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Keywords = columnar jointed rock mass

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27 pages, 14949 KiB  
Article
Experimental Study on Strength and Deformation Moduli of Columnar Jointed Rock Mass—Uniaxial Compression as an Example
by Zhenbo Xu, Zhende Zhu, Chao Jiang and Xiaobin Hu
Symmetry 2024, 16(10), 1380; https://doi.org/10.3390/sym16101380 - 17 Oct 2024
Viewed by 1176
Abstract
The irregular joint network unique to columnar joints separates the rock mass into several irregular polygonal prisms. Similar physical model specimens of columnar jointed rock mass (CJRM) were fabricated using a rock-like material. The effect of the irregularity of the joint network was [...] Read more.
The irregular joint network unique to columnar joints separates the rock mass into several irregular polygonal prisms. Similar physical model specimens of columnar jointed rock mass (CJRM) were fabricated using a rock-like material. The effect of the irregularity of the joint network was considered in the horizontal plane, and the effect of the dip angle of the joint network was considered in the vertical plane. The strength and deformation moduli of the specimen were investigated using uniaxial compression tests. A total of four failure modes of regular columnar jointed rock mass (RCJRM) and irregular columnar jointed rock mass (ICJRM) were identified through the tests. The peak stress of the irregular columnar jointed rock mass specimen is reduced by 56.65%. The strength and deformation moduli of RCJRM were greater than those of ICJRM, while the anisotropic characteristics of ICJRM were stronger. The failure mode of CJRM was determined by the dip angle. With the increase in the dip angle, the strength and deformation moduli of irregular columnar jointed rock mass are a symmetrical “V” type distribution, 45° corresponds to the minimum strength, and 30° obtains the minimum deformation modulus. With the increase in the irregularity coefficient, the strength and deformation moduli of CJRM decreased first and then increased gradually. When the irregularity coefficient is 0.1, the linear deformation modulus reaches the minimum value. When the irregularity coefficient is 0.7, the median deformation modulus reaches the minimum value. The fitting function proposed in the form of the cosine function managed to predict the strength value of CJRM and showed the strength of the anisotropic characteristics caused by the change in the dip angle. Compared with the existing physical model test results, it is determined that the strength of the specimen is positively correlated with the addition amount of rock-like material and the loading rate, and negatively correlated with the water consumption. Full article
(This article belongs to the Section Engineering and Materials)
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Figure 1

Figure 1
<p>Pictures of CJRM.</p>
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<p>Irregular polygon Voronoi diagrams.</p>
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<p>Normalized area and side length distribution of polygons.</p>
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<p>Specimen fabrication process.</p>
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<p>ICJRM specimens with different joint dip angles.</p>
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<p>Uniaxial compression test system and specimen loading diagram.</p>
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<p>Stress–strain curves of CJRM specimens.</p>
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<p>Stress–strain curves of CJRM specimens.</p>
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<p>The influence of the irregularity coefficient and the inclination angle on peak stress.</p>
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<p>Failure modes of RCJRM specimens with different dip angles.</p>
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<p>Failure modes of ICJRM specimens with different dip angles.</p>
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<p>Effect of the irregularity coefficient on the peak stress.</p>
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<p>Schematic diagram of the value-taking methods for the specimen deformation modulus.</p>
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<p>Effect of the dip angle on the deformation modulus.</p>
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<p>Effect of the irregularity coefficient on the deformation modulus.</p>
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<p>Specimen cracking evolution.</p>
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<p>Effect of the irregularity coefficient on the anisotropy ratio coefficient.</p>
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<p>Effect of the irregularity coefficient on the area of the anisotropy region. (<b>a</b>) Area of the anisotropy region when the irregularity coefficient was 0.1 (<math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mrow> <mn>0.1</mn> </mrow> </msub> </mrow> </semantics></math>); (<b>b</b>) Areas of the anisotropy regions when the irregularity coefficients were 0.3 (<math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mrow> <mn>0.3</mn> </mrow> </msub> </mrow> </semantics></math>), 0.5 (<math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mrow> <mn>0.5</mn> </mrow> </msub> </mrow> </semantics></math>), and 0.7 (<math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mrow> <mn>0.7</mn> </mrow> </msub> </mrow> </semantics></math>).</p>
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<p>Summary of the anisotropy index values. It is compared with the data in the references [<a href="#B11-symmetry-16-01380" class="html-bibr">11</a>,<a href="#B22-symmetry-16-01380" class="html-bibr">22</a>,<a href="#B24-symmetry-16-01380" class="html-bibr">24</a>,<a href="#B34-symmetry-16-01380" class="html-bibr">34</a>,<a href="#B37-symmetry-16-01380" class="html-bibr">37</a>,<a href="#B38-symmetry-16-01380" class="html-bibr">38</a>].</p>
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<p>Peak stress fitting curves.</p>
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<p>Comparison between the fitted values and the test values under the optimum fitting condition.</p>
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<p>MAPE values of the simplified fitting functions.</p>
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<p>Comparison between the fitted values and the test values under the simplified fitting condition.</p>
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<p>Comparison between the fitted values obtained by two fitting methods and test values.</p>
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25 pages, 29463 KiB  
Article
Numerical Simulation of Failure Modes in Irregular Columnar Jointed Rock Masses under Dynamic Loading
by Yingjie Xia, Bingchen Liu, Tianjiao Li, Danchen Zhao, Ning Liu, Chun’an Tang and Jun Chen
Mathematics 2023, 11(17), 3790; https://doi.org/10.3390/math11173790 - 4 Sep 2023
Cited by 2 | Viewed by 1378
Abstract
The mechanical properties and failure characteristics of columnar jointed rock mass (CJRM) are significantly influenced by its irregular structure. Current research on CJRMs is mainly under static loading, which cannot meet the actual needs of engineering. This paper adopts the finite element method [...] Read more.
