1 Introduction

Periodic loading caused by ocean waves, earth tides or earthquakes are complicated dynamic natural movements connected with dynamic stresses (Stein 1999; Julian 2000; Candela et al. 2014; Wei et al. 2013, 2015; Zhou et al. 2020; Xie et al. 2021). Under certain conditions, shear and normal loads on surrounding fractures are altered and slipping along rock fractures under dynamic normal stresses happens (Stein 1999; Candela et al. 2014). To investigate the slip behavior of rock joints/fractures exposed to dynamic normal load disturbance, several researchers performed corresponding shear experiments (Linker and Dieterich 1992; Cochard et al. 2000, 2003, 2021; Rice et al. 2001; Hong and Marone 2005; Konietzky et al. 2012; Smith et al. 2015; Kilgore et al. 2012, 2017; Molinari and Perfettini, 2017; Dang et al. 2016, 2018; Meng et al. 2018; Webber et al. 2018; Aiken et al. 2018; Mighani et al. 2019; Gong et al. 2020, 2021; Ji et al. 2021).

Normal stress disturbance weakens the shear stress of rock fractures (Marone, 1998; Rice, 2006; Sheng et al. 2020; Mei et al. 2021), and the evolution of friction due to a sudden increase in normal stress obeys a three-stage development. Hobbs and Brady (1985), Olsson (1988) carried out double shear experiments on gabbro and welded tuff responding to sudden variations in normal load. The shear stress alters significantly at the beginning followed by an exponential pattern. Linker and Dieterich (1992) performed double shear tests on Westerly granite under a step-wise load variation. They observed that the changes in shear stress are divided into three stages, an instantaneous increase in the first stage, fast linear increase in the second stage, and reaching a constant value in the third stage. Hong and Marone (2005) conducted slip experiments on Westerly granite with different gouge fillings in the fractures. The shear stress also changes when normal stress alters according to a multi-stage behavior. Kilgore et al. (2012, 2017) investigated the shear behavior on granite exposed to dynamic disturbance. A gradual, approximately exponential variation in shear stress is reported. Moreover, the memory effect is pronounced (Prakash, 1998; Bureau et al. 2000). Dang et al. (2021) preformed direct shear tests on basalt samples with different slip rates. A critical slip rate is reported, which induces normal load oscillation-induced weakening or normal load oscillation-induced strengthening under dynamic normal load conditions.

Boettcher and Marone (2004), and Dang et al. (2017, 2018) investigated the frictional response of oscillating force amplitudes acting on planar fracture surfaces. Boettcher and Marone (2004) varied the normalized oscillation amplitude from 1 to 10%. Their work proved that larger oscillation amplitude produces a pronounced reduction in shear strength. Dang et al. (2018) varied the normalized oscillation amplitude from 11 to 67%, and reported that the minimum friction coefficient decreases with increasing oscillation amplitude. Moreover, a distinct phase difference between normal load and shear load with normal load leading is documented, where the relative phase difference increases with rising oscillation amplitude. Dang et al. (2020) investigated the cyclic shear response of artificial smooth fractures exposed to normal load oscillations. They found that normal force oscillation amplitudes, shear displacement amplitudes, normal force oscillation frequencies, shear direction reverse frequencies, and initial normal load levels are not linked to the maximum apparent coefficient of friction.

Natural rock fractures have typically rough surfaces. Planar/flat rock fracture surfaces are very rare in nature. However, most of the earlier studies on the slip behavior under dynamic normal forces concentrate on planar rock fractures (Hobbs and Brady, 1985; Hong and Marone, 2005; Dang et al. 2016, 2017, 2018). For planar rock fractures, the patterns of shear force and dilation are nearly the same in the stable slipping stage. Without consideration of the roughness of fracture surfaces, the understanding of the dynamic fracture strengthening/weakening is quite limited.

2 Experimental Methodology

The slip experiments were performed on a rough basalt rock joint using a dynamic shear equipment (Fig. 1a, Konietzky et al. 2012) at temperature of approximately 20 ℃ and nominal humidity of approximately 40%. The dynamic shear equipment consists of two hydraulic loading systems, which provides a constant slip rate and a periodic normal force. Shear and normal displacements are measured by linear variable differential transformers (LVDTs) with an accuracy of 0.001 mm (Fig. 1e, f). All measured data are sampled with rate of 100 Hz.

