Study on Time-Dependent Closure Behavior of Rock Fractures Subject to Normal Stress
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摘要: 基于不同高径比微凸体的受压时效性试验，根据赫兹接触理论，得到了不同微凸体弹性模量随时间的衰减规律. 对红砂岩和石灰岩新鲜裂隙面开展了不同法向应力条件下的受压时变闭合试验，结合小波分析法、区域生长算法和参考面法，提出了一种岩石裂隙细观尺度微凸体形貌的识别方法，对比了试验前后微凸体数量、高度和高径比的差异. 建立了考虑微凸体间相互作用的接触力学模型，求解Boussinesq方程并考虑弹性模量随时间的衰减，对两种岩石裂隙开展了逐级增加应力条件下的受压时变闭合计算. 通过对比计算和试验得到的损伤面积和蠕变变形验证了模型的有效性，阐明了微凸体的应变、接触面积和接触应力随时间的演化规律，揭示了不同细观形貌特征的微凸体在裂隙受压时变闭合过程中的关键性作用.Abstract: This study conducted time-dependent compression tests on asperities with different height-to-radius ratios using ultra-hard gypsum. According to Hertz contact theory, the attenuation laws of the elastic modulus of different asperities over time were fitted. Time-dependent closure tests were performed on fresh fracture surfaces of red sandstone and limestone under varying normal stresses. By integrating wavelet analysis, region growth algorithms, and the reference surface method, a novel approach was developed for identifying the mesoscale asperity morphology of rock fractures, and compared the differences in the number, height, and height-to-radius ratio of asperities before and after the experiment. Utilizing Boussinesq's solution, an influence matrix was constructed to account for interactions between asperities. Based on the law of the elastic modulus decaying over time, enabling time-dependent closure calculations for different rock fractures under variable stress conditions. This approach precisely analyzes the temporal evolution of strain, contact area, and contact stress for individual asperity, with simulation results matching experimental data in terms of damage area and creep deformation. The study reveals the pivotal role of asperities with distinct mesoscale morphological features in the time-dependent closure process of rock fractures under compression.
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Key words:
- rock fracture /
- asperity /
- morphological characteristics /
- time-dependent closure /
- Hertzian contact /
- rock mechanics
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图 2 A-1~A-4在统一荷载下的受压时变试验
Fig. 2. The time-dependent compression experiments of A-1 to A-4 under an identical load
图 3 岩石裂隙受压时变闭合试验
a. 红砂岩受压时变闭合试验；b. 石灰岩受压时变闭合试验；c. 法向变形随时间的变化
Fig. 3. Experiments on time-dependent closure of rock fractures under compression
图 5 不同高径比的微凸体模型及识别效果（单位：mm）
Fig. 5. Asperities models and identification results of A-1~A-4(unit: mm)
图 6 不同岩石裂隙表面微凸体识别结果（单位：mm）
a.裂隙面微凸体识别效果（以红砂岩为例）；b.微凸体概化为半椭球体；c.分水岭方法的局限性（Wen et al.，2022）；d.红砂岩和石灰岩裂隙面微凸体分布图
Fig. 6. Identification results of asperities on rock fracture surfaces (unit: mm)
图 7 试验前裂隙面微凸体高度分布（单位：mm）
a.红砂岩裂隙；b.石灰岩裂隙
Fig. 7. Height distribution of asperities on the fracture surfaces of red sandstone and limestone (unit: mm)
图 8 试验前裂隙面微凸体高径比分布
a. 红砂岩裂隙；b.石灰岩裂隙
Fig. 8. Height-to-radius ratios distribution of asperities on the fracture surfaces of red sandstone and limestone
图 9 受压裂隙时效变形计算流程
Fig. 9. Calculation process for time-dependent fracture deformation subject to compression
图 10 红砂岩损伤区域
上行为刚达到某应力时的损伤区域；下行为达到某应力并蠕变72小时后的损伤区域；单位：mm
Fig. 10. Damage zone in sandstone
图 11 石灰岩损伤区域
上行为刚达到某应力时的损伤区域；下行为达到某应力并蠕变72 h后的损伤区域；单位：mm
Fig. 11. Damage zone in limestone
图 12 不同法向应力下裂隙面损伤面积演化趋势
a.红砂岩裂隙损伤面积演化趋势；b.石灰岩裂隙损伤面积演化趋势
Fig. 12. Evolution of fracture damage area under different normal stresses
图 13 数值和试验损伤区域对比
a. 红砂岩损伤区域；b. 石灰岩损伤区域；1为数值计算结果；2为试样三维扫描结果；3为试验后实拍照片
Fig. 13. Comparison of numerical similation and experimental damage areas
图 14 不同岩石裂隙法向变形-时间曲线的试验数据与数值结果对比
a.法向变形-时间曲线（红砂岩裂隙）；b.法向变形-时间曲线（石灰岩裂隙）
Fig. 14. Comparison of experimental data and numerical results of normal deformation-time curves of different rock fractures
图 15 红砂岩裂隙面微凸体的变形和受力特性
a. ε-t；b. ω-t；c. A-t；d. σn-t
Fig. 15. The The deformation characteristics mechanical properties of asperities on the fracture surface of red sandstone
图 16 石灰岩裂隙面微凸体变形和受力特性
a. ε-t；b. ω-t；c. A-t；d. σn-t
Fig. 16. The deformation characteristics mechanical properties of asperities on the fracture surface of limestone
图 17 试验后裂隙面微凸体高度分布（单位：mm）
a.红砂岩裂隙；b.石灰岩裂隙
Fig. 17. Height distribution of asperities on the fracture surfaces after experiment (unit: mm)
图 18 试验后裂隙面微凸体高径比分布
a.红砂岩裂隙；b.石灰岩裂隙
Fig. 18. Height-to-radius ratios distribution of asperities on the fracture surfaces after experiment
图 19 S-No.5和L-No.23的对比
a. ε-t；b. ω-t；c. A-t；d. σn-t
Fig. 19. The comparison between S-No. 5 and L-No. 23
图 20 S-No.5(a)和L-No.23(b)的位置（用浅蓝色标识）
Fig. 20. The position of S-No.5(a) and L-No.23(b) (marked with blue)
表 1 弹性模量衰减拟合参数
Table 1. Parameters of different types of rock samples
A(103 GPa) B(103 GPa) R2 A-1 1.718 3.152 0.933 A-2 8.509 6.463 0.997 A-3 22.916 13.250 0.995 
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表 2 不同类型岩石试样参数
Table 2. Parameters of different types of rock samples
名称 密度(g/cm3) 单轴抗压强度(MPa) 泊松比 弹性模量(GPa) 抗拉强度(MPa) 硬度(MPa) 相对硬度 红砂岩 2.5 45.8 0.30 26.8 3.1 29.5 0.02 石灰岩 2.7 190.0 0.25 35.5 12 74.7 0.04
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