A joint shear model incorporating small-scale and large-scale irregularities

International Journal of Rock Mechanics & Mining Sciences 76 (2015) 78–87

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International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms

A joint shear model incorporating small-scale and large-scale irregularities J. Oh a, E.J. Cording b, T. Moon c,n a

School of Mining Engineering, The University of New South Wales, Sydney 2052, NSW, Australia Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA c Geotechnical and Tunneling Division, HNTB, Five Penn, New York, NY 10119, USA b

art ic l e i nf o

a b s t r a c t

Article history: Received 7 March 2014 Received in revised form 20 February 2015 Accepted 23 February 2015 Available online 19 March 2015

The strength and dilation of rock joints in the field cannot be evaluated solely on the basis of parameters scaled from laboratory data, but also requires assessment of large-scale irregularities not present in the laboratory sample. A constitutive model for rock joints has been developed that considers the dilation and strength along both small-scale joint roughness scaled from laboratory data, and large-scale waviness determined from geologic observations. The model’s performance is illustrated by providing its correlation with experimental results taken from literature. The degradation in dilation and post-peak strength along small-scale irregularities is modeled using the plastic work done in shear, and the degradation along large-scale irregularities is modeled using a sinusoidal function. A dimensionless product of plastic work, rock strength, and wavelength of irregularities has been developed which fits the direct shear test results. An approach to scaling shear strength and shear displacement from laboratory to field-scale is also suggested. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Rock joint Dilation Irregularities Scale effect Constitutive model

1. Introduction The behavior of rock joints dominates the behavior of rock masses by providing planes of weakness along which shear and dilation can occur. A number of experimental studies have been conducted to understand the behavior of rock joints and many joint constitutive models have been proposed to predict their mechanical behavior. Patton’s model [1], one of the earliest and most fundamental models for peak shear strength, was developed from the basic mechanics of sliding up the asperity with inclination angle or shearing through the asperity depending on the normal stress level. Later, Ladanyi and Archambault [2] proposed a semi empirical model which featured the curved failure envelopes. Barton [3] developed a useful empirical model by introducing a morphological parameter known as the joint roughness coefficient (JRC) and using the concept of roughness mobilization. A significant advanced theoretical model was developed by Plesha [4], in which asperity degradation is a function of the plastic work during shear. More recently, approaches such as fractal [5] and geostatistical analysis [6,7] have been proposed to evaluate the mechanical behavior of rock joints under shear. A large body of literature (e.g., [1,8–17]) indicates that strength and shear behavior of rock joints vary both qualitatively and

n

Corresponding author. Tel.: þ 1 201 916 7558; fax: þ 1 212 947 4030. E-mail address: [email protected] (T. Moon).

http://dx.doi.org/10.1016/j.ijrmms.2015.02.011 1365-1609/& 2015 Elsevier Ltd. All rights reserved.

quantitatively as a result of a change in sample- or in situ blocksize. Ignoring scale effect may lead to overestimation or underestimation of field shear strength of joints if the peak strength obtained from laboratory joint shear test is used. Shear behavior of rock joints in the field should be evaluated by considering the dilation and strength along both small-scale joint roughness scaled from laboratory data, and large-scale waviness determined from geologic observations. However, most rock joint constitutive models proposed in literature have been developed on the basis of data obtained from laboratory tests on natural or model rock joints. Thus, they do not fully represent the behavior of rock joints in the field. This paper describes a rock joint constitutive model which can generate shear stress–displacement–dilation curves for both small-scale and large-scale joints. The model is incorporated in 3DEC [18] using the built-in programming language, FISH and is correlated with experimental results of direct shear tests taken from literature.

2. Description of a constitutive model for small-scale joints 2.1. Mobilized shear strength The shear stress–displacement–dilation curves generated by the proposed joint model can be characteristically divided into five stages: (1) elastic region, (2) pre-peak softening, (3) mobilized

J. Oh et al. / International Journal of Rock Mechanics & Mining Sciences 76 (2015) 78–87

Fig. 1. Schematic shear stress–displacement–dilation curves from the joint model. Curves of simulation results using the same material properties as used in [19] in the direct shear tests as illustrated in Fig. 6. Superscript e and p indicate elastic and plastic, respectively.

