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Breakage and deformation mechanism of crushable blocky materials
International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
Contents lists available at SciVerse ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Note
Breakage and deformation mechanism of crushable blocky materials EnLong Liu a,n, Shengshui Chen b, Guoying Li b a b
State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu 610065, PR China Nanjing Hydraulic Research Institute, Nanjing 210029, PR China
a r t i c l e i n f o Article history: Received 16 September 2010 Received in revised form 26 August 2011 Accepted 11 November 2011 Available online 1 December 2011
1. Introduction Many geological materials encountered in engineering are particulate materials, which are composed of distinct particles interacting with each other and associated voids. Upon loading, the particulates may slide, rotate and crush or break. The deformation and breakage mechanism of particulate materials is complicated and many experimental studies have attempted to characterize this process. Rowe [1] formulated the relationship between the rate of dilatancy and the maximum stress ratio for any ideal packing of an assembly of particles using a theoretical and experimental approach. Drescher [2] reported experiments on optically sensitive granular materials consisting of crushing glass particles using a double shear apparatus in order to observe the pattern of principal stress trajectories occurring at large plastic deformations. Chappell [3] performed tests on photo-elastic blocks to examine both the deformational response and stress distribution as a total system within particulate materials and concluded that deformation in a discontinuum, whether it is rotund or blocky, is related not only to material deformation but also mechanisms such as slip and rotation. Oda et al. [4–6] conducted biaxial compression tests on assemblies of oval cross-sectional rods in an effort to evaluate the effects of inter-particle friction, particle shape, and initial fabric on the overall strength of granular materials. The variation in the spatial arrangement of the particles (fabric) and particle rolling and sliding are monitored by taking photo-elastic pictures at various stages during the course of deformation. Bathurst and Rothenburg [7] presented a series of theoretical developments and numerical experiments directed at quantifying important features of the micromechanical behavior of granular media by introducing average characteristics of fabric anisotropy and certain statistical averages of contact forces. Calvetti et al. [8] carried out several
n
Corresponding author. Tel./fax: þ 86 28 85405121. E-mail addresses: [email protected], [email protected] (E. Liu).
laboratory tests on Schneebeli material specimens composed of wooden roller stacks to analyze the material micromechanical behavior under complex stress conditions, which involved loading– unloading cycles and principal axes rotations. Lanier [9] performed tests on an assembly of wooden rods using a specially designed shear apparatus to study the micro-mechanisms of deformation in granular materials, mainly concerned with macroscopic behavior (boundary conditions) and local kinematics (namely, rods displacements and rotations). Wolf et al. [10] presented three series of sandbox experiments of extensional deformation in granular material to investigate the dependence of the developing shear band system on different boundary conditions and material properties. Delenne et al. [11] investigated the stress–strain behavior and failure of a cohesive granular material by experiments on the assembly of aluminum rods glued together by means of an epoxy resin under simple loading conditions. Alshibli and Roussel [12] tested on the spherical beads to study load oscillation in granular materials under axisymmertric triaxial loading condition at 25, 100, 250 and 400 kPa confining pressures. Furthermore, some researchers use theoretical and numerical methods to study the failure mechanism of particulate materials. McDowell et al. [13,14] investigated the fractal crushing of granular materials using DEM at the micro-scale. Cecconi et al. [15] developed a constitutive model for granular materials within the framework of strain-hardening elastoplasticity, aiming at describing some of the macroscopic effects of the degradation processes associated with grain crushing. Anandarajah [16] proposed a constitutive model for granular materials based on the consideration of their microstructure, in which two primary failure mechanisms of the microelements, one based on particle sliding and another based on particle rolling, are identified. Chang and Hicher [17] derived an elasto-plastic stress– strain relationship by extending granular mechanics approach. The aforementioned experimental and theoretical studies focused primarily on the microscopical deformational mechanisms such as sliding and rolling or rotations of particulates that occur in stiff materials. Upon loading, however, the particles of an
1365-1609/$ - see front matter Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2011.11.008
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E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
Fig. 1. Sketch of loading cell employed: (a) vertical view; (b) side view—(1) sample cell, (2) supporting block at the bottom, (3) bolt hole, (4) loading plate at the top, (5) glass plate, (6) air cell, (7) confining steel plate, (8) a thin piece of reducing friction, (9) base plate, (10) transferring loading block, (11) air vent, and (12) the high pressure triaxial loading system.
σa
σl
σl
σa
σa
σl
σa
σl
σl
σl
σa σa
Fig. 2. Pattern of arrangement: (a) arrangement a, (b) arrangement b, and (c) arrangement c.
