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Numerical Simulation of Size Effect of Laminated Rock
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ScienceDirect Procedia Engineering 191 (2017) 984 – 991
Symposium of the International Society for Rock Mechanics
Numerical Simulation of Size Effect of Laminated Rock Yuting Xue, Brijes Mishra* Mining Engineering Department, West Virginia University, Morgantown, WV 26505, USA
Abstract
Roof and rib falls are still a leading cause of injuries in underground coal mines. When numerical simulation method is adopted to study this phenomenon, the influence of regular laminations on strength and size effect should be carefully considered because rock mass of roof is usually laminated. In order to get an insight into the influence of different lamination factors on size effect, cubic rock specimens with different sizes, stress directions, lamination orientations and lamination spacing were tested under true tri-axial conditions with 3DEC. The results show that regular lamination can cause size effect even without considering size effect of the intact rock and rock joints. The influence of lamination orientation on size effect is distinct and is orientation-dependent while the influence of lamination thickness is minimal. Changing dip or stress direction will affect the influence of orientation and size effect as well. © Published by Elsevier Ltd. This © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017. Peer-review under responsibility of the organizing committee of EUROCK 2017 Keywords: Size effect; laminated rock; orientation; spacing; stress direction
1. Introduction Improvements in roof control technology have led to significant decrease in accidents related to roof and rib falls. This brought the number of injury from roof and rib falls from 377 in 2012 down to 265 in 2013 [1]. But roof fall is still a leading cause of injuries in underground coal mines. And numerical simulation method has been widely used to investigate the mechanism and prevention of such problems [2–4]. Proper rock mechanics property, however, is crucial to obtain practical results in numerical simulation. Size effect is an important factor when laboratory-determined strength value is used to determine rock mass strength value for simulation.
* Corresponding author. Tel.: +1-304-293-3872. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017
doi:10.1016/j.proeng.2017.05.270
Yuting Xue and Brijes Mishra / Procedia Engineering 191 (2017) 984 – 991
985
Due to the importance of size effect, studies have been conducted to investigate the relationship between rock strength and their size. Bieniawski [5] conducted extensive underground tests on cubical coal specimens ranging from 0.75 inches to 6.6 ft in size. Singh and Huck [6] performed tri-axial compression tests on specimens with diameters ranging from 2 to 36 inches and approximate 50 percent strength reduction was observed. Medhurst and Brown [7] performed tri-axial compression tests on coal specimens with diameters of 61, 101, 146 and 300 mm to investigate size effect on mechanical behaviour of coal. Besides, numerical simulation was widely used to investigate size effect. Adey and Pusch [8] simulated size effect with boundary element method by considering larger discontinuities with increasing specimen volume. The discrete element program Ca2 was used [9] to simulate size effect and the result shows that initial crack length must be increased faster than specimen size to capture size effect. Scholtès et al. [10] studied size effect of coal by simulating true tri-axial compression tests with a discrete element model, which shows the capability to reproduce the dependency of strength on specimen size. Zhang et al. [11] investigated size effect with bonded particle model, where the initial fractures were considered as randomly orientated and located disks.
Fig. 1. Laminated roof condition.
From these literatures, we can see there are still two problems. First, simulations of size effect mostly focus on randomly distributed defects. Rock mass, especially shale formation in roof of coal mine openings, is usually laminated. As failure of such rock mass mostly occurs in the form of sliding along lamination, especially under low confining stress conditions, deformability and strength of laminated rock mass are strongly controlled by these laminations. For size effect, specimens with various sizes will have different numbers or different lengths of lamination, which will affect stress distribution in specimens and thus affect the strength. Second, for a specific geological condition, layouts of the underground excavation will lead to different stress directions acting on the laminated rock mass. And failure along laminations is very sensitive to stress direction. Therefore, it is crucial to study size effect under true tri-axial condition. However, laboratory and numerical tests are focused on uniaxial compression test and true tri-axial tests for size effects are quite rare in literatures. The reason is that true tri-axial test machine is difficult to design due to the interaction between loading platens and friction effects between platens and specimens [12, 13]. However, numerical simulation, as an alternative, overcomes some of these difficulties. In this paper, 3DEC was used to simulate cubic rock specimens under different conditions for the purpose of analysing their influence on size effect of rock. The scheme for the numerical experiments was first introduced, which included the method for simulation of true tri-axial test and all the conditions considered in the experiment. A series of simulations were then conducted to investigate the influence of different factors on strength and size effect.