Fig. 1
figure 1

a Scheme of equipment (Konietzky et al. 2012). b Schematic of slip experiment via periodical load with a constant slip rate (Dang et al. 2016). c Basalt rock showing the rough fracture surfaces. d Scanned fracture topography before the experiment, e vertical LVDTs f horizontal LVDT

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The rough basalt fracture was obtained by splitting an intact basalt block with a size of 150 × 150 × 100 mm3 (length × width × height). A pre-cut groove with a depth of 10 mm was created at the middle part of the sample using a motor saw. Then the pre-cut groove block is sheared using a uniaxial test machine, and the fracture surface is created nearly at the middle part of the rock sample. The effective contact area was 130 × 130 mm2 (length × width) (Fig. 1c, d). Upper and lower surfaces matched almost completely. The intact basalt samples showed a uniaxial compressive strength of 220 MPa, and a uniaxial tensile strength of 16 MPa.

To carry out repeated experiments with the same rough surface, we adopted a relatively low initial normal force to guarantee the surface asperities are maintained almost unbroken during these tests. Tests were performed with an initial normal force of 50 kN, a slip velocity of 5.0 mm per minute, and a maximum slip length (umax) of 5.0 mm. Each test lasted for 60 s. As shown in Fig. 1b, the superimposed oscillating normal load was governed by a sinusoidal wave (Eq. 1), with oscillating amplitudes of 0, 10, 20, 25, 30, and 40 kN, respectively. The oscillating amplitude of 0 kN means that the slip experiment was performed under constant normal load conditions. The normal load oscillation frequency was 0.5 Hz. The experimental plan is shown in Table 1. All six experiments were performed with dynamic normal load amplitude changes ranging from 0 to 80% (A/FN). The fracture surfaces were scanned before and after the tests using a 3D scanner. The whole experiments lasted for several hours, during which temperature and humidity were kept nearly constant, so that temperature-humidity effects were neglected.

$$F_{d} = A{\text{sin}}\left( {2\pi ft} \right)$$
(1)
Table 1 Laboratory test programs
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3 Experimental Results

The surfaces of the rock fracture were scanned before and after the test by the 3D scanner “z-Snapper” (Fig. 2a), with an accuracy of 0.001 mm. The topography of the rough surface before and after the experiment of the fractures is shown in Fig. 2b. According to Tse and Crude (1979) law (Eqs. 2 and 3), joint roughness coefficient (JRC) and root mean square slope Z2 were 0.260 and 13.2 before the test, and 0.257 and 13.0 after the test, respectively, which proves that the asperities of the fractures experience only very small degeneration. Based on the shear strength criterion proposed by Barton and Choubey (1977), the slight reduction of JRC has little effect on the shear strength.

$$Z_{2} = \left[ {\frac{1}{{\left( {n - 1} \right)\left( {\Delta x} \right)^{2} }}\mathop \sum \limits_{i = 1}^{n - 1} \left( {Z_{i + 1} - Z_{i} } \right)^{2} } \right]^{\frac{1}{2}}$$
(2)
$${\text{JRC}} = 32.2 + 32.47{\text{log}}Z_{2}$$
(3)
Fig. 2
figure 2

a 3D scanner “z-Snapper”. b Scanned topography of the rough fracture surface before and after the experiment. The scanned z- coordinate of the upper fracture was converted. Red lines indicate the shear direction