B
(σa-σl)/2
element of soil can be crushed or broken [18–22]. This, in turn, profoundly affects the strength and stress–strain behavior of soil elements greatly. Although a few theoretical methods have been employed to study the crushing mechanism of granular materials, the simulation results need to be confirmed using experimental data. To formulate the constitutive model of particulate materials, which can take into account the influence of microstructure deformation mechanisms, tests on crushable blocky materials at the meso scale were performed and are the basis for this manuscript. We tracked the mesoscopical breakage process of particulate materials. Biaxial compressional tests with three types of arrangements at 0.045 MPa, 0.15 MPa and 0.45 MPa lateral stresses were performed on an assembly of blocky materials that were
A O
45°
(σa+σl)/2 Fig. 3. Stress path.
E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
crushable. Finally, we comprehensively investigated the deformation mechanism and breakage processes.
2. Test methods and procedures 2.1. Test set-up We devised an apparatus (Fig. 1) that could be axially loaded using a sample cell from one high pressure triaxial
127
equipment, which was laterally loaded by air pressure. Two transparent rigid glass plates were set in parallel 3 cm apart to keep the specimen under biaxial loading conditions. The size of the specimen chamber was 20 cm 10 cm 3 cm. During the loading process, a series of pictures was taken to track the breakage processes of the specimens at the meso scale. When performing the tests, we smeared some silicone oil on the surfaces of the upper and bottom platens uniformly to reduce the friction and induced stress concentration on the interfaces.
Fig. 4. Breakage processes (arrangement a): (a) sl ¼ 0.045 MPa, (b) sl ¼0.15 MPa, and (c) sl ¼0.45 MPa.
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E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
2.2. Test materials and procedures The blocky materials used in the tests consisted of chalk and purified water, which were mixed uniformly together in a mass ratio of 1:1. After the specimens were formed, they were air-dried under identical conditions. The size of the block was 1 cm 1 cm 3 cm, the density was 0.92 103 kg/m3, the
unconfined compression strength was 2.1 MPa and the tensile strength was 0.525 MPa. The arrangements of the blocky materials are shown in Fig. 2, having pattern a, pattern b and pattern c. The total number of blocks used for each test was 200. Axial loading was applied by displacement control, and the corresponding loading velocity was set at 0.125 mm/min. During the loading process, the axial displacement and corresponding axial force
Fig. 5. Breakage processes (arrangement b): (a) sl ¼0.045 MPa, (b) sl ¼ 0.15 MPa, and (c) sl ¼ 0.45 MPa.
E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
Fig. 6. Breakage processes (arrangement c): (a) sl ¼0.045 MPa, (b) sl ¼ 0.15 MPa, and (c) sl ¼ 0.45 MPa.
Fig. 7. Breakage modes.
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were recorded by the high-pressure triaxial apparatus. The lateral displacements were measured using a micrometer, and the corresponding lateral stress was recorded using a pressure gage. The stress path depicted in Fig. 3 was used to carry out the tests. In Fig. 3, the OA loading stage denotes the axial loads and lateral loads being applied equally, and the AB stage denotes that the lateral pressures remain unchanged while the axial loads are applied until the sample fails. In the following, sa ,ea ,sl and el represent the axial stress, the axial strain, the lateral stress and the lateral strain, respectively.
3. Test results and discussions 3.1. Breakage processes Figs. 4–6 depicts the breakage processes of the block assembly under the loading stress path given in Fig. 3, which correspond to the arrangements a, b and c in Fig. 2, respectively. Three sets of lateral pressures, 0.045 MPa, 0.15 MPa and 0.45 MPa, were applied. For each set of lateral pressures, the processes of breakage of the assembly of blocky materials were different as shown in Figs. 4–6, and they mostly depend on their initial arrangement. When the lateral pressure is lower, initially the blocks move toward each other and become closer. At this point, voids appear between blocks, accompanied by cracking of some of them. The breakage bands are formed gradually. Finally, some new blocks crack, leading to additional conjugated breakage bands. When the lateral pressure is high, the blocks move toward each other. Some of the blocks break or are crushed and the breakage bands are formed step by step. The breakage band is formed by shearing under low lateral pressures and by shearing and crushing under high lateral pressures.
Fig. 8. Breakage modes (a) breakage mode A (- - - - crack resulted by vertically splitting), (b) breakage mode B (- - - - slide plane by shear failure), and (c) breakage mode C (- - - - the profile of the block after locally crushed).