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2. Numerical simulation scheme
Fig. 2. Simulated situation.
Various size effects, including size effect of intact rock, size effect of laminations or discontinuities and size effect caused by laminations, are involved in the study on size effect of laminated rock. Simulation of size effect of intact rock is usually achieved by including random defects into rock specimens; size effect of discontinuities can be simulated with asperities. However, these are excluded in the current study. The main purpose is to investigate the third aspect. Different sizes of specimen will have different numbers of laminations, as shown in Fig. 2, which will not only provide potential failure planes, but also affect the stress distribution in the whole specimens. The rock between laminations is treated as intact rock and laminations are treated as planes without asperity. Configuration of the numerical simulation is shown in Fig. 3 [14]. Cubic specimens with laminations are placed in centre. Boundary stress is applied to the external surface of the four platens to provide confining pressure on the specimens. The bottom of specimen is fixed in Z-direction and a velocity of 0.001 m/s is applied from the specimen’s top.
Fig. 3. Configuration of numerical simulation.
Intact rock is treated as an elastic/plastic material while platens are elastic. Drucker-Prager failure criterion is used to predict failure. Besides, laminations in the specimens are elastic/plastic with Coulomb slip failure criterion. The mechanical properties of intact rock and laminations are provided in Table 1 [15]. For the interfaces between specimens and platens, the friction angle, cohesion, tension and shear are set to zero to eliminate the friction effect. And lamination length, thickness and orientation parameters used in the numerical models are listed in Table 2.
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Table 1. Mechanical properties for the materials used in simulations.
Lamination
Interfaces between specimens and platens
Friction Angle (°)
11
0
Tension (MPa)
0.52
0
Cohesion (MPa)
1.00
0
Normal Stiffness (GPa)
12.9
50
Shear Stiffness (GPa)
7.10
0
Specimen Density (kg/m3)
2500
Young's Modulus (GPa)
16
Poison Ratio
0.38
Material Constant qϕ
0.35
Material Constant kϕ (MPa)
3.6
Table 2. Situations considered in numerical simulations.
Size-side length(m)
Lamination thickness (m)
Dip
Dip
Confining stress
direction (°)
angle (°)
(X: 5.79 MPa; Y: 3.45 MPa)
0.2
0.1
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.1
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.2
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.3
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.1
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.2
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.3
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.1
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.2
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.3
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.5
Y Z
n tio ec dir Dip
le ang Dip X
1.0
2.0
3. Results and discussion The simulation results of a laminated specimen is first analysed to demonstrate the influence of laminations on deformation, strength and mode of failure. And then the influence of factors-dip angle and dip direction of laminations and lamination thickness-on size effect is examined. The results of a 2.0 m specimen with 0.3 m-thickness laminations at 90° dip direction and various dip angles is selected as a case study in Fig. 4. First, it can be found that the laminations decrease specimens’ overall stiffness. In addition, the existence of laminations, more or less, decreases specimens’ strength. An exemption is the specimen with dip angle of 90° whose stress-strain curve overlaps with that of the specimen without laminations. This is because the specimen with dip angle of 90° can be considered as a stack of smaller intact cuboid specimens. Finally, strength reduction varies with dip angle or orientation. The strength reduction for specimens with 30°, 45° and 60° dip angles is more significant than those with dip angles of 15° and 75°, resulted from different failure modes associated with different dip angles. For the specimens with dip angles of 30°, 45° and 60°, failure usually occurs as sliding along laminations.
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Yuting Xue and Brijes Mishra / Procedia Engineering 191 (2017) 984 – 991
Fig. 4. Stress-strain curves for specimens with 2.0 m side length and with laminations of 90° dip direction, various dip angles and 0.3 m thickness.
The stress of the specimen with 45° dip angle directly drops to its residual value, demonstrating failure of the whole lamination plane. Failure of specimens with 30° and 60° dip angles is shown by an initial stress drop and then an increase in stress. This is caused by partial failure of the lamination planes. While failure of specimens of dip angles 15° and 75° occurs as gradual yielding, possibly due to yielding of intact rock. 3.1. The influence of lamination orientation on size effect
Fig. 5. Strength variations with 0.1 m thickness and various orientations and sizes.
