Modeling shear behavior and strain localization in cemented sands by two-dimensional distinct element method analyses

Computers and Geotechnics 38 (2011) 14–29

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Modeling shear behavior and strain localization in cemented sands by two-dimensional distinct element method analyses M.J. Jiang a,⇑, H.B. Yan a, H.H. Zhu a, S. Utili b,1 a b

Dept. of Geotechnical Engineering, Tongji University, Shanghai 200092, China Dept. of Engineering Science, Oxford University, Oxford OX1 3PJ, UK

a r t i c l e

i n f o

Article history: Received 25 November 2009 Received in revised form 7 September 2010 Accepted 8 September 2010 Available online 12 October 2010 Keywords: Cemented sand Bond breakage Strain localization Numerical analyses Distinct element method

a b s t r a c t This paper presents a numerical investigation of shear behavior and strain localization in cemented sands using the distinct element method (DEM), employing two different failure criteria for grain bonding. The first criterion is characterized by a Mohr–Coulomb failure line with two distinctive contributions, cohesive and frictional, which sum to give the total bond resistance; the second features a constant, pressureindependent strength at low compressive forces and purely frictional resistance at high forces, which is the standard bond model implemented in the Particle Flow Code (PFC2D). Dilatancy, material friction angle and cohesion, strain and stress fields, the distribution of bond breakages, the void ratio and the averaged pure rotation rate (APR) were examined to elucidate the relations between micromechanical variables and macromechanical responses in DEM specimens subjected to biaxial compression tests. A good agreement was found between the predictions of the numerical analyses and the available experimental results in terms of macromechanical responses. In addition, with the onset of shear banding, inhomogeneous fields of void ratio, bond breakage and APR emerged in the numerical specimens. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Most natural soils are characterized by a bonded structure arising from various processes, for example, the solution and deposition of silica at particle contacts [1]. Soils are also sometimes artificially cemented by chemical agents in ground-treatment processes. Experimental results in the literature have shown that cemented soils have peculiar behaviors different from uncemented ones, such as stiffening at low pressure followed by yielding in a manner similar to overconsolidated soils [2,3], enhanced strength [4–8], a relatively brittle stress–strain response and a more dilative volumetric response [6,9–11]. These findings have encouraged extensive research into constitutive models of cemented soils. Several continuum constitutive models have been suggested to describe some important features of structured soils by previous researchers [12–21]. Although the models differ in mathematical details, they are all based on the principle that the size of the state-boundary surface increases with interparticle bonding (for instance, see [20]). However, concerning cemented soils, only the macroscopic response can be observed in the laboratory, while the mechanisms taking place at the micromechanical level remain mostly unknown. One of the motivations for the present study was ⇑ Corresponding author. Tel./fax: +86 21 65980238. 1

E-mail address: [email protected] (M.J. Jiang). Formerly at Strathclyde University, Glasgow, UK.

0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.09.001

to bridge the gap between macro- and micromechanics for these types of soils. In the past decades, a number of theoretical, experimental and numerical works have been carried out on strain localization, particularly in granulates [22–26]. Regarding experimental works, the 1c2e apparatus developed in Grenoble in the 1990s was the first intended to relate micromechanics to macromechanics by running biaxial tests on Schneebeli wooden rods [27,28]. However, only a few reports can be found on strain-localization analysis in clays due to the theoretical, numerical and technical difficulties related to this research field [29–33]. For example, it is very difficult to gather sufficient microscopic data from specimens in the laboratory even with advanced technologies such as X-ray techniques [34–36], stereophotogrammetric techniques [37], or particle-image velocimetry [38]. Such an unsatisfactory situation extends to the analysis of strain localization in cemented sands, which constituted another strong motivation for this study. It is authors’ opinion that the distinct element method (DEM) presents an effective method to investigate the global mechanical behavior, strain localization, and associated micromechanisms occurring in cemented sands. The DEM was first proposed by Cundall and Strack [39], in which a detailed description of the method can be found. The main objective of this study was to provide insight into the shear behavior and strain localization taking place in cemented sands by DEM analyses. For this purpose, a series of biaxial compression tests was run on assemblies of 2D

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bonded disks. Two different bond models were employed, which will be introduced in the following section. All the simulations were run using PFC2D [40].

2. DEM numerical modeling Two bond models were used in this study. The first, hereafter referred to as Jiang’s model, was initially proposed by Jiang et al. [41,42] to investigate the yielding of microstructured sands, whereas the second model is a standard model offered by PFC2D, the so-called ‘‘contact-bond” model (see [40]). The common physical features of both models are illustrated in Fig. 1. Both bond models are made by a combination of a spring, a dashpot and a divider in the normal direction, whereas a spring, a dashpot and a slider are present in the tangential direction. However, in all the simulations performed in this study, damping was set to either zero or to a very small value; this was verified not to influence the results obtained. Hence, no dashpot was actually present in the bonds employed in the performed simulations. The global behavior of a geomaterial depends on the type of breakage it undergoes, which can be either fragile or ductile (see [43,44]). It is generally accepted that the breakage process is mainly fragile in cemented sands. This is also underpinned by tests recently performed on glued steel rods, as reported in [45]. Therefore, the bonds used in the presented simulations break in a purely fragile fashion, as illustrated in Fig. 2. Here, the main mechanical features of the employed bond models are shown in terms of the normal and tangential relative displacements versus the respective contact forces, Fn and Fs. Fig. 2a2 and b2 show the relationships between force and displacement achieved by experimental tests carried out on pairs of Schneebeli steel rods glued together. These experimental tests were meant to mimic the mechanics of two-dimensional bonded disks. According to the results of these experiments, the relationship between force and displacement is linear until the bond breaks. For uncemented granular geomaterials, several researchers have proposed nonlinear relationships to take into account the dependency of the material stiffness on the magnitude of normal forces exchanged at the contacts. However, in the case of bonded materials, the force–displacement relationship can be satisfactorily approximated by a simpler linear one, which requires the calibration of fewer parameters, as the bonding material is responsible for most of the compliancy (i.e., compressive stresses remain sufficiently low to avoid significant compression of the grains), as also indicated by the available experimental evidence [45,46]. Therefore, the parameters determining bond behavior (see Fig. 2) are Rn, the normal bond strength, Rt, the tangential bond strength, Kn, the normal contact stiffness, Ks, the tangential contact stiffness, and /l, the interparticle friction angle. Fig. 2a shows that the two models display the same perfor-

