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ScienceDirect Soils and Foundations 58 (2018) 1492–1510 www.elsevier.com/locate/sandf
Experimental investigation of inter-particle contact evolution of sheared granular materials using X-ray micro-tomography Zhuang Cheng, Jianfeng Wang ⇑ Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong Received 14 December 2017; received in revised form 17 July 2018; accepted 20 August 2018 Available online 29 September 2018
Abstract The inter-particle contact evolution of two sheared granular materials, i.e., a spherical glass bead (GB) specimen and an angular Leighton Buzzard Sand (LBS) specimen, is investigated non-destructively using X-ray micro-tomography. A miniature triaxial apparatus is developed for the testing with in-situ scanning. Full-field X-ray CT images of the two specimens are obtained at different shearing stages. A series of image processing and analysis techniques in combination with a particle-tracking approach is developed to detect the inter-particle contacts and to determine the contact gain, the contact loss, and the contact movement during each shear increment. It is found that the average coordination number (CN) experiences a strong change in the pre-peak shearing stage, and tends to reach a steady value after the peak. As the shear progresses, the average CN of the particles with different sizes follows the same trend as the overall average CN. Additionally, as the shear progresses, the branch vectors of the specimens, which, prior to shearing, are nearly isotropically distributed for the rounded GB and concentrated along the horizontal direction for the angular LBS, are found to show a directional preference towards the loading direction. The contact gain and the contact loss, which contribute to this directional preference, and the contact movement, which leads to the attenuation of the directional preference, are shown to be the two competing factors determining the evolution of the fabric anisotropy of granular materials. The higher degree of fabric anisotropy in the shear bands is shown to be mainly attributed to the higher percentages of contact gain and contact loss when compared to that of the entire samples. Ó 2018 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.
Keywords: Inter-particle contact; Granular materials; In-situ triaxial test; Shear-induced anisotropy; Fabric; Shear band; X-ray micro-tomography
1. Introduction Recent decades have witnessed an increasing interest in the interpretation of the macro-scale mechanical behavior of granular materials from the viewpoint of the grain scale. The important role of the inter-particle contacts on the macro-scale mechanical behavior of granular materials under shear has been well recognized. It has been reported in previous numerical studies that the average number of Peer review under responsibility of The Japanese Geotechnical Society. ⇑ Corresponding author at: Department of Architecture and Civil Engineering, B6409, Academic 1, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. E-mail address:
[email protected] (J. Wang).
contacts per particle (i.e., the average coordination number) is closely related to the void ratio of granular materials. This average coordination number changes and reaches a stable value as the shearing progresses (Thornton, 2000; Rothenburg and Kruyt, 2004). The fabric, which can be defined in terms of contact normals, branch vectors, or particle orientation, etc., has been found to have a significant influence on the mobilized strength and dilatancy of granular materials (Oda, 1972; Yimsiri and Soga, 2010). Studies using DEM simulations (e.g., Rothenburg and Bathurst, 1989; Thornton, 2000) or photo-elastic pictures (Oda et al., 1985) have also demonstrated that granular materials exhibit an anisotropic contact network under shear in which the contact vectors (i.e., particle contact normals
https://doi.org/10.1016/j.sandf.2018.08.008 0038-0806/Ó 2018 Production and hosting by Elsevier B.V. on behalf of The Japanese Geotechnical Society.
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and branch vectors) show a directional preference towards the loading direction. The rapidly developing X-ray micro-tomography and the accompanying image processing techniques have been increasingly used as powerful tools for investigating the grain-scale behavior of granular materials. These tools have provided valuable insights into the characterization of particle size, orientation, and morphology (e.g., Fonseca et al., 2012; Zhou et al., 2015; Zhao and Wang, 2016), as well as an understanding of the grain-scale behavior of granular materials, including particle crushing (e.g., Zhao et al., 2015), particle kinematics (e.g., Ando` et al., 2012; Watanabe et al., 2012; Alshibli et al., 2016), void ratio evolution (e.g., Frost and Jang, 2000), and strain localization (e.g., Desrues et al., 1996; Oda et al., 2004; Hall et al., 2010; Higo et al., 2011, 2013; Alikarami et al., 2015), etc. The increasing spatial resolution of X-ray microtomography also provides possibilities for the characterization of the inter-particle contacts of granular materials. Despite the fact that an inherent problem of X-ray micro-tomography (i.e., the partial volume effect) generally leads to an over-estimation of the inter-particle contacts, X-ray micro-tomography provides reliable experimental results consistent with those obtained from numerical studies. For example, Fonseca et al. (2013a, b, 2016) quantified the fabric evolution in terms of the branch vectors and contact normals of intact and reconstructed samples of Reigate sand under shear. The contact orientation was also studied through the implementation of watershed algorithms on binary 3D images (e.g., Viggiani et al., 2013; Ando` et al., 2013) and the use of mathematical level set functions on grey-scale 3D images (e.g., Vlahinic´ et al., 2014; Vlahinic´ et al., 2017) obtained from X-ray micro-tomography. In particular, it is found that through the use of a local threshold image segmentation and an increase in the spatial resolution of X-ray micro-tomography, the over-estimation of the inter-particle contacts can be significantly reduced (Wiebicke et al., 2017a). These studies have provided deep insights into the evolution of inter-particle contacts during the shearing of granular materials. The aim of the current study is to further explore the potential use of X-ray micro-tomography and advanced image processing techniques to investigate the interparticle contact evolution (i.e., the contact gain, the contact loss, and the contact movement) of granular materials. The novel contribution of this paper lies in the experimental quantification of the contact gain, the contact loss, and the contact movement, and its effects on the fabric evolution of two granular materials under shear through the combined application of a series of image processing and analysis techniques and a particle-tracking approach (Cheng and Wang, 2017). In the following, the experimental setup and the materials employed in the experiments are firstly introduced. Then, the image processing and analysis techniques used for the extraction of particles from raw CT images, the detection of inter-particle contacts, and the
