# Quantitative Evaluation of Soil Structure and Strain in Three Dimensions under Shear Using X-ray Computed Tomography Image Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Method

#### 2.1. Ellipsoid Fitting Method

^{−5}between ${\overline{\mathsf{\epsilon}}}^{\mathrm{n}=\mathrm{k}}$ (which is the average of ${\mathsf{\epsilon}}_{\mathrm{i}}$ at the present iteration) and ${\overline{\mathsf{\epsilon}}}^{\mathrm{n}=\mathrm{k}~\mathrm{k}-5}$ (which is the average of ${\mathsf{\epsilon}}_{\mathrm{i}}$ for the five iterations prior to the present iteration).

#### 2.2. Materials

^{3}. GBs are commercially available glass beads with a smooth surface and an almost spherical shape. The particle size of a GB is 0.71–1.00 mm, and the true density is 2500 kg/m

^{3}. KS is a natural dry sand with a rough surface used to indicate angularity. The density of KS is 2640 kg/m

^{3}and the particle size distribution is 1–3 mm. The photographs of the samples used in this study are shown in Figure 3.

#### 2.3. Validation of the Ellipsoid Fitting Method

#### 2.3.1. CT Imaging and Particle Shape Evaluation

_{25}, 25%), the second quartile value (Q

_{50}, 50%), and the third quartile value (Q

_{75}, 75%), were calculated for ${\mathrm{d}}_{1}$, ${\mathrm{d}}_{2}$, ${\mathrm{d}}_{3}$, and ${\mathrm{S}}_{\mathrm{k}}$. Furthermore, the evaluation accuracy for KS and the GBs were validated to obtain the cumulative curve for ${\mathrm{d}}_{1}$, ${\mathrm{d}}_{2}$, and ${\mathrm{d}}_{3}$; histogram of ${\mathrm{d}}_{3}/{\mathrm{d}}_{1}$, ${\mathrm{d}}_{2}/{\mathrm{d}}_{1}$, and ${\mathrm{d}}_{3}/{\mathrm{d}}_{2}$; and histogram contour plot of a Zingg diagram [39].

#### 2.3.2. Particle Measurements Using a Particle Image Analyzer

#### 2.4. Direct Shear Experiment with CT Imaging and Image Analysis

#### 2.4.1. Direct Shear Experiment

#### 2.4.2. Examination of Representative Volume Elements

#### 2.4.3. Analysis of Strain Localization by the Digital Image Correlation Method

#### 2.4.4. Evaluation of the Particle Direction

## 3. Results

#### 3.1. Validation of the Ellipsoid Fitting Method

#### 3.1.1. CT Images and Segmented Images

#### 3.1.2. Ellipsoid Fitting

#### 3.1.3. Evaluation of the Particle Shape Characteristics

_{50}of ${\mathrm{d}}_{1}$, ${\mathrm{d}}_{2}$, and ${\mathrm{d}}_{3}$ for RP1 and RP2 were approximately equal to the manufacturing conditions, with a maximum error of 0.17 mm (4 voxels). Q

_{50}of ${\mathrm{d}}_{1}$, ${\mathrm{d}}_{2}$, and ${\mathrm{d}}_{3}$ for the GBs were within the range of 0.7–1.0 mm, which is the specification of the particle size of the glass beads. Furthermore, Q

_{50}of ${\mathrm{d}}_{1}$, ${\mathrm{d}}_{2}$, and ${\mathrm{d}}_{3}$ for KS evaluated from the CT images were not significantly different from Q

_{50}of ${\mathrm{d}}_{\mathrm{m}}$ measured using PIA. Q

_{50}of ${\mathrm{S}}_{\mathrm{k}}$ calculated by Equation (6) was 0.94 for the GBs since the glass beads represented a sphere and were expected to have a sphericity close to 1. Figure 4(c1) shows that some glass beads were not sphere, but more ellipsoid. This may have affected the sphericity, which was not exactly equal to 1.

#### 3.2. Direct Shear Experiment

#### 3.2.1. Experimental Results, CT Images, and Segmented Images

_{h}) reached ~1.3 mm. After the peak of shear stress, the shear stress gradually decreased, and the material shifted to the strain softening process. Until D

_{h}reached 0.6 mm, the vertical displacement was decreasing and the whole specimen was shrinking in volume. When D

_{h}reached 1.8 mm, the vertical displacement began to increase and the whole specimen expanded in volume. In particular, the vertical displacement increased rapidly from S1 (D

_{h}= 0.3 mm) to S2 (D

_{h}= 3 mm) and from S2 to S3 (D

_{h}= 5 mm).

