- Email: [email protected]
Breakage and deformation mechanisms of crushable granular materials
Computers and Geotechnics 37 (2010) 723–730
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Technical Communication
Breakage and deformation mechanisms of crushable granular materials Enlong Liu State Key Laboratory of Hydraulics and Mountain River Engineering, School of Hydraulic and Hydropower Engineering, Sichuan University, Chengdu 610065, PR China
a r t i c l e
i n f o
Article history: Received 18 September 2009 Received in revised form 20 April 2010 Accepted 25 April 2010 Available online 18 May 2010 Keywords: Biaxial tests Crushable granular materials Round rods Breakage mechanism
a b s t r a c t Biaxial compressional tests with two types of stress paths were carried out on an assembly of round bars, which can be crushed to investigate the breakage and deformation mechanisms of granular materials at the mesoscale. The following was found experimentally: (1) upon loading, the crushable rods slide, rotate and break, and finally the breakage band forms for the two types of stress paths and different stress states; and (2) for the axial loading stress path, the round rods mainly fail in the vertically split mode and laterally crushed mode. However, for the lateral unloading stress path, the round rods fail with the combination mode of locally crushed and vertically split. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Granular materials are composed of distinct particles interacting with each other and associated voids. Upon loading, the particles may slide, rotate and crush or break. The deformation and breakage mechanism of granular materials is complicated and many experimental and theoretical studies have been carried out on these materials. Rowe [17] 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 [8] reported experiments on optically sensitive granular materials consisting of crushed glass particles using a double shear apparatus in order to observe the pattern of principal stress trajectories occurring at large plastic deformations. Chappell [5,6] 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. [14–16] 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 [2] presented a series of theoretical developments and numerical experiments directed at quantifying important features of the micromechanical behavior of granular media by introducing E-mail address: [email protected] 0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.04.009
average characteristics of fabric anisotropy and certain statistical averages of contact forces. Calvetti et al. [4] carried out several 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. Additionally, Lanier [11] 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). Furthermore, Wolf et al. [19] 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. [7] investigated the stress–strain behavior and failure of a cohesive granular material by experiments on the assembly of aluminium rods glued together by means of an epoxy resin under simple loading conditions. Finally, Alshibli and Roussel [1] tested on the spherical beads to study load oscillation in granular materials under axisymmetric triaxial loading condition at 25, 100, 250 and 400 kPa confining pressures. The experimental studies mentioned above mainly focused on the microscopical deformation mechanisms of sliding and rolling or rotations of particles occurring in the materials with great stiffness. Upon loading, however, the particles of an element of soil can be crushed or broken [9,10,13,12,3], which affects the strength and stress–strain behavior of soil elements greatly. Although a few numerical methods have been employed to study the crush mechanism of granular materials, the simulation results need to be verified by experimental results. To formulate the constitutive
724
E. Liu / Computers and Geotechnics 37 (2010) 723–730
model of granular materials, which can take into account the influence of microstructure deformation mechanisms, tests on granular materials at the mesoscale are required and carried out here. An apparatus that can track the mesoscopical breakage process of granular materials is employed here. Biaxial compressional tests with two different types of stress paths were conducted on an assembly of round bars that are crushable. Finally, we investigated the deformation mechanism and breakage processes of granular materials comprehensively.
2. Test methods and procedures 2.1. Test set-up Fig. 1 presents the experimental apparatus employed to carry out the biaxial compressional tests. By modifying the sample cell of the high pressure triaxial equipment, an apparatus was devised
σa (a)
σl
σl
σa Fig. 2. Pattern of arrangement.
(a)
(b)
a
l
/2
D
A
45
( a+ l)/2
O
(b)
l
/2
C
a
45 B
A
45
( a+ l)/2
O
Fig. 3. Stress paths: (a) axial loading and (b) the lateral unloading.
Table 1 The stress states at the beginning of unloading for samples LUT1 and LUT2. 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, (12) the high pressure triaxial loading system.
No. of the samples tested
ra (MPa)
rl (MPa)
LUT1 LUT2
2.100 2.450
0.600 0.630
E. Liu / Computers and Geotechnics 37 (2010) 723–730
that can be axially loaded using the loading system of high pressure triaxial equipment and laterally loaded by air pressure. Two transparent rigid glass plates in parallel set with 3-cm distance were to keep the specimen under biaxial loading conditions. The size of the specimen chamber was 20 cm 10 cm 3 cm. During the process of loading, a series of pictures could be taken to track the breakage processes of the specimens at the mesoscale.
