shaped’’ and ‘‘wavy
shaped’’ as shown in Fig. 1 [1].
In spite of many attempts made in the past to
delineate the engineering behavior of intrinsically
anisotropic rocks still their nature is not adequately
understood. Prediction of strength of anisotropic rocks
through various criteria proposed by Jaeger [22], Walsh
and Brace [23], McLamore and Gray [4] and Hoek and
Brown [24] requires a large amount of pre-evaluated
experimental data, loosing the simplicity in their form
and hence the practical application. Adoption of these
failure criteria necessitates performance of at least three
triaxial tests at different confining pressures and at three
different orientations of plane of weakness.
Keeping in view the inadequate knowledge of transverse anisotropy of deformational and strength responses of schists over the entire range of b; an attempt
has been made through a comprehensive investigation
on four varieties of schists obtained from two Hydroelectrical power project sites in the Himalayas. The main
objectives of the present investigation are, therefore:
(i) To study the transverse anisotropic behavior of
schists in terms of compressive strength and
deformational responses in uniaxial and in triaxial
compression up to high confining pressures.
(ii) To evaluate the applicability of the non-linear
strength criterion proposed by Ramamurthy and
co-workers for the prediction of triaxial compressive strength of schists.
(iii) To develop a methodology to predict the tangent
modulus and stress-strain response with minimum
pre-evaluation of experimental data.
where sci is the uniaxial compressive strength of intact
rock without a weak plane, s1 and s3 are the principal
stresses, ai is the slope of the plot between ÐS1 — s3Þ=s3
and (sci=s3) on log-log scale and B i (G\ 1 <x3)/cr3
when (ffci/aj) = 1; ai and Bi are considered as strength
parameters. The authors had suggested a constant value
of 0.8 for ai at all orientations (b) even for intact
anisotropic rocks. Owing to the fact that Bi parameter
did not vary much, in their analysis, a constant value of
Bi as well could have been assumed. The variation in the
value of Bi was calculated corresponding to a constant
average value of ai = 0:8. Ramamurthy et al. [15], on the
basis of the results obtained from the triaxial compressive strength on three anisotropic rocks viz. quartzitic,
carbonaceous and micaceous phyllites and plots between log ðs1 — s3Þ=s3 and log ðsci=s3Þ for different
orientations, have concluded that even for intact
anisotropic rocks, the strength parameters denoted by
aj and Bj cannot be taken as constants and these
parameters showed systematic variation with scj of
anisotropic rock and orientation angle b: Subscript i
refers to intact rock without weak plane andj with weak
planes. Analyzing the experimental data of more than 13
various other intact anisotropic rocks, following two
equations have been developed [15] for the evaluation of
aj and Bj; the strength parameters at any orientation
other than b = 901; with scj as the compressive strength
at the corresponding orientation,
(a;/«9o) =
(2)
5
Bj=B90 = ða90=ajÞ0:
:5:
ð3Þ
The empirical strength criterion for an anisotropic
rock at any orientation is given by
(ffi - <x3)/<x3 = Bj(acj - oif,
(4)
where sc90 is the compressive strength at b = 901; a90
and B90 the strength parameters aj; and Bj; at b = 901
obtained by a minimum of two triaxial tests at this
orientation.
Minimum requirement to evaluate, the strength from
the criterion in its new form is from two triaxial tests at
/? = 901 orientation and the uniaxial compressive
strength values at three orientations of b = 01, 301and
901, because of the prevalent U-shaped anisotropy.
2. Non-linear criterion for anisotropic rocks
3. Modulus in unconfined state
Based on the test results of four intact sandstones and
from the analyses of available test data from more than
80 intact rocks, Ramamurthy et al. [14] and Rao et al.
It has been pointed out by Chenevert and Gatline [3],
that only two elastic constants i.e. E (Young's modulus)
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
and n (Poisson's ratio) are required for the linear theory
of elasticity for isotropic homogeneous body, whereas
evaluation of nine elastic constants are required to
completely describe the elastic behavior of an orthotropic anisotropic body. They have however reduced the
number of elastic constants to five to describe the
deformational behavior of a transverse anisotropic rock
through the analysis of their experimental data. In an
attempt made by Read et al. [25], to evaluate the elastic
constants for Hast schist, they found that modulus of
elasticity parallel to foliation Ey = Ex = E1 =
65;000 MPa and normal to foliation i.e. Ez = E2 =
45;000 MPa; The value of Poisson's ratio for the planes
nxy = nyx = n1 = 0:3 whereas Poisson's ratio for the
planes nzx = nzy = n2 = 0:15: Shear modulus, G; can be
calculated through the following equation
All the aforementioned calculations were made
assuming that the direction of foliation in schist
becomes parallel to the x2y co-ordinate plane, along
which there exist a transverse isotropy. It has been
demonstrated by Amadei [26] that for transversely
isotropic rocks, the modulus G in the plane normal to
the plane of transverse isotropy can be defined in terms
of E1; E2; n1 and n2 using the following empirical
equation
\/G=
(6)
where E1 and E2 are Young's moduli in the plane of
transverse isotropy and in direction normal to it,
respectively, n1 and n2 are Poisson's ratios characterizing
the lateral strain response in the plane of transverse
isotropy to a stress acting parallel or normal to it,
respectively and G is the shear modulus in planes normal
to the plane of transverse isotropy.