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determination of the contact gain, the contact loss, and the contact movement are presented. Finally, the results of the stress-strain response, the evolution of particle displacement, the coordination number, the branch vector orientation, the contact gain, the contact loss, and the contact movement are presented and analyzed. 2. Experimental setup and materials 2.1. Experimental setup The triaxial compression tests in the current study are carried out by means of synchrotron micro-tomography at the BL13W beam-line of the Shanghai Synchrotron Radiation Facility (SSRF). This X-ray imaging device is composed of parallel beams, a motor-controlled rotation stage, and a detector. The setup can supply the parallel X-ray beams with energy ranging from 8 to 72.5 keV and the detector with different pixel sizes (e.g., 13 lm, 9 lm, 6.5 lm, 3.25 lm, 0.65 lm, and 0.325 lm, etc.), and can be chosen depending on the scanned object (i.e., its optical properties), spatial resolution, and scanning-size requirements. Hence, it is capable of obtaining 3D high resolution images. In the present tests, an X-ray beam with energy of 25 keV is used to scan the specimens; it produces a good contrast between the grain particles and the void spaces. The detector has a spatial resolution of 6.5 lm. In order to conduct the in-situ tests, a miniature triaxial compression apparatus is specially fabricated to supply confining stress and deviatoric stress to the samples, as shown in Fig. 1(a). The apparatus is light in weight and small in size to facilitate its mounting onto the rotation stage, so that the whole apparatus can be rotated 180 degrees at a constant rate during scanning. The axial load device of this apparatus is composed of a servo-controlled stepping motor and a screw jack driven by a worm and a worm gear. The confining stress is provided by a pressure controller. The apparatus allows for triaxial compression testing of a sample with a confining pressure up to 2 MPa and a constant loading rate in the range of 1– 1000 lm=min. In the tests, the samples are firstly isotropically compressed to a stress of 500 kPa and then sheared at a constant rate of 0.2%/min. The shearing is paused at different levels of axial strain (e.g., 0%, 1%, 2%, and 4%, etc.) to acquire synchrotron micro-tomography scans. Owing to the limited scanning size, i.e., 13.31mmðwidthÞ 5:33mm (height), the 16-mm-high samples need to be divided into four sections for each scan. It should be noted that there is an overlap of about 1.33 mm between any two consecutive sections, which is needed for image stitching. The scanning of a whole sample during each pause takes about 10 min. For the easy presentation of the results, Fig. 1(b) shows the orientation of the sample, where the YZ plane is chosen such that the normal vector of the shear band lies within the YZ plane. The XZ plane is orthogonal to the YZ plane and both planes contain the sample axis.
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samples, as the detection of inter-particle contacts relies highly on the high spatial resolution. In the current study, the GB sample and the LBS sample have initial porosities of 0.331 (relative density Dr = 125.6%) and 0.343 (relative density Dr = 127.7%), respectively. The initial porosities are acquired through the measurement of the 3D CT images of the samples after the confining stress has been applied.
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(b) Fig. 1. Miniature triaxial compression apparatus and sample orientation (a) triaxial apparatus (b) orientation of the sample.
2.2. Materials tested Two granular materials are tested in this study. One consists of glass beads (GBs), with a nearly spherical shape, from Zhaotong Micro Glass Bead Business, Guangzhou, China. The other is Leighton Buzzard Sand (LBS), with an angular shape, which has been studied previously by Zhao et al. (2015) in single particle compression tests. These two materials are used to examine whether or not the particle shape affects the inter-particle contacts. Fig. 2 shows the grading curves of the two materials. As is seen in this figure, the GB grains have a diameter of 300–600 lm and the LBS grains have a diameter of 400–800 lm. Both materials are made into dry cylindrical samples, with a diameter of about 8 mmand a height of about 16 mm, by air pluviation. The GB sample has about 16,000 grains and the LBS sample has about 4600 grains. The large grain diameters and the small sample size are used in this study to achieve a high spatial resolution (e.g., a grain of d 50 has about 174,000 voxels) and a full-field scanning of the
3. Image processing and analysis 3.1. Image reconstruction, smoothing, thresholding, and particle segmentation The raw projection images obtained from the synchrotron micro-tomography scans must be reconstructed into CT slices, then transformed into binary images, and eventually transformed into labelled images for which each particle and contact are individually identified. These labelled images are used to detect the inter-particle contacts and to compute the particle properties (e.g., particle volume, centroid, etc.). The reconstruction of the CT slices is performed on PITRE (Chen et al., 2012) from SSRF, where a series of raw projections obtained at different angles is transformed into a stack of grey-scale CT slices. A series of image processing techniques is then implemented on these CT slices. Fig. 3 illustrates the image processing procedure for typical GB and LBS slices in the isotropic compression state. It should be noted that the image processing procedure is performed on 3D images, i.e., stacks of 2D images. Firstly, an anisotropic diffusion filter (Perona and Malik, 1990) is applied to remove the noise from the grey-scale raw CT images shown in Fig. 3(a). The anisotropic diffusion filter has the advantage of removing noise from features and backgrounds of the images while preserving the boundaries and enhancing the contrast between them. This is achieved by setting a diffusion stop threshold
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GB LBS (a)
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Fig. 3. Illustration of image processing procedure on typical CT slices (a) grey-scale raw CT image (b) smoothed grey-scale image (c) binary image (d) labelled image.
