An accurate measurement method of crack fabric is proposed. Considering the density, geometry and spatial distribution in cracks, a novel method suggests the formula of high-order fabric tensor in the orthogonal space to describe cracks quantitatively. The crack is automatically identified by morphology and determined by stereo scans of image processing techniques. The fabric tensor is defined based on the idea of probability and statistics. The zero-order tensor describes the average effect of the density of rectangular cracks, the second-order tensor describes the two dominant directions of spatial cracks, and the higher-order tensor more accurately describes the anisotropy of cracks. With the increase of the tensor order, the more accurate direction and degree of the anisotropy can be described, and the morphological automatic identification of crack and its physical meaning of quantitative determination are clear. The plane description of the crack can be described by the invariants of the plane tensor. One invariant describes the degree of the anisotropy, and the other invariant describes its direction. The scan line is rotated and scanned on the rock plane image to obtain the accurate distribution of plane cracks, and the distribution law can be accurately described by the proposed high-order fabric tensor. In this paper, the rationality and effectiveness of the proposed measurement method are well verified by CT image analyses.
Highlights
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An accurate measurement method of crack fabric is proposed.
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The expression of crack fabric tensor is redefined by the normalized idea.
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The plane crack fabric is automatically identified by morphology, determined by stereo scans, and described by the its invariants equivalently.
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On the basis of stereology, the rationality and effectiveness of the proposed measurement method are well verified by SEM and CT image analyses.
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Abbreviations
- a :
-
Side length of the cube rock
- \(a_k\),\(b_k\) :
-
Scalar parameters of \({{N}_{ij\cdots l}}\)
- \(c_k\) :
-
Size invariant of crack fabric tensor on the plane
- \(E\left( \alpha ,\beta ,r \right)\) :
-
Spatial distribution function of cracks
- \({{F}_{0}}\) :
-
Average crack density of rock
- \({{F}_{ij}}\),\({{F}_{ijkl}}\),\({{F}_{ij\cdots l}}\) :
-
Crack fabric tensors with the normalized idea
- \({{m}^{\left( V \right) }}\) :
-
Number of centroids of cracks
- \({{n}_{i}}(i=1,2,3)\) :
-
Unit vector relative to the orthogonal coordinate system
- \({{n}_{i}}\),\({{n}_{j}}\),\({{n}_{k}}\),\({{n}_{l}}\) :
-
Cosine values of the coordinate direction
- \({{N}_{ij}}\),\({{N}_{ijkl}}\),\({{N}_{ij\cdots l}}\) :
-
Crack fabric tensors with zero trace
- r :
-
Radius of each penny-shape crack
- 2r :
-
Crack diameter
- \({{r}^{k}}\) :
-
Radius of the kth crack
- s :
-
Crack area
- T :
-
Thickness of the crack
- V :
-
Volume of the rock mass
- \(\alpha\) :
-
Angle between the scan line and the \({{x}_{3}}\) axis on the \({{x}_{2}}-{{x}_{3}}\) plane
- \(\beta\) :
-
Angle between the scan line and the \({{x}_{1}}\) axis
- \(\theta\) :
-
Angle variable of the plane coordinate system
- \(\theta _{k}\) :
-
Direction invariant of crack fabric tensor on the plane
- \(\rho\) :
-
Number of cracks per unit volume
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Acknowledgements
This work was financially supported by the Projects for Leading Talents of Science and Technology Innovation of Ningxia (No. KJT2019001), the National Natural Science Foundation of China (No. 12162028), and The innovation team for multi-scale mechanics and its engineering applications of Ningxia Hui Autonomous Region (2021), and these supports are gratefully acknowledged.
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XL: methodology, funding acquisition, supervision, writing—review & editing. CD: data curation, writing—original draft. XW: software, validation. JZ: visualization, writing—review & editing.
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Li, X., Du, C., Wang, X. et al. Quantitative Determination of High-Order Crack Fabric in Rock Plane. Rock Mech Rock Eng 56, 5029–5038 (2023). https://doi.org/10.1007/s00603-023-03319-x
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DOI: https://doi.org/10.1007/s00603-023-03319-x