Shear strength and micro-descriptors of bidisperse ellipsoids under different loading paths

Mechanics of Materials 41 (2009) 748–763

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Shear strength and micro-descriptors of bidisperse ellipsoids under different loading paths Tang-Tat Ng * Department of Civil Engineering, MSC01 1070, University of New Mexico, Albuquerque, NM 87131, USA

a r t i c l e

i n f o

Article history: Received 28 January 2009

a b s t r a c t The paper examines the macroscopic and microscopic behaviors of a granular material, systems of bidisperse ellipsoids. The larger ellipsoids of these systems have an axial length ratio (major axial length to minor axial length) of 1.5 and the shorter ellipsoids have an axial length ratio of 1.2. The weights of these two kinds are identical in each sample. Samples are subjected to various confining pressures ranging from 30 to 3000 kPa. Then they are loaded following three stress paths including axial extension condition, triaxial extension condition, and axial compression condition. These drained tests are stopped till the critical state is reached. A linear critical state line in the void ratio and mean stress space was observed for each stress path. The slopes of these critical state lines are different. The critical state lines for the axial extension and triaxial extension tests are above that of the axial compression tests. We also examined different microscopic descriptors at peak and critical states. They are based on the micro-structural tensor of contact normals or normal contact forces. Excellent relationship was found between these descriptors and the peak shear strength in the compression tests. Good relationship was observed for the extension tests. The relationships between these descriptors and the critical shear strength are fair for the compression tests. However, these descriptors do not capture the trends of axial extension tests in critical state. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Under loading, soils arrive at a final state such that the deviator stress, the mean stress, and the void ratio will not change with further straining. This final state is called critical state. If the initial void ratio of the material is above the critical void ratio, there will be contraction under drained loading (development of positive pore pressure under undrained loading). Potential of liquefaction can be accessed by comparing the initial state and the critical state (Casagrande, 1975). The behavior of cohesionless materials in critical state is more complex than that of cohesive materials. In addition, particle breakage complicates the behavior. The laboratory result has revealed a nonlinear critical state line in the void ratio and logarith* Tel.: +1 505 277 4844; fax: +1 505 277 1989. E-mail address: [email protected] 0167-6636/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2009.01.031

mic mean stress space (e  log rm). Non-unique critical state line has been observed for different loading paths in physical experiments (Riemer and Seed, 1997). The critical state was found to be a function of the deformation mode. Here, we perform numerical true triaxial tests on arrays of bidisperse ellipsoids to examine the critical state without the alternation of gradation due to particle crushing. Also, the effect of loading path can be investigated without data scattering due to multiple samples. Furthermore, the micromechanics information at contact level can be retrieved. The usage of non-spherical particles allows us to investigate additional microscopic parameters since the direction of branch vector (vector connecting centers of the adjacent particles) may not be the same as that of the contact normal. In micromechanics study of granular materials, different descriptors (parameters) have been suggested to quantify the distribution of contact forces (e.g. Satake, 1982,

T.-T. Ng / Mechanics of Materials 41 (2009) 748–763

Rothenburg and Bathurst, 1989, Radjai et al., 1998, Ng, 2001). The linkage between micro-descriptors and macroscopic arguments has been suggested for systems of disks,

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ellipses, spheres and ellipsoids. These micro descriptors, as well as a new micro-descriptor, will be examined using the current numerical result.

Fig. 1. Configurations at different stages while particles settling due to gravity. (a) Initial stage after particles randomly generated; (b) intermediate stage; and (c) final stage.

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2. Discrete element method and the program Discrete element method originally developed by Cundall and Strack (1979) has been wildly used to study the behavior of granular materials for more than a quarter century. Samples contains many individual particles are loaded either by boundary displacement or prescribed strain field. The numerical method tracks the movement of each particle and the interaction between particles. The corresponding state of stress is obtained from the contact forces between particles. The state of stress or boundary force can be controlled through a feedback loop between stress (force) and deformation. Spheres and periodic boundaries were implemented in the 3D original program (Strack and Cundall, 1984). Nonlinear contact laws according to the Hertz and Mindlin theories were then implemented (Ng, 1989). Later, Lin and Ng (1997) has incorporated 3D ellipsoidal particles with periodic boundaries. Pressure boundaries were introduced to simulate true triaxial tests (Ng, 2002). The program is used to perform the numerical true triaxial tests presented in this paper.

