# Study of Dynamic Evolution of the Shear Band in Triaxial Soil Samples Using Photogrammetry Technology

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory of 3D Model Reconstruction

#### 2.1. Establishment and Transformation of the Coordinate System

_{1}, n

_{1}) is the coordinate of the space point in the pixel coordinate system, (x

_{I}′, y

_{I}′) is the coordinate of the space point I in the pixel physical coordinate system, (M, N) is the pixel unit of the image plane, and (F

_{x}, F

_{y}) is the actual physical length and width of a single image.

_{I}, y

_{I}, z

_{I}) is the coordinate of space point I in the camera coordinate system, p

_{x}and p

_{y}are the image physical coordinates from the camera optical center to the projection point of the image unit along the optical axis, and f is the distance from the camera optical center to the image plane.

_{I}, Y

_{I}, Z

_{I}) refers to the coordinates of space point I in the world coordinate system, R(ω, φ, κ) is the rotation matrix, ω, φ, κ are the rotation angles of X, Y and Z axis of the camera coordinate system, respectively. The translation matrix (X

_{s}, Y

_{s}, Z

_{s}) is used to shift the coordinate origin of the camera coordinate system to the origin of the world coordinate system.

#### 2.2. Determination and Correction of Depth Information

_{a}and n

_{c}are the refractive index of air and pressure chamber materials, $\overrightarrow{{i}_{1}}$ is the direction vector of the incident light, and $\overrightarrow{{r}_{1}}$ is the direction vector of light after refraction.

_{Ci}, Y

_{Ci}, Z

_{Ci}) is the three-dimensional coordinate of the ith point of the same marking point that intersects point C

_{i}on the inner surface of the glass cover, (α

_{r2i}, β

_{r2i}, γ

_{r2i}) represents the cosine of the refracted light C

_{i}P in the pressure chamber, (X

_{p}, Y

_{p}, Z

_{p}) is the P coordinate of the coding point obtained from the one-time shooting information, and the calculated S coordinate is considered as the “real” coordinate of coding point P.

## 3. Experiment Scheme

#### 3.1. Soil Sample

^{3}, 1.5 g/cm

^{3}and water content of 23% for standard triaxial tests, with a height of 80mm high and a diameter of 39.1 mm.

#### 3.2. Test Procedure

## 4. Results and Discussion

#### 4.1. Processing and Analysis of Soil Sample Deformation Data

#### 4.2. Analysis of Stress–Strain Curve

#### 4.3. Dynamic Evolution Analysis of Shear Band Using Displacement Field

^{3}and confining pressure of 200 kPa.

## 5. Conclusions

- The image measurement method is applied to the unconsolidated undrained test of Guilin red clay. The whole measurement process does not need to contact the soil sample and affect the normal data collection of the conventional triaxial test. This method mainly collects the deformation information of the pre-marked coding points on the soil sample surface to obtain the local deformation information, which is a challenge for the conventional triaxial test method.
- Digital image technology avoids the influence of end constraints in the triaxial test. The stress–strain curves of different regions of soil obtained by this method can be used to determine the formation process of shear bands in soil samples. By comparing the stress–strain curves measured by this method with those measured by conventional methods, it is known that the actual triaxial shear band occurs before the axial stress reaches the peak value. The stress–strain curve measured by the new method can more accurately reflect the time and process of shear failure of soil samples.
- Polynomial interpolation of the three-dimensional spatial coordinates of the code points on the surface of the soil sample can be transformed into three-dimensional spatial displacement fields, which enables observations of the formation, development and penetration of the soil sample shear band and therefore provides a new measurement method for the study of the formation and evolution of the soil shear band.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Acknowledgments