_{h}reached a certain displacement. Hence, the specimen was considered to be affected by the shear stress relaxation to a certain degree, as shown in Figure 7. However, the vertical load was still applied during the CT imaging, and the affection on the internal structure was considered to be insignificant.

_{h}= 3 mm), and S4 (D

_{h}= 8 mm). The voxel size was 91 μm/voxel, and the image size was 1024 × 1024 × 600 voxels. In this experiment, the shear stress reached its peak when D

_{h}reached ~1.3 mm, so the images after S2 are the CT images during the strain softening process. In S0 (Figure 8(a1)), the voids appeared to be homogeneous over the sample, but after S2 (Figure 8(b1,c1)), the voids near the shear plane increased.

#### 3.2.2. Evaluation of the Porosity and Contact Surface between Particles

#### 3.2.3. The Change of the Porosity and the Contact Surface between Particles

_{h}= 0.3 mm) to S2 (D

_{h}= 3 mm), the porosity increased over a wide region in the vertical direction around the shear plane (Figure 12(a1)). As for the CSR, the decreasing tendency near the shear plane was remarkable (Figure 12(b1)). This indicated that the contact area decreased due to the loss of interlocking between the particles as a result of shearing. In addition, the increase in the porosity and the decrease in the CSR generally correspond to each other near the shear plane.

_{h}= 5 mm) to S4 (D

_{h}= 8 mm), the porosity near the shear plane increased (Figure 12(a2)) similar to the change from S1 to S2. However, the region of increase in the porosity is smaller than the change from S1 to S2 and remained near the shear plane. The CSR was a mixture of increasing and decreasing regions near the shear plane (Figure 12(b2)). In addition, the increase in the porosity and the decrease in CSR generally did not necessarily correspond to each other near the shear plane.

#### 3.2.4. Evaluation of the Volumetric Strain and the Shear Strain

#### 3.2.5. Comparison of the Porosity, the Contact Surface Ratio, the Volumetric Strain, and the Shear Strain versus Distance from the Shear Plane

#### 3.2.6. Evaluation for the Particle Direction

_{h}= 3 mm), some particles were visually observed to rotate (Figure 8(b1,b2)), but the direction of most particles did not change significantly from S0. Even when the shear progressed to reach S4 (D

_{h}= 8 mm), there was no significant change in the overall direction of the particles (Figure 8(c1,c2)).

_{h}= 3 mm), but it appeared to have become increasingly rapid after S3. As shown in Figure 18a, which is the temporal change of the horizontal particle long axis, G4 and G5 showed a different tendency from G1–G3 and G6–G8, and the ratio of the horizontal long axes of the particles decreased monotonically, with a maximum decrease of ~6%. As shown in Figure 18b, which is the temporal change of the vertical particle long axis, the ratio of the vertical long axes of the particles increased monotonically in G4 and G5, with a maximum increase of ~4%. In the regions other than G4 and G5, except for G3, the ratio of the vertically directed long axes of the particles remained almost unchanged or showed a decreasing tendency. In the histogram for the particle long axis of G4 in S4 (Figure 16(b3)), peaks were also observed in the 90° and 270° directions. The increase in the ratio of the vertically directed long axes of the particles can be considered as one of the changes in soil structure due to shearing.

## 4. Discussion

#### 4.1. Validation of the Ellipsoid Fitting Method

#### 4.2. Relationship between the Changes in Soil Structure and Strain Due to Shearing

_{h}of ~1.3 mm, followed by a strain softening process. In this study, shearing was stopped, and CT scanning was performed at the initial state (S0) and when D

_{h}reached 0.3, 3, 5, and 8 mm (S1 to S4, respectively).