3. Numerical samples The samples are composed of two different types of ellipsoids. The minor axis lengths of these two types (I and II) are identical. The major axis length of Type II is greater than that of Type I. Type I is smaller and shorter with an axial length ratio (major axis length/minor axis length) of 1.2. Type II particles have an axial length ratio of 1.5. The weights of Type I and Type II particles are the same in each sample. The total number of particles is 1170 (620 Type I and 550 Type II particles). The contact relationships between particles are the nonlinear, force dependent contact laws (Lin and Ng, 1997). The gravitational constant is set to zero during shearing. The properties of these ellipsoids are shear modulus = 29 GPa (Simmons and Brace, 1965), Poisson’s ratio = 0.15, and friction coefficient (l) = 0.5 (Lambe and Whitman, 1969). The sample was created in three steps including particle generation, particle settling, and the application of confining stress. First, particles were generated by randomly assigning location and orientation inside a rectangular prism. No initial contacts are allowed. Then, gravity was introduced to densify the sample (similar to the raining technique, one of the conventional sample preparation methods). To create a denser packing the friction coefficient is reduced to 0.01 during deposition phase. Fig. 1 shows the snap shots at different stages during particle deposition. The darker particles are Type II. After particles were in equilibrium under gravity, the gravity field was removed and l was set to 0.5. Systems were allowed to reach equilibrium before the top boundary was activated. Then, the top, front, and right boundaries were moved inward and compressed the sample. A servomechanism was used to achieve the prescribed initial confining pressure. The sample was consolidated isotropically to eight different confining pressures (rc = 30, 100, 400, 800, 1200, 1600, 2000, and 3000 kPa). The configurations of the samples at rc = 30

Fig. 2. Configurations of the samples (rc = 30 kPa and 3 MPa).

and 3000 kPa are shown in Fig. 2. No excessive particle overlap is observed since the maximum particle overlap is less than 0.6% of the minor axis length when rc = 3000 kPa. As rc increases from 30 to 3000 kPa, the void ratio (e) decreases by 0.105 from 0.638 and the coordination number increases by 1.479 from 4.326. There is only minor difference in the appearance of these two samples. The lack of particle crushing may play a role. Checking the contact force of each particle reveals that there are 83 floating particles (zero particle contact) when rc = 30 kPa and only nine floating particles when rc = 3000 kPa. The e, rc, and coordination number of these samples are plotted in Fig. 3. Second-order polynomial trend lines are observed. 4. Numerical triaxial tests Three drained tests were applied to the samples where

rc = 30, 100, 400, 800, 1600, and 2000 kPa. They are the axial compression (AC) test, the axial extension (AE) test, and the triaxial extension (TE) test. The loading paths of these three tests are plotted in Fig. 4. The axial extension test is often referred to the triaxial extension test in the literature since the test can be achieved by pulling in the axial

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b Coordination Number

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Void Ratio Fig. 3. Void ratio, confining pressure, and coordination number of the samples.

Deviator stress

Axial Compression

3

Axial Extension

Triaxial Extension 3

1 3

2 1

Mean stress Fig. 4. Stress paths of the three drained simulations.

direction of a conventional cylindrical sample. The true triaxial extension test can only be performed in a true triaxial test apparatus. The deviator stress is defined as the differ-

ence between major and minor principal stresses. The mean stress (rm) is the average of the three principal stresses. In AC tests, rz (vertical stress) > rx = ry (lateral

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a

3.5

3

σ1 / σ 3

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2

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1 0

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40

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Applied Strain (%)

b

0.9

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CD 0.6

0

10

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TE 40

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Applied Strain (%) Fig. 5. Results of drained tests of the sample at rc = 30 kPa.