Degree of modulus anisotropy as measured by the
ratio E1=E2 has been extensively worked out through
analysis of 98 measurements of elastic properties of
rocks by Amadei et al. [27]. It has been shown by them
that for most of the intact transversely isotropic rocks,
the ratio of E1=E2 varies between 1 and 4. Few cases
where E1 =E2 observed to be less than 1, but did not fall
below 0.7. The ratio G=G0 varied between 0.1 and 0.7.
According to Worotnicki [28] degree of modulus
anisotropy for quartzo-feldspathic and basic/lithic
groups of rocks is less than 1.3 for 70% and less than
1.5 for 80% of the rocks analyzed. For the abovementioned class of rocks, the said ratio does not exceed
3.5. The rocks classified by him under pelitic clay and
pelitic micas group exhibit the greatest degree of
anisotropy with Emax=Emin less than 1.5 for about 33%
and 2 for 50% of the rocks analyzed. For this group of
rocks modulus ratio was not more than six with most of
the cases falling below four. His fourth anisotropic
group of rocks being carbonate rocks which revealed
Emax= Emin n °t being more than 1.7.
Prediction of apparent Young's modulus through
determination of elastic constants for transversely
isotropic rock samples involves strain measurements
along x; y and z coordinate systems in rock specimens,
cored with b along 01, 451 and 901. Adopting the same
methodology Amadei [26] attempts to predict the
apparent Young's modulus and Poisson's ratio for a
diatomite and finds that these values vary widely as b
changes from 01 to 901.
In the present study an attempt is made to categorize
the variation of modulus of elasticity in the unconfined
condition with b ranging from 01 to 901. Analysis of the
data for three schists viz. Graywake schist I, Graywake
schist II (Pinto, [29]) and Hasst schist (Read et al. [25]),
three shales, viz. Moszczenica shale, Borynia shale and
Wilchwy silty shale (Kwasniewski and Nguyen [30]),
Barnsley hard coal (Pomeroy et al. [19]) and Diatomite
(Allirote and Boehler [20]) reveal the fact that such
variation can be categorized under two broad types, i.e.
"U-shaped" and ‘‘decreasing order-shaped’’. Both types
are characterized with higher value of modulus at b = 01
than at b = 901 (Fig. 2).
4. Modulus in confined state
Investigators working on the effect of confining
pressure on modulus demonstrated that modulus
increase with increase of confining pressures. This
increase has been related to the closure of the micro
cracks and pore spaces in response to confinement.
Preferential orientation of micro cracks along the
foliation planes is considered to be another cause of
anisotropy at microscopic level, [31]. Hobbs [32], in an
attempt to study the effect of confining pressure on
Young's modulus of seven coals cored parallel to the
bedding plane found that with the exception of one coal
(Barnsley Hards), Young's modulus of other coals
increased with confining pressure up to 40 MPa nonlinearly. Evaluation of modulus of elasticity as a
function of confinement through measurements of
ultrasonic propagation in a triaxial cell by Homand
et al. [33], suggests that Young's modulus and shear
modulus can be expressed as power function of ð1 þ s3Þ
as confining pressure increases up to 40 MPa for
specimens tested at b = 01; 451and 901.
In spite of the fact that many researchers have tried to
study the effect of confinement on the modulus of
anisotropic rocks, the data available over the entire
range of b in the confined and the unconfined states in
the literature is scanty. Application of linear elastic
theory to model the variation of modulus with respect to
confinement is of limited value and requires nonlinearity to be considered.
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
(d)
90
Fig. 2. (A) U-shaped variation of modulus with b for (a) Hasst schist, (b) graywake schist-I, (c) Wilchwy silty shale and (d) Moszczenica shale. (B)
Decreasing order-shaped variation of modulus with b for (a) Barnsley hard coal, (b) graywake schist-II, (c) diatomite and (d) Borynia shale.
5. Geology of the site and rock tested
Keeping the objectives in view four anisotropic
schistose rocks have been collected from the tectonically
active and complex geological sequence of rocks in the
foothills of the Himalayas for laboratory investigation.
Quartzitic and chlorite schists were collected from the
Uri hydroelectrical project in Baramula District, Kashmir. This project envisages construction of a concrete
dam on the river Jhelum with a 10 km long and 8 m
diameter of head race tunnel in quratazitic schist
belonging to the Tanawal series of Precambrian age.
An underground powerhouse with a tailrace tunnel is
being constructed in chlorite schist belonging to the
Panjal traps (Precambrian to Eocene). In the eastern
side, the Tanawals are overlaid by the Panjal lava flows,
which have been chloritized and developed schistosity.
