Open-pit slopes contain numerous nonpenetrating, intermittent joints which maintain stability under blasting operations. The tip dynamic response coefficient (DRC) of parallel cracks in a typical rock mass under combined dynamic and static loading conditions was calculated in this study based on the superposition principle. The dynamic response law of the intermittent joint in the slope under blasting was determined accordingly. The influence of many factors (the disturbance amplitude of dynamic load, the lateral confining pressure, the length of rock bridge, the length between cracks, the staggered distance between cracks, and the crack inclination angle) on the dynamic response was theoretically analyzed as well. The ABAQUS numerical assessments were conducted on simulation models with parallel cracks under combined dynamic and static loading conditions. The results show that a larger dynamic load amplitude and smaller crack inclination angle/confining pressure result in greater Type II dynamic strengthening effect on the crack tip. When the length of the rock bridge between cracks (
The fracture failure characteristics of the deep rock mass in an open-pit mine slope are different from those of the shallow rock mass under blasting conditions. Deep rock mass failure is the result of the joint action of high ground stress and explosion load. The role of ground stress, in particular, is very important [
There have been many other valuable contributions to the literature. Zhou and Chen [
However, there have been few previous studies on the propagation and evolution of parallel cracks and the dynamic response between cracks under dynamic and static loading conditions. Additionally, the research methods on the propagation law of cracks under combined dynamic and static loading are mainly focused on rock specimen testing and numerical simulation while the relevant theoretical quantitative researches are yet lacking. There is no fully feasible, effective, straightforward methods for calculating the stress intensity factor (SIF) at the crack tip of rock masses with multiple cracks. The dynamic response of parallel cracks under the combined action of dynamic and static loads was investigated in this study based on the superposition principle. Variations in the self-defined dynamic response coefficient (DRC) of the crack tip were investigated in detail as per the disturbance amplitude of dynamic load, the lateral confining pressure, the length of rock bridge, the length between cracks, the staggered distance between cracks, and the crack inclination angle. A numerical dynamic/static load crack propagation simulation was conducted in ABAQUS, and the results were compared against the theoretical results to validate them.
Three failure forms may occur in the rock mass under the combined action of in situ stress and explosion stress: shear failure, tensile crack failure, and compression-shear failure, where the failure is dominated by both I-II tension-shear mixed mode failure (when the stress acted on the crack surface is tensile stress) and I-II compression-shear mixed mode failure (when the stress acted on the crack surface is compression stress) along the structural plane [
Figure
Sketch of mechanical models of “key block” in the slope with intermittent joints.
As shown in Figure
According to the superposition principle of fracture mechanics theory, the total stress field caused by two or more different loading systems near the crack tip can be obtained by the algebra of each SIF on the basis of linear elastic mechanics under the same boundary conditions [
Superposition method of stress state of rock mass with double cracks under dynamic and static loading conditions.
The key block in the jointed rock mass of the slope can be regarded as an infinite plate with parallel double cracks [
Equivalent sketch of stress State B under biaxial compression.
According to the stress circle theory in material mechanics, the stress State B of the rock mass falls under biaxial compression [
According to the superposition principle, the stress State B1 of the rock mass under biaxial compression can then be decomposed into the stress State B2 and the stress State B3 (Figure
Superposition sketch of stress State B1 under biaxial compression.
According to the stress-circle theory, the vertical stress and shear stress acting on the crack surface in Figure
For the State B2 with vertical stress, the cracks are closed under compression, and there is only SIFs of Type II (
The stress State C of the rock mass under dynamic loading can be considered equivalent to the stress State C1, as shown in Figure
Equivalent sketch of stress State C under dynamic loading.
Superposition sketch of stress State C1 under dynamic loading.
According to references [
The dynamic SIFs at the crack tips can be obtained as based on Formulas (
It is assumed that the blasting dynamic load on the key block is a simple harmonic wave in the form of a sinusoidal function (Figure
Dynamic disturbance wave.