stresses). In AE tests, rz < rx = ry. In TE tests, rz = rx > ry. ry = rc in all simulations. AC simulations (rc = 100, 400, and 800 kPa) are previous results (Ng, 2007). The principal stress-ratio (r1/r3) versus strain curve and volume change behavior of these three drained simulations of two confining pressures (rc = 30 and 2000 kPa) are plotted in Figs. 5 and 6. The strains at the location of the maximum shear strength (er) and at the location of minimum void (ee) are shown in Fig. 6. These particular strains are of interest that will be discussed later. The initial slopes of the r1/r3 curves of different confining pressures are similar to the experimental observation (Yamamuro and Lade, 1996). However, only slight drop in r1/r3 after the initial peak has been reported for sands at elevated pressure (e.g. Colliat-Dangus et al., 1986, Yamamuro and Lade, 1996). A greater drop is observed in these numerical simulations (see Fig. 6a). The alteration of grain size distribution at elevated pressure in physical testing may be the factor. There is no dilation in the TE

tests at elevated pressure. The numerical simulations show that strain softening can occur without dilation at elevated confining pressure. This contraction is not observed in AE tests. Therefore, this particular behavior is related to the loading path and the initial state of a sample (rc and e). The peak friction angles of these drained simulations are plotted in Fig. 7. The friction angle (Mohr–Coulomb se1 cant friction angle) is defined as / ¼ sin ½ðr1 þ r3 Þmax . Neglecting the effect of initial confining pressure, both AE and AC tests indicate that the peak friction angle decreases slightly with the increase of initial void ratio. The behavior of TE tests (diamond symbols) is different at lower initial void ratios (elevated confining pressure). The peak friction angle of a granular material is contributed by the internal friction (particle to particle friction) and the friction due to shear locking (dilatancy). The increase of mean stress suppresses the dilatancy which in turn yields a lower peak friction angle. However, unless we can distinguish the

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a

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σ1 / σ 3

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Applied Strain (%)

b

0.8 0.7

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Min e 0.3

0

CD

10

AE 20

TE

30

40

50

Applied Strain (%) Fig. 6. Results of drained tests of the sample at rc = 2000 kPa.

Peak Friction Angle

35

30

AE TE AC

25

20 0.54

0.56

0.58

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0.62

Initial void ratio Fig. 7. Peak friction angles of the simulations.

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effect due to particle arrangement without volume change, we may not really understand the exact reason for the decrease in friction angle at elevated pressure. Fig. 8 shows the residual friction angles of these simulations. There are slight ups and downs in the stress–strain

curves even the critical state is reached (see Figs. 5a and 6a). The residual friction angles are calculated based on the data of the final 5% strain. The symbols are the average values. The top and bottom bars represent ± one standard deviation. For these average residual friction angles, the

(a) AC tests Residual Friction Angle

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(b) TE tests Residual Friction Angle

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Initial void ratio

(c) AE tests Residual Friction Angle

25

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Initial void ratio Fig. 8. Residual friction angles of numerical simulations.

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lowest and highest values are 19.4° (the AC test, rc = 1200 kPa) and 22.8° (the TE test, rc = 30 kPa). The range is less than 3.4°. Unlike the peak friction angle, no clear trend can be identified for these three stress paths. Therefore, the relationship between residual friction angle and rc (or e) is more complicate than that of peak friction angle. The er and ee plotted against rc for AC and TE tests are shown in Fig. 9. The result of AE tests was not included since there is no contraction (ee = 0 for all cases). It is evident that er and ee are related to rc. In general, greater rc yields greater er (or ee). Second-order polynomial trends are found between ee and rc for both AC and TE tests. The maximum r1/r3 occurs at or after the sample starts dilation following the initial contraction (ee 6 er). Similar trend has been observed in physical experiments. The critical void ratio versus mean stress in the e  log rm space is shown in Fig. 10. The symbols connected with lines are the critical state lines for the three drained simulations. The squares, circles, and diamonds are for AC, AE, and TE tests, respectively. The solid circles