value, which is determined by a parametric study. Each voxel in the image is diffused unless the difference in intensity between the voxel and its six face-centered neighboring voxels exceeds the threshold value. The resulting images are the smoothed grey-scale images shown in Fig. 3(b). Secondly, global thresholding is applied to the filtered grey-scale images to transform them into binary images (Fig. 3(c)). It should be noted that the use of a global threshold in the image binarization may result in the over-detection of contacts because of the partial volume effects of the CT images, in which the interface voxels between the grains and the voids have lower intensities than those within the grains (e.g., Cnudde and Boone, 2013; Wiebickea et al., 2015; Weis and Schro¨ter, 2017). However, these effects can be alleviated through the use of a local threshold, an increase in the spatial resolution of the X-ray micro-tomography, or the use of CT images of objects with sharp edges (Wiebicke et al., 2017a, b). It is believed that the effects were minimized in the current study because of the high spatial resolution (i.e., 6.5 lm (0.014d 50 )) of the X-ray synchrotron device and the sharpness of the grain edges in the CT images. Meanwhile, a parametric study on the over-estimation of inter-particle contacts was carried out. The results show that a slight over-estimation of the inter-particle contacts does not have a significant influence on the experimental results (Cheng, 2018). Thirdly, in the particle segmentation, an opening operation with a 7 7 7 (voxels) ball-shaped structural element is applied to the binary images to clear the isolated noisy voxels generated in the thresholding. It should be noted that the opening operation modifies the measured X-ray attenuation field at particle contacts during the removal of the noise. A parametric study is performed, and it is found that the use of an opening operation does
not have a significant effect on the final contact detection results in the current study (Cheng, 2018). A chamfer distance transformation (Borgefors, 1986) is performed on the filtered binary images and a marker-based watershed algorithm (Wa¨hlby, 2004) is applied to them to obtain the watershed lines used to separate the individual particles and the extracted contacts. The separated particles and the extracted contacts are obtained by performing the logical operation ‘NOT’ and the logical operation ‘AND’ between the filtered binary images and the images of the watershed lines, respectively. The binary images with separated particles (or contacts) are converted into labelled images for which each particle (or contact) corresponds to an isolated region and has a unique intensity value. It can be seen in Fig. 3(d) that individual grains are well identified for both images of GB and LBS. It should be noted that the contact voxels are removed in Fig. 3(d). Fig. 4 illustrates the extraction of the inter-particle contacts from the 2D slices. The watershed lines pass through the inter-particle contacts between any two overlapping particles (if they are successfully separated in the end) in the binary images. If any two neighboring particles are isolated, however, the watershed lines pass through the interparticle voids. As the touching particles are successfully separated, the watershed lines are believed to have successfully detected the locations of the inter-particle contacts (i.e., if there is a contact between two particles) of the granular materials. There may be a certain degree of errors in the shapes of the extracted contacts of the granular materials, which can result in significant errors in the determination of the contact orientations of the granular materials (Ando` et al., 2013; Jaquet et al., 2013). However, in the current study, we use branch vectors, whose orientations are not sensitive to these errors, to examine the fabric
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Watershed line passing through inter-particle void Extracted contact
Watershed line passing through inter-particle contact (a) Watershed line passing through inter-particle void
Extracted contact
Watershed line passing through inter-particle contact (b) Fig. 4. Illustration of extraction of inter-particle contacts in 2D slices (a) GB (b) LBS.
evolution of the granular system. Thus, the influence of these errors is largely avoided. Meanwhile, it is worth mentioning that the contacts of a sample under loading can generally be categorized into load-bearing contacts, that form strong force chains, and a complementary weak network of contacts that provides lateral supports to the strong force chains (e.g., Muthuswamy and Tordesillas, 2006; Barreto and O’Sullivan, 2012). The load-bearing contacts are the contacts whose contact forces are above the average contact force. As these contacts carry the majority of the external loads, they largely affect the mechanical behavior of granular materials. For example, the failure of granular materials under shear is generally associated with the collapse of the load-bearing contacts (e.g., Tordesillas et al., 2010). For the extracted contacts in this study, we do not distinguish between load-bearing contacts and weak contacts, due to the lack of the measurement of the contact forces. However, recent studies using contact network topology to estimate the load-bearing contacts of granular materials (e.g., Tordesillas et al., 2010; Fonseca et al., 2016) and the estimation of contact forces based on particle kinematics (e.g., Tolomeo et al., 2017) have provided possible ways to estimate the load-bearing contacts using X-ray micro-tomography imaging techniques.
ume of a particle is calculated as the number of voxels multiplied by the voxel size (i.e., (6.5 lm)3). The particle centroid coordinates are expressed as the mean value of the coordinates of all voxels composing the particle. These particle properties can be calculated using the intrinsic MATLAB (Mathworks, 2013) function regionprops (i.e., Area.regionprops and Centroid.regionprops). A particle-tracking method, similar to the one proposed by Ando` et al. (2012), has been developed by Cheng and Wang (2017) to track particles at different shearing stages (i.e., at different scans). This particle-tracking method makes use of the lists of particle volume, particle surface area, and particle centroid coordinates obtained from the labelled images of two consecutive scans to match the particles in the deformed configuration (i.e., the earlier scan) to the reference configuration (i.e., the later scan). The tracking results are used to determine the particle displacement and, more importantly, the inter-particle contact evolution. The displacement of a particle is calculated as the vector from the centroid of the reference particle, i.e., the particle in the reference configuration, to the centroid of the deformed particle, i.e., its match particle in the deformed configuration.
3.2. Characterization of particle properties
A code is developed to detect the inter-particle contacts based on the labelled image of the particles (denoted as Lp ) and the labelled image of the contacts (denoted as Lc ). The labelled image of the contacts is dilated using a 3 3 3
The particles’ volume and centroid coordinates are obtained from the labelled images of the particles. The vol-
3.3. Detection of inter-particle contacts
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Fig. 5. Illustration of inter-particle contact detection process (a) labelled image of particles Lp (b) labelled image of contacts Lc (c) dilated contact image Lc 0 (d) intersection between Lc and.Lp
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cubic structuring element mask (i.e., Lc ¼ Lc SE, where Lc is the resulting image after the dilation and SE is the structuring element). The SE is a small matrix of pixels, each with a value of one or zero. In this study, all of the 27 pixels in the 3 3 3 matrix have the value of one. The dilation process moves the structuring element across the image, similar to convolution, to determine whether or not to set the pixel in question (i.e., the pixel in the image located at the centroid of the structuring element) to one in the output image. This pixel is set to one if, for all pixels in the structuring element with the value of one, there is at least one pixel in the image whose value is also one at the same position (Serra and Soille, 1994). Two particles are determined to be in contact with each other if both particles have a nonempty intersection with the same dilated contact region. Fig. 5 illustrates a 2D example of the inter-particle contact detection process. In the labelled image of particles Lp shown in Fig. 5(a), the ith and jth particles are denoted by the green region with the intensity value of pi and the yellow region with the intensity value of pj , respectively. The kth contact in the labelled image of contacts Lc is denoted by the grey region in Fig. 5(b). A dilated image
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of contact Lc , shown in Fig. 5(c), is obtained by dilating the labelled image of contacts Lc with a 3 3 square structuring element, SE (i.e., a 3 3 matrix of pixels all having the value of one). Contact k is determined to be a contact between particle i and particle j as both particles have a nonempty intersection with the dilated contact region (denoted by ck ) in Fig. 5(c), as seen in the green and yellow regions shown in Fig. 5(d). This code is implemented for both GB and LBS and is found to work with high efficiency (Tens of thousands of contacts are detected within only several minutes.). Fig. 6 (a)–(d) show typical inter-particle contacts between two GB grains and two LBS particles, respectively. It should be noted that for particles with irregular shapes (e.g., concave particles), more than one isolated contact region could be assigned to the same particle pair (e.g., those LBS grains shown in Fig. 6(d)). In this case, these isolated regions are regarded as one contact. This simplification is made because of the lack of techniques in tracking multiple contacts between two particles at different loading states. For this reason, we propose the estimation of the contact evolution of granular materials in terms of branch vectors through a combined use of the contact detection technique
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Fig. 6. Illustration of typical inter-particle contacts (particles are shown in green and blue and contacts are shown in red) (a) an overall view of a GB contact (b) a close-up view of a GB contact (c) an overall view of a LBS contact (d) a close-up view of a GB contact.