The mechanical properties and failure characteristics of columnar jointed rock mass (CJRM) are significantly influenced by its irregular structure. Current research on CJRMs is mainly under static loading, which cannot meet the actual needs of engineering. This paper adopts the finite element method (FEM) to carry out numerical simulation tests on irregular CJRMs with different dip angles under different dynamic stress wave loadings. The dynamic failure modes of irregular CJRMs and the influence law of related stress wave parameters are obtained. The results show that when the column dip angle α is 0°, the tensile-compressive-shear failure occurs in the CJRMs; when α is 30°, the CJRMs undergo tensile failure and a small amount of compressive shear failure, and an obvious crack-free area appears in the middle of the rock mass; when α is 60°, tensile failure is dominant and compressive shear failure is minimal and no crack area disappears; and when α is 90°, the rock mass undergoes complete tensile failure. In addition, in terms of the change law of stress wave parameters, the increase in peak amplitude will increase the number of cracks, promote the development of cracks, and increase the proportion of compression-shear failure units for low-angle rock mass. The changes in the loading and decay rate only affect the degree of crack development in the CJRMs, but do not increase the number of cracks. Meanwhile, the simulation results show that the crack expansion velocity of the CJRMs increases with the increase in dip angle, and the CJRMs with dip angle α = 60° are the most vulnerable to failure. The influence of the loading and decay rate on the rock mass failure is different with the change in dip angle. The results of the study provide references for related rock engineering. Full article
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Figure 1
<p>Structural characteristics of CJRMs: (<b>a</b>) left bank of Baihetan hydropower station; (<b>b</b>) columnar joints; (<b>c</b>) fissures.</p>
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<p>Constitutive relation of rock under uniaxial stress: (<b>a</b>) uniaxial compression; (<b>b</b>) uniaxial tension [<a href="#B63-mathematics-11-03790" class="html-bibr">63</a>].</p>
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<p>Numerical model under dynamic loading: (<b>a</b>) numerical model of the CJRMs; (<b>b</b>) load settings; (<b>c</b>) mesh generation; (<b>d</b>) loading plans.</p>
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<p>Physical laboratory experiments: (<b>a</b>) uniaxial compression test system; (<b>b</b>) columnar joint shear test [<a href="#B22-mathematics-11-03790" class="html-bibr">22</a>].</p>
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<p>Numerical results of models with different column dip angles <span class="html-italic">α</span> at the peak amplitude of 12 MPa.</p>
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<p>Single-step AE energy curves of models with different column dip angles <span class="html-italic">α</span> at the peak amplitude of 12 MPa: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>Numerical results of the CJRM with <span class="html-italic">α</span> = 30° under loading plan I.</p>
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<p>Stress nephogram of the CJRMs under the loading plan I: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 60°; (<b>c</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>AE diagram of the CJRMs under the loading plan I: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>AE events under different peak amplitudes: (<b>a</b>) Loading plan I; (<b>b</b>) Load plan II.</p>
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<p>Stress nephogram of the CJRMs under the loading plan II: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>AE diagram of the CJRMs under the loading plan II: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>Stress nephogram of the CJRMs under the loading plan III: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>AE diagram of the CJRMs under the loading plan III: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>Single step AE and energy curves of loading plan III: (<b>a</b>) peak amplitude of 5 μs; (<b>b</b>) peak amplitude of 10 μs; (<b>c</b>) peak amplitude of 15 μs; (<b>d</b>) peak amplitude of 20 μs.</p>
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<p>Stress nephogram of the CJRMs under loading plan IV: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>AE diagram of the CJRMs under the loading plan IV: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>Sketch diagram of cracks of the CJRMs at the peak value of 12 MPa: (<b>a</b>) crack development process of <span class="html-italic">α</span> = 30° rock mass; (<b>b</b>) cracks in rock mass under different dip angles. Red lines represent the activated joints.</p>
Full article ">Figure 18 Cont.
<p>Sketch diagram of cracks of the CJRMs at the peak value of 12 MPa: (<b>a</b>) crack development process of <span class="html-italic">α</span> = 30° rock mass; (<b>b</b>) cracks in rock mass under different dip angles. Red lines represent the activated joints.</p>
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<p>AE events and energy curves of models at the peak value of 14 MPa with different ascending and descending rates and the same loading period: (<b>a</b>) <span class="html-italic">α</span> = 0°; (<b>b</b>) <span class="html-italic">α</span> = 30°; (<b>c</b>) <span class="html-italic">α</span> = 60°; (<b>d</b>) <span class="html-italic">α</span> = 90°.</p>
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<p>AE event curves of rock mass with changes in different dynamic stress wave parameters at different dip angles: (<b>a</b>) Loading plan I; (<b>b</b>) Loading plan II; (<b>c</b>) Loading plan III and loading plan IV.</p>
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19 pages, 10677 KiB  
Article
A Case Study on Tunnel Excavation Stability of Columnar Jointed Rock Masses with Different Dip Angles in the Baihetan Diversion Tunnel
by Luxiang Wang, Zhende Zhu, Shu Zhu and Junyu Wu
Symmetry 2023, 15(6), 1232; https://doi.org/10.3390/sym15061232 - 9 Jun 2023
Cited by 4 | Viewed by 1587
Abstract
Columnar jointed rock mass (CJRM) formed by intact rock divided by special symmetrical columnar joints is a special type of rock with poor mechanical properties, strong anisotropy, and weak self-supporting ability, severely affecting the excavation safety and stability of underground tunnels. In this [...] Read more.