Substituting Formulas (
The effects of dynamic loading among multiple cracks on the SIF at the crack tip may be reinforcing, null, or shielding [
Due to space constraints, only the inner tip of Crack 1 (position (1) in Figure
Calculation results of
Time (s) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.1 | 1.1034 | 1.2069 | 1.3103 | 1.4138 | 1.5172 | 1.0974 | 1.1947 | 1.2921 | 1.3895 | 1.4868 |
0.3 | 1.3062 | 1.6124 | 1.9186 | 2.2248 | 2.5309 | 1.2882 | 1.5764 | 1.8646 | 2.1529 | 2.4411 |
0.5 | 1.4967 | 1.9934 | 2.4902 | 2.9869 | 3.4836 | 1.4676 | 1.9351 | 2.4027 | 2.8702 | 3.3378 |
0.7 | 1.6674 | 2.3349 | 3.0023 | 3.6697 | 4.3372 | 1.6283 | 2.2565 | 2.8848 | 3.5130 | 4.1413 |
0.9 | 1.8115 | 2.6230 | 3.4346 | 4.2461 | 5.0576 | 1.7639 | 2.5278 | 3.2916 | 4.0555 | 4.8194 |
1.1 | 1.9232 | 2.8464 | 3.7697 | 4.6929 | 5.6161 | 1.8690 | 2.7380 | 3.6071 | 4.4761 | 5.3451 |
1.3 | 1.9981 | 2.9962 | 3.9943 | 4.9923 | 5.9904 | 1.9395 | 2.8790 | 3.8185 | 4.7580 | 5.6975 |
1.5 | 2.0331 | 3.0663 | 4.0994 | 5.1325 | 6.1656 | 1.9725 | 2.9450 | 3.9174 | 4.8899 | 5.8624 |
1.7 | 2.0270 | 3.0539 | 4.0808 | 5.1078 | 6.1347 | 1.9667 | 2.9333 | 3.9000 | 4.8666 | 5.8333 |
1.9 | 1.9798 | 2.9596 | 3.9394 | 4.9192 | 5.8990 | 1.9223 | 2.8445 | 3.7668 | 4.6891 | 5.6114 |
2.1 | 1.8935 | 2.7871 | 3.6806 | 4.5742 | 5.4677 | 1.8411 | 2.6822 | 3.5233 | 4.3643 | 5.2054 |
2.3 | 1.7716 | 2.5433 | 3.3149 | 4.0866 | 4.8582 | 1.7263 | 2.4527 | 3.1790 | 3.9054 | 4.6317 |
2.5 | 1.6190 | 2.2379 | 2.8568 | 3.4758 | 4.0947 | 1.5826 | 2.1652 | 2.7478 | 3.3304 | 3.9131 |
2.7 | 1.4416 | 1.8831 | 2.3247 | 2.7662 | 3.2078 | 1.4156 | 1.8313 | 2.2469 | 2.6625 | 3.0782 |
2.9 | 1.2466 | 1.4931 | 1.7397 | 1.9862 | 2.2328 | 1.2321 | 1.4642 | 1.6962 | 1.9283 | 2.1604 |
3 | 1.1449 | 1.2897 | 1.4346 | 1.5794 | 1.7243 | 1.1364 | 1.2727 | 1.4091 | 1.5454 | 1.6818 |
As shown in Figure
To observe the DRC variations of Type II of the inner tip of Crack 1 with confining pressure (
Calculation results of
Time (s) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.1 | 1.0616 | 1.0775 | 1.1046 | 1.1606 | 1.3461 | 1.0599 | 1.0752 | 1.1011 | 1.1539 | 1.3222 |
0.3 | 1.1823 | 1.2294 | 1.3095 | 1.4754 | 2.0244 | 1.1774 | 1.2227 | 1.2991 | 1.4554 | 1.9538 |
0.5 | 1.2957 | 1.3722 | 1.5021 | 1.7711 | 2.6618 | 1.2878 | 1.3613 | 1.4853 | 1.7388 | 2.5473 |
0.7 | 1.3973 | 1.5001 | 1.6746 | 2.0362 | 3.2329 | 1.3867 | 1.4855 | 1.6521 | 1.9928 | 3.0791 |
0.9 | 1.4831 | 1.6081 | 1.8202 | 2.2599 | 3.7149 | 1.4701 | 1.5903 | 1.7928 | 2.2071 | 3.5279 |