without line are the isotropic compression line (ICL). The critical state line of sands involving elevated pressure is commonly modeled by means of a bilinear critical state line (Been et al., 1991). The steepening of the critical state line is suggested as the onset of grain crushing (change of grain size distribution). Here, we found that grain crushing is not the necessary condition for the steepening of critical state lines. Fig. 11 depicts the critical state lines of our simulations in the e  rm space. The lower straight line (ICL) is also shown in the figure. The ranking of these critical state lines is ICL, AE, AC, and TE in increasing order of slope steepness. We expect the critical state lines of AE, AC, and TE tests will cross the ICL at some points. Different critical state lines for AE and TE tests indicate that the loading or unloading stress path affects the critical state. If an initial state (r, e) of a soil falls between the two critical state lines (the shaded areas in Fig. 12), different behaviors (contraction or dilation) will be shown under AE or TE conditions. In the case of undrained loading, liquefaction will occur along one stress path but not the other stress path. This

(a) AC tests 15

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Strain (%)

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Confining pressure (kPa) Fig. 9. Strains at the maximum principal stress-ratio or minimum void ratio.

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Critical void ratio

0.9

TE

AE

0.8

AC 0.7

ICL 0.6 0.5 10

100

1000

10000

Mean Stress (kPa) Fig. 10. Critical state lines and isotropic compression line.

Critical void ratio

0.9 0.8 0.7

AE

AC 0.6

ICL 0.5

0

500

1000

1500

2000

TE 2500

3000

3500

4000

Mean Stress (kPa) Fig. 11. Critical state lines in void ratio-mean stress space.

Fig. 12. Critical state lines of extension tests.

new discovery makes the analysis of liquefaction potential more challenge. However, since both critical void ratios for AE and TE tests are above those of AC tests. Using critical void ratios determined under AC condition for design will always be on the safe side.

5. Microscopic parameters at the contact level Micromechanical information at particle level includes particle orientation, branch vector (vector connecting the centers of adjacent particles), contact normal, and contact

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force. When spheres are used, particle orientation has no physical significance. Branch vector is identical to contact normal. Only contact normal (or branch vector) and contact force can be investigated. Researchers have developed micro-macro relationships based on these micromechanical information. It is well known that contact forces are directly related to the macroscopic stress (Satake, 1982, Rothenburg and Bathurst, 1989, Radjai et al., 1998, Thornton, 2000, Ng, 2001). In the present paper, we will investigate three different descriptions of this micromechanical information and their relationships with macroscopic shear strength at peak and critical states. Satake (1982) defined a microstructural tensor (/) of contact normal to describe the structural anisotropy which can be related to the stress anisotropy. M 1 X /ij ¼ na na ; M a¼1 i j

ð1Þ

where M is the number of contacts, na is the unit normal vector of contact a. Different vectors such as normal contact force (normalized to average normal contact force) and tangential contact force (normalized to average tangential contact force) can be used to obtain different microstructural tensors. In this paper, a superscript will be used to identify the vectors used to calculate the microstructural tensor. For example, /NV represents the microstructural tensor of unit normal contact vectors identical to the Satake’s definition. Different scalar quantities derived from the microstructural tensor have been used to relate the shear strength of the material. Three eigenvalues can be calculated from these microstructural tensors (/1 P /2 P /3; /1 + /2 + /3 = 1). A micro-descriptor for contact normals, ANV, is defined as the difference between the maximum and minimum eigenvalues of the microstructural tensor based on contact normals: NV ANV ¼ /NV 1  /3 :

ð2Þ

ANV has been linked to the deviator stress of assemblies of disks and spheres (Rothenburg and Bathurst, 1989, Thornton, 2000). Rothenburg and Bathurst (1989) proposed a micro– macro relationship for 2D assemblages of circles subjected to biaxial tests by using three micro-descriptors (ANV, ANF, and AT).

r1  r2 1 ANV þANF þAr : ¼ r1 þ r2 2 1 þ ANV ANF

ð3Þ

2

ANF and AT describe the distribution of normal contact forces and the distribution of the tangential contact forces, respectively. Discrete element simulations of disks have shown that the micro–macro relationship can be developed by using portion of the contacts (Radjai et al., 1998). The concept of strong and weak contacts was introduced. The strong contact is defined as the collection of contacts that the contact force (Fc) is greater than the average contact force (Fa). The microstructural tensor, /NV(1), is defined for the contact normals of the strong contact. If the calculation in-