Columnar jointed rock mass (CJRM) formed by intact rock divided by special symmetrical columnar joints is a special type of rock with poor mechanical properties, strong anisotropy, and weak self-supporting ability, severely affecting the excavation safety and stability of underground tunnels. In this study, taking the Baihetan hydropower station as the engineering background, CJRM geological numerical models with different dip angles that combined well with the natural CJRM were generated based on the geological statistical parameters of the engineering site and were verified to have high rationality and accuracy. Tunnel excavation and overloading tests were carried out on these numerical models, and the results showed that the stress and displacement distributions after excavation exhibited strong anisotropic characteristics under different dip angles, and the positions where engineering safety problems are most likely to occur are the side walls, which are prone to stress-structure-controlled failure mode. The self-supporting ability at different dip angles after excavation from weak to strong are 45°, 60°, 75°, 90°, 30°, 0°, and 15°. The safety factors assessed by overloading for CJRM with dip angles of 0–90° degrees were 2.5, 2.6, 2.6, 1.8, 2.1, and 2.2, respectively, providing a valuable reference for the construction safety and support measures of CJRM excavation. Full article
(This article belongs to the Section Engineering and Materials)
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Figure 1

Figure 1
<p>Location and components of the Baihetan hydropower station. (<b>a</b>) Location of the station; (<b>b</b>) Layout of the station; (<b>c</b>) The Yangtze River Basin and surrounding hydropower stations.</p>
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<p>Stratigraphic distribution of the station and layout of diversion tunnels.</p>
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<p>Jointed rock mass drilled in the Baihetan area.</p>
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<p>Generation process of CJRM geometric model. (<b>a</b>) Voronoi diagram with the target average prism diameter and the irregular factor; (<b>b</b>) Process of sectioning; (<b>c</b>) The final CJRM geometric model.</p>
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<p>Constitutive joint model of contact surface.</p>
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<p>Basalt CJRM geological numerical models with different dip angles. (<b>a</b>) Dip angle of 0°; (<b>b</b>) Dip angle of 15°; (<b>c</b>) Dip angle of 30°; (<b>d</b>) Dip angle of 45°; (<b>e</b>) Dip angle of 60°; (<b>f</b>) Dip angle of 75°; (<b>g</b>) Dip angle of 90°.</p>
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<p>Basalt CJRM geological numerical models with different dip angles. (<b>a</b>) Dip angle of 0°; (<b>b</b>) Dip angle of 15°; (<b>c</b>) Dip angle of 30°; (<b>d</b>) Dip angle of 45°; (<b>e</b>) Dip angle of 60°; (<b>f</b>) Dip angle of 75°; (<b>g</b>) Dip angle of 90°.</p>
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<p>Tunnel excavation in CJRM geological numerical model. (<b>a</b>) Tunnel spatial position; (<b>b</b>) Tunnel size and layout of monitoring points.</p>
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<p>In situ rigid bearing plate test on the numerical model.</p>
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<p>Comparison of the in situ rigid bearing plate experiment results and numerical simulation results.</p>
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<p>XX-stress distribution contour map of excavated geological numerical models. (<b>a</b>) Dip angle of 0°; (<b>b</b>) Dip angle of 15°; (<b>c</b>) Dip angle of 30°; (<b>d</b>) Dip angle of 45°; (<b>e</b>) Dip angle of 60°; (<b>f</b>) Dip angle of 75°; (<b>g</b>) Dip angle of 90°.</p>
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<p>ZZ-stress distribution contour map of excavated geological numerical models. (<b>a</b>) Dip angle of 0°; (<b>b</b>) Dip angle of 15°; (<b>c</b>) Dip angle of 30°; (<b>d</b>) Dip angle of 45°; (<b>e</b>) Dip angle of 60°; (<b>f</b>) Dip angle of 75°; (<b>g</b>) Dip angle of 90°.</p>
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<p>X-displace distribution contour map of excavated geological numerical models. (<b>a</b>) Dip angle of 0°; (<b>b</b>) Dip angle of 15°; (<b>c</b>) Dip angle of 30°; (<b>d</b>) Dip angle of 45°; (<b>e</b>) Dip angle of 60°; (<b>f</b>) Dip angle of 75°; (<b>g</b>) Dip angle of 90°.</p>
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<p>Z-displace distribution contour map of excavated geological numerical models. (<b>a</b>) Dip angle of 0°; (<b>b</b>) Dip angle of 15°; (<b>c</b>) Dip angle of 30°; (<b>d</b>) Dip angle of 45°; (<b>e</b>) Dip angle of 60°; (<b>f</b>) Dip angle of 75°; (<b>g</b>) Dip angle of 90°.</p>
Full article ">Figure 13 Cont.
<p>Z-displace distribution contour map of excavated geological numerical models. (<b>a</b>) Dip angle of 0°; (<b>b</b>) Dip angle of 15°; (<b>c</b>) Dip angle of 30°; (<b>d</b>) Dip angle of 45°; (<b>e</b>) Dip angle of 60°; (<b>f</b>) Dip angle of 75°; (<b>g</b>) Dip angle of 90°.</p>
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<p>Collapse failure of surrounding rock of the side wall in the diversion tunnel at the Baihetan hydropower station.</p>
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<p>Displacements of each monitoring position versus the overloading ratio.</p>
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19 pages, 8019 KiB  
Article
Study on the Anisotropy of Strength Properties of Columnar Jointed Rock Masses Using a Geometric Model Reconstruction Method Based on a Single-Random Movement Voronoi Diagram of Uniform Seed Points
by Zhende Zhu, Luxiang Wang, Shu Zhu and Junyu Wu
Symmetry 2023, 15(4), 944; https://doi.org/10.3390/sym15040944 - 20 Apr 2023
Cited by 7 | Viewed by 1694
Abstract
The unique structural characteristics and special symmetry of columnar jointed rock mass result in its complex mechanical properties and strong anisotropy, which seriously affects the safety of engineering construction. To better simulate natural columnar jointed rock mass, a geometric model reconstruction method based [...] Read more.