79

that elastic shear displacement (δes ) is 0.3 times peak shear displacement (δpeak). By performing direct shear tests on different sized replicas casts from various natural joint surfaces, Bandis et al. [10] concluded that for practical purposes a peak shear displacement (δpeak) can be taken as approximately equal to 1% of the joint length for a large range of block sizes and types of roughness. Since no general relationship for peak shear displacement (δpeak) is yet available, as the preceding discussion indicates, a peak shear displacement (δpeak) is treated as an input parameter in the proposed joint model, as ‘n’ times of elastic shear displacement (δes ), where δes is determined by the combination of stress level and joint shear stiffness (Ks). For the initial analysis or with limited test data available, the value of ‘n’ can be selected to be 3, based on direct shear test results frequently observed in the literature [3,24]. After peak shear strength, a mobilized shear stress is gradually decreased until it reaches a residual value. The mobilized shear stress during plastic region is calculated as

τmob ¼ σ n tan ðϕr þ αmob Þ

ð2Þ

where σn is a normal stress, τmob is a mobilized shear stress, ϕr is a residual friction angle, and αmob is a mobilized asperity angle that degrades as plastic work increases. 2.2. Asperity degradation The model proposed here simulates the progressive degradation of a joint asperity under shear. It is modeled by assuming that degradation is a function of the plastic work, Wp and its relationship is given by Eq. (3), which was suggested by Plesha [4].
 αmob ¼ α0 exp  cW p ð3Þ Fig. 2. Schematic shear stress–displacement curves. Bold one represents shear stress–displacement curve for joint in the field.

peak strength, (4) post-peak softening, and (5) residual strength. Schematic curves in Fig. 1 shows there are five stages in the shear stress–displacement–dilation curves. It is frequently observed from experimental results of direct shear tests in the literature that the shear stress–displacement curves show almost linear elastic behavior to a stress approximately equivalent to the residual strength of the joint. The proposed joint model accounts for these observations. Therefore, during elastic region, the shear stress is mobilized as a function of joint shear stiffness (Ks) and elastic shear displacement (δes ). The shear stress increment (Δτe) is calculated as

Δτe ¼ K s Δδes

ð1Þ

After the elastic region, the joint starts to slide and dilation takes place, which means the plastic shear displacement occurs from this point on. The degradation in asperity and also dilation is modeled as a function of the plastic work done in shear. This is discussed in detail in the following sections. Most rock joints show that peak shear strength is mobilized at very small deformation. A peak shear displacement is sometimes considered to be a material constant, not affected significantly by changes of normal stresses, which is supported by experimental results such as Leichnitz [20] and Herdocia [21]. On the other hand, as shown in some direct shear test results, such as Jaeger [22], Schneider [23], and Flamand et al. [19], this parameter is not always constant for a given joint. On the basis of results from shear tests on model tension fractures, Barton and Choubey [8] suggested that a peak shear displacement is dependent on sample scale and occurs after a shear displacement equal to 1% of the joint sample length up to some limiting size. Later, Barton [3] assumed

where α0 is an initial asperity angle, c is a asperity degradation P constant, and Wp is expressed as W p ¼ Δδps τ. Although Eq. (3) possessed good qualitative and quantitative agreements with experimental observations, it is difficult to relate the asperity degradation constant, c to other properties of rock joints [4]. From the cyclic shear test results on some thirty real rock granite and limestone joints, Hutson and Dowding [25] proposed an advanced relationship for the asperity degradation constant, c given by   c ¼  0:141 α0 N=σ c cm2 =J ð4Þ where α0 is an initial asperity angle, N is a normal stress, and σc is an unconfined compressive strength of rock. This relation indicates that asperity degradation is a function of material strength as well as stress level, which has been mentioned by many researchers [2,8,13]. While the asperity degradation constant, c, in Eq. (4) describes general behavior of a rock joint, it still has some questions to be considered. First, it is not a dimensionless relation. The constant c has a unit of [cm2/J] with a dimensional constant of 0.141. Thus Eq. (4) is valid in only one consistent system of units. Second, Eq. (4) is valid for limited range of scale of irregularities. The Hutson–Dowding model was determined from tests on rock joints having wavelengths of 3.2 and 5.0 cm, and is not expected to apply to the typical laboratory samples. Hutson mentioned in his thesis [26] that large laboratory models, such as the largest used by Bandis [9], should produce similar results. Finally, the effect of the applied normal stress in Eq. (4) is duplicated since this constant is multiplied by the plastic work, Wp as shown in Eq. (3). In order for the asperity degradation constant to apply to any scaled laboratory samples, Eq. (4) appears to be modified. A geometric parameter, that is, a wavelength is introduced and thus the wavelength, λ with the initial asperity angle, α0 describes the joint shape of typical laboratory samples. By removing the normal

