Mechanical characterisation of jointed rock-like material with non-persistent rough joints subjected to uniaxial compression

Engineering Geology 260 (2019) 105224

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Mechanical characterisation of jointed rock-like material with nonpersistent rough joints subjected to uniaxial compression

T

Mostafa Asadizadeha, , Mohammad Farouq Hossainib,d, Mahdi Moosavib, Hossein Masoumic, , P.G. Ranjithc ⁎

⁎

a

Department of Mining Engineering, Hamedan University of Technology, Hamedan 65155-579, Iran School of Mining Engineering, College of Engineering, University of Tehran, Tehran 1439957131, Iran c Department of Civil Engineering, Monash University, Melbourne, VIC 3800, Australia d School of Minerals and Energy Resources Engineering, UNSW Sydney, NSW 2052, Australia b

ARTICLE INFO

ABSTRACT

Keywords: Non-persistent rough joint Jointed rock Crack coalescence Joint Roughness Coefficient (JRC) Joint angle Bridge angle Bridge length

Understanding the mechanical behaviour of jointed rock particularly under uniaxial compression is so important for an appropriate design of structures on or within a rock mass. Jointed rocks may consist of persistent or nonpersistent joints. Typically, the persistent joints can dominate the whole mechanical behaviour of jointed rocks while the effects of non-persistent joints on the rock masses require careful consideration during the characterisation process. Majority of the earlier studies on the non-persistent jointed rocks have included the joints with open smooth surfaces while the behaviour of jointed rocks with non-persistent rough joints has been explored very limitedly. Therefore, in this study, a number of artificial jointed rocks with non-persistent rough joints were tested under uniaxial compression. With the aid of 3D printing technology a wide range of joint roughness has been accommodated inside the artificial samples. The joints were parallel or coplanar. The influences of four different parameters including Joint Roughness Coefficient, bridge length, bridge angle and joint angle on the uniaxial compressive strength, the deformation modulus and the crack coalescence stress of jointed rocks were investigated through multivariate statistical analysis using Response Surface Methodology. In total, 30 experiments were conducted and the resulting failure patterns were classified into six different categories based on their crack coalescence conditions. A number of samples revealed a specific failure pattern known as “fish eye” due to the development of “asperity interlocking cracks” which were associated with the high range of joint roughness. From Response Surface Methodology it was found that all the four parameters had individual and interactive effects on the uniaxial compressive strength and the deformation modulus of jointed rock. Also, it was concluded that the overall mechanical behaviour of a jointed rock with non-persistent rough joints is mainly controlled by the joint angle under uniaxial compression.

1. Introduction Discontinuities are the most important component of rock masses which can appear in different forms including natural fractures, joints, bedding planes and faults. These discontinuities can be either persistent or non-persistent. Majority of the earlier studies have focused on the jointed rocks with persistent discontinuities (Einstein et al., 1983; Ramamurthy and Arora, 1994; Sitharam et al., 2001; Singh et al., 2002; Grasselli, 2006; Mas Ivars et al., 2011; Sherpa et al., 2013; Bahaaddini et al., 2014; Bahaaddini et al., 2016b; Dang et al., 2016; Alejano et al., 2017) while non-persistent discontinuities are as important as the persistent ones which have particular effects on the initiation and the

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propagation of new cracks in the jointed rocks (Asadizadeh and Rezaei, 2019Lajtai, 1969; Prudencio and Van-Sint-Jan, 2007; Ghazvinian et al., 2013; Sarfarazi et al., 2014; Zhou et al., 2014; Bahaaddini et al., 2016a; Cheng et al., 2016b; Vergara et al., 2016; Guo et al., 2017; Lee et al., 2017; Asadizadeh et al., 2018a; Asadizadeh et al., 2018b). Understanding the mechanical properties of jointed rock (e.g. uniaxial compressive strength and deformation modulus) is so important mainly for the design of structures on or within the rock masses. In most of the underground operations (e.g. hard rock or coal mining), pillar design is a common practice as its role for the stability of underground structures is critical. Some portions of these pillars are generally under uniaxial compression whose mostly contain non-persistent joints. Also, the

Corresponding authors. E-mail addresses: [email protected] (M. Asadizadeh), [email protected] (H. Masoumi).

https://doi.org/10.1016/j.enggeo.2019.105224 Received 29 October 2018; Received in revised form 10 June 2019; Accepted 5 July 2019 Available online 06 July 2019 0013-7952/ © 2019 Elsevier B.V. All rights reserved.