Fig. 1. Physical analogue of the employed bond models.

mance in compression and tension, but with different shear strengths. Fig. 2b shows that the shear strength in the Jiang model is pressure dependent and increases linearly with the normal force according to the Mohr–Coulomb criterion. The bond strength can be thought of as the sum of frictional and cohesive contributions. The shear strength of the standard bond model in the PFC2D code (PFC model) is instead pressure independent at low stresses, while it increases linearly with the normal pressure at high stresses. Therefore, this bond strength can be assumed to be provided by a purely cohesive contribution when the normal contact force acting on the bond is low (Fn < Rt/tan /l), and a purely frictional one when the force is high (Fn > Rt/tan /l). With the current experimental apparatus, an accurate determination of the failure envelope for cementitious bonds is a difficult task. Moreover, there exists no consensus among researchers on what bond model best represents reality. Therefore, both models were used in this work. In summary, the failure envelope according to the Jiang model is:

F n;f ¼ Rn

F s;f ¼ Rt þ F n tan /l

ð1Þ

and for the PFC model is:

( F n;f ¼ Rn

F s;f ¼

Rt

if F n 6 Rt = tan /l

F n tan /l

if F n > Rt = tan /l

ð2Þ

It may be argued that the sharp transition between tensile and shear resistance employed in both bond models is not realistic because there is a kink point along the strength envelope (point B in Fig. 2c), whereas it would be reasonable to expect a smooth transition, for instance, given by a curve starting from point A and tangentially reaching the inclined straight line Fs = Fntan /l (see Fig 2c). However, the shape of such a curve is uncertain and the adoption of a nonlinear failure envelope rather than a linear one is likely to have a small influence on the global behavior of the granular material. Therefore, it can be concluded that the adoption of a nonlinear failure envelope for bonds is not justified given the current state of knowledge of bonded granulates. For the same reason, Rn was taken as equal to Rt. Concerning the contact-stiffness parameters, the values used in all the performed simulations were independent of the amount of cement. Although the small strainelastic stiffness of cemented soils depends on the cement content (see [47]), the values of the contact stiffnesses, Kn and Ks, employed in the simulations were constant, as the soil elastic response at small strains was not the focus of this study. The same choice of constant contact stiffnesses was utilized by Wang and Leung (see [54]). The particle-size distribution shown in Fig. 3, featuring ten different particle radii, was used to generate the DEM specimens. The numerical specimens were 800 mm wide and 1680 mm high. The planar void ratio at the beginning of the biaxial tests was 0.27, which is representative of a loose sample. The total number of sand particles employed in each numerical specimen was around 24,000. In some preliminary simulations, fewer particles (6000) were used, with the macromechanical response showing no significant differences. However, if accurate information on shear-band width and strain and rotation fields within the shear bands is sought, such a small number of particles would be insufficient. With the adoption of 24,000 particles per specimen, the resulting ratios of shear-band width to average particle diameter were approximately 14 at an axial strain of 6% and approximately 17 at an axial strain of 13%, which are large enough to obtain reliable information on the shear bands. This can be regarded as a outstanding feature of the simulations performed here, as in previous DEM studies on other geomaterials the number of particles lying within the shear band appeared to be too small to draw reliable conclusions about the kinematic mechanisms taking place,

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10

Normal force (kN)

8 6

3

(a2) Jiang model (R=1500N) PFC model (R=1500N) Jiang model (R=5000N) PFC model (R=5000N)

2

Normal force (kN)

(a1)

4 2 0

1

Compression

Jiang et al, 2007

0

Delenne et al, 2004 -1

-2

Tension

-2 -4 -6 -0.06 -0.04 -0.02

0.00

0.02

0.04

0.06

0.08

-3 -0.20 -0.15 -0.10 -0.05

0.10

(b2)

10

Shear force (kN)

Shear force (kN)

5

0

-10 -0.10

Jiang model (R=1500N) PFC model (R=1500N) Jiang model (R=5000N) PFC model (R=5000N) -0.05

0.00

0.15

0.20

Normal force: 4kN

0.05

Jiang et al, 2007

Normal force: 2kN 2.0

Normal force: 0kN Delenne et al, 2004

1.5

Normal force: 0kN

1.0 0.5 0.0 0.0

0.10

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Shear displacement (mm)

Shear displacement (mm)

(c)

0.10

3.0 2.5

Normal force: 0 kN

-5

0.05

Normal displacement (mm)

Normal displacement (mm)

(b1)

0.00

2.0

Fn=-0.5 Fs= 0.5 3 Fs= Fntanφμ

1

Shear strength (kN)