1.1 | 1.5496 | 1.6918 | 1.9332 | 2.4333 | 4.0886 | 1.5349 | 1.6715 | 1.9020 | 2.3732 | 3.8759 |
1.3 | 1.5942 | 1.7479 | 2.0088 | 2.5495 | 4.3391 | 1.5782 | 1.7260 | 1.9751 | 2.4846 | 4.1091 |
1.5 | 1.6150 | 1.7741 | 2.0442 | 2.6039 | 4.4563 | 1.5985 | 1.7515 | 2.0093 | 2.5367 | 4.2182 |
1.7 | 1.6114 | 1.7695 | 2.0380 | 2.5943 | 4.4356 | 1.5949 | 1.7470 | 2.0033 | 2.5275 | 4.1990 |
1.9 | 1.5833 | 1.7342 | 1.9903 | 2.5211 | 4.2779 | 1.5676 | 1.7127 | 1.9572 | 2.4574 | 4.0521 |
2.1 | 1.5319 | 1.6695 | 1.9032 | 2.3872 | 3.9893 | 1.5177 | 1.6499 | 1.8730 | 2.3291 | 3.7834 |
2.3 | 1.4594 | 1.5782 | 1.7799 | 2.1979 | 3.5815 | 1.4470 | 1.5613 | 1.7539 | 2.1478 | 3.4037 |
2.5 | 1.3685 | 1.4638 | 1.6256 | 1.9609 | 3.0707 | 1.3586 | 1.4502 | 1.6047 | 1.9206 | 2.9280 |
2.7 | 1.2629 | 1.3309 | 1.4463 | 1.6855 | 2.4772 | 1.2558 | 1.3212 | 1.4314 | 1.6568 | 2.3755 |
2.9 | 1.1468 | 1.1847 | 1.2492 | 1.3828 | 1.8248 | 1.1428 | 1.1793 | 1.2409 | 1.3667 | 1.7680 |
3 | 1.0862 | 1.1085 | 1.1464 | 1.2249 | 1.4846 | 1.0839 | 1.1054 | 1.1415 | 1.2155 | 1.4512 |
As shown in Figure
The above analysis indicates that as the confining pressure of the key block located far away from the slope surface (“Y” in Figure
To observe the DRC variations of Type II of the inner tip of Crack 1 with the length ratio of the rock bridge (
Calculation results of
0.0 | 2.3261 | 2.6642 | 2.4720 | 0.15 | 1.7967 | 1.9443 | 1.8604 |
0.5 | 1.9568 | 2.1339 | 2.0332 | 0.2 | 1.7965 | 1.9440 | 1.8602 |
1.0 | 1.8387 | 1.9940 | 1.9057 | 0.25 | 1.7908 | 1.9372 | 1.8540 |
1.5 | 1.8340 | 1.9891 | 1.9003 | 0.3 | 1.7850 | 1.9304 | 1.8477 |
2.0 | 1.8292 | 1.9843 | 1.8950 | 0.35 | 1.7847 | 1.9299 | 1.8473 |
3.0 | 1.8246 | 1.9772 | 1.8905 | 0.4 | 1.7844 | 1.9296 | 1.8470 |
4.0 | 1.8288 | 1.9823 | 1.8951 | 0.45 | 1.7841 | 1.9293 | 1.8467 |
5.0 | 1.8331 | 1.9874 | 1.8997 | 0.75 | 1.8113 | 1.9615 | 1.8761 |
6.0 | 1.8355 | 1.9906 | 1.9027 | 1 | 1.8384 | 1.9937 | 1.9054 |
7.0 | 1.8379 | 1.9934 | 1.9052 | 1.5 | 1.9106 | 2.0792 | 1.9834 |
As shown in Figure
As shown in Figure
It appears that the staggered distance between the nonpenetrating joints and the length of the intermediate rock bridge in the actual slope have an important influence on the initiation of the crack tip and the penetration between the cracks during blasting operations. When the length of the rock bridge between the cracks is small and the staggered distance between cracks is large, the crack tip is more strongly affected by the dynamic load and cracks are more likely in general.
