Unloading effect of the shear resistance of rock joints

To investigate the stress path dependent of rock joints, a series of tangential loading tests and normal unloading tests were conducted using 32 cement mortar replicas of artificially split rock joints with 4 different joint roughness coefficients (JRC). The critical normal stresses of the unloading tests were recorded, and a comparative study was conducted by referencing the results under different loading paths and joints conditions. The test results indicate that the shear resistance has a distinct unloading effect, which is mainly influenced by the morphologic characteristics, normal stress levels and stress paths. By analyzing the results, it can be concluded that the variation trend of shear/normal stress ratio against the normal stress and JRC of the two test conditions were identical. However, under low normal stress condition, the stress ratio of the joints under normal unloading stress path is the higher one; while under higher normal stress, the relationship becomes converse. Compared to that of the tangential loading condition, shear/normal stress ratio of the unloading stress path reduces rapidly as the increasing of normal stress, and the influence of the morphology is masked under lower normal stress.


Introduction
A natural rock mass contains numerous structural planes from faults, joints to microcracks. These structural planes of different sizes can greatly affect the stability of natural or engineered rock structures (Barton 2013). Due to the control role on the safety of rock engineering, the mechanical performance of rock joints is one of the hottest topics in field of rock mechanics and rock engineering.
In the excavation of underground engineering, the stress of specific direction will be unloaded. The reduction in the normal stress of the structural plane may easily cause serious engineering disasters, such as unloading splitting of wall rock and structural controlled rockburst (Pusch and Standfors 1992;Manouchehrian and Cai 2015). Particularly, in situations involving large depths and high initial in situ stress, the structural planes are weak planes that control the failure forms and the damage severity of the wall rock (Liu et al. 2016). Estimating the shear resistance of natural rock discontinuities under unloading circumstance is crucial for the stability analysis and support design in rock engineering.
The rockburst accident of the diversion tunnel of Jinping II hydropower station is an example of such disasters. Owing to the unloading of the normal stress of a rigid fault, the stress redistribution during chamber excavation led to a serious tectonic rockburst and great economic loss (Zhang et al. 2012(Zhang et al. , 2013. In the case of the Baihetan underground caverns, the contact shear slip of highly stressed staggered zones is one of the main failure mode under conditions of excavation unloading disturbances (Duan et al. 2017).
According to literatures (Zhou et al. 2016Meng et al. 2019;Han et al. 2020;Hu et al. 2020;Li et al. 2020aLi et al. , 2020b, the shear resistance and stability of rock mass discontinuity is mainly controlled by three factors, namely, the morphology of discontinuity, the normal stress of the joint plane and strength characteristics of materials. Considering that the basic friction parameters of materials and asperities before losing shear stability is commonly assumed to be constant, the shear performance of a structural plane or rock joint is governed by the coupling between the normal 1 3 292 Page 2 of 10 and shear stresses. A higher normal stress is favorable for improving the shear capacity of a structural plane. When a sufficiently high normal stress is applied, the joint maintains its shear stability. However, if the normal stress is unloaded to a relatively low level, the rock joint that is subjected to shear may fails. Lau and Chandler (2004) considered that using the mechanical parameters of rock mass under the unloading path is more accurate in engineering practice. Experiments and in situ observations have already indicated that the mechanical and deformation performance of rock mass is obviously influenced by the stress path (You 2002;Diederichs et al. 2004;Li et al. 2013Li et al. , 2019Fan et al. 2018). At the same time, the unloading effect on the rock joints' shear resistance has not been well investigated. The major of the existing researches on the behavior of the rock discontinuities focused on the peak shear resistance under tangential loading conditions. Literatures devoted on the unloading effect of rock joints or jointed rocks are relatively rare, and most of the works are qualitative. Zhu and Huang (2019) investigated the shear mechanics characteristics of rock under unloading conditions, and described the relationship between the initial normal stress and the failure normal stress using the exponential model. Zhou et al. (2003) explored the mechanical