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Figure 3 presents details of the slip behavior under constant and dynamic normal forces. Shear force, friction coefficient, and dilation showed periodical changes along with periodical changes in normal force. The changing patterns of shear force, friction coefficient and dilation were significantly influenced by normal load oscillation amplitude and shear displacement. Taking A = 30 kN for example: Fig. 4 illustrates the methodology used to analyze the measured frictional response. Six quantities were investigated for each cycle: (1) shear force variation during normal load oscillations, ΔFs1 and ΔFs2; (2) coefficient of friction variation during normal load oscillations, Δμd1 and Δμd2; (3) normal displacement variation during normal load oscillations, Δd1 and Δd2; (4) phase lag, D1 and D1’, which are defined as phase lag between peaks and valleys in normal force and shear force; (5) phase lag, D2 and D2’, which are defined as phase lag between peaks and valleys in normal force and actual coefficient of friction; and (6) phase lag, D3 and D3’, which are defined as phase lag between peaks and valleys in normal force and normal displacement. Figure 5 shows the patterns of ΔFs1, ΔFs2, Δμd1, Δμd2, Δd1, Δd2, D1, D2 and D3 versus shear displacement under different normalized oscillation amplitude Ɛ.

Fig. 3
figure 3

Experimental results of normal and shear force, actual friction coefficient, and normal displacement as a function of shear displacement for slip rate of 5.0 mm/min, oscillation frequency of 0.5 Hz, initial normal force of 50 kN; dynamic sinusoidal normal forces are ± 10, ± 20, ± 25, ± 30, and ± 40 kN, respectively. ± 0 kN means shear test under constant normal load. The figures on the right are zoom-in of the left ones

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Fig. 4
figure 4

Quantification of frictional response to oscillating normal loading in the case of A = 30 kN. a Changes in normal force. b Changes in shear force, ΔFs1 and ΔFs2 are the peak-to-valley amplitudes of shear load response to normal load vibrations in the loading and unloading stage. Phase lag D1 and D1’ are the shear displacement difference between the peaks and valleys of normal and shear force, respectively. c) Variation in apparent dynamic friction coefficient, Δμd1 and Δμd2, are the peak-to-valley amplitudes of the actual coefficient linked to normal load vibrations in the loading and unloading stage. Phase lag D2 and D2’ are the shear displacement difference between the peaks and valleys of normal force and coefficient of friction, respectively. d Changes of normal displacement, Δd1 and Δd2, are the peak-to-valley amplitude normal displacements in relation to normal load vibrations in the loading and unloading stage. Phase lag D3 and D3’ are the shear displacement difference between the peaks and valleys of the normal force and normal displacement, respectively

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Fig. 5
figure 5

Influence of normal load oscillation amplitude on shear strength, coefficient of friction, normal load displacement and phase lag. a Apparent dynamic friction coefficient (ΔFs1/2A, ΔFs2/2A) versus shear displacement, solid lines denote ΔFs1/2A, dot lines denote ΔFs2/2A. b Peak-to-valley amplitudes of actual coefficient of friction (Δμd1 and Δμd2) versus shear displacement, solid lines denote Δμd1, dot lines denote Δμd2. c Peak-to-valley amplitudes of normal displacement versus shear displacement, solid lines denote Δd1, dotted lines denote Δd2. d Details of the time difference between normal force and shear force (D1) versus shear displacement. e Details of the time difference between normal force and apparent friction coefficient (D2) versus shear displacement. f Details of the time difference between normal force and normal displacement (D3) versus shear displacement

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3.1 Variation of Shear Force ΔFs1 and ΔFs2

The measured peak-to-valley value of the shear force fluctuations ΔFs1 and ΔFs2 show complex patterns influenced by normal load oscillation amplitude and shear displacement. In Fig. 3, the maximum and minimum values of the shear forces decrease with increasing normal load oscillation amplitude. Under large normal load oscillation amplitude, the measured shear force can even have negative values, e.g., negative shear force is measured when A = 40 kN. ΔFs1 is larger than ΔFs2 in each cycle in an early shearing stage. With ongoing shear displacement, peak and valley values of shear force, ΔFs1 and ΔFs2, increase gradually, finally reaching a constant value, where ΔFs1 is nearly identical to ΔFs2. Figure 5a shows the relationship between the apparent dynamic coefficients of friction (ΔFs1/2A, ΔFs2/2A) and the shear displacement. ΔFs1/2A and ΔFs2/2A increase with shearing and increasing normal force oscillation amplitude until A = 30 kN, while, they show a decreasing pattern when A = 40 kN. Besides, the cyclic changes in the shear forces are more serrated instead of sinusoidal.