In general the strike of the formations is N301E-S301W
dipping 70-801 in a northerly direction.
The other two less competent schistose rocks, i.e.,
quartz mica and biotite schists were obtained from the
foundation of the underground powerhouse site at
Nathpa-Jhakri hydroelectrical project in Kinaur District, Himachal Paradesh. The scheme envisages construction of a 60.5 m high concrete dam on the river
Sutlej at Nathpa with 27.78 km long and 10.15 m
diameter head race tunnel and underground powerhouse
at Jhakri, and a 280 m long trail race tunnel. The rocks
of the area are quartz mica, biotite schist, granite gneiss,
amphibolite and pegmatite belonging to the Wangatu
Jeori gneissic complex of Precambrian age. The foliation
trend of these metamorphic rocks generally varies from
N701W-S701E to N701E-S701W having an average dip
of the order of 351 in the northerly direction.
6. Experimental work
In accordance with the objective mentioned earlier
various tests were conducted on the four schistose rocks
following ISRM [34] and IS (Indian Standard) practices
[35]. The tests are classified under the following three
major categories: (i) petrography and petro-fabric, (ii)
physical properties and (iii) geotechnical properties.
6.1. Specimen preparation
Large size blocks were trimmed with their sides
perpendicular to each other to facilitate coring at
different inclinations, using a special frame fitted to
the base of the conventional laboratory drilling
machine. About 500 specimens of length to diameter
of 2 having 3.8 cm diameter at different orientation
angles b (01, 151, 301, 451, 601, 751, and 901) were cored
from the four rock blocks. Fig. 3 shows such specimens
for uniaxial and triaxial compression tests at different
orientations.
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
11
*
3C)
o
o
1
nii
'fi
a
A
|
ieo 0
o
!?;'••
'1
: •
1
£~
••:-.
| QUARTZITIC SCHIST
CHLORITE SCHIST
QUARTZ MICA SCHIS
BIOTITE SCHIST
Fig. 3. Specimens prepared at different orientation angles for the schists.
An attempt was made to drill cores coring the range
of b from a single block to minimize the lithological
differences. The specimens meeting the tolerance limits
were first oven dried at 1051C for 24 h and kept in a
desiccator for cooling. For measuring axial (ea) and
diametral (ed) strains under uniaxial and triaxial
compression, electrical resistance strain gauges were
fixed, two axially and two diametrically on opposite
sides at the mid-height of the specimens. Uniaxial
compressive strength of the schists was determined as
per ISRM test procedure. Triaxial compressive tests
were carried out using a 150MPa capacity triaxial
cell placed in a 5MN capacity loading frame. Five
different confining pressures applied during the triaxial
tests were 5, 15, 35, 50, and 100 MPa. The specimens
were first subjected to the required confining pressure
and then the axial load was applied until the specimen
failed.
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
Table 1
Physical and engineering properties of schists tested
Properties
Schists tested
3
Dry density (g/cm )
Specific gravity
Porosity (%)
Permeability, k(cm/s), b = 01
UCS (s c90 ) (MPa), b = 901
Tangent modulus, Et90; (GPa) a
Poisson's ratio, b = 90 1a
Shear modulus, G90 (GPa)
Tensile strength, b = 901
Brazilian, stb (MPa),
Point load axial, stpa (MPa)
Cohesion, c (MPa), b = 901
Angle of internal friction, f1901
a
Quartzitic
Chlorite
Quartz mica
Biotite
2.63
2.66
0.81
3.5 x 1O~10
190.0
20.0
0.25
8.0
2.88
2.90
0.26
4.2 x 10" 11
110.0
12.0
0.2
5.7
2.72
2.83
1.70
2 x 10~6
50.0
4.0
0.14
2.20
2.74
2.85
0.76
1 x 10~7
50.0
5.0
0.1
1.4
29.0
15.0
35.0
56.0
24.0
10.0
25.0
53.0
12.0
4.0
15.0
50.0
8.9
3.8
11.5
46.0
Tangent modulus and Poisson's ratio measured at 50% of failure stress.
O
Experimental
Predicted
200
125
175
100'
150
2
\i25
- 75
100
50
75
50
15
30
45
60
75
90
15
30
45
60
75
90
(a)
60 -
50 r
£40
0
0
15
30
45
60
75
90
(c)
Fig. 4. Variation of experimental and predicted scj for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite schists.
7. Results and discussion
7.1. Petrography
X-ray diffraction and scanning electron micrograph
(SEM) studies revealed the following:
(a) Quartzitic schist is predominantly made up of
crypto-crystalline to fine grained flaky micaceous
minerals, preferably oriented with fine grained
recrystallized quartz, which are in abundance.
Quartz constitutes about 43%, mica 15% and
feldspar 12.6% of the rock, with clay minerals
such as kaolinite, illite and chlorite forming the
rest.
(b) Chlorite schist is very fine grained, highly chloritized basaltic rock with well-developed schistosity.