2 1.5

1 1.0

2

Jiang model PFC model

Fn=-0.5 Fs= Fntanφμ+0.5

2 3

0.5

2

B 1

0.0 -1.0

A

Residual strength of both models

1

-0.5

ϕu =° 26.6

0.0

0.5

1.0

1.5

2.0

Normal force (kN) Fig. 2. Mechanical responses of the contact-bond models used in the DEM analyses. (The data in (a2) and (b2) come from [45,56]). (a1) Model prediction in normal direction. (a2) Experimental data in normal direction. (b1) Model prediction in tangential direction. (b2) Experimental data in tangential direction. (c) Relationships between shear strength and normal force. (The solid line represents peak strength while dashed line residual strength).

rotation rates, bond-breakage rates and strain field inside and outside the shear bands. To simulate the soft rubber membrane used to confine soil samples in geotechnical tests, flexible side boundaries consisting of bonded particles were employed, as originally proposed in [48]. The stress-controlled flexible boundary implemented in the code is the same as that used by Wang and Leung [49], to which the reader is referred to for details of the boundary implementation in DEM code. As already shown in [50,51,58], the use of flexible membrane boundaries allows the model to capture the deformation characteristics of developing shear bands with a good degree of accuracy. The input parameters for sand grains and membrane particles are summarized in Table 1. The top and bottom boundaries were simulated

by rigid walls having the same normal and tangential contact stiffnesses as the sand particles. The coefficient of friction between walls and particles was set to zero to reproduce ideal experimental conditions. The multilayer undercompaction method proposed by Jiang et al. [52] was used to generate the packing of particles so as to obtain loose and homogeneous specimens. Details on the generation procedure can be found in [52]. Five horizontal layers were used during specimen generation, with each layer containing 4800 particles randomly distributed in a rectangular area 800 mm wide and 437 mm high. Particles were compacted to the target planar void ratio, ep = 0.27, by moving the top rigid wall downward at a constant speed of 5.0 m/s with the lateral and bottom walls fixed.

M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

17

Percentage of finer (%)

100

80

60

40

20

0 5

6

7

8

9

10

Grain diameter (mm) Fig. 4. DEM specimen after generation. Fig. 3. Particle size distribution used in the DEM analyses.

Table 1 Sample size and material parameters used in the DEM simulations. Samples Width of sample (mm) Height of sample (mm) Initial void ratio Sand particles Total number in sample Density (kg/m3) Diameter (mm) Normal contact stiffness for test (N/m) Tangential contact stiffness for test (N/m) Interparticle coefficient of friction for test Coefficient of friction between wall and particle Normal contact stiffness for specimen generation (N/m) Tangential contact stiffness for specimen generation (N/m) Interparticle coefficient of friction for specimen generation Normal contact stiffness between sand and membrane particles (N/m) Local damping coefficient Viscous damping coefficient In normal and tangential directions Membrane particles Diameter (mm) Density (kg/m3) Normal bond strength in normal direction (N) Bond strength in tangential direction (N) Normal contact stiffness (N/m) Tangential contact stiffness (N/m)

800 1680 0.27 24,000 2600 Gradation in Fig. 3 7.5  107 5.0  107 0.5 0.0 7.5  109 5.0  109 1.0 3.75  106 0.5 0.0 2.0 1000 1.0  10100 1.0  10100 3.75  106 2.5  106

In order to generate a sufficiently loose sample, a coefficient of interparticle friction, tan /l = 1.0, was used during the generation process. The numerical specimen obtained is shown in Fig. 4. After specimen generation, the coefficient of interparticle friction was set to tan /l = 0.5. The specimens were then consolidated in 1D under a small vertical pressure of 12.5 kPa with the side walls kept fixed. In this stage, contacts between particles were found to be distributed uniformly within the specimens. Then bonds were formed at all particle contacts. The specimens were subsequently subjected to an isotropic constant confining pressure. Finally, the specimens were vertically compressed, with the top wall moving downward and the bottom wall moving upward, both at a constant speed, while the lateral pressure acting on the particle membranes was kept unchanged. The strain rate adopted in all the simulations was 6.0% axial strain per minute. This low strain rate ensured that quasistatic conditions were always present during testing. Moreover, this low rate also ensured that the pressures measured at the top and bottom walls remained similar throughout the test. These conditions imply that the stress field within

the specimen can be considered uniform on average until the occurrence of localization. Note that, although the numbers of contacts was the same when bonds were assigned to particles, the number of bonds at the beginning of the biaxial test differed depending on the bond model employed. This is due to the difference in implementation of the force-relative-displacement relationship in the normal direction between the Jiang and PFC models. In the Jiang bond model (see [41,42]), particles always overlap at the bonded contacts. In the PFC bond model (see [40]), bonds can exist with particles separated by a certain distance proportional to the exchanged tension force Fn. When the contact normal force is zero, the overlap is nil. However, such a difference does not affect the main conclusions in this study. A series of biaxial compression tests was performed with different bond strengths, i.e., 0 N, 1.5 kN, and 5 kN, and under different confining pressures, i.e., 50 kPa, 100 kPa, 200 kPa, 400 kPa, and 800 kPa. To describe the macroscopic mechanical responses of the specimens, two-dimensional stress invariants were employed: the mean effective stress, s = (ry + rx)/2, and the deviatoric stress, t = (ry  rx)/2.

