Powder Technology 328 (2018) 436–451
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A particle-tracking method for experimental investigation of kinematics of sand particles under triaxial compression Zhuang Cheng, Jianfeng Wang ⁎ Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong
a r t i c l e
i n f o
Article history: Received 24 August 2017 Received in revised form 17 November 2017 Accepted 22 December 2017 Available online 5 January 2018 Keywords: Particle kinematics Particle tracking Granular materials X-ray micro-tomography Tracking criteria
a b s t r a c t Particle kinematics including particle translation and particle rotation, plays a very important role in the shearing behavior of granular soils. In this study, a particle-tracking method based on particle volume (or particle surface area) is presented in detail and applied to the acquisition of particle kinematics of a uniformly graded sands undergoing shearing in a mini-triaxial apparatus using the X-ray synchrotron radiation technique. The effectiveness of this method, and the effects of different tracking criteria (i.e., particle volume and particle surface area) and tracking parameters on its tracking performance are examined. It is found that the particle tracking based on the two tracking criteria provides consistent results of particle kinematics with high accuracy, given that appropriate tracking parameters are used. The presented particle-tracking approach and the research findings will be useful for the measurement and quantification of particle-scale kinematics of granular materials. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Particle kinematics, i.e., particle translation, and particle rotation, play an important role in the macro-scale mechanical response of granular materials. Its importance has been well recognized in the discrete element method (DEM) modeling of granular materials in the past few decades. In earlier years, researchers used the circle or sphere to model individual particles and this has resulted in the over-rotation of particles during shearing, leading to a lower peak strength behavior [1]. To obtain a more realistic and accurate modeling of particle kinematics, many authors incorporated rolling resistance models [2–5], or irregular particle shapes using ‘clumps’, ‘clusters’ [6–9], ellipsoids [10] or super quadrics [11] in DEM simulation. As a result, more realistic stress-strain and particle kinematic behaviors have been acquired. In spite of the great development of DEM modeling skills, however, modeling of particle kinematics of real granular materials requires the reproduction and incorporation of real particle shapes, and the development of sophisticated contact models, all of which requires extremely high computational capability. In addition to the DEM modeling, the non-destructive X-ray imaging techniques such as X-ray radiography, X-ray stereography, and X-ray particle tracking velocimetry (PTV) have been widely applied to the investigation of the kinematics of granular materials. These techniques generally require the placement of lead markers within the granular materials and acquire the kinematics information by analysis of the ⁎ Corresponding author at: B6409, Academic 1, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong. E-mail address:
[email protected] (J. Wang).
https://doi.org/10.1016/j.powtec.2017.12.071 0032-5910/© 2018 Elsevier B.V. All rights reserved.
changing positions of the markers. Because of their high temporal resolutions, they have been widely used for the measurement of rapid granular flows [12–16] and fluid flows [17–19]. However, these markerbased techniques provide relatively sparse data as the grids for the location of the markers are usually somewhat coarse. With the ability of distinguishing grains within granular materials, the recently developed X-ray computed tomography and X-ray micro-tomography techniques offer us an unprecedented opportunity for the measurement of the true kinematics of particles with a high level of accuracy. Using these techniques, a large number of studies have been carried out to quantitatively study grain-scale behaviors of granular materials, including void ratio and fabric evolution [20–23], shear banding and strain localization [24–29], particle crushing [30,31], and particle morphology characterization and evolution [32–35]. Particularly, in the examination of particle kinematics of sands, a discrete Digital Image Correlation (discrete DIC) technique was proposed by Hall et al. [27] based on grey-scale images. It has proven to offer a high level of accuracy of particle tracking results when it is used in conjunction with the high-resolution X-ray microtomography. Alternatively, a binary image-based approach called ‘ID track’ [36,37] has been developed recently to track individual particles for measuring kinematics of Caicos Ooids and Hostun sand grains. Similar particle-tracking approaches have also been used in the measurement of 3D kinematics of other granular soils [38–40]. In the ‘ID track’ approach, particles are first extracted from grey-scale images and stored in binary images. Then particles' volume and centroid coordinates, etc., are derived from the binary images and recorded as their ‘ID’ information. As the ‘ID track’ technique only needs the ‘ID’ information for particle tracking, it is highly efficient in the acquisition and quantification of particle kinematics.
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the device is rotated back to the original position and lifted up for the scanning of the next section, which takes about 1 min. Benefiting from the short exposure time (80 ms), a scan of the whole sample takes only about 10 min.
Based on the previous studies, this paper reexamines the particletracking method proposed by Andò et al. [36] by applying the method to the tracking of sand particles in a triaxial test and assessing the effects of different particle-tracking criteria and tracking parameters on the tracking performance. An extension of the original method is made by including particle surface area as an alternative tracking criterion in this study. In the following part of the paper, the CT images used for particle-tracking are firstly obtained from a mini-triaxial compression test of a Leighton Buzzard Sand (LBS) using an X-ray synchrotron radiation facility. Then, the image processing and analysis skills used for the acquisition of particle ‘ID’ information, the framework of particletracking, and the calculation of particle kinematics are presented in detail. Next, the results of particle displacement and rotation evolution during the triaxial compression test are presented and analyzed. Finally, the influences of different particle-tracking criteria (i.e., particle volume and particle surface area) and tracking parameters on the tracking performance are discussed.