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
600-
15
30
45
60
75
90
15
30
45
60
75
90
Fig. 5. Failure strength versus b at different s3 for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite schists.
It appears that the quartz bands are formed due to
filling of the foliation planes. The rock contains
29% quartz, 25% chlorite, 11% mica, with clay
minerals forming the rest of the constituents.
(c) Quartz mica schist is a coarse grained rock with
well-defined schistose texture. Recrystallized and
elongated quartz grains with sutured boundaries
show strong preferred orientation. This rock contained 31% quartz, 26% chlorite, 22% mica, with
clay minerals forming the rest.
(d) Biotite schist is characterized by thicker foliation
planes and coarser than quartz mica schist. This
rock contained 56% biotite, 10% quartz, 20%
sepiolite; kaolinite and illite form the other constituents of this rock.
7.2. Physical and engineering properties
Determination of physical and engineering properties
of the schists have been carried out based on ISRM and
IS specification and the results are presented in Table 1.
Engineering properties are shown at ‘‘representative
orientation’’ i.e. (b = 901) for the schists.
7.3. Uniaxial compressive strength
The variation between compressive strength (sc) and
the corresponding orientation is presented in Fig. 4 on
the basis of average experimental data obtained from
three to five tests for each orientation. Compressive
strength values for all the schists at b = 901 are found to
be more than at other orientations. It is further observed
that a number of compressive strength tests conducted
at this orientation, under similar conditions, exhibit
least scattering in their results as compared to all other
orientations. This may be due to the averaging effect of
the planes of anisotropy with b = 901 as compared to
when the foliations are inclined. The highest compressive strength is found to be at b = 901; it is generally
designated as the ‘‘representative compressive strength’’
value for the anisotropic rocks and denoted as sc90;
while the strength at other orientations is generally
designated simply as scj: The variation of scj with b
10
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
Predicted
Experimental
C
MPa
3
o
'
„„„,_
c
Ramamurthy etal. [14]
Hoek& Brown [24]
5
Fig. 6. Comparison between observed and predicted strength, through second method for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite
schists.
shows minimum strength between 301 and 451 orientations for all the schists.
7.4. Strength prediction in unconfined state
On the basis of the observation made for the variation
of uniaxial compressive strength with b varying from 01
to 901 exhibiting a "U-shaped" pattern, following
equation has been used to predict the variation of scj
over the entire range of b; for the schists
scj = A-Dcos 2ðx -
(7)
The minimum requirement to evaluate compressive
strength scj at any value of b is to conduct uniaxial
compressive strength at 01, 301 and 901 in Eq. (7), scj is
the uniaxial compressive strength other than 01, 301 and
901; x the orientation angle that has minimum value of
uniaxial compressive strength, usually x = 301; A and D
the constants that describe the variance over the range
of 0 1 p b p x a n d over the range x p b p 9 0 1 respectively.
Fig. 4 shows the variation of experimental and predicted
uniaxial compressive strength (scj) with b; for the
schists. The predicted value obtained through Eq. (7)
appears to agree well with the experimental values.
7.5. Strength behavior in triaxial condition
Triaxial tests were conducted at different orientations
ðbÞ on the four schistose rocks up to high confining
pressure. The plots of compressive strength between s1
and b at different s3 for the four schists are presented in
Fig. 5. The plots have been drawn taking the average
experimental results out of three to five tests into
consideration. The overall strength behavior of quartzitic, chlorite and quartz mica schists is similar as far as
the shape of the anisotropy curve in the confined state is
concerned. The shape of the anisotropy curves for these
three rocks is towards "U-shaped" over the entire range
of s3 adopted, whereas biotite schist, although exhibiting a "U-shaped" anisotropic trend in the lower range
of confining pressure, does not retain this shape but
shows flattening of anisotropy curves at higher confinement.
It is clear from the plots that the maximum strength
values are observed at b = 901 for quartzitic and chlorite
schists, throughout the range of confining pressure.
Though in general the minimum strength values are
obtained at b = 301: It has been observed that the
minimum of the curve shifted to b = 451 at confining
pressures greater than 15 MPa. A similar observation
was reported by McLamore and Gray [4] for slate
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
Experr mental
G 3 , MPa
5
•
15
35 „
50 *
100 b:
•
Predicted
Ramamurthy et al. [ 15]
500
0
D
11
400
300' _
•
200
100
o
C
(a)
1
15
1
1
1
1
1
30
45
60
75
9C
60
75
(b)
S.
\
if
200
\5
0
J
15
L
30
45
60
J
75
L
90
0
15
30
45
90
(d)
(C)
Fig. 7. Comparison between observed and predicted strength, through third method for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite
schists.
beyond 276 MPa of confining pressure and by Singh
et al. [36] for quartzitic and carbonaceous phyllites
beyond 70 MPa of confining pressure.
At b = 01 quartzitic schist shows a 30% strength
improvement as a result of confinement (at
s3 = 100 MPa). This improvement for chlorite schist is
15% and for quartz mica schist is 10%. Quartz mica
schist exhibits similar strength in unconfined state when
compared with biotite schist, but shows higher rates of
increase in compressive strength with increase of
confining pressure owing to it's higher quartz and lower
mica contents.