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stress, an asperity degradation constant, c, which has dimensionless relation along with the plastic work, is proposed as c ¼ kα0 =λσ c

ð5Þ

where σc is an unconfined compressive strength of rock, λ is a wavelength of asperity, and k is a constant. It should be noted that the degradation constant, Eq. (5) is independent of the applied stress but the effect of stress on the asperity degradation is included in the plastic work, Wp as shown in Eq. (3). The constant k is determined which fits the direct shear test results for both natural and model joints. It has a range of 0.4 to 0.9 in most cases as illustrated in Section 4. 2.3. Dilation The mobilized dilation is calculated from the mobilized asperity angle. A dilation increment, Δδv is calculated, based on the plastic shear displacement increment, Δδps , as

Δδv ¼ Δδps tan ðdn Þ

ð6Þ

where Δδv is a normal displacement, Δδ is a plastic shear displacement, and dn is a dilation angle. The dilation angle is calculated from the relationship given by p s

dn;mob ¼ Dα0 expð  cW p Þ

ð7Þ

where D is a coefficient for the joint surface damage and Eq. (5), proposed here is used for the asperity degradation constant, c. The coefficient D ¼0.5 in Eq. (7) is to be a reasonably conservative estimation of dilation angle with limited test data available, based on Barton and Choubey’s test results [8]. Through the test data obtained from 130 model fractures, Barton concluded that the coefficient 1.0 could be used for joints which suffer relatively little damage during shear while the coefficient 0.5, for further accumulated damage of asperity [8,24]. A summary of experimental results obtained by Bandis [9] indicates that for small samples (5 or 6 cm), average value of ratio of dn to (ϕp,mob  ϕr) is about 0.4 and for large samples (36 or 40 cm), average value of ratio is about 0.5. Therefore, it is suggested that the coefficient values in Eq. (7) can be in a range between 0.4 and 1.0, which provide good agreements with the experimental results as illustrated in Section 4.

field, therefore, can be expressed in the following equation as a general form:

τmob ¼ σ n tan ðϕr þ αn;mob þ imob Þ

ð8Þ

where ϕr and σn are obtained from laboratory test and i can be determined from geologic observations in the field. The main question that remains to be addressed is: how could the field-scale small roughness (αn) be determined from laboratory test? During the shearing process of the joint in the field, large-scale irregularities generally lead to gradual dilation and thus the mobilized shear strength due to large-scale irregularities is also gradual. At the same time, shearing resistance contributed from small-scale irregularities is mobilized over small displacement at the contact portions. Before the peak shear strength of discontinuity in the field is mobilized, shearing resistance due to small irregularity interlocking has already been fully mobilized and it will degrade rapidly. This component of shear strength (αn) can be practically obtained from laboratory direct shear test for a typical size of sample as a fraction of small-scale roughness (e.g., 0.6(ϕp,mob  ϕr)), which would provide the value similar to shearing-through component (sn) of small-scale joint; Bandis’s test results [9] show the average value of dilation component (dn) is 0.37(ϕp,mob  ϕr). The field-scale small roughness (αn) is, however, not solely shearing-through component (sn) because αn would contribute to some dilatancy for the strength of small-scale roughness. αn,mob at peak in the field will be less than 0.6(ϕp,mob  ϕr)