## Conflicts of Interest

## References

- Desrues, J.; Viggiani, G. Strain Localization in Sand: An Overview of the Experimental Results Obtained in Grenoble Using Stereophotogrammetry. Int. J. Numer. Anal. Methods Geomech.
**2004**, 28, 279–321. [Google Scholar] [CrossRef] - Rechenmacher, A.L. Grain-Scale Processes Governing Shear Band Initiation and Evolution in Sands. J. Mech. Phys. Solids
**2006**, 54, 22–45. [Google Scholar] [CrossRef] - White, T.G.; Patten, J.R.W.; Wan, K.-H.; Pullen, A.D.; Chapman, D.J.; Eakins, D.E. A Single Camera Three-Dimensional Digital Image Correlation System for the Study of Adiabatic Shear Bands. Strain
**2017**, 53, e12226. [Google Scholar] [CrossRef] [Green Version] - Macari, E.J.; Parker, J.K.; Costes, N.C. Measurement of volume changes in triaxial tests using digital imaging techniques. Geotech. Test. J.
**1997**, 20, 103–109. [Google Scholar] [CrossRef] - Oelschlägel, F. Intervallmathematische Zuverlässigkeitsanalyse von Reihen-/Parallelschaltungen bei ungenauen Primärdaten. Forsch. Ing.
**2000**, 66, 94–100. [Google Scholar] [CrossRef] - Gachet, P.; Klubertanz, W.; Vulliet, L.; Laloui, L. Interfacial behavior of unsaturated soil with small-scale models and use of image processing techniques. Geotech. Test. J.
**2003**, 26, 12–21. [Google Scholar] [CrossRef] [Green Version] - Lin, H.; Penumadu, D. Strain Localization in Combined Axial-Torsional Testing on Kaolin Clay. J. Eng. Mech.
**2006**, 132, 555–564. [Google Scholar] [CrossRef] - Bhandari, A.R.; Powrie, W.; Harkness, R.M. A digital image-based deformation measurement system for triaxial tests. Geotech. Test. J.
**2012**, 35, 209–226. [Google Scholar] [CrossRef] - Laloui, L.; Péron, H.; Geiser, F.; Rifa’i, A.; Vulliet, L. Advances in Volume Measurement in Unsaturated Soil Triaxial Tests. Soils Found.
**2006**, 46, 341–349. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; Li, L.; Xia, X. Recent Advances in Volume Measurements of Soil Specimen during Triaxial Testing. JGS Spec. Publ.
**2019**, 7, 31–37. [Google Scholar] [CrossRef] - Wang, H.; Sato, T.; Koseki, J.; Chiaro, G.; Tan Tian, J. A System to Measure Volume Change of Unsaturated Soils in Undrained Cyclic Triaxial Tests. Geotech. Test. J.
**2016**, 39, 20150125. [Google Scholar] [CrossRef] [Green Version] - Ng, C.W.W.; Zhan, L.T.; Cui, Y.J. A New Simple System for Measuring Volume Changes in Unsaturated Soils. Can. Geotech. J.
**2002**, 39, 757–764. [Google Scholar] [CrossRef] - Chen, W.B.; Yin, J.H.; Feng, W.Q. A New Double-Cell System for Measuring Volume Change of a Soil Specimen under Monotonic or Cyclic Loading. Acta Geotech.
**2019**, 14, 71–81. [Google Scholar] [CrossRef] - Cabarkapa, Z.; Cuccovillo, T. Automated triaxial apparatus for testing unsaturated soils. Geotech. Test. J.
**2006**, 29, 2921–2929. [Google Scholar] [CrossRef] - Laudahn, A.; Sosna, K.; Bohdp, J. A simple method for air volume change measurement in triaxial tests. Geotech. Test. J.
**2005**, 28, 313–318. [Google Scholar] [CrossRef] - Chen, C.B.; Ye, G.L. Development of small-strain triaxial apparatus using LVDT sensors and its application to soft clay test. Rock Soil Mech.
**2018**, 39, 2304–2310. [Google Scholar] [CrossRef] - Belmokhtar, M.; Delage, P.; Ghabezloo, S.; Conil, N. Drained Triaxial Tests in Low-Permeability Shales: Application to the Callovo-Oxfordian Claystone. Rock Mech. Rock Eng.
**2018**, 51, 1979–1993. [Google Scholar] [CrossRef] - Alikarami, R.; Andò, E.; Gkiousas-Kapnisis, M.; Torabi, A.; Viggiani, G. Strain Localisation and Grain Breakage in Sand under Shearing at High Mean Stress: Insights from in Situ X-Ray Tomography. Acta Geotech.
**2015**, 10, 15–30. [Google Scholar] [CrossRef] - Cheng, Z.; Wang, J.; Coop, M.R.; Ye, G. A Miniature Triaxial Apparatus for Investigating the Micromechanics of Granular Soils with in Situ X-Ray Micro-Tomography Scanning. Front. Struct. Civ. Eng.
**2020**, 14, 357–373. [Google Scholar] [CrossRef] - Sachan, A.; Penumadu, D. Strain Localization in Solid Cylindrical Clay Specimens Using Digital Image Analysis (DIA) Technique. Soils Found.
**2007**, 47, 67–78. [Google Scholar] [CrossRef] - Salazar, S.E.; Coffman, R.A. Consideration of Internal Board Camera Optics for Triaxial Testing Applications. Geotech. Test. J.
**2015**, 38, 20140163. [Google Scholar] [CrossRef] - Salazar, S.E.; Barnes, A.; Coffman, R.A. Development of an Internal Camera–Based Volume Determination System for Triaxial Testing. Geotech. Test. J.