The materials used in the in-situ triaxial compression test is a uniformly graded (0.60–1.18 mm) Leighton Buzzard Sand (LBS), which is an angular quartzitic sand. The breakage behavior of a single LBS particle has been studied previously by using a Nano-focused laboratory scanner [31]. The sample tested is 8 mm in diameter and 16 mm in height, which contains about 10 LBS grains in its diameter direction, and about 1500 grains in the whole sample. The small-size sample is adopted to facilitate the implementation of the ID-track method to the particles. There is no intension in this paper to investigate the strain localization or shear band development within the sample based on the tracking results.
2. Experimental setup
2.3. Acquisition of testing results
2.1. X-ray synchrotron radiation facility and mini-triaxial apparatus
During the test, the stress and displacement data are acquired through a data logger of the triaxial compression device, while the scanning images (32-bit projection images) are obtained from the detector. Fig. 2 shows the stress-strain curve of the tested LBS sample, on which three representative scanning points (i.e., one prior to the shear, one around the peak of deviatoric stress, and one after the peak) are marked. Only the results from the three scans are presented for the demonstration of the particletracking methods. It should be noted that more scans along the stressstrain curve can be taken for acquisition of more insights into the particle kinematics. It is seen in Fig. 2 that a sudden decrease of the deviatoric stress at each scan occurs, and this is caused by the stress relaxation of the sample during the scan. Overall the stress-strain curve appears oscillatory, especially as it approaches to and after the peak of the deviatoric stress (i.e., around the axial strain of 5%). This is caused by the limited number of LBS particles within the sample.
The X-ray source used for this study is a synchrotron microtomography setup at the BL13W beam-line of the Shanghai Synchrotron Radiation Facility (SSRF). Synchrotron micro-tomography, featured by its high spatial resolution and rapid scanning, is more efficient than laboratory scanners [41,42]. The X-ray imaging device is composed of a parallel beam, a rotation stage, and a detector, as shown in Fig. 1. The beam energy is 25 Kev. The size of the detector is 12.66 mm (width) × 4.888 mm (height). It has a spatial resolution of 6.5 μm, which allows for imaging, for example, of a particle having a diameter of 0.8 mm with about 1,000,000 voxels. The triaxial compression apparatus is fixed on the rotation stage located between the X-ray beam and the detector, which enables the whole apparatus to be rotated across 180° with a constant rate of 2.083°/s during scanning. In the test, the sample is sheared (compressed axially and confined with a stress of 1.5 MPa laterally) at a constant rate of 0.2%/min and the shearing is paused at 0%, 5%, and 10% axial strain to acquire the synchrotron micro-tomography scans. In each scan, the 16 mm-high sample is divided into 4 sections (with an overlap of about 0.888 mm between each two adjacent sections), and each section is scanned with about 1080 projection images. Each time when the scanning of a section is finished,
2.2. Material tested
3. Image processing and analysis The ‘ID’ track method requires particles' ‘ID’ information, including particle centroid coordinate, particle volume, particle size, particle principal axes' orientations and particle surface area, to be determined. This
X-ray beam Detector Rotation stage
Triaxial apparatus
Fig. 1. Synchrotron micro-tomography setup and triaxial apparatus at the BL13W beam-line of SSRF.
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Deviatoric stress /MPa
5
4
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3rd scan
3
2
LBS, 0.6~1.18 mm, σ3=1.5 MPa
1
1st scan 0 0
3
6
9
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Axial strain % Fig. 2. Stress-strain curve of the LBS sample.
15
section describes the application of image processing and analysis to prepare the CT images for the acquisition of particles' ‘ID’ information. Fig. 3 presents the flow chart of the image processing and analysis procedure on the projection images. As the first step, the reconstruction of grey-scale CT slices from projection images is illustrated in Fig. 4. For a projection set containing M projection images with a size of N × P (pixels), N sinograms are first obtained with a size of M × P (pixels). Gridrec algorithm [43] is then performed on these sinograms to reconstruct CT slices. An example of a projection image, a sinogram, and a reconstructed horizontal slice are shown in Fig. 4(a), (b) and (c), respectively. It should be noted that in order to alleviate the ‘ring artifact’ in CT slices (Fig. 4(c)) arising from vertical ripples in the corresponding sonograms (Fig. 4(b)), a ‘ring artifact’ removal algorithm [44] is adopted. Fig. 4(d) shows the slice in Fig. 4(c) after the removal of ‘ring artifact’. The reconstruction of CT slices is performed on PITRE [45] from SSRF, in which the Gridrec algorithm and ‘ring artifact’ removal algorithm are available. In slice reconstruction, a stack of 32-bit slices are obtained, and are converted into 8-bit grey-scale images for the subsequent processing. A 3D median filter with the size of 3 × 3 × 3 (voxels) is first implemented as a low-pass filter for three times in order to reduce the random noise. Thresholding is then performed, according to the
Image processing
32 bit projection image Gridrec slice reconstruction Image conversion from 32bit to 8bit 8bit Grey-scale CT image
3D median filtering Image analysis
Thresholding Binary image
Particle volume Particle surface area
3D median filtering Segmentation
Particle size Labelled image
Particle ‘ID’ information
Particle centroid Particle principal axes’ orientations
Fig. 3. Flow chart of image processing and analysis of X-ray CT images.