3.2. Breakage modes of the blocky materials The breakage modes of the blocky materials are shown in Fig. 7a–d upon loading. Takei et al. [23] proposed models for breakage modes of chalk bars under one-dimensional compression conditions. For the crushable blocks under biaxial compressions presented in Figs. 4–6, three types of breakage modes are shown in Fig. 8a–c. From the breakage processes presented in Figs. 4–6, we observe that: more of the blocks are vertically split (Breakage Mode A) or failed by shear (Breakage Mode B) when the laterally pressures are low, whereas more blocks are locally crushed (Breakage Mode C) when the laterally pressures are high upon loading. The different breakage modes result from the local stress states of the blocks within the sample. For some sample tested, the three breakage modes A, B and C could be observed simultaneously. Because of differences in loading conditions, the local stresses experienced by individual blocks are different, and thus the blocks fail according to unique breakage modes. 3.3. Stress–strain behavior The stress and strain herein (compression being positive) refer to the average stress and the average strain of the specimen consisting of the assembly of blocky materials. Figs. 9–11 show the stress–strain and volumetric strain–axial strain relationship curves for tests, which correspond to the arrangements a, b and c in Fig. 2, respectively. Deformational features are summarized as follows: (1) for arrangements a and b, the specimen exhibit strain softening under low lateral pressure and strain hardening under high lateral pressure, but for arrangement c the specimen exhibit strain hardening under the lateral pressure ranging from lower value to higher one; and (2) when the lateral
Fig. 9. Stress–strain and axial strain–volumetric strain relationship curves (arrangement a): (a) stress–strain curves and (b) axial strain–volumetric strain curves.
pressure is lower, the volumetric strain is compressive and then expansive with the increase of axial strain. When the lateral pressure is higher, the volumetric strain is compressive all the time.
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4. Breakage mechanism analysis 4.1. Rotation analysis During the loading processes, we observed that the blocks slid, rotated and were crushed, because of the different lateral pressures. When the lateral pressures are low, the blocks located along the breakage bands slide, rotate and are crushed (shown in Fig. 12a corresponding to Fig. 4a, in Fig. 13a corresponding
Fig. 12. Slide, rotation and crush (the ‘‘þ ’’ lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading) (arrangement a): (a) sl ¼0.045 MPa, sa ¼ 2.59 MPa and (b) sl ¼0.45 MPa, sa ¼ 5.04 MPa.
Fig. 10. Stress–strain and axial strain–volumetric strain relationship curves (arrangement b): (a) stress–strain curves, and (b) axial strain–volumetric strain curves.
Fig. 13. Slide, rotation and crush (the ‘‘þ ’’ lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading) (arrangement b): (a) sl ¼0.045 MPa, sa ¼ 1.85 MPa and (b) sl ¼0.45 MPa, sa ¼ 5.31 MPa.
Fig. 11. Stress–strain and axial strain–volumetric strain relationship curves (arrangement c): (a) stress–strain curves, and (b) axial strain–volumetric strain curves.
Fig. 14. Slide, rotation and crush (the ‘‘þ ’’ lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading) (arrangement c): (a) sl ¼0.045 MPa, sa ¼ 1.35 MPa and (b) sl ¼0.45 MPa, sa ¼ 5.46 MPa.
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Table 1 Stress state, axial strain and cumulative crushing ratio (%): arrangement a. Test no.
Axial stress, strain and cumulative crushing ratio (%)
sl ¼ 0.045 MPa sl ¼ 0.15 MPa sl ¼ 0.45 MPa
2.54 MPa, 3.30% and (1.5%) 3.59 MPa, 7.09% and (9.0%) 4.34 MPa, 8.05% and (6.0%)
1.92 MPa, 6.30% and (9.5%) 3.46 MPa, 11.30% and (22.5%) 4.86 MPa, 14.65% and (53.5%)
2.03 MPa, 13.35% and (14.5%) 3.12 MPa, 18.82% and (34.0%) 5.04 MPa, 18.14% and (88.0%)
Table 2 Stress state, axial strain and cumulative crushing ratio (%): arrangement b. Test no.
Axial stress, strain and cumulative crushing ratio (%)
sl ¼ 0.045 MPa sl ¼ 0.15 MPa sl ¼ 0.45 MPa
1.58 MPa, 5.10% and (9.5%) 3.73 MPa, 7.28% and (10.5%) 4.06 MPa, 6.44% and (11.5%)
1.51 MPa, 6.40% and (19.5%) 3.40 MPa, 10.30% and (31.0%) 4.76 MPa, 11.50% and (56.0%)
1.53 MPa, 12.80% and (22.5%) 3.92 MPa, 17.70% and (44.0%) 5.30 MPa, 19.90% and (96.0%)
Table 3 Stress state, axial strain and cumulative crushing ratio (%): arrangement c. Test no.