3. Mechanical behavior of bonded DEM specimens 3.1. Stress–strain and volumetric responses Figs. 5 and 6 present the numerical results obtained from biaxial compression tests on the Jiang and PFC models under confining pressures of 50 kPa, 200 kPa and 800 kPa. For reference, the responses of uncemented specimens having the same initial void ratio are also plotted in the figures. Figs. 5 and 6 show that with both the Jiang and PFC bond models specimens were characterized by strain softening and shear dilatancy, unlike the uncemented case. Moreover, peak deviatoric stress and dilatancy increased with bond strength. These trends are in agreement with the available experimental data, shown in Fig. 7, reported by [49]. From these data emerges evidence that grain cementation significantly alters the stress–strain response of loose sands, which here changed from strain hardening to strain softening with an increasing degree of cementation. Analogously, the volumetric response switched from contractive to dilative. At a micromechanical level, this type of response is due to the formation of particle arches within the shear band. This in turn occurs because of the breakage of some bonds, causing the formation of clusters of bonded particles which are free to rotate, thus contributing to volumetric dilation. A detailed description of the kinematics of cluster formation in loose bonded granulates and the consequent dilative behavior can be found in Wang and Leung [54].

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

(b)

350

Jiang sample (R=5000N) Jiang sample (R=1500N) PFC sample (R=5000N) PFC sample (R=1500N) Uncemented sample

Deviatoric stress (kPa)

300 250

Deviatoric stress (kPa)

(a)

200 150 100

500

Jiang sample (R=5000N) Jiang sample (R=1500N) PFC sample (R=5000N) PFC sample (R=1500N) Uncemented sample

400

300

200

100

50 0

0 0

2

4

6

8

10

12

14

0

2

4

(c)

1000

Deviatoric stress (kPa)

Axial strain (%)

800

6

8

10

12

14

Axial strain (%)

600

400

Jiang sample (R=5000N) Jiang sample (R=1500N) PFC sample (R=5000N) PFC sample (R=1500N) Uncemented sample

200

0 0

2

4

6

8

10

12

14

Axial strain (%) Fig. 5. Stress–strain responses of cemented sands for various bond strengths under different confining pressures from DEM simulations: (a) r3 = 50 kPa; (b) r3 = 200 kPa; r3 = 800 kPa.

Comparing the stress–strain response obtained from the Jiang model with that from the PFC model, it was observed that, at low confining stresses, the specimen in the PFC bond model exhibited a higher peak stress and a more pronounced strain softening than the specimen in Jiang’s model. For instance, the response obtained at R = 5 kN and r3 = 50 kPa was characterized by a peak deviatoric stress of 325 kPa in the Jiang model and 175 kPa in the PFC model. However, at high confinement, the specimen in the Jiang model exhibited the higher peak deviatoric stress and more pronounced strain softening. This can be explained by considering the strength envelopes of the two bonds, which are shown in Fig. 2. The tensile strength of the two bonds is identical but the shear strength is different. Therefore, although bonds may fail in either tension or shear, the different macromechanical behavior obtained was mainly due to the different bond breakage under shear. In the tension zone (e.g., Point B in Fig. 2c), the shear strength of the specimen in the Jiang model was lower than that in the PFC model, whereas for bonds in compression it was the opposite. This explains why the peak deviatoric stress obtained from the PFC model with R = 5 kN was higher than that in the Jiang model at r3 = 50 kPa but was the same at r3 = 200 kPa and lower at r3 = 800 kPa. These results, especially those obtained from the Jiang model, show a good agreement with the available experimental data from a qualitative viewpoint (see Fig. 7). In particular, examining the experimental curves corresponding to various cement contents, the marked increase of peak deviatoric stress and softening taking place when the cement content changed from 2% to 3% was captured well by the Jiang model. In addition, Fig. 6 shows that specimens with Jiang-model bonds demonstrated more dilation than those with PFC-model bonds.

Finally, it can be noted that in case of the PFC model the resulting stress–strain curve became progressively closer to the curve of the unbonded specimen with increasing confinement; these ultimately coincide in the case of R = 1.5 kN and r3 = 800 kPa (see Fig. 5c). To explain the observed trend, it is necessary to consider that, according to the PFC model, the peak and the residual bond strengths coincide for high normal contact forces whereas they are distinctively different when the normal forces are low. This means that a higher confining stress leads to a higher number of bonds in the residual state and thus a smaller difference between bonded and unbonded specimens. Therefore, the fact that the bonded and unbonded curves coincide means that the majority of the PFC bonds reached their residual states. 3.2. Failure envelopes Fig. 8 shows the envelopes of the peak and residual strengths obtained by the numerical simulations in the Jiang and PFC bond models and in the uncemented case. It can be seen here that both the peak and residual strengths of the cemented specimens are larger than the strengths exhibited by the uncemented ones. This is due to bonding effects. The apparent cohesion increases with bond strength in both bond models. However, the obtained friction angles varied in different ways. The peak friction angle, /peak, obtained from the Jiang specimens increased with bond strength; conversely, /peak obtained from the PFC specimens decreased with bond strength. This difference can be linked to the different peakstrength envelopes present in the two adopted bond models. The obtained residual friction angle, /res, increased slightly with bond strength in both bond models, the increase being higher in the case

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

(b)

-8

Jiang sample (R=5000N) Jiang sample (R=1500N) PFC sample (R=5000N) PFC sample (R=1500N) Uncemented sample

-6

-4

-2.5

Jiang sample (R=5000N) Jiang sample (R=1500N) PFC sample (R=5000N) PFC sample (R=1500N) Uncemented sample

-2.0

Volumetric strain (%)

Volumetric strain (%)

(a)

-2

0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

2 0

2

4

6

8

10

12

0

14

2

4

Axial strain (%)

(c)

8

10

12

14

0.0

Jiang sample (R=5000N) Jiang sample (R=1500N) PFC sample (R=5000N) PFC sample (R=1500N) Uncemented sample

0.2

Volumetric strain (%)

6

Axial strain (%)

0.4 0.6 0.8 1.0 1.2 0

2

4

6

8

10

12

14

Axial strain (%) Fig. 6. Volumetric vs. axial strain of cemented sands for various bond strengths under different confining pressures from DEM simulations: (a) r3 = 50 kPa; (b) r3 = 200 kPa; r3 = 800 kPa.

