Visualization of Failure and the Associated Grain-Scale Mechanical Behavior of Granular Soils under Shear using Synchrotron X-Ray Micro-Tomography

1. Designing the experiment well in advance

1. Determine test material, particle size, sample size and sample initial porosity.
   NOTE: Leighton Buzzard sand with a diameter of 0.15~0.30 mm and a sample size of 8 x 16 mm (Diameter x Height) is used as an example to demonstrate the protocol of this study. Other sands such as Fujian sand, Houston Sand, Ottawa sand and Caicos ooids, etc. and similar sample sizes can also be used.
2. Choose an appropriate detector (Figure 1A) according to the required spatial resolution and scanning area, which are determined according to the predetermined particle size and sample size. For example, a detector with a spatial resolution of 6.5 μm is used in this study. It has an effective scanning area of 2048 x 860 pixels (i.e.,13.3 × 5.6 mm).
   NOTE: During a triaxial compression test, the deformed sample should remain in the scanning region of the detector. A high-spatial resolution detector should be used so that individual particles contain sufficient voxels for the appropriate extraction of particle properties.
3. Determine the required energy of the X-ray source (Figure 1A) and exposure time according to the test material and sample size. Generally, a higher energy should be used for a larger sample composed of a denser material. Use an X-ray energy of 25 keV and an exposure time of 0.05 s for the sand samples in this study.
   NOTE: The required X-ray energy and exposure time can be determined by trial and error using a scanned projection of the sample. The ratio of the minimum grey-scale intensity of the projection to its maximum value should not be lower than 0.2. Otherwise, a higher X-ray energy or longer exposure time should be used.
4. Determine the required rotation speed ω (degrees per second) for the rotation stage (Figure 1A) of the X-ray device. The rotation speed ω is calculated according to the required number of projections N (e.g., N = 1,080) for CT slice reconstruction.
   NOTE: ω=180 V_s/N. Here, V_s is the scanning speed of the X-ray device, i.e., the number of radiographs scanned and recorded per second. V_s is mainly affected by the performance of the detector and the hardware associated with the detector such as the computer.
5. Fabricate a triaxial loading apparatus (Figures 1B,C, see also reference 33) to be used in conjunction with the X-ray μCT device. The apparatus should have the same main functions as a conventional triaxial compression apparatus. The design should consider the requirement of sample size, the range of confining stresses and loading rates.
   NOTE: The apparatus should be able to fit into the X-ray μCT device and be light to facilitate its rotation using the rotation stage. The triaxial cell should be transparent to X-rays. Considering the transparency requirement, acrylic and polycarbonate might be used to fabricate the triaxial cell.
6. Carry out a test with the same confining pressure, loading speed and sample properties (i.e., material, sample size and initial porosity) outside of the X-ray CT scanner to plan when to pause the loading for CT scanning.