Axial stress, strain and cumulative crushing ratio (%)
sl ¼ 0.045 MPa sl ¼ 0.15 MPa sl ¼ 0.45 MPa
0.44 MPa, 2.50% and (2.5%) 1.65 MPa, 10.53% and (4.0%) 4.62 MPa, 9.67% and (6.0%)
0.74 MPa, 6.15% and (9.5%) 2.10 MPa, 14.85% and (18.0%) 5.03 MPa, 13.84% and (53.5%)
to Fig. 5a and in Fig. 14a corresponding to Fig. 6a, which were recorded from another surface of the biaxial compressional apparatus through a transparent rigid glass plate). The other blocks which are located outside of the breakage bands only slide and move closer, having very few rotations. This happens because some of the blocks slide and are split first and then are crushed. This is then accompanied by the emergency of big voids between the blocks under low lateral pressure. The blocks are then able to rotate and form the breakage bands, whereas the other blocks outside of the breakage bands only slide and barely rotate. When the lateral pressures are high, the blocks mainly slide and are crushed, undergoing very few rotations (shown in Fig. 12b corresponding to Fig. 4c, in Fig. 13b corresponding to Fig. 5c and in Fig. 14b corresponding to Fig. 6c). This occurs because many of the blocks slide and are crushed in a manner that forces them to move until no room is present between them with high lateral pressure. Finally, the breakage bands, which are wider under low lateral pressure forms. 4.2. Breakage analysis Tables 1–3 list the stress state, axial strain and cumulative crushing ratio of the tested samples corresponding to the breakage processes given in Figs. 4–6, respectively. The cumulative crushing ratio [23], Rcc, is defined as follows: Rcc ¼
ncc , nALL
1.13 MPa, 12.40% and (16.5%) 2.66 MPa, 19.81% and (39.0%) 5.32 MPa, 18.20% and (94.5%)
upon loading. Thus, the energy produced by the external loads is dissipated by the rotation and slip of the blocks, and only a fraction of the energy is dissipated by crushing them.
5. Conclusions An apparatus that can track the breakage process of particulate materials at the meso scale was used here to investigate the breakage and deformation mechanism of crushable blocky materials under biaxial compressional conditions with three types of arrangement. The following conclusions can be drawn: (1) the blocky materials slide, rotate and are crushed under different stress states and, as a result of this process, the breakage bands are formed; (2) the specimens show strain hardening or strain softening due to the loading conditions and stress state, and the volume and lateral deformation are different for different arrangements and loading stress state; (3) the breakage modes of the blocky materials are primarily vertically split, with shear failure and local crushing occurring in response to the local stress states induced; (4) upon loading, under lower lateral pressures the blocky materials mainly slide, rotate and are crushed, whereas under higher lateral pressures they primarily slide, are crushed and barely rotate.
ð1Þ
where ncc is the cumulative number of crushed blocks at each loading stage, and nALL is equal to 200, which is the total number of blocks in the specimen. The numbers of crushed blocks at the each loading stage were visually counted and recorded. From Tables 1–3, we can conclude that upon loading, the cumulative crushing ratios at failure increase greatly with the increase in lateral pressure. For instance (see Table 2), the cumulative crushing ratio at failure is only 22.5% when the lateral stress is 0.045 MPa, whereas it is increased to 96.0% when the lateral stress is 0.45 MPa. This occurs because when the lateral pressure is low, the crushable blocks can easily rotate and slip
Acknowledgments The authors thank the anonymous reviewer and Prof. Robert W. Zimmerman for their careful review, contributions and critics, which led to the improvement of the manuscript. The authors are deeply indebted to the late Prof. Zhujiang Shen of Tsinghua University and Prof. Tielin Chen of Beijing Jiaotong University for their valuable discussion and help in performing the tests. This work was funded by National Science Foundation of China (Grant nos. 51009103 and 90815024) and The Ministry of Water Resources of the People’s Republic of China (Grant no. 200801014).
E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
References [1] Rowe PW. The stress–dilatancy relation for static equilibrium of an assembly of particles in contact. Proc R Soc London A 1962;269:500–27. [2] Drescher A. An experimental investigation of flow rules for granular materials using optically sensitive glass particles. Geotechnique 1976;26(4):591–601. [3] Chappell BA. Load distribution and redistribution in discontinua. Int J Rock Mech Min Sci Geomech Abstr 1979;16:391–9. [4] Oda M, Konishi J, Nemat-Nasser S. Some experimentally based fundamental results on the mechanical behavior of granular materials. Geotechnique 1980;30(4):479–95. [5] Oda M, Konishi J, Nemat-Nasser S. Experimental micromechanical evaluation of strength of granular materials: effects of particle rolling. Mech Mater 1982;1:269–83. [6] Oda M, Kazama H. Microstructure of shear bands and its relation to the mechanisms of dilatancy and failure of dense granular soils. Geotechnique 1998;48(4):465–81. [7] Bathurst RJ, Rothenburg L. Observations on stress–force–fabric relationships in idealized granular materials. Mech Mater 2000;9:65–80. [8] Calvetti F, Combe G, Lanier J. Experimental micromechanical analysis of a 2D granular material: relation between structure evolution and loading path. Mech Cohes Frict Mater 1997;2:121–63. [9] Lanier L. Micro-mechanisms of deformation in granular materials: experiments and numerical results. In: Vermeer PA, et al., editors. LNP 568; 2001. p. 163–72. + [10] Wolf H, Konig D, Triantafyllidis T. Experimental investigation of shear band patterns in granular material. J Struct Geol 2003;25:1229–40.