2.2. Test materials and stress paths The round rods used in the tests were made of gesso and purified water, which were mixed uniformly together according to a mass ratio of 1:1. After the specimens were formed, they were air-dried under the same conditions. The diameter of the round rod was 1 cm, the density was 0.92 103 kg/m3, and the uncon-
(a)
εa= 3.10% σa=0.77MPa
εa=11.56%
σa= 0.76MPa
εa= 19.20% σa= 1.09MPa
εa= 8.57% σa=1.82MPa
εa=16.00% σa= 2.24MPa
εa= 22.00% σa= 2.52MPa
εa= 15.10% σa= 3.40MPa
εa= 22.00% σa= 3.36MPa
(b)
(c)
εa= 7.86% σa= 2.56MPa
725
Fig. 4. Breakage processes (the axial loading tests): (a) rl = 0.03 MPa, (b) rl = 0.15 MPa and (c) rl = 0.45 MPa.
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E. Liu / Computers and Geotechnics 37 (2010) 723–730
(a)
εa= 7.05%
σl= 0.60MPa
εa= 13.35%
σl= 0.12 MPa
εa= 21.53%
σl= 0.09 MPa
εa= 13.30%
σl= 0.18MPa
εa= 22.83%
σl= 0.09MPa
(b)
εa= 6.77%
σl= 0.63MPa
Fig. 5. Breakage processes (the lateral unloading tests): (a) LUT1 ra = 2.1 MPa and (b) LUT2 ra = 2.45 MPa.
fined compression strength was 2.1 MPa. The arrangement of the round rods was simple cubic packing, as shown in Fig. 2. The total number of round rods used for each test was 200. The axial loading was applied by displacement-control, and the corresponding loading velocity was set to be 0.125 mm/min. During the process of loading, the axial displacement and the corresponding axial force could be recorded by the high pressure triaxial apparatus, the lateral displacements could be measured by a micrometer, and the corresponding lateral stress could be recorded by the pressure gauge. Two different types of stress paths, demonstrated in Fig. 3a and b were employed to carry out the tests. In Fig. 3a, the OA loading stage denotes the axial loads and lateral loads applied equally, and the AD stages denote that the lateral pressures remain unchanged while the axial loads are applied, until the sample fails. In Fig. 3b, the OA loading stage also denotes the axial loads and lateral loads applied equally, and the AB stage denotes that the lateral pressures remain unchanged while the axial load is applied to some value at which the sample has initial lateral and axial stresses and the lateral stress reduction phase takes place. The BC stage denotes that the axial loads are kept constant while the lateral pressures are unloaded until the sample fails. In the following,
Fig. 6. Breakage modes (the axial loading test): (a–d).
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E. Liu / Computers and Geotechnics 37 (2010) 723–730
ra ; ea ; rl and el represent the axial stress, the axial strain, the lat-
Fig. 7. Breakage modes (the lateral unloading test): (a and b).
(a)
eral stress and the lateral strain, respectively. For the sake of brevity, the axially loading test refers to the one loaded along the stress path presented in Fig. 3a in which the sample is loaded along the paths OA and AD until failure, and the lateral unloading test refers to the one loaded along the stress path presented in Fig. 3b in which the sample is loaded along the paths OA, AB and BC until failure. For the axial loading stress path given in Fig. 3a, three sets of lateral pressures, 0.03 MPa, 0.15 MPa and 0.45 MPa, were applied. For the lateral unloading stress path given in Fig. 3b, two sets of tests were conducted, called LUT1 and LUT2, and the stress states (B point in Fig. 3b) when the lateral pressures started to unload are presented in Table 1. 3. Test results and discussion 3.1. Breakage processes Fig. 4a–c shows the breakage processes of the assembly of round rods under the axial loading stress path given in Fig. 3a. For the three sets of lateral pressures, the processes of breakage of the assembly of round rods are different under different lateral pressures as shown in Fig. 4. When the lateral pressure is low, the round rods first slide and rotate, accompanied by the emergence of some big pores between the rods. Then, the induced pores gradually become smaller with the crushing of some rods. Finally, the shear bands or breakage bands are formed. When the lateral pressure is high, the round rods first slide slightly and move close to each other. Then, the round rods break or are crushed, and the breakage bands are formed at last. The breakage band is formed by shearing under lower lateral pressures and by shearing and crushing under higher lateral pressures. Fig. 5a and b shows the breakage processes of the assembly of round rods under the lateral unloading stress path given in Fig. 3b. For the tests of LUT1 and LUT2 shown in Fig. 5, when unloading the lateral pressure, first the round rods slide and slightly rotate, and a few of them crack. Then the round rods slide and crack with the crushing of the round rods. Finally, the breakage bands are formed. The breakage bands are mainly formed by shearing.
(b)
(c)
Fig. 8. Breakage modes: (a) breakage mode A, (b) breakage mode B and (c) breakage mode C.
3.1.1. Breakage modes of the round rods The breakage modes of the round rods are shown in Fig. 6a–d under the axial loading stress path and in Fig. 7a and b under the lateral unloading stress path. Takei et al. [18] proposed some breakage modes of chalk bars under one-dimensional compression conditions. For the round rods under biaxial compression, we can have three types of breakage modes, which are shown in Fig. 8a– c. From the breakage processes provided in Figs. 4 and 5, the amount of failed rods corresponding to the breakage models is summarized in Table 2. From these data, we can conclude that
Table 2 The amount of failed rods corresponding to breakage models. Test no.