3.2 Variation of Actual Friction Coefficient Δμd1 and Δμd2

The actual friction coefficient follows the same variation pattern like the shear force, which changes periodically with changes in normal loads. Peak and valley data of the actual friction coefficient grow with ongoing shear displacement. The plot of the coefficient of friction has a fairly regular shape (sinusoidal) until A = 30 kN. A ‘dynamic overshooting’ happens when A = 40 kN and the sinusoidal friction pattern changes (see Sect. 4.3). Figure 5b illustrates the peak-to-valley amplitudes Δμd1 and Δμd2 determined according to Fig. 4. Δμd1 is larger than Δμd2 in the early stage of shearing. The variation in apparent friction coefficient increases gradually until Δμd1 = Δμd2. For all dynamic test cases, the peak values of the actual friction coefficient are similar to that obtained under CNL conditions, i.e., fdmax = μss. The valley values of the actual friction coefficient rise with decreasing normal load oscillation amplitude. Δμd1 and Δμd2 increase with increasing normal load oscillation amplitude.

3.3 Variation of Normal Displacement Δd1 and Δd2

As shown in Figs. 3d and 5c the normal displacement under constant normal load shows a nearly linear increase with ongoing shear displacement. In general, the dilation is larger under periodical normal force than that under constant normal force. Both, unloading of normal force and the asperities of the fracture surface promote dilation during the shearing process. For every cycle of loading and unloading, the fracture is compressed and separated in a cyclic manner. During loading, the normal displacement (dilation) is restrained, the separation of the fracture declines and causes enlarged strain at the contact area (Greenwood and Williamson, 1966). Only the asperity of the fracture surface promotes dilation, which makes Δd1 larger than Δd2 during the cyclic loading. Some irreversible plastic deformations at the contacts also lead the apparent asymmetry between Δd1 and Δd2. Moreover, Δd1 and Δd2 increase with increasing normal load oscillation amplitude, which has a slight increasing trend with ongoing shear displacement. Δd1 and Δd2 consist of three parts: elastic deformation of the fracture as well as the elastic and plastic deformation of the rock matrix (Goodman 1976; Kilgore et al. 2017).

3.4 Phase Lag D1(D1’), D2(D2’) and D3(D3’)

Perfettini et al. (2001) are the first who theoretically found a time difference between horizontal shear load and vertical load during dynamic oscillations. Then, this theory was proved by laboratory experiments on planar fractures. Boettcher and Marone (2004) observed a time difference between normal and shear load under normal load oscillation frequencies between 0.025 and 2.0 Hz. Dang et al. (2018) found that the time difference is velocity-frequency-amplitude related. However, the above mentioned phase lag pattern changes when fracture roughness is considered.

As illustrated in Fig. 5d–f, the time difference also exists in this study. We observed a time difference between (1) peak vertical force and peak shear force (D1), (2) valley vertical force and valley shear force(D1’), (3) peak vertical force and peak actual friction coefficient (D2), (4) valley vertical force and valley actual friction coefficient (D2’), (5) peak vertical force and peak normal displacement (D3), and (6) valley vertical force and valley normal displacement (D3’). D1 has a slight trend of reduction with ongoing shear displacement, while D2 and D3 remain nearly constant. For different shear positions, the actual coefficient of friction is different (Fig. 3c, the actual friction coefficient increases with ongoing shear displacement), which indicates a phase lag decrease of D1 with increasing actual coefficient of friction. Figure 5d–f show that the relative phase lag of D1 increases with growing normal load oscillation amplitude and decrease with ongoing shear displacement. The relative phase lag of D2 is constant at about 30% in all the tests, which is independent of normal load oscillation amplitude and shear displacement. The relative phase lag of D3 is 50% for all the tests, which means that the variation of vertical settlement changes synchronously with the variation of normal force, consistent with the observations of Kilgore et al. (2017). Time differences between the valleys of normal and shear load, coefficient of friction, and vertical settlement are small, whereas the relative phase lags for D1’, D2’and D3’ are 0, about 30% and 50%, respectively, nearly constant for all the tests. The magnitude of phase lag of D1 is influenced by the magnitude of shear stiffness (Dang et al. 2016). The shear stiffness increases with increasing joint roughness and stress level (Bandis et al. 1983, Amiri Hossaini et al. 2014). Consequently, the phase lag pattern is different compared to Boettcher and Marone (2004), Dang et al. (2017), when fracture roughness is taken into consideration.