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volves every contact, that is /NV. A micro–macro relationship for assemblies of disks was also proposed using the concept of strong contact (Radjai et al., 1998):

r1  r2 1 ¼ ½A ð1Þ þ ANF ð1Þ þ AT ð1Þ: r1 þ r2 2 NV

ð4Þ

ANF(1) and AT(1) describe the distribution of normal contact forces and the distribution of the tangential contact forces of the strong contact, respectively. Recent numerical results of assemblies of spheres and ellipses suggested that ANF(1) and AT(1) do not have the same degree of significance as ANV(1). The shear strength can be related solely to ANV(1) (Thornton, 2000, Antony, 2003).

r1  r3 / ðr1 þ r3 ÞANV ð1Þ:

ð5Þ

Although good micro–macro relationships were found using the concept of strong contact, the physics behind the definition of strong contact is not very clear. The natural logarithmic of the ratio of the maximum and minimum eigenvalues of /NF has been related successfully to r1/r3 for assemblies of ellipsoids subjected to different stress paths up to the peak r1/r3 (Ng, 2004). The micro–macro relationship is a power law as shown in Eq. (6). A linear relationship (b ¼ 1) was found for isotropic samples while b > 1 for anisotropic samples.

r1 /NF ¼ a ln 1NF r3 /3

!b :

ð6Þ

In this paper, another form will be used to relate mobilized shear strength with ANF:

r1  r3 ¼ a þ bANF : r1 þ r3

ð7Þ

In addition, we propose a new micro-descriptor, AMF, which is based on the directions of the maximum contact normal force of each particle (MCFP). An individual particle may have many contacts. We choose the contact normal which has the maximum force to represent this particle. The microstructural tensor, /MF, is calculated based on these contact normals. The same contact between particles A and B is not allowed to be counted as MCFP for both particles. If the MCFP is accounted for particle A, the second greatest MCFP of particle B will be the MCFP for particle B. It is expected that the new definition represents the strongest force chains through all load-bearing particles of the system. The micro-descriptors including ANF, ANV(1), and AMF will be investigated here. They will be directly linked to the mobilized shear strength. Fig. 13 shows these microdescriptors at peak state (maximum shear stress). ANV(1) and AMF capture the decrease trend between peak friction angle and initial void ratio of AC tests (square symbols in Fig. 13a and c). ANF values are higher at greater initial void ratios (see Fig. 13b). For AE tests, AMF and ANF capture the decrease trend of AE tests but not ANV(1). Therefore, AMF is slightly better than ANF and ANV(1). All three micro-descriptors cannot reproduce the trend of TE tests. In addition, the relative position of these three curves is not right. The AC curve should be below the AE

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Initial void ratio Fig. 13. Micro-descriptors ANF(1), ANF, AMF at peak state.

and TE tests as shown in Fig. 7. The reason is that the intermediate principal stress plays an important role in these extension tests (r2 = r1) and these three micro-descriptors only involve the major and minor eigenvalues of the microstructural tensor. Thus, a new micro-descriptor (BNV) is introduced including /2 as NV NV BNV ¼ /NV 1 þ /2  2/3 :

ð8Þ

As shown in Fig. 14, improvement is evident using these new micro-descriptors (BNF, BNV(1), and BMF). The curve of AC tests is now below the curves of extension tests for all three micro-descriptors. Comparing Figs. 14c and 7, the micro-descriptor BMF yields slightly better result. Fig. 15 shows the average value of the micro-descriptors (BNF, BNV(1), and BMF) in critical state. Similar to the average residual friction angle these values are the average

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B MF

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Initial void ratio Fig. 14. Micro-descriptors BNF(1), BNF, BMF at peak state.