The result for tests under true tri-axial condition where the dip direction is 0° is shown in Fig. 5. The strength of different specimens (σ) is normalized to the strength of intact specimens (σint). It should also be noted that no specimen with orientation or dip angle of 0° was simulated and the strength for 0° in Fig. 5 represents the result of intact specimens. Each curve in Fig. 5 shows a trend similar to previous results [16, 17] from conventional triaxial tests. Also the distances between the four points corresponding to four different sizes for each dip angle are different, which demonstrates that size effect varies with different orientations. This means size effect can be affected by lamination orientation. The data in Fig. 5 is then plotted in Fig. 6. The strength reduction for specimens with 30°, 45° and 60° dip angles is higher than that of the specimens with 15° and 75° dip angles which means that size effect for specimens with 30°, 45° and 60° dip angles is more significant than that with 15° and 75° dip angles. In Summary, size effect varies with orientation in laminated rock mass. Two variables are needed to quantify the relation between size effect and orientation, one to describe the significance of lamination influence on strength and the other to quantify size effect. Anisotropic Effect factor (Ae), introduced by Ghazvinian et al. [17] is adopted to quantify the first factor. It is defined as (1) where σia and σja are the compressive strengths of the intact and jointed rock mass, respectively, and a is the confining pressure. This equation shows that the value of Ae reflects the relative strength reduction caused
Yuting Xue and Brijes Mishra / Procedia Engineering 191 (2017) 984 – 991
by inclined laminations and can quantify orientation influence. Besides, size effect is used to describe strength reduction with increasing size. The difference in rock strength for specimens between 0.3 m and 2.0 m in length (or size) in our simulations can be treated as the consequence of size effect and their value can reflect the significance of size effect. Their relation is plotted in Fig. 7. It is clear that relative strength difference increases linearly with Anisotropic Effect factor, which means size effect is significant if Anisotropic Effect factor is large. In other words, if the rock strength is significantly decreased from intact rock strength by the laminations with a specific orientation, resulting in large Anisotropic Effect factor value, size effect should be profound for rock mass with laminations of same orientation.
Fig. 6. Strength variation with orientations and sizes.
Fig. 7. Relation between relative strength difference and Anisotropic Effect factor (Ae).
3.2. The influence of lamination thickness on size effect
Fig. 8. Strength-thickness relationship for specimens with various sizes and orientations.
Previous conventional tri-axial tests [13, 18] on cylindrical specimens with horizontal joints showed that strength decreases as the number of joint increases. Baecher and Einstein [13] interpreted that asperity on joint surface served as stress concentration loci, whose number increased in proportion to the number of joint. This asperity-related failure cannot be achieved in this study because laminations are treated as smooth planes. Instead, its influence is based on the fact that small lamination thickness can induce more lamination into the specimen, resulting in more potential failure planes. The relationship between lamination thickness and strength for specimens with 45° and 60° dip angles are shown in Fig. 8. For the same orientation, the strength of specimens reduces with the increase of specimen size and their relationships for different lamination thickness follow the same trend and are very close to each other. This indicates that the lamination thickness has limited influence on size effect. 3.3. The influence of dip or stress direction on strength and size effect Change in stress direction can affect strength of laminated rock because failure of laminated specimens is highly controlled by laminations and sliding along laminations is sensitive to confining stress [13]. In this study, the confining stress was kept constant and dip direction varied from 0° to 30°, 60° and 90°. In such a way,
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the influence of dip or stress direction on strength and size effect is investigated. For comparison purpose, the relation between strength and orientation for the specimens with 90° dip direction and different sizes is plotted in Fig. 9. On one hand, as the stresses in X and Y directions are 5.79 MPa and 3.45 MPa, respectively, varying dip direction from 0° to 90° would increase the confining stress to the lamination planes. By comparing the results plotted in Fig. 9 with Fig. 5, change of dip or stress direction definitely affects the strength.
Fig. 9. Strength variations with 90° dip direction, 0.1 m thickness and various orientations and sizes.