The unique structural characteristics and special symmetry of columnar jointed rock mass result in its complex mechanical properties and strong anisotropy, which seriously affects the safety of engineering construction. To better simulate natural columnar jointed rock mass, a geometric model reconstruction method based on a single-random movement Voronoi diagram of uniform seed points using the feasible geological parameters of horizontal polygon density, irregular factor, dip angle, strike angle, transverse joint spacing, and transverse joint penetration rate is proposed in this paper. Based on this method, numerical simulation of CJRM models with varying strike angles, dip angles, and irregular factors under uniaxial compression were conducted. The results show that the uniaxial compression strengths versus strike angle and dip angle both decrease with the increase in the irregular factor, showing a U-shape and a gentle W-shape, respectively. The strength anisotropy of the strike angle decreases from 1.1057 to 1.0395 with the increase in the irregular factor, indicating relatively isotropy. With the increase int the irregular factor, the strength anisotropy of the dip angle increases from 4.3381 to 6.7953, indicating an increasing strong anisotropy at a high degree, and the effect of the irregular factor on strength behavior has the strongest and weakest impact at the dip angles of 60° and 90°, respectively. Full article
(This article belongs to the Topic Mathematical Modeling)
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Figure 1

Figure 1
<p>Baihetan Hydropower Station. (<b>a</b>) Location of the hydropower station. (<b>b</b>) Layout of the project.</p>
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<p>Geological environment of the Baihetan dam foundation. (<b>a</b>) Typical engineering geological section of the dam foundation. (<b>b</b>) Vertical cross-section of the CJRM (parallel to the column axis direction). (<b>c</b>) Horizontal cross-section of the CJRM (perpendicular to the column axis direction).</p>
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<p>Voronoi diagram. (<b>a</b>) The Voronoi diagram generated by random seed points. (<b>b</b>) The Voronoi polygon with adjacent seed points.</p>
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<p>Single-random movement with range constraint Voronoi diagram of uniform seed points. (<b>a</b>) Expanded region (2 m × 2 m) with uniformly distributed seed points. (<b>b</b>) Voronoi diagram generated by uniform distributed seed points (<span class="html-italic">I<sub>r</sub></span> = 0). (<b>c</b>) Single-random movement with range constraint diagram. (<b>d</b>) Single-random moved seed points. (<b>e</b>) Voronoi diagram generated by single-random moved seed points. (<b>f</b>) Voronoi diagram with <span class="html-italic">I<sub>r</sub></span> = 35%. (<b>g</b>) Existence and elimination of relative short edges. (<b>h</b>) Voronoi diagram with target polygon density and irregular factor without relative short edges.</p>
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<p>CJRM geometric model reconstruction method. (<b>a</b>) Generation of transverse joint. (<b>b</b>) Generation of the coordinate axis. (<b>c</b>) Rotation of the cube. (<b>d</b>) The final CJRM geometric model with six geological statistical parameters.</p>
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<p>The entire procedure of the CJRM geometric model reconstruction method.</p>
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<p>CJRM geometric model reconstruction method. (<b>a</b>) Special mold for sample preparation. (<b>b</b>) Generation of the real project geometric model.</p>
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<p>Failure criterion of Mohr–Coulomb criterion in the 3DEC software.</p>
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<p>Comparison of the physical model test and numerical simulation under uniaxial compression.</p>
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<p>Comparison of the uniaxial compressive strength between experimental data [<a href="#B15-symmetry-15-00944" class="html-bibr">15</a>] and numerical results.</p>
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<p>Variation in uniaxial compression strength of the numerical CJRM model with different strike angles and irregular factors. (<b>a</b>) Varying strike angles. (<b>b</b>) Varying irregular factors.</p>
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<p>Variations in strength anisotropy of the strike angle with irregular factor.</p>
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<p>Variation in the uniaxial compression strength of the numerical CJRM model with different dip angles and irregular factors. (<b>a</b>) Varying dip angles. (<b>b</b>) Varying irregular factors.</p>
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<p>Variations in strength anisotropy of the dip angle with irregular factor.</p>
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<p>Visualization of three-dimensional anisotropy characteristics in the CJRM with varied irregular factors using Polar Coordinate System Transformation. (<b>a</b>) <span class="html-italic">I<sub>r</sub></span> = 0; (<b>b</b>) <span class="html-italic">I<sub>r</sub></span> = 10%; (<b>c</b>) <span class="html-italic">I<sub>r</sub></span> = 20%; (<b>d</b>) <span class="html-italic">I<sub>r</sub></span> = 30%; (<b>e</b>) <span class="html-italic">I<sub>r</sub></span> = 40%.</p>
Full article ">Figure 15 Cont.
<p>Visualization of three-dimensional anisotropy characteristics in the CJRM with varied irregular factors using Polar Coordinate System Transformation. (<b>a</b>) <span class="html-italic">I<sub>r</sub></span> = 0; (<b>b</b>) <span class="html-italic">I<sub>r</sub></span> = 10%; (<b>c</b>) <span class="html-italic">I<sub>r</sub></span> = 20%; (<b>d</b>) <span class="html-italic">I<sub>r</sub></span> = 30%; (<b>e</b>) <span class="html-italic">I<sub>r</sub></span> = 40%.</p>
Full article ">
15 pages, 6057 KiB  
Article
Model Test Study on the Anisotropic Characteristics of Columnar Jointed Rock Mass
by Zhende Zhu, Xiangcheng Que, Zihao Niu and Wenbin Lu
Symmetry 2020, 12(9), 1528; https://doi.org/10.3390/sym12091528 - 16 Sep 2020
Cited by 6 | Viewed by 2123
Abstract
Because of its special structure, the anisotropic properties of columnar jointed rock mass (CJRM) are complicated, which brings difficulty to engineering construction. To comprehensively study the anisotropic characteristics of CJRM, uniaxial compression tests were conducted on artificial CJRM specimens. Quadrangular, pentagonal and hexagonal [...] Read more.