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Vertical ripple
(a)
(b)
‘ring’ artifact
(d)
(c)
Fig. 4. Illustration of slice reconstruction (a) a projection image (b) a sinogram (c) a reconstructed slice (d) the reconstructed slice in (c) after removal of the ‘ring’ artifact.
intensity histogram of the images based on Otsu's threshold method [46], on the grey-scale images using the intrinsic functions, graythresh and im2bw, from MATLAB (Mathworks, Natick, MA, USA). The thresholding converts the grey-scale images with multiple intensity levels into binary images with only two intensity values (i.e., 1 and 0). Fig. 5 shows the 3D binary images of the sample at the first scan and the third scan acquired after thresholding, in which grains (including
sand grains and porous stone grains) are in grey and background in white. Following thresholding, image segmentation is used to separate individual particles, which is essential for the quantitative analysis of individual particles. Just prior to the segmentation, the same 3D median filter used before thresholding is applied again on the binary images for three times to clear isolated noise voxels generated in the thresholding.
Porous stone
LBS grains
Porous stone
3rd scan
1st scan Fig. 5. Binary images obtained through thresholding.
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A morphological watershed algorithm [47] is used for segmentation. The watershed algorithm is implemented on the inverted chamfer distance map of the binary images to determine the watershed lines needed for particle separation. Generally speaking, the watershed lines, which are related to the watershed depth, determine the number of particles separated, but are sensitive to intensity variations in the distance map. This sometimes causes over-segmentation. In order to alleviate the oversegmentation, a marker-based approach similar to that used in a previous study by the second author [31] is used to control the minimum watershed depth. Note that this approach is applied to 3D images. To examine the performance of the segmentation, the segmented slices of a vertical cross section at the first and the third scans are compared with the corresponding grey-scale slices in Fig. 6. For a better visualization, different particles in the segmented slices are labelled in different colors. It can be seen that most of the particles are well separated. It should be noted that in the slices shown in Fig. 6(a) and (c) the porous stones are removed. The subsequent process converts the segmented binary images into 3D labelled images, in which each particle corresponds to an isolated region and has a unique intensity value. Based on these labelled images, the determination of the particle ‘ID’ information, including particle centroid, volume, size, principal axes' orientations and surface area, is carried out using the image analysis techniques used by Fonseca [48].
(a)
(c)
To create the particle ‘ID’ information, an ID number is defined for each particle as the intensity value of the voxels comprising it. Particles' centroid coordinates and volume are acquired using the intrinsic MATLAB function regionprops. Specifically, for each particle denoted by a region, functions Centroid.regionprops and Area.regionprops give the global coordinates of each particle centroid and the number of voxels comprising the particle, respectively. The volume of the particle is then calculated by the voxel number multiplied by the voxel size (i.e., (6.5 μm)3). The particle size is calculated as the equivalent particle diameter d according to Eq. (1): d¼2
3V 4π
13
;
ð1Þ
where V is the particle volume. The principal axes' orientations of each particle, i.e., the principal inertia axes' orientations, are described by three orthogonal unit vectors in a 3 × 3 matrix A. The matrix can be determined by a principal component analysis (PCA) method according to the local coordinates of the particle centroid and boundary voxels. Each particle is first extracted and stored in a logic matrix using another intrinsic function Image.regionprops. The local coordinates of the centroid (x0, y0, z0), and the boundary voxels (x1, y1, z1),(x2, y2, z2),…, (xN, yN, zN), of the particle
(b)
(d)
Fig. 6. Illustration of segmentation results: (a), (c) are segmented images at the first and third scan and (b), (d) are the corresponding grey-scale images.
Z. Cheng, J. Wang / Powder Technology 328 (2018) 436–451
are then derived using intrinsic functions Centroid.regionprops and bwprim, respectively. The orientation matrix A, as expressed in Eq. (2), is then determined by another intrinsic function pca with the input of a N × 3 matrix P given in Eq. (3): A ¼ ðn1 ; n2 ; n3 Þ; 0 B P¼B @
x1 −x0 ; y1 −y0 ; z1 −z0 x2 −x0 ; y2 −y0 ; z2 −z0 ⋮ ⋮ ⋮ xN −x0 ; yN −y0 ; zN −z0
ð2Þ
441
the deformed configuration in a later scan). Section 4.1 describes the framework of the ‘ID track’ technique for identifying the particles that are originally located in the reference configuration from the deformed configuration. Section 4.2 describes the calculation of the kinematics, i.e., displacement and rotation, of the particles that are identified in the deformed configuration.