On the other hand, the influence of dip or stress direction on strength for different orientations is shown in Fig. 10 which showing the strength-orientation relationship for 0.5 m specimens with various dip directions. We find that higher strength for all orientations is achieved with higher confining stress along the laminations. This means that increasing the confining stress along laminations could offset some of the strength reduction caused by inclined laminations. Therefore, this change reduces the influence of laminations [16, 17]. Besides, the influence of dip or stress direction on strength varies with orientations. For specimens with 30°, 45° and 60° dip angles whose failure usually occurs as fully or partial sliding along lamination planes, the influence of dip or stress direction is more significant than those with 15° and 75° dip angles whose failure is caused by yield of intact rock.
Fig. 10. Strength-orientation relation for 0.5 m specimens with various dip directions.
Fig. 11. Strength-size relations for specimen with 60° dip angle and various dip direction.
Although increasing confining stress prevents the strength reduction caused by laminations, size effect behaviour was expected to be unaffected by this change because the confining stress was increased for specimens with different sizes. As shown in Fig. 11, contrary to the expectations, the strength reduction from 0.2 m to 2.0 m decreases for dip directions from 90° to 0°. In addition, the strength-size relationship for 0° and 30° dip directions and for 60° and 90° dip directions are close and have similar trend, respectively. 4. Conclusions Influence of regular laminations on strength and size effect of rock was examined with numerical simulation method in this paper. The factors considered include orientation, thickness of lamination and dip or stress direction.
Yuting Xue and Brijes Mishra / Procedia Engineering 191 (2017) 984 – 991
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From this study, we can see that even without considering size effect of intact rock and rock joint, there is size effect for rock specimens with regular laminations. This is obvious for specimens with lamination orientation of 30°, 45° and 60° and is not obvious for the specimens with lamination orientated 15° and 75°, which means size effect varies with lamination orientations. Therefore, size effect for specimen with regular laminations is orientation-dependent, which can be quantified by Anisotropic Effect factor. In addition, the influence of lamination thickness on size effect was achieved by including different numbers of laminations into specimens, which would have different numbers of potential failure plane. The results demonstrate that the influence of lamination spacing on size effect is limited. Finally, the variation in dip or stress direction gradually changes the confining stress along lamination planes. It is found that the strength of laminated rock specimens is affected and this influence is orientationdependent. As a result, the variation in dip or stress direction reduces or increases the influence of orientation on strength of laminated specimen. And size effect can also be affected by change of dip or stress direction because strength reduction 0.2 m to 2.0 m is affected by this change. References [1]
A. Louviere, MSHA news release: roof fall accidents remain a leading cause of coal mining injuries 2014. https://www.dol.gov/newsroom/releases/msha/msha20141062 (accessed January 1, 2016). [2] M. Gadde, S. Peng, Numerical simulation of cutter roof failure under weak roof conditions, in: Got Mining, SME Annual Meeting and Exhibit, Salt Lake City, 2005, Preprint 05–64. [3] A.K. Ray, Influence of cutting sequence and time effects on cutters and roof falls in underground coal mine – numerical approach, West Virginia University, Morgantown, 2009. [4] J. Coggan, F. Gao, D. Stead, D. Elmo, Numerical modelling of the effects of weak immediate roof lithology on coal mine roadway stability, Int. J. Coal Geol. 90-91 (2012) 100–109. [5] Z.T. Bieniawski, The effect of specimen size on compressive strength of coal, Int. J. Rock Mech. Min. Sci. Geo. Abst. 5 (1968) 325–335. [6] M.M. Singh, P.J. Huck, Large scale triaxial tests on rock, Rock Mechanics, in: H.R. Hardy, Jr. & R. Stefanko (Eds.), New Horizons in Rock