_{h}reached 8 mm, changes in the porosity were observed in the range of G3–G6, or ~14.6 mm. This corresponds to 7.1 times the median particle size of ${\mathrm{d}}_{\mathrm{m}}$ (2.05 mm) measured by PIA. Oda and Kazama (1998) reported that the width of the shear zone is 7 to 8 times larger than the median grain size by microscopically observing the 2D cross section near the shear zone of specimens after shear experiments using Toyoura sand (D

_{50}= 0.206 mm) and Ticno sand (D

_{50}= 0.527 mm) [1]. In this study, shear experiments using KS (D

_{50}= 2.05 mm) were conducted to evaluate the change in porosity in 3D space, and the width of the shear zone was generally similar. However, it was clarified that the porosity changes slowly as the shear progresses, and that the porosity inside the shear zone increased in variability. Similar to the porosity, a decrease in CSR was observed in the range of G3–G6, indicating a loss of interlocking between particles due to shearing. The volume strain and the shear strain also changed in the range of G3–G6. In particular, the regions where the porosity changed (Figure 12(a1,a2)) and the volumetric strain occurred (Figure 13(a1,a2)) generally correspond to each other, suggesting that the shear changed the porosity and caused the volumetric change. Previous studies have reported that the volume change caused by the progress of shear is largely due to the effect of particle rotation, which occurs when a particle rides up or rides down on a neighboring particle [1]. The relationship between the volume change and particle rotation in the shear zone has been pointed out by numerical simulations using DEM by considering the resistance between particles [44] and particle shape [45]. On the other hand, in this study, the evaluation of soil structure inside the actual shear zone showed that the particles rotated in the range of the volumetric strain, suggesting that the volume change and particle rotation are closely related, similar to the previous study. However, the region where the change in particle direction occurred was narrower than the region where the change in the volume strain occurred.

_{h}= 3 mm), volume expansion was dominant inside the shear zone. On the other hand, some particles were visually observed to have rotated on the CT image, but there was no significant change in the histogram showing particle direction. This suggests that there were only a few particles that produced rotation that caused volume expansion.

_{h}= 8 mm), the minimum volumetric strain decreased further, and the maximum volumetric strain also increased further. However, the average volumetric strain approached zero and the state reached equilibrium (or steady state). In contrast, according to the change of particle direction with time, the change of particle direction became sharper after S3 (D

_{h}= 5 mm) when the volume expansion inside the shear zone exceeded the peak. Finally, the ratio of horizontally directed long axes of the particles decreased by ~6%, and the ratio of vertically directed short axes of the particles decreased by a maximum of ~8%. In addition, the ratio of vertical long axes of the particles increased by ~4%, as well as the ratio of the horizontal short axes of the particles increasing by ~4%. These changes in the soil structure were not observed in the region away from the shear plane, which is considered to be a characteristic of the soil structure near the shear plane.

## 5. Conclusions

- The ellipsoid fitting method proposed in this study is less affected by the irregularities on the particle surface and local shape changes; thus, it can fit particles with complex shapes such as average ellipsoids. The proposed method can accurately fit ellipsoids not only for spherical or ellipsoidal particles produced by glass beads or 3D printers, but also for natural soil particles with surface roughness and complex shapes without changing the calculation method.
- The specimen of direct shear experiment used in this study was filled by free fall into a shear box placed on a horizontal table, so that the long axis of most of the particles were directed in the horizontal direction and the short axis in the vertical direction in the initial state. Even if shearing occurred, the overall tendency in the direction of the particles is sustained. However, it was clarified that the direction of the particles partially changed when the volume expansion inside the shear zone exceeded the peak. The ratio of both the horizontally directed long axes and vertically directed short axes of the particles decreased by 6~8%. On the other hand, the ratio of both the vertical long axes and horizontal short axes of the particles increased by ~4%. Since no such change was observed in the region away from the shear plane, it was suggested that the change is characteristic of the soil structure near the shear plane.
- The porosity, contact–surface ratio, volumetric strain, and shear strain changed significantly in the range of ~7.1 times the median grain size of the sand used in this study. On the other hand, it is obvious that the change in particle direction occurs within an even narrower range than the change in porosity, contact–surface ratio, volumetric strain, and shear strain, and is restricted to the vicinity of the shear plane.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CSR | ratio of contact surface between particles on RVE |

CT | computed tomography |

DEM | distinct element method |

D_{h} | horizontal displacement of lower box of direct shear experiment’s apparatus |

DIC | digital image correlation |

GBs | glass beads |

KS | Kashima–Keisa sand |

PIA | particle image analyzer |

RP | resin particles |

RVE | representative volume element |

## Appendix A

## Appendix B

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**Figure 1.**Schematic illustration of a three-dimensional ellipsoid with its center of gravity located at the origin.