over the final 5% strain. The solid curves without symbols are the trend of the average residual friction angle from Fig. 8. The agreement between micro-descriptors and mobilized shear strength is not as good as that of peak state. They are only in fair agreement to the solid curves for AC tests and TE tests. For AE tests, the shape is different. This may due to the greater volume increase in critical state. It implies that these micro-descriptors with the con-

sideration of contact force are not sufficient to describe the mobilized shear strength in some cases. To develop a better relationship between microscopic and macroscopic behaviors, we may need to identify the micro-descriptors that describe the distribution of voids. Fig. 16 shows the configurations of the sample (rc = 100 kPa) at initial, at peak, and in critical states subjected to AC condition. The corresponding strong contact

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Residual micro-descriptors

(a) AC tests 0.6 0.5 0.4 0.3 0.2 0.1 0 0.54

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BMF

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(c) AE tests 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.54

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Initial void ratio Fig. 15. Average micro-descriptors BNF(1), BNF, BMF in critical state.

is plotted on the left side. The thickness and color of the lines represent the magnitude of the contact forces. Thicker lines represent greater contact forces. Initially, the distribution of strong contact is random. At maximum mobilized shear strength, the direction of most contacts

of the strong contact (Fc P Fa) is very close to the major principal direction (vertical). In critical state, the degree of alignment is not as close as that of peak state. However, the contacts are more aligned in the vertical direction than that of initial state. Comparing Fig. 16b and c, some

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Fig. 16. Configurations of samples and distribution of normal forces at initial state, at peak shear state, and in critical state (the AC test, rc = 100 kPa).

extreme strong force contacts (Fc P 4Fa) disappear. For all tests, the coordination number at peak state is greater than that in critical state. The degree of decrease in the number of contacts is greater for the very strong contacts (Fc P 3Fa) than other strong contacts (Fa 6 Fc < 3Fa). It implies that these very strong contacts have more effect on r1/r3 than other contacts.

Fig. 17 shows the contact normals of MCFP. The three figures on the left side are MCFP with contact force greater than Fa. In general, 33–37% of MCFP with force less than Fa. They are plotted on the right side of Fig. 17. No preferred alignment of contact normals can be identified at initial state which is similar to that of strong contact as shown in Fig. 16a. At peak state, the concentration of MCFP that

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Fig. 17. Distribution of maximum normal force of each particle at initial state, peak shear state, and critical state (the AC test, rc = 100 kPa).

is also classified as strong contact (Fc P Fa) is along the vertical direction. The ‘‘weak” MCFP (Fc < Fa) does not show the same degree of concentration even they are MCFP. Similar to the result of strong contact in critical state, the degree of concentration of ‘‘strong” MCFP along the vertical direction decreases. The current result shows that similar macroscopic and microscopic linkage were observed when using BNV(1) or

BMF although BMF is slightly better. There is a fundamental difference between these two descriptors. BNV(1) is based solely on force magnitude. It may not include every particle. It ignores the contribution of the weak contacts (Fc < Fa). BMF involves both force magnitude and each particle. However, it ignores the contribution of other strong forces that are not MCFP. Force chains should be the network that transmits the boundary load from one side to

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the other side. The network may include some weak contacts in some cases. Further investigation is needed to identify the reason that similar results were generated by these two descriptors. Better descriptors may be developed through the study. 6. Conclusions Numerical true triaxial tests have been carried out to investigate the behavior of a granular material at very large strain subjected to low to elevated confining pressures. Three different drained simulations (AC, AE, and TE tests) were performed. The macroscopic and microscopic results were presented. It was found that the isotropic compression line is linear in e  rm space. Ignore the effect of confining pressure, the coordination number increases with the decrease of void ratio at a decreasing rate. A second-order polynomial trend was observed between the coordination number and void ratio for our samples. For each stress path, similar trends of macroscopic behavior were observed for samples with different confining pressures. It is due to the lack of particle crushing at elevated confining pressures. For the AC and AE tests, the peak friction angle decreases with the increase of void ratio as expected. However, the behavior of TE tests is more complex which may relate to the more profound increase of mean stress during shearing. The mean stress restrains the development of shear locking due to dilation. Linear critical state lines in e  rm space are found for all three stress paths. The critical state lines for extension (TE and AE) tests are above the critical state line of AC tests. The slopes of these critical state lines are different. The paper also examined the microstructure of the samples at contact level. Six different micro-descriptors have been investigated. When intermediate principal stress is the same as the minor principal stress, the micro-descriptors involve the major and minor eigenvalues related well with the mobilized shear strength. However, when the intermediate principal stress is equal to the major principal stress (extension tests), the micro-descriptors including all three eigenvalues provide better results. The result also shows that these micro-descriptors are better related to the mobilized shear strength at peak state than the mobilized shear strength in critical state. Good correlation with mobilized shear strength were found for two descriptors BNV(1) or BMF although there are some fundamental differences. To reveal the behind

