Effects of Displacement Velocity on Rock Fracture Shear Strengths under Large Confinements

Triaxial shear tests are performed to assess the effects of displacement velocity and confining pressure on shear strengths and dilations of tension-induced fractures and smooth saw-cut surfaces prepared in granite, sandstone and marl specimens. A polyaxial load frame is used to apply confining pressures between 1 and 18 MPa with displacement velocities ranging from 1.15×10 to 1.15×10 mm/s. The results indicate that the shearing resistances of smooth saw-cut surfaces tend to be independent of the displacement velocity and confining pressure. Under each confinement the peak and residual shear strengths and dilation rates of rough fractures increase with displacement velocities. The sheared-off areas increase when the confining pressure increases, and the displacement rate decreases. The velocity-dependent shear strengths tend to act more under high confining pressures for the rough fractures in strong rock (granite) than for the smoother fractures in weaker rocks (sandstone and marl). An empirical criterion that explicitly incorporates the effects of shear velocity is proposed to describe the peak and residual shear strengths. The criterion fits well to the test results for the three tested rocks.


Introduction
Understanding of the frictional behavior of rock fractures is important for the prediction of natural geologic hazards (e.g., fault movements and landslides), and for the stability evaluation of geo-engineering structures (e.g., tunnels, mine openings, foundations and waste repositories).The frictional resistance of rock fracture is conventionally determined by the direct shear testing (e.g., ASTM D5607-08) which provides the peak and residual fracture shear strengths under constant normal load (CNL) or under constant normal stiffness (CNS) (Usefzadeh et al., 2013;Shrivastava & Rao, 2015).The direct shear test configurations have some disadvantages.The magnitudes of the applied normal stress are limited by the uniaxial compressive strength of the rock, and the fractures are sheared under unconfined conditions.Their results may not truly represent the friction behavior and movement of the fractures at great depth where the fractures are under high normal stresses and confinements.The triaxial shear test method (Barton, 1976;Brady & Brown, 2006;Jaeger et al., 2007;Li et al., 2012) has therefore been developed to simulate the frictional resistance of rock fractures under confinements.The cylindrical rock core containing an inclined fracture or weakness plane can be axially loaded in a triaxial pressure cell with a wide range of applied confining pressures.The normal stress at which the shear strengths are measured can be controlled by the applied axial stress and confining pressures.Barton (1976) states that under higher confining pressures the behavior of rock ceases to be brittle as the brittle-ductile transition is reached.The Mohr envelopes describing failure of intact rock eventually reach a point of no gradient on crossing a certain line, defined as the critical state line.This critical state is associated with a critical effective confining pressure for each rock.It appears that the dilation normally associated with the shearing of non-planar joints and faults may be completely suppressed if the applied stress reaches the level of the critical effective confining pressure.The triaxial shear test results obtained by Li et al. (2012) suggest that the back-predicted joint compressive strength (JCS) from triaxial shear tests under high confining pressures is larger than the rock uniaxial compressive strength which might be attributed to the size effect.Barton (1976) and Barton & Chouby (1977) propose a shear strength criterion for the fractures under high confining stresses.They replace the joint compressive strength (JCS) parameter by the difference between the maximum and minimum principal stresses at failure of intact rock (σ 1σ 3 ).The criterion agrees well with the shear test results of fractures in Weber sandstone.Results from true triaxial shear testing obtained by Kapang et al. (2013) also suggest that the confining stress parallel to the strike of the fracture plane can also cause localized stress concentration and dilation of the asperities on the fracture wall, and hence weaken the rock asperities and reduce the peak shear strengths of the fracture.
Different displacement velocities may impose different behavior to the shearing resistance of rock fractures.Crawford & Curran (1981) state that the frictional resistance of rock joints depends on the rate of shear displacement.For hard rocks, the resistance decreases with increasing shear displacement rates greater than a variable critical velocity.For soft rocks, the resistance increases with increasing shear velocity, up to a critical shear displacement rate, and thereafter remains essentially constant.Laboratory test results by Curran and Leong (1983) suggest that the fracture shear strengths are sensitive to a certain range of shear velocities.Above and below this range the fracture shear strengths are independent of the shear velocity.Li et al. (2012) perform cyclic shear tests (with shear velocities ranging from 0.5 to 50 mm/minute) on artificial rock fractures and conclude that the residual shear strength increases with displacement velocity.The effect of the shear velocity on the peak stress remains inconclusive due to the complex variability of the asperities.
The objective of this study is to develop a fracture shear strength criterion that can explicitly incorporate the effects of confining pressure and displacement velocity.The task involves performing triaxial shear tests to obtain the strengths and dilations of rock fractures under confining pressures up to 18 MPa.Tension-induced fractures and smooth saw-cut surfaces are prepared in granite, marl and sandstone specimens.Under each confinement the fractures are sheared with constant displacement velocities ranging from 1.15×10 -2 to 1.15×10 -5 mm/s.The joint roughness coefficients (JRC) are determined prior to and after shearing.
The basic mechanical properties of the rocks are determined by performing uniaxial and triaxial compressive strength tests.The test procedure and calculation methods are in accordance with the ASTM (D7012-07) standard practice.Based on the classification suggested by the International Society for Rock Mechanics (ISRM - Brown, 1981) the granite, sandstone and marl are classified as very strong, medium strong and strong rocks.Table 1 summarizes the results of the mechanical testing.
For the triaxial shear testing the specimens are prepared to obtain rectangular blocks with nominal dimensions of 50×50×87 mm 3 .Two types of fractures are artificially made in the laboratory: tension-induced fractures and smooth saw-cut surfaces.They have nominal areas of 50×100 mm 2 .The normal to the fracture plane makes an angle (β) of 59.1° with the specimen vertical axis.The tension-induced fractures are clean and well mated.The asperity amplitudes are measured from the laser-scanned profiles along the shear direction.The readings are made to the nearest 0.01 mm. Figure 1 shows examples of laser scanned images for the three rock types.The maximum amplitudes are used to determine the joint roughness coefficients (JRC) of each fracture based on the Barton's chart (Barton, 1982).Each rock type tends to show a consistent degree of fracture roughness.The means and standard deviations of the JRC's are 13±1.0,10±0.5 and 11±0.5 for the granite, sandstone and marl, respectively.All fractures show both first and second order asperities.
It is recognized that there have been several systems and approaches proposed to describe the rock fracture roughness (e.g., Lee et al., 1990;Odling, 1994;Belem et al., 2000;Tang et al., 2012).The JRC is used here primarily because it is relatively simple and quick to determine from the two-dimensional fracture profile along the shear direction.It has been widely applied in the analyses of fracture and fault shear strengths.