3. Description of a constitutive model for large-scale joints 3.1. Evaluation of peak shear strength in the field Patton [1] and Cording and Mahar [11,13] obtained an estimate of the peak shear strength in the field by using the residual angle of friction obtained from laboratory direct shear tests and adding a component, i of the large-scale waviness to the angle of friction. McMahon [15] reported the correlation of back-calculated effective friction angles from eight rockslides with results from laboratory direct shear tests and field measurements of roughness. He showed that the effective friction angle of potential rockslide surfaces could be estimated as the sum of the mean laboratory ultimate friction angles and the mean of large scale roughness angles. These approaches (i.e., [1,11,13,15]), however, may result in conservative estimates of the peak shear strength, especially in the case of rough and tensional fractured joints. Peak shear strength in the field is generally composed of not only components of residual (ϕr) and field-scale irregularity (i) angles, but also a component of field-scale small roughness, as illustrated in Fig. 2. In this study the primary reduction in shear strength with increasing block size in the field is assumed to be due to reduction in a geometrical component (dilation angle) of shear strength. Shear strength in the

Fig. 3. Comparison of peak displacements from test results [9] with the values derived from equations. Sample no. 1–5: strongly undulating; 6–8: moderately undulating; 9–11: almost planar joints.

Fig. 4. Normalized Sinusoidal Function from Eq. (12).

J. Oh et al. / International Journal of Rock Mechanics & Mining Sciences 76 (2015) 78–87

81

Fig. 5. Direct shear test models and boundary conditions. a) 3-D model for direct shear test. and, (b) boundary conditions.

E (MPa)

Kn (MPa/m)

Ks (MPa/m)

σc (MPa)

φb (degree)

α0 (degree)

λ (mm)

k

32200

315000

63000

82.0

37.0

22.0

3.0

0.45

JRC = 8.5 to 10 estimated by Flamand et al. Fig. 6. Comparison between model simulations and laboratory experimental results; experimental data from [19]. (a) shear stress-displacement curves. (a) shear stressdisplacement curves. and, (c) profile of surface geometries of joint.

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J. Oh et al. / International Journal of Rock Mechanics & Mining Sciences 76 (2015) 78–87

5

normal displacement (mm)

shear stress (MPa)

2.5 2 1.5 1 0.5 0

4 3 2 1 0

0

2

4 6 8 shear displacement (mm)

10

0

Simulated

2

4 6 8 shear displacement (mm)

10

Experimental

E (MPa)

Kn (MPa/m)

Ks (MPa/m)

σc (MPa)

φr (degree)

α0 (degree)

λ (mm)

k

2208

21000

1300

7.36

35.0

30.0

17.0

0.4

Fig. 7. Comparison between model simulations and laboratory experimental results; experimental data from [30]. (a) shear stress-displacement curves, (b) normal-shear displacement curves , (c) profile of surface geometries after shear test and, (d) input parameters used in simulation.

because of the degradation before peak. Increasing peak shear displacement (δpeak) will cause more degradation before peak and lower αn,mob. The true scale reduction in the field, therefore, will depend on the amount of degradation that occurs prior to peak displacement. As shown in the examples later, simulation of direct shear tests with 0.6(ϕp,mob  ϕr) for the initial value of αn shows a reasonably good correlation with experimental results taken from literature. A similar approach was used by Hencher and Richards [27] to evaluate field shear strength of discontinuity. They performed laboratory direct shear tests in order to get dilation-corrected friction angles; the stress measured in the horizontal and vertical plane is resolved tangentially and normally to the plane along which shearing is actually taking place. These corrected stresses may then be plotted to give a strength envelope reflecting the shear strength of non-dilating, naturally textural surfaces. After determining a dilation-corrected friction angle, the angle, i of large-scale irregularity can be added to it to represent field shear

strength. Richards and Cowland [28] successfully applied this procedure to slope stability design in Hong Kong. 3.2. Evaluation of peak shear displacement in the field Barton and Bandis [13], after reviewing a large number of shear tests for peak shear displacements reported in the literature (about 650 data points), proposed the following empirical relation for estimation of the peak shear displacement in the field.   L JRC 0:33 δp ¼ ½m ð9Þ 500 L where δp is a slip magnitude required to mobilize peak strength, or that occurring during unloading in an earthquake and L is a length of a joint or a faulted block in meter. Removing difficulty in selecting the appropriate value of JRC for in situ block, Eq. (9) is modified in the form, expressed with the ratio of block size and the angle of inclination of the large-scale

