1. INTRODUCTION

Rock strength and its elastic modulus are important geotechnical engineering parameters. However, it is

difficult to obtain completely mechanical behaviors if

rocks belong to highly heterogeneous media, i.e., mélange and fault breccia, which are considered as a

type of bimrock, and defined as “a mixture of rocks,

composed of geotechnically significant blocks within a bonded matrix of finer texture” [1]. Because bimrock

has chaotic block dispersion, it is difficult to sample the

representative specimen. In addition, its engineering characteristics and mechanical behaviors are more

complicated than other geomaterials. Volume fraction

(Vf) plays an important role in bimrock mechanical behaviors [2-6].

Bimrock spatial variability could cause uncertainties in

physical and mechanical measurements. Medley [7] manufactured physical synthetic bimrocks to investigate

the uncertainty of Vf estimation using the scanline

method. Tien et al. [8-10] proposed analytical and numerical solutions for quantifying the uncertainty of Vf

measurements using the scanline method and a sampling

window. Kahraman and Alber [3, 11-12] tested various sizes of fault breccia with a uniaxial compressive test;

the results show that the variability of fault breccia

strength decreases with increasing size.

Numerous experimental studies only qualitatively

analyze uncertainties of bimrock mechanical behaviors , as quantitative analyses are insufficient. To establish

quantitative theory, examining a large number of

specimens is required. However, controlling every inclusion, geometry, and nature of each specimen is not

possible; thus, many studies have used synthetic physical

or numerical bim-models to investigate their mechanical behaviors [4,13-16]. Recently, PFC2D have been widely

used to simulate heterogeneous media (faulted rocks,

layered rocks, bimrock, etc.) mechanical behaviors [16-21], and simulation results compared very well with

experimental results.

Thus, this paper used PFC2D to simulate bimrock mechanical behaviors with uniaxial compressive tests, to

quantify (1) the variances of Young’s modulus (E) and

Poisson’s ratio (υ) measurements using a strain gauge and (2) the variances of uniaxial compressive strength

(UCS), Young’s modulus, and Poisson’s ratio affected

by volume fraction. In addition, (3) the results will be compared with micro-mechanic frames, and (4)

document crack propagation during the simulations.

2. METHODOLOGY

2.1. Bim-model generation In PFC2D, rock generation followed the methods of

Potyondy and Cundall’s [21] bonded particle model

generation (BPM). The generation procedure in this paper includes (a) compact initial assembly, (b) blocky

material determination, (c) installation of specified

isotropic stress, (d) reduction of the number of ‘‘floating’’ particles, (e) installation of parallel bonds,

and (f) removal from the material vessel. “Blocky

material determination” is followed by clump logic [16], which is used to create blocky material in BPM.

Additionally, using this method can more naturally allow

for crack propagation. The main procedure of “blocky material determination” is shown in Fig. 1, and block

particle distribution was determined by Eq. (1).

ARMA 15-614

Variability of mechanical properties of bimrock

Tien, Y.M., Lu, Y.C., and Cheng, H.H.

Department of Civil Engineering, National Central University, Taoyuan City, Taiwan

Copyright 2015 ARMA, American Rock Mechanics Association

This paper was prepared for presentation at the 49th US Rock Mechanics / Geomechanics Symposium held in San Francisco, CA, USA, 28 June- 1 July 2015.

This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or

members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: This paper used PFC2D to simulate bimrock mechanical behaviors under uniaxial compressive tests and presents the

means and variances of uniaxial compressive strength (UCS), Young’s modulus (E), and Poisson’s ratios (υ) for various volume

fractions and strain gauge lengths. The results show that UCS, E, and υ increase as Vf increases, and simulated E and υ were

consistent with the differential scheme and Hashin-Shtrikman bounds. In quantifying the uncertainty of Young’s modulus and

Poisson’s ratio measurements using a strain gauge, the measurement uncertainties initially increase and then decrease with

increasing Vf; when Vf =0.3~0.4 (0 or 1), Young’s modulus and Poisson’s ratio reach a maximum (minimum) value. In addition,

crack propagation was also observed and discussed in this paper.

Established bim-models for various Vf are shown in Fig.

2. In this paper, specimen size is 5 cm × 10 cm.

2

max

11(%)

block

block

d

dP (1)

Fig. 1. Blocky material determination, including (a) compact initial matrix assembly, (b) stamp blocks location, and (c)

block mechanical properties.

Fig. 2. Bimrock specimens in PFC2D for various Vf.

The bim-model’s micro-mechanical parameters, including particle and parallel bond, are listed in Table 1

and Table 2, respectively. These micro-mechanical

parameters were calibrated with macro experimental data [22] obtained from the uniaxial compressive test.

This paper assumed a perfect bonding between the block

and matrix interface. Hence, the mechanical behaviors of the block-matrix bond ranged in block-block and matrix-

matrix bonds (obtained from their means).