(a)

(b)

Fig. 7. Experimental data obtained from laboratory tests on sands with various degrees of cementation at 50 kPa confining pressure (after Wang and Leung [49]): (a) stress– strain relationship, and (b) void ratio-strain relationship.

of the Jiang bond model. Fig. 9 shows the peak and residual strength envelopes observed experimentally by Wang and Leung [49] for cemented sands; here, apparent cohesion, peak and residual friction angles all increased with cement content. Comparing our numerical results with these experimental data, the Jiang model proved to be better than the PFC model in capturing the material behavior. The degree of agreement with the experimental data is quite remarkable considering that the bond models employed in these analyses were fairly simple, as they assumed linear strength envelopes and did not account for any exchange of moments.

4. Strain localization in bonded DEM specimens In this section, both the Jiang and PFC bond models were investigated with regard to strain localization. For brevity’s sake, in the following, the results from only two specimens were selected to present and discuss: one each with bonds in the Jiang model and PFC models, both specimens being confined at a pressure of r3 = 100 kPa and with a bond strength of R = 5 kN. However, the choice of restricting the description of the results obtained to just these two samples was justified by the fact that the behaviors

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

(a) 600

(b) Jiang sample (R=5000N) Jiang sample (R=1500N) PFC sample (R=5000N) PFC sample (R=1500N) Uncemented sample

400 300 200

400 300 200 100

100 0

Jiang sample (R=5000N) Jiang sample (R=1500N) PFC sample (R=5000N) PFC sample (R=1500N) Uncemented sample

500

Shear strength (kPa)

Shear strength (kPa)

500

600

0 0

200

400

600

800

1000

1200

1400

0

Mean normal stress (kPa)

200

400

600

800

1000

1200

1400

Mean normal stress (kPa)

Fig. 8. Achieved strength envelopes for various bond strengths: (a) peak strength envelopes, and (b) residual strength envelopes.

(a)

(b)

Fig. 9. Strength envelopes of cemented sand observed experimentally at different cement contents (after Wang and Leung [49]): (a) peak strength envelopes, and (b) residual strength envelopes.

observed in tests run for different bond strengths and confining pressures were similar (data not shown). In Fig. 10, the obtained stress–strain and volumetric responses for the aforementioned specimens are shown. The test can be divided into five stages, with starting and ending points as indicated by points O, A–E. Point O indicates the initial state before any vertical load is applied. Point A can be defined as a ‘‘yielding point”, as this is where some bonds begin to break. This point also marks the transition between dilative and contractive behaviors. Point B marks the peak of the deviatoric stress and point C the occurrence of maximum dilatancy. At point D, the volumetric strain becomes nearly constant. Finally, point E signals the end of the test. These stages will be used in the next subsections to describe the features of the observed strain localization in the two specimens. 4.1. Deformed specimens The deformation patterns of the two numerical specimens observed at different axial strains are shown in Fig. 11. Square grids of particles with different colors were employed to illustrate the characteristics of the strain field in the specimens [53]. Fig. 11a and f shows that the strain field at the yielding (point A in Fig. 9) was homogeneous in both specimens. Fig. 11b and g shows that the development of a shear band when the peak of the deviatoric stress is reached, i.e., point B in Fig. 10, was more pronounced in the specimen in the Jiang model than in the specimen in the PFC model. Fig. 11c–e and h–j shows that the evolution of the strain

fields in the two numerical specimens was similar in the remaining stages, with shear bands developing from the top-left to the bottom-right corner of the specimens and showing a similar inclination of about 52° to the horizontal. 4.2. Contact-force chains and stress fields In Fig. 12, the formation of contact force chains can be observed in the two bonded specimens at different stages. The thickness of the lines in the figure is proportional to the magnitude of the contact forces. A webbed pattern of force chains, with thick chains distributed mainly in the vertical direction, can be observed until the peak of the deviatoric stress is reached. This is in agreement with previously published works. Then, some thicker, columnar chains progressively developed with increasing axial strain, as illustrated in Fig. 12c–e and h–j. Moreover, it was found that these columnar force chains, which were initially oriented along the vertical direction, gradually rotated away from the shear band inclination but not so much as to become perpendicular to it. One way of describing the state of stress within the specimens at the onset of shear banding is to calculate the average stresses. This can be done by considering a suitable number of particles within a representative area and calculating the equivalent stress tensor from contact orientations and forces [39]. In this study, a simplified method included in the PFC2D code (see [40]) was used to obtain a stress tensor by averaging stresses over a circular area inside the specimen. Preliminary analyses

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

(b) 400

400

Deviatoric stress (kPa)

Volumetric strain: P: 0.38% Q: -0.74%

B

350

A

300

B

250

V o lu m e tric s tra in : P : 0 .1 9 % Q : -1 .4 %

C

A

200

S tre s s -s tra in c u rv e D

150 E

100

V o lu m e tric re s p o n s e Q

50 0

O

0

2

4

6

8

10

12

A

300

Stress-strain curve

250

C

14

Volumetric response

200

Bonding breakage

D

150

E Q

100 50

O

0

B o n d in g b re a k a g e

P

Volumetric strain: P: 0.38% Q: -0.74%

B

350

Deviatoric stress (kPa)

(a)

P 0

2

4

6

8

10

12

14

Axial strain (%)

Axial strain (%)

Fig. 10. Mechanical responses for the cemented samples used to describe strain localization (R = 5 kN, r3 = 100 kPa): (a) Jiang sample, and (b) PFC sample.