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[11] Delenne JY, Youssoufi MSE, Cherblanc F, et al. Mechanical behavior and failure of cohesive granular materials. Int J Numer Anal Methods Geomech 2004;28:1577–94. [12] Alshibli KA, Roussel LE. Experimental investigation of slip-stick behavior in granular materials. Int J Numer Anal Methods Geomech 2006;30:1391–407. [13] Mcdowell GR, Bolton MD, Robertson D. The fractal crushing of granular materials. Int J Mech Phys Solids 1996;44:2079–102. [14] Mcdowell GR, Bolton MD. On the micromechanics of crushable aggregates. Geotechnique 1998;48(5):667–79. [15] Cecconi M, DeSimone A, Tamagnini C, et al. A constitutive model for granular materials with grain crushing and its application to a pyroclastic soil. Int J Numer Anal Methods Geomech 2002;26:1531–60. [16] Anandarajah A. Sliding and rolling constitutive theory for granular materials. J Eng Mech 2004;130(6):73–7. [17] Chang CS, Hicher PY. An elasto-plastic model for granular materials with microstructural consideration. Int J Solids Struct 2005;42:4258–77. [18] Hardin BO. Crushing of soil particles. J Geotech Eng 1984;111(10):1176–92. [19] Hagerty MM, Hite DR, Ullrich CR, et al. One-dimensional high-pressure compression of granular media. J Geotech Eng 1991;119(1):11–8. [20] Nakata Y, Hyodo M, Hyde AFL, et al. Microscopic particle crushing of sand subjected to high pressure one-dimensional compression. Soils Found 2001; 41(1):69–82. [21] Liu E. Deformation and breakage properties of crushable blocky materials. Rock Mech Rock Eng 2011;44:373–7. [22] Bartake PP, Singh DN. A generalized methodology for determination of crushing strength of granular materials. Geotech Geol Eng 2007;25:203–13. [23] Takei M, Kusakabe O, Hayashi T. Time-dependent behavior of crushable materials in one-dimensional compression tests. Soils Found 2001;41(1):97–121.
Contents lists available at SciVerse ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Note
Breakage and deformation mechanism of crushable blocky materials EnLong Liu a,n, Shengshui Chen b, Guoying Li b a b
State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu 610065, PR China Nanjing Hydraulic Research Institute, Nanjing 210029, PR China
a r t i c l e i n f o Article history: Received 16 September 2010 Received in revised form 26 August 2011 Accepted 11 November 2011 Available online 1 December 2011
1. Introduction Many geological materials encountered in engineering are particulate materials, which are composed of distinct particles interacting with each other and associated voids. Upon loading, the particulates may slide, rotate and crush or break. The deformation and breakage mechanism of particulate materials is complicated and many experimental studies have attempted to characterize this process. Rowe [1] formulated the relationship between the rate of dilatancy and the maximum stress ratio for any ideal packing of an assembly of particles using a theoretical and experimental approach. Drescher [2] reported experiments on optically sensitive granular materials consisting of crushing glass particles using a double shear apparatus in order to observe the pattern of principal stress trajectories occurring at large plastic deformations. Chappell [3] performed tests on photo-elastic blocks to examine both the deformational response and stress distribution as a total system within particulate materials and concluded that deformation in a discontinuum, whether it is rotund or blocky, is related not only to material deformation but also mechanisms such as slip and rotation. Oda et al. [4–6] conducted biaxial compression tests on assemblies of oval cross-sectional rods in an effort to evaluate the effects of inter-particle friction, particle shape, and initial fabric on the overall strength of granular materials. The variation in the spatial arrangement of the particles (fabric) and particle rolling and sliding are monitored by taking photo-elastic pictures at various stages during the course of deformation. Bathurst and Rothenburg [7] presented a series of theoretical developments and numerical experiments directed at quantifying important features of the micromechanical behavior of granular media by introducing average characteristics of fabric anisotropy and certain statistical averages of contact forces. Calvetti et al. [8] carried out several
n
Corresponding author. Tel./fax: þ 86 28 85405121. E-mail addresses: [email protected], [email protected] (E. Liu).