Mechanics, 14th U.S. Symposium on Rock Mechanics, Am. Soc. Civil Eng., University Park, Pennsylvania, 1973. [7] T.P. Medhurst, E.T. Brown, A study of the mechanical behaviour of coal for pillar design, Int. J. Rock Mech. Min. Sci. 35 (1998) 1087–105. [8] R.A. Adey, R. Pusch, Scale dependency in rock strength, Eng. Geol. 53 (1999) 251–258. [9] A. Fakhimi, Y. Akbarzadeh, Numerical simulation of size effect in rock, in: D.P Yale (ed.), (Golden Rocks 2006)-50 Years of Rock Mechanics, 41st US Symposium on Rock Mechanics, Golden, Colorado, USA, 2006. [10] L. Scholtès, F.V. Donzé, M. Khanal, Scale effects on strength of geomaterials, case study: Coal, J. Mech. Phy. Sol. 59 (2011) 1131–1146. [11] Q. Zhang, H. Zhu, L. Zhang, X. Ding, Study of scale effect on intact rock strength using particle flow modeling, Int. J. Rock Mech. Min. Sci. 48 (2011) 1320–1328. [12] L. Shi, X. Li, B. Bai, Q. Li, X. Feng, Numerical analysis of loading boundary effects in Mogi-type true triaxial tests, in: M. Takahashi (ed.), True triaxial testing of rocks, CRC Press, Florida, 2012, pp. 19–34. [13] G.B. Baecher, H.H. Einstein, Size effect in rock testing, Geop. Res. Let. 8 (1981) 671–674. [14] G. Cruz, M.E. Pierce, A 3DEC Model for Heavily Veined Massive Rock Masses. in: J.F. Labuz (ed.), 48th US Rock Mechanics/Geomechanics Symposium, Curran, Minnesota, 2014. ARMA-2014-7660. [15] X. Lu, Study on a model of blocks for rock mass of deep roadway and its application in engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, 2010. [16] T. Ramamurthy, V.K. Arora, Strength predictions for jointed rocks in confined and unconfined states, Int. J. Rock Mech. Min. Sci. Geom. Abst. 31 (1994) 9–22. [17] A. Ghazvinian, M.R. Hadei, B. Madani, Behavior of mechanical anisotropic specimens under triaxial testing, in: 42nd US Rock Mechanics Symposium and 2nd US-Canada Rock Mechanics Symposium, Curran Associates, CA, 2008, ARMA-08-144. [18] M. Sherpa, P. Hagan, H. Masoumi, The effect of joint frequency on a discontinuous rock mass under experimental compressive strength testing conditions, in: L.J. Pyrak-Nolte (ed.), 47th US Rock Mechanics/Geomechanics Symposium, Curran Associates, California, 2013, ARMA-2013-240.
ScienceDirect Procedia Engineering 191 (2017) 984 – 991
Symposium of the International Society for Rock Mechanics
Numerical Simulation of Size Effect of Laminated Rock Yuting Xue, Brijes Mishra* Mining Engineering Department, West Virginia University, Morgantown, WV 26505, USA
Abstract
Roof and rib falls are still a leading cause of injuries in underground coal mines. When numerical simulation method is adopted to study this phenomenon, the influence of regular laminations on strength and size effect should be carefully considered because rock mass of roof is usually laminated. In order to get an insight into the influence of different lamination factors on size effect, cubic rock specimens with different sizes, stress directions, lamination orientations and lamination spacing were tested under true tri-axial conditions with 3DEC. The results show that regular lamination can cause size effect even without considering size effect of the intact rock and rock joints. The influence of lamination orientation on size effect is distinct and is orientation-dependent while the influence of lamination thickness is minimal. Changing dip or stress direction will affect the influence of orientation and size effect as well. © Published by Elsevier Ltd. This © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017. Peer-review under responsibility of the organizing committee of EUROCK 2017 Keywords: Size effect; laminated rock; orientation; spacing; stress direction
1. Introduction Improvements in roof control technology have led to significant decrease in accidents related to roof and rib falls. This brought the number of injury from roof and rib falls from 377 in 2012 down to 265 in 2013 [1]. But roof fall is still a leading cause of injuries in underground coal mines. And numerical simulation method has been widely used to investigate the mechanism and prevention of such problems [2–4]. Proper rock mechanics property, however, is crucial to obtain practical results in numerical simulation. Size effect is an important factor when laboratory-determined strength value is used to determine rock mass strength value for simulation.