Table 1. Particle parameters in PFC2D.

block Matrix

density, ρ（kg/m3） 1860 1480

Young’s modulus, Ec（GPa） 10.8 2.68

stiffness ratio, kn/ks 1.70 1.20

dmax/dmin 1.40 1.40

friction coefficient, μ（kg/m3） 0.550 0.550

Table 2. Parallel bond parameters in PFC2D.

block-

block

matrix-

matrix

block-

matrix

Young’s modulus,

Ecp（GPa） 10.8 2.68 6.74

stiffness ratio, knp/ksp 1.70 1.20 1.45

bonded normal

strength, σp（MPa） 55.5 25.7 40.6

bonded shear

strength, τp（MPa） 55.5 25.7 40.6

magnified factor, λ 1.00 1.00 1.00

2.2. Elastic modulus measurement To avoid the “boundary effect”, axial and lateral strain

gauges were located in the specimen center, shown in

Fig. 3. In this paper, three pairs of measuring particles presented three different distance sets for each strain

gauge, and only the mean value was recorded in the stain

calculation.

Fig. 3. Strain gauge measurement and its location.

A linear stress-stain relation had been observed in each

simulation before failure. Hence, using the tangent

modulus, average modulus, and secant modulus had the same result. The secant modulus was regarded as

Young’s modulus.

2.3. Simulation variables (i) Strain gauge length

Due to bimrock spatial variability, the location dependency of the strain gauge measurement cannot be

avoided; if the strain gauge is not long enough, the

measurement will be non-representative. To investigate how long strain gauge should be, this paper regarded

gauge length (GL) as a simulation variable, where GL =

0.1 D, 0.2 D, 0.4 D, 0.6 D, 0.8 D, and 1.0 D (where D is the specimen diameter).

(ii) Volume fraction

Bimrock mechanical behaviors are highly dependent on their volume fraction. As a result, the volume fraction

was also a simulation variable, where Vf = 0%, 5%, 10%,

15%, 20%, 30%, 35%, 40%, 45%, 50%, 65%, and 100%.

2.4. Micromechanics models Two micromechanics models of Hashin-Shtrikman bounds (HS bounds) and the differential scheme (DS)

were compared with the simulation results. These two

models were used for estimating the macroscopic elastic moduli of the heterogeneous media, in which HS bounds

predicted a range under a specific Vf (Eq. (2)~ Eq. (5)),

and DS predicted one solution for a specific Vf (Eq. (6)~ Eq. (9)).

The Hashin-Shtrikman bounds equation is as follows

[23]:

11112

21

)(

43/3/1 GKCKK

CKK

(2)

22221

12

)(

43/3/1 GKCKK

CKK

(3)

11111112

21

)(

435/2/6/1 GKGGKCGG

CGG

(4)

22222221

12

)(

435/2/6/1 GKGGKCGG

CGG

(5)

where 21 KK , 21 GG . Here, )(

K is the minimum

bulk modulus of composite prediction; )(

K is the

maximum bulk modulus of composite prediction; )(

G is

the minimum shear modulus of composite prediction; )(

G is the maximum shear modulus of composite

prediction; 1C is the matrix volume / total volume; 2C

is the block volume / total volume, which is the same as

Vf; 1K is the bulk modulus of the matrix; 2K is the bulk

modulus of the block; 1G is the shear modulus of the

matrix; and 2G is the shear modulus of the block.

The Differential scheme is as follows [24]:

*

2

*

2

2

2 1 KK

KK

C

KK

dC

Kd (6)

*

2

*

2

2

2 1 GG

GG

C

GG

dC

Gd (7)

where GK3

4* (8)

GK

GKGG

26

89*

(9)

where K is the bulk modulus of the composite; and G

is the shear modulus of the composite.

3. SIMULATION RESULTS

3.1. Gauge length influence The simulation results are shown in Fig. 4 and Fig. 5.

Fig. 4 shows the mean and coefficient of variance (COV) of E by using various strain gauge lengths for E

measurement, and each data point obtained from

different 30 specimen simulations (using the same Vf). From Fig. 4, it can be observed that the mean of E is

constant with variable Vf, but the COV of E decreases

with increasing gauge length (presented in normalized length, GL/D).

Fig. 4. (a) Mean of E vs. GL/De and (b) COV of E vs. GL/ De.

Fig 5 shows the mean and COV of υ using various strain

gauge lengths for υ measurement, and each data point also obtained from 30 different specimen simulations.

The results are the same as the E measurement results; the mean of υ is constant with variable Vf, but the COV

of υ decreases with increasing gauge length.

Fig. 5. (a) Mean of υ vs. GL/ De and (b) COV of υ vs. GL/ De.

Fig. 6. Using the Central Limit Theorem to quantify the relation of “COV of E vs. GL/De” for various Vf.

Fig. 7. COVP of E (denoted as COVEP) vs. Vf. Fig. 8. COVP of υ (denoted as COVυP) vs. Vf.

The relationships between gauge length and COV of E or

COV of υ, can be quantified by the Central Limit Theorem (CLT),

N

COVCOV P (10)

where COV is the measurement (sampling) COV; COVP

is the population COV, which is based on the spatial probability density function of Vf; N is the sampling size,

herein, GL/ De was used. De is the equivalent diameter

[10].