characteristics of a fractured rock mass under loading and unloading conditions, proposed the stress intensity factors of each stage and analyzed the similarities and differences of stress intensity factors under loading and unloading conditions. Huang et al. (2020) studied the shear mechanical properties of rock mass with single prefabricated crack during unloading, established five failure modes of fractured rock mass through analysis of crack propagation and mechanical properties. To understand the fracture mechanism of rock mass, Hanson et al. (2004) simulated the closed crack process under unloading condition. Zhang et al. (2009) explored the change in the mechanical properties of structural planes in the unloading state, and they found that the climbing effect is more sufficient due to normal unloading. Through in situ investigation, Yin et al. (2021) studied the shear behavior of fractured rock mass under different initial shear stress and normal stress unloading and found that high shear stress promotes dilatancy deformation, while high normal stress inhibits dilatancy deformation. Zhu et al. (2020) studied the shear mechanical properties of rock mass with parallel pre-faults during unloading, and found that the fracture surface of unloading test is less damaged than that of direct shear test. Chen et al. (2018) experimentally investigated the influence of jointed rock bridge on the bearing capacity and failure mode of rock slopes under excavation unloading and found that the unloading failure is more brittle compared with uniaxial loading. Through in situ investigation, Li et al. (2017) investigated the failure process of the surrounding rock during excavation and found that the unloading results from excavation-induced numerous microseismic events. Tu and Deng (2021) through the geological survey of the Southwest River Valley, considered that the unloading effect caused by the rapid river cutting affected the deep collapse failure of the rock mass, and proposed an empirical method that estimated tensile unloading depths of a rock masses.
Up to now, the influence and mechanism of normal unloading on the shear performance of rock joints are still unclear. In this study, a total 32 replicas of artificially split rock joints were casted and tested by direct shear apparatus. Among them, 16 samples were tested under tangential loading stress path, and the rest 16 samples, were tested under normal stress unloading stress path. A comparative study of the tangential loading (with fixed normal stress levels) and normal unloading tests (with fixed tangential stress levels) was conducted. The experiment results revealed a considerable stress path effect of the shear performance of rock joints.

Sample preparation
Four groups of structural planes, namely, S 1 to S 4 , were prepared through splitting tests of granite cuboids. To ensure the repeatability, cement mortar was used as the analogue material for rocks.
The joints samples were prepared following the next procedure. First, model gypsum was poured on one of the surface of the prototype joint. As the gypsum solidified, the characteristics of the surface were reproduced. Thereafter, cement mortar was poured on the gypsum moulds to reproduce the replicas of the surfaces of prototype rock joints. This procedure was repeated until sufficient replicas were produced. The fabrication and quality evaluation process of the replicas of structural planes is shown in Fig. 1.
The cement mortar samples were cast with the mix ratio of 1:2.3:5.0 by weight of water, cement and sand. In total, 32 artificial joints of dimensions 200 × 100 × 100 mm were casted. At the same times, standard cubic samples (100 × 100 × 100 mm) also were prepared for determining the basic material parameters.
The results of the splitting test and the uniaxial compression test indicated that the average tension strength and compression strength of the analogue material were 2.10 MPa and 40.82 MPa, respectively. The basic friction angle of rock joints was determined as 32° by a tilt test. First, place structural planes into the groove. Then observe whether the upper block slides down within 10 s by raising the slope to a certain height. If the upper block slides, record the angle of the slope. Otherwise, continue to raise slowly until the upper surface slides down (Fig. 2).
To examine the accuracy of the replicas, a fidelity analysis was conducted (Jiang et al. 2018). Five cement Page 3 of 10 292 mortar samples were randomly selected, and the morphological features of the structural surfaces were scanned by a 3D scanner. The scanning accuracy is 0.2 mm. Then, the morphological characteristics of the original surface of the rock joints and its cement mortar replicas were compared using 3D point cloud processing software Cloudcompare. The fidelity analysis results showed that the average fidelity of the replicas exceeds 98.4%, indicating a very good reproduction quality and uniformity.