1 C C; A
ð3Þ
where n1, n2, n3 are three unit vectors denoting the orientations of the major, intermediate, and minor axes of the particle, respectively. The particle surface area is calculated as the number of boundary voxels multiplied by the 2D voxel size, i.e., (6.5 μm)2. Note that similar methods were also used by other authors [42,49–50] for the determination of particle surface areas. 4. Grain tracking based on particle ‘ID’ information Based on the particle ‘ID’ information obtained for each particle in Section 3, this section presents the ‘ID track’ technique for the quantification of particle kinematics during a loading increment between two configurations (i.e., the reference configuration in an earlier scan and
4.1. Framework of ‘ID track’ The main idea of the ‘ID track’ in this paper is based on three assumptions. The first assumption is that particles experience no or very little crushing, thus their volume, or surface area would not change significantly (a small variation is allowed because of image processing errors) during the loading increment between the reference configuration and the deformed configuration. For LBS studied in this paper, very little crushing means that sand particles do not undergo major splitting events. Kinematics of significantly crushed particles (i.e., particles undergoing major splitting) is out of the scope of this paper. The second assumption is that particles would only move within a certain region with radius D (D, for example, is about 2–4 times the average grain diameter d50 of the sample) away from the particle centroid during each loading increment. This assumption actually limits the maximum displacement of the particles, which depends on the increment of the loading. Note that this assumption might not be valid for the tiny fragments generated
Fig. 7. Flow charts of the algorithms used in ‘ID track’ (a) step 1 (b) step 2.
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Fig. 7 (continued).
R15
D15
R14
D14
R16 R4
D16 R2
R6 R13
R1
R3 R10
R8 R11
R9
D2
D6 D13
R7 R12
D4
D7 D12
D1 D10
D8
R5
D5 D9
D11 (a)
D3
(b)
Fig. 8. Illustration of matching candidate lists in 2D condition (a) particles from ‘R1’ to ‘R16’ in the reference configuration (b) particles from ‘D1’ to ‘D16’ in the deformed configuration.
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during the test, as these small-size particles may travel through the pores between the large-size particles. For the two loading increments of the uniformly graded sample in this study, this assumption is valid. The third assumption is that particles within a small volume (for instance, a spherical region with a diameter about 2–4 d50 and centered at the centroid of the reference particle) move harmoniously so that the displacement of the reference particle is close to the median displacement of the particles within the small volume. Based on these assumptions, the ‘ID track’ method takes two steps to track the particles, as illustrated by the flow charts in Fig. 7(a) and (b), respectively.
The first step determines an initial matching list in which particles in the deformed configuration are matched to those in the reference configuration according to particle volume. Specifically, a matching candidate list is first determined for each reference particle by comparing the distance between the reference particle centroid and the deformed particle centroid in the deformed configuration with a threshold distance D (D, for example, is 400 voxels). Those particles with the distance smaller than D are saved as the matching candidates for the reference particle. For the ease of visualization, an example of the determination of a matching candidate list of a reference particle in 2D condition is given in Fig. 8. Note that the determination of matching candidate lists 1.0
After isotropic compression Shearing to 5% Shearing to 10%
Percentage finer in mass
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(c) 1.0 After isotropic compression Shearing to 5% Shearing to 10%
Percentage finer in number
Percentage finer in number
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Particle equivalent diameter/mm
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Particle surface area/voxels
Particle volume/voxels
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Particle surface area/voxels
(e)
100000
0.8
After isotropic compression Shearing to 5% Shearing to 10%
0.6
0.4
0.2
0.0 0.1
1
Particle equivalent diameter/mm
(f)
Fig. 9. Particle volume, particle surface area, and particle size distribution evolution of LBS during triaxial compression under 1.5 MPa: (a) (b) and (c) by mass; (d) (e) and (f) by number.