* Corresponding author. Tel.: +1-304-293-3872. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017
doi:10.1016/j.proeng.2017.05.270
Yuting Xue and Brijes Mishra / Procedia Engineering 191 (2017) 984 – 991
985
Due to the importance of size effect, studies have been conducted to investigate the relationship between rock strength and their size. Bieniawski [5] conducted extensive underground tests on cubical coal specimens ranging from 0.75 inches to 6.6 ft in size. Singh and Huck [6] performed tri-axial compression tests on specimens with diameters ranging from 2 to 36 inches and approximate 50 percent strength reduction was observed. Medhurst and Brown [7] performed tri-axial compression tests on coal specimens with diameters of 61, 101, 146 and 300 mm to investigate size effect on mechanical behaviour of coal. Besides, numerical simulation was widely used to investigate size effect. Adey and Pusch [8] simulated size effect with boundary element method by considering larger discontinuities with increasing specimen volume. The discrete element program Ca2 was used [9] to simulate size effect and the result shows that initial crack length must be increased faster than specimen size to capture size effect. Scholtès et al. [10] studied size effect of coal by simulating true tri-axial compression tests with a discrete element model, which shows the capability to reproduce the dependency of strength on specimen size. Zhang et al. [11] investigated size effect with bonded particle model, where the initial fractures were considered as randomly orientated and located disks.
Fig. 1. Laminated roof condition.
From these literatures, we can see there are still two problems. First, simulations of size effect mostly focus on randomly distributed defects. Rock mass, especially shale formation in roof of coal mine openings, is usually laminated. As failure of such rock mass mostly occurs in the form of sliding along lamination, especially under low confining stress conditions, deformability and strength of laminated rock mass are strongly controlled by these laminations. For size effect, specimens with various sizes will have different numbers or different lengths of lamination, which will affect stress distribution in specimens and thus affect the strength. Second, for a specific geological condition, layouts of the underground excavation will lead to different stress directions acting on the laminated rock mass. And failure along laminations is very sensitive to stress direction. Therefore, it is crucial to study size effect under true tri-axial condition. However, laboratory and numerical tests are focused on uniaxial compression test and true tri-axial tests for size effects are quite rare in literatures. The reason is that true tri-axial test machine is difficult to design due to the interaction between loading platens and friction effects between platens and specimens [12, 13]. However, numerical simulation, as an alternative, overcomes some of these difficulties. In this paper, 3DEC was used to simulate cubic rock specimens under different conditions for the purpose of analysing their influence on size effect of rock. The scheme for the numerical experiments was first introduced, which included the method for simulation of true tri-axial test and all the conditions considered in the experiment. A series of simulations were then conducted to investigate the influence of different factors on strength and size effect.
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2. Numerical simulation scheme
Fig. 2. Simulated situation.
Various size effects, including size effect of intact rock, size effect of laminations or discontinuities and size effect caused by laminations, are involved in the study on size effect of laminated rock. Simulation of size effect of intact rock is usually achieved by including random defects into rock specimens; size effect of discontinuities can be simulated with asperities. However, these are excluded in the current study. The main purpose is to investigate the third aspect. Different sizes of specimen will have different numbers of laminations, as shown in Fig. 2, which will not only provide potential failure planes, but also affect the stress distribution in the whole specimens. The rock between laminations is treated as intact rock and laminations are treated as planes without asperity. Configuration of the numerical simulation is shown in Fig. 3 [14]. Cubic specimens with laminations are placed in centre. Boundary stress is applied to the external surface of the four platens to provide confining pressure on the specimens. The bottom of specimen is fixed in Z-direction and a velocity of 0.001 m/s is applied from the specimen’s top.
Fig. 3. Configuration of numerical simulation.
Intact rock is treated as an elastic/plastic material while platens are elastic. Drucker-Prager failure criterion is used to predict failure. Besides, laminations in the specimens are elastic/plastic with Coulomb slip failure criterion. The mechanical properties of intact rock and laminations are provided in Table 1 [15]. For the interfaces between specimens and platens, the friction angle, cohesion, tension and shear are set to zero to eliminate the friction effect. And lamination length, thickness and orientation parameters used in the numerical models are listed in Table 2.
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Table 1. Mechanical properties for the materials used in simulations.
Lamination
Interfaces between specimens and platens
Friction Angle (°)
11
0
Tension (MPa)
0.52
0
Cohesion (MPa)
1.00
0
Normal Stiffness (GPa)
12.9
50
Shear Stiffness (GPa)
7.10
0
Specimen Density (kg/m3)
2500
Young's Modulus (GPa)
16
Poison Ratio
0.38
Material Constant qϕ
0.35
Material Constant kϕ (MPa)
3.6
Table 2. Situations considered in numerical simulations.