The quantification of surface roughness has taken many different forms (Magsipoc et al. 2020). The joint roughness coefficient (JRC) is a widely used two-dimensional surface roughness characterisation parameter which provides a simple and objective index of surface morphology (Barton 1978). In this study, the JRC was determined from the average value of ten evenly spaced profiles of the surface (Fig. 3). The root mean square of the slope, proposed by Tes and Cruden (1979), was adopted to calculate the roughness values of the four groups of the joints replicas. The calculated JRCs are listed in the following Table 1.

Test procedure
The tests were carried out on a microcomputer controlled electronic rock direct shear apparatus. The apparatus comprises a loading device, a microcomputer servo control system, a stress-strain measuring device, etc. The test equipment could be controlled in the force or displacement mode. The tangential loading capacity is 1000 kN, and the normal loading capacity is 300 kN.
In the conventional direct shear tests, the normal load was controlled by force, while the tangential load was controlled by displacement. and the tangential loading speed was 0.02 mm/s. The tangential load was loaded several seconds after the normal load reached the target value. Four levels of normal stress-1.0, 3.0, 5.0 and 10.0 MPawere considered in the shear stress loading tests.
In the normal stress unloading tests, both the normal and tangential loads were controlled by force. First, the normal and tangential stresses were loaded to the predetermined value successively, and then, the normal stress was unloaded step by step. During the unloading process, only the normal load was reduced, while the tangential load force was maintained constant. The specimen was considered to have failed when the tangential force decreased or the tangential displacement increased rapidly. The test was terminated when the joint failed.
The tangential loading speed was 0.2 kN/s, and the target value of the tangential load varied with the initial normal stress. After the tangential load reached the target value for a certain period of time, normal unloading was performed in steps of − 0.4 kN/s. Under each normal stress level, the difference in the normal unloading steps and the holding time after reaching the target value were identical. The conceptual stress path of the samples is shown in Fig. 4.
The initial shear force (or stress) of the unloading test was selected according to the shear resistance peak of the conventional loading test. To avoid joint failures before the unloading procedure starts, the initial shear stress was set as 60-80% of the peak value of shear resistance determined according to the results of loading tests. The difference of the normal stress at each level was selected according to the initial normal stress. The parameters of the normal stress unloading tests are given in Table 2.
During the test, the time, normal force, normal deformation, normal displacement, tangential force and tangential displacement were collected by displacement sensors and servo control system.
When the normal stress equals to 10.0 MPa, the total tangential loading time and normal unloading time were relatively long, and the structural plane tended to be damaged during the pre-loading stage. Therefore, the holding time of each step unloading was set relatively small.

Results of shear tests under the normal stress unloading condition
The typical stress-time curves and displacement-time curves obtained from the conducted normal stress unloading tests are given in Fig. 5.
From Fig. 5, a general deformation mode of the normal stress unloading test can be found. The shear displacement of the joints increases linearly as the shear stress increases from zero to the pre-determined value, and then, it remains stable. When the normal stress is unloaded stepwise, the shear deformation increases gradually. If the normal stress is unloaded to a critical value below which the joint cannot resist the pre-loaded shear stress, the joint fails, and the shear displacement increases rapidly and loses its stability. Meanwhile, the shear stress decreases.  The critical displacement for joints under different initial normal stress conditions ranges from 1 to 6 mm. The joints with the least JRC (S 1 group) have the least critical displacements. A joint with higher JRC tends to endure larger critical displacement. However, the JRC and critical displacement are weakly correlated.
The critical normal stresses under which the joints lose their stability are listed in Table 3.