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reason may lead to a better descriptor for force chains with physical significance. References Antony, S.J., 2003. Interlink between micro fabrics and shear strength in granular materials. In: Quasi-static Deformations of Particulate Materials, Budapest, Hungary, pp. 25–32. Been, K., Jefferies, M.G., Hachey, J., 1991. The critical state of sands. Géotechnique 41 (3), 365–381. Casagrande, A., 1975. Liquefaction and cyclic deformations of sands A critical review. In: Proc. of the 5th Pan-American Conf. on Soil Mechanics and Foundation Engineering, Beunos Aires 5, pp. 80–133. Colliat-Dangus, J.L., Desrues, J., Foray, P., 1986. Triaxial testing of granular soil under elevated cell pressure. In: Proc. of the the Symposium on Advanced Triaxial Testing of Soil and Rock, 18–20 June, Louisville, KY, pp. 290–310. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Géotechnique 29 (1), 47–65. Lambe, T.W., Whitman, R.V., 1969. Soil Mechanics. Wiley, New York. Lin, X., Ng, T.-T., 1997. A three dimensional element model using arrays of ellipsoids. Géotechnique 47 (2), 319–329. Ng, T.-T., 1989. Numerical simulation of granular soils under monotonic and cyclic loading: a particulate mechanics approach. Ph.D. Dissertation, Department of Civil Engineering, Rensselaer Polytechnics Institute, Troy, NY. Ng, T.-T., 2001. Fabric evolution of ellipsoidal arrays with different particle shapes. Journal of Engineering Mechanics, ASCE 125 (10), 994–999. Ng, T.-T., 2002. Hydrostatic boundaries in discrete element methods. In: Discrete Element Methods: Numerical Modeling of Discontinua. Geotechnical Special Publication, No. 117, ASCE, pp. 47–51. Ng, T.-T., 2004. Macro- and micro behavior of granular materials under different sample preparation methods and stress paths. International Journal of Solids and Structures 41 (21), 5871–5884. Ng, T.-T., 2007. Ultimate state of a granular material. In: Proc. of the International Workshop on Constitutive model of Geomaterials, January 12–13, Hong Kong, pp. 188–193. Radjai, F., Wolf, D.E., Jean, M., Moreau, J.-J., 1998. Bimodal character of stress transmission in granular packings. Physical Review Letters 80 (1). Riemer, M.F., Seed, R.B., 1997. Factors affecting apparent position of steadstate line. Journal of Geotechnical and Geoenvironmental Engineering ASCE 123 (3), 281–288. Rothenburg, L., Bathurst, R.J., 1989. Analytical study of induced anisotropy in idealized granular materials. Géotechnique 39 (4), 601–614. Satake, M., 1982. Fabric tensor in granular materials. In: Deformation and Failure of Granular Materials, pp. 63–68. Simmons, G., Brace, W.F., 1965. Comparison of static and dynamic measurements of compressibility of rocks. Journal of Geophysical Research 70, 5649–5656. Strack, O.D.L., Cundall, P.A., 1984. Fundamental studies of fabric in granular materials. Interim Report to National Science Foundation, CEE-8310729, University of Minnesota, Minneapolis. Thornton, C., 2000. Numerical simulations of deviator shear deformation of granular media. Géotechnique 50 (1), 43–53. Yamamuro, J.A., Lade, P.V., 1996. Drained sand behavior in axisymmetirc tests at high pressure. Journal of Engineering Mechanics 122 (2), 109– 119.