Size-side length(m)
Lamination thickness (m)
Dip
Dip
Confining stress
direction (°)
angle (°)
(X: 5.79 MPa; Y: 3.45 MPa)
0.2
0.1
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.1
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.2
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.3
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.1
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.2
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.3
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.1
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.2
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.3
0, 30, 60, 90
15, 30, 45, 60, 75, 90
0.5
Y Z
n tio ec dir Dip
le ang Dip X
1.0
2.0
3. Results and discussion The simulation results of a laminated specimen is first analysed to demonstrate the influence of laminations on deformation, strength and mode of failure. And then the influence of factors-dip angle and dip direction of laminations and lamination thickness-on size effect is examined. The results of a 2.0 m specimen with 0.3 m-thickness laminations at 90° dip direction and various dip angles is selected as a case study in Fig. 4. First, it can be found that the laminations decrease specimens’ overall stiffness. In addition, the existence of laminations, more or less, decreases specimens’ strength. An exemption is the specimen with dip angle of 90° whose stress-strain curve overlaps with that of the specimen without laminations. This is because the specimen with dip angle of 90° can be considered as a stack of smaller intact cuboid specimens. Finally, strength reduction varies with dip angle or orientation. The strength reduction for specimens with 30°, 45° and 60° dip angles is more significant than those with dip angles of 15° and 75°, resulted from different failure modes associated with different dip angles. For the specimens with dip angles of 30°, 45° and 60°, failure usually occurs as sliding along laminations.
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Yuting Xue and Brijes Mishra / Procedia Engineering 191 (2017) 984 – 991
Fig. 4. Stress-strain curves for specimens with 2.0 m side length and with laminations of 90° dip direction, various dip angles and 0.3 m thickness.
The stress of the specimen with 45° dip angle directly drops to its residual value, demonstrating failure of the whole lamination plane. Failure of specimens with 30° and 60° dip angles is shown by an initial stress drop and then an increase in stress. This is caused by partial failure of the lamination planes. While failure of specimens of dip angles 15° and 75° occurs as gradual yielding, possibly due to yielding of intact rock. 3.1. The influence of lamination orientation on size effect
Fig. 5. Strength variations with 0.1 m thickness and various orientations and sizes.
The result for tests under true tri-axial condition where the dip direction is 0° is shown in Fig. 5. The strength of different specimens (σ) is normalized to the strength of intact specimens (σint). It should also be noted that no specimen with orientation or dip angle of 0° was simulated and the strength for 0° in Fig. 5 represents the result of intact specimens. Each curve in Fig. 5 shows a trend similar to previous results [16, 17] from conventional triaxial tests. Also the distances between the four points corresponding to four different sizes for each dip angle are different, which demonstrates that size effect varies with different orientations. This means size effect can be affected by lamination orientation. The data in Fig. 5 is then plotted in Fig. 6. The strength reduction for specimens with 30°, 45° and 60° dip angles is higher than that of the specimens with 15° and 75° dip angles which means that size effect for specimens with 30°, 45° and 60° dip angles is more significant than that with 15° and 75° dip angles. In Summary, size effect varies with orientation in laminated rock mass. Two variables are needed to quantify the relation between size effect and orientation, one to describe the significance of lamination influence on strength and the other to quantify size effect. Anisotropic Effect factor (Ae), introduced by Ghazvinian et al. [17] is adopted to quantify the first factor. It is defined as (1) where σia and σja are the compressive strengths of the intact and jointed rock mass, respectively, and a is the confining pressure. This equation shows that the value of Ae reflects the relative strength reduction caused
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by inclined laminations and can quantify orientation influence. Besides, size effect is used to describe strength reduction with increasing size. The difference in rock strength for specimens between 0.3 m and 2.0 m in length (or size) in our simulations can be treated as the consequence of size effect and their value can reflect the significance of size effect. Their relation is plotted in Fig. 7. It is clear that relative strength difference increases linearly with Anisotropic Effect factor, which means size effect is significant if Anisotropic Effect factor is large. In other words, if the rock strength is significantly decreased from intact rock strength by the laminations with a specific orientation, resulting in large Anisotropic Effect factor value, size effect should be profound for rock mass with laminations of same orientation.
Fig. 6. Strength variation with orientations and sizes.