Because of its special structure, the anisotropic properties of columnar jointed rock mass (CJRM) are complicated, which brings difficulty to engineering construction. To comprehensively study the anisotropic characteristics of CJRM, uniaxial compression tests were conducted on artificial CJRM specimens. Quadrangular, pentagonal and hexagonal prism CJRM models were introduced, and the dip direction of the columnar joints was considered. Based on the test results and the structural features of the three CJRM models, the deformation and strength characteristics of CJRM specimens were analyzed and compared. The failure modes and mechanisms of artificial specimens with different dip directions were summarized in accordance with the failure processes and final appearances. Subsequently, the anisotropic degrees of the three CJRM models in the horizontal plane were classified, and their anisotropic characteristics were described. Finally, a simple empirical expression was adopted to estimate the strength and deformation of the CJRM, and the derived equations were used in the Baihetan Hydropower Station project. The calculated values are in good agreement with the existing research results, which reflects the engineering application value of the derived empirical equations. Full article
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<p>Columnar jointed rock mass (CJRM) at Baihetan Hydropower Station in China: (<b>a</b>) CJRM in the vertical plane; (<b>b</b>) CJRM in the horizontal plane.</p>
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<p>Structural characteristics of the three CJRM models: (<b>a</b>) 4P-CJRM model; (<b>b</b>) 6P-CJRM model; (<b>c</b>) 5P-CJRM model.</p>
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<p>CJRM specimens with different dip directions <span class="html-italic">α</span>: (<b>a</b>) 4P-CJRM specimens; (<b>b</b>) 5P-CJRM specimens; (<b>c</b>) 6P-CJRM specimens.</p>
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<p>Manufacturing process of the artificial CJRM specimens: (<b>a</b>) Rock-like material with a mixing ratio of m<sub>g</sub>:m<sub>s</sub>:m<sub>w</sub> = 3:1:2.4; (<b>b</b>) Acrylic molds for making the columns with different cross-sectional shapes; (<b>c</b>) Removing the column bars from the molds; (<b>d</b>) Checking the size of the three types of columns; (<b>e</b>) Gluing the columns with the same cross-sectional shape using the cement slurry to form a model block; (<b>f</b>) Standard cubic CJRM specimens.</p>
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<p>RMT-150B servo-controlled test system.</p>
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<p>Stress-strain curves of the CJRM specimens: (<b>a</b>) 4P-CJRM specimens; (<b>b</b>) 5P-CJRM specimens; (<b>c</b>) 6P-CJRM specimens.</p>
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<p>Variations in elastic modulus and uniaxial compression strength (UCS) with dip direction <span class="html-italic">α</span>: (<b>a</b>) Elastic modulus anisotropy curves of the three CJRM models; (<b>b</b>) UCS anisotropy curves of the three CJRM models.</p>
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<p>Final failure appearances of three types of artificial CJRM specimens with different dip directions: (<b>a</b>) 4P-CJRM specimens; (<b>b</b>) 5P-CJRM specimens; (<b>c</b>) 6P-CJRM specimens.</p>
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<p>Typical failure processes of the CJRM specimens: (<b>a</b>) Failure process and mechanism of Mode I; (<b>b</b>) Failure process and mechanism of Mode II; (<b>c</b>) Failure process and mechanism of Mode III.</p>
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<p>Elastic modulus and UCS of the three CJRM models in the polar coordinate system: (<b>a</b>) Elastic modulus <span class="html-italic">E<sub>cj</sub></span>; (<b>b</b>) UCS <span class="html-italic">σ<sub>cj</sub></span>.</p>
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<p>Theoretical curves of the normalized mechanical parameters: (<b>a</b>) Normalized UCS <span class="html-italic">σ<sub>cr</sub></span>; (<b>b</b>) Normalized elastic modulus <span class="html-italic">E<sub>cr</sub></span>.</p>
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16 pages, 3410 KiB  
Article
Anisotropic Constitutive Model of Intermittent Columnar Jointed Rock Masses Based on the Cosserat Theory
by Wenbin Lu, Zhende Zhu, Xiangcheng Que, Cong Zhang and Yanxin He
Symmetry 2020, 12(5), 823; https://doi.org/10.3390/sym12050823 - 17 May 2020
Cited by 10 | Viewed by 2388
Abstract
In this work, an anisotropic constitutive model of hexagonal columnar jointed rock masses is established to describe the distribution law of deformation and the failure of columnar joint caverns under anisotropic conditions, and is implemented to study the columnar jointed rock mass at [...] Read more.
