Calculating changes in fractal dimension of surface cracks to quantify how the dynamic loading rate affects rock failure in deep mining

Abstract

The split-Hopkinson pressure bar (SHPB) and digital image correlation (DIC) techniques are combined to analyze the dynamic compressive failure process of coal samples, and the box fractal dimension is used to quantitatively analyze the dynamic changes in the coal sample cracks under impact load conditions with different loading rates. The experimental results show that the fractal dimension can quantitatively describe the evolution process of coal fractures under dynamic load. During the dynamic compression process, the evolution of the coal sample cracks presents distinct stages. In the crack propagation stage, the fractal dimension increases rapidly with the progress of loading, and in the crack widening stage, the fractal dimension increases slowly with the progress of loading. The initiation of the crack propagation phase of the coal samples gradually occurs more quickly with increasing loading rate; the initial cracks appear earlier. At the same loading time point, when the loading rate is greater, the fractal dimension of the cracks observed in the coal sample is greater.

Keywords

fractal dimension / loading rate / impact load / coal crack

References

[1] MandelbrotB BThe fractal geometry of nature [M], 1982, New York, W H Freeman

[2] CampbellP, AbhyankarS. Fractals form chance and dimension [J]. Mathematical Intelligencer, 1978, 1(1): 35-37

[3] XieH-P. Fractal kinematics of crack propagation in geomaterials [J]. Journal of China University of Mining & Technology, 1995, 5(1): 1-8(in Chinese)

[4] JiangY-D, PanY-S, JiangF-X, DouL-M, JuY. State of the art review on mechanism and prevention of coal bumps in China [J]. Journal of China Coal Society, 2014, 39(2): 205-213(in Chinese)

[5] DouL-M, JiangY-D, CaoA-Y, LiuH-S, GongS-Y, CaiW, ZhuG-A. Monitoring and pre-warning of rockburst hazard with technology of stress field and wave field in underground coalmines [J]. Chinese Journal of Rock Mechanics and Engineering, 2017, 36(4): 803-811(in Chinese)

[6] AhamedM A A, PereraM S A, LiD-Y, RanjithP G, MatthaiS K. Proppant damage mechanisms in coal seam reservoirs during the hydraulic fracturing process: A review [J]. Fuel, 2019, 253: 615-629

[7] KiyamaT, NishimotoS, FujiokaM, XueZ-Q, IshijimaY, PanZ-J, ConnellD L. Coal swelling strain and permeability change with injecting liquid/supercritical CO₂ and N₂ at stress- constrained conditions [J]. International Journal of Coal Geology, 2011, 85(1): 56-64

[8] XuJ, SuX-P, ChengL-C, WangL, LiuJ, FengD. Evolution characteristics of meso-crack of gas-filled raw coal under compression-shear stress [J]. Chinese Journal of Rock Mechanics and Engineering, 2014, 33(3): 458-467(in Chinese)

[9] ZhaoY-X, LiuS-M, ZhaoG-F, DerekE, JiangY-D, HanJ-L. Failure mechanisms in coal: Dependence on strain rate and microstructure [J]. Journal of Geophysical Research Solid Earth, 2015, 119(9): 6924-6935

[10] LiZ-T, LiuD-M, CaiY-D, RanjithP G, YaoY-B. Multi-scale quantitative characterization of 3-D pore-fracture networks in bituminous and anthracite coals using FIB-SEM tomography and X-ray μ-CT [J]. Fuel, 2017, 209: 43-53

[11] GongF-Q, YeH, LuoY. Rate effect on the burst tendency of coal-rock combined body under low loading rate range [J]. Journal of China Coal Society, 2017, 42(11): 2852-2860(in Chinese)

[12] GongF-Q, SiX-F, LiX-B, WangS-Y. Dynamic triaxial compression tests on sandstone at high strain rates and low confining pressures with split Hopkinson pressure bar [J]. International Journal of Rock Mechanics and Mining Sciences, 2019, 113: 211-219

[13] GongF-Q, ZhaoG-F. Dynamic indirect tensile strength of sandstone under different loading rates [J]. Rock Mechanics & Rock Engineering, 2013, 47(6): 2271-2278

[14] KlepaczkoJ R, BassimM N, HsuT. Fracture toughness of coal under quasi-static and impact loading [J]. Engineering Fracture Mechanics, 1984, 19(2): 305-316

[15] ZipfR K, BieniawskiZ T. Mixed-mode fracture toughness testing of coal [J]. International Journal of Rock Mechanics & Mining Sciences & Geomechanics Abstracts, 1990, 27(6): 479-493

[16] ShanR-L, ChengR-Q, GaoW-J. Study on dynamic constitutive model of anthracite of Yunjialing coal mine [J]. Chinese Journal of Rock Mechanics and Engineering, 2006, 25(11): 2258-2263(in Chinese)