and the particle tracking technique. It is expected that the evolution of these branch vectors will provide new insights into the fabric evolution of granular materials. Once all the particle pairs in contact are determined, the coordination number (CN) of each particle is calculated as the number of particles in contact with it. The branch vector of a particle pair in contact is expressed as the vector from one particle centroid to the other particle centroid. The inter-particle contact evolution of GB and LBS is investigated by combining the inter-particle contact detection and the particle-tracking technique. Fig. 7 illustrates three typical types of inter-particle contact evolution that may occur in a loading increment, which includes contact loss (Fig. 7(a)), contact gain (Fig. 7(b)), and contact movement (Fig. 7(c)). In a real scenario, contact movement includes both contact rotation and contact translation. Both are considered in the contact detection algorithm. Additionally, the contact gain, contact loss, and contact movement occur continuously and there may be several times of contact update (i.e., the creation or separation of
a contact) between two particles during a strain interval (e.g., Hanley et al., 2014). In this regard, an analysis based on smaller-size strain intervals generally gives a better interpretation of the inter-particle contact evolution. However, as the change in the number of inter-particle contacts is a gradual process during the shear (e.g., Thornton, 2000; Hanley et al., 2014), the adoption of a relatively larger-size strain interval is believed to have little effect on the overall results. Thus, the four typical strain increments (i.e., two strain increments at the pre-peak stage and two strain increments at the post-peak stage) during the shear are adopted for the illustration, and the inter-particle contact evolution is determined purely according to the start and the end of each shear increment. Fig. 8 demonstrates the process of the determination of the type of inter-particle contact evolution during a loading increment. In each loading increment (e.g., from loading state 1 to loading state 2), a particle pair list (i.e., a list of particle pairs where each particle in loading state 1 is matched to one in loading state 2) is firstly determined
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(c) Fig. 7. Illustration of typical types of inter-particle evolution (a) contact loss (b) contact gain (c) contact movement. (Note: The purple dashed lines denote the orientation of the branch vectors).
Contact detection
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ment because their match particles (i.e., j5 and j6 ) are not in contact with particle j0 . Similarly, the type of evolution for the contacts between j2 and j0 is contact gain as the match particle of j2 (i.e., i6 ) is not in the contact list of particle i0 . 4. Results 4.1. Stress-strain behavior
List of particles in contact with
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List of particles in contact with
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Comparison Loading g Loading state 1 state 2 nt Contact movemen movement
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Fig. 8. Illustration of the process of the determination of inter-particle contact evolution type.
using the particle-tracking algorithm, as seen in Fig. 8. Subsequently, for each particle pair in the list (e.g., particle i0 and particle j0 ), a contact list of each particle is determined. The two contact lists are then compared with the particle pair list to determine the type of inter-particle contact evolution. The contact lists of particle i0 (i.e., i1 ; i2 ; i3 ; i4 ; and i5 ) and particle j0 (i:e:; j1 ; j2 ; j3 and j4 ) are taken as examples. For particles i1 ; i2 ; and i5 in loading state 1, as their match particles in loading state 2 (i.e., j3 ; j4 and j1 ) are in the contact list of particle j0 , the type of evolution for their contacts with particle i0 are determined as the contact movement. The contacts of particles i3 and i4 with particle i0 are lost during the loading incre-
Fig. 9(a) and (b) show the deviatoric stress vs. the axial strain, and the volumetric strain vs. the axial strain of the GB sample and the LBS sample, respectively, sheared under a confinement of 500 kPa. Each sample is scanned 5 times during the shear (i.e., at axial strains e1 of 0%, 2.02%, 3.96%, 8.06%, and 12.14% for GB, and 0%, 0.98%, 4.94%, 10.40%, and 15.34% for LBS). The scanning points are marked for both GB and LBS in Fig. 9(a). It can be seen in Fig. 9(a) and (b) that both samples show strain softening and volumetric dilation, which is expected for dense granular materials. The peak deviatoric stress of both samples occurs around the third scan (i.e., e1 ¼ 3.96% for GB and e1 = 4.94% for LBS). The critical state friction angles of LBS and GB from the tests are about 32° and 23°, respectively. These values are very close to those obtained from the conventional triaxial tests on large samples (e.g., Cavarretta et al., 2010; Yang and Luo, 2015; Fu et al., 2017). 4.2. Particle displacement Through particle tracking, particles in a later scan are matched to those in an earlier scan. The displacement of particles during each two consecutive scans is calculated. Figs. 10 and 11 present the vertical slices located in the YZ plane (Fig. 1(b)) of the particle displacement of GB and LBS, respectively, occurring during the loading
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Fig. 9. Stress strain curves of the samples sheared under 500 kPa (a) deviatoric stress v.s. axial strain (b) volumetric strain v.s. axial strain.