laboratory tests on Schneebeli material specimens composed of wooden roller stacks to analyze the material micromechanical behavior under complex stress conditions, which involved loading– unloading cycles and principal axes rotations. Lanier [9] performed tests on an assembly of wooden rods using a specially designed shear apparatus to study the micro-mechanisms of deformation in granular materials, mainly concerned with macroscopic behavior (boundary conditions) and local kinematics (namely, rods displacements and rotations). Wolf et al. [10] presented three series of sandbox experiments of extensional deformation in granular material to investigate the dependence of the developing shear band system on different boundary conditions and material properties. Delenne et al. [11] investigated the stress–strain behavior and failure of a cohesive granular material by experiments on the assembly of aluminum rods glued together by means of an epoxy resin under simple loading conditions. Alshibli and Roussel [12] tested on the spherical beads to study load oscillation in granular materials under axisymmertric triaxial loading condition at 25, 100, 250 and 400 kPa confining pressures. Furthermore, some researchers use theoretical and numerical methods to study the failure mechanism of particulate materials. McDowell et al. [13,14] investigated the fractal crushing of granular materials using DEM at the micro-scale. Cecconi et al. [15] developed a constitutive model for granular materials within the framework of strain-hardening elastoplasticity, aiming at describing some of the macroscopic effects of the degradation processes associated with grain crushing. Anandarajah [16] proposed a constitutive model for granular materials based on the consideration of their microstructure, in which two primary failure mechanisms of the microelements, one based on particle sliding and another based on particle rolling, are identified. Chang and Hicher [17] derived an elasto-plastic stress– strain relationship by extending granular mechanics approach. The aforementioned experimental and theoretical studies focused primarily on the microscopical deformational mechanisms such as sliding and rolling or rotations of particulates that occur in stiff materials. Upon loading, however, the particles of an
1365-1609/$ - see front matter Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2011.11.008
126
E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
Fig. 1. Sketch of loading cell employed: (a) vertical view; (b) side view—(1) sample cell, (2) supporting block at the bottom, (3) bolt hole, (4) loading plate at the top, (5) glass plate, (6) air cell, (7) confining steel plate, (8) a thin piece of reducing friction, (9) base plate, (10) transferring loading block, (11) air vent, and (12) the high pressure triaxial loading system.
σa
σl
σl
σa
σa
σl
σa
σl
σl
σl
σa σa
Fig. 2. Pattern of arrangement: (a) arrangement a, (b) arrangement b, and (c) arrangement c.
B
(σa-σl)/2
element of soil can be crushed or broken [18–22]. This, in turn, profoundly affects the strength and stress–strain behavior of soil elements greatly. Although a few theoretical methods have been employed to study the crushing mechanism of granular materials, the simulation results need to be confirmed using experimental data. To formulate the constitutive model of particulate materials, which can take into account the influence of microstructure deformation mechanisms, tests on crushable blocky materials at the meso scale were performed and are the basis for this manuscript. We tracked the mesoscopical breakage process of particulate materials. Biaxial compressional tests with three types of arrangements at 0.045 MPa, 0.15 MPa and 0.45 MPa lateral stresses were performed on an assembly of blocky materials that were
A O
45°
(σa+σl)/2 Fig. 3. Stress path.
E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
crushable. Finally, we comprehensively investigated the deformation mechanism and breakage processes.
2. Test methods and procedures 2.1. Test set-up We devised an apparatus (Fig. 1) that could be axially loaded using a sample cell from one high pressure triaxial
127
equipment, which was laterally loaded by air pressure. Two transparent rigid glass plates were set in parallel 3 cm apart to keep the specimen under biaxial loading conditions. The size of the specimen chamber was 20 cm 10 cm 3 cm. During the loading process, a series of pictures was taken to track the breakage processes of the specimens at the meso scale. When performing the tests, we smeared some silicone oil on the surfaces of the upper and bottom platens uniformly to reduce the friction and induced stress concentration on the interfaces.
Fig. 4. Breakage processes (arrangement a): (a) sl ¼ 0.045 MPa, (b) sl ¼0.15 MPa, and (c) sl ¼0.45 MPa.
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E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
2.2. Test materials and procedures The blocky materials used in the tests consisted of chalk and purified water, which were mixed uniformly together in a mass ratio of 1:1. After the specimens were formed, they were air-dried under identical conditions. The size of the block was 1 cm 1 cm 3 cm, the density was 0.92 103 kg/m3, the
unconfined compression strength was 2.1 MPa and the tensile strength was 0.525 MPa. The arrangements of the blocky materials are shown in Fig. 2, having pattern a, pattern b and pattern c. The total number of blocks used for each test was 200. Axial loading was applied by displacement control, and the corresponding loading velocity was set at 0.125 mm/min. During the loading process, the axial displacement and corresponding axial force
Fig. 5. Breakage processes (arrangement b): (a) sl ¼0.045 MPa, (b) sl ¼ 0.15 MPa, and (c) sl ¼ 0.45 MPa.
E. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 49 (2012) 125–133
Fig. 6. Breakage processes (arrangement c): (a) sl ¼0.045 MPa, (b) sl ¼ 0.15 MPa, and (c) sl ¼ 0.45 MPa.
Fig. 7. Breakage modes.
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were recorded by the high-pressure triaxial apparatus. The lateral displacements were measured using a micrometer, and the corresponding lateral stress was recorded using a pressure gage. The stress path depicted in Fig. 3 was used to carry out the tests. In Fig. 3, the OA loading stage denotes the axial loads and lateral loads being applied equally, and the AB stage denotes that the lateral pressures remain unchanged while the axial loads are applied until the sample fails. In the following, sa ,ea ,sl and el represent the axial stress, the axial strain, the lateral stress and the lateral strain, respectively.