4 Discussion

4.1 Normal Load Oscillation-Induced Strengthening/Weakening

We define ΔFyield as the difference between the maximum friction under the oscillatory normal force and the friction under the constant normal force for the same shear distance. General understanding is that normal load oscillation reduces the shear strength. However, laboratory test results show that ΔFyield (Fig. 6a) can be both: greater than zero or less than zero considering different normal load oscillation amplitudes. Figure 6b shows the influence of oscillation amplitude and shear displacement on ΔFyield. For small oscillation amplitude, Ɛ < 50%, ΔFyield is positive, the peak shear force enhances up to 10% compared to the case of constant normal load. When Ɛ is over 50%, ΔFyield starts to decrease with increasing oscillation amplitude. The peak shear load can be reduced by nearly 40% at an oscillation amplitude of 80%. A critical vibration amplitude A*c is defined when ΔFyield/Fs = 0 (Fig. 6c), which separates the entire zone into dynamic fracture weakening (A > A*c), and dynamic fracture strengthening (A < A*c). In this study, A*c nearly equals 55%. In general, vibrations with large amplitude easily cause normal load oscillation-induced weakening.

Fig. 6
figure 6

a Definition of ΔFyield. ΔFyield shows positive or negative values, representing strengthening or weakening, respectively. b Variation in peak yield strength (ΔFyield) versus shear displacement with different normal force oscillation amplitude. c Dynamic fracture strengthening/weakening linked to normal force oscillation amplitude. A critical vibration amplitude A*c defined as ΔFyield/Fs = 0 (A > A*c indicates induced normal load oscillation-induced weakening, A < A*c indicates induced normal load oscillation-induced strengthening)

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The apparent weakening phenomenon documented by the shear strength data, can well be explained using the rate-state friction law (Ruina 1983; Perfettini et al. 2001), which has been effectively applied to simulate frictional sliding of solids and for earthquake events (firstly established for CNL conditions). Linker and Dieterich (1992), Perfettini et al. (2001) extended the rate-state friction law to include variations in normal load, and provided an analytical equation to predict the critical oscillation amplitude. In Perfettini’s assumption, the critical amplitude is linked to state variables and fracture stiffness. The normal load oscillation-induced strengthening/weakening is controlled by the loading/unloading rate of normal force, slip rate and fracture stiffness.

4.2 Friction coefficient reduction and shear work

We calculated energy consumption during shearing using the shear force—shear displacement plots, \(E = \int_{0}^{u\max } {F_{s} du}\), where u is the slip length and Fs is the shear force. The elastic energy of the fracture is not taken into consideration in the present work. The ratio of friction coefficient reduction is the ratio between the minimum friction coefficient under periodical vertical force and the quasi-static friction coefficient, μdmin/μss. Figure 7 shows the shear work and the ratio of friction coefficient reduction under various normal load oscillation amplitudes. A large normal load oscillation amplitude leads to a more serious vibration of the upper block of the sample, which is not beneficial to store the strain energy (Li et al. 2020) in the asperities during shearing. Consequently, the shear work decreases with increasing normal load oscillation amplitude. μd/μss shows a cyclic behavior, whereby peak and valley values are nearly constant at the same normal load oscillation amplitude. The peak values of μd/μss are nearly constant during all the tests with a value close to 1, i.e., peak μd nearly equals μss. The minimum of μdmin/μss increases with reducing the normal load oscillation amplitude. The above results indicate that a high normal load oscillation amplitude leads to a lower energy consumption and a large reduction in coefficient of friction, which easily destabilizes the slip. These changing patterns are similar to those documented for planar joints (Dang et al. 2017).