process. It can be seen that both specimens exhibit failure along a well-defined single shear band. From Fig. 10(b), the particle displacement field of the GB specimen shows the onset of a localized shear zone at the axial strain increment of e1 = 2.02–3.96%, where the deviatoric stress shown in Fig. 9(a) reaches the peak. The displacement fields in Fig. 10(c) for GB and Fig. 11(c) for LBS both become clearly localized after the peak (i.e., e1 = 3.96–8.06% for GB and e1 = 4.94–10.40% for LBS). A well-developed single shear band can be seen in the last strain increments for both GB (i.e., e1 = 8.06–12.14%) in Fig. 10(d) and LBS (i.e., e1 = 10.40–15.34%) in Fig. 11(d). The shear bands of GB and LBS have thicknesses of about 6 d 50 and 5 d 50 , respectively, which are smaller than those observed in large sample tests (Han and Drescher, 1993; Desrues et al., 1996). This may be related to the limited number of grains within the samples. It should be noted that for both materials, the particle displacement at the top left corner is slightly higher than that at the top right corner within each shear increment. This is because of the free top pedestal, which allows for a rotation of the pedestal during the shearing. It was reported that the rotation of the top pedestal led to a lower initial stiffness at small axial strains (e.g., e1 < 0:1%), but it is not expected to have large effects on the overall stress and strain behavior of the samples at large axial strains (Gasparre et al., 2014; Ye et al., 2015). The objective of this paper is to explore the inter-particle contact evolution during triaxial shearing, which is believed to be very slightly influenced by the free pedestal. 4.3. Coordination number (CN) and branch vector orientation (BV) Fig. 12(a) and (b) present the frequency distributions of the coordination number (CN) of the GB grains and the LBS grains, respectively, at different axial strains. The porosities of the samples at each axial strain are also pre-
sented. As shown in these figures, the peak frequency occurs at CN = 6 in the isotropic compression state for both materials. The peak frequency moves gradually to CN = 5 as the porosities of the samples increase during the shear. Prior to the peak deviatoric stress (i.e., e1 = 0– 3.96%), the GB displays an apparent increase and decrease in the frequency of the CN lower and higher than 6, respectively (Fig. 12(a)). In the post-peak stage (i.e., e1 = 3.96– 12.14%), the frequency distribution tends to stabilize. In Fig. 12(b), the LBS displays a behavior similar to that of GB, except that in the pre-peak stage (i.e., e1 = 0–0.98%), where a slight decrease in sample porosity occurs, the frequency distribution of the CN shows no noticeable change. Fig. 13(a)–(d) plot the average coordination number (CN) vs. the axial strain and the average coordination number (CN) vs. the particle diameter for GB and LBS, respectively. The average CN for particles with a specific diameter (size-dependent average CN) is calculated as the total CN of the grains divided by the number of grains. It can be seen in Fig. 13(a) that the overall average CN (i.e., the average CN for all particles) of GB, starting from 6.43 in the isotropic compression, decreases noticeably to 5.83 in the pre-peak shear (i.e., e1 = 0–3.96%), and then drops slightly to 5.67 in the post-peak shear to e1 = 12.14%. As shown in Fig. 13(b), except for a slight increase in the early shear stage (i.e., e1 = 0–0.98% where a slight decrease of sample porosity occurs), the overall average CN of LBS exhibits a similar behavior to that of GB in the post-peak shear. This is in agreement with the previous observations (e.g., Thornton, 2000; Rothenburg and Kruyt, 2004; Fonseca, et al., 2013b; Gu et al., 2014) that dense granular materials display a significant change in CN in the pre-peak shear and tend to reach a stable CN in the critical state of shearing. In addition, it is interesting to note that for both GB in Fig. 13(a) and LBS in Fig. 13(b), the size-dependent average CN follows the same trend as the overall average CN. The decrease in the
Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510
1501
Fig. 10. Particle displacement of GB sheared under 500 kPa during the axial strain increments of (a) 0–2.02% (b) 2.02–3.96% (c) 3.96–8.06% (d) 8.06– 12.14%.
average CN is associated with the increase in sample porosity throughout the shear. This can be seen clearly from the evolution of the size-dependent average CN in Fig. 13(c) and (d). Furthermore, the size-dependent average CN decreases almost linearly with the decrease in particle diameter during the whole shearing process. The distributions of branch vector orientations of GB and LBS are shown in Fig. 14(a), (b) and Fig. 13(c), (d), respectively, in the form of frequencies plotted in spherical coordinates. The bin interval for the frequency calculation is 7.2° for GB and 15° for LBS, because of the different numbers of particles in the samples. The results in the XZ plane and the YZ plane (Fig. 1(b)) are presented. It can be seen in Fig. 14(a) and (b) that, for GB prior to
the shear, the branch vectors show a nearly isotropic distribution. In the pre-peak shear (i.e., e1 = 0–3.96%), the distribution experiences a significant loss in frequency in the horizontal direction and a slight gain in the vertical direction. This anisotropic distribution of the branch vectors undergoes little change in the post-peak shear (i.e., e1 = 3.96–12.14%). For LBS, the distribution of branch vectors exhibits a bias towards the horizontal direction in the isotopic compression (Fig. 14(c) and (d)). This is expected as the LBS grains are angular in shape and the particles tend to lie in the most stable position when they are deposited. However, as the shear progresses, the distribution changes to show a directional preference towards the vertical direction similar to that of GB. These findings
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Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510
Fig. 11. Particle displacement of LBS sheared under 500 kPa during the axial strain increments of (a) 0–0.98% (b) 0.98–4.94% (c) 4.94–10.40% (d) 10.40– 15.34%.
are in agreement with those reported in previous studies on sheared sand (e.g., Fonseca et al., 2013a; Druckrey et al., 2016). The realignment of these vectors along the loading direction is attributed to the requirement of the granular materials to resist the vertical load. 4.4. Evolution of inter-particle contacts The evolution of the percentages of contact gain, contact loss, and contact movement of the tracked GB and LBS grains within the samples and the shear bands (see
Figs. 10(d) and 11(d)) during the shearing process is presented in Fig. 15(a) and (b), respectively. For both materials, with all the increments in shear for the samples, the percentage of contact movement is much higher than that of contact loss or contact gain. The percentage of contact loss is always higher than that of contact gain throughout the shear, except for LBS in the shear increment of e1 = 0– 0.98%, where the contact gain percentage is higher. This is expected as the higher percentage of contact loss than contact gain leads to a decrease in the average coordination number, as shown in Fig. 13(a), (b). For both materials,
Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510 4000
1503
1000
Axial strain Porosity =0% (0.343) 1
Axial strain Porosity
2000
1
=0%
(0.331)
1
=2.02%
(0.335)
1
=3.96%
(0.346)
1
=8.06%
(0.353)
1
=12.14% (0.359)
800
=0.98%
(0.342)
=4.94%
(0.346)
1
Frequency/number
Frequency/number
3000
1000
1
600
=10.40% (0.368) 1 =15.34% (0.376)
1
400
200
0
0
0
2
4
6
8
10
12
14
16
0
2
4
6
8
10
12
14
16
Coordination number (CN)
Coordination number (CN)
(b)
(a)
Fig. 12. Coordination number distribution evolution in the shear of (a) GB (b) LBS.