Engineering Geology 260 (2019) 105224

M. Asadizadeh, et al.

sidewalls of underground excavations are typically under uniaxial compression which highlights the significance of assessing a jointed rock with non-persistent joints under uniaxial compression (Brady and Brown, 2006). The earliest study on the characterisation of jointed rock subjected to uniaxial compression was conducted by Erdogan and Sih (1963) who casted some artificial rocks with a single crack. The cracks were made by a cutter inside the samples to replicate the jointed rocks with a single non-persistent joint. Since then, many similar experimental studies have been carried out to understand the mechanical behaviour and cracking process of jointed rocks with non-persistent joints under uniaxial compression (Wong and Chau, 1998; Lee and Jeon, 2011; Zhou et al., 2014; Cao et al., 2015; Cheng et al., 2016b; Huang et al., 2016; Afolagboye et al., 2017; Yang et al., 2017). In most of these studies, the artificial rock samples with pre-exiting cracks or joints were investigated through considering the effects of different joint geometrical parameters such as joint orientation or angle, joint step angle, joint length, bridge length, joint spacing, joint aperture and joint tip to tip angle on the uniaxial compressive strength, the deformation modulus and the cracking process of jointed rocks (Bahaaddini et al., 2013; Bahaaddini et al., 2016a). Due to the advancement of computer technology and numerical software some recent works have examined the mechanical behaviour of jointed rock masses under uniaxial compression through building some complex rock mass models known as Synthetic Rock Mass (SRM) (Bahaaddini et al., 2013; Bahaaddini et al., 2016a; Cao et al., 2016; Cheng et al., 2016a; Yang et al., 2016). Such models typically contain a rock mass with a multiple patterns of non-persistent joints for a better replication of field conditions (Yang et al., 2014; Zhang et al., 2015; Lee et al., 2017). Despite the extensive studies on the mechanical behaviour of nonpersistent jointed rocks, still our understanding regarding the effects of non-persistent rough joints on the rock masses is very limited. In fact, most of the earlier works in this regard have focused on the non-persistent joints with open smooth surfaces. As a result, in this study, a unique set of laboratory experiments were carried out to comprehensively investigate the mechanical behaviour of jointed rock-like material with non-persistent rough joints (coplanar and stepped) under uniaxial compression. This was feasible through applying the 3D printing technology leading to an accurate accommodation of the nonpersistent joints with a wide range of roughness inside the casted samples as one the important novelties of this research. Perhaps the only study with a minor similarity to the scope of this work was conducted by Wong and Chau (1998) who attempted to examine the behaviour of artificial jointed rocks with non-persistent joints at very low range of roughness levels. The effects of Joint Roughness Coefficient (JRC), bridge length (L), bridge angle (γ) and joint angle (θ) (see Fig. 1) on the uniaxial compressive strength (σcm), deformation modulus (Ecm) and crack coalescence stress (σcs) of jointed rocks were assessed extensively. This was followed by an examination of the cracking process of tested samples along with some comparative analysis with the earlier studies (Wong and Chau, 1998; Cao et al., 2015; Zhang et al., 2015). A range of JRC values was selected according to Barton (1973) shear strength model who then proposed his well-known JRC profiles (Barton and Choubey, 1977). Response Surface Methodology (RSM) (Box and Wilson, 1951) was used for the design of experiments as well as the post-data processing (Asadizadeh et al., 2019) where the interactive effects of defined independent parameters (JRC, L, γ and θ) on the dependent ones (σcm, Ecm and σcs) were included as a second objective of this study.

Fig. 1. Schematic representation of a jointed rock-like material under uniaxial compression illustrating some geometrical parameters as well as the loading and boundary conditions.

2. Experimental study The artificial slab-shaped rock samples were casted with the dimensions of 300 × 300 × 120 mm3. The samples were made of plaster (48% mass), cement (24% mass) and water (28% mass) leading to a material with sufficient brittleness to properly represent the natural jointed rocks. The samples were cured for 14 days at room temperature. The final product was achieved by trial and error. Five uniaxial compressive tests were carried out on the intact slab-shaped samples to obtain their mechanical properties for the comparative purposes leading to the mean uniaxial compressive strength (UCS) of 22.97 MPa, the mean Young's modulus (E) of 3.78 GPa and the mean tensile strength (σt) of 3.43 MPa. Such an artificial rock with the reported mechanical properties can be classified as a weak to medium rock according to Brady and Brown (2006) classification system. Also, a number of uniaxial compressive tests was performed on the cylindrical artificial samples according to International Society for Rock Mechanics (ISRM, 2007) suggested methods. The resulting UCS and E from the cylindrical samples were 23.70 MPa and 10.53 GPa, respectively. The change in UCS and E from the cylindrical shape to the slabshaped can be associated with size and shape effects in the brittle materials (Adey and Pusch, 1999; Masoumi et al., 2016). The tensile strength was obtained from the Brazilian test conducted on a number of cylindrical samples according to (ISRM, 2007). A number of sheets with different JRC profiles were designed and made using 3D printing technology with the dimensions of 150 × 100 × 1 mm3 (see Fig. 2). All the JRC sheets were made of VeroGray (RGD850) material. A unique and novel mold was designed and fabricated (see Fig. 3) with the same dimensions as that explained for the slab-shaped samples to cast the jointed artificial rocks with nonpersistent rough joints. The joints were designed in the stepped coplanar forms as demonstrated in Fig. 4.

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Fig. 2. Example of a 3D printing sheet used during the casting of jointed rock to make the non-persistent joints with rough surfaces.