Eq. (2) and regression analysis were used to obtain

COVP for each Vf, and the results are shown in Fig. 6

(used COV of E for example, and COV of υ show similar results). Furthermore, regression analysis was employed

to quantify the relation of COVP and Vf for E and υ, as

shown in Fig. 7 and Fig. 8, respectively. From Fig. 7 and 8, both of the maximum COVP occur at approximately Vf

=0.2~0.4, suggesting that the greatest spatial variability

and heterogeneity occur at Vf=0.2~0.4; the minimum COVPs occur at Vf =0 and 1. Because in this situation,

they belong to a single material, the heterogeneity is

lowest.

3.2. Volume fraction influence UCS, E, and υ were observed in this section. Notice that, only the sampling size GL=1.0D was used here, and

each data were obtained from the mean of 30 specimen

tests. The simulation results are shown in Fig. 9 ~ Fig. 12.

Fig. 9 shows the relation of UCS and Vf, which have a

positive correlation because the blocky material has greater strength. In addition, the simulations showed that

the bimrock UCS was closed to the matrix when Vf <

50%, and dramatically increased when Vf >50%. Fig. 10 shows COV of UCS is not correlated with Vf.

Fig. 11 shows the simulated E, Hashin-Shtrikman

bounds (HS bounds), and differential scheme (DS) predictions. From Fig. 11, the simulations results are

consistent with HS bounds and DS prediction, and most

of the simulated data are located in HS bounds. In addition, simulated υ has similar results (Fig. 12),

consistent with HS bounds and DS. These results

illustrate that BPM has a very continuous mechanical demonstration. COV of E and COV of υ were discussed

in the previous section; the maximum values occur at

approximately Vf =0.2~0.4, and the minimum values occur at Vf =0 and 1.

Fig. 9. Mean of UCS vs. Vf.

Fig. 10. COV of UCS vs. Vf.

Fig. 11. Simulation results (E) compared with the differential

scheme (DS) and Hashin-Shtrikman bounds (HS bounds).

Fig. 12. Simulation results (υ) compared with the differential

scheme (DS) and Hashin-Shtrikman bounds (HS bounds).

3.3. Crack propagation Crack propagation was also observed in this paper, and typical cases for various Vf are shown in Fig. 12~ Fig. 14.

In these figures, it can be observed that initial micro

cracks appeared at a stress state of 80%~90% UCS before the peak, randomly dispersed in the bim-model.

After peak stress, these micro cracks rapidly propagated

to macro fractures.

Three different bonded materials, matrix-matrix bonds,

block-block bonds, and block-matrix bonds, were also

monitored during the tests, and can explain which material will influence bimrock strength. For instance,

lower Vf bimrock strength was mainly controlled by

matrix-matrix bonds, which had the most broken bounds (Fig. 13). For higher Vf (>30%), bimrock strength was

also still controlled by matrix-matrix bonds, and hence,

their strength still approached matrix material (compared with block material).

a b c d e

Tensile crack Shear crack

Fig. 13. Crack propagation in a low Vf specimen (Vf = 10%) in the uniaxial compressive test.

A b C d e

Tensile crack Shear crack

Fig. 14. Crack propagation in a medium Vf specimen (Vf = 30%) in the uniaxial compressive test.

A b C d e

Tensile crack Shear crack

Fig. 15. Crack propagation in a high Vf specimen (Vf = 65%) in

the uniaxial compressive test.

Moreover, this paper uses statistics of three types of cracks (matrix-matrix, matrix-block, and block-block

cracks) at the peak and ultimate states, as shown in Fig 16. This figure can also explain why the high Vf bimrock

strength is still similar to matrix material; weaker bonds

will break first.

Fig. 16. Statistics of various types of cracks at the peak and

ultimate state (no pure matrix and block material are

considered here).

4. CONCLUSIONS

This paper used PFC2D to simulate bimrock mechanical

behaviors under uniaxial compressive tests and reached the following conclusions:

(i) UCS, E, and υ of bimrock increased with

increasing Vf because blocks have greater strength

and higher stiffness. In addition, the simulation

results (E and υ) are consistent with the

differential scheme and Hashin-Shtrikman bounds.

(ii) The relationship between gauge length and the uncertainties of strain gauge measurement can be

quantified using the Central Limit Theorem.

Furthermore, “COV of E vs. Vf” and “COV of υ vs.

Vf” were quantified using regression analysis.

(iii) For COV of E and COV of υ using scanline measurements, the maximum values occur at

approximately Vf =0.2~0.4, and the minimum

values occur at Vf =0 and 1. COV of UCS is not

correlated with Vf.

(iv) PFC2D successfully demonstrates crack

propagation, which can help us to understand the

failure mechanism of bimrock under compressive stress.

5. ACKNOWLEDGEMENTS

This study was supported by the Ministry of Science and Technology of ROC (Taiwan) through Grant No. MOST

102-2221-E-008-080-. This support is greatly

appreciated.

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