Z. Cheng, J. Wang / Powder Technology 328 (2018) 436–451
in 3D condition follows exactly the same way. For particle ‘R1’ in the reference configuration (Fig. 8(a)), the particles ‘D1’, ‘D2’, ‘D3’, ‘D4’, ‘D6’, ‘D7’, ‘D8’, ‘D10’, ‘D12’, ‘D13’, ‘D16’ in the deformed configuration (Fig. 8(b)) are listed as its matching candidates, as the distances from their centroids to the reference particle centroid are within the threshold D. Then, for each reference particle, deformed particles whose volume differences from the reference particle are within a tolerance X (e.g., X = 10% of the reference particle volume) are selected from its matching candidate list. Among them the one that has the volume closest to the reference particle is temporarily assigned as the best match to the reference particle and the particle ‘ID’ is stored in the initial matching list. If no particles in the matching candidate list are within the tolerance, the reference particle is regarded as “lost”. This process is repeated for each reference particle in the reference configuration. If a particle is found to be assigned to two reference particles, the reference particle with a volume closer to the candidate particle is regarded as a better match to it. The other reference particle is then released from the initial matching list and made available for rematching again. The final matching list is determined in the second step considering not only the particle volume, but also the particle displacement. Similar to the first step, a matching candidate list is first determined for each reference particle. The best match to the reference particle is then determined from the candidate list not only according to the particle volume difference, but also the difference between the displacement of the reference particle and the median displacement of particles within the spherical volume with a radius of D (D = 400 voxels), which is centered at the centroid of the reference particle. These differences are equally weighted and the particle that minimizes the combination of both is chosen as the best match and the ‘ID's of the match pairs are then stored in an updated matching list. The above process is repeated for each reference particle to update the whole matching list. It should be noted that in the calculation of particle displacement, the current matching list is used and the particle centroid movement is taken as the displacement, which will be described in Section 4.2. To minimize the effects of mistakes in the initial matching list, in which particle displacement is not involved, the matching list is updated three times according to the above process before being output as the final matching list. 4.2. Calculation of particle displacement and particle rotation In a granular system where particle crushing and particle deformation is not involved, a particle motion can be decomposed as the translation of the mass center and the rotation around a certain axis passing through the mass center. In the following derivation, the centroid of a 3D particle image is taken as its mass center, because the LBS tested, mainly consisting of silica, is a rather homogenous material. The displacement of particle i from the reference configuration to the deformed configuration is determined by the vector δi(x, y, z) from the centroid of reference particle i, Ri(x, y, z), to the centroid of its match particle Di(x, y, z) in the deformed configuration, expressed by Eq. (4): δi ðx; y; zÞ ¼ Di ðx; y; zÞ−Ri ðx; y; zÞ;
ð4Þ
The rotation of particle i can be determined based on the rotation axis orientation unit vector wi and rotation angle θi, through the rotation matrix Mi. Given the orientation matrix Ai(n1, n2, n3) of particle i in the reference configuration, and the orientation matrix Bi(n1′, n2′, n3′) of its match particle in the deformed configuration, the rotation matrix Mi(m1, m2, m3) of particle i describing its rotation can be calculated by Eq. (5): M i ¼ Bi A−1 i ;
ð5Þ
Then wi and θi can be calculated according to Eqs. (6)–(9): 8 9 < m11 m12 m13 = M i ðm1 ; m2 ; m3 Þ ¼ m m m ; : 21 22 23 ; m31 m32 m33 i
ð6Þ
ui ¼ ðm32 −m23 ; m13 −m31 ; m21 −m12 ÞTi ;
ð7Þ
wi ¼
ui ; kui k
ð8Þ
kui k ; θi ¼ arcsin 2
ð9Þ 1
where kui k ¼ ½ðm32 −m23 Þ2 þ ðm13 −m31 Þ2 þ ðm21 −m12 Þ2 2 is the magnitude of ui. Note that θi always lies between (0°, 90°) as the orientation vectors for particle i can also be represented by their opposite vectors. 5. Test results and discussion 5.1. Evolution of particle size distribution (PSD) Particle size distribution of the tested LBS sample is obtained using the image processing and analysis techniques presented in the previous sections. Fig. 9 presents the particle volume, particle surface area, as well as particle diameter distribution of the LBS sample at the three scans shown in Fig. 2 in terms of percentage in mass and percentage in number, respectively. The particle diameter in Fig. 9(c) and (f) is determined according to Eq. (1). As can be seen in Fig. 9(a), (b) and (c), no significant particle crushing occurs in terms of percentage in mass during the shearing increments of 0–5% and 5–10% when particle volume, surface area, or particle diameter is examined. There is only a very slight change of the mass of the small-size fragments (i.e., the particle equivalent diameter smaller than 0.8 mm in Fig. 9(c)) during the whole test. Note that the smallsize fragments in Fig. 9(a) and (b) corresponds to the particles with particle volume smaller than 1,000,000 voxels and surface area smaller than 55,000 voxels, respectively. A similar trend can also be found in Fig. 9(d), (e) and (f) in terms of percentage in number for large-size particles, although the increase of the small-size fragments is obvious. To indicate further the degree of particle crushing of the large-size particles (i.e., the particles with diameter larger than 0.8 mm) throughout the test, we illustrate in Fig. 10 the frequency distributions of the
200 1st scan 2nd scan 3rd scan
180 160 140 Frequency
444
120 100 80 60 40 20 0 1.0
1.5
2.0
2.5
3.0
3.5
6
Particle volume (×10 voxels) Fig. 10. Particle volume distribution of the large-size particles (particle diameter N 0.8 mm) in the three scans.
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(a)
445
(b)
Fig. 11. Randomly selected particles for tracking accuracy estimation (a) the first scan (b) the second scan.
Table 1 Results of sampling survey from the tracked particles. PVD
0–0.125%
0.125%–0.25%
0.25%–5%
5%–10%
SAD 0–0.125% 0.125%–0.25%
74.9%/75/75 14.5%/15/14
0.25%–5%
particle volume of the large-size particles during the two loading increments. Note that a particle with a diameter of 0.8 mm has a particle volume of around 1,000,000 voxels. Fig. 10 indicates that there are no significant differences of the particle volume distribution between the three scans. Based on the results shown in Figs. 9 and 10, we can conclude that there is only a very slight degree of crushing (i.e., very little major particle splitting) of the large-size particles within the tested sample. Therefore, the three assumptions for particle tracking are valid for the large-size particles.