The total number of failed rods
The number of failed rods (breakage mode A)
The number of failed rods (breakage mode B)
The number of failed rods (breakage mode C)
The axial loading stress path
rl = 0.03 MPa rl = 0.15 MPa rl = 0.45 MPa
21 66 182
15 32 8
4 28 167
2 6 7
The lateral unloading stress path
LUT1 LUT2
165 163
5 7
2 4
158 152
Note: the total number of round rods composing the specimen is 200.
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E. Liu / Computers and Geotechnics 37 (2010) 723–730
Fig. 9. Stress–strain and axial strain–volumetric relationship curves (the axial loading tests): (a) stress–strain curves and (b) axial strain–volumetric curves. Fig. 11. Slide, rotation and crushing (the ‘‘+” lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading): (a) ra = 1.26 MPa, rl = 0.03 MPa and (b) ra = 3.92 MPa, rl = 0.45 MPa.
Fig. 10. Stress–strain and axial strain–volumetric relationship curves (the lateral unloading tests): (a) stress–strain curves and (b) axial strain–volumetric curves.
more of the round rods are vertically split (breakage mode A) when the lateral pressures are low and more of them are laterally crushed (breakage mode B) when the lateral pressures are high under the axial loading stress path. More of the round rods fail with the combination of local crushing and vertical splitting (breakage mode C) under the lateral unloading stress path. The different breakage modes result from the local stress states of the round rods within the sample. For some samples tested, the three breakage modes A–C could be observed simultaneously. Occurring to the differences in the loading conditions, the local stresses of different round rods are different, and thus, the round rods fail according to different breakage modes.
Fig. 12. Slide, rotation and crushing (the ‘‘+” lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading): (a) LUT1 ra = 2.1 MPa, rl = 0.09 MPa (b) LUT2 ra = 2.45 MPa, rl = 0.12 MPa.
3.1.2. Stress–strain behavior Because strain localization can happen during the course of loading, the stress and strain here refer to the average stress and
E. Liu / Computers and Geotechnics 37 (2010) 723–730 Table 3 Stress state, axial strain and cumulative crushing ratio (%) of the axially loading stress path. Test no. (MPa)
Axial stress, strain and cumulative crushing ratio (%)
rl = 0.03
0.77 MPa, 3.10% (1%)
0.76 MPa, 11.56% (6%)
1.09 MPa, 19.20% (10.5%)
rl = 0.15
1.82 MPa, 8.57% (12.5%)
2.24 MPa, 16.0% (28.5%)
2.52 MPa, 22.0% (33%)
rl = 0.45
2.56 MPa, 7.86% (5.5%)
3.40 MPa, 15.10% (59.5%)
3.36 MPa, 22.00% (91%)
Table 4 Stress state, axial strain and cumulative crushing ratio (%) of the laterally unloading stress path. Test no.
Lateral stress, strain and cumulative crushing ratio (%)
LUT1
ra = 2.1 MPa
0.60 MPa, 7.05% (2.5%)
0.12 MPa, 13.35% (76%)
0.09 MPa, 21.53% (82.5%)
LUT2
ra = 2.45 MPa
0.63 MPa, 6.77% (3%)
0.18 MPa, 13.30% (69%)
0.09 MPa, 22.83% (81.5%)
the average strain of the specimen consisting of the assembly of round rods. Fig. 9a and b shows the stress–strain and volumetric strain–axial strain relationship curves of the axial loading tests performed, respectively. Some deformational features can be summarized as follows: (1) the specimen will show strain hardening under low lateral pressure and strain softening under high lateral pressure; and (2) when the lateral pressure is low, the volumetric strain is compressive and then expansive with the increase of axial strain, but when the lateral pressure is high, the volumetric strain is compressive all the time. Fig. 10a and b shows the stress–strain and volumetric strain– axial strain relationship curves, respectively, of the lateral unloading tests conducted. Some deformational features can be summarized as follows: (1) after the lateral pressures are unloaded, the specimens show strain hardening; and (2) when the lateral pressures are unloaded, the volume of the specimens will contract. 3.2. Breakage mechanism analysis 3.2.1. Rotation analysis During the loading processes under the axial loading conditions, it was observed experimentally that the round rods slid, rotated and were crushed, which was influenced greatly by the different lateral pressures. When the lateral pressures are low, the round rods mainly slide and rotate (shown in Fig. 11a corresponding to Fig. 4a, which was recorded from another surface of the biaxial compressional apparatus through a transparent rigid glass plate), which results because the specimens have enough strength to resist the crushing or cracking of the round rods with relative low lateral pressures but can still easily slide and rotate. When the lateral pressures are high, the round rods will mainly be crushed and barely rotate (shown in Fig. 11b corresponding to Fig. 4c, which was also recorded from the other surface of the biaxial compressional apparatus through a transparent rigid glass plate). Finally, the breakage bands form, which are wider with a low lateral pressure than with the high lateral one. During unloading of the lateral pressure, it was also observed experimentally that the round rods slide, rotate and break or are crushed, as shown in Fig. 12a of sample LUT1 corresponding to Fig. 5a, and in Fig. 12b of sample LUT2 corresponding to Fig. 5b.