7.6. Strength prediction in confined state
Three methods have been adopted to predict the
failure strength of schistose rocks under triaxial conditions.
(i) The first method adopted was through Hoek and
Brown [24] empirical failure criterion based on Griffith's
crack theory. The criterion is as follows
(7i = cr3 + (maca3 + sa^f-5,
(8)
where s1 and s3 are the major and minor principal
stresses respectively; sc is the uniaxial compressive
strength; m and s are the dimensionless parameters
characterizing the intact rock.
To obtain the material constant m and s for schists,
the experimental uniaxial compressive strength and
expression suggested by them have been used. To
evaluate this criterion for anisotropic rocks it is
necessary to conduct triaxial tests at various orientations and confining pressures.
(ii) The second method adopted for prediction of
failure strength of the schists was through strength
criterion suggested by Ramamurthy et al. [15] based on
non-linear criterion for intact anisotropic rocks, Eq. (4).
This method involves plotting of values of ÐS1 — s3Þ=s3
and scj=s3 on log-log scale for all orientations of schists
on the basis of triaxial experimental data obtained for
different orientations and confining pressures. The
variation of aj and Bj thus obtained for schists is
confined to a closer range when it is compared with the
values of m and s as per Hoek and Brown criterion.
Fig. 6 shows the comparison between observed and
predicted strength through the second method for the
schist.
(iii) The third approach to predict the failure strength
of the schists is through the use of Eqs. (2)-(4) as per
Ramamurthy et al. [15]. Eqs. (2) and (3) have been
12
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
OjMl'a
0.4
0.0
0.4
0.8
1.2
0.4
0.0
0.4
0.8
1.0
I
04
1 ^W\
o!o
I
04
.
—(Ui
a
I
.
ti
'/.
Fig. 8. Stress-strain curves at different S3 and orientation b for quartzitic schist.
adopted for evaluation of aj and Bj parameters at any
orientations other than b = 901: According to this
method the minimum pre-evaluation experimental data
required is only two triaxial tests at b = 901 orientation
and uniaxial compressive strength values at only three
different orientations (01, 301, and 901) for the rocks
which exhibits U-shaped anisotropy. Fig. 7 exhibits the
comparison between the experimental data and predicted strength values obtained from the third method
for the schists. In general a reasonable agreement is
observed between the experimental and predicted values
from the present study on schists and also from other
studies [15]. Out of the three methods adopted to predict
the strength behavior of anisotropic schist, the third one
proves to be simpler and involves minimum experimental data.
7.7. Modulus in unconfined state
From the stress-strain (Figs. 8-11) diagrams the
tangent modulus, Et; and the Poisson's ratio, n; were
estimated at 50% of the peak strength for all the
orientations of the four rocks. The variation of modulus
in the unconfined state with b is presented in Fig. 12 for
the four schists. It is obvious from the figure that the
modulus for all the four rocks at b = 01 is higher than at
other orientations, as a result of interlocking of
vertically oriented foliation planes when they are
subjected to compression, whereas the values of
modulus at b = 901 are less than those at b = 01;
especially for the two weaker schists i.e. quartz mica and
biotite schists.
7.8. Modulus in confined state
The tangent modulus, Et and strain ratio, n determined at 50% of peak strength of the schistose rocks
were obtained from their respective stress-strain curves
shown in Figs. 8-11. A study of deformation modulus
values (Fig. 13) reveals that their variation is similar to
that of triaxial compressive strength as a function
of b at all confining pressures especially for the two
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
KMPa
13
OJMPa
OJ,MPa
ioo|
50
\-
50 i \
35
351
I
0.4
0.0
0.4
.
1
.
1
,
1
•
I
.
0.8
50
I
0.8
I
I
0.4
I
0.0
1
0.4
,
1
,
0.8
1
,
1.2
1
1.6
Fig. 9. Stress-strain curves at different s3 and orientation b for chlorite schist.
stronger rocks i.e. quartzitic and chlorite schists.
Fig. 13a and b show that the variation of modulus with
jS at different confining pressures for quartzitic and
chlorite schists follow a U-shaped trend similar to their
strength anisotropy curves. This behavior can be
attributed to the effect of foliation planes in the
regions 151 o bo601 where failure takes place along
foliation planes and as a result of low strength, the
corresponding modulus also gets affected. Quartz mica
and biotite schists do not show a trend towards Ushaped variation of modulus with b as a function of
confining pressures, similar to their strength anisotropy
curves. Quartz mica schist exhibits minimum modulus at
all orientations and confining pressures in comparison
14
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
3, MPa
-1
O^MPa
100
100
50
0.4
0.8
1.2
I
0.4
0.0
0.4
0.8
0.4
0.0
0.4
0.8
1.2
Oj.MPa
100
50
0.0
0,4
0.8
0.0
0.4
0.8
1-2
1.6
O^.MPa
O^.MPa
04
0,4
1.2
0.0
0.4
0-8
1.2
1.6
0.8
0.4
0.0
0.4
0.8
1.2
1.6
2.0
Fig. 10. Stress-strain curves at different s3 and orientation b for quartz mica schist.
to other three schists as observed in Fig. 13c due
to its thin foliation plane and higher degree of porosity.