Here, note that the unloading test of S 3 group under 5.0 MPa initial normal stress was a trial test to determine the appropriate test settings. The normal stress holding time of the specimen was 60 s, which was longer than that of the other specimens. Because of the different test settings, the corresponding result, i.e., 3.78 MPa, was excluded as an outlier in the following analysis and regressions.
The data in Table 3 indicate that the larger the shear stress, the higher is the critical normal stress. Under identical initial normal stresses, the critical normal stress of complex joints (S 3 and S 4 ) is lower than that of simple joints (S 1 and  S 2 ), and the shear resistance of complex joints are higher than that of simple joints when the other conditions are the same.
In the normal stress unloading test, under the condition of low normal stress, the shear strength is higher than the applied normal stress. If the normal stress increases to high stress domain, the peak value of shear resistance is lower than the normal stress.

Unloading effect of shear resistance and influence of morphology
According to the data of loading and unloading tests, Fig. 6 shows the relationship between the normal stress and shear resistance. Figure 6 indicates that the shear resistance of joints increases approximately linearly with increasing surface complexity and normal stress, both for the shear loading stress path and normal unloading stress path. However, the data points are very closely distributed, and it is difficult to perform a detailed analysis.
To compare the results of loading and unloading paths intuitively, a non-dimensional index, namely, the stress ratio, was defined. For the loading cases, the stress ratio was defined as the ratio of the shear strength to the predetermined normal stress, and for the unloading cases, it was defined as the ratio of the pre-determined shear stress to the critical normal stress corresponding to the failure of the joint.
The stress ratios are listed in Table 4. From the results listed in Table 4, the following observations are made: 1. The stress ratio of joints decreases as the normal stress increases. Under tangential loading stress, the highest stress ratio is 1.21, corresponding to the case of the lowest normal stress, and the lowest stress ratio is 0.74, corresponding to the case of the highest normal stress. As the normal stress increases from 1.0 to 10.0 MPa, the average stress ratio of the joints decreases from 1.01 to 0.77, which is a reduction of about 24%. Under normal unloading stress, when the normal stress increases from 1.0 to 10.0 MPa, the average stress ratio of joints decreases from 1.12 to 0.89, which is a reduction of about 21%. 2. The stress ratio of joints increases with the complexity of the surface morphology of the joints. Under tangential loading stress, the highest stress ratio is 1.21, corresponding to the case of the joint with the highest JRC, while the lowest stress ratio is 0.74, corresponding to the case of the joint with the second-lowest JRC (which is very close to the stress ratio of the joint with the lowest JRC). As the JRC increases from 7.02 (S 1 ) to 11.02 (S 4 ), the average stress ratio of joints increases from 0.81 to 1.04 by about 28%. Under normal unloading stress, when the normal stress increases from 7.02 (S 1 ) to 11.02 (S 4 ), the average stress ratio of joints increases from 0.90 to 1.14 by about 27%.

Effect of stress path on shear strength
The variations in the stress ratio under the loading and unloading stress paths are quite similar. However, the failure cannot be predicted in the normal stress unloading tests and the normal stress levels are different. Hence, the stress ratios of the loading tests and unloading tests cannot be compared directly.
To reveal the variation tendency of the shear resistance of joints, the non-dimensional stress ratios were linearly regressed according to the normal stress. The regression results are shown in Fig. 7. Figure 7 shows that the relationship between the stress ratio and normal stress can be described by lines with negative slopes. The lowest correlation index R is 0.812, and half of the indices are larger than 0.95, indicating strong linear correlation between the regression parameters.
The slopes of the regression lines are closely related to the morphology of the joint surface. From S 1 to S 4 , the slopes of the regression lines decrease from − 0.008 to − 0.046, while under normal unloading condition, the slopes decrease from − 0.025 to − 0.092. The more complex the morphology, the faster does the stress ratio decrease with the normal stress. Figure 7 also shows the predicted curves of modified Barton model (Xing et al. 2021). It can be seen that, on the whole, the variation trend of modified Barton model is basically consistent with the test results under both tangential loading and normal unloading path. However, there are still some differences between the prediction of modified Barton model and the actual data. Under low normal stress condition, the predicted values of modified Barton model are relatively high. And the convergence normal stress of modified Barton model is higher than the linear regression.