J. Oh et al. / International Journal of Rock Mechanics & Mining Sciences 76 (2015) 78–87

83

Model No. 1 from fitting to laboratory test

60

1.5

normal displacement (mm)

Sample size = 6 cm

shear stress (kPa)

50 40 30 Sample size = 36 cm

20

from Eq. (11) 10 0

Sample size = 6 cm 1.0 0.8

from Eq. (15)

0.6

from Eq. (17)

0.4

Sample size = 36 cm

0.2 0

0

5 2 3 6 4 shear displacement (mm)

1

7

8

Simulated

0

1

2

5 3 6 4 shear displacement (mm)

7

8

Experimental

Fig. 8. Comparison between model simulations and laboratory experimental results—Category I; experimental data from [9]. (a) shear stress-displacement curves and normal-shear displacement curves, (b) profiles of surface geometries of joint, (c) sample size; [M] ¼ model, [P] ¼prototype and , (d) input parameters used in simulations.

waviness, i.  0:11 δp;n Ln ¼ i0:33 L0 δp;0  0:33  0:67

δp;n Ln ¼ L0 δp;0

ð10Þ

i0:33

ð11Þ

where δp,0 and L0 are the laboratory-scale peak shear displacement and joint length and δp,n and Ln are the field-scale peak shear displacement and joint length, respectively. Eq. (10) is derived by correlating with Bandis’s test results [9] as shown in Fig. 3. The values of δp in the figure are for largest samples (36 or 40 cm). Analyses are performed by considering 36 or 40 cm samples as models of the field-scale, on the other hand,

smallest samples (5 or 6 cm) as models of the lab-scale. At first sight Eq. (10) appears sound. It provides a good agreement with the peak displacements estimated by Bandis [9] and results in the similar estimation with Eq. (9) as shown in Fig. 3. However, the value of exponent, 0.11 in Eq. (10) seems to be small. In Barton and Bandis’s analyses [9,13], the peak displacements (δp) for largest sample or in situ block were determined from the displacement at peak dilation, not at peak strength. Peak strength is often mobilized later than peak dilation for large blocks as clearly shown in Bandis’s test results [9]. Therefore, Eq. (10) could result in smaller value of the peak displacement for rock joints in the field (increase in δp by a factor of 1.66 for the sample ratio of 100). Eq. (11) thus uses a larger value of exponent, 0.33 to 0.67 for the size ratio and provides more reasonable value of the peak displacement that corresponds to peak strength of in situ block.

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J. Oh et al. / International Journal of Rock Mechanics & Mining Sciences 76 (2015) 78–87

Model No. 7 70

normal displacement (mm)

shear stress (kPa)

60

Sample size = 5 cm

50 40 30 Sample size = 40 cm

20

from Eq. (11)

10

Sample size = 5 cm

0.6

from fitting to laboratory test

from Eq. (15)

0.5 0.4 0.3

Sample size = 40 cm

0.2 from Eq. (17)

0.1 0

0 0

1

5 2 3 6 4 shear displacement (mm)

7

0

8

1

5 2 6 3 4 shear displacement (mm)

7

8

Experimental

Simulated

Fig. 9. Comparison between model simulations and laboratory experimental results—Category II; experimental data from [9]. (a) shear stress-displacement curves and normal-shear displacement curves, (b) profiles of surface geometries of joint, (c) sample size; [M] ¼ model, [P] ¼ prototype and , (d) input parameters used in simulations.

3.3. Degradation in dilation and post-peak strength along large-scale irregularity

Function (fs): " s

f ¼ Field observations indicate that the large-scale irregularities are little sheared through and the strength component of the largescale irregularities will not be lost with small shearing displacements. Also, they have low angle of inclinations (usually less than 101) and sinusoidal shapes. Based on these observations, the following assumptions were made to model the degradation in dilation and post-peak strength along a large-scale irregularity. First, the angle of inclination, i will contribute to dilation as well as strength component, second, the degradation of a dilation angle is determined from sine curve, and third, a dilation angle will be zero (i.e. residual strength) after a shear displacement is a half of wavelength of the large-scale irregularity. The sine curve is expressed in the following equation [29] and is called Sinusoidal

π

4

þ tan  1  sin π

!