within the shear bands, the contact gain, the contact loss, and the contact movement follow the same trend as those within the samples. However, there are much higher percentages of contact gain and contact loss within the shear bands than the percentages within the entire samples, especially in the post-peak shear increments. Figs. 16 and 17 present the orientation frequencies of the branch vectors from different types of inter-particle contact evolution (i.e., contact gain, contact loss, and movement) within the entire samples and the shear bands throughout the shearing process for both GB and LBS materials, respectively. For the contact gain and contact loss, the branch vector orientation frequencies of the gained contacts and the lost contacts, at the end and at the start of each shear increment, respectively, are plotted. For the contact movement, the frequency is plotted at both the start and the end of each shear increment. The bin intervals used for the entire samples of GB and LBS are the same as those used in Fig. 14. In the calculation of the branch vector orientation frequencies in the shear bands, a bin interval of 14.4° for GB and 30° for LBS is adopted because of the smaller amount of branch vectors. Only the results in the YZ plane (Fig. 1(b)) are presented due to the limited space. As illustrated in Figs. 16(a), (b) and 17(a), (b), for the entire samples and the shear bands of both GB and LBS, the branch vectors of the gained contacts in the early shear increments (i.e., e1 = 0–2.02% for GB and e1 = 0– 0.98% for LBS) exhibit a strong bias to the loading direction, and then the bias is gradually weakened afterwards. Correspondingly, the branch vectors of the lost contacts display a directional preference towards the horizontal direction in the early shear, which also gradually decreases in the subsequent shear (Figs. 16(c), (d) and 17(c), (d)). It is interesting to note that the branch vectors of the moved contacts exhibit a lower bias in the loading direction at the end than at the start of each shear increment in both
the shear bands and the entire samples (Figs. 16(e) and 17(e)). For contact movement only, the results from the final shear increments are presented because of the limited space. 4.5. Fabric tensor analysis Many fabric tensors (Kanatani, 1984; Oda, 1982; Satake, 1982) have been proposed to characterize granular materials in terms of various kinds of directional data, such as contact normal vectors, branch vectors, particle orientations, and void spaces. A second-order fabric tensor, following Satake (1982), is used to quantitatively investigate the preferred orientation of the branch vectors. In particular, the branch vectors of the gained, lost, and moved contacts of GB and LBS during the shear are also examined. The tensor is calculated as Eq. (1): F ij ¼
N 1 X nk nk ; N k¼1 i j
ð1Þ
where N is the total number of vectors and nki is the component of the kth unit orientation vector along direction i. Referring to Yimsiri and Soga (2010), a single parameter a can be used to quantify the anisotropy of the distribution of the vectors’ orientation in axis-symmetric systems. Parameter a is related to a Fourier series expression of the orientation distribution function of the vectors and can be obtained by equating the fabric tensor determined by Eq. (1) with the expression given in Eq. (2): 3 2 3 2 3a5 0 0 5ða3Þ F 11 F 12 F 13 7 3a5 6 7 6 0 0 7: ð2Þ 4 F 21 F 22 F 23 5 ¼ 6 5ða3Þ 4 5 F 31
F 32
F 33
0
0
ð5þaÞ 5ða3Þ
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Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510 11
13 For all particles For particles with diameter of 0.30~0.36 mm For particles with diameter of 0.36~0.42 mm For particles with diameter of 0.42~0.48 mm For particles with diameter of 0.48~0.54 mm For particles with diameter of 0.54~0.60 mm
12 11
10 9 8
Average CN
Average CN
10
For all particles For particles with diameter of 0.40~0.48 mm For particles with diameter of 0.48~0.56 mm For particles with diameter of 0.56~0.64 mm For particles with diameter of 0.64~0.72 mm For particles with diameter of 0.72~0.80 mm
9 8 7
7 6 5
6
4
5
3
4 0
4
8
Axial strain
0
12
4
(%) 1
8
Axial strain
12 1
16
((%)
(b)
(a) 9
10
Average CN
9
Porosity
Axial strain
=0% 1
(0.331)
=0% 1
=2.02% 1
(0.335)
1
=3.96%
(0.346)
=8.06% 1
(0.353)
Axial strain
8
7
Average CN
11
=12.14% (0.359)
1
8 7
Porosity (0.343)
1
=0.98%
(0.342)
1
=4.94%
(0.346)
1
=10.40% (0.368)
1
=15.34% (0.376)
6
Increase in sample porosity
5
Increase in sample porosity 6
4 5 0.3
0.4
0.5
Particle diameter (mm)
(c)
0.6
3 0.4
0.5
0.6
0.7
0.8
Particle diameter (mm)
(d)
Fig. 13. Average coordination number (CN) evolution in the shear of (a) (c) GB (b) (d) LBS.