729
During the stage of loading the axial loads, the round rods slide, move close together and almost rotate, and the lateral deformation and volume of the specimens are contractive. After unloading the lateral pressures, first the round rods continue to slide and rotate, and the lateral deformation contracts. Then some of the round rods crack and fail. Continuing to unload the lateral pressure, the round rods rotate increasingly, and the pores between the round rods appear. The lateral deformation begins to expand, and many round rods break and fail. At last, the lateral pressure is reduced to a very small value, the specimen shows strain hardening, and the breakage bands are formed. 3.2.2. Breakage analysis Tables 3 and 4 list the stress state, axial strain and cumulative crushing ratio of the tested samples corresponding to the breakage processes given in Figs. 4 and 5 under the axial loading stress path and axial unloading stress path, respectively. The cumulative crushing ratio [18], Rcc, is defined as follows:
Rcc ¼
ncc ; nALL
ð1Þ
where ncc is the cumulative number of crushed rods at each loading stage, and nALL is equal to 200, which is the total number of the round rods composing the specimen. The numbers of crushed rods at the each loading stage were visually counted with the recorded visual data. From Table 3 we can conclude that for the axial loading stress path test, the cumulative crushing ratios at failure increase greatly with the increase of lateral pressure. For example, the cumulative crushing ratio at failure is only 10.5% when the lateral stress is 0.03 MPa, whereas the cumulative crushing ratio at failure is increased to 91% when the lateral stress is 0.45 MPa, the reason for which is that when the laterally pressure is low, the round rods easily rotate and slip upon loading; thus, the energy produced by the external loads is mainly dissipated by the rotation and slip of the round rods, and only a small amount of the energy is dissipated by the crushing of the round rods. From Table 4, we conclude that for the lateral unloading stress path test, the cumulative crushing ratio at failure can have a higher value of 82.5% and 81.5% for the two tests, LUT1 and LUT2, respectively, the reason for which is that before unloading the lateral pressure, many of the round rods slip and move close to each other, while during the unloading of the lateral pressure, some of the rods rotate, slip and break, forming the breakage band or shear band, and some other rods are crushed to dissipate the energy produced by external loads. 4. Conclusions An apparatus that can track the breakage process of granular materials at the mesoscale was employed here to investigate the breakage and deformation mechanism of crushable materials under biaxial compressional conditions under two types of loading stress paths. The following conclusions can be drawn: (1) the round rods will slide, rotate and be crush under different stress paths and stress states, and finally, the breakage bands can be formed; (2) the specimens will show strain hardening or strain softening due to the loading conditions and stress state, and the volume and lateral deformation are different for different stress paths and loading stress states; (3) the breakage modes of the round rods are mainly vertically split and laterally crushed for the axial loading stress path and are both locally crushed and vertically split for the lateral unloading stress path; and (4) for the axial loading conditions with low lateral pressures, the round rods mainly slide and rotate, and under high lateral pressures, the round
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E. Liu / Computers and Geotechnics 37 (2010) 723–730
rods are mainly crushed and barely rotate. During the unloading of the lateral pressure conditions, the round rods first slide, move close together and almost do not rotate during the stage of loading the axial loads. Then, they continue to slide and rotate after unloading the lateral pressures until many of them break or are crushed. Acknowledgments The author is deeply indebted to Prof. Zhujiang Shen of Tsinghua University for his valuable discussion and help in performing the tests and also thanks Dr. H.L. Xing of The University of Queensland for his great help in writing this manuscript. This work was funded by The Ministry of Water Resources of the People’s Republic of China (Grant No. 200801014) and National Science Foundation of China (Grant No. 10272062). References [1] Alshibli KA, Roussel LE. Experimental investigation of slip-stick behavior in granular materials. Int J Numer Anal Methods Geomech 2006;30:1391–407. [2] Bathurst RJ, Rothenburg L. Observations on stress–force–fabric relationships in idealized granular materials. Mech Mater 1990;9:65–80. [3] Bartake PP, Singh DN. A generalized methodology for determination of crushing strength of granular materials. Geotech Geol Eng 2007;25:203–13. [4] 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. [5] Chappell BA. Deformational response in discontinua. Int J Rock Mech Min Sci Geomech Abstr 1979;16:377–90.