(a) 1.2%

(b) 2%

(f) 1.2%

(g) 1.7%

(c) 6%

(d) 8%

(h) 6%

(i) 8%

(e) 12%

(j) 12%

Fig. 11. The deformed DEM specimens controlled by two bond models at different axial strain that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample: (a–e), PFC sample: (f–j)).

showed that the accuracy of the obtained stresses depended on the ratio between the radius of the chosen circular area and the radii of the soil particles. If the radius of the circle was sufficiently large, e.g., 10–12 times the average particle radius, the calculated stress values were practically unaffected by the size of the radius of the circle. In Fig. 13, vectors of the principal stresses obtained in this way are shown at various stages of loading for both numerical specimens. Note here that the stress

fields remained homogeneous until the onset of shear banding, with the direction of the major principal stress almost vertical everywhere until the peak of the deviatoric stress was reached (point B in Fig. 10). Once the deviatoric stress exceeded the peak, the direction of the major principal stress inside the shear band started to rotate gradually away from the shear-band direction. In contrast, outside the shear band the principal stresses remained unchanged.

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

(a) 1.2%

(b) 2%

(f) 1.2%

(g) 1.7%

(c) 6%

(h) 6%

(d) 8%

(i) 8%

(e) 12%

(j) 12%

Fig. 12. Contact force chains obtained in two different bonded samples at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample: (a–e), PFC sample: (f–j)).

4.3. Bond-breakage fields It is generally accepted, although not yet confirmed either experimentally or numerically, that the formation of shear bands in cemented sands is associated with bond breakage. To investigate this aspect, the relationship between bond-breakage rate and axial strain obtained for the two analyzed numerical specimens is shown in Fig. 14. The bond-breakage rate g was defined as:

g¼

ðN1  N2 Þ=N e2  e1

ð3Þ

where N is the total number of bonds at the beginning of the compression test and N1 and N2 represent the number of intact bonds at strains e1 and e2, respectively. Note that N amounted to 25,365 for the specimen in the Jiang model and 33,543 for the specimen in the PFC model. This is due to the fact that bonds were assigned to particle contacts in different ways, as described in Section 2. As shown in Fig. 14, most bonds remained intact until the two specimens reached point A, previously defined as the ‘‘yielding point”, where the volumetric response changed from contractive to dilative. Then, bond breakages occurred differently in the two specimens. In the case of the Jiang model, the peak value of the bond-breakage rate was quite large and occurred soon after the peak of the deviatoric stress was reached. After this peak, the bond-breakage rate was significantly reduced; from this much lower rate, it then decreased gradually until the end of the test

(see CDE in the figure). Conversely, the specimen in the PFC model showed a lower peak value but a higher rate in the remaining part of the test, with a non-negligible number of bonds still breaking at constant volumetric strain at the end of the test. Considering the results shown for both specimens, it was concluded that bond breakage started at the material yielding point and reached its maximum value during strain softening. Fig. 15 illustrates the distribution of bond breakages observed in the two bonded specimens at different values of axial strain. Bond breakage was concentrated within a narrow region, tending to coincide with the shear bands forming in the two specimens. In the specimen in the Jiang model, only one shear band developed continuously during the test. Conversely, several shear bands formed along different conjugate lines in the specimen in the PFC model, with an inclination of about 52° to the horizontal. Among the several shear bands taking place, only one became significantly thick, with all the other bands remaining relatively thin. Measuring the bond breakages occurring in five circular regions, it was observed that only a few bonds broke outside the shear bands, as can also be seen in the images of Fig. 15. This indicates a strong link between bond breakage and the formation of either permanent or transient shear bands. To assess the cumulative effect of bond breakage over time, the evolution of the number of bonds within each circular region is shown in Fig. 16 for both bond models. In the case of circles lying entirely outside a shear band, the number of bonds remained prac-

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

Fig. 13. Stress fields in the two bonded samples measured numerically at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample: (a–e), PFC sample: (f–j).

Volumetric strain: P: 0.38% Q: -0.74%

300

6 Stress-strain curve

250

C

5

Volumetric response

200

Bonding breakage

D

150

4 3

E Q

100 50

2 1

O

0

0

P 0

2

4

6

8

10

12

3

B

250

7

A

14

Axial strain (%)

Deviatoric stress (kPa)

B

350

Deviatoric stress (kPa)

(b) 300

8

Volumetric strain: P: 0.19% Q: -1.4%

C

A 200

Stress-strain curve

2

D 150

E Volumetric response

100

1

Q 50 0

O Bonding breakage

P 0

2

4

6

8

10

12

Bonding breakage rate

400

Bonding breakage rate

(a)

0

14

Axial strain (%)

Fig. 14. Bonds breakage rate against axial strain observed in the DEM cemented samples (R = 5 kN, r3 = 100 kPa): (a) Jiang sample, and (b) PFC sample.

tically unchanged, as expected. On the contrary, in the case of circles lying inside a shear band, it is worth noting that the evolution of the bond breakages was different for the two specimens (see curve 5 in Fig. 16a and b). In fact, in the case of the Jiang bond model, a roughly linear decrease of intact bonds over time was observed, whereas in the case of the PFC bond model the decrease was parabolic, with more bonds breaking at small strains than at large strains. Moreover, the final number of broken and intact bonds at the end of the test (ea = 12%) also differed: in the case of the Jiang bond model, 79% of bonds were still intact, whereas in

the case of the PFC bond model the percentage of intact bonds was only 63%. These results are different from those shown in [49], where a significant number of bond breakages occurred outside the shear bands as well (see Fig. 17). Such a difference may be attributed to the features of the bond model employed by Wang and Leung, which make use of several small particles one order of magnitude smaller than the sand grains to reproduce the bonding cement. At present, it would be premature to conclude which bond model is more realistic, as simplifying assumptions were introduced in both

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

Fig. 15. Distributions of bond breakage in the two bonded samples at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa). Jiang sample: (a–e); PFC sample: (f–j).