3. Test results and discussions 3.1. Breakage processes Figs. 4–6 depicts the breakage processes of the block assembly under the loading stress path given in Fig. 3, which correspond to the arrangements a, b and c in Fig. 2, respectively. Three sets of lateral pressures, 0.045 MPa, 0.15 MPa and 0.45 MPa, were applied. For each set of lateral pressures, the processes of breakage of the assembly of blocky materials were different as shown in Figs. 4–6, and they mostly depend on their initial arrangement. When the lateral pressure is lower, initially the blocks move toward each other and become closer. At this point, voids appear between blocks, accompanied by cracking of some of them. The breakage bands are formed gradually. Finally, some new blocks crack, leading to additional conjugated breakage bands. When the lateral pressure is high, the blocks move toward each other. Some of the blocks break or are crushed and the breakage bands are formed step by step. The breakage band is formed by shearing under low lateral pressures and by shearing and crushing under high lateral pressures.
Fig. 8. Breakage modes (a) breakage mode A (- - - - crack resulted by vertically splitting), (b) breakage mode B (- - - - slide plane by shear failure), and (c) breakage mode C (- - - - the profile of the block after locally crushed).
3.2. Breakage modes of the blocky materials The breakage modes of the blocky materials are shown in Fig. 7a–d upon loading. Takei et al. [23] proposed models for breakage modes of chalk bars under one-dimensional compression conditions. For the crushable blocks under biaxial compressions presented in Figs. 4–6, three types of breakage modes are shown in Fig. 8a–c. From the breakage processes presented in Figs. 4–6, we observe that: more of the blocks are vertically split (Breakage Mode A) or failed by shear (Breakage Mode B) when the laterally pressures are low, whereas more blocks are locally crushed (Breakage Mode C) when the laterally pressures are high upon loading. The different breakage modes result from the local stress states of the blocks within the sample. For some sample tested, the three breakage modes A, B and C could be observed simultaneously. Because of differences in loading conditions, the local stresses experienced by individual blocks are different, and thus the blocks fail according to unique breakage modes. 3.3. Stress–strain behavior The stress and strain herein (compression being positive) refer to the average stress and the average strain of the specimen consisting of the assembly of blocky materials. Figs. 9–11 show the stress–strain and volumetric strain–axial strain relationship curves for tests, which correspond to the arrangements a, b and c in Fig. 2, respectively. Deformational features are summarized as follows: (1) for arrangements a and b, the specimen exhibit strain softening under low lateral pressure and strain hardening under high lateral pressure, but for arrangement c the specimen exhibit strain hardening under the lateral pressure ranging from lower value to higher one; and (2) when the lateral
Fig. 9. Stress–strain and axial strain–volumetric strain relationship curves (arrangement a): (a) stress–strain curves and (b) axial strain–volumetric strain curves.
pressure is lower, the volumetric strain is compressive and then expansive with the increase of axial strain. When the lateral pressure is higher, the volumetric strain is compressive all the time.
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4. Breakage mechanism analysis 4.1. Rotation analysis During the loading processes, we observed that the blocks slid, rotated and were crushed, because of the different lateral pressures. When the lateral pressures are low, the blocks located along the breakage bands slide, rotate and are crushed (shown in Fig. 12a corresponding to Fig. 4a, in Fig. 13a corresponding
Fig. 12. Slide, rotation and crush (the ‘‘þ ’’ lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading) (arrangement a): (a) sl ¼0.045 MPa, sa ¼ 2.59 MPa and (b) sl ¼0.45 MPa, sa ¼ 5.04 MPa.
Fig. 10. Stress–strain and axial strain–volumetric strain relationship curves (arrangement b): (a) stress–strain curves, and (b) axial strain–volumetric strain curves.
Fig. 13. Slide, rotation and crush (the ‘‘þ ’’ lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading) (arrangement b): (a) sl ¼0.045 MPa, sa ¼ 1.85 MPa and (b) sl ¼0.45 MPa, sa ¼ 5.31 MPa.
Fig. 11. Stress–strain and axial strain–volumetric strain relationship curves (arrangement c): (a) stress–strain curves, and (b) axial strain–volumetric strain curves.
Fig. 14. Slide, rotation and crush (the ‘‘þ ’’ lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading) (arrangement c): (a) sl ¼0.045 MPa, sa ¼ 1.35 MPa and (b) sl ¼0.45 MPa, sa ¼ 5.46 MPa.
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Table 1 Stress state, axial strain and cumulative crushing ratio (%): arrangement a. Test no.
Axial stress, strain and cumulative crushing ratio (%)
sl ¼ 0.045 MPa sl ¼ 0.15 MPa sl ¼ 0.45 MPa
2.54 MPa, 3.30% and (1.5%) 3.59 MPa, 7.09% and (9.0%) 4.34 MPa, 8.05% and (6.0%)
1.92 MPa, 6.30% and (9.5%) 3.46 MPa, 11.30% and (22.5%) 4.86 MPa, 14.65% and (53.5%)
2.03 MPa, 13.35% and (14.5%) 3.12 MPa, 18.82% and (34.0%) 5.04 MPa, 18.14% and (88.0%)
Table 2 Stress state, axial strain and cumulative crushing ratio (%): arrangement b. Test no.
