# A Dual DIC System for Analysis of Dynamic Mechanical Properties of Large Sandstone under Uniaxial Compression Load

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Specimen Preparation and Compression Experimentation

^{3}. The sandstone sample is processed into cuboid specimens of 500 mm × 500 mm × 100 mm (length × width × thickness), and a prefabricated hole with the diameter of 100 mm through the thickness direction is cut by a water knife in the center of the sandstone, as shown in Figure 1.

#### 2.1. Speckle Preparation and Assessment

_{f}) is shown in Equation (1):

_{ij}; f

_{x}(x

_{ij}), and f

_{y}(x

_{ij}) are the gray-scale derivatives of pixel x

_{ij}in x and y directions, respectively. We adopted Roberts cross gradient operator to replace gray-scale derivatives in two directions, and its formula is Equation (2):

_{e}) and standard deviation (σ

_{u}). Two error calculation formulas are Equations (3) and (4):

_{imp}is pre-added real sub-pixel displacement. Yu [18] proposed that the MISD (represented by ω

_{f}) can be used to compare the speckle quality when the MIG is not different enough to intuitively reflect the quality of two speckle patterns. For speckle patterns with almost the same MIG, the smaller MISD it has, the smaller the displacement measurement error. The formula of MISD is Equation (5):

_{xx}(x

_{ij}) and f

_{yy}(x

_{ij}) are the second derivative of the gray-scale value of pixel x

_{ij}in the x and y directions, which can be used to substitute calculation by second order difference shown in Equation (6):

_{s}) it has, the smaller the measurement error. SE formula is shown by Equation (7):

_{i}) is the probability of occurrence of each gray-scale level.

#### 2.2. Compression Experiment

## 3. Results Analyses

#### 3.1. Analysis on the Images Taken by Low-Speed Camera

_{y}around the two cracks is not obvious. We know that new stress equilibrium will be sought after a crack occurs. If a new equilibrium point is found, the crack remains at a “plateau” for some time. After the new stress equilibrium is broken again, the crack will spread further. If the stress equilibrium point is not found or the equilibrium is broken again, the crack will continue to expand. At the same time, even stripping, fragmentation, rock ejection and other damage will appear. Therefore, the stress around crack I does not change significantly from t = 5015 s to t = 5400 s. However, crack II appeared at about t = 5315 s and rapidly expanded within a few seconds. When t = 5400 s, crack II stopped expanding. According to the strain diagram, the x-directional stress σ

_{x}on both sides of crack II increased from t = 5015 s to t = 5400 s, but not significantly. However, when t = 5015 s there was no significant shear stress τ

_{xy}on both sides of the crack II, while when t = 5400 s there was already a significantly larger shear stress τ

_{xy}on both sides of the crack II. Therefore, crack II was mainly caused by normal stress, which is consistent with the characteristics of opening mode crack.

#### 3.2. Analysis on the Images Taken by High-Speed Camera

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Variations in displacement field. The unit of displacement in this figure is pixel. (

**a**) t = 1100 s, F = 549.34 kN. (

**b**) t = 2000 s, F = 660.02 kN. (

**c**) t = 2800 s, F = 770 kN. (

**d**) t = 3700 s, F = 879.99 kN. (

**e**) t = 4500 s, F = 990.01 kN. (

**f**) t = 5015 s, F = 1035.13 kN. (

**g**) t = 5400 s, F = 1100 kN. (

**h**) t = 6179 s, F = 1208.9 kN.

**Figure 8.**Strain in x-direction (ε

_{x}), y-direction (ε

_{y}) and shear (γ

_{xy}). Where (

**a**–

**c**) are at time t = 5015 s, when F = 1035.13 kN. (

**d**–

**f**) are at time t = 5400 s, when F = 1100 kN. (

**g**–

**i**) are at time t = 6179 s, when F = 1208.9 kN.

**Figure 9.**Displacement of specimen surface captured by high-speed camera at t = 6179 s. The unit of displacement in this figure is pixel.

**Figure 10.**Strain in x-direction (ε

_{x}), y-direction (ε

_{y}) and shear (γ

_{xy}). Where (

**a**–

**c**) are at time t = 6175.3 s, when F = 1208.27 kN. (

**d**–

**f**) are at time t = 6178.9 s, when F = 1208.87 kN. (

**g**–

**i**) are at time t = 6179 s, when F = 1208.9 kN, (

**j**–

**l**) are at time t = 6179.02 s, when F = 1208.87 kN.

**Figure 11.**Displacement, x-directional strain (ε

_{x}) and shear strain (γ

_{xy}) of multiple groups sandstone specimens. When these groups specimens G1, G2, G3, G4 were destroyed, they were captured by the low-speed camera. The results of each group displacement, ε

_{x}and γ

_{xy}are given. Where (

**a**–

**c**) belongs to group G1, which is destroyed at t = 6275 s and F = 1198 kN, (

**d**–

**f**) belongs to group G2, which is destroyed at t = 5503 s and F = 1105 kN, (

**g**–

**i**) belongs to group G3, which is destroyed at t = 6105 s and F = 1210 kN, (

**j**–

**l**) belongs to group G4, which is destroyed at t = 5905 s and F = 1111 kN.

Speckle Pattern | MIG | MISD | SE |
---|---|---|---|

A | 17.2696 | 15.3053 | 9.8501 |

B | 19.9866 | 16.4400 | 7.1012 |

C | 23.2605 | 21.7038 | 11.4230 |

D | 22.1941 | 20.6816 | 11.3406 |

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**MDPI and ACS Style**

Zhong, Y.; Chen, F.; Gao, X.; Guo, Z.; Sun, J.; Zhang, L.; Wang, Y.; Liu, Y.; Li, C. A Dual DIC System for Analysis of Dynamic Mechanical Properties of Large Sandstone under Uniaxial Compression Load. *Sustainability* **2023**, *15*, 2623.
https://doi.org/10.3390/su15032623

**AMA Style**

Zhong Y, Chen F, Gao X, Guo Z, Sun J, Zhang L, Wang Y, Liu Y, Li C. A Dual DIC System for Analysis of Dynamic Mechanical Properties of Large Sandstone under Uniaxial Compression Load. *Sustainability*. 2023; 15(3):2623.
https://doi.org/10.3390/su15032623

**Chicago/Turabian Style**

Zhong, Yichen, Fanxiu Chen, Xinya Gao, Zhanwei Guo, Jie Sun, Liming Zhang, Yuan Wang, Yuxin Liu, and Changtai Li. 2023. "A Dual DIC System for Analysis of Dynamic Mechanical Properties of Large Sandstone under Uniaxial Compression Load" *Sustainability* 15, no. 3: 2623.
https://doi.org/10.3390/su15032623