The modulus at b = 01 being maximum and gradually
decreases as b varies form 01 to 901 for this rock.
Similar trend is observed for confining pressure
up to 35 MPa. Whereas the variation of modulus with
jS at higher confining pressures i.e. 50 and 100 MPa
changes towards U-shaped variation for quartz mica
schist.
Fig. 13d depicts variation of modulus as a function of
jS and s3 for biotite schist. Modulus for this rock at
jS = 01 for all confining pressures is more than other
orientations. Modulus values at 100 MPa of s3 exhibit a
marked decrease unlike other schistose rocks in the
region b > 451: Such a decrease of modulus after certain
confining pressure was also reported by McLamore and
Gray [4] at b = 901 for slate, b = 301 for Green river
shale-I and at b = 601 for Green river shale-II.
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
15
0°
0.8
0.4
0.0
0.4
0.8
1.2
0.0
0.4
0.8
1.2
Fig. 11. Stress-strain curves at different s3 and orientation b for biotite schist.
7.9. Effect of confinement on modulus
The degree of anisotropy in the modulus gets
suppressed at higher s3 for different orientation angles.
Variation of Et=s3 versus b at different s3 is shown in
Fig. 14a-d. This ratio is highest at b = 01 for the schists
at lower s3 and being highest for quartzitic schist i.e.
0.6 xlO 4 , 0.3 x l O 4 and 0 . 5 x l 0 4 M P a at s 3 = 5 M P a
and at b = 01; 301 and 901. Whereas these values
decreases to 0.03 x 10 4 MPa at all orientations angles
for 100 MPa of confining pressure. This reduction being
minimum for quartz mica schist as it shows least values
of modulus, varying from 0.2 x 104, 0.15 x 104 and
0.1 x 10 4 MPa at b = 01; 301 and 901 respectively at
5 MPa of s3 to a value of 0.03 x 104 MPa at almost all
the orientation under s3 = 100 MPa. The respective
variation of Et=s3 versus b for chlorite and biotite
schists fall in between the two extremes with similar
tendency of suppressed anisotropy at higher confinement. The ratio of Et01=Et301 in the unconfined state for
quartzitic, chlorite, quartz mica and biotite schists is 1.6,
3, 2.1 and 1.85, respectively and this ratio at confining
pressure of 100 MPa, for the schists changes to 1.6, 1.9,
1.5 and 1, respectively. These results indicate that biotite
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
16
strength property of the schistose rock is quite complex
and depends on various parameters such as mineralogy,
nature of foliation planes, grain size, porosity and bond
along the grain boundaries.
schist shows maximum anisotropy decrease with an
increase of confining stress, whereas such a phenomenon
is reversed in the case of quartzitic schist. This may
indicate that the rate of decrease in modulus anisotropy
as a function of the confining stress is different for
different schistose rocks. Basically how effective are the
foliation planes in determining the deformation and
2.5
7.10. Prediction of modulus in confined state
In the present study a simple approach has been used
in predicting the failure strength through minimum pre
evaluation experimental data. It has been observed that
the modulus values vary non-linearly with the orientation of weak plane as shown in Fig. 15a-d. The
following equations have been proposed to represent
the non-linear behavior of moduli of schistose rocks
Quartzitic schist
Chlorite schist
Quartz mica schist
Biotite schist
2.0
8.
2
1.5' -
/ CT3)'f-sSO
where Et90 is the modulus at 50% of failure strength at
jS = 901; sc90 the uniaxial compressive strength at b =
901; s3 the minor principal stress; ae90 the slope of the
plot between log (Et90=s3) and log ðsc90=s3Þ; Be90 =
Et90=s3 when SC90=S3 = 1:
The evaluation of these modulus parameters aej and
Bej at any orientation is achieved through the following
two equations
1.0
0.5
0.0
(9)
30
15
A5
60
75
90
ft°
j=ae90
ð10Þ
=
Fig. 12. Variation of modulus with b for the schists in unconfined
state.
=
ðae90=aejÞ 1 to 3
°3 ,MPa
o
•
•
A
A
5
i.O
15
35
3.0
50
100 >,
I
2.0
W"
1.0 -
0.0
(b)
90
(c)
Fig. 13. Variation of Et versus b at different s3 for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite schists.
ð11Þ
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
15
30
45
60
75
17
90
Fig. 14. Variation of Et=s3 versus b at different s3 for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite schists.
where scj is the compressive strength, at the desired
orientation other than b = 901; aej a n d Bej the modulus
parameters at the desired orientation.
The minimum pre-evaluation experimental data
required is only two triaxial tests at b = 901 orientation
and uniaxial compressive tests at b = 01, 301 and 901.