To observe the DRC variations of Type II of the inner tip of Crack 1 with the inclination angle (
Calculation results of
15 | 1.9002 | 2.0669 | 1.9721 | 1.7302 | 1.8969 | 1.8021 |
20 | 1.8840 | 2.0477 | 1.9547 | 1.7140 | 1.8777 | 1.7847 |
25 | 1.8744 | 2.0363 | 1.9443 | 1.7044 | 1.8663 | 1.7743 |
30 | 1.8680 | 2.0287 | 1.9373 | 1.6980 | 1.8587 | 1.7673 |
35 | 1.8633 | 2.0231 | 1.9322 | 1.6933 | 1.8531 | 1.7622 |
40 | 1.8596 | 2.0188 | 1.9283 | 1.6896 | 1.8488 | 1.7583 |
45 | 1.8567 | 2.0153 | 1.9252 | 1.6867 | 1.8453 | 1.7552 |
50 | 1.8543 | 2.0125 | 1.9225 | 1.6843 | 1.8425 | 1.7525 |
55 | 1.8522 | 2.0100 | 1.9203 | 1.6822 | 1.8400 | 1.7503 |
60 | 1.8504 | 2.0078 | 1.9183 | 1.6804 | 1.8378 | 1.7483 |
65 | 1.8487 | 2.0058 | 1.9165 | 1.6787 | 1.8358 | 1.7465 |
70 | 1.8472 | 2.0040 | 1.9149 | 1.6772 | 1.8340 | 1.7449 |
75 | 1.8458 | 2.0024 | 1.9133 | 1.6758 | 1.8324 | 1.7433 |
80 | 1.8444 | 2.0008 | 1.9119 | 1.6744 | 1.8308 | 1.7419 |
As shown in Figure
The above analysis indicates that a smaller inclination angle of the nonpenetrating joint in the slope makes the crack tip more significantly affected by the dynamic load, in which case cracks are generally more likely to occur.
To validate the theoretical analyses presented above, numerical simulations of 20 groups of calculation models (Table
Numerical models. (a) Model 1. (b) Model 5. (c) Model 13. (d) Model 15.
Geometric parameters of cracks in models.
Model number | Inclination angle ( | Length of Crack 1 ( | Length of Crack 2 ( | Length of rock bridge ( | Staggered distance ( | Lateral confining pressure ( |
---|---|---|---|---|---|---|
1 | 45 | 20 | 20 | 0 | 8 | 1 |
2 | 45 | 20 | 20 | 2 | 8 | 1 |
3 | 45 | 20 | 20 | 4 | 8 | 1 |
4 | 45 | 20 | 20 | 6 | 8 | 1 |
5 | 45 | 20 | 20 | 8 | 8 | 1 |
6 | 45 | 20 | 20 | 10 | 8 | 1 |
7 | 45 | 20 | 20 | 12 | 8 | 1 |
8 | 45 | 20 | 20 | 0 | 2 | 1 |
9 | 45 | 20 | 20 | 0 | 4 | 1 |
10 | 45 | 20 | 20 | 0 | 6 | 1 |
11 | 45 | 20 | 20 | 0 | 10 | 1 |
12 | 45 | 20 | 20 | 0 | 12 | 1 |
13 | 30 | 20 | 20 | 0 | 8 | 1 |
14 | 60 | 20 | 20 | 0 | 8 | 1 |
15 | 75 | 20 | 20 | 0 | 8 | 1 |
16 | 45 | 20 | 20 | 0 | 8 | 1.2 |
17 | 45 | 20 | 20 | 0 | 8 | 1.4 |
18 | 45 | 20 | 20 | 0 | 8 | 1.6 |
19 | 45 | 20 | 20 | 0 | 8 | 1.8 |
20 | 45 | 20 | 20 | 0 | 8 | 2 |
In the simulation, considering the effect of the original rock stress on the crack propagation in the slope, the static load was preapplied to simulate the initial stress field before the dynamic load was applied to the model. The multiload step technique in ABAQUS was used for this purpose. The stress application sequence and path are shown in Figure
Sketch of the calculation stress path.
Due to the space constraints, only the initiation and propagation processes of cracks in Models 1, 5, 13, and 15 under combined dynamic and static loading are shown here (Figures
Model 1 stress distribution in numerical simulation. (a)
Model 5 stress distribution in numerical simulation. (a)
Model 13 stress distribution in numerical simulation. (a)
Model 15 stress distribution in numerical simulation. (a)
As shown in Figures
The SIFs (
In order to validate the dynamic response law of crack propagation obtained by theoretical analysis (Sections
DRC calculation results and dynamic initiation SIFs (DISs) of crack tip.
