1 INTRODUCTION

Anisotropy is one of the key points of jointed rock masses, and has been studied from the early development stages of rock mechanics (Salamon 1968, Jaeger 1972, Hoek 1983). Anisotropic characteristics generally originate from mineral foliation in metamorphic rocks, stratification in sedimentary rocks, and discontinuities in rock masses (Cho et al. 2012). Many rock engineering projects can be significantly affected by rock mass anisotropy, such as tunneling, drilling bore-holes for petroleum recovery and in situ rock stress measurement, etc. Previously, the anisotropy of rock masses has been studied based on experiment, analytical approach and numerical ap-proach. When an equivalent continuum approach base on Finite Element Method (FEM) or Fi-nite Different Method (FDM) is used for regularly-fractured rock masses, anisotropy must be considered due to major deformations expected along the discontinuities. Discrete Element Method (DEM) techniques have also been applied to investigate the stability of rock masses where the rock mass is simulated with blocks and discontinues. The blocks in DEM need to be assigned a constitutive model just as in the numerical approach of FEM and FDM. Analytical approaches mainly focuses on periodic joint sets rock masses, and the non-elastic response after peak strength has been considered (Lee et al. 2012).

Schistose rock and stratified rock can be described as transversely isotropic material. In gen-eral, the most anisotropic elastic behavior of material is described by 21 independent elastic constants. The transversely isotropic model is characterized by 5 independent elastic constants. Transversely model is the most popular model for schistose rock like schist, gneiss, shale, etc.

(Gonzaga et al. 2008, Nunes 2002, Tien et al. 2006, Tien & Tsao 2007). Although significant achievements have been made over the past 40 years, there is currently

no standard method to determine the elastic constants of schistose rock (ISRM 2007). Experi-mental results of schistose rock indicate that strength characteristics vary between different rock types. In this paper, we first review the strength and deformation characteristics of schistose ma-terial and then discuss our experiments on Danba Schist. The main objective of this study is to

Numerical simulation of deformation and strength anisotropy of Danba Schist

W.-J. Chu HydroChina Itasca R&D Center, Hangzhou China

C.-S. Zhang & J. Hou HydroChina Huadong Engineering Corporation, Hangzhou, China

ABSTRACT: The first objective of the work is to investigate the strength and deformation anisotropy of

Danba Schist, the second objective is to develop a suitable numerical approach to simulate anisotropy of

schistose rock and produce the similar response of laboratory test result of Danba schist. In order to get

better understanding of anisotropy of schistose rock, the theory background of strength and deformation

of transversely isotropic material is reviewed, and previous UCS and triaxial compressive test results of

schistose rock are also reviewed. The difference of UCS and Young’s modulus between Danba schist and

previous rock types is described and the reason for the difference is analyzed. A numerical approach

based on PFC and smooth joint model can reproduce a similar laboratory response of UCS and Young’s

modulus at each different schistosity plane angle and specimen failure mode consistent with laboratory

specimen failure mode. The numerical approach proposed is suitable for investigation the anisotropy of

schistose rock.

investigate the strength and deformation anisotropy of Danba schist by simulating a series standard unconfined compression tests with PFC2D (Itasca 2008). The results of the numerical experiments are compared to laboratory test results of Danba Schist.

2 THEORETICAL BACKGROUND AND EXPERIMENT REVIEW OF SCHISTOSE ROCK

2.1 Strength of schistose rock

According the earlier notes (Salamon 1968, Jaeger 1972, Hoek 1983), the axial strength of a triaxial specimen containing an inclined weak plane is given by the following equation (Jaeger, 1972):

Where is the minimum principal stress or confining pressure, is the cohesion of the weak plane, is the friction of the weak plane, is the angle between the axial loading direc-tion and orientation of the weak plane.

Equation 1 can only be solved when less than intact strength, while value of close to 0º or 90º, the slip on discontinuity will not occur, and failure strength will controlled by the intact rock strength. A typical plot of axial strength versus angle is given in Figure 1.

Figure 1. Strength of specimen predicted by means of Equation (1) (Jaeger 1972).

2.2 Elastic constants of Schistose Rock

For a transversely isotropic material, the generalized Hooke’s law in local coordinate system is given by following equation:

Where is strain, is elastic compliance and is stress. Equation 2 can be given in matrix form as:

'

'

'

' '

'

'

'

'

10 0 0

1 10 0 0

10 0 0

10 0 0 0 0

2(1 )0 0 0 0 0

10 0 0 0 0

x x

y y

z z

yz yz

zx zx

xy xy

v v

E E E

v

E E E

v v

E E E

G

v

E

G

(3)

There are 5 independent elastic constants in the compliance matrix, and are the elastic moduli in the plane of transverse isotropy and in a direction normal to the plane, and are the Poisson’s ratios that characterize the lateral strain response in the plane of transverse isotro-py and normal to it, respectively (Cho et al. 2012), is the shear modulus in the plane normal to the plane of transverse isotropy. The general Hooke’s law in global coordinates is expressed as follows:

Equation (4) can be expressed in a matrix form as:

' ' ' ' ' '' 11 12 13 14 15 16

' ' ' ' ' '' 21 22 23 24 25 26

' ' ' ' ' '' 31 32 33 34 35 36

' ' ' ' ' '' ' 41 42 43 44 45 46

' ' ' ' ' '' ' 51 52 53 54 55 56

' ' ' ' ' '' ' 61 62 63 64 65 66

x

y

z

y z

z x

x y

S S S S S S

S S S S S S

S S S S S S

S S S S S S

S S S S S S

S S S S S S

'

'

'

' '

' '

' '

x

y

z

y z

z x

x y

(5)

The compliance matrix in Equation (5) can be obtained through the tensorial transformation which involves angle as shown in left of Figure 2. To determine the five independent elastic constants of transversely isotropic rock, at least three specimens are needed to carried out uniax-ial compression tests with and a inclined angle between 0º to 90º (Amadei 1996), as shown in Figure 2.

Figure 2. Global and local coordinates system and three specimens for determine 5 independent elastic constants of transversely isotropic rock.

Independent equations are extracted from the uniaxial compression tests on the three speci-

mens respectively. From specimen (a) two equations were obtained as shown in Equation (6) while three equations were obtained from specimens (b) and (c), as shown in Equation 7 and Equation 8.

'

'

'

1

x z

y y

y

y

v

E

E

(6)

'

'

1y

y

x

x

z

y

E

v

E

v

E

(7)

' 2 '2 2

' ' ' '

' 4 4 2 '

' ' ' '

' '2 2

' '

sin 2 1 1 1( ) (cos sin )

4

sin cos sin 2 1( )

4

sin cos

x

y

y

y

z

y

v

E E G E

v

E E E G

v v

E E

(8)

2.3 Experiment results review

Experimental studies of strength and deformation anisotropy can be found in literature (Hoek 1983, Cho et al 2012, Autio et al. 2000). The uniaxial compressive test results of gneiss speci-mens indicate that the strength and deformation properties are significantly anisotropic and de-pend on the angle of the schistosity plane (Autio 2000). Obtained Young’s modulus and UCS tests show that strength is at a maximum when the specimen is tested parallel to the schistosity plane ( ). When is 38-43º, measured uniaxial compressive strength is approximately equal to the value obtained when specimens are loaded perpendicular to the schistosity plane ( ). Measured values of Young’s modulus exhibit similar behavior as uniaxial strength with respect to the schistosity plane angle. Value of Young’s modulus are at a maximum when specimens are oriented parallel to the schistosity plane, and drop 20% when the orientation is close to perpendicular.

Cho et al.’s (2012) uniaxial compressive test results of gneiss, shale and schist show that the

obtained uniaxial strength at is slightly lower than that at , and the minimum uniaxial strength occurred at for all three rocks. According to Cho, an initial crack generally started from the weak plane near the upper or lower boundary in the case , while the specimen of failed after the vertical cracks were fully connected which lead to uniaxial strength at slightly lower than that at .

Triaxial test results of slate with different angle between loading direction and normal vector of schistosity plane is obtained by McLamore & Gray (1967). The axial strength increase with the confining pressure and the axial strength at is higher than that at , the mini-mum uniaxial strength occurred at .

3 EXPERIMENTAL RESULTS OF DANBA SCHIST

The Danba Power Station Project includes two headrace tunnels, 16 km in length and 13~14 m in diameter. The key stability issue of the two headrace tunnels is a schist region that is 2 km in length and has 1200 m in overburden above it. The mineral ingredients of Danba Schist is 40~50% quartz, 30~40% mica and 2~10% plagioclase. The high ratio of mica leads to the rela-tively low UCS strength which makes the tunnel stability issue more critical than for the other rock type regions. Due to a high density schistosity plane in Danba Schist, the rock exhibits a significant strength and deformation anisotropy. The UCS and Young’s modulus of Danba Schist at different angles is given in Table 1.

Based on the discussion in Section 2.2 three load directions have been tested: parallel to the schistosity plane, perpendicular to the schistosity plane and . At each orientation, three specimens have been tested. The test results of UCS and Young’s modulus of Danba Schist are given in Figure 3. As we

can see from the figure, the maximum uniaxial strength occur in case of and the mini-mum uniaxial strength occur in case of approximately equal to 60~70. Obtained uniaxial strength at is slightly higher than the minimum uniaxial strength. The test results of Young’s modulus exhibit similar behavior with respect to the results of UCS. The anisotropic behavior of Danba Schist is similar the test results of gneiss Autio (2000) while significantly dif-ferent to Cho et al. (2012)’s test result of gneiss, shale and schist. The Danba Schist is general weaker than Autio’s (2000) test specimens since, when the schistosity plane of Danba Schist is approximately perpendicular, cracks can more easily propagate along the weakness planes.

Figure 4 provides the failure photographs of the Danba Schist specimens. The failure mode of schistosity plane horizontal ( ) and vertical are significantly different. The spec-imen cracks along the schistosity planes and volume dilation is observed in case of .