Test Apparatus and Method
A polyaxial load frame (Fuenkajorn & Kenkhunthod, 2010) is used to apply constant and uniform lateral stresses (confining pressures, σ 2= σ 3 ) and vertical (axialσ 1 ) stress to the block specimen.Figure 2 shows the directions of the applied stresses with respect to fracture orientation.The confining pressures are maintained constant at 1, 3, 7, 12 and 18 MPa for tension-induced fractures, and at 1, 7 and 12 MPa for smooth saw-cut surfaces.They are applied by using four cantilever beams arranged in mutually perpendicular directions on the polyaxial load frame.The axial stress is applied under constant displacement rates varying from 10 -5 to 10 -2 mm/s using a 100-ton hydraulic cylinder.During loading the pressure of the hydraulic cylinder is monitored to calculate the variations of the axial stress.The lateral displacements of the fracture are obtained by monitoring the vertical movement of the cantilever beams.
The specimen is first loaded under hydrostatic stress equivalent to the selected confining pressure.Neoprene sheets are used to minimize the friction at the interfaces between the loading platen and the rock surface.The peak and residual shear strengths are recorded.The test is terminated when a total vertical displacement of 5 mm is reached.After shearing the fractures are examined, laser-scanned and photographed.The frictional resistance at the interfaces between the loading platens and the lateral neoprene sheets are determined by vertically loading an intact specimen with the same dimensions while the constant lateral stresses parallel to the fracture are applied.The results are shown in Figure 3.They are used to correct the magnitude of the axial stress (σ 1 ) under its corresponding lateral stress (σ 3 ) during the fracture shearing tests.