2.1. Testing procedure The uniaxial compressive tests were performed using a servo-controlled loading frame with the maximum loading capacity of 400 ton. The axial displacement rate was constant at 0.005 mm/s in all the experiments. Such a loading frame was used to perform the uniaxial compressive tests on both intact and jointed samples. To minimize the end surfaces effects, two brushed steel platens were used in addition to two 2 mm thick Teflon sheets at both end surfaces (see Fig. 5). 2.2. Mechanical properties In total, 30 jointed samples were tested (see Table 1) under uniaxial compression and the uniaxial compressive strengths (σcm), the crack coalescence stresses (σcs) and the deformation moduli (Ecm) were determined from the resulting stress-strain curves. It is noteworthy that the defined levels for each JRC profiles are required to conduct the Multiple Regression Modelling (MRM). The average of the minimum and maximum range of each JRC profile was defined as a corresponding level for that particular profile. These values are a set of representative numbers for MRM and can be defined differently without any change in the analysis. The σcs was defined when the first crack coalescence occurred during the test as shown by the circles in Fig. 6. Lee and Jeon (2011) demonstrated that when a uniaxial compressive test is performed on a jointed rock with the non-persistent joints, a sudden drop of load (step) during the elastic zone is associated with the crack coalescence. The sudden stress reduction during the elastic zone or the crack coalescences are marked by circles A, B and C for the samples U4, U6 and U23, respectively in Fig. 6. Out of 30 experiments, 25 samples revealed such a drop. 3. Design of experiments based on Response Surface Methodology (RSM) For the optimum selection of number of experiments that provides a sound understanding of the effects of independent parameters on the dependent ones in an interactive manner, the Response Surface Methodology (RSM) was adopted (Box and Wilson, 1951). RSM is a combination of statistical and mathematical approaches which has been developed to model a process as well as exploring the interaction of factors or the independent parameters that impact on the responses or the dependent parameters of a system. In other words, RSM is a

Fig. 3. Illustration of the designed mold for the casting of artificial jointed rock samples with non-persistent rough joints: 1) box for the casting of sample, 2) sliding rail for controlling θ, 3) steel rod for linking the upper and the lower platforms, 4) protractor for controlling γ, 5) T segment for controlling L, 6) head of the L segment, 7) sliding rail of L segment for controlling its movement along the X and Y directions, 8) the holders of JRC 3D printing sheet, 9) JRC 3D printing sheet (Asadizadeh et al. 2018).

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Fig. 4. Example of a jointed rock with non-persistent rough joints a) casted with JRC profile of 14–16 and b) failed after the experiment.

dependent parameters or the responses which can be expressed using a quadratic model (Noshadi et al., 2012) as follows: 3

y=

0

+

3 i Xi

3 2 ii Xi

+

i=1

3

+

i=1

ij Xi Xj

(1)

i =1 j =i+1

where y is a response or a dependent variable (e.g. σcm, Ecm or σcs), βii, βij, βi, and β0 are the regression coefficients and Xi and Xj are the values of independent parameters (e.g. θ, γ, L and JRC) coded in a program according to:

Xi =

xi

x0

(2)

x

where x0 is the value of xi at the centre point and Δx is the variation interval. In analysing an experiment, the response or the dependent parameters were related to a set of controllable variables. For the continuous control of variables, the linear, factorial or quadratic models are often used as follows:

Linear model Y =

0

+

Factorial model Y =

0

Quadratic model Y =

1 X1

+ 0

+

+

1 X1

2 X2

+

1 X1

+

+

2 X2

+

2 X2

+

12 X1 X2

+

12 X1 X2

+

2 11 X 1

+

2 22 X 2

+

Apart from the intercept, the terms in these models fall into one of the following categories: a) Linear terms (main effects) of form βiXi which can model the average effect of a variable control; b) Two-factor interactions of form βijXiXj which allows the effect of change in one control according to the setting of another control; c) Quadratic terms of form βiiX2i which allows for the curvature in the effect of a control according to the response or the dependent parameter.

Fig. 5. The loading frame with a sample ready for the uniaxial compressive test.

multivariate statistical method which can assess the common effects of independent parameters on the dependent ones in a system with an optimum number of experiments. In this study, the Central Composite Design (CCD) was utilized to model RSM (Asghar et al., 2014). Here, the independent parameters are JRC, θ, L and γ. σcm, σcs and Ecm are the

Models for the categorical controls often involve a set of terms to represent a single main effect or two-factor interaction, but the interpretation of the effects is similar. The codes and the levels of independent parameters in this study are

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Table 1 Experiments and the independent and dependent parameters. Sample code

JRC (Level)

θ (degree)

L (mm)

γ (degree)

σcm (MPa)

σcs (MPa)

Ecm (GPa)

U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 U16 U17 U18 U19 U20 U21 U22 U23 U24 U25 U26 U27 U28 U29 U30