Axial stress, strain and cumulative crushing ratio (%)
sl ¼ 0.045 MPa sl ¼ 0.15 MPa sl ¼ 0.45 MPa
1.58 MPa, 5.10% and (9.5%) 3.73 MPa, 7.28% and (10.5%) 4.06 MPa, 6.44% and (11.5%)
1.51 MPa, 6.40% and (19.5%) 3.40 MPa, 10.30% and (31.0%) 4.76 MPa, 11.50% and (56.0%)
1.53 MPa, 12.80% and (22.5%) 3.92 MPa, 17.70% and (44.0%) 5.30 MPa, 19.90% and (96.0%)
Table 3 Stress state, axial strain and cumulative crushing ratio (%): arrangement c. Test no.
Axial stress, strain and cumulative crushing ratio (%)
sl ¼ 0.045 MPa sl ¼ 0.15 MPa sl ¼ 0.45 MPa
0.44 MPa, 2.50% and (2.5%) 1.65 MPa, 10.53% and (4.0%) 4.62 MPa, 9.67% and (6.0%)
0.74 MPa, 6.15% and (9.5%) 2.10 MPa, 14.85% and (18.0%) 5.03 MPa, 13.84% and (53.5%)
to Fig. 5a and in Fig. 14a corresponding to Fig. 6a, which were recorded from another surface of the biaxial compressional apparatus through a transparent rigid glass plate). The other blocks which are located outside of the breakage bands only slide and move closer, having very few rotations. This happens because some of the blocks slide and are split first and then are crushed. This is then accompanied by the emergency of big voids between the blocks under low lateral pressure. The blocks are then able to rotate and form the breakage bands, whereas the other blocks outside of the breakage bands only slide and barely rotate. When the lateral pressures are high, the blocks mainly slide and are crushed, undergoing very few rotations (shown in Fig. 12b corresponding to Fig. 4c, in Fig. 13b corresponding to Fig. 5c and in Fig. 14b corresponding to Fig. 6c). This occurs because many of the blocks slide and are crushed in a manner that forces them to move until no room is present between them with high lateral pressure. Finally, the breakage bands, which are wider under low lateral pressure forms. 4.2. Breakage analysis Tables 1–3 list the stress state, axial strain and cumulative crushing ratio of the tested samples corresponding to the breakage processes given in Figs. 4–6, respectively. The cumulative crushing ratio [23], Rcc, is defined as follows: Rcc ¼
ncc , nALL
1.13 MPa, 12.40% and (16.5%) 2.66 MPa, 19.81% and (39.0%) 5.32 MPa, 18.20% and (94.5%)
upon loading. Thus, the energy produced by the external loads is dissipated by the rotation and slip of the blocks, and only a fraction of the energy is dissipated by crushing them.
5. Conclusions An apparatus that can track the breakage process of particulate materials at the meso scale was used here to investigate the breakage and deformation mechanism of crushable blocky materials under biaxial compressional conditions with three types of arrangement. The following conclusions can be drawn: (1) the blocky materials slide, rotate and are crushed under different stress states and, as a result of this process, the breakage bands are formed; (2) the specimens show strain hardening or strain softening due to the loading conditions and stress state, and the volume and lateral deformation are different for different arrangements and loading stress state; (3) the breakage modes of the blocky materials are primarily vertically split, with shear failure and local crushing occurring in response to the local stress states induced; (4) upon loading, under lower lateral pressures the blocky materials mainly slide, rotate and are crushed, whereas under higher lateral pressures they primarily slide, are crushed and barely rotate.
ð1Þ
where ncc is the cumulative number of crushed blocks at each loading stage, and nALL is equal to 200, which is the total number of blocks in the specimen. The numbers of crushed blocks at the each loading stage were visually counted and recorded. From Tables 1–3, we can conclude that upon loading, the cumulative crushing ratios at failure increase greatly with the increase in lateral pressure. For instance (see Table 2), the cumulative crushing ratio at failure is only 22.5% when the lateral stress is 0.045 MPa, whereas it is increased to 96.0% when the lateral stress is 0.45 MPa. This occurs because when the lateral pressure is low, the crushable blocks can easily rotate and slip
Acknowledgments The authors thank the anonymous reviewer and Prof. Robert W. Zimmerman for their careful review, contributions and critics, which led to the improvement of the manuscript. The authors are deeply indebted to the late Prof. Zhujiang Shen of Tsinghua University and Prof. Tielin Chen of Beijing Jiaotong University for their valuable discussion and help in performing the tests. This work was funded by National Science Foundation of China (Grant nos. 51009103 and 90815024) and The Ministry of Water Resources of the People’s Republic of China (Grant no. 200801014).
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