**2015**, 38, 20140249. [Google Scholar] [CrossRef] [Green Version] - Rui, S.; Wang, L.; Guo, Z.; Cheng, X.; Wu, B. Monotonic Behavior of Interface Shear between Carbonate Sands and Steel. Acta Geotech.
**2021**, 16, 167–187. [Google Scholar] [CrossRef] - Rui, S.; Wang, L.; Guo, Z.; Zhou, W.; Li, Y. Cyclic Behavior of Interface Shear between Carbonate Sand and Steel. Acta Geotech.
**2021**, 16, 189–209. [Google Scholar] [CrossRef] - Shen, J.; Wang, X.; Liu, W.; Zhang, P.; Zhu, C.; Wang, X. Experimental Study on Mesoscopic Shear Behavior of Calcareous Sand Material with Digital Imaging Approach. Adv. Civ. Eng.
**2020**, 2020, e8881264. [Google Scholar] [CrossRef] - Wang, X.; Wu, Y.; Cui, J.; Zhu, C.-Q.; Wang, X.-Z. Shape Characteristics of Coral Sand from the South China Sea. J. Mar. Sci. Eng.
**2020**, 8, 803. [Google Scholar] [CrossRef] - Shao, L.T.; Liu, G.; Zeng, F.T.; Guo, X.X. Recognition of the Stress-Strain Curve Based on the Local Deformation Measurement of Soil Specimens in the Triaxial Test. Geotech. Test. J.
**2016**, 39, 20140273. [Google Scholar] [CrossRef] - Wang, P.; Guo, X.; Sang, Y.; Shao, L.; Yin, Z.; Wang, Y. Measurement of Local and Volumetric Deformation in Geotechnical Triaxial Testing Using 3D-Digital Image Correlation and a Subpixel Edge Detection Algorithm. Acta Geotech.
**2020**, 15, 2891–2904. [Google Scholar] [CrossRef] - Xu, J.; Wu, Z.; Chen, H.; Shao, L.; Zhou, X.; Wang, S. Triaxial Shear Behavior of Basalt Fiber-Reinforced Loess Based on Digital Image Technology. KSCE J. Civ. Eng.
**2021**, 25, 3714–3726. [Google Scholar] [CrossRef] - Li, L.; Zhang, X. A New Triaxial Testing System for Unsaturated Soil Characterization. Geotech. Test. J.
**2015**, 38, 20140201. [Google Scholar] [CrossRef] - Li, L.; Zhang, X. Factors Influencing the Accuracy of the Photogrammetry-Based Deformation Measurement Method. Acta Geotech.
**2019**, 14, 559–574. [Google Scholar] [CrossRef] - Fayek, S.; Xia, X.; Li, L.; Zhang, X. Photogrammetry-Based Method to Determine the Absolute Volume of Soil Specimen during Triaxial Testing. Transp. Res. Rec.
**2020**, 2674, 206–218. [Google Scholar] [CrossRef] - Li, L.; Lu, Y.; Cai, Y.; Li, P. A Calibration Technique to Improve Accuracy of the Photogrammetry-Based Deformation Measurement Method for Triaxial Testing. Acta Geotech.
**2021**, 16, 1053–1060. [Google Scholar] [CrossRef] - Xia, X.; Zhang, X.; Mu, C. A Multi-Camera Based Photogrammetric Method for Three-Dimensional Full-Field Displacement Measurements of Geosynthetics during Tensile Test. Geotext. Geomembr.
**2021**, 49, 1192–1210. [Google Scholar] [CrossRef] - Fan, L.Y.; Mu, C.M.; Yang, J.; Wu, H.J. Refraction Error Correction in Digital Measurement of Triaxial Samples. J. Univ. Jinan Sci. Technol.
**2022**, 36, 571–579. [Google Scholar] [CrossRef] - Zhang, X.; Li, L.; Chen, G.; Lytton, R. A Photogrammetry-Based Method to Measure Total and Local Volume Changes of Unsaturated Soils during Triaxial Testing. Acta Geotech.
**2015**, 10, 55–82. [Google Scholar] [CrossRef] - Li, L.; Li, P.; Cai, Y.; Lu, Y. Visualization of Non-Uniform Soil Deformation during Triaxial Testing. Acta Geotech.
**2021**, 16, 3439–3454. [Google Scholar] [CrossRef] - Li, W.J.; Ye, K.; Xia, Y.; Mu, C.M. Research on deformation measurement of triaxial specimen based on photographic reconstruction technology. China Meas. Test
**2022**, 1–10. Available online: https://kns.cnki.net/kcms/detail/51.1714.TB.20220831.1726.004.html (accessed on 1 September 2022). - Wu, H.J. Research on Deformation Characteristics of Triaxial Soil Samples Based on Image Measurement. Master’s Thesis, Guilin University of Technology, Guilin, China, April 2021. [Google Scholar] [CrossRef]
- Li, L.; Zhang, X.; Chen, G.; Lytton, R. Measuring Unsaturated Soil Deformations during Triaxial Testing Using a Photogrammetry-Based Method. Can. Geotech. J.
**2016**, 53, 472–489. [Google Scholar] [CrossRef] [Green Version] - Shao, L.T.; Liu, G.; Guo, X.X. Effects of strain localization of triaxial samples in post-failure state. Chin. J. Geot. Eng.
**2016**, 38, 385–394. [Google Scholar] [CrossRef] - Zhang, B.W.; Wang, X.B.; Dong, W. Measurement of Shear Band Widths Based on the Digital Image Correlation Method Considering Splitted Subsets Surrounding the Monitored Point. Acta Metrol. Sin.
**2022**, 43, 40–47. Available online: http://jlxb.china-csm.org:81/Jwk_jlxb/EN/Y2022/V43/I1/40 (accessed on 1 September 2022).