7.10.1. Basis for the proposition
On account of similar variation of modulus as a
function of b at all confining pressures to that of triaxial
compressive strength for the schists, an attempt was
made to study the variation of aej a n d Bej with
orientation angle, b (Fig. 16a-d). The actual variation
in the parameters has been explored through the plots
between log ðEt=s3Þ and log ðscj=s3Þ for different
orientations of the schists.
The value of aej at b = 01 is higher than at other
orientations. This variation is similar to that observed
for Etj at b = 01: The analysis shows that the power of
ðae90=aejÞ in Eq. (11), is varying from 1 to 3 in general; it
is 3 for the two stronger quartzitic and chlorite schists
and 1.5 for quartz mica schist and 1.1 for biotite schist.
Figs. 17a to 18b show the comparison of observed
tangent moduli at different confining pressures and b
with the predicted values through the proposed criterion
for the four schists. There exists a good agreement
between the observed and the predicted moduli for the
schists.
7..11. Stress-strain response in triaxial compression
Figs. 8-11 demonstrate the variations between axial
strain (ea), diametral strain (ed) and deviator stress
(S1 — s 3 ) at different confining pressures i.e. s3 = 5, 15,
35,50 and 100 MPa, at different orientation angles, b1 :
Stress-strain curves corresponding to uniaxial compression tests are also presented in these figures. Stressstrain curves for quartzitic and chlorite schists show
plastic-elastic behavior at all confining pressures as per
the Miller's classification of stress-strain curves,
whereas quartz mica and biotite schists exhibit plasticelastic-plastic nature especially for the specimens tested
perpendicular to foliations. The two latter schists show
comparatively higher degree of plastic deformation at
lower confining pressures due to the closure of micro
cracks associated with preferred orientation of minerals
along the foliation planes. The S-shaped stress-strain
curves for the latter rock change to hyperbolic shape at
higher confinement as the initial concave upward
portion change to linear portion and gradually show
18
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
1=0
B=15
5 = 30 • 3 = 45
3 = 60
3 = 75
' 3 = 90
40
60
3.
MPa
Fig. 15. Non-linear variation of modulus with s3 at different b for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite schists.
varying degree of inelastic yielding as failure is
approaching.
7.11.1. Strain ratio in triaxial state
The study of strain ratio at 50% of failure stress
(diametral strain/axial strain), ed=ea at various confining
pressures reveals that there exists a systematic variation
of the strain ratio with b: As it is shown in Fig. 19 for
quartzitic and chlorite schists, this ratio in the regions of
bo151 and b > 151shows a large variation from 0.1 to
0.4. On the other hand, values of strain ratio in the
region 1 5 1 o b > 6 0 1 vary between 0.1 and 0.25. Lower
value of strain ratios for the later orientations is due to
unconstrained freedom to failure along these orientations. Fig. 20 depicts the variation of strain ratio with b
at different s3 for quartz mica and biotite schists. The
scatter in the results of strain ratio at b = 01 is small
compared to those at b = 901: The trend shows that this
ratio increases with b and varies from 0.1 to 0.4 as b
changes form 01 to 901. Higher values of strain ratio at
jS = 901 for these rocks are as a result of large lateral
deformation in response to axial stress. Thinly laminated layers in these two schists register higher lateral
strains at b = 901 those at b = 01:
7.12. Prediction of stress-strain response in triaxial
compression
In the present study the original Eq. (12) proposed by
Kondner and Zelasko [37], has been effectively used in
representing the non-linear stress-strain response observed for schistose rocks at different b and s 3 :
£a/(ffi - ff3) = a + b£a,
(12)
where ea is the axial strain; S1 and s3 are the major and
minor principal stresses respectively, a the reciprocal of
initial tangent modulus Ei; and b the reciprocal of
asymptotic value of deviator stress i.e. reciprocal of
(s1 - s 3 ) ult .
Figs. 21 and 22 represent transformed hyperbolic
stress-strain plots for the schists at 100 MPa of
confining pressure. It is clear in all the cases, the data
points lie fairly along straight lines, indicating that
hyperbolic relation can be extended to schistose rocks.
Another important parameter, which has been evaluated
for stress-strain behavior is, Rf; which is the ratio of
deviator stress at failure to the ultimate deviator stress
i.e. ÐS 1 G-S){/(G\ 1 <x3)ult. In the present study it is
found that the value of Rf varies from 0.25 to 0.5 as
confining pressure increase form 15 to 100 MPa for both
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
0.05
(c)
(d)
Fig. 16. Systematic variation of aej and Bej versus b for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite schists.
1.0
90
(a)
Fig. 17. Comparison between observed and predicted modulus for (a) quartzitic, (b) chlorite schists.