The average of the a values calculated from the three diagonal elements of the fabric tensor (i.e., F11 , F22 , and F33 ) is used to characterize the degree of anisotropy of the vector distribution. a ¼ 0 denotes completely isotropic; the vectors tend to concentrate in the vertical direction (i.e., direction 3) when a > 0, and they tend to concentrate in the horizontal direction (i.e., direction 1 and direction 2) when a < 0. Fig. 18(a) and (b) show the variations in anisotropy coefficient a of the overall branch vectors, and the branch vectors of the gained, lost, and moved contacts for GB and LBS, respectively, as the shear progresses. The overall branch vectors for both GB, which initially are nearly isotropically distributed with a 0, and LBS, which are initially concentrated along the horizontal direction with a < 0, experience an increase in the degree of anisotropy along the vertical direction during the shear. Correspondingly, the branch vectors of the gained contacts and the lost
contacts undergo a decrease in the degree of anisotropy along the vertical and horizontal directions, respectively. On the one hand, the contact gain and contact loss clearly contribute to the increase in the degree of anisotropy of the overall branch vectors along the vertical direction. On the other hand, the movement of the contacts leads to the decrease in the degree of anisotropy, because the contacts before movement show a higher degree of anisotropy along the vertical direction than those after the movement throughout the shear. Therefore, it can be concluded that the evolution of the anisotropy of granular materials is a combined result of the enhancement of anisotropy along the major principal stress direction (i.e., the vertical direction), caused by the contact gain and the contact loss, and the anisotropy attenuation caused by the contact movement. Fig. 19 shows the evolution of anisotropy degree a of the entire samples and the shear bands. It can be seen from this
Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510 90
90 1200
120
=2.02% 30
150
=8.06%
300 0
300 600
=3.96%
900
30
150
=12.14%
0 180
0
300 330
210
900
240
1200
300
240
1200
270 YZ plane
(a)
(b) 90
90 0%
60
600
120
0
200 330
210
400
Frequency/Number
=15.34%
0 180
400
=10.40%
200
=0.98% =4.94%
=4.94% 30
150
0%
60
600
=0.98%
400
300
270 XZ plane
120
Frequency/Number
=8.06%
300
600
330
210
0%
60
=2.02%
600
=12.14%
0 180
120
900
=3.96%
Frequency/Number
Frequency/Number
1200
0%
60
900 600
1505
30
150
=10.40% =15.34%
200 0 180
0
200 400
330
210
600
600 240
240
300
300 270 YZ plane p
270 XZ plane p
(d)
(c)
Fig. 14. Branch vector orientation frequency distribution in the shear of (a) GB in XZ plane (b) GB in YZ plane (c) LBS in XZ plane (d) LBS in YZ plane.
100
100 Contact movement (overall) Contact movement (in shear band) 80
60
60
Percentage/%
80
Contact gain(overall) Contact loss (overall) Contact gain (in shear band) Contact loss(in shear band)
40
20
.3 0~ 15 .4
40
%
10
94 4.
0.
98
~1
~4
.9
Shear increment
0.
4%
%
98 0.
2. ~1 06 8.
~8 3.
96
2. 02
(a)
0
% 14
6%
.9 6 ~3
.0
%
%
02 2.
Shear increment
4%
20
0
0~
Contact gain (overall) Contact loss (overall) Contact gain (in shear band) Contact loss (in shear band)
40
0~
Percentage/%
Contact movement (overall) Contact movement (in shear band)
(b)
Fig. 15. Percentage of contact gain, contact loss, and contact movement in different shear increments (a) GB (b) LBS.
Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510 0~2.02%
90
Frequency/Number
250
120
60
200
=3.96~8.06%
150
=8.06~12.14% 30
150
100 50 0 180
0
50
0~2.02%
90 200
=2.02~3.96%
120
60
100
=2.02~3.96% =3.96~8.06%
150
Frequency/Number
1506
=8.06~12.14% 30
150
50 0 180
0
50
100 100
330
210
150
330
210
150
200
300 270 Branch vectors' orientation in YZ plane for gained contacts in shear band
240
300 270 Branch vectors' orientation in YZ plane for gained contacts
250
240
200
(b)
(a) 0~2.02%
90 120
60
=2.02~3.96%
200
=3.96~8.06%
150
=8.06~12.14% 30
150
100 50 0
0 180 50
0~2.02%
90 200
120
60
100
=2.02~3.96% =3.96~8.06%
150
Frequency/Number
Frequency/Number
250
=8.06~12.14% 30
150
50 0 180
0
50
100 330
210
150
330
210
150
200 250
100
240
300 270 Branch vectors' orientation in YZ plane for lost contacts
(c)
8.06% (overall)
90 1000
120
300 270 Branch vectors' orientation in YZ plane for lost contacts in shear band
200
60
240
(d)
12.14% (overall) 8.06% (in shear band)
750
Frequency/Number
500
12.14% (in shear band) 30
150
250 0 180
0
250 500
330
210
750 1000
240
300 270 Branch vectors' orientation ion in YZ pla plane for moved contacts
(e) Fig. 16. Orientation frequency of branch vectors with different types of inter-particle evolution of GB (a) (c) contact gain and contact loss within the sample (b) (d) contact gain and contact loss within the shear band (e) contact movement during the shear increment of 8.06–12.14%.
figure that for both GB and LBS, the degree of anisotropy of the shear bands is always higher than that of the entire samples throughout the shear, which indicates a higher
degree of bias of the branch vectors to the loading direction within the shear bands than in the entire samples. This is mainly because of the higher percentages of contact gain
Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510 0~0.98%
90 100
120
60
60
150
0 180
0
25
150 25
0 180
0
25 330
210
330
210
=0.98~4.94% =10.40~15.34% 30
50
Frequency/Number
Frequency/Number
120
=4.94~10.40%
=10.40~15.34% 30
25
50
0~0.98%
90 75
=0.98~4.94% =4.94~10.40%
75 50
1507
50
75 100
300 270 Branch vectors' orientation in YZ plane for gained contacts in shear band
240
300 270 Branch vectors' orientation in YZ plane for gained contacts
240
75
(a)
(b) 0~0.98%
90 120
200
60
60
50 0 180
0
50
=10.40~15.34% 30
150
Frequency/Number
100
=0.98~4.94% =4.94~10.40%
80
=10.40~15.34% 30
150
Frequency/Number
120
=4.94~10.40%
150
0~0.98%
90 120
=0.98~4.94%
40
0 180
0
40
100
200
80 300 270 Branch vectors' orientation in YZ plane for lost contacts in shear band
240
300 270 Branch vectors' orientation in YZ plane for lost contacts
120
10.40% (overall)
90 120
450
Frequency/Number
240
(d)
(c)
300
330
210
330
210 150
60
15.34% (overall) 10.40% (in shear band) 15.34% (in shear band) 30
150
150 0 180
0
150 300 450
330
210
240
300 270 Branch vectors' orientation in YZ plane for moved contacts
(e) Fig. 17. Orientation frequency of branch vectors with different types of inter-particle evolution of LBS (a) (c) contact gain and contact loss within the sample (b) (d) contact gain and contact loss within the shear band (e) contact movement during the shear increment of 10.40–15.34%.
4%
%
15 0~
Concentrated Concentrated horizontally vertically
10
.4
Shear increment
4.
94
~1
0.
.3
40
0. 98 ~4 .9 4%
0~ 0. 98 %
14 ~1 06
Shear increment
8.
2. 02
3. 96
~3
~8
2.