6.0%/6/6
5%–10%
4.6%/4/3
5.2. Tracking of particles experiencing little crushing
>10%
Note: PVD denotes particle volume difference and SAD denotes surface area difference. For a/b/c in the table, a is the percentage of particle pairs that are within the difference range, b is the number of particle pairs selected for sampling, and c is the number of correctly tracked particle pairs.
The ‘ID track’ is applied to the LBS particles with particle volume larger than 1,000,000 voxels. Here we only present the results of particle tracking using particle volume as the tracking criterion. The volume difference tolerance X is set as 10% (i.e., 10% of the reference particle volume), and the search region radius D set as 400 voxels. Note that the
Table 2 Typical image segmentation errors resulting in incorrect tracking. Typical errors
Examples
Local error
2nd-1678
3rd-1913
Volume difference 2.44%
Surface area difference 0.67%
Over-segmentation
1st-1537
2nd-1712
Volume difference 5.6%
Surface area difference 2.5%
Under-segmentation
1st-34
2nd-11
Volume difference 0.26%
Surface area difference 2.59%
Note: particles shown in the table are not in the randomly selected particle list. 3rd-1913, for example, means particle ‘ID’ 1913 at the third scan.
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2500
2500
120
100
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120
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2000
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80
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1500 60
60 1000
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40
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20
0
0
500
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0
0
0
500
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0
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Voxels
(b)
(a)
Fig. 12. Vertical displacements (voxels) of tracked particles during shearing increments (a) 0%–5% (b) 5%–10%.
‘ID track’ based on particle surface area is also possible, which is discussed in Section 5.4. A high percentage of particles were tracked using the ‘ID track’. The number of particles with volume larger than 1,000,000 voxels at the three scans is 875, 853, and 819, respectively, within which 854 and 831 particles are tracked in the shearing strain increments of 0%–5% (i.e., scan 1-scan 2) and 5%–10% (i.e., scan 2-scan 3), respectively. In addition, 816 of them are tracked throughout the two shear increments. In order to evaluate the accuracy of the tracking results, a sampling survey of 100 pairs of particles from the 816 particle pairs in the shearing increment of 0%–5% is made. The 100 particle pairs are randomly selected and divided into four groups according to the magnitudes of particle volume difference and particle surface area difference. Fig. 11 shows the 100 particle pairs that are randomly selected at the first and second scans. Note that a particle pair with smaller differences in
particle volume and particle surface area normally has a higher possibility to be correctly tracked. For each selected particle pair, a visual comparison is made to check the tracking result. The particle volume difference and particle surface area difference, number of particle pairs selected in each group, and the number of particle pairs that are correctly matched are listed in Table 1. It can be seen from Table 1 that only two particle pairs are incorrectly matched. The ‘ID track’ technique provides highly accurate matching results in almost all the cases, except when a significant image segmentation error occurs, such as over-segmentation, under-segmentation, or a local error in particle separation. Most of these errors result in relatively large differences of particle volume, or particle surface area between particle pairs, and lead to a lower probability of correct matching. Table 2 shows examples of the typical image segmentation errors which may result in incorrect tracking.
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Fig. 13. Rotation angles (degrees) of tracked particles during shearing increments (a) 0%–5% (b) 5%–10%.
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Fig. 14. Orientation of the projection plane.
5.3. Particle kinematics Particle displacements and rotations of the tracked particles in the two shearing increments of 0%–5% and 5%–10% are calculated according to Eqs. (4)–(9) and the results are shown in Figs. 12 and 13. The results are plotted with colored scatter points in a projection plane parallel to section A (a vertical cross section, which contains the axis of the sample in Fig. 14). Each scatter point corresponds to a particle centroid location
in the projection plane, and the color of the scatter point represents the magnitude of the vertical displacement of the particle, shown in Fig. 12, or the rotation of the particle, shown in Fig. 13. It can be seen in Fig. 12 that there is an overall trend of increasing displacements of particles from the bottom upwards, although several particles show a sudden change of vertical displacement, which is normally caused by the incorrect tracking. Note that in the test, the bottom of the sample is fixed and the sample is loaded by pushing the top of the
0.8 0.7
Probability density/×102
0.6 0.5 0.4 0.3 0.2 0.1 0.0 -15.0 -12.5 -10.0 -7.5 -5.0 -2.5
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5.0
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Surface area difference/ % Fig. 15. PDF of surface area difference of the tracked particles in the load increment of 0%–5%.