In this work, an anisotropic constitutive model of hexagonal columnar jointed rock masses is established to describe the distribution law of deformation and the failure of columnar joint caverns under anisotropic conditions, and is implemented to study the columnar jointed rock mass at the dam site of the Baihetan Hydropower Station on the Jinsha River. The model is based on the Cosserat theory and considers the mesoscopic bending effect on the macroscopic mean. The influences of joint plane inclination on equivalent anisotropic elastic parameters are discussed via the introduction of an off-axis transformation matrix and the analysis of an example. It is also pointed out that the six-prism columnar jointed rock mass changes from transverse isotropy to anisotropy under the influence of the angle. A numerical calculation program of the Cosserat constitutive model is developed and is applied to the simulation calculation of a Baihetan diversion tunnel to compare and analyze the respective plastic zones and stress distributions after tunnel excavation under both isotropic and anisotropic conditions. The results reveal that, compared with the isotropic model, the proposed Cosserat anisotropic model better reflects the state of stress and asymmetric distribution of the plastic zone after tunnel excavation, and the actual deformation of the surrounding rock of the tunnel is greater than that calculated by the isotropic method. The results aid in a better understanding of the mechanical properties of rock masses. Full article
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<p>Baihetan columnar jointed basalt.</p>
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<p>Rock mass decomposition diagram of a six-prism columnar jointed rock mass.</p>
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<p>Element model of a single set of intermittent jointed rock masses.</p>
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<p>Stress and couple stress on the equivalent model of the Cosserat element.</p>
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<p>Schematic diagram of the columnar joint of a regular hexagonal prism.</p>
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<p>Off-axis diagram of the columnar jointed rock mass (hexagonal prism).</p>
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<p>Variations of the elastic parameters with the inclination angle <span class="html-italic">β</span>.</p>
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<p>Schematic diagram of the finite element calculation of the tunnel.</p>
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<p>Plastic zone distributions calculated by the two methods.</p>
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<p>Z-direction stress cloud maps calculated by the two methods.</p>
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15 pages, 2748 KiB  
Article
Constitutive Model of Stress-Dependent Seepage in Columnar Jointed Rock Mass
by Zihao Niu, Zhende Zhu and Xiangcheng Que
Symmetry 2020, 12(1), 160; https://doi.org/10.3390/sym12010160 - 13 Jan 2020
Cited by 9 | Viewed by 2833
Abstract
Columnar jointed rock mass (CJRM) is a highly symmetrical natural fractured structure. As the rock mass of the dam foundation of the Baihetan Hydropower Station, the study of its permeability anisotropy is of great significance to engineering safety. Based on the theory of [...] Read more.
Columnar jointed rock mass (CJRM) is a highly symmetrical natural fractured structure. As the rock mass of the dam foundation of the Baihetan Hydropower Station, the study of its permeability anisotropy is of great significance to engineering safety. Based on the theory of composite mechanics and Goodman’s joint superposition principle, the constitutive model of joints of CJRM is derived according to the Quadrangular prism, the Pentagonal prism and the Hexagonal prism model; combined with Singh’s research results on intermittent joint stress concentration, considering column deflection angles, the joint constitutive model of CJRM in three-dimensional space is established. For the CJRM in the Baihetan dam site area, the Quadrangular prism, the Pentagonal prism and the Hexagonal prism constitutive models were used to calculate the permeability coefficients of CJRM under different deflection angles. The permeability anisotropy characteristics of the three models were compared and verified by numerical simulation results. The results show that the calculation results of the Pentagonal prism model are in good agreement with the numerical simulation results. The variation of permeability coefficient under different confining pressures is compared, and the relationship between permeability coefficient and confining pressure is obtained, which accords with the negative exponential function and conforms to the general rule of joint seepage. Full article
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<p>Typical columnar joint, pictured (<b>a</b>) parallel to the column axis direction; (<b>b</b>) perpendicular to the column axis direction.</p>
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<p>Schematic diagram of three-direction stress of single fractured rock mass.</p>
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<p>Typical columnar joint: (<b>a</b>) Quadrangular prism model; (<b>b</b>) Hexagonal prism model; (<b>c</b>) Pentagonal prism model.</p>
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<p>Element model of discontinuous jointed rock mass.</p>
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<p>Composite solution method for joint flexibility matrix of CJRM: (<b>a</b>) calculation method of Quadrangular flexibility matrix; (<b>b</b>) calculation method of Hexagonal flexibility matrix; (<b>c</b>) calculation method of Pentagonal flexibility matrix. <span class="html-italic">s, and n</span> are parallel and perpendicular to the direction of joint extension, respectively.</p>
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<p>Simplified cross-section of Hexagonal and Pentagonal models: (<b>a</b>) simplified Hexagonal prism model; (<b>b</b>) simplified Pentagonal prism model.</p>
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<p>Columnar joints with different joint deflection angles from Xiong: (<b>a</b>) deflection angle of 0°; (<b>b</b>) deflection angle of 30°; (<b>c</b>) deflection angle of 60°; (<b>d</b>) deflection angle of 90°.</p>
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<p>Comparison of the permeability coefficient theoretical curve and numerical simulation curve.</p>
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<p>Comparison of the permeability coefficient theoretical curve and fitted curve: (<b>a</b>) deflection angle of 0°; (<b>b</b>) deflection angle of 18°; (<b>c</b>) deflection angle of 36°; (<b>d</b>) deflection angle of 54°; (<b>e</b>) deflection angle of 72°; (<b>f</b>) deflection angle of 90°.</p>
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<p>Comparison of the permeability coefficient theoretical curve and fitted curve: (<b>a</b>) deflection angle of 0°; (<b>b</b>) deflection angle of 18°; (<b>c</b>) deflection angle of 36°; (<b>d</b>) deflection angle of 54°; (<b>e</b>) deflection angle of 72°; (<b>f</b>) deflection angle of 90°.</p>
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24 pages, 25758 KiB  
Article
Mechanical Anisotropy and Failure Characteristics of Columnar Jointed Rock Masses (CJRM) in Baihetan Hydropower Station: Structural Considerations Based on Digital Image Processing Technology
by Yingjie Xia, Chuanqing Zhang, Hui Zhou, Chunsheng Zhang and Wangbing Hong
Energies 2019, 12(19), 3602; https://doi.org/10.3390/en12193602 - 20 Sep 2019
Cited by 8 | Viewed by 2913
Abstract
The columnar joints in Baihetan hydropower station are primary tensile joints since they were formed during the process of lava condensation. Understanding the influence of columnar jointed rock mass (CJRM) on the mechanical response and failure modes is the basis for designing of [...] Read more.