.0 6
% .9 6
% 2. 02 0~
1.0
%
Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510
%
1508
Degree of anisotropy /a
0.5
0.0
-0.5
For all contacts For gained contacts For lost contacts For contacts before movement For contacts after movement
-1.0
-1.5 0
2
4
6
8
10
12
14
Degree of anisotropy /a
Concentrated Concentrated horizontally vertically
0.5
0.0
-0.5
For all contacts For gained contacts For lost contacts For contacts before movement For contacts after movement
-1.0
-1.5 0
2
4
6
8
10
12
14
16
18
Axial strain %
Axial strain %
(a)
(b)
Fig. 18. Evolution of anisotropy degree a of the branch vectors of overall, gained, lost, and moved contacts during shear (a) GB (b) LBS.
LBS (overall) LBS (in shear band) GB (overall) GB (in shear band)
0.3
Degree of anisotropy/ a
0.2
0.1
0.0 2
-0.1
4
6
8
10
12
14
16
Axial strain /%
-0.2
Fig. 19. Evolution of anisotropy degree a of the entire samples and the shear bands.
and contact loss (see Fig. 15(a) and (b)) within the shear bands than in the entire samples. 5. Conclusions This paper has presented the use of X-ray microtomography to quantify the inter-particle contact evolution of two granular materials, i.e., nearly spherical-shaped glass beads (GBs) and angular-shaped Leighton Buzzard sand (LBS), under shear. The image processing and analysis techniques used detect the inter-particle contacts have been presented. The inter-particle contact evolution, i.e., the contact gain, the contact loss, and the contact movement, during the shearing process has been quantitatively investigated by combining the inter-particle contact detection and a particle-tracking approach. The main conclusions are as follows:
The frequency distribution of the coordination number (CN) tends to shift toward the direction that decreases the overall average coordination number in the shear. The shift is obvious in the volumetric dilation process of the pre-peak shear stage, and gradually tends to a stabilized state in the post-peak shear stage. The average coordination number of particles with different sizes (i.e., the sizedependent average CN) is found to follow the same trend as the overall average coordination number during the shear, with larger-size particles having higher average coordination numbers. The overall branch vectors of the spherical GB show a nearly isotropic distribution prior to loading, while those of the angular LBS tend to have, on average, a higher frequency oriented in the horizontal plane. For both materials, the branch vectors are found to develop a directional preference towards the loading direction as the shear progresses, which is clearly observed in the pre-peak shear and found to reach a relatively steady state in the postpeak shear. The contact gain and the contact loss in a shear increment, which only account for a small fraction of the total number of contacts, are found to play a very important role in the shear-induced anisotropy. This factor and the contact movement are shown to be the two competing factors in the determination of the evolution of the anisotropy of granular materials. The contact gain and the contact loss drive the overall branch vectors to show a directional preference towards the major principal stress direction, whereas the contact movement is found to attenuate this directional preference. In the pre-peak shear, the anisotropy of the overall branch vectors clearly increases because the contact gain and the contact loss show a strong anisotropy. It reaches a relatively steady state in the postpeak shear when the anisotropy degree values of contacts
Z. Cheng, J. Wang / Soils and Foundations 58 (2018) 1492–1510
(i.e., the gained contacts, lost contacts and moved contacts) gradually become stable. The higher percentages of contact gain and contact loss throughout the shear lead to a higher degree of fabric anisotropy of the shear bands when compared to those of the entire samples. The above findings provide a novel insight into the inter-particle contact evolution of granular materials and should serve as experimental evidence of previous findings in DEM simulations. Acknowledgements This study was supported by the General Research Fund No. CityU 11272916 from the Research Grant Council of the Hong Kong SAR, Research Grant No. 51779213 from the National Science Foundation of China, and the BL13W beam-line of Shanghai Synchrotron Radiation Facility (SSRF). The authors would like to thank Prof. Matthew R. Coop of University College London (formerly City University of Hong Kong) for his help with the development of the triaxial apparatus. The authors also appreciate Prof. Mingjing Jiang in Tongji University for his help with the in-situ tests for this study. References Alikarami, R., Ando`, E., Gkiousas-Kapnisis, M., Torabi, A., Viggiani, G., 2015. Strain localisation and grain breakage in sand under shearing at high mean stress: insights from in situ X-ray tomography. Acta Geotech. 10 (1), 15–30. Alshibli, K.A., Jarrar, M.F., Druckrey, A.M., Al-Raoush, R.I., 2016. Influence of particle morphology on 3D kinematic behavior and strain localization of sheared sand. J. Geotech. Geoenviron. Eng., 04016097 Ando`, E., Hall, S.A., Viggiani, G., Desrues, J., Be´suelle, P., 2012. Grainscale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotech. 7 (1), 1–13. Ando`, E., Viggiani, G., Hall, S.A., Desrues, J., 2013. Experimental micromechanics of granular media studied by x-ray tomography: recent results and challenges. Ge´otech. Lett. 3 (3), 142–146. Barreto, D., O’Sullivan, C., 2012. The influence of inter-particle friction and the intermediate stress ratio on soil response under generalised stress conditions. Granul. Matter 14 (4), 505–521. Borgefors, G., 1986. Distance transformations in digital images. Comput. Vision, Graphics, Image Process. 34 (3), 344–371. Cavarretta, I., Coop, M., O’Sullivan, C., 2010. The influence of particle characteristics on the behaviour of coarse grained soils. Ge´otechnique 60 (6), 413–423. Chen, R.C., Dreossi, D., Mancini, L., Menk, R., Rigon, L., Xiao, T.Q., Longo, R., 2012. PITRE: software for phase-sensitive X-ray image processing and tomography reconstruction. J. Synchrotron Radiat. 19 (5), 836–845. Cheng, Z. Investigation of the grain-scale mechanical behavior of granular soils under shear using X-ray micro-tomography. (PhD thesis), City University of Hong Kong, 2018. Cheng, Z., Wang, J., 2017. A particle-tracking method for experimental investigation of kinematics of sand particles under triaxial compression. Powder Technol. https://doi.org/10.1016/j.powtec.2017.12.071. Cnudde, V., Boone, M.N., 2013. High-resolution X-ray computed tomography in geosciences: a review of the current technology and applications. Earth Sci. Rev. 123, 1–17. Desrues, J., Chambon, R., Mokni, M., Mazerolle, F., 1996. Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography. Ge´otechnique 46 (3), 529–546.
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