Fractal correction of dynamic fracture parameters of black sandstone under impact loads
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摘要: 基于分形理论研究了偏折裂纹扩展路径对动载荷作用下黑砂岩的动态断裂力学参数的测试误差影响作用，采用传统的分离式霍普金森压杆（split Hopkinson pressure bar, SHPB）实验装置对修正侧开单裂纹半孔板（improved single cleavage semi-circle specimen, ISCSC）试样进行动态冲击实验，随后采用裂纹扩展计进行裂纹起裂时间与裂纹扩展速度等动态断裂力学参数测试，采用分形理论对测试的裂纹扩展速度与动态应力强度因子进行修正，利用实验-数值法对黑砂岩的动态断裂韧度进行计算。研究结果表明，ISCSC构型构件能够有效应用于岩石材料动态裂纹扩展行为的研究，并发生了止裂现象，经分形修正的裂纹扩展速度与动态断裂韧度更接近实际裂纹动态扩展情况，修正前后得到黑砂岩材料的裂纹扩展速度误差为33.51%，动态断裂韧度最大误差为7.68%，说明利用分形理论对动态断裂韧度等动态断裂参数计算更合理。Abstract: When studying the dynamic fracture behavior of cracked rock mass, dynamic fracture toughness is an important mechanical parameter to study the fracture characteristics of cracks, which can accurately reflect the energy required in the crack initiation and propagation stage. However, compared with the static fracture problem, it is difficult to obtain an analytical solution for dynamic fracture toughness. Therefore, many scholars measure the crack propagation speed by using crack propagation gauges, and then calculate the dynamic fracture toughness according to the universal function. In this way, the crack propagation speed plays a leading role in the calculation accuracy, but in the experiment, the crack propagation speed cannot be measured accurately due to the measuring instrument. In this paper, the fractal theory is used to correct this error. According to the fractal theory, the effects of deflected crack propagation trajectories on dynamic fracture properties of black sandstone under impact loads were studied. A traditional modified split Hopkinson pressure bar (SHPB) test device was used to conduct a dynamic impact test by using an improved single cleavage semi-circle (ISCSC) specimen, crack propagation speed and other fracture mechanics parameters were measured using crack propagation gauge (CPG). Subsequently, the fractal theory was applied to correct dynamic crack propagation speed and dynamic stress intensity factor, and the dynamic fracture toughness of black sandstone was also calculated using the experimental-numerical method. The research results indicate that the ISCSC specimen can be effectively applied to study the crack arrest behavior of rock materials. Crack propagation speed and dynamic fracture toughness after fractal correction are closer to the actual dynamic crack propagation characteristics. Comparisons between before and after the correction, the maximum error of the crack propagation speed of black sandstone material is 33.51%, and the maximum error of dynamic fracture toughness is 7.68%, indicating that it is more reasonable to use fractal theory to calculate dynamic fracture parameters such as crack propagation speed and dynamic fracture toughness.
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图 4 试件裂纹扩展路径
Figure 4. Crack propagation trajectories of black sandstone ISCSC specimens
图 5 试件1中裂纹起裂、扩展、止裂阶段的应力波状态
Figure 5. Stress wave states during crack initiation, propagation and crack arrest in the specimen 1
图 11 裂纹扩展速度分形修正前后的比较
Figure 11. Comparison of crack propagation speeds before and after fractal correction
图 12 有限单元法模型示意图及位移外推法
Figure 12. Finite element method model and displacement extrapolation method
图 14 裂纹动态应力强度因子随裂纹扩展位移的变化
Figure 14. Crack dynamic stress intensity factors varied with displacement of propagation crack
图 15 修正动态扩展韧度随vc/cR的变化
Figure 15. Variation of corrected crack propagation toughness with vc/cR
表 1 分形修正前后的裂纹扩展速度和动态扩展韧度
Table 1. Crack propagation speeds and dynamic crack propagation toughnesses before and after fractal correction
样品 v0/(m·s−1) vc/(m·s−1) ev/% $ K_{\text{I}}^{\text{d}} $/(MPa·m1/2) $ {K_{{\text{I,c}}}^{\text{d}}} $/(MPa·m1/2) eK/% 1 723.56 781.25 7.97 3.956 3.760 4.94 476.19 518.27 8.84 4.355 4.236 2.72 400.00 502.40 25.60 4.412 4.137 6.22 357.14 388.54 8.79 4.681 4.597 1.81 400.00 423.04 5.76 4.420 4.359 1.38 294.12 310.91 5.71 4.828 4.783 0.91 238.10 296.06 24.35 4.827 4.681 3.03 434.78 504.68 16.08 4.613 4.412 4.37 555.56 638.65 14.96 4.399 4.140 5.89 357.14 394.39 10.43 4.681 4.581 2.15 312.50 376.07 20.34 4.296 4.144 3.54 263.16 317.04 20.47 4.462 4.334 2.87 322.58 335.76 4.09 4.473 4.440 0.73 212.77 246.41 15.81 4.403 4.327 1.71 128.21 158.57 23.68 4.589 4.523 1.45 400.00 429.20 7.30 4.412 4.335 1.75 200.00 229.67 14.83 4.610 4.541 1.50 555.56 594.27 6.97 4.399 4.279 2.72 384.62 486.32 26.44 4.452 4.181 6.09 344.83 434.55 26.02 4.267 4.046 5.17 151.52 164.63 8.66 4.538 4.510 0.63 50.25 53.93 7.33 5.258 5.242 0.30 2 625.00 653.59 4.57 4.230 4.138 2.16 588.24 668.14 13.58 4.298 4.046 5.86 370.37 428.20 15.61 4.794 4.532 5.48 588.24 692.50 17.72 3.907 3.606 7.68 400.00 471.78 17.95 4.132 3.953 4.34 285.71 317.43 11.10 4.521 4.443 1.72 555.56 635.54 14.40 4.399 4.150 5.66 250.00 283.37 13.35 4.607 4.527 1.75 76.92 100.54 30.71 4.917 4.634 5.76 2 370.37 417.47 12.72 4.444 4.322 2.76 123.46 153.54 24.37 4.624 4.558 1.43 370.37 429.51 15.97 4.312 4.162 3.47 625.00 678.86 8.62 4.183 4.012 4.10 714.29 792.18 10.91 4.087 3.816 6.62 232.56 310.49 33.51 4.492 4.309 4.07 625.00 700.12 12.02 3.910 3.685 5.74 588.24 635.83 8.09 3.899 3.764 3.47 256.41 290.27 13.21 4.752 4.667 1.79 555.56 649.17 16.85 4.066 3.795 6.64 322.58 353.43 9.56 4.321 4.246 1.73 263.16 293.77 11.63 4.620 4.545 1.63 217.39 272.11 25.17 4.393 4.269 2.81 333.33 395.52 18.65 4.319 4.167 3.53 156.25 170.84 9.34 4.740 4.706 0.71 312.50 342.41 9.57 4.345 4.273 1.66 86.96 100.09 38.11 4.659 4.574 1.82 65.79 80.47 37.51 5.637 5.584 0.93 8.73 9.60 9.94 5.346 5.344 0.05 4.98 5.40 8.46 4.762 4.761 0.02 
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