Model number | SIF (State A) (MPa m1/2) | SIF (State B) (MPa m1/2) | SIF (State C) (MPa m1/2) | SIF (State D) (MPa m1/2) | DRC (State B) | DRC (State C) |
---|---|---|---|---|---|---|
1 | 3.21 | 7.76 | 8.05 | 9.16 | 2.42 | 2.51 |
2 | 3.24 | 7.33 | 7.56 | 9.27 | 2.26 | 2.33 |
3 | 3.29 | 7.22 | 7.42 | 9.42 | 2.19 | 2.25 |
4 | 3.38 | 7.08 | 7.35 | 9.59 | 2.11 | 2.19 |
5 | 3.41 | 6.73 | 7.14 | 9.71 | 1.98 | 2.10 |
6 | 3.44 | 6.47 | 6.95 | 9.88 | 1.87 | 2.01 |
7 | 3.48 | 6.42 | 6.80 | 9.91 | 1.85 | 1.96 |
8 | 3.25 | 7.15 | 7.35 | 9.23 | 2.20 | 2.26 |
9 | 3.28 | 7.08 | 7.28 | 9.33 | 2.16 | 2.22 |
10 | 3.23 | 7.20 | 7.46 | 9.25 | 2.23 | 2.31 |
11 | 3.18 | 7.92 | 8.20 | 9.07 | 2.49 | 2.58 |
12 | 3.15 | 7.88 | 8.13 | 9.02 | 2.50 | 2.58 |
13 | 3.14 | 7.91 | 8.26 | 8.95 | 2.52 | 2.63 |
14 | 3.33 | 7.59 | 7.89 | 9.47 | 2.28 | 2.37 |
15 | 3.42 | 7.22 | 7.63 | 9.89 | 2.11 | 2.23 |
16 | 3.21 | 7.51 | 7.93 | 9.22 | 2.34 | 2.47 |
17 | 3.28 | 7.38 | 7.87 | 9.35 | 2.25 | 2.40 |
18 | 3.34 | 7.05 | 7.45 | 9.51 | 2.11 | 2.23 |
19 | 3.41 | 6.92 | 7.40 | 9.76 | 2.03 | 2.17 |
20 | 3.52 | 6.84 | 7.23 | 10.07 | 1.94 | 2.05 |
State A: axial pressure and lateral static confining pressure both reach preset value before dynamic load is applied. State B: dynamic load lasts for 1.0 s (
DRC and dynamic initiation SIF variations in crack tip as per various factors. (a) Variations with length of rock bridge (
Figures
As shown in Figure
Suppose the number of elements of the comparison column
The interval relative valuing is used to dedimensionalize the matrices of
The elements in the dedimensionalized matrices of
Then, the calculation Formula of the correlation coefficient can be expressed as follows:
Select the change values of the influencing factors in Table
According to the Grey correlation theory, the sensitivity of DRC to influencing factors is in the following order:
The dynamic response laws of parallel cracks under combined dynamic and static loading conditions were analyzed in this study based on the multiple superposition principle and SIF calculations under fracture mechanics theory. A series of ABAQUS numerical simulations were performed to validate the theoretical analysis results. The conclusions can be summarized as follows.
The dynamic load amplitude, confining pressure, and crack inclination angle all markedly influence the crack dynamic response law. A larger load amplitude, smaller inclination angle, and smaller confining pressure result in greater dynamic strengthening effect of Type II on the crack tip. That is, crack initiation and propagation in the tip are significantly affected by dynamic loading The length of the rock bridge and the staggered distance between cracks also significantly influence the crack dynamic response law. When the length of the rock bridge between cracks is small ( The crack propagation under combined dynamic and static loading is the most sensitive to the lateral confining pressure ( The ABAQUS simulation produced results in close agreement with the theoretical analysis results, which validates the dynamic response laws of parallel double cracks under combined dynamic and static loading conditions reported in this paper.
As shown in Figures
Values of
Values of
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This research was supported by the National Key R&D Programs of China (grant nos. 2016YFC0801602 and 2017YFC1503103).