10–12 (11) 0–2 (1) 14–16 (15) 14–16 (15) 18–20 (19) 4–6 (5) 4–6 (5) 4–6 (5) 14–16 (15) 4–6 (5) 10–12 (11) 10–12 (11) 4–6 (5) 10–12 (11) 10–12 (11) 4–6 (5) 14–16 (15) 10–12 (11) 10–12 (11) 10–12 (11) 10–12 (11) 14–16 (15) 4–6 (5) 14–16 (15) 14–16 (15) 14–16 (15) 10–12 (11) 4–6 (5) 10–12 (11) 10–12 (11)

0.0 45.0 67.5 67.5 45.0 67.5 67.5 22.5 22.5 22.5 45.0 45.0 22.5 45.0 45.0 67.5 22.5 45.0 90.0 45.0 45.0 22.5 67.5 67.5 67.5 22.5 45.0 22.5 45.0 45.0

25.0 25.0 17.5 17.5 25.0 32.5 17.5 17.5 17.5 32.5 25.0 25.0 32.5 25.0 25.0 17.5 17.5 25.0 25.0 25.0 10.0 32.5 32.5 32.5 32.5 32.5 25.0 17.5 40.0 25.0

135.0 135.0 112.5 157.5 135.0 157.5 157.5 112.5 157.5 112.5 135.0 135.0 157.5 180.0 135.0 112.5 112.5 135.0 135.0 90.0 135.0 112.5 112.5 112.5 157.5 157.5 135.0 157.5 135.0 135.0

21.66 15.00 16.04 18.47 19.03 15.99 15.61 17.50 18.90 18.03 17.01 16.21 18.57 18.88 16.44 16.19 18.61 16.04 17.60 13.90 15.50 18.34 16.19 14.12 16.37 19.34 16.55 18.92 18.52 16.78

NA 8.59 4.83 5.90 5.56 4.04 5.65 7.75 NA 15.00 10.94 9.10 NA 7.24 9.50 1.50 6.00 9.75 0.80 3.70 8.00 15.00 6.67 4.82 NA 10.00 10.50 NA 13.00 11.00

4.44 3.12 3.18 3.61 3.53 3.35 3.21 3.40 4.01 3.32 3.41 3.52 3.63 3.52 3.19 3.34 3.24 3.21 3.54 3.10 3.01 4.08 2.80 3.86 4.04 4.07 3.35 3.61 3.60 3.21

given in Table 2 according to CCD. The distances of boundary points from the centre is denoted by α and is always specified in terms of coded values. α depends on the certain properties related to the design of experiments such as whether it is orthogonally blocked or not. In other words, it needs to be clear whether or not, a design is separated into a number of blocks which their characteristics do not affect the estimation of coefficients in a second order model. It is noteworthy that such characteristics also depend on a number of contributing factors or the independent parameters (e.g. JRC, θ, L and γ). Here for the four independent parameters, five levels were defined in which the focus of RSM is within the range of −1 to 1 for each independent parameter. The levels corresponding to each independent parameter were defined by trial and error as well as a pre-assessment of RSM to ensure a success in the analysis. Also, based on the past experiences (Barton, 1973; Bahaaddini et al., 2016a) the substantial effect of each level on the responses or the dependent parameters was considered during the levels determination. For example, the range of levels for joint angle varies between 0° to 90° that covers a suitable range for observing significant change in the responses (σcm, σcs and Ecm). For JRC, as pointed earlier, the average of maximum and minimum range of each JRC profile was attributed to each profile for running the RSM. Determination of the corresponding values to each JRC profile is as follows (see Table 1):

Fig. 6. Examples of the stress–strain curves obtained from the uniaxial compressive tests on the tested jointed samples exhibiting crack coalescence during the elastic limit. Table 2 Coded independent parameters and their corresponding levels obtained from CCD. Variable/parameter/factor

Joint roughness coefficient θ (degree) L (mm) γ (degree)

Code

JRC J.A L B.A

Level -α

-1

0

1

+α

1.0 0.0 10.0 90.0

5.0 22.5 17.5 112.5

11.0 45.0 25.0 135.0

15.0 67.5 32.5 157.5

19.0 90.0 40.0 180.0

JRC 0–2 = 1.0. JRC 4–6 = 5.0. JRC 10–12 = 11.0. JRC 14–16 = 15.0. and JRC 18–20 = 19.0.

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Table 3 ANOVA for the polynomial models of non-persistent jointed rock-like material based on RSM. Description

Ecm

σcs

σcm

Statistical parameter

Models are significant Models can be used to navigate the design space The lack of fit is not significant. Good agreement between the predictive models and the data sets based on high R2

32.15 24.05 0.61 0.9353

26.85 18.639 3.17 0.9171

48.37 30.97 1.05 0.9752

F-value Adequate precision Lack of Fit to F-value R2

Table 4 ANOVA results for the uniaxial compressive strength (σcm).

Table 6 ANOVA results for the deformation modulus (Ecm).