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tend to localize in an inclined band in the post-peak shearing increment of 5%–10%.
sample downward. Particles located in the upper left corner in the two shearing increments have larger vertical displacements than those in the upper right corner, due to the tilting of the porous stone during loading. An inclined band with localized displacement is observed in the middle of the sample in Fig. 12(b), which includes a wide range of displacement magnitude varying from around 20 to 80 voxels. In Fig. 13, we can see that particle rotations exhibit a disordered distribution in the pre-peak shearing increment of 0%–5%, while the rotations
14
5.4. Discussion In the previous sections, we have shown the successful application of the ‘ID track’ technique to the tracking of LBS experiencing little crushing based on particle volume with a high level of accuracy. In the
Track using particle surface area Track using particle volume
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(b) Fig. 16. Comparison of particle kinematics between ‘ID’ track by particle surface area and particle volume at the shearing increment of 0%–5% (a) displacement magnitude probability distribution (b) rotation angle probability distribution.
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different types of boundary voxels [33,35]. More accurate calculation of the particle surface areas can be achieved by assigning different surface area weights to different boundary voxel configurations [31,51], or reconstructing the surfaces with triangular surface meshes using Marching Cubes algorithm [35,52]. Despite the above fact, very close values of surface area will be obtained for the same
following paragraphs, we show the results using particle surface area as the tracking criterion. The particle surface areas used for the particle tracking are determined based on the total number of voxels on the particle boundaries, as stated in Section 3. Note that this may overestimate the real areas of the particles because of the lack of consideration of the
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X: Particle volume difference tolerance
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D=400
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X: Particle surface area difference tolerance
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(b) Fig. 17. Effects of difference tolerance (a) particle volume difference tolerance (b) particle surface area difference tolerance.
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particle at different scans as long as a consistent voxel-counting method is used. Fig. 15 shows the probability density function (PDF) of surface area difference of the tracked particles in the load increment of 0%–5%. The tracking results are obtained from the particle volume-based tracking with the particle difference tolerance of 10% (i.e., X = 10%). The surface area difference is calculated by: surface area at the second scan Surface area difference ¼ 1− Particle . From Fig. 15 Particle surface area at the first scan
we can see that for the vast majority of the tracked particles, the surface area differences between the two scans are within ± 2.5%. The ‘ID track’ based on particle surface area is applied to LBS particles with surface area larger than 55,000 voxels with a tolerance X = 10% and search radius D = 400 voxels. Note that the particles involved in tracking are not exactly the same as those used previously. Only the results in the first shearing increment (i.e., 0–5% strain) are presented and compared with those obtained by the volume-based ‘ID track’, as those have been sampled and investigated before. As can be seen in Fig. 16(a) and (b), the particle displacement and particle rotation probability distributions obtained by the two ‘ID track’ techniques agree with each other very well, although there is a slight difference at the tails. The results indicate that surface area-based ‘ID track’ is equally effective as the volume-based ‘ID track’. Next, we show the influences of the difference tolerance X of the particle ‘ID’ information (i.e., particle volume or particle surface area), and the search region radius D on the tracking results. Fig. 17(a) and (b) show probability distributions of the displacement magnitude of the tracked particles in the first shearing increment (0%–5%) against the variations of difference tolerance X of particle volume and particle surface area, respectively. It is found that for both ‘ID track’ approaches, X values in the range of 7.5% to 20% basically yield consistent results, while great changes in the results occur when X is lower than 5% (e.g., 1% and 2.5%). This is because for a significant proportion of reference particles, small values of difference tolerance exclude the real match particles from their candidate lists, which, however, would have been included if a larger difference tolerance was used. A larger difference tolerance (for example, X = 7.5%, 10% and 20%) means more match candidates to each reference particle, which generally
increases the possibility of mismatching in the first tracking step. However, a compensation is made in the second tracking step by the ‘harmonious displacement’ rule, which largely reduces the cases of mismatching. Eventually, the use of X = 7.5%, 10% and 20% leads to overall consistent results. Similar trends can also be found in Fig. 18 for the effects of search region radius D. The use of D smaller than 200 (e.g., D = 100 and D = 150) leads to remarkable differences in the probability density distribution of the displacement magnitude, from the cases with D = 300, D = 400 and D = 500. This situation generally occurs when there is a large amount of particles within the sample whose actual displacements during the shearing increment is larger than the search region radius D. 6. Conclusions A particle-tracking method (‘ID track’) is presented for the measurement of kinematics (i.e., particle displacement and particle rotation) of sand particles sheared in a mini-triaxial apparatus and scanned by an X-ray synchrotron radiation source. The particle-tracking approach is based on particles' ‘ID’ information (i.e., particle volume or particle surface area) which would not change significantly for particles experiencing no or very little crushing. Sufficient details of the method have been provided to facilitate the convenient use of the method by the readers. In addition to particle volume, the use of particle surface area as an alternative particle-tracking criterion is examined. Both particle volume-based tracking and particle surface area-based tracking are applied to the LBS sample. It is found that they are equally effective and provide tracking results with high accuracy, given that the assumptions required for ‘ID track’ are basically valid. These assumptions are the preconditions of particle volume (or particle surface area) consistency needed for tracking. Factors leading to the inconsistency of the particle volume (or particle surface area) between the reference configuration and the deformed configuration may result in loss of tracking, or even tracking errors. These factors include the inappropriate image thresholding and image segmentation (such as those given in Table 2, which is shown to be the main reason leading to incorrect tracking) in the image processing.