[6] Chappell BA. Load distribution and redistribution in dsicontinua. Int J Rock Mech Min Sci Geomech Abstr 1979;16:391–9. [7] 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. [8] Drescher A. An experimental investigation of flow rules for granular materials using optically sensitive glass particles. Geotechnique 1976;26(4):591–601. [9] Hardin BO. Crushing of soil particles. J Geotech Eng 1984;111(10):1176–92. [10] Hagerty MM, Hite DR, Ullrich CR, et al. One-dimensional high-pressure compression of granular media. J Geotech Eng 1991;119(1):11–8. [11] Lanier L. Micro-mechanisms of deformation in granular materials: experiments and numerical results. In: Vermeer P.A. et al. editors. LNP 568; 2001. p. 163–72. [12] Mcdowell GR. On the yielding and plastic compression of sand. Soils Found 2002;42(1):139–45. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] Takei M, Kusakabe O, Hayashi T. Time-dependent behavior of crushable materials in one-dimensional compression tests. Soils Found 2001;41(1):97–121. }nig D, Triantafyllidis T. Experimental investigation of shear band [19] Wolf H, Ko patterns in granular material. J Struct Geol 2003;25:1229–40.
Contents lists available at ScienceDirect
Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Technical Communication
Breakage and deformation mechanisms of crushable granular materials Enlong Liu State Key Laboratory of Hydraulics and Mountain River Engineering, School of Hydraulic and Hydropower Engineering, Sichuan University, Chengdu 610065, PR China
a r t i c l e
i n f o
Article history: Received 18 September 2009 Received in revised form 20 April 2010 Accepted 25 April 2010 Available online 18 May 2010 Keywords: Biaxial tests Crushable granular materials Round rods Breakage mechanism
a b s t r a c t Biaxial compressional tests with two types of stress paths were carried out on an assembly of round bars, which can be crushed to investigate the breakage and deformation mechanisms of granular materials at the mesoscale. The following was found experimentally: (1) upon loading, the crushable rods slide, rotate and break, and finally the breakage band forms for the two types of stress paths and different stress states; and (2) for the axial loading stress path, the round rods mainly fail in the vertically split mode and laterally crushed mode. However, for the lateral unloading stress path, the round rods fail with the combination mode of locally crushed and vertically split. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Granular materials are composed of distinct particles interacting with each other and associated voids. Upon loading, the particles may slide, rotate and crush or break. The deformation and breakage mechanism of granular materials is complicated and many experimental and theoretical studies have been carried out on these materials. Rowe [17] 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 [8] reported experiments on optically sensitive granular materials consisting of crushed glass particles using a double shear apparatus in order to observe the pattern of principal stress trajectories occurring at large plastic deformations. Chappell [5,6] 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. [14–16] 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 [2] presented a series of theoretical developments and numerical experiments directed at quantifying important features of the micromechanical behavior of granular media by introducing E-mail address: [email protected] 0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.04.009
average characteristics of fabric anisotropy and certain statistical averages of contact forces. Calvetti et al. [4] carried out several 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. Additionally, Lanier [11] 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). Furthermore, Wolf et al. [19] 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. [7] investigated the stress–strain behavior and failure of a cohesive granular material by experiments on the assembly of aluminium rods glued together by means of an epoxy resin under simple loading conditions. Finally, Alshibli and Roussel [1] tested on the spherical beads to study load oscillation in granular materials under axisymmetric triaxial loading condition at 25, 100, 250 and 400 kPa confining pressures. The experimental studies mentioned above mainly focused on the microscopical deformation mechanisms of sliding and rolling or rotations of particles occurring in the materials with great stiffness. Upon loading, however, the particles of an element of soil can be crushed or broken [9,10,13,12,3], which affects the strength and stress–strain behavior of soil elements greatly. Although a few numerical methods have been employed to study the crush mechanism of granular materials, the simulation results need to be verified by experimental results. To formulate the constitutive
724
E. Liu / Computers and Geotechnics 37 (2010) 723–730
model of granular materials, which can take into account the influence of microstructure deformation mechanisms, tests on granular materials at the mesoscale are required and carried out here. An apparatus that can track the mesoscopical breakage process of granular materials is employed here. Biaxial compressional tests with two different types of stress paths were conducted on an assembly of round bars that are crushable. Finally, we investigated the deformation mechanism and breakage processes of granular materials comprehensively.
2. Test methods and procedures 2.1. Test set-up Fig. 1 presents the experimental apparatus employed to carry out the biaxial compressional tests. By modifying the sample cell of the high pressure triaxial equipment, an apparatus was devised
σa (a)
σl
σl
σa Fig. 2. Pattern of arrangement.
(a)
(b)
a
l
/2
D
A
45
( a+ l)/2
O
(b)
l
/2
C
a
45 B
A
45
( a+ l)/2
O
Fig. 3. Stress paths: (a) axial loading and (b) the lateral unloading.
Table 1 The stress states at the beginning of unloading for samples LUT1 and LUT2. 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, (12) the high pressure triaxial loading system.
No. of the samples tested
ra (MPa)
rl (MPa)
LUT1 LUT2
2.100 2.450
0.600 0.630
E. Liu / Computers and Geotechnics 37 (2010) 723–730
that can be axially loaded using the loading system of high pressure triaxial equipment and laterally loaded by air pressure. Two transparent rigid glass plates in parallel set with 3-cm distance were to keep the specimen under biaxial loading conditions. The size of the specimen chamber was 20 cm 10 cm 3 cm. During the process of loading, a series of pictures could be taken to track the breakage processes of the specimens at the mesoscale.