**Figure 6.**Comparison of stress–strain curves of soil samples under confining pressures of 100 kPa and 200 kPa. (

**a**) 1.4 g/cm

^{3}, 100 kPa; (

**b**) 1.4 g/cm

^{3}, 200 kPa; (

**c**) 1.5 g/cm

^{3}, 100 kPa; (

**d**) 1.5 g/cm

^{3}, 200 kPa.

**Figure 9.**Comparison of the shear deformations under different axial displacements. (

**a**) 10 mm axial displacement; (

**b**) 14 mm axial displacement.

Property | Value |
---|---|

Natural moisture content (%) | 28 |

Maximum dry density (g/cm^{3}) | 1.60 |

Optimum moisture content (%) | 23 |

Specific gravity | 2.65 |

Liquid limit | 56.5 |

Plastic limit | 31.8 |

Plastic index | 24.7 |

Dry Density | Chamber Pressure | Conventional Measurement | Photogrammetry | ||
---|---|---|---|---|---|

Part A | Part B | Part C | |||

g/cm^{3} | kPa | % | % | % | % |

1.4 | 100 | 9.216 | 7.581 | 12.041 | 9.015 |

1.4 | 150 | 9.269 | 7.223 | 12.353 | 8.951 |

1.4 | 200 | 9.271 | 7.711 | 12.248 | 8.845 |

1.5 | 100 | 9.331 | 9.625 | 14.721 | 11.132 |

1.5 | 150 | 9.274 | 10.323 | 14.561 | 10.851 |

1.5 | 200 | 9.273 | 10.421 | 14.075 | 10.366 |

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**MDPI and ACS Style**

Xia, Y.; Mu, C.; Li, W.; Ye, K.; Wu, H.
Study of Dynamic Evolution of the Shear Band in Triaxial Soil Samples Using Photogrammetry Technology. *Sustainability* **2022**, *14*, 14660.
https://doi.org/10.3390/su142114660

**AMA Style**

Xia Y, Mu C, Li W, Ye K, Wu H.
Study of Dynamic Evolution of the Shear Band in Triaxial Soil Samples Using Photogrammetry Technology. *Sustainability*. 2022; 14(21):14660.
https://doi.org/10.3390/su142114660

**Chicago/Turabian Style**

Xia, Yi, Chunmei Mu, Wenjie Li, Kai Ye, and Haojie Wu.
2022. "Study of Dynamic Evolution of the Shear Band in Triaxial Soil Samples Using Photogrammetry Technology" *Sustainability* 14, no. 21: 14660.
https://doi.org/10.3390/su142114660