δps π  2 0:5 U λlarge

!!#

π

=

2

ð12Þ

where λlarge is the wavelength of large-scale irregularity observed in the field. Fig. 4 depicts the normalized Sinusoidal Function and the Sinusoidal Function (fs) is used to obtain the mobilized dilation angle from large-scale irregularity as in Eq. (13) dn;mob ¼ i0 f

s

ð13Þ

where i0 is an initial angle of inclination of a large-scale irregularity. 3.4. Summary of the proposed joint models The proposed joint model has been developed by modifying some parameters of the existing models or adding new features to

J. Oh et al. / International Journal of Rock Mechanics & Mining Sciences 76 (2015) 78–87

85

Model No. 10

0.25

40

from fitting to laboratory test

normal displacement (mm)

Sample size = 6 cm

Sample size = 6 cm

shear stress (kPa)

30

20 Sample size = 36 cm from Eq. (11) 10

0.2 from Eq. (15) 0.15

Sample size = 36 cm

0.1 from Eq. (17)

0.05

0

0 0

1

2

6 3 5 4 shear displacement (mm)

7

8

0

1

2

3

4

5

6

7

8

shear displacement (mm)

Simulated

Experimental

Fig. 10. Comparison between model simulations and laboratory experimental results—Category III; experimental data from [9]. (a) shear stress-displacement curves and normal-shear displacement curves, (b) profiles of surface geometries of joint, (c) sample size; [M] ¼ model, [P] ¼ prototype and, (d) input parameters used in simulations.

the existing models. This section thus describes what has been adopted in the proposed model from the previous work first and then what is proposed to the model in the current study. The complete form of the proposed joint model for typical laboratoryscale samples is expressed in the following equation:   τ ¼ σ n tan ϕr þ fα0 expð  cW p Þg ð14Þ This model was originally suggested by Plesha [4] as mentioned in Section 2.2. The mobilized dilation is also calculated from the mobilized asperities of small scale. By combining Eqs. (6) and (7), the formulation is given by  Δδv ¼ Δδps tan Dα0 expð  cW p Þ ð15Þ This equation is the result from a combination of the asperity degradation model by Plesha [4] and the asperity damage coefficient by Barton and Choubey [8].

In the current work, the asperity degradation constant, c ¼ kðα0 =λσ c Þ, Eq. (5) is proposed. As shown in Eqs. (14) and (15), by introducing the new asperity degradation constant, the dimensionless product of plastic work, rock strength, and wavelength of irregularities has been developed. The complete form of the joint model for field-scale asperities is given by    τ ¼ σ n tan ϕr þ αn expð  cW p Þ þ i0 f s ð16Þ The mobilized dilation is calculated from the mobilized asperities of both small-scale and large-scale irregularities as  Δδv ¼ Δδps tan Dαn expð  cW p Þ þ i0 f s ð17Þ In this study, two new parameters are proposed to represent the shear strength and dilation behavior of a rock joint in the field. αn is a field-scale small roughness obtained by laboratory shear test (e.g., 0.6  (ϕp,mob  ϕr)) and fs is a sinusoidal function, expressed in Eq. (12). Finally, the study proposes an approach to

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scaling the peak shear displacement from laboratory to field-scale as in Eq. (11).