2.2. Test materials and stress paths The round rods used in the tests were made of gesso and purified water, which were mixed uniformly together according to a mass ratio of 1:1. After the specimens were formed, they were air-dried under the same conditions. The diameter of the round rod was 1 cm, the density was 0.92 103 kg/m3, and the uncon-
(a)
εa= 3.10% σa=0.77MPa
εa=11.56%
σa= 0.76MPa
εa= 19.20% σa= 1.09MPa
εa= 8.57% σa=1.82MPa
εa=16.00% σa= 2.24MPa
εa= 22.00% σa= 2.52MPa
εa= 15.10% σa= 3.40MPa
εa= 22.00% σa= 3.36MPa
(b)
(c)
εa= 7.86% σa= 2.56MPa
725
Fig. 4. Breakage processes (the axial loading tests): (a) rl = 0.03 MPa, (b) rl = 0.15 MPa and (c) rl = 0.45 MPa.
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(a)
εa= 7.05%
σl= 0.60MPa
εa= 13.35%
σl= 0.12 MPa
εa= 21.53%
σl= 0.09 MPa
εa= 13.30%
σl= 0.18MPa
εa= 22.83%
σl= 0.09MPa
(b)
εa= 6.77%
σl= 0.63MPa
Fig. 5. Breakage processes (the lateral unloading tests): (a) LUT1 ra = 2.1 MPa and (b) LUT2 ra = 2.45 MPa.
fined compression strength was 2.1 MPa. The arrangement of the round rods was simple cubic packing, as shown in Fig. 2. The total number of round rods used for each test was 200. The axial loading was applied by displacement-control, and the corresponding loading velocity was set to be 0.125 mm/min. During the process of loading, the axial displacement and the corresponding axial force could be recorded by the high pressure triaxial apparatus, the lateral displacements could be measured by a micrometer, and the corresponding lateral stress could be recorded by the pressure gauge. Two different types of stress paths, demonstrated in Fig. 3a and b were employed to carry out the tests. In Fig. 3a, the OA loading stage denotes the axial loads and lateral loads applied equally, and the AD stages denote that the lateral pressures remain unchanged while the axial loads are applied, until the sample fails. In Fig. 3b, the OA loading stage also denotes the axial loads and lateral loads applied equally, and the AB stage denotes that the lateral pressures remain unchanged while the axial load is applied to some value at which the sample has initial lateral and axial stresses and the lateral stress reduction phase takes place. The BC stage denotes that the axial loads are kept constant while the lateral pressures are unloaded until the sample fails. In the following,
Fig. 6. Breakage modes (the axial loading test): (a–d).
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ra ; ea ; rl and el represent the axial stress, the axial strain, the lat-
Fig. 7. Breakage modes (the lateral unloading test): (a and b).
(a)
eral stress and the lateral strain, respectively. For the sake of brevity, the axially loading test refers to the one loaded along the stress path presented in Fig. 3a in which the sample is loaded along the paths OA and AD until failure, and the lateral unloading test refers to the one loaded along the stress path presented in Fig. 3b in which the sample is loaded along the paths OA, AB and BC until failure. For the axial loading stress path given in Fig. 3a, three sets of lateral pressures, 0.03 MPa, 0.15 MPa and 0.45 MPa, were applied. For the lateral unloading stress path given in Fig. 3b, two sets of tests were conducted, called LUT1 and LUT2, and the stress states (B point in Fig. 3b) when the lateral pressures started to unload are presented in Table 1. 3. Test results and discussion 3.1. Breakage processes Fig. 4a–c shows the breakage processes of the assembly of round rods under the axial loading stress path given in Fig. 3a. For the three sets of lateral pressures, the processes of breakage of the assembly of round rods are different under different lateral pressures as shown in Fig. 4. When the lateral pressure is low, the round rods first slide and rotate, accompanied by the emergence of some big pores between the rods. Then, the induced pores gradually become smaller with the crushing of some rods. Finally, the shear bands or breakage bands are formed. When the lateral pressure is high, the round rods first slide slightly and move close to each other. Then, the round rods break or are crushed, and the breakage bands are formed at last. The breakage band is formed by shearing under lower lateral pressures and by shearing and crushing under higher lateral pressures. Fig. 5a and b shows the breakage processes of the assembly of round rods under the lateral unloading stress path given in Fig. 3b. For the tests of LUT1 and LUT2 shown in Fig. 5, when unloading the lateral pressure, first the round rods slide and slightly rotate, and a few of them crack. Then the round rods slide and crack with the crushing of the round rods. Finally, the breakage bands are formed. The breakage bands are mainly formed by shearing.
(b)
(c)
Fig. 8. Breakage modes: (a) breakage mode A, (b) breakage mode B and (c) breakage mode C.