Results of Tension-Induced Fractures
The shear stress-displacement (τ-d s ) curves obtained under all displacement velocities are shown in Figure 4.For all rock types the peak stresses are reached within 2 mm of shearing.The residual stresses tend to remain constant up to 5 mm of the displacement.Under each confining stress (σ 3 ), the differences between the peak and residual stresses notably reduced when the fractures are subjected to lower shear velocities.These differences tend to be smaller for fractures with lower JRC values (marl and sandstone) as compared to those with higher JRC values (granite).The major principal stresses for the peak (σ 1,P ) and residual (σ 1,R ) increase with displacement velocities (Figure 5).
The velocity-dependent shear strength appears for all confining stresses.It tends to act more for the fractures in granite (with high JRC and strength) than in sandstone and marl (with lower JRC's and strengths).This is suggested by a larger separation of the major principal stresses under different displacement velocities found in granite as compared to those in sandstone and marl.Using Equations ( 1) and ( 2) the peak and residual shear strengths and their corresponding normal stresses can be calculated.The shear and normal (dilation) displacements of the tested fractures can be calculated using Equations ( 3) and (4).Figure 6 shows the results for σ 3 = 1 and 18 MPa in the forms of d n -d s diagrams.The dilations measured prior to and after the peak stresses notably decrease with the displacement velocities.The dilations tend to increase with the displacement until the peak stresses are reached, and remain constant in the residual region.This agrees reasonably with the constant residual shear stresses observed during the test, implying that the fracture roughness remains relatively unchanged within the residual region.Larger confining pressures also induce lower fracture dilations.Granite with high strength and high JRC fractures tends to show larger dilations than do the lower strengths and lower JRC's sandstone and marl, particularly under low confining stresses.This is probably because the joint walls tend to climb over the asperities under low confining stresses while they tend to be sheared off under high confining stresses.This behavior can also be observed from the increase of dilation rates (d n /d s ) measured before the peak stresses, as shown in Figure 7.The diagrams show that the dilation rates also increase with the displacement velocity.Table 2 shows examples of some post-test fractures obtained under the highest and lowest displacement velocities and confining stresses for the three rock types.The light areas on the fracture surfaces represent the sheared-off asperities.The increase of confining stresses and the decrease of displacement velocities enhance the sheared-off areas.The scanning images of the post-test fractures show that the second order asperities are sheared-off for all tested rocks.The JRC's of the sheared fractures are determined from the laser scanning profile along the shear direction.Figure 8 plots the post-test JRC's as a function of σ 3 .They support the previous observations that higher confining stresses and lower displacement velocities induce larger sheared-off areas, and result in a smoother fracture surface (lower JRC value).

Results of Smooth Saw-Cut Surfaces
The shear strengths of smooth saw-cut surfaces are determined under the confining stresses between 1 and 12 MPa with displacement velocities from 1.15×10 -4 , 1.15×10 -3 to 1.15×10 -2 mm/s.The test procedure and calculation method are identical to those of the tension-induced fractures.The Coulomb criterion is applied to represent the peak shear strengths under various shear velocities and confinements: where φ and c are friction angle and cohesion of the saw-cut surfaces.The shear stress (τ) and normal stress (σ n ) are calculated from the major principal (σ 1 ) and confining (σ 3 ) stresses, using Equations ( 1) and ( 2).The criterion is fitted to the test results in the forms of τ−σ n diagrams as shown in Figure 9. Good correlations (R 2 ≥ 0.9) are obtained.The friction angles for granite, sandstone and marl are determined as 30°, 33° and 31°, and the cohesions are 1.36, 1.67 and 1.28 MPa, respectively.For all rock types the shearing resistances for the smooth saw-cut surfaces tend to be independent of the displacement velocity and confining stress.

Empirical Criterion
The empirical shear strength criterion for fractures under high confining pressures proposed by Barton (1976) and Barton & Choubey (1977) can not be applied to the test results obtained here because as the confining stresses increase the τ-σ n relations for the three rock types tend to be non-linear.An alternative empirical criterion is proposed to represent the fracture shear strengths as a function of normal stress, as follows: where α and λ are empirical parameters, depending on rock types.Regression analyses are performed on Equation (7) using the peak and residual shear strength data given in Table 3. Good correlations are obtained (R 2 ≥ 0.9).The parameters α and λ determined for each shear velocity are summarized in Table 4.The parameter λ tends to be independent of the shear velocity.It probably relates to the fracture roughness.This suggests by the fact that for each rock type λ for residual shear strength is higher than that of the peak shear strength (Table 4).
For smooth saw-cut surface λ would be equal to 1.0.The parameter α increases with shear velocity ( s d

•
), which can be best represented by: where η and ω are empirical constants.Their numerical values are given in Figure 10.For smooth saw-cut surface (λ = 1.0) α is equal to tan φ b , and is independent of the shear velocity.This means that α is also dependent of fracture roughness, as evidenced by that α's for the peak shear strengths are higher than those for the residual shear strengths.