2. Carrying out in situ triaxial compression testing

1. Place the triaxial loading equipment and the test material on site.
   NOTE: The loading apparatus and confining pressure offering device (see the Table of Materials) are placed in the X-ray CT scanning room, while the data acquisition and controlling devices are located outside. Triaxial loading and CT scanning of the sample are then operated outside the scanning room.
2. Fix a lifting stage on the board of the X-ray micro CT device (Figure 1B). Fix a tilting stage on the lifting stage and a rotation stage on the tilting stage, respectively (Figure 1B).
   NOTE: The lifting stage and tilting stage should have sufficient loading capacity to move the relevant equipment placed on them.
3. Adjust the position and orientation of the rotation stage via the tilting stage such that any single X-ray passes through the same points within the sample when it is rotated across 180 degrees around the axis of the rotation stage.
   NOTE: Steps 2.2 to 2.3 are applicable to the X-ray micro CT device at Shanghai Synchrotron Radiation Center (SSRF). For X-ray micro CT devices specifically used for in situ triaxial testing, these steps can be omitted after the careful positioning and fixation of the rotation stage.
4. Prepare a soil sample on the board according to the following procedures.
   1. Add a small amount of silicone grease around the lateral surface of the top end of the base plate and place a porous stone on its upper surface. Put a membrane around the lateral surface of the top end (Figure 2A).
   2. Add a small amount of silicone grease on the contact surfaces between the two parts of the sample maker and lock it. Place the sample maker on the base plate and allow the membrane to pass through it (Figure 2B).
   3. Create suction (e.g., 25 kPa) inside the sample maker through its nozzle using a vacuum pump. Fix the membrane to the lateral surface of its upper end. Ensure that the membrane is attached to the inner surface of the sample maker (Figure 2C).
   4. Drop the test granular material from a certain height into the sample maker using a funnel until it is completely filled. The upper surface of the soil sample should be the same level as the upper edge of the sample maker (Figure 2D).
   5. Place another porous stone on top of the soil sample, and a stainless-steel cushion plate on top of the porous stone. Apply some silicone grease around the lateral surface of the cushion plate. Remove the top side of the membrane from the sample maker and fix it to the cushion plate (Figure 2E).
   6. Remove the suction inside the sample maker nozzle and create suction inside the valve on the base plate. Finally, remove the sample maker. A miniature dry sample is produced, as seen in Figure 2F.
      NOTE: This step demonstrates the procedure of producing a miniature soil sample using the air pluviation method. The traditional dry compaction method can also be used to produce the sample.
5. Fix the confining cell on the base plate and fix the chamber top plate on the top of the confining cell (Figure 1C).
6. Fix the piston shaft of the cell on the chamber top plate (Figure 1C).
7. Position the base plate together with the confining cell and chamber top plate on the rotation stage. A frame is used to adjust the height of the sample for CT scanning (Figure 1B).
   NOTE: This frame is used due to the limited movement range of the lifting stage at SSRF. There is no need to use a frame if a lifting stage with a large movement range is used.
8. Affix the rest of the loading apparatus on the chamber top plate.
9. Install the linear variable differential transformer (LVDT), load cell and stepping motor and activate them (Figure 1C).
10. Fill the cell with de-aired water through the cell pressure (CP) valve (see Figure 1C) using the water supplied from a confining pressure offering device (see Table of Materials). Close the water exit (WE) valve (see Figure 1C) when the water starts to flow out of the valve.
    NOTE: Set the confining pressure offering device to the constant pressure mode with a very low constant pressure value (e.g., 10 kPa).
11. Add a constant confining pressure of 25 kPa to the sample and remove the suction inside the sample.
12. Gradually increase the confining pressure to a pre-determined value using the confining pressure offering device.
13. Carry out the first scan of the sample. For a high-spatial resolution CT scanner (e.g., with a pixel size of 6.5 μm), a full scan of the sample (e.g., with a height of 16 mm) usually requires the sample to be scanned at several different heights (i.e., the scan is divided into several sections).
    NOTE: If a low spatial resolution detector and a small size sample are used, the scanning area might be sufficient to acquire a full-field scan of the sample using a single section.
    1. Scan a section of the sample. Set the CT scanner to Image capture mode and then start the rotation stage to rotate the entire apparatus across 180 degrees at a pre-determined constant rotation rate (e.g., 3.33 degrees/s) to capture CT projections of the sample at different angles.
       NOTE: It is suggested that the sample is scanned from its bottom upwards (i.e., the first section contains all the particles located at the bottom of the sample).
    2. Turn off the Image capture mode when the rotation is finished. Rotate the apparatus back to the initial position.
    3. Lift the sample together with the entire apparatus up using the lifting stage (Figure 1B) by a certain height (e.g., 4 mm) for scanning the next section of the sample.
       NOTE: The lifting should ensure that there is an overlap between the current section and the last section (i.e., there is an overlap between any two consecutive sections). The overlap should be at least 10 pixels to facilitate the stitching of them.
    4. Repeat steps 2.13.1-2.13.3 until the last section of the sample is scanned.
14. Apply an axial load on the sample with a constant loading rate. Here, a loading rate of 0.2%/min is used in this study. Users can set a different loading rate according to the experiment requirement.
15. Pause the axial loading at a pre-determined axial strain. Wait until the measured axial force reaches a steady value (generally within 2 min) and carry out the next scan. The scan procedures are the same as demonstrated in step 2.13.
16. Repeat steps 2.14 and 2.15 until the end of loading.
17. Unload the test and remove the sample from the triaxial apparatus.
18. Install the base plate and the confining cell on the rotation stage to acquire several flat projections (generally 10 projections) from the detector. Shut down the X-ray source to acquire the same number of dark projections from the detector.
    NOTE: Flat and dark projections are used for the phase retrieval of raw CT projections. The implementation of flat and dark correction enhances the contrast between the sample and the surrounding background in the reconstructed CT slices. It also helps to alleviate the ring artifacts resulting from defective pixels of the detector.