Parameters

Sum of Squares

Degree of freedom

Mean Square

Weighting contribution (%)

Noise (%)

Parameters

Sum of Squares

Degree of freedom

Mean Square

Weighting contribution (%)

Noise (%)

Model JRC L γ θ JRC × L L×θ θ2 JRC × γ × θ JRC × L JRC2× γ JRC × L2 Residual

83.53277 8.12045 4.5602 12.4002 31.1676 1.215506 1.434006 0.995006 15.57907 1.204506 2.348556 4.921602 3.397352

13 1 1 1 1 1 1 1 1 1 1 1 16

6.425598 8.12045 4.5602 12.4002 31.1676 1.215506 1.434006 0.995006 15.57907 1.204506 2.348556 4.921602 3.397352

6.61 8.35 4.69 12.75 32.05 1.25 1.47 1.02 16.02 1.24 2.41 5.06

< 0.01 < 0.01 < 0.01 < 0.01 < 0.01 0.80 0.47 1.46 < 0.01 0.83 0.07 < 0.01

Model JRC L γ θ JRC × L θ2 JRC × L × γ L×γ×θ JRC × L2 Residual

3.691135 0.08405 0.310538 0.413438 0.592204 0.381306 0.892531 0.204756 0.077006 0.066752 0.255131

9 1 1 1 1 1 1 1 1 1 20

0.410126 0.08405 0.310538 0.413438 0.592204 0.381306 0.892531 0.204756 0.077006 0.066752 0.012757

11.95 2.45 9.05 12.04 17.25 11.11 26.00 5.96 2.24 1.94

< 0.01 1.84 < 0.01 < 0.01 < 0.01 < 0.01 < 0.01 0.07 2.33 3.32

confidence level on the resulting functions (Eqs. 3 to 5) obtained from MRM (Kirmizakis et al., 2014) (see Table 3). The estimated F-values for all the three dependent parameters confirm the high confidence level on Eqs. (3) to (5). There is only a 0.01% chance that such a large “FValues” occur due to noise. The “Adeq precision” is the signal-to-noise ratio and the ratio > 4 is desirable (Kirmizakis et al., 2014). For all the three dependent parameters, this ratio is much > 4. The “Lack of Fit Fvalue” implies that the lack of fit is not significant compared to the pure error. Table 3 also indicates that the resulting coefficient of determinations (R2) for all the three dependent parameters (σcm, Ecm and σcs) are > 0.9 confirming a very good agreement between the polynomial models and the data sets. ANOVA was also performed to quantify the weighting contribution of individual and interactive effects of the independent parameters (e.g. JRC, L, γ and θ) on the dependent ones (σcm, σcs and Ecm). Such an analysis was dependent on the resulting polynomial models presented in Eqs. (3) to (5). The weighting contributions of JRC, θ, L and γ on σcm, σcs and Ecm are given in Tables 4 to 6, respectively. These contributions include both individual and interactive effects. From Tables 4 and 5 it is evident that the maximum impact on σcm and σcs was due to θ at about 32% and 39% weighting contributions, respectively and it had the second highest weighting contribution on Ecm at about 17% according to Table 6. Thus, it is true to state that the overall mechanical behavior of a jointed rock with non-persistent rough joints is mainly controlled by the joint angle (θ) under uniaxial compression.

3.1. Multiple Regression Modelling (MRM) From the given coded values in Table 2, the polynomial models for the dependent parameters (σcm, σcs and Ecm) were estimated as the functions of the independent parameters (JRC, L, γ and θ) through the best fits to Eq. (1) resulting in: cm

= 16.62 + 1.01 × JRC + 0.75 × L + 1.24 × 0.25 × L ×

+ 0.74 × 2

0.96 × JRC2 × L cs = 10.25 + 1.2 × L

0.80 × JRC2 ×

3.03 ×

1.14 ×

0.27 × JRC × L ×

1.5L ×

0.28 × JRC × L + 0.3 × JRC ×

+ 0.38 × JRC ×

×

(3)

0.81 × JRC × L2 1.08L ×

0.77 × JRC2

1.17 × 2

0.79 × 2

(4) Ecm = 3.34 + 0.10 × JRC + 0.11 × L + 0.13 × 0.16 × + 0.15JRC × L + 0.18 × 2 0.11 × JRC × L × + 0.069L × × + 0.11 × JRC × L2

(5)

It is evident from Eqs. (3) and (5) that all the four independent parameters had individual and interactive effects on σcm and Ecm while Eq. (4) confirms that only L and θ had individual effects on σcs along with some interactive effects by all the four independent parameters. 3.2. Analysis of Variance (ANOVA) Analysis of variance (ANOVA) was performed to assess the Table 5 ANOVA results for the crack coalescence stress (σcs).