Fig. 7
figure 7

a Dynamic friction coefficient and quasi-static friction coefficient versus shear displacement under normal load oscillation amplitude of 0, ± 10, ± 20, ± 25 and ± 30 kN, respectively. b and c Ratio of dynamic friction coefficient reduction and energy consumption versus normalized normal load oscillation amplitude

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4.3 Interpretation of Negative Shear Force

For direct shear tests with constant slip rate, a negative shear force under a positive normal load seems unreasonable at first glance (shown in Fig. 3, A =  ± 40 kN). However, negative shear forces were reliably measured in the case of large dynamic load amplitude (Fig. 8a). Moreover, negative cutting forces are also monitored in Cerchar scratch tests (Zhang 2020).

Fig. 8
figure 8

a Additional shear test: shear force versus shear displacement, normal force of 50 ± 40 kN with slip rate of 100 mm/min and loading frequency of 1.0 Hz, b Scratch test: cutting force versus scratch length under normal load of 70 N with scratch rate of 20 mm/min. Both of them show temporarily negative shear/cutting forces

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The apparent friction can reach negative values during tests linked to negative values of shear force. When the rock block is exposed to normal and shear loads, elastic energy is stored due to the elastic deformation. As shown in Fig. 9, the stored energy changes with increasing/decreasing normal load at different shear positions. In the loading stage, (Fig. 9a, b, normal load is FN and FN + A), elastic energy is stored in the lower block, which reaches the peak value (E2) under normal load of FN + A. During unloading, the stored elastic energy releases, the frictional resistance decreases, and the lower block expands (Fig. 9c). The stored elastic energy reaches the minimum value (E3) under normal load of FN – A. Under normal load of 50 ± 40 kN, the unloading rate of normal force is up to 4800 kN/min (changing from 90 to 10 kN lasts about one second in the laboratory test). This more or less sudden drop in normal force leads to a fast release of elastic energy, a kind of ‘shooting’ of the lower block combined with a quick drop of the shear force. The expanding rate of the specimen is about 9 mm/min under normal load of 50 ± 40 kN, which is nearly two times larger than the slip rate (5 mm/min). Due to the fast expanding rate, whenever the frictional resistance becomes too low, the lower block may shoot away (‘dynamic overshooting’, Aboudi and Ryvkin 2014). Consequently, the lower specimen moves faster than the desired (fixed) slip rate, thus the piston has to produce a counterforce (negative force) to maintain the constant velocity. Earthquakes and blasting may induce a fast release/storage of the elastic energy in the rock matrix. If they act on a steady creeping rough fracture, the stress state of the fracture surface becomes very complicated. In some specific cases, apparent friction can become negative.

Fig. 9
figure 9

Schematics of asperity deformation, elastic energy storage and release of the lower block under different normal load conditions. a Under normal load of FN, the lower block is compressed to some extend, the elastic energy is stored, b with increasing normal load, the lower block is further compressed, elastic deformation is further accumulated, and stored elastic energy is accumulated reaching the peak values under normal load of FN + A, c when normal load becomes smaller, dilation develops, and stored elastic energy is released, a “dynamic overshooting” occurs, stored elastic energy reaches a minimum and dilation reaches the peak value under normal load of FN – A

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5 Conclusions

Normal load oscillation amplitude has a distinct influence on the slip behavior of rough rock fractures. In general, a small oscillation amplitude increases the shear strength and a large oscillation amplitude reduces the shear strength. A critical normal load oscillation amplitude is confirmed to judge the normal load oscillation-induced strengthening or normal load oscillation-induced weakening of a slipping rock fracture. The reduction of the friction coefficient is larger under large normal load oscillationn amplitude. The energy consumption decreases with increasing normal load oscillation amplitude. A time difference between peak vertical force and peak shear force is observed. It rises with rising vertical force oscillation amplitude and decreases with ongoing shear displacement. Our results imply that slipping joints or fractures which experience higher normal load oscillation amplitude react more sensitively.

A limitation of the present research is that damage or plastic deformation of the fracture asperities is not considered. Besides, normal load oscillation frequency, slip rate, water and temperature effect are not yet taken into account, although they may have an influence on the frictional behavior under dynamic normal loading (Boettcher and Marone 2004; Beeler et al. 2019; Dang et al. 2019, 2021).