(a)

Number of bonds

Number of bonds

(b)

Axial strain (%)

Axial strain (%)

Fig. 16. Number of bonds within and outside shear bands (cycles 1–5 in Fig. 14) in the two bonded samples at different axial strains (R = 5 kN, r3 = 100 kPa). Jiang sample (a); PFC sample (b).

[49] and the present models. In [49], the PFC parallel bond model was employed at the contacts between the small particles representing the bonding cement and the large particles representing the sand grains. In the present study, a simpler numerical modeling

of the cementing material was used to make an investigation of the influence of bond strength and microscopic failure criteria on both macroscopic mechanical response and the onset of shear banding more feasible.

M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

4.4. Void-ratio distribution In Fig. 18, maps of the distribution of void ratios in the two cemented specimens obtained at different axial strains, corresponding to points A–E in Fig. 10, are shown. The results obtained from both specimens show that the void ratio remained homogeneous until the peak of the deviatoric stress was reached. After this

(a)

25

point, the void ratio became increasingly larger (i.e., dilation) within the shear bands as the axial strain increased. In contrast, the void ratio outside the shear bands changed only slightly during the test. This indicates that the volumetric dilation of the whole specimen was almost entirely due to the dilation occurring within the shear bands. This observation is in good agreement with the available experimental evidence on structured soils [30].

(b)

Fig. 17. Numerical results obtained from biaxial test on cemented sand with different bond model (after [49]): (a) bonding network and location of shear band, and (b) distribution of bond breakage.

Fig. 18. Distributions of void ratio in the two bonded samples at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample: (a–e), PFC sample: (f–j)).

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

In addition, Fig. 18 shows that the void ratio within the shear band at large axial strain was larger in case of the Jiang bond model than in the case of the PFC bond model. This is also in agreement with the results presented in Section 3.1, namely, that the specimen in the Jiang bond model demonstrated a larger overall dilation than the specimen in the PFC bond model (see Fig. 6). It is interesting to compare the maps of void-ratio distribution in Fig. 18 with the results reported in [55] relative to biaxial tests run on uncemented sand specimens using flexible boundaries. In [55], Bardet and Proubet found that the determining the distribution of the void ratio was not helpful in identifying shear bands

and therefore concluded that ‘‘in two-dimensional materials, the (the use of the) volumetric strain calculated from the void ratio is inappropriate to detect shear bands.” However, they acknowledged that ‘‘this observation is in disagreement with the radiographic measurements of shear bands on real sands”, referring to those reported in [56,57]. In our other simulations (data not shown), we found that strain localization did not occur in the case of an uncemented specimen with the same initial void ratio as the cemented specimens studied in this work. On the contrary, shear banding did occur in the case of a dense packing of uncemented particles, as reported in [58]. Therefore, we concluded that the

0.1 0.0 -0.1 -0.2 1 2 3 4 5

-0.3 -0.4 -0.5

A B

0

2

C

D

4

6

8

Axial strain (%)

10

12

14

(b)

0.2

Averaged pure rotation rate

(a) Averaged pure rotation rate

Fig. 19. Distributions of APR in the two bonded samples at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample: (a–e), PFC sample: (f–j)).

0.1 0.0 -0.1 -0.2 -0.3

1 2 3 4 5

-0.4 -0.5 -0.6

A B 0

2

C

D 4

6

8

10

12

14

Axial strain (%)

Fig. 20. Relationships between APR and axial strain observed in the two cemented samples at measurement circles 1–5 shown in Fig. 14 (R = 5 kN, r3 = 100 kPa): (a) Jiang sample, and (b) PFC sample.

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

presence of bonds substantially changes the characteristics of strain localization because persistent shear bands were detected in the void-ratio distribution maps at axial strains as low as 1.8% in the Jiang specimen and 2.8% in the PFC2D specimen. 4.5. Distributions of averaged pure rotation rate (APR) The first author [59] has recently proven that the energy dissipation between sand particles is related to their relative sliding displacement and can be further expressed in terms of their sliding rotation rate. The sliding rotation rate consists of two parts: one related to particle translation and the other to particle rotation and radius. The second part, hereafter termed the pure rotation rate, h_ p (see [59]), can be expressed by:

1 h_ p ¼ ðr 1 h_ 1 þ r 2 h_ 2 Þ r

ð4Þ

shear band developed in the specimens. Fig. 21 displays the evolution of both average void ratio and APR for each band at the different stages of the test. In Fig. 21a, the adopted local coordinate system and measurement bands are shown. It can be observed that at yielding (point A) both the average void ratios and APRs in each measurement band were very similar, with the APR values being almost zero. After point B was reached, the average void ratio and APR inside the persistent shear band drifted away from the values measured outside the shear band. A comparison between Fig. 21b and c shows that: (1) the thickness of the shear band in the Jiang specimen was larger than in the PFC specimen; and (2) the thickness of the shear band depended on which variable was used to identify it. For instance, if the void ratio was used, this

(a)

where r1 and r2 are the radii and h_ 1 and h_ 2 are the rotation rates of the two particles in contact; here, r is the equivalent radius of the two particles in contact, defined as:

r¼

2r 1 r 2 r1 þ r2

ð5Þ

Therefore, the averaged pure rotation rate x (APR) can be expressed by [59–61]:

ð6Þ

4.6. Thickness of shear bands To investigate the thickness of the shear bands, the specimens were divided into 25 measurement bands, all parallel to the main

0.3

0.35

0.2

0.30 0.25

0.1 Shear band

0.20

0.0 APR A B C D E

-0.1 -0.2 -0.3 0.0

0.1

0.2

Void ratio A B C D E 0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.15

Void ratio

(b)

0.10 0.05 0.00 1.0

Distance from origin (m)

(c)

0.3

0.35

0.2

0.30 0.25

0.1

Shear band

0.20

0.0

APR A B C D E

-0.1 -0.2 -0.3 0.0

0.1

0.2

Void ratio A B C D E 0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.15

Void ratio

where N is the total number of contacts and rk is the equivalent radius for the two particles at the kth contact, which can be calculated from Eq. (5). APR is thus a local variable linking the macro- and micromechanics of sand motion which does not exist in classical continuum mechanics; it was first introduced in [59–61]. In Fig. 19 are shown maps of the APR distributions in the two cemented specimens obtained at different axial strains, corresponding to points A–E in Fig. 10. Measurement circles with a radius equal to 10–12 times the average particle radius were chosen to carry out APR measures. From Fig. 19 it is evident that the APR was almost zero in the whole specimen until the peak of the deviatoric stress was reached (point B in Fig. 10); after point B, it grew continuously within the shear bands. However, the APR was nearly zero and changed only slightly outside the shear bands during the test. These observations are consistent with the maps of void ratios shown in Fig. 18. In general, at large strains the PFC specimen manifested a larger APR within the shear bands than did the Jiang specimen. Fig. 20 displays the APR values measured at different strain levels from five measurement circles (see Fig. 15). Here, the APR in all the circles was almost zero until yielding (point A) and increased later. The APR in circle 5 was the largest for both specimens. These observations are consistent with the results shown in Figs. 15 and 19. The APRs calculated for circles 1–4 in the case of the PFC specimen became relatively small after the specimen had experienced maximum dilatancy (point C in Fig. 10), while the APRs calculated for circles 3 and 4 in the case of the Jiang specimen assumed significantly larger values than their initial ones after the peak of the deviatoric stress (point B in Fig. 10) was reached. This is because circles 3–4 are quite close to the shear band in the Jiang specimen and the enhanced APR zone was wider in the Jiang specimen than in the PFC specimen.

Averaged pure rotation rate (APR)

 N N  1X 1 X 1 _k k _k k h_ p ¼ h r þ h r 2 2 N k¼1 N k¼1 r k 1 1

Averaged pure rotation rate (APR)

x¼

0.10 0.05 0.00 1.0

Distance from origin (m) Fig. 21. Averaged APR and void ratio obtained from different bands for two bond models (R = 5000 N, r3 = 100 kPa): (a) definition of local coordinates and measurement bands, (b) Jiang sample, and (c) PFC sample.

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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29

resulted in the thickness of the detected shear band being larger than the thickness measured employing the APR. 5. Conclusions This study presented an insight into the mechanical behavior and strain localization of cemented sands by means of DEM analyses. Two types of specimens characterized by two different bond models were investigated by performing numerical compressive biaxial tests. One bond model was originally proposed by Jiang et al. [41,42] (Jiang’s model), while the other is the standard contact-bond model available in PFC2D (PFC model). The multilayer undercompaction technique was employed to generate loose bonded specimens. Stress–strain relationships, overall material strength, stress fields, bond-breakage fields, and APR fields, as well as microscopic responses were analyzed. The main conclusions of the study are summarized as follows: (1) In both the Jiang and PFC bond models, numerical specimens exhibited strain softening and shear dilatancy. The opposite behavior was exhibited by an unbonded specimen with the same initial void ratio. The peak of the deviatoric stress and the angle of dilatancy increased with increasing bond strength. In the case of the PFC bond, a larger strain softening and a higher deviatoric stress peak were observed at low confinement. The opposite was observed at high confinement. In addition, the obtained global friction angle in the case of the PFC bond model decreased with increasing bond strength. The opposite was true in the case of the Jiang bond model. In general, the obtained numerical results were in good agreement with the experimental data reported in [49] relative to Portland-cemented sands, especially in case of the Jiang bond model. (2) Although the mechanical responses of the two investigated bonded granulates were different, the main features of the observed strain-localization processes were the same. Both granulates developed shear bands with an inclination of approximately 52° to the horizontal during biaxial compression. Bond breakage, void ratio and averaged pure rotation rate exhibited higher values inside the developed shear bands than outside. Finally, it was shown that during biaxial compression contact-force directions and principal stresses rotate within the shear bands. (3) There were also some differences observed between the two bonded granulates concerning the mechanisms taking place during shear banding. The maximum rate of bond breakage and the void ratio inside the shear bands were larger for the Jiang bond model than for the PFC bond model; the opposite is true for the APR. The thickness exhibited by the main shear band in the case of the Jiang bond model was wider than the one developed by specimens with the PFC bond model. Such differences are likely due to the different failure criteria adopted in the two bond models.

Acknowledgments This research was financially supported by the National Science Foundation, China, with Grant No. 10972158, China National Funds for Distinguished Young Scientists with Grant No. 521025932, the Fund for Chinese Researchers Returning from Overseas, Ministry of Education, China, Grant No. 2007-1108, and a Travel Grant from the Royal Society, Grant No. 2008/R2 for the fourth author. The support of the Itasca Consulting Group is also gratefully acknowledged.

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