18 16
Particle volume difference tolerance=10% Probability density ×10−3
14 12 10
D=500 D=400 D=300 D=200 D=150 D=100
8 6 4 2 0 0
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Displacement magnitude (voxels) Fig. 18. Effects of search region radius.
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The performance of ‘ID track’ may also be affected by some parameters used in the particle-tracking. In the ‘ID track’ process, difference tolerance X and search region radius D lower than a threshold value (e.g., 5% for X, 200 for D) may lead to a higher probability of incorrect matching or loss of matching. The threshold of X is hinged on the scope and degree of image processing errors such as incorrect thresholding, under-segmentation, over-segmentation, as well as local segmentation errors, while the threshold of D is determined by the magnitude of the shearing increment. As the performance of the particle-tracking method highly relies on the consistency of particle volume (or particle surface area) during a shearing increment, a limitation of the current ‘ID track’ technique is the lack of capability in tracking of particles that are significantly crushed. Improved ‘ID track’ techniques capable of tracking of crushed particles should utilize particle features that are consistent even if crushing occurs as a tracking criterion, such as particle local morphology, and particle local curvature, etc. The incorporation of these particle features into the ‘ID track’ technique is currently underway. Acknowledgement 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. 51379180 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 in 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 the help with the in-situ test for this study. References [1] L. Rothenburg, R.J. Bathurst, Micromechanical features of granular assemblies with planar elliptical particles, Géotechnique 42 (1) (1992) 79–95. [2] K. Iwashita, M. Oda, Rolling resistance at contacts in simulation of shear band development by DEM, J. Eng. Mech. 124 (3) (1998) 285–292. [3] M.J. Jiang, H.S. Yu, D. Harris, A novel discrete model for granular material incorporating rolling resistance, Comput. Geotech. 32 (5) (2005) 340–357. [4] J. Ai, J.F. Chen, J.M. Rotter, J.Y. Ooi, Assessment of rolling resistance models in discrete element simulations, Powder Technol. 206 (3) (2011) 269–282. [5] B. Zhou, R. Huang, H. Wang, J. Wang, DEM investigation of particle anti-rotation effects on the micromechanical response of granular materials, Granul. Matter 15 (3) (2013) 315–326. [6] T. Matsushima, H. Saomoto, Discrete element modeling for irregularly-shaped sand grains, Proc. NUMGE2002: Numerical Methods in Geotechnical Engineering 2002, pp. 239–246. [7] M. Price, V. Murariu, G. Morrison, Sphere clump generation and trajectory comparison for real particles, Proceedings of Discrete Element Modelling, 2007. [8] J. Ferellec, G. McDowell, Modelling realistic shape and particle inertia in DEM, Géotechnique 60 (3) (2010) 227–232. [9] J. Wiącek, M. Molenda, J. Horabik, J.Y. Ooi, Influence of grain shape and intergranular friction on material behavior in uniaxial compression: experimental and DEM modeling, Powder Technol. 217 (2012) 435–442. [10] T.T. Ng, Fabric study of granular materials after compaction, J. Eng. Mech. 125 (12) (1999) 1390–1394. [11] P.W. Cleary, The effect of particle shape on simple shear flows, Powder Technol. 179 (3) (2008) 144–163. [12] P.L. Bransby, P.M. Blair-Fish, R.G. James, An investigation of the flow of granular materials, Powder Technol. 8 (5–6) (1973) 197–206. [13] J. Lee, S.C. Cowin, J.S. Templeton III, An experimental study of the kinematics of flow through hoppers, Trans. Soc. Rheol. 18 (2) (1974) 247–269. [14] T.A. Kingston, T.B. Morgan, T.A. Geick, T.R. Robinson, T.J. Heindel, A cone-beam compensated back-projection algorithm for X-ray particle tracking velocimetry, Flow Meas. Instrum. 39 (2014) 64–75. [15] T.A. Kingston, T.A. Geick, T.R. Robinson, T.J. Heindel, Characterizing 3D granular flow structures in a double screw mixer using X-ray particle tracking velocimetry, Powder Technol. 278 (2015) 211–222. [16] T.B. Morgan, T.J. Heindel, X-ray particle tracking of dense particle motion in a vibration-excited granular bed, Proceedings of the 2010 ASME International Mechanical Engineering Congress and Exposition, Vancouver, British Columbia, Canada 2010, pp. 12–18. [17] Y.G. Guezennec, R.S. Brodkey, N. Trigui, J.C. Kent, Algorithms for fully automated three-dimensional particle tracking velocimetry, Exp. Fluids 17 (4) (1994) 209–219. [18] A. Seeger, K. Affeld, L. Goubergrits, U. Kertzscher, E. Wellnhofer, X-ray-based assessment of the three-dimensional velocity of the liquid phase in a bubble column, Exp. Fluids 31 (2) (2001) 193–201.
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