19
20
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
o,, MPa
o 5
• 15
Predicted
x
uS
15
30
45
60
75
90
0
(a)
15
30
45
60
75
90
(b)
Fig. 18. Comparison between observed and predicted modulus for (a) quartz mica and (b) biotite schists.
a,, MPa
° 5
0.4
a
•
A
A
0.3
03,
o
o
•
MPa
5
15
35
A
50
A 100
0.5
15
35
50
100
0.4
O
~ 0.2
o
i_
8
'5 0.2
c
A
A..
55
0.1
0.0
0
(a)
A
A
0.1
I
15
30
45
60
75
15
90
ft"
H
(b)
30
45
ft
•
A
60
75
90
|3°
Fig. 19. Variation of strain ratio with b at different S3 for (a) quartzitic and (b) chlorite schists.
quartzitic and chlorite schists. This ratio for quartz mica
and biotite schists is observed to vary from 0.22 to 0.4
with S3 increasing from 35 to 100 MPa. It has also been
found that the ratio of the initial tangent modulus ðEiÞ
to that of modulus at 50% of strength ðEi=EtÞ decrease
form 0.9 to 0.7 when confining pressure increases form
15 to 100 MPa for quratazitic and chlorite schists. The
variation of this ratio with s3 for quartz mica and biotite
schists is from 0.9 to 0.8 as confining pressure increases
from 35 to 100 MPa. An average value of 0.8 can be
suggested for the ratio of ðEi=EtÞ:
Using the predicted strength from the strength
criterion (Eqs. (1)-(3)) and the predicted modulus value
(Eqs. (9)-(1 1)), an attempt has been made to predict
stress-strain behavior at high confining pressures.
Fig. 23 show the comparison between the predicted
and observed stress-strain curves for b = 01 orientation
of the four schists.
8. Conclusions
In the past, many researchers have studied the
anisotropic behavior of the schistose rocks, but only in
a very few studies, systematic tests were performed to
observe and analyze the strength and deformational
responses of schistose rocks in uniaxial and triaxial
states over the entire range of orientation angle ðbÞ
varying from 01 to 901 with respect to vertical. In the
present study strength and deformational behavior of
four schists, transversely anisotropic, are affected by the
schistose planes of weakness at different rates depending
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
0.5-
o
f
u
0.4 a
0.4
A
1 0.3 a
'5
m
A
0.2 -
0
n
•
A
Q
0
o
B
T
i
i
I
15
30
45
5
15
35
50
100
.£ 0.3
'5
a
c
w
o
T
•
a
A
15
35
50
100
A
o
0^ MPa
0.5
5
M
JQ
c
E
21
0.2
E
•
*
A
60
75
0
I
T
O.l1
.D
0.0
(a)
60
75
P
J
o.o.
1
9C
i
I
15
30
45
(b)
90
p
Fig. 20. Variation of strain ratio with b at different s3 for (a) quartz mica and (b) biotite schists.
S
s.
i
—
2xl o
5
?
.—x
0.0
10
•
•
-
a-—*
I
0.002
•
—,
—
*
•
—
1
0.004
-3
—»•
—I—*
.
,30"
"
«
15°
=^
7 5 °
—*-90c
*~
.
2x10
*
1
0.006
1
0.010
1
0.008
1
0.012
0.0
0.002
0.004
0.006
0.008
0.010
0.012
0 010
0.012
(b)
(a)
Fig. 21. Variation of ea=s1 2s3 with ea for various b for (a) quartzitic and (b) chlorite schists.
6x10 5 |
0.0
(a)
6x10"-
0.002
0.004
0.006
0.008
0 010
0.012
0.01-4
£<x
0.0
0.002
0.004
0.006
0.008
0.014
(b)
Fig. 22. Variation of ea=s1 2s3 with ea for various b for (a) quartz mica and (b) biotite schists.
upon the type and mineralogical composition of the
foliation planes. Quartzitic and chlorite schists, physically and mechanically being stronger demonstrate
pronounced "U-shaped" anisotropy compared to
quartz mica and biotite schists. Based on the non-linear
variation of strength and modulus as a function of
confining pressure for the four schists simple expressions
have been proposed to predict their variation with
minimum pre-evaluated experimental data over the
entire arrange of b:
The values of parameters for strength prediction i.e. aj
and Bj and for modulus i.e. aej and Bej can be estimated
for various orientations, by performing only two triaxial
tests at orientation angle b = 901 and uniaxial compressive strength test at b= 01, 301 and 901. Using the
predicted strength and modulus value it has been shown
that the predicted stress-strain curves match reasonably
well with the experimental stress-strain responses even
at higher confining pressures.
The relationships presented for strength, modulus and
prediction of stress-strain behaviour in transversely
anisotropic rocks will enable analysis of the response of
such rock masses. These can then be utilized to estimate
more realistically the applied stress states due to
construction of structures with a minimum of preevaluated experimental data.
M.H.B. Nasseri et al. / International Journal of Rock Mechanics & Mining Sciences 40 (2003) 3-23
400r-
1.2
Fig. 23. Comparison between observed and predicted stress-strain curves at b = 01 for (a) quartzitic, (b) chlorite, (c) quartz mica and (d) biotite
schists.
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