The columnar joints in Baihetan hydropower station are primary tensile joints since they were formed during the process of lava condensation. Understanding the influence of columnar jointed rock mass (CJRM) on the mechanical response and failure modes is the basis for designing of associated engineering works. Hence, the structural characteristics of Baihetan CJRM were analyzed by carrying out a geological survey at first. Three groups of numerical models capable of reflecting the structural characteristics of CJRM were then established to analyze the mechanical and failure characteristics. The results in this study showed that: (1) Irregularity of columnar basalt restricted crack propagation on columnar joints and also led to stress concentration in the distorted parts, and thus, damage of basalt columns; (2) when the included angle between direction of concentrated defect structures in CJRM and uniaxial stress was large, the defect structures can prevent crack propagation on columnar joints, and the failure of defect structure can cause the overall failure of the rock mass; and (3) under the condition of same columnar structure and included angle, the peak strength of models with microcracks and structural plane was low and the irregular shape of columnar joints decreased the anisotropy of mechanical parameters. Full article
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<p>Structural characteristics of columnar jointed basalt in drainage tunnel #1-1: (<b>a</b>) Sampling window for columnar jointed basalt; (<b>b</b>) analysis of various structures in the sampling window.</p>
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<p>Calculation of joint roughness coefficient (<span class="html-italic">JRC</span>) for columnar joints: (<b>a</b>) Primary columnar jointed rock (CJRM) with columnar joints; (<b>b</b>) acquisition of JRC across a longitudinal section.</p>
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<p>The occurrence of microcracks in columnar jointed basalt in the drainage tunnel #1-1: (<b>a</b>) Contour diagram; (<b>b</b>) rosette plot.</p>
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<p>Establishment of numerical model: (<b>a</b>) Digital image containing basalt columns, microcracks and structural planes; (<b>b</b>) numerical model based on recognition by DIP technology.</p>
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<p>Three groups of sampling windows for numerical simulation: (<b>a</b>) Sampling window containing microcracks and structural planes; (<b>b</b>) sampling window without microcracks and structural planes; (<b>c</b>) generalized model for the sampling window of CJRM.</p>
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<p>Elasto-brittle damage constitutive law of element subject to uniaxial stress: (<b>a</b>) Under uniaxial tensile stress; (<b>b</b>) under uniaxial compressive stress.</p>
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<p>Stress-strain curves and failure modes of columnar jointed basalt specimens: (<b>a</b>) Laboratory test results; (<b>b</b>) calibration of parameters for numerical simulation between stress-strain curves and failure process obtained by numerical simulation and laboratory test.</p>
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<p>Shear properties of columnar joints by numerical simulation.</p>
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<p>Shear stress-shear displacement curve and displacement field of the numerical model for columnar joints.</p>
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<p>Simulation results of the model containing microcracks and structural planes in the case of <span class="html-italic">α</span> = 10°: (<b>a</b>) Elastic modulus (MPa); (<b>b</b>) max principal stress (MPa); (<b>c</b>) acoustic emission (AE) events (blue: Tensile failure; red: Shear failure).</p>
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<p>Simulation results of the model containing microcracks and structural planes in the case of <span class="html-italic">α</span> = 30°: (<b>a</b>) Elastic modulus (MPa); (<b>b</b>) max principal stress (MPa); (<b>c</b>) AE events (blue: Tensile failure; red: Shear failure).</p>
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<p>Simulation results of the model containing microcracks and structural planes in the case of α = 50°: (<b>a</b>) Elastic modulus (MPa); (<b>b</b>) max principal stress (MPa); (<b>c</b>) AE events (blue: Tensile failure; red: Shear failure).</p>
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<p>Simulation results of the model containing microcracks and structural planes in the case of <span class="html-italic">α</span> = 90°: (<b>a</b>) Elastic modulus (MPa); (<b>b</b>) max principal stress (MPa); (<b>c</b>) AE events (blue: Tensile failure; red: Shear failure).</p>
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<p>Numerical simulation results of the model without microcracks and structural planes: (<b>a</b>) Elastic modulus (MPa); (<b>b</b>) max principal stress (MPa); (<b>c</b>) AE events (blue: Tensile failure; red: Shear failure).</p>
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<p>Numerical simulation results of the generalized model: (<b>a</b>) Elastic modulus (MPa); (<b>b</b>) max principal stress (MPa); (<b>c</b>) AE events (blue: Tensile failure; red: Shear failure).</p>
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<p>Stress-strain curves of numerical models: (<b>a</b>) The model containing microcracks and structural planes; (<b>b</b>) the model without microcracks and structural planes; (<b>c</b>) the generalized model.</p>
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<p>Anisotropy of mechanical parameters: (<b>a</b>) Peak strength; (<b>b</b>) elastic modulus; (<b>c</b>) axial strain corresponding to peak strength; (<b>d</b>) lateral strain corresponding to peak strength; (<b>e</b>) Poisson’s ratio.</p>
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<p>Anisotropy of mechanical parameters: (<b>a</b>) Peak strength; (<b>b</b>) elastic modulus; (<b>c</b>) axial strain corresponding to peak strength; (<b>d</b>) lateral strain corresponding to peak strength; (<b>e</b>) Poisson’s ratio.</p>
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<p>Failure modes of columnar jointed basalt during in situ testing [<a href="#B12-energies-12-03602" class="html-bibr">12</a>]: (<b>a</b>) Preparation of in situ test specimens; (<b>b</b>) the overall failure mode of CJRM during in situ testing; (<b>c</b>) failure characteristics of transverse section; (<b>d</b>) failure characteristics of the longitudinal section.</p>
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<p>Failure modes of columnar jointed basalt during in situ testing [<a href="#B12-energies-12-03602" class="html-bibr">12</a>]: (<b>a</b>) Preparation of in situ test specimens; (<b>b</b>) the overall failure mode of CJRM during in situ testing; (<b>c</b>) failure characteristics of transverse section; (<b>d</b>) failure characteristics of the longitudinal section.</p>
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<p>Failure modes of reconstructed CJRM specimens with different dips angles [<a href="#B33-energies-12-03602" class="html-bibr">33</a>].</p>
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<p>The failure mode of CJRM obtained by 3 dimensional distinct element code (3DEC) modeling [<a href="#B12-energies-12-03602" class="html-bibr">12</a>] (N for normal joint displacement, S for joint shear displacement).</p>
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15 pages, 12121 KiB  
Article
Generation of Numerical Models of Anisotropic Columnar Jointed Rock Mass Using Modified Centroidal Voronoi Diagrams
by Qingxiang Meng, Long Yan, Yulong Chen and Qiang Zhang
Symmetry 2018, 10(11), 618; https://doi.org/10.3390/sym10110618 - 9 Nov 2018
Cited by 33 | Viewed by 7586
Abstract
A columnar joint network is a natural fracture pattern with high symmetry, which leads to the anisotropy mechanical property of columnar basalt. For a better understanding the mechanical behavior, a novel modeling method for columnar jointed rock mass through field investigation is proposed [...] Read more.