Table 1. UCS and Young’s modulus of Danba Schist. __________________________________________________________________________________________________________

Diameter Height E UCS Number (deg.) (mm) (mm) (GPa) (MPa) __________________________________________________________________________________________________________

#1 0 50.00 101.20 18.60 52.50 #2 0 50.00 103.10 9.79 30.40 #3 0 50.00 102.25 9.99 28.60 #4 0 50.00 110.00 8.60 25.20 #5 60 50.00 98.40 2.74 17.54 #6 60 50.00 98.49 3.26 12.98 #7 60 50.00 97.01 3.86 21.35 #8 90 49.95 100.45 5.76 24.70 #9 90 50.00 103.00 4.74 18.20 #10 90 50.00 101.90 4.58 17.70 __________________________________________________________________________________________________________

Figure 3. The USC and Young’s modulus test result of Danba schist.

Figure 4. Failure specimens after uniaxial compressive test.

4 ANISOTROPY SIMULATION WITH PFC

When FEM or FDM are used to simulate axial compressive test of schistose rock, two difficul-ties arise: (1) high density schistosity planes can’t be simulated properly and (2) sliding along schistosity planes and crack propagation can’t be simulated. PFC3D (Itasca, 2008) was used to simulate axial compressive tests of schistose rock in earlier literature (Wanne 2002). In these tests the schistosity planes were simulated with weak bonds between the particles. This is sig-nificantly different from the real distribution of weak plane in schistose rock. When the smooth joint logic was developed and introduced into PFC (Mas Ivars et al. 2008), it enhanced the abil-ity to simulate schistosity planes. The smooth joint model simulates the behavior of an interface regardless of the local particle contact orientations along the interface. Figure 5 shows the high

density of schistosity planes in Danba Schist specimens simulated with smooth joints in PFC. A series of uniaxial compressive tests with angles ranging from 0 to 90º with an interval 10º have been completed. During loading of these samples, the smooth joints can slip and the rock host between the joint planes may crack.

The properties of the particles and smooth joints used for the simulation of Danba Schist are given in Table 2. A bonded particle model is selected to describe the intact response. In order to match the test UCS and Young’s modulus of Danba Schist, calibration of the sample response has been carried out. The properties in Table 2 represent the best fit of laboratory test results based on considerable quantities of numerical calibration analyses.

Figure 5. The schistosity plane was simulated by smooth joint.

Table 2. Particles and smooth joint Properties for simulation axial compressive test with PFC2D

. __________________________________________________________________________________________________________

Particle Properties Bond Properties Smooth Joint Properties __________________________________________________________________________________________________________

Rmin [mm] 1.22 1 1 Dmax/Dmin 1.66 Ec [GPa] 11 k

n [GPa/m] 4e3

[kg/m3] 2730 Kn/ks 2.5 K

s [GPa/m] 0.5e3

Ec [GPa] 12 c [MPa] 28±4 M 3 kn/ks 2.5 c [MPa] 28±4

c [MPa] 12

0.5 c [MPa] 35

b

[degree] 0 0.85 0

__________________________________________________________________________________________________________

The numerical simulated UCS and Young’s modulus values along with the laboratory test re-

sults are given in Figure 6. The numerical approach generates a similar response in uniaxial compression, i.e. the maximum uniaxial strength and Young’s modulus occur in case of and the minimum uniaxial strength and Young’s modulus occur in case of approximately equal to 60~70. In addition, the uniaxial compressive strength and Young’s modulus at is slightly higher than the minimum value respectively.

The axial stress versus axial strain curves for each test is given in Figure 7. It can be seen that the different schistosity plane angles of the specimens produce the different axial stress – axial strain responses. The UCS and Young’s modulus are obtained from the axial stress – axial strain curves.

The failed specimens from the numerical simulations are provided in Figure 8. When the an-gle of schistosity plane is low (<10

o), cracking in the host rock are the dominant mode of fail-

ure. While slip along schistosity planes become dominant with the increase of angle in the schistosity plane. As we can see from Figure 8, volumetric dilation can be observed in the fail-ure specimen at , which is consistent with the laboratory failure specimen present in Fig-ure 4.

Figure 6. Simulated UCS and Young’s Modulus values compared to the laboratory test results.

Figure 7. The axial stress versus axial strain obtained from simulation with PFC.

Figure 8. The simulated failure specimens with different .

5 CONCLUSION

This paper reports on the numerical experiment investigation of the strength and deformation anisotropy of Danba Schist. The smooth joint model has been used to simulate schistosity planes in Danba Schist specimens.

A significant anisotropy of Danba Schist is observed. The maximum uniaxial compressive strength occurs in case of . The minimum uniaxial compressive strength occurs in case of approximately equal to 60~70 degrees. The uniaxial compressive strength at is slight-ly higher than the minimum axial strength. The Young’s modulus exhibits similar behavior with respect to the results of the UCS testing. The numerical approach produced the similar response of laboratory axial stress – axial strain tests. The failure modes of the numerical simulations were consistent with laboratory specimen failure modes. The numerical approach proposed in this paper is suitable for investigation the anisotropy of schistose rock.

ACKNOWLEDGEMENTS

The authors thank Loren Lorig of Itasca Consulting Group, Minneapolis for his suggestions and discussion of numerical simulation of the axial compressive tests in PFC.

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