Discussions and conclusions
The angle β which is maintained constant at 59.1° limits the lower ends of the τ-σ n curves for the peak and residual shear strengths (Figure 12).This angle is selected primarily because it gives the length-to-width ratio of the block specimens of about 2.0 which is most suitable for the polyaxial loading device.Different angles will provide different magnitudes of σ 1 and σ 3 that correspond to the same peak and residual shear strengths.As a result the fracture shear strengths obtained from the triaxial shear testing would be independent of this angle.
For all rock types the fracture dilations measured prior to and after the peak strengths significantly decrease with increasing confining pressures and decreasing displacement velocities (Figure 7).This is supported by the visual observations and the JRC measurements of the post-test fractures that the reduction of the shear velocity notably increases the sheared-off areas, particularly when the fractures are under high confining pressures (Table 2).
A quantitative assessment of the relations between the fracture roughness and the displacement velocity cannot be made.Only a narrow range of JRC's is obtained from the tension-inducing method for each rock type.Nevertheless, the granite with rougher fractures tends to show larger time-dependent effect than those obtained from the smoother fractures in sandstone and marl.This is also evidenced by the fact that for all rock types the shear strengths of smooth saw-cut surfaces are independent of the displacement velocity.
The velocity-dependent shear strengths can be observed more clearly in stronger granite than in the softer sandstone and marl (Figure 5).This agrees with the conclusions drawn by Crawford & Curran (1981).The velocity-dependent strengths are probably governed by the time-dependent propagation of fissures and micro-cracks of rock forming minerals on the fracture wall.Each rock type may be sensitive to a certain range of the displacement velocities.The range of the displacement velocities used in this study (1.15×10 -5 -1.15×10 -2 mm/s) may reflect the time-dependent shear strengths of the granite fractures better than those of the sandstone and marl fractures.The time-dependent shear strength of fractures in sandstone and marl would be more clearly observed if a higher range of displacement velocities was used (i.e., higher than 10 -2 mm/s).
The fracture areas tested here (50×100 mm 2 ) are relatively small even though they comply with the ASTM standard practice and ISRM suggested method.Barton & Bandis (1980), Fardin et al. (2001), Ohnaka (2003), and Johansson & Stille (2014) conclude from their experimental and observational results that fracture shear strengths decrease with increasing scale.As a result the shear strengths obtained here would likely overestimate those of the larger fractures under in-situ condition.
The triaxial shear testing has clear advantages over the direct shear testing.It allows simulating the shearing resistances of fractures under significantly larger confining pressures and normal stresses than those of the direct shear test where it is limited by the unconfined compressive strength of the rock.The triaxial test configurations and loading paths would be similar to those of the in-situ conditions where σ 1 that induces relative displacement is mostly not parallel to the fracture or fault planes.As a result a constitutive law and model describing fracture shear strengths and dilations under triaxial stress states, such as the one proposed in this study, would likely represent the shearing behavior of in-situ fractures or faults at great depths better than that obtained under low normal stresses and unconfined conditions.

Figure 1 .
Figure 1.Some fracture images obtained from laser scanning for granite (a), sandstone (b) and marl (c)

Figure 2 .
Figure 2. Directions of applied stresses by polyaxial load frame, with respect to the fracture orientation

Figures 4 .
Figures 4. Shear stresses (τ) as a function of shear displacement (d s ) for fractures in granite (a), sandstone (b) and marl (c)

Figure 7 .
Figure 7. Dilation rates (d n /d s ) as a function of the shear velocity ( s d • ) for granite (a), sandstone (b) and marl (c)

Figure 9 .
Figure 9. Shear strengths of smooth saw-cut surfaces in granite (a), sandstone (b) and marl (d)

Figures 10 .
Figures 10.Parameter α as a function of shear velocity

Figures 11 .
Figures 11.Proposed empirical criterion for peak and residual shear strengths compared with test data for granite (a, b), sandstone (c, d) and marl (e, f)

Table 1 .
Mechanical properties of rock samples