4. Cracking behaviour and discussion

Parameters

Sum of Squares

Degree of freedom

Mean Square

Weighting contribution (%)

Noise (%)

Model L θ L×γ L×θ JRC2 γ2 θ2 Residual

300.2687 24.92853 104.1132 22.92724 12.74031 16.07868 37.2793 8.713489 27.15596

7 1 1 1 1 1 1 1 17

42.89553 24.92853 104.1132 22.92724 12.74031 16.07868 37.2793 8.713489 1.59741

15.90 9.25 38.61 8.50 4.73 5.96 13.82 3.23

< 0.01 0.10 < 0.01 0.15 1.17 0.56 0.02 3.20

Overall, six different failure modes were identified for the samples tested in this study based on Wong and Einstein (2009) (see Fig. 7) classification system who considered the crack coalescence condition as the main contributing factor in their proposed cracking process. The following steps were taken to sketch the resulting failure patterns. 1. A digital camera with 18MPix resolution and capability of capturing 22 frames per seconds was used to record the cracking process. 2. Tensile cracking was considered when the resulting cracks at the tips of the joints were opened. 3. Those newly created cracks at the end of tests were categorized as 6

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Category

Failure mode

1

Pure tensile

Crack coalescence pattern in each specimen

U6

2

U10

U8

U16

U17

U23

U22

U26

Pure tensile and Pure shear U2

Subcategory 1

3

Tensile-Shear

U3

4

U5

U7

U11

U12, 15, 18, 27

U19

U29

U30

U20

Pure shear U14

6

U24

Pure Tensile and Tensile-Shear U4

5

Subcategory 2

U21

No coalescence U1

U9

U13

U25

U28

Fig. 7. Different failure modes observed from the uniaxial compressive tests on the non-persistent jointed rocks with various rough surfaces. It is noteworthy that the sketches do not reveal the real JRC profiles due to simplicity in drawing.

shear cracks if they were filled with the crushed materials or the trace of slickensides was observed. 4. According to Broek, 2012, those cracks with the square fractures were categorized as the tensile cracks.

combination of tensile and shear cracking. It is believed that due to the application of low level of joint roughness no trace of “fish eye” failure pattern was identifiable from Wong and Chau (1998) test results. Cao et al. (2015) conducted a number of uniaxial compressive testes on the slab-shaped artificial jointed rock samples with non-persistent open smooth joints where the joint angle and the bridge angle were the variables. Cao et al. (2015) accommodated two joints inside the samples similar to this study. Comparison between the resulting failure modes in here and those reported by Cao et al. (2015) indicates that the effect of joint roughness on the type of cracking and the final failure pattern is quite important. Cao et al. (2015) concluded two types of failure modes being tensile and shear failures while the tested samples in here led to the failure modes with more varieties. Zhang et al. (2015) conducted an extensive numerical modelling on 2D samples based on Discrete Element Code. Zhang et al. (2015) assessed the effects of change in the joint angle on the mechanical behaviour of jointed rocks subjected to axial loading. Zhang et al. (2015) included two joints for the tested samples and each joint had a different joint angle leading to two different joint sets. Both joints were non-persistent with open smooth surfaces. While the resulting failure patterns from the tested samples by Zhang et al. (2015) were more than those reported by Cao et al. (2015), there is a clear distinction between the ones resulted in this study and those reported by Zhang et al. (2015). Considering the earlier findings in this regard (e.g. Wong and Chau, 1998; Lee and Jeon, 2011; Zhou et al., 2014; Cao et al., 2015; Cheng et al., 2016b; Huang et al., 2016; Afolagboye et al., 2017; Yang et al., 2017) and that reported above, it can be suggested that the “fish eye” failure mode is a characteristic pattern associated with the high range

From each category the resulting failure mode of a representative sample along with its detailed sketch of failure pattern is presented in Fig. 8. A number of samples revealed “fish eye” failure mode which was originally suggested by Wong et al. (2001). Such a failure pattern occurred in those samples with mixed mode of tensile-shear failure. Wong et al. (2001) performed a number of shear tests on the slab-shaped artificial jointed rocks with non-persistent open smooth joints where all the joints where parallel to the direction of shearing. However, in here, the joint angle in those samples with “fish eye” failure mode was not parallel to the loading direction. Definitely shearing along the joints was a significant factor in such a failure which can be attributed to a high range of joint roughness utilized in this study. For instance in sample U24 with “fish eye” failure mode, the JRC was relatively high (14–16) and its joint angle was 67.5°. It is believed that such a failure mode is a result of ‘asperity interlocking cracks’ due to the presence of joints with JRC profiles of > 10–12 and θ bigger than 45 degrees (e.g. samples U3, U4, U5, U20 and U24). The objective of Wong and Chau (1998) study found to be similar to the scope of this research where they utilized the stainless steel sheets with a low range of roughness to create non-persistent rough joints inside the artificial rock samples. Wong and Chau (1998) only considered the change in the joint and bridge angles leading to nine different failure patterns including, tensile cracking, shearing or 7

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a)

d)

b)

e)

c)

f)

Fig. 8. Illustration of a range of failed samples along with their detailed sketches of final failure pattern for samples a) U6, b) U2, c) U19, d) U20, e) U14 and f) U1.

of joint roughness. The “asperity interlocking cracks” found to be the main cause of “fish eye” failure mode during the shearing of jointed rocks with relatively high JRC profile and joint angle. It is noteworthy that although the above findings obtained from the uniaxial compressive tests on the artificial jointed rocks, due to the similarity between the mechanical behaviour of natural rocks and the tested samples; the results can be applicable to rock masses. It is worth mentioning that for the accommodation of such rough joints inside the sample, the use of artificial rock-like material was essential as with the current technology, it would be very difficult and costly to make the non-persistent rough joints inside the natural rock.