[17] LundbergB. A split Hopkinson bar study of energy absorption in dynamic rock fragmentation [J]. International Journal of Rock Mechanics & Mining Sciences & Geomechanics Abstracts, 1976, 13(6): 187-197

[18] ZhaoY-X, GongS, ZhangC-G, ZhangZ-N, JiangY-D. Fractals characteristics of crack propagation in coal under impact loading [J]. Fractals-complex Geometry Patterns & Scaling in Nature & Society, 2018, 26(2): 1840014

[19] ZhaoY-X, GongS, HaoX-J, PengY, JiangY-D. Effects of loading rate and bedding on the dynamic fracture toughness of coal: Laboratory experiments [J]. Engineering Fracture Mechanics, 2017, 178: 375-391

[20] XiaK W, AresJ R, HirooK, JamesR R. Laboratory earthquakes along inhomogeneous faults: Directionality and supershear [J]. Science, 2005, 308(5722): 681-684

[21] XiaK W, AresJ R, HirooK, JamesR R. Laboratory earthquakes: The sub-Rayleigh-to-supershear rupture transition [J]. Science, 2004, 303(5665): 1859-1861

[22] ZhangX-H, HaoH, MaG-W. Laboratory test and numerical simulation of laminated glass window vulnerability to debris impact [J]. International Journal of Impact, 2013, 5549-62

[23] ZhangX-H, HaoH, MaG-W. Dynamic material model of annealed soda-lime glass [J]. International Journal of Impact, 2015, 77: 108-119

[24] ZhaoH-B, WangZ-W, ZhangH, LiW. Effects of dynamic loads on development of internal microstructure and distribution of new surface fractures of coal [J]. Chinese Journal of Rock Mechanics and Engineering, 2016, 35(5): 971-979(in Chinese)

[25] HanX-H, LiC-W, XingT-Z, GaoP-B. Experimental study on coal impact fracturing based on temporal and spatial evolution of deformation field [J]. Journal of China Coal Society, 2016, 41(11): 2743-2755(in Chinese)

[26] LiuX-H, DaiF, LiuJ-F, ZhangR. Brazilian splitting tests on coal rock considering bedding direction under static and dynamic loading rate [J]. Chinese Journal of Rock Mechanics and Engineering, 2015, 10: 2098-2105(in Chinese)

[27] AiD-H, ZhaoY-C, WangQ-F, LiC-W. Experimental and numerical investigation of crack propagation and dynamic properties of rock in SHPB indirect tension test [J]. International Journal of Impact Engineering, 2019, 126(2019): 135-146

[28] XieH-PFractal: An introduction to rock mechanics [M], 1996, Beijing, Science Press

[29] MandelbrotB B. How long is the coast of Britain? statistical self-similarity and fractional dimension [J]. Science, 1967, 156(3775): 636-638

[30] RolphS. Fractal geometry: Mathematical foundations and applications [J]. Mathematical Gazette, 1990, 74(469): 288-317

[31] XingH Z, ZhangQ B, BrathwaiteC H, PanB, ZhaoJ. High-speed photography and digital optical measurement techniques for geomaterials: Fundamentals and applications [J]. Rock Mechanics & Rock Engineering, 2017, 50(6): 1611-1659

[32] HuangS, XiaK-W, YanF, FengX-T. An experimental study of the rate dependence of tensile strength softening of long you sandstone [J]. Rock Mechanics & Rock Engineering, 2010, 43(6): 677-683

[33] DaiF, HuangS, XiaK-W, TanZ-Y. Some fundamental issues in dynamic compression and tension tests of rocks using split Hopkinson pressure bar [J]. Rock Mechanics & Rock Engineering, 2010, 43(6): 657-666

[34] XiaK-W, YaoW, WuB-B. Dynamic rock tensile strengths of Laurentian granite: Experimental observation and micromechanical model [J]. Journal of Rock Mechanics & Geotechnical, 2017, 9(1): 116-124

[35] XuP, YangR-S, JuY, XiaK-W, GuoY. Experimental study on influences of inclined weak interface on the dynamic crack propagation behavior [J]. Chinese Journal of Geotechnical Engineering, 2019, 41(9): 1645-1652(in Chinese)

[36] GradyD, KippM E. Continuum modelling of explosive fracture in oil shale [J]. International Journal of Rock Mechanics & Mining Sciences & Geomechanics Abstracts, 1980, 17(3): 147-157

[37] NickB. Non-linear shear strength for rock, rock joints, rockfill and interfaces [J]. Innovative Infrastructure Solutions, 2016, 302370

[38] GongF-Q, LuD-H, LiX-B, RaoQ-H. Experimental research of sandstone dynamic strength criterion under different strain rates [J]. Rock and Soil Mechanics, 2013, 34(9): 2433-2441(in Chinese)

[39] LiuX-H, ZhangR, LiuJ-F. Dynamic test study of coal rock under different strain rates [J]. Journal of China Coal Society, 2012, 37(9): 1528-1534(in Chinese)