4. Correlation with experimental data In this section, the verification of the model is illustrated by providing its correlation with experimental results taken from literature. The simulation of direct shear test, which consists of a single horizontal joint that is first subjected to a normal confining stress and then to a unidirectional shear displacement, is performed using the three-dimensional distinct element code, 3DEC [18]. The model is incorporated in 3DEC using the built-in programming language, FISH. The average normal and shear stresses and normal and shear displacements along the joint are measured using FISH. The joint is defined by one contact that is composed of ten sub-contacts. Fig. 5 shows the direct shear test model and boundary conditions. The first example shows the model’s performance on the typical laboratory sample. In the second example, correlation is given for an artificial joint with a single tooth-shaped asperity. Finally, examples show the model’s performance on both smallscale and large-scale joints. 4.1. Simulation of Flamand et al.’s test Flamand et al. [19] conducted direct shear tests with identical replicas of a natural fracture in granite, modeled from the original sample of a fracture in the Gueret Granite (France), drilled perpendicular to the fracture plane under three different normal stresses (σn) applied on the shear plane. The samples have a circular section with a shear surface of 90 mm in diameter. The characteristics of the mortar used are: σc ¼ 82 MPa, E ¼32.2 GPa, and basic friction angle (ϕb) is 371. Joint profiles recorded parallel to the shear direction with a constant step, 0.5 mm are presented in Fig. 6. Fig. 6 shows a comparison between simulation and experimental test results under constant normal stress condition. Based on joint profile shown in Fig. 6(c), the average value of wavelength (λ) is determined. Fig. 6(d) presents input parameters used in simulations. Since all material parameters were not provided in their study, some of them are estimated on the basis of experimental results. Based on the slopes of a straight line from the origin to the end of the elastic region of experimental results shown in Fig. 6(a), a joint shear stiffness (Ks) is estimated to be Ks ¼ 63 GPa/m. Without sufficient information, a joint normal stiffness (Kn) is assumed to be Kn ¼315 GPa/m. Based on the joint profile shown in Fig. 6(c) and the back-calculation from test results, an initial asperity angle (α0) is estimated to be 221 (in [19] an initial asperity angle (α0) to it was estimated to be 15–301). As shown in Fig. 6, the agreement between simulation and experimental result is very good. 4.2. Simulation of Yang and Chiang’s test An experimental study was done by Yang and Chiang [30] on the progressive shear behavior of rock joints with tooth-shaped asperities. For a basic understanding of shear behavior, direct shear tests were performed for the artificial joint with a single tooth-shaped asperity. The model material consisted of plaster and water mixed by weight ratios of 1: 0.6 and its unconfined compressive strength (σc) was 7.36 MPa and the basic friction angle (ϕb) was about 351. The single tooth-shaped asperity has a base length of 1.7 cm and an inclined angle of 301. With no data available, the elastic modulus (E) was assumed to be 2208 MPa, based on Deere’s classification; for most rocks, the

ratio E/σc lies in the range from 200 to 500 and averages 300 [31]. Joint shear stiffness (Ks) was approximated to be 1300 MPa/m from test result by taking the slope from the origin to the end of the elastic region and a joint normal stiffness (Kn) is again assumed to be 21 GPa/m. Fig. 7 shows a comparison between model simulation and experimental result. Unlike natural joint surface, the test specimen has only one single triangular tooth-shaped asperity with large wavelength (λ ¼1.7 cm), compared to typical laboratory samples and the model still provides a reasonable correlation. 4.3. Simulation of Bandis’s test Bandis [9] conducted systematic studies for the effect of scale on the shear behavior of rock joints by performing direct shear tests on different sized replicas cast from various natural joint surfaces. Direct shear tests were performed under the constant normal stress of 24.5 kPa. The model joint compressive strength (JCS ¼ σc) was set at 2 MPa for all the types of surfaces tested. The E/σc ratio value was about 400 for all joint block sizes. Basic friction angle (ϕb) of joint surfaces was about 321. Description of joint surface according to visual appearance can be classed as belonging to one of three categories: category I—strongly undulating, rough to moderately rough; category II—moderately undulating, very rough; category III—moderately undulating to almost planar, moderately rough to almost smooth. Typical examples of joint profile of each category will be shown with experimental and simulation results. Figs. 8–10 show the performance of the joint model for scale effect on shear behavior of rock joints as well as the comparison with experimental results for each category. One of the main purposes of this study is, as previously mentioned, to propose a method of evaluating joint shear behavior in the field from laboratory test. To evaluate this, the largest (36 or 40 cm) samples are considered as a model of the field-scale and smallest (5 or 6 cm) samples are considered as a model of the lab-scale. Input parameters used in simulations and joint profiles for each category are shown in Figs. 8–10. Each joint profile was constructed on the basis of a 0.7 mm sampling interval. Irregularities (α0 and i0) and wavelength (λ and λlarge) for both small (5 or 6 cm) and large (36 or 40 cm) samples were estimated by visual measurements and obtaining the average values from a set of three surface profiles for each model. Figs. 8–10 also show the characteristics of joint asperities and wavelengths at different scales. Joint shear stiffness (Ks) of the small samples was back calculated from the secant line of the elastic region of the shear stress–displacement curve obtained from tests. For lack of information on a joint normal stiffness (Kn), it is assumed to be about 15  Ks. Since both experimental tests and simulations are performed under the constant-normal-stress boundary condition, the results of shear behavior are independent of the particular value chosen for Kn. The peak displacement (δp) for the lab-scale was determined from fitting to the lab test. Using that value (δp for labscale) and Eq. (11) with exponent, 0.33, the peak displacement (δp) for the field-scale was determined. The value of 0.6(ϕp,mob  ϕr) obtained from the laboratory test of the lab-scale sample was used to obtain the value of the field-scale small roughness (αn). Dilation of both lab-scale and field-scale samples is modeled using Eqs. (15) and (17) respectively, where D¼ 0.4 is used. As shown in the correlation with experimental results for both small and large samples, there are very good agreements between simulation and experimental results regarding the shear stressshear displacement relation of the three different cases but less agreement for dilatancy relation. This is partly due to the difficulty in estimating material parameters, especially the joint normal stiffness (Kn) as mentioned above. The main reason lies in the limitation of numerical modelling and can be explained as follows.