3. Image processing and analysis

1. Image processing
   1. Implement phase retrieval (Figure 3B) of raw CT projections (Figure 3A) of the sample using the free software PITRE³⁴. Load projections (including the flat and dark projections) into PITRE from the menu Load image. Click the icon PPCI. Enter the relevant scanning parameters and click Single to implement the phase retrieval.
      NOTE: The implementation of phase retrieval provides enhancement of interfaces between different phases (i.e., the void phase and the solid phase) in the reconstructed CT slices, which is of significant importance to the subsequent image-based analysis of inter-particle contacts.
   2. Reconstruct CT slices of the sample using PITRE based on the CT projections after phase retrieval (Figure 3C). Load the projections into PITRE from the menu Load image. Click the icon ProjSino. Enter relevant parameters in the appeared window and click Single to reconstruct a CT slice.
      NOTE: Check horizontal slices to ensure that there are no heavy beam hardening artefacts or ring artefacts. Otherwise change of the current scanning parameters and rescan of the sample are required. Check vertical slices. If the sample is severely tilted prior to the shear, the test is considered unsuccessful.
   3. Implement image filtering on the CT slices. An anisotropic diffusion filter is used to perform image filtering (Figure 3D).
   4. Perform image binarization on the filtered CT slices. Implement the image binarization (Figure 3E) by applying an intensity value threshold to the CT slices, which is determined according to the intensity histogram of the CT slices using Otsu’s method³⁵.
      NOTE: For CT slices with a grey-scale intensity histogram exhibiting a significant overlap of intensities between the solid phase and the void phase, a validation of the image binarization is required using the mass of the solid phase³⁶.
   5. Separate individual particles from the binarized CT slices using a marker-based watershed algorithm and store the results in a 3D labelled image (Figure 3F). Validate the results by comparing the calculated particle size distribution from the CT image to those from a mechanical sieving test.
      NOTE: The module Separate Objects of the software Avizo Fire can be used to implement this algorithm. Remove the porous stones from the binarized CT slices using the module Border Kill of Avizo Fire. To acquire a reliable particle separation results, readers are suggested to try different particle segmentation algorithms^37,38,39.
2. Image analysis
   1. Extract particle properties from the labelled image. A MATLAB script is used to extract particle properties including particle volume, particle surface area, particle orientation and particle centroid coordinates.
      NOTE: The intrinsic MATLAB functions regionprops, bwprim and pca are used to acquire these properties of each particle. A more detailed description of these procedures can be found in the work of Cheng and Wang²⁸.
   2. Extract contact voxels from the binarized CT slices by implementation of a logical operation AND between the binary image of the CT slices (Figure 4) and a binary image of watershed lines acquired from the implementation of the marker-based watershed algorithm³¹.
      NOTE: Over-detection of contact voxels could occur due to the partial volume effect and the random noise of CT images^40,41. However, a slight over-detection of inter-particle contacts would not have significant effects on the overall trend of inter-particle contact evolution behavior⁴².

4. CT image-based exploration of grain-scale mechanical behavior of soils

NOTE: The following image-based analysis is not applicable to idealistically spherical particles or samples with very narrow grading ranges (i.e., monodisperse samples). However, for particles with high roundness and poor grading (e.g., 0.3~0.6 mm glass beads), the methodology yields good results (see Cheng and Wang³¹).

1. Quantify particle kinematics of the sample. Use a particle tracking method to track individual particles within the sample at different scans based on either particle volume or particle surface area. A detailed description of this method is given in Cheng and Wang²⁸.
   1. Calculate the translation of each particle during any two consecutive scans. It is calculated as the difference in the particle centroid coordinates between the two scans.
   2. Determine the rotation angle of each particle according to the difference in its major principal axis orientations between the two scans.
2. Quantify the strain field of the sample. Use a grid-based method to calculate the strain field during any two consecutive scans based on the particle translation and particle rotation.
   NOTE: The method requires the labeled images of the sample from both scans and the particle kinematics results. Readers are referred to a previous work²⁴ for a detailed description.
3. Analyze inter-particle contact evolution of the sample. Based on the extracted contact voxels, the labeled images of particles and the particle tracking results, analyze the branch vector orientation of the lost contacts and the gained contacts within the sample during each shear increment.
   NOTE: A full description of this method is given in Cheng and Wang³¹.