3.1.1. Breakage modes of the round rods The breakage modes of the round rods are shown in Fig. 6a–d under the axial loading stress path and in Fig. 7a and b under the lateral unloading stress path. Takei et al. [18] proposed some breakage modes of chalk bars under one-dimensional compression conditions. For the round rods under biaxial compression, we can have three types of breakage modes, which are shown in Fig. 8a– c. From the breakage processes provided in Figs. 4 and 5, the amount of failed rods corresponding to the breakage models is summarized in Table 2. From these data, we can conclude that
Table 2 The amount of failed rods corresponding to breakage models. Test no.
The total number of failed rods
The number of failed rods (breakage mode A)
The number of failed rods (breakage mode B)
The number of failed rods (breakage mode C)
The axial loading stress path
rl = 0.03 MPa rl = 0.15 MPa rl = 0.45 MPa
21 66 182
15 32 8
4 28 167
2 6 7
The lateral unloading stress path
LUT1 LUT2
165 163
5 7
2 4
158 152
Note: the total number of round rods composing the specimen is 200.
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Fig. 9. Stress–strain and axial strain–volumetric relationship curves (the axial loading tests): (a) stress–strain curves and (b) axial strain–volumetric curves. Fig. 11. Slide, rotation and crushing (the ‘‘+” lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading): (a) ra = 1.26 MPa, rl = 0.03 MPa and (b) ra = 3.92 MPa, rl = 0.45 MPa.
Fig. 10. Stress–strain and axial strain–volumetric relationship curves (the lateral unloading tests): (a) stress–strain curves and (b) axial strain–volumetric curves.
more of the round rods are vertically split (breakage mode A) when the lateral pressures are low and more of them are laterally crushed (breakage mode B) when the lateral pressures are high under the axial loading stress path. More of the round rods fail with the combination of local crushing and vertical splitting (breakage mode C) under the lateral unloading stress path. The different breakage modes result from the local stress states of the round rods within the sample. For some samples tested, the three breakage modes A–C could be observed simultaneously. Occurring to the differences in the loading conditions, the local stresses of different round rods are different, and thus, the round rods fail according to different breakage modes.
Fig. 12. Slide, rotation and crushing (the ‘‘+” lines are mutually orthogonal in the vertical and horizontal directions at the beginning of loading): (a) LUT1 ra = 2.1 MPa, rl = 0.09 MPa (b) LUT2 ra = 2.45 MPa, rl = 0.12 MPa.
3.1.2. Stress–strain behavior Because strain localization can happen during the course of loading, the stress and strain here refer to the average stress and
E. Liu / Computers and Geotechnics 37 (2010) 723–730 Table 3 Stress state, axial strain and cumulative crushing ratio (%) of the axially loading stress path. Test no. (MPa)
Axial stress, strain and cumulative crushing ratio (%)
rl = 0.03
0.77 MPa, 3.10% (1%)
0.76 MPa, 11.56% (6%)
1.09 MPa, 19.20% (10.5%)
rl = 0.15
1.82 MPa, 8.57% (12.5%)
2.24 MPa, 16.0% (28.5%)
2.52 MPa, 22.0% (33%)
rl = 0.45
2.56 MPa, 7.86% (5.5%)
3.40 MPa, 15.10% (59.5%)
3.36 MPa, 22.00% (91%)
Table 4 Stress state, axial strain and cumulative crushing ratio (%) of the laterally unloading stress path. Test no.
Lateral stress, strain and cumulative crushing ratio (%)
LUT1
ra = 2.1 MPa
0.60 MPa, 7.05% (2.5%)
0.12 MPa, 13.35% (76%)
0.09 MPa, 21.53% (82.5%)
LUT2
ra = 2.45 MPa
0.63 MPa, 6.77% (3%)
0.18 MPa, 13.30% (69%)
0.09 MPa, 22.83% (81.5%)
the average strain of the specimen consisting of the assembly of round rods. Fig. 9a and b shows the stress–strain and volumetric strain–axial strain relationship curves of the axial loading tests performed, respectively. Some deformational features can be summarized as follows: (1) the specimen will show strain hardening under low lateral pressure and strain softening under high lateral pressure; and (2) when the lateral pressure is low, the volumetric strain is compressive and then expansive with the increase of axial strain, but when the lateral pressure is high, the volumetric strain is compressive all the time. Fig. 10a and b shows the stress–strain and volumetric strain– axial strain relationship curves, respectively, of the lateral unloading tests conducted. Some deformational features can be summarized as follows: (1) after the lateral pressures are unloaded, the specimens show strain hardening; and (2) when the lateral pressures are unloaded, the volume of the specimens will contract. 3.2. Breakage mechanism analysis 3.2.1. Rotation analysis During the loading processes under the axial loading conditions, it was observed experimentally that the round rods slid, rotated and were crushed, which was influenced greatly by the different lateral pressures. When the lateral pressures are low, the round rods mainly slide and rotate (shown in Fig. 11a corresponding to Fig. 4a, which was recorded from another surface of the biaxial compressional apparatus through a transparent rigid glass plate), which results because the specimens have enough strength to resist the crushing or cracking of the round rods with relative low lateral pressures but can still easily slide and rotate. When the lateral pressures are high, the round rods will mainly be crushed and barely rotate (shown in Fig. 11b corresponding to Fig. 4c, which was also recorded from the other surface of the biaxial compressional apparatus through a transparent rigid glass plate). Finally, the breakage bands form, which are wider with a low lateral pressure than with the high lateral one. During unloading of the lateral pressure, it was also observed experimentally that the round rods slide, rotate and break or are crushed, as shown in Fig. 12a of sample LUT1 corresponding to Fig. 5a, and in Fig. 12b of sample LUT2 corresponding to Fig. 5b.