among those independent parameters that had substantial influences on σcm, Ecm and σcs were selected for the visual presentation below. Also, while Eqs. (3) to (5) were obtained from the codded values, the RSM graphs are presented based on the decoded or the real values according to Table 2. 5.1. Effects of independent parameters on σcm From MRM (Eq. 3) only three combinations of the independent parameters found to be effective on σcm including the combination of L and θ, the combination of JRC and γ and the combination of L and JRC. Fig. 9a and b show the interactive influences of L and θ on σcm indicating that when θ = 22.5°, an increase in L within the range of 17.5 mm to 32.5 mm leads to 11.49% increase in σcm (from 17.49 MPa to 19.50 MPa). Similarly, when θ = 67.5°, an increase in L within the same range results in an increase in σcm at 6.43%. Such an increase in σcm can be related to an increase in the load bearing capacity of the jointed rock in which with an increase in L the intact zone in the bridge

5. Multivariate statistical analysis on the resulting data The RSM results are presented along with the detailed explanations on the potential physical reasons for the effects of each independent parameter (θ, JRC, γ and L) on the dependent ones (σcm, Ecm and σcs). It is noteworthy that from Eqs. (3) to (5), only the interactive effects

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Fig. 9. 2D and 3D RSM results for σcm visualizing the interactive effects among four independent parameters when: (a and b) L and θ are the variables, (c and d) JRC and γ are the variables and (e and f) L and JRC are the variables. 9

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area increases leading to a higher load bearing capacity and subsequently greater σcm. Inversely, when L is constant, an increase in θ leads to decrease in σcm. For example at L = 17.5 mm, an increase in θ from 22.5° to 67.5° results in 10.18% decrease in σcm and similarly at L = 32.5 a drop in σcm is evident when θ increases within the same range. While Fig. 9a demonstrates the 2D interactive effects of L and θ on σcm, Fig. 9b visualizes these interactions in a 3D format. It is important to note that JRC and γ were constant at their middle points when the multivariate statistical analysis was conducted on L and θ. From Fig. 9c and d, the interactive influences of JRC and γ on σcm can be interpreted in the same way as that explained for L and θ. Interestingly, it is identifiable that at γ = 122.5°, the range of increase in σcm (13.25%) due to an increase in JRC profile is greater than that where γ = 157.5 degrees (12.55%). This is also the case when JRC is constant and γ is variable. Increase in σcm due to an increase in JRC profile can be associated with the further interlocking of asperities that can lead to an increase in the frictional resistance during the shearing process of joint surfaces. For the case where the interaction among L and JRC is considered, it is evident from Fig. 9e and f that when L is constant, an increase in JRC leads to an increase in σcm. It is important to note that the range of increase in σcm (5.81%) due to an increase in JRC is significantly different to that when L = 17.5 mm (0.91%). Also, when JRC is constant at 4–6 (level 5.00), with an increase in L from 17.5 mm to 32.5 mm, 0.86% increase is observable in σcm. As it can be seen in Fig. 9e while L is constant, an increase in the JRC has a positive effect on σcm. It is important to note that when L = 17.5 mm, an increase in the JRC from 4 to 6 profile to 14–16 profile causes 5.81% increase in σcm (from 16.35 MPa to 17.30 MPa); however, when L = 32.5 mm, 0.91% decrease is caused by an increase in the JRC profile from 4 to 6 to 14–16. Furthermore, when the JRC is constant at 4–6 profile, an increase in the bridge length from 17.5 to 32.5 mm causes 0.86% increase in σcm (from16.35 MPa to 16.49 MPa) and when JRC is 14–16, σcm decreases by 5.55% (from 17.3 MPa to 16.34 MPa). Such a behaviour can be attributed to the asperity interlocking during the loading stage which can trigger mixed mode cracking that usually initiates in shear and then continues under tensile failure. These types of cracks are considered as “asperity interlocking cracks” which are formed due to an increase in JRC profiles. Initiation and propagation of “asperity interlocking cracks” significantly weaken the mechanical stability of the sample before reaching the peak stress. The resulting effects of γ on σcm are consistent with the earlier studies on rock masses with non-persistent open smooth joints (Park and Bobet, 2009; Wong and Einstein, 2009; Lee and Jeon, 2011; Huang et al., 2016; Yang et al., 2016). From the failure patterns of the tested samples, it is evident that the failure mode of bridge zone changes from pure tension to pure shear when γ varied from 90° to 180°.