Fig. 7. Relation between relative strength difference and Anisotropic Effect factor (Ae).
3.2. The influence of lamination thickness on size effect
Fig. 8. Strength-thickness relationship for specimens with various sizes and orientations.
Previous conventional tri-axial tests [13, 18] on cylindrical specimens with horizontal joints showed that strength decreases as the number of joint increases. Baecher and Einstein [13] interpreted that asperity on joint surface served as stress concentration loci, whose number increased in proportion to the number of joint. This asperity-related failure cannot be achieved in this study because laminations are treated as smooth planes. Instead, its influence is based on the fact that small lamination thickness can induce more lamination into the specimen, resulting in more potential failure planes. The relationship between lamination thickness and strength for specimens with 45° and 60° dip angles are shown in Fig. 8. For the same orientation, the strength of specimens reduces with the increase of specimen size and their relationships for different lamination thickness follow the same trend and are very close to each other. This indicates that the lamination thickness has limited influence on size effect. 3.3. The influence of dip or stress direction on strength and size effect Change in stress direction can affect strength of laminated rock because failure of laminated specimens is highly controlled by laminations and sliding along laminations is sensitive to confining stress [13]. In this study, the confining stress was kept constant and dip direction varied from 0° to 30°, 60° and 90°. In such a way,
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the influence of dip or stress direction on strength and size effect is investigated. For comparison purpose, the relation between strength and orientation for the specimens with 90° dip direction and different sizes is plotted in Fig. 9. On one hand, as the stresses in X and Y directions are 5.79 MPa and 3.45 MPa, respectively, varying dip direction from 0° to 90° would increase the confining stress to the lamination planes. By comparing the results plotted in Fig. 9 with Fig. 5, change of dip or stress direction definitely affects the strength.
Fig. 9. Strength variations with 90° dip direction, 0.1 m thickness and various orientations and sizes.
On the other hand, the influence of dip or stress direction on strength for different orientations is shown in Fig. 10 which showing the strength-orientation relationship for 0.5 m specimens with various dip directions. We find that higher strength for all orientations is achieved with higher confining stress along the laminations. This means that increasing the confining stress along laminations could offset some of the strength reduction caused by inclined laminations. Therefore, this change reduces the influence of laminations [16, 17]. Besides, the influence of dip or stress direction on strength varies with orientations. For specimens with 30°, 45° and 60° dip angles whose failure usually occurs as fully or partial sliding along lamination planes, the influence of dip or stress direction is more significant than those with 15° and 75° dip angles whose failure is caused by yield of intact rock.
Fig. 10. Strength-orientation relation for 0.5 m specimens with various dip directions.
Fig. 11. Strength-size relations for specimen with 60° dip angle and various dip direction.
Although increasing confining stress prevents the strength reduction caused by laminations, size effect behaviour was expected to be unaffected by this change because the confining stress was increased for specimens with different sizes. As shown in Fig. 11, contrary to the expectations, the strength reduction from 0.2 m to 2.0 m decreases for dip directions from 90° to 0°. In addition, the strength-size relationship for 0° and 30° dip directions and for 60° and 90° dip directions are close and have similar trend, respectively. 4. Conclusions Influence of regular laminations on strength and size effect of rock was examined with numerical simulation method in this paper. The factors considered include orientation, thickness of lamination and dip or stress direction.
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From this study, we can see that even without considering size effect of intact rock and rock joint, there is size effect for rock specimens with regular laminations. This is obvious for specimens with lamination orientation of 30°, 45° and 60° and is not obvious for the specimens with lamination orientated 15° and 75°, which means size effect varies with lamination orientations. Therefore, size effect for specimen with regular laminations is orientation-dependent, which can be quantified by Anisotropic Effect factor. In addition, the influence of lamination thickness on size effect was achieved by including different numbers of laminations into specimens, which would have different numbers of potential failure plane. The results demonstrate that the influence of lamination spacing on size effect is limited. Finally, the variation in dip or stress direction gradually changes the confining stress along lamination planes. It is found that the strength of laminated rock specimens is affected and this influence is orientationdependent. As a result, the variation in dip or stress direction reduces or increases the influence of orientation on strength of laminated specimen. And size effect can also be affected by change of dip or stress direction because strength reduction 0.2 m to 2.0 m is affected by this change. References [1]
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