Since the stress ratio of joint with higher roughness has a larger descending rate, the regression lines gradually come closer to each other as the normal stress increases. It rational to speculate that they will converge to one point or approach a certain limitation, under both tangential loading and normal unloading conditions. In other words, the influence of joint morphology weakens as the normal stress increases, and if the normal stress is sufficiently high, the influence of joints morphology tends to disappear.
To intuitively compare the variations in stress ratios and reveal the unloading effect, the regression lines were plotted in one coordinate system, as shown in Fig. 8.
From Fig. 8, one can see that the stress ratio declines with an increase in the normal stress for both tangential loading stress path and normal unloading stress path. Under low normal stress condition, the stress ratios of joints under the normal unloading stress path are significantly higher (about 10%) than those of the joints under the tangential loading stress path. However, the stress ratios of the joints under the normal unloading stress path decline more rapidly as the normal stress increases. Consequently, when the  normal stress is higher, the relationship is reversed; in other words, the stress ratios of joints under tangential loading are greater than those of the joints under normal unloading. Compared to the joints under tangential loading, those under the unloading stress experience lower normal stress such that the influence of the joint morphology is totally masked. The intersection of the regression lines of the stress ratios of joints under tangential loading is larger than 10 MPa, and the corresponding values are estimated as 5.5-6.0 MPa.
According to the comparative analysis, the shear strength of rock joints has an obvious unloading effect. This stress path dependence has not been well investigated and the mechanism is still not clear. A comparison of the joint surface morphology before and after the action of normal stress leads us to the deduction that the stress path dependence of joints arises partly from the imperfect matching of the two surfaces of one joint, which is likely to cause the damage of asperities during normal stress loading.
To verify this conjecture, two normal loading tests (purely loaded by normal stress and without any shearing) were conducted to compare the variations in the joint surface morphology before and after the action of normal stress. The joint surfaces after the action of 10.0 MPa normal stress were scanned by the 3D scanner. The scanning accuracy was selected as 0.1 mm. The comparison results are shown in Fig. 9.
As seen from Fig. 9, only about 94% of the data points of sample 1 were unaffected by the normal loading process, and about 96% of the data points of sample 2 were unaffected. The damage of joint surface asperities weakens the influence of the initial morphology and may result in an earlier convergence of the regression lines.

Conclusion
To investigate the unloading effect of rock joints, tangential loading tests with pre-determined normal stress and normal unloading tests with pre-determined shear stress were performed on cement mortar replicas. From the results, the following conclusions were drawn: The shear strength or resistance of joints increases approximately linearly as the normal stress and JRC increase, under both tangential loading and normal unloading conditions. The relationship between the stress ratio and normal stress can be expressed by a cluster of negative slope lines (related with JRC) converging to a certain point. The slopes of the regression lines decrease as the complexity of joint surfaces increase.
A notable unloading effect was observed through the test results. Under the low normal stress, the unloading path had a higher stress ratio. Besides, stress ratios of the joints subjected to the normal unloading stress path decline as the normal stress increases, which are more rapid than the tangential loading path. Therefore, the relationship between the stress ratio under the two stress paths was reversed under a specified normal stress.
When the normal stress is high enough, the influence of the joint morphology will be masked and rock joints with different JRCs tend to have the same shear strength ratio. The critical normal stress of rock joints under normal unloading stress path is lower than that of the rock joints under tangential loading stress path. The estimated intersection of the regression lines of stress ratio decreases from > 10 MPa to about 5.5-6.0 MPa when the stress path switches from tangential loading to normal unloading.
The comparative study revealed a previously unknown unloading effect on the mechanical behavior of rock joints and will aid the estimation of the rock joints' stability in a complex stress environment.