two different trends are observable. At the lower level of bridge angle i.e. γ = 112.5°, the tensile coalescence is dominant in which with an increase in the bridge length the coalescence stress also increases (see Fig. 8). When γ = 157.5°, an increase in the bridge length can lead to a slight decrease in σcs. Such a phenomenon can be pertained to ‘asperity interlocking cracks’ whose play an important role in forming the ‘fish eye’ failure mode in the bridge area. When L = 17.5 mm, with an increase in the bridge angle, the coalescence stress also increases. It is worth mentioning that as the bridge angle increases, the failure mode is changing from pure tensile to pure shear and thus, it justifies an increase in the crack coalescence stress. On the other hand, when L = 32.5 mm the initiation and propagation of ‘asperity interlocking cracks’ and formation of ‘fish eye’ failure mode affect this process. In Figs. 10c and d the influences of L and θ on σcs are visualized in 2D and 3D formats. When θ = 22.5°, a rise in L from 17.5 mm to 32.5 mm results in 44.63% increase in σcs (from 10.21 MPa to 14.77 MPa). By contrast, when L is constant (e.g. 17.5 mm or 32.5 mm), an increase in θ leads to the substantial decrease in σcs. Such a decrease can be associated with a direction of bridge zone area with respect to the loading condition during the uniaxial compressive test. When θ increases, the bridge area will be parallel to the direction of axial loading resulting in faster coalescence process at the lower stress level. 5.3. Effects of independent parameters on Ecm Similar to σcm and σcs the interactive effects of independent parameters on Ecm was assessed through the multivariate statistical analysis. The resulting trends are presented in 2D and 3D graphs for the comparative purposes (see Fig. 11). When any two variables were selected for the analysis (e.g. JRC and L), the other two (e.g. γ and θ) were kept constant at their middle points. Figs. 11a and b show the influences of L and JRC on Ecm when γ and θ are unchanged at their middle values. When L = 17.5 mm, an increase in JRC from profile 4–6 (Level 5.00) to 14–16 (Level 15.00) results in 3.82% increase in Ecm. If L is constant at 32.5 mm such an increase is about eight times greater than that when L was 17.5 mm (from 3.09 GPa to 3.83 GPa). By contrast, when JRC is constant at profile 14–16, Ecm reveals some increase if L increases from 17.5 mm to 32.5 mm. 6. Conclusions A unique set of uniaxial compressive tests were performed on 30 jointed rock-like material (artificial) samples with non-persistent rough joints to assess their mechanical and cracking behaviours. The interactive effects of four different parameters including Joint Roughness Coefficient (JRC), bridge length (L), bridge angle (γ) and joint angle (θ) on the uniaxial compressive strength (σcm), the crack coalescence stress (σcs) and the deformation modulus (Ecm) of artificial jointed rocks were extensively examined from the experimental viewpoint. The Response Surface Methodology (RSM) was used for the sound understanding of interactive effects of defined parameters on σcm, σcs and Ecm through multivariate statistical analysis. Overall six different failure modes were identified based on the resulting crack coalescences including: (1) pure tensile, (2) pure tensile and shear, (3) mixed mode of tensile-shear, (4) pure tensile and mixed mode of tensile-shear, (5) pure shear and (6) failure mode with no crack coalescence. The third category was classified into two sub-categories exhibiting “fish eye” failure mode as a characteristic pattern associated with the jointed rocks with high range of joint roughness. The trigger

5.2. Effects of independent parameters on σcs From Eq. (4) only three independent parameters (L, γ and θ) were found to have significant effects on σcs (see Fig. 10). The first multivariate analysis was carried out when the interactive effects of L and γ on σcs was considered and in the second analysis, the interactive effects of L and θ were included. From Figs. 10a and b it is evident that when γ =112.5°, an increase in L from 17.5 mm to 32.5 mm leads to 86.12% increase in σcs (from 6.38 MPa to 11.87 MPa), while when γ =157.5°, an increase in L along the same range as that for γ =112.5° results in 6.47% reduction in σcs (from 9.38 MPa to 8.78 MPa). Similarly, for the case when L is constant,

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Fig. 10. 2D and 3D RSM results for σcs visualizing the interactive effects among the three independent parameters when: (a and b) γ and L are the variables and (c and d) L and θ are the variables.

for such a characteristic pattern suggested to be the “asperity interlocking cracks”. From RSM it was found that all the four independent parameters (JRC, L, γ and θ) had substantial effects on σcm through different interactive effects including: combinations of JRC and L, L and θ as well as JRC and γ. From Analysis of Variance (ANOVA) it was found that θ had the maximum effect on σcm with about 32% weighting contribution similar σcs which was mostly affected by θ at about 39% weighting contribution. For Ecm, θ revealed the second highest impact at about 17% weighting contribution. Based on the ANOVA results, it was

concluded that the mechanical behaviour of a jointed rock with nonpersistent rough joints is mainly controlled by the joint angle (θ) under uniaxial compression. The findings from this study and the proposed multivariate statistical methodology (RSM) can assist the geological engineers to better analysis the complex mechanical behaviour of rock masses during the design stage. This is due to the similarity between the mechanical behaviour of tested artificial jointed rocks and the natural jointed rocks with non-persistent rough joints.

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Fig. 11. 2D and 3D RSM results for Ecm visualizing the interactive effects among two independent parameters when L and JRC are the variables.

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