**Figure 3.**Acrylic glass columns filled with the materials: (

**a**) RP1 and (

**b**) RP2 (resin particles made by a 3D printer), (

**c**) GBs (glass beads), and (

**d**) KS (Kashima–Keisa natural sand).

**Figure 4.**CT images and segmented images of four materials poured into acrylic glass columns: (

**a1**,

**a2**) RP1 and (

**b1**,

**b2**) RP2 (resin particles made by a 3D printer), (

**c1**,

**c2**) GBs (glass beads), and (

**d1**,

**d2**) KS (Kashima–Keisa natural sand). The left columns are CT images, and the right columns are segmented images.

**Figure 5.**Ellipsoid fitting results: (

**a1**,

**a2**) RP1 and (

**b1**,

**b2**) RP2 (resin particles made by a 3D printer), (

**c1**,

**c2**) GBs (glass beads), and (

**d1**,

**d2**) KS (Kashima–Keisa natural sand). The left columns are the two-dimensional views, and the right columns are the three-dimensional views.

**Figure 6.**Evaluation results of particle shape. (

**a1**,

**a2**) are the cumulative ratio curve of ${\mathrm{d}}_{1}$, ${\mathrm{d}}_{2}$, ${\mathrm{d}}_{3}$, and ${\mathrm{d}}_{\mathrm{m}}$ (only for KS). (

**b1**,

**b2**) are the histograms of ${\mathrm{d}}_{3}/{\mathrm{d}}_{1}$, ${\mathrm{d}}_{2}/{\mathrm{d}}_{1}$, ${\mathrm{d}}_{3}/{\mathrm{d}}_{2}$, and ${\mathrm{e}}_{\mathrm{w}}/{\mathrm{e}}_{\mathrm{l}}$ (only for KS). (

**c1**,

**c2**) are Zingg diagrams. The left columns are evaluation results of the GBs, and the right columns for KS. GBs, glass beads; KS, Kashima–Keisa natural sand.

**Figure 7.**Experimental results of direct shear experiment. CT scanning was performed when horizontal displacement was 0.0 (S0), 0.3 (S1), 3.0 (S2), 5.0 (S3), and 8.0 mm (S4).

**Figure 8.**CT images and segmented images of the direct shear experiment on the vertical plane (XZ plane). (

**a1**,

**a2**) are the results at S0 (initial sate). (

**b1**,

**b2**) are the results at S2 (D

_{h}= 3.0 mm). (

**c1**,

**c2**) are the results at S4 (D

_{h}= 8.0 mm). D

_{h}, horizontal displacement. The left columns are CT images, and the right columns are segmented images. The boundary surface between the upper and lower boxes is at a position where the Z-axis is about 27 mm. The lower box is displaced horizontally from the +X to the −X direction.

**Figure 10.**Histogram of the porosity. The porosity was calculated as the grid size of the representative volume element every 10 voxels between 10 and 100 voxels for S0.

**Figure 11.**Evaluation results of the porosity and the ratio of contact-surface between particles on the vertical plane (XZ plane). (

**a1**,

**a2**) are the results at S0 (initial sate). (

**b1**,

**b2**) are the results at S2 (D

_{h}= 3.0 mm). (

**c1**,

**c2**) are the results at S4 (D

_{h}= 8.0 mm). D

_{h}, horizontal displacement. The left columns are the porosity, and the right columns are the ratio of contact-surface between particles.

**Figure 12.**Change in the porosity and the ratio of contact surface between particles on the vertical plane (XZ plane). (

**a1**,

**a2**) show the change in the porosity. (

**b1**,

**b2**) show the change in the ratio of contact surface between particles. The left columns represent the change between S1 (D

_{h}= 0.3 mm) and S2 (D

_{h}= 3 mm), and the right columns represent the change between S3 (D

_{h}= 5 mm) and S4 (D

_{h}= 8 mm). D

_{h}, horizontal displacement.

**Figure 13.**Digital image correlation (DIC) analysis results on the vertical plane (XZ plane). (

**a1**,

**a2**) are the volumetric strain, while (

**b1**,

**b2**) are the shear strain. The left columns show the analysis results based on the CT images of S1 (D

_{h}= 0.3 mm) and S2 (D

_{h}= 3 mm), and the right columns show the analysis results based on the CT images of S3 (D

_{h}= 5 mm) and S4 (D

_{h}= 8 mm). D

_{h}, horizontal displacement.

**Figure 15.**Statistical analysis (represented as box plots) for each group: (

**a**) porosity, (

**b**) ratio of contact surface between particles, (

**c**) volumetric strain, and (

**d**) shear strain.

**Figure 16.**Histograms of particle direction of the long axis on the vertical plane (XZ plane): (

**a1**–

**a3**) are G5; (

**b1**–

**b3**) are G4. The left columns represent S0 (initial state), the center columns S2 (D

_{h}= 3 mm), and the right columns S4 (D

_{h}= 8 mm). D

_{h}, horizontal displacement.