A columnar joint network is a natural fracture pattern with high symmetry, which leads to the anisotropy mechanical property of columnar basalt. For a better understanding the mechanical behavior, a novel modeling method for columnar jointed rock mass through field investigation is proposed in this paper. Natural columnar jointed networks lies between random and centroidal Voronoi tessellations. This heterogeneity of columnar cells in shape and area can be represented using the coefficient of variation, which can be easily estimated. Using the bisection method, a modified Lloyd’s algorithm is proposed to generate a Voronoi diagram with a specified coefficient of variation. Modelling of the columnar jointed rock mass using six parameters is then presented. A case study of columnar basalt at Baihetan Dam is performed to demonstrate the feasibility of this method. The results show that this method is applicable in the modeling of columnar jointed rock mass as well as similar polycrystalline materials. Full article
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<p>Typical columnar jointed rock masses: (<b>a</b>) Devil’s Postpile, Yosemite in California; (<b>b</b>) Fingal’s Cave, Staffa in Scotland; and (<b>c</b>) Giant’s Causeway, Antrim in Northern Ireland.</p>
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<p>A colorized map of about 200 columns at the Giant’s Causeway made by O’Reilly in 1879 [<a href="#B1-symmetry-10-00618" class="html-bibr">1</a>,<a href="#B21-symmetry-10-00618" class="html-bibr">21</a>].</p>
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<p>Diagrammatic drawing of the joints in columnar basalt: (<b>a</b>) vertical columnar joint, and (<b>b</b>) horizontal transverse joint.</p>
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<p>An illustration of Voronoi tessellation with 10 generators: (<b>a</b>) a set of generators, and (<b>b</b>) the resulting Voronoi diagram.</p>
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<p>Illustration of the stages of the constrained Voronoi tessellation algorithm: (<b>a</b>) initial seeds, (<b>b</b>) classical Voronoi tessellation, (<b>c</b>) open Voronoi cells, (<b>d</b>) symmetry seeds, (<b>e</b>) new Voronoi tessellation, (<b>f</b>) constrained Voronoi diagram.</p>
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<p>Illustration of the stages of the constrained Voronoi tessellation algorithm: (<b>a</b>) initial seeds, (<b>b</b>) classical Voronoi tessellation, (<b>c</b>) open Voronoi cells, (<b>d</b>) symmetry seeds, (<b>e</b>) new Voronoi tessellation, (<b>f</b>) constrained Voronoi diagram.</p>
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<p>Random and centroidal Voronoi diagram: generators (white dots) and centroids (black dots): (<b>a</b>) random Voronoi diagram, and (<b>b</b>) centroidal Voronoi diagram.</p>
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<p>Discretization of n-polygon: (<b>a</b>) polygon with n vertices, and (<b>b</b>) <span class="html-italic">n</span> − 2 discrete triangles.</p>
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<p>Illustration of the sampling method: (<b>a</b>) Voronoi tessellation, and (<b>b</b>) random sampling points.</p>
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<p>Implementation of the constrained centroidal Voronoi algorithm: (<b>a</b>) initial condition, (<b>b</b>) step 5, (<b>c</b>) step 20, (<b>d</b>) step 50, (<b>e</b>) change of energy, and (<b>f</b>) change of coefficient of variation.</p>
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<p>Procedure for modelling columnar joints.</p>
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<p>Site conditions of the Baihetan Hydropower Station: (<b>a</b>) construction site, and (<b>b</b>) typical columnar jointed basalt.</p>
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<p>Typical <span class="html-italic">P</span><sub>2</sub><span class="html-italic">β</span><sub>3</sub> columnar basalt at Baihetan Hydropower Station: (<b>a</b>) geological photo, and (<b>b</b>) joints skeleton.</p>
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<p>Numerical specimen generation: (<b>a</b>) Voronoi diagram with CV = 56.45%, (<b>b</b>) Voronoi diagram with CV = 44.18%, (<b>c</b>) extrude 2-D Voronoi diagram with direction, (<b>d</b>) cut columnar rock with transverse joint, (<b>e</b>) block model of columnar rock, and (<b>f</b>) particle model of columnar rock.</p>
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<p>Columnar joints with different coefficient of variation: (<b>a</b>) block model with CV = 40%, (<b>b</b>) block model with CV = 30%, (<b>c</b>) block model with CV = 20%, (<b>d</b>) block model with C = 10%, (<b>e</b>) particle model with CV = 40%, (<b>f</b>) particle model with CV = 30%, (<b>g</b>) particle model with CV = 20%, and (<b>h</b>) particle model with CV = 10%.</p>
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<p>Columnar joints with different joint dip angles: (<b>a</b>) block model with dip = 0°, (<b>b</b>) block model with dip = 30°, (<b>c</b>) block model with dip = 60°, (<b>d</b>) block model with dip = 90°, (<b>e</b>) particle model with dip = 0°, (<b>f</b>) particle model with dip = 30°, (<b>g</b>) particle model with dip = 60°, and (<b>h</b>) particle model with dip = 90°.</p>
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