J. Oh et al. / International Journal of Rock Mechanics & Mining Sciences 76 (2015) 78–87

The built-in programming language, FISH has been used to incorporate the developed model into 3DEC. For this, FISH modifies the built-in joint model, Mohr–Coulomb which is an elastoperfectly plastic constitutive model. This joint model allows the dilation to take place at the peak shear displacement, which is quite different characteristic of dilation behaviour from the developed model as explained in the previous chapter. Therefore, the discrepancy results from when the joint starts to dilate. Nevertheless, the joint model reasonably well represents shear stress– displacement–dilation relation of various natural joint surfaces and scales. The simulation results also support that the proposed approach to evaluating rock joint behavior in the field from laboratory data and field observations, is reasonably reliable.

5. Discussion and conclusions The most important feature of the model is that it considers the dilation and strength along both small-scale and large-scale irregularities. The study also includes how the scale effect is employed in the model. In literature there is considerable discussion on the scale effect of rock joint behavior and the extrapolation of the shear strength of lab-scale joints to large-scale joints in the field. Many studies show that there is a reduction in the joint shear strength with increasing length of the joint (i.e., [9,10,12], but the opposite behavior has also been observed (i.e., [32,33]). Another investigation reported that the scale-dependency is limited to a certain size and for larger than the stationarity limit the joint strength remains almost constant (i.e., [17]). Although the problem of scale effect is very complex and under debate, it seems to be true that changing the scale involves a variation of the asperity size, which contributes to the change of its mechanical behavior. It is also true that the evaluation of strength of large-scale joint requires assessment of field-scale irregularity determined from geologic observations. It is often observed in the field that as the length of joint increases, the size of asperity controlling its strength increases. This controlling asperity has a relatively gentle slope and a large wavelength compared to that of a small-scale joint, thereby reducing its strength and brittle behavior. This paper presents a rock joint constitutive model that represents the behavior of both a small-scale joint in the laboratory and a large-scale joint in the field. A dimensionless product of plastic work, rock strength, and wavelength of irregularities has been developed to represent the degradation in strength and dilation along small-scale irregularities. On the other hand, a sinusoidal function is used to model the degradation in strength and dilation along large-scale irregularities. The comparisons of the model with experimental data taken from literature show that the proposed model has very good overall agreement, although there is some discrepancy attributed to a lack of information on some parameters. The model can be incorporated into discrete element computer codes using a program language in order to solve boundary value problems. Although some simplifications are involved in the modeling, especially related to the scale effect of rock joints, the model predicts reasonably well the shear behavior of both lab-scale and field-scale joints.

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