729
During the stage of loading the axial loads, the round rods slide, move close together and almost rotate, and the lateral deformation and volume of the specimens are contractive. After unloading the lateral pressures, first the round rods continue to slide and rotate, and the lateral deformation contracts. Then some of the round rods crack and fail. Continuing to unload the lateral pressure, the round rods rotate increasingly, and the pores between the round rods appear. The lateral deformation begins to expand, and many round rods break and fail. At last, the lateral pressure is reduced to a very small value, the specimen shows strain hardening, and the breakage bands are formed. 3.2.2. Breakage analysis Tables 3 and 4 list the stress state, axial strain and cumulative crushing ratio of the tested samples corresponding to the breakage processes given in Figs. 4 and 5 under the axial loading stress path and axial unloading stress path, respectively. The cumulative crushing ratio [18], Rcc, is defined as follows:
Rcc ¼
ncc ; nALL
ð1Þ
where ncc is the cumulative number of crushed rods at each loading stage, and nALL is equal to 200, which is the total number of the round rods composing the specimen. The numbers of crushed rods at the each loading stage were visually counted with the recorded visual data. From Table 3 we can conclude that for the axial loading stress path test, the cumulative crushing ratios at failure increase greatly with the increase of lateral pressure. For example, the cumulative crushing ratio at failure is only 10.5% when the lateral stress is 0.03 MPa, whereas the cumulative crushing ratio at failure is increased to 91% when the lateral stress is 0.45 MPa, the reason for which is that when the laterally pressure is low, the round rods easily rotate and slip upon loading; thus, the energy produced by the external loads is mainly dissipated by the rotation and slip of the round rods, and only a small amount of the energy is dissipated by the crushing of the round rods. From Table 4, we conclude that for the lateral unloading stress path test, the cumulative crushing ratio at failure can have a higher value of 82.5% and 81.5% for the two tests, LUT1 and LUT2, respectively, the reason for which is that before unloading the lateral pressure, many of the round rods slip and move close to each other, while during the unloading of the lateral pressure, some of the rods rotate, slip and break, forming the breakage band or shear band, and some other rods are crushed to dissipate the energy produced by external loads. 4. Conclusions An apparatus that can track the breakage process of granular materials at the mesoscale was employed here to investigate the breakage and deformation mechanism of crushable materials under biaxial compressional conditions under two types of loading stress paths. The following conclusions can be drawn: (1) the round rods will slide, rotate and be crush under different stress paths and stress states, and finally, the breakage bands can be formed; (2) the specimens will show strain hardening or strain softening due to the loading conditions and stress state, and the volume and lateral deformation are different for different stress paths and loading stress states; (3) the breakage modes of the round rods are mainly vertically split and laterally crushed for the axial loading stress path and are both locally crushed and vertically split for the lateral unloading stress path; and (4) for the axial loading conditions with low lateral pressures, the round rods mainly slide and rotate, and under high lateral pressures, the round
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rods are mainly crushed and barely rotate. During the unloading of the lateral pressure conditions, the round rods first slide, move close together and almost do not rotate during the stage of loading the axial loads. Then, they continue to slide and rotate after unloading the lateral pressures until many of them break or are crushed. Acknowledgments The author is deeply indebted to Prof. Zhujiang Shen of Tsinghua University for his valuable discussion and help in performing the tests and also thanks Dr. H.L. Xing of The University of Queensland for his great help in writing this manuscript. This work was funded by The Ministry of Water Resources of the People’s Republic of China (Grant No. 200801014) and National Science Foundation of China (Grant No. 10272062). References [1] Alshibli KA, Roussel LE. Experimental investigation of slip-stick behavior in granular materials. Int J Numer Anal Methods Geomech 2006;30:1391–407. [2] Bathurst RJ, Rothenburg L. Observations on stress–force–fabric relationships in idealized granular materials. Mech Mater 1990;9:65–80. [3] Bartake PP, Singh DN. A generalized methodology for determination of crushing strength of granular materials. Geotech Geol Eng 2007;25:203–13. [4] 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. [5] Chappell BA. Deformational response in discontinua. Int J Rock Mech Min Sci Geomech Abstr 1979;16:377–90.
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