**Figure 17.**Histograms of particle direction of the short axis on the vertical plane (XZ plane): (

**a1**–

**a3**) are G5; (

**b1**–

**b3**) are G4. The left columns represent S0 (initial state), center columns represent S2 (D

_{h}= 3 mm), and right columns represent S4 (D

_{h}= 8 mm). D

_{h}, horizontal displacement.

**Figure 18.**Variation in the ratio of the particle’s long axis from S0 (initial state). (

**a**) 337.5–22.5° and 157.5–202.5°, and (

**b**) 67.5–112.5° and 247.5–292.5°.

**Figure 19.**Variation in the ratio of the particle’s short axis from S0 (initial state). (

**a**) 337.5–22.5° and 157.5–202.5°, and (

**b**) 67.5–112.5° and 247.5–292.5°.

Material | Parameter | Unit | Mean | Std | Q_{25} | Q_{50} | Q_{75} |
---|---|---|---|---|---|---|---|

RP1 | ${\mathrm{d}}_{1}$ | [mm] | 15.63 | 0.58 | 15.67 | 15.87 | 15.94 |

${\mathrm{d}}_{2}$ | 11.72 | 0.42 | 11.79 | 11.85 | 11.90 | ||

${\mathrm{d}}_{3}$ | 3.91 | 0.13 | 3.84 | 3.88 | 3.90 | ||

${\mathrm{S}}_{\mathrm{k}}$ | [-] | 0.57 | 0.015 | 0.57 | 0.57 | 0.57 | |

RP2 | ${\mathrm{d}}_{1}$ | [mm] | 15.52 | 0.71 | 15.47 | 15.84 | 15.90 |

${\mathrm{d}}_{2}$ | 7.85 | 0.24 | 7.83 | 7.93 | 7.97 | ||

${\mathrm{d}}_{3}$ | 3.87 | 0.13 | 3.82 | 3.83 | 3.85 | ||

${\mathrm{S}}_{\mathrm{k}}$ | [-] | 0.50 | 0.017 | 0.49 | 0.50 | 0.50 | |

GBs | ${\mathrm{d}}_{1}$ | [mm] | 0.88 | 0.14 | 0.81 | 0.87 | 0.96 |

${\mathrm{d}}_{2}$ | 0.78 | 0.10 | 0.74 | 0.79 | 0.84 | ||

${\mathrm{d}}_{3}$ | 0.76 | 0.10 | 0.72 | 0.76 | 0.82 | ||

${\mathrm{S}}_{\mathrm{k}}$ | [-] | 0.92 | 0.06 | 0.89 | 0.94 | 0.97 | |

KS | ${\mathrm{d}}_{1}$ | [mm] | 2.55 | 0.45 | 2.24 | 2.48 | 2.78 |

${\mathrm{d}}_{2}$ | 1.90 | 0.22 | 1.75 | 1.89 | 2.05 | ||

${\mathrm{d}}_{3}$ | 1.39 | 0.21 | 1.26 | 1.39 | 1.54 | ||

${\mathrm{d}}_{\mathrm{m}}$ | 2.02 | 0.40 | 1.91 | 2.05 | 2.20 | ||

${\mathrm{S}}_{\mathrm{k}}$ | [-] | 0.75 | 0.08 | 0.70 | 0.75 | 0.80 |

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**MDPI and ACS Style**

Nohara, S.; Mukunoki, T.
Quantitative Evaluation of Soil Structure and Strain in Three Dimensions under Shear Using X-ray Computed Tomography Image Analysis. *J. Imaging* **2021**, *7*, 230.
https://doi.org/10.3390/jimaging7110230

**AMA Style**

Nohara S, Mukunoki T.
Quantitative Evaluation of Soil Structure and Strain in Three Dimensions under Shear Using X-ray Computed Tomography Image Analysis. *Journal of Imaging*. 2021; 7(11):230.
https://doi.org/10.3390/jimaging7110230

**Chicago/Turabian Style**

Nohara, Shintaro, and Toshifumi Mukunoki.
2021. "Quantitative Evaluation of Soil Structure and Strain in Three Dimensions under Shear Using X-ray Computed Tomography Image Analysis" *Journal of Imaging* 7, no. 11: